PATTERN RECOGNITION TECHNIQUES FOR INTEGRATION OF GEOPHYSICS, REMOTE SENSING, GEOCHEMISTRY AND

GEOLOGY

A Thesis submitted for the degree of DOCTOR OF PHILOSOPHY

in the FACULTY OF SCIENCE OF

THE UNIVERSITY OF LONDON

by

JONG NAM PARK M.Sc (London)

Geophysics Section Royal School of Mines Imperial College of Science & Technology London SW7

APRIL 1983

ii

A B S T R A C T

An evaluation of pattern recognition techniques applied to

filtered multivariate data has been carried out on selected geo-

physical ,remote sensing, and geochemical test data from the Bodmin

Moor area, S.W. England. A total of 14 data sets was used.

Unsupervised classification methods, factor analysis and

cluster analysis, and supervised classification techniques,

empirical discirminant analysis and characteristic analysis, were

adopted. Their usefulness in geological mapping and outlining

potential mineralization has been compared.

Generally the supervised classification techniques proved

fast in classifying the multivariate data based on an initial template

of known data. The unsupervised classification techniques did not

require such initial information but needed more computing time.

It was found that regional lithological mapping could be

best achieved using transformed data, while the untransformed data

were more suited to defining potential mineralization zones. However,

the Bodmin Moor granite rock type was consistently identified by all of

the methods.

Much emphasis was put on feature extraction techniques,

a different technique being applied to each of the 14 data sets

depending on the characteristic property, either physical, chemical

or spectral.

During the course of the research, several new techniques of

data analysis have been developed, with associated computer programs,

especially data extrapolation in convolution filtering. This

technique has been effectively applied in geophysical data processing

in order to avoid loss of information at the edges of data sets.

Also, a new pseudogravity filter has been designed for use on equally

spaced data. This technique facilitates the interpretation of

complex subsurface geological features by comparing gravity data

with the pseudogravity transform of the corresponding magnetic data.

iv

ACKNOWLEDGEMENTS

I wish to express most gratitude to Dr D.R. Cowan for suggesting the research topic, and for his help, encouragement and advice throughout the research. My immense grati tude also goes to Dr R.J. Howarth, for providing useful computer programs on pattern recognition techniques and graphic plots, and for his help and advice particularly in the multivariate data analysis. Their cri t ical reading of the manuscript was also most grateful .

I am also very grateful to the British government for providing a grant for the duration of the research and in particular Miss Celia Wood of the British Council for her help.

I wish to thank the Korean Government, particularly, Professor B.K. Hyun, President of Korea Institute of Energy and Resources (KIER) and Dr J . H. Koo, Head of Geophysics Department of KIER, for allowing me to have leave of absence from KIER to undertake this research.

I am also much indebted to Professor R.G. Mason, Head of Geophysics Department of Imperial College (IC) for his help in obtaining the British Council grant and Dr Anna Thomas-Betts of IC for her help in providing the allocation of the computer units and also for useful discussions on certain mathematical aspects. Many thanks are also due to M Hale of IC for providing useful information on geology and geochemistry.

I am also very grateful to many people for helping in various ways during the research. Particular mention must be made of Dr R.B. Evans and Dr J. Hawkes from the Institute of Geological Sciences (IGS). Many thanks to Mr P. Jarvis of the Imperial College Computer Centre for his help in colour plotting and allowing access to his software program for Landsat MSS data manipulation. Immense thanks also goes to Mrs S. Cowan for the colour plot of the geo-chemical data over the Bodmin Moor area in Chapter

I also wish to thank T. Richardson for typing the thesis. Finally special thanks are due to my wife and family for

their patience, encouragement and great support throughout the period of the research.

This thesis is dedicated to the memory of my father W.K. Park who supported me in all aspects of my life and in particular my further education.

J.N. Park

V

LIST OF CONTENTS

Page

Abstract Acknowledgements iv List of Contents v

List of Figures ix List of Tables xiii List of Appendices xv Summary of Abbreviations xvi

CHAPTER 1: INTRODUCTION 1

CHAPTER 2: GEOLOGICAL SETTING 6

2.1 Geomorphology and Soil Types 6 2.2 Geology 7 2.2.1 The Geology of Cornwall 7

(a) Devonian and Carboniferous Rocks 9 (b) Grani tes and Minor Intrusions 9 (c) Mineralization 10 (d) The Structure 12

2.2.2 The Geology of the Bodmin Moor Area 15

(a) Devonian Rocks (b) Carboniferous Rocks (c) Devonian Volcanic Rocks and Greenstone 18 (d) Granites and Later Intrusive Rocks 19 (e) Elvans 2 0

(f) Aureole of Thermo-Metamorphism surrounding 20 the Granites

(g) Mineralization 21

CHAPTER 3: DATA PREPARATION AND QUALIFICATION ' 23 3.1 General Considerations in Data Preparation 23

and Qualification 3.2 Interpolation and Extrapolation 24 3.2.1 Interpolation 25 3.2.2 Extrapolation 26

(a) Extrapolation in Space Domain 26 (b) Extrapolation in Frequency Domain 27

3.3 Digitization of Two-Dimensional Data 28 3.3.1 Digitization of Geophysical Data 28

(a) Gravity Data 32 (b) Magnetic Data 32

3.3.2 Digitization of Landsat MSS Data 33 3.3.3 Digitization of Geochemical Data 35 3.4 Noise Evalaution 36 3.4.1 Estimates of Noise Contributions 36 3.4.2 Noise Filtering 39 3.5 Data Qualification 44 3.5.1 Stationarity 44 3.5.2 Normality 48 3.6 Discussions 49

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CHAPTER 4: FEATURE EXTRACTION 51

4 . 1 Introduction 51 4 .2 Geophysical Data Processing 52 4 . 2 . 1 Filter Operators for Analysis of 52

Potential Field Data 4 . 2 .2 Review of Filter Operators 55

(a) Derivatives 55 (b) Upward and Downward Continuations 56 (c) Lowpass and Highpass Filtering 58 (d) Reduction to the Pole 59 (e) Pseudogravity 60

4, .2. .3 Theoretical Background of Filter Operators 61 4, .2, .4 The Description of GEOPAK Program 64 4, .2, .5 Qualitative Interpretation of Potential Field 66

Data (a) Description of Filtered Gravity and Magnetic 66

Maps (b) First- and Second-Derivative Maps 68 (c) Upward and Downward continuation Maps 74 (d) Lowpass and Highpass Filtered Maps 78 (e) Reduction to the Pole Map 81 (f) Pseudogravity Map 82

4. .3 Landsat MSS Data Processing 84 4. .3. . 1 Introduciton 84 4. .3. .2 Basic Principles in Landsat MSS Data 89

Processing 4. .3. ,3 Feature Extraction and Interpretation 93

(a) Black-and-White MSS Images of Cornwall 93 (b) False-Colour Composite of Cornwall 100 (c) Colour-Ratio Composite of Cornwall 101 (d) Description of Surface Maps for Bodmin Moor 103

Area (e) False-Colour Composite of the Bodmin Moor Area 106 (f) Colour-Ratio Composite of the Bodmi'n Moor Area 108

4. 4 Geochemical Data Processing 1 11 4. 4. 1 Principles of Regional Geochemical Data 1 13

Processing (a) Analysis of the Regional Distribution Patterns 113

of Geochemical Elements (b) Detection of Geochemical Haloes using the 1 14

Probability Plot 4. 4. 2 Regional Distribution of Geochemical 1 15

Elements 4. 4. 3 Probability Analysis 1 18 4. 5 Conclusions 126

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CHAPTER 5: TREND SURFACE ANALYSIS 127

5.1 Introduciton 127 5.2 Mathematical Procedures 128 5.3 Applications to the Bodmin Moor Area Data 131 5.3.1 Geophysical Data 132

(a) Gravity Data 132 (b) Magnetic Data 132

5.3.2 Landsat MSS Data 136 5.3.3 Geochemical Data 142 5.3.4 Residuals from the Third-Degree Trend 144

Surface (a) Geophysical Data (Gravity & Magnetics) 144 (b) Landsat MSS Data 148 (c) Geochemical Data 149

5.4 Discussion 151

CHAPTER 6: SIMILARITY ANALYSIS 155

6.1 Introduction 155 6.2 Principles of the Procedures 155 6.2.1 Overall Similarity 155 6.2.2 Spatial Similarity 157 6.2.3 Coherence Analysis 159 6.3 Applications of Similarity Analysis 162 6.3.1 Overall Similarity 162 6.3.2 Similarity Map 171 6.3.3 Coherence Analysis 175 6.4 Discussion 181

CHAPTER 7: CLASSIFICATION AND IDENTIFICATION OF 185 MULTIVARIATE DATA

7.1 Introduction 185 7.2 Transformation of Data ' 186 7.3 Selection of Variables in Multivariate Data 193

Analysis 7.4 Unsupervised Classification 198 7.4.1 Factor Analysis 198

(a) Factor Analysis Procedures 198 (b) Applications of Factor Analysis 204

7.4.2 Cluster Analysis 228 (a) Cluster Analysis Procedures 228 (b) Testing the ISODATA Program 236 (c) Applications of Cluster Analysis 245

7.5 Supervised Classification 259 7.5.1 Discriminant Analysis 259

(a) Discriminant Analysis Procedures 259 (b) Selection of the Initial Condition for EDF 264

Program (c) Applications of EDF 269

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7.5.2 Characteristic Analysis 275 (a) Characteristic Analysis Procedures 275 (b) Application of Characteristic Analysis 279

7.6 Conclusions 282

CHAPTER 8 : CONCLUSIONS AND RECOMMENDATIONS 287

8.1 Some Concluding Remarks 287 8.2 Summary of Recommendations for Further Work 289

REFERENCES 292

IX

LIST OF FIGURES Page

1.1 A pattern recognition system 2

2.1 Geology of Cornwall (S.W. England) 8

2.2 The structural geology of S.W. England 13

2.3 Geology of the Bodmin Moor area 16

3.1 Contour map of raw gravity data in gravity units 30 (128 by 128 array size)

3.2 Contour maps of potential field data 31 (64 by 64 array size)

3.3 MSS Scanning Arrangement 34

3.4. One-dimensional representation of two-dimensional 37 power spectrum by radial averaging

3.5 Two-dimensional power spectra of the filtered 41 data in log scale

4.1 Contour maps of the filtered potential fields of the 67 Bodmin Moor area with a cutoff wavelength of 800m.

4.2 Derivative maps of the gravity data 70

4.3 Derivative maps of the magnetic data 72

4.4 Upward and downward continuation maps of the 75 gravity data

4.5 Upward and downward continuation maps of the magnetic 76 data

4.6 Upward continuation maps of the gravity and magnetic 77 data (h = 3)

4.7 Lowpass and highpass filtered maps of the gravity data 79

4.8 Lowpass and highpass filtered maps of the magnetic data 80

4.9 Reduction to the Pole map of the magnetic data 83

4.10 Pseudogravity map of the magnetic data 83

4.11 Contrast stretched black-and-white band images 94

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4.12 Frequency distributions of black-and-white 97 MSS band images

4.13 False Colour composite of MSS bands 4, 5 and 7. 102

4.14 Colour-composite of MSS band ratios, R5/4, R6/5 and 102 R7/6 in blue green and red, respectively

4.15 Surface maps of MSS bands 4 and 7 over the Bodmin 104 Moor area, Cornwall

4.16 Surface maps of MSS bands 4 and 7 over Bodmin Moor 105 granite

4.17 Colour representation of MSS bands 4, 5 and 7 by 107 normal slicing over the Bodmin Moor area and their false colour-composite

4.18 Colour representation of MSS band ratios R5/4, R6/5 and 110 R7/6 and their colour-composite

4.19 Geochemical data over Bodmin Moor area 116

4.20 Probability plots of 8 geochemical elements 120

4.21 Spatial distribution of geochemical haloes of Cu, Pb, 125 Sn and Zn

5.1 Comparison of trend surfaces of degree 1 for 14 133 variables in the Bodmin Moor area

5.2 Comparison of trend surfaces of degree 3 for 14 137 variables in the Bodmin Moor area

5.3 Contoured residual maps of the third degree poly- 145 nomial

5.4 Linear directions of the data and 'predominant1 153 direction of the linear trends

6.1 Dendrogram - clustering with absolute correlation 167 coefficients (untransformed data)

6.2 Dendrogram - clustering with absolute correlation 170 coefficients (transformed data)

6.3 Similarity maps 172

6.4 Coherence vs. Frequency - Bodmin Moor Area 176

6.5 Bandpass filtered maps of coherence region A 178

6.6 Bandpass filtered maps of coherence region B 180

6.7 Bandpass filtered maps of incoherent region C 182

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7.1 Histograms of the filtered data 189

7.2 Histograms of X transformed data except the gravity 191 data (a) in which case the arc sine transform was applied

7.3 Cattell's Scree Test for determining the correct 197 number of principal components calculated from 16 variables

7.4 Cattell's Scree Test for determining the correct 205 number of principal components of four different variable sets

7.5 FA (Factor analysis) of the untransformed 6-MSS data 208 set

7.6 FA of the transformed 6-MSS data set 209

7.7 FA of the untransformed 10-MSS variable set 212

7.8 FA of the transformed 10-MSS variable set 213

7.9 FA of the untransformed 8-geochemical element set 216

7.10 FA of the transformed 8-geochemical element set 220

7.11 FA of the untransformed 9-mixed variable set 224

7.12 FA of the transformed 9-mixed variable set 227

7.13 Flowchart for ISODATA 234

7.14 Results of clustering by ISODATA for 8-geochemical 247 element sets

7.15 Results of clustering by ISODATA for 9-mixed variable 252 sets

7.16 Results of clustering by ISODATA for 8-PCA Score sets 257

7.17 Two overlapped bivariate distribution showing the 261 effective classification by projecting onto the discriminant function line

7.18 Interpolated one-dimensional probability density 262 function for five training set samples with increasing values of smoothing parameter a.

7.19 Correct classification performance rates of the training 268 set for the untransformed and transformed variable sets

7.20 Results of classification by EDF for the 8-geochemical 270 element sets

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Page

7.21 Results of classification by EDF for the 9-mixed 273 variable sets

7.22 Results of classification by EDF for the 8-PCA 274 score sets

7.23 Hypothetical data profile showing areas above local 277 inflexion points (second derivative negative) labelled

and other locations labelled '0'

7.24 Results of characteristic analysis for a variable set 281 of As, Cu, Pb, Sn and Zn

xiii

LIST OF TABLES Page

3.1 Noise filter coefficients 40

3.2 Test result of stationarity 47

4.1 Comparison of the number of multiplications 54 required between ordinary convolution, 8-folded convolution, discrete Fourier Transform and Fast Fourier Transform

4.3 Means and standard devl ations of 8 geochemical 119 elements from the Bodmin Moor area in lithologic units

4.4 Statistical results of probability analysis 123

5.1 Trend surface coefficients of degree 1 and 140 degree 3

5.2 Summary of the test values of goodness-of-fit and 141 F-test

6.1 Correlation matrix (untransformed data) 166

6.2 Correlation matrix (transformed data) 169

7.1 Estimation of mean X values using Dunlap and Duffy, 187 Skewness/Kurtosis 2/1, and maximum likelihood schemes

7.2 Statistical results of factor analysis for 6- 210 Landsat MSS variable sets

7.3 Statistical results of factor analysis for 214 10-Landsat MSS variable sets

7.4 Statistical results of factor analysis for 217 8-geochemical element sets

7.5 Statistical results of factor analysis for 225 9-mixed variable sets

7.6 An example of the input parameters for ISODATA 237

7.7 Test results of ISODATA of 8-geochemical element 239 sets by varying the spherical factor with the rest of the input parameters constant

7.8 Test results of ISODATA of 9-mixed variable sets by 240 varying the spherical factor with the rest of the input parameters constant

XIV

Page

7.9 Test of ISODATA of 8-PCA score sets by varying the 241 spherical factor with the rest of the input parameters constant

7.10 Test of ISODATA by varying the Euclidean distance 243 criterion (8 ) with the rest of the input parameters constant

7.11 Means and standard deviations of clustering for 248 8-geochemical element sets

7.12 Means and standard deviations of clustering for the 253 9-mixed variable sets

7.13 Eigenvectors of the first 8-principal components 256 of PCA with 16 variables

7.14 Means and standard deviations of clustering for the 258 8-PCA score sets

7.15 Characteristic weights of five elements (As, Cu, Pb, Sn 282 and Zn) for the model selected

XV

LIST OF APPENDICES Page

Appendix A New Extrapolation Techniques 315 A.1 Extrapolation in Frequency Domain using 315

Bicubic Spline Method A.2 Extrapolation in Space Domain 319

Appendix B Sampling of Landsat MSS Data for the 322 Bodmin Moor Area

Appendix C Calculation of Filter Operators 326 C.l Calculation of Continuations 326 C.2 Calculation of Reduction to the Pole 327 C.3 Calculation of First Vertical Derivative 328 C.4 Calculation of First Horizontal Derivative 329

in Gravity C.5 Calculation of Lowpass Filter 330 C.6 Derivation of Pseudogravity 331 C.7 First Horizontal Derivative of the 336

Magnetic Field C.8 Second Vertical Derivative 341

Appendix D Descriptions of GEOPAK Program 349

Appendix E Contrast Stretching using the Probability Denstiy 353 Function of Gaussian Distribution for PICPAC Gray Scale Plotting

Appendix F Trend Surface Analysis of First- and Third- 357 IXgree Polynomials

F.l Least Square Fit for a Flat Plane using 357 First-Degree Polynomials

F.2 Least Square Fit using Third-Degree 358 Polynomials

Appendix G Empirical Discriminant Function • 360

Microfiche Listing (rear folder)

GEOPAK

COHAN AND BPFILT

FACTOR

xv i

SUMMARY OF ABBREVIATIONS

BNG British National Grid

CCT Computer Compatible Tape

CHARAN CHARacteristic ANalysis

DFT Discrete Fourier Transform

EDT Empirical Discriminant Function

FA Factor Analysis

FFT Fast Fourier Transform

IGRF International Geomagnetic Reference Field

IGS the Institute of Geological Sciences

LDF Linear Discriminant Function

MSQ Mean SQuare Value

MSS MultiSpectral Scanner

PCA Principal Component Analysis

RSU Remote Sensing Unit

SLM Single Linkage Method

ULCC University of London Computer Centre

USGS United States Geological Survey

WPGM Weighted Pair Group average Method

1

CHAPTER ONE

INTRODUCTION

In the Earth Sciences there is a growing trend to change

from a univariate or bivariate data analysis to multivariate

data analysis in almost every aspect of the subject. This tendency

is due to several factors; firstly, in the study of Earth Sciences,

often no single variable is sufficient to extract useful geological

information from complex geological provinces or mineralized zones.

Often, a combination of several variables is more significant in

defining and delineating these areas. Secondly, modern methods in

Earth Sciences are providing a variety of data sets from the same

area, and thirdly, modern computing methods enable the manipulation

of complex data sets.

In this respect, pattern recognition techniques have been

successfully applied in various fields of multivariate data analysis.

Since the application of pattern recognition techniques to

Earth Sciences began in the 1960's, there has been increasing interest

in the utilization of the techniques for the extraction of

useful geological information, particularly for geological mapping and

evaluation of potential mineralization. In particular, the use of

Computer Compatible Tape (CCT) from Landsat Multi-Spectral Scanner

(MSS) data has increased the usefulness of the techniques in various

fields of study.

What then is pattern recognition and how is it useful in

Earth Sciences?

2

Pattern recognition has developed mostly as statistical

classification techniques (Fu, 1976). Tou (1974) has defined

pattern recognition techniques as "a process of categorization of input

data into identifiable classes via extraction of any features of

significance in the input data from a background of irrelevant details",

These techniques are involved with the extraction of features from

the input data and then classifying each feature into a class based

on certain criteria or a set of selected measurements extracted from

the input data. These criteria or selected measurements are

supposed to be invariant or less sensitive with respect to the commonly

encountered variations and distortions, and also supposed to contain

less redundancies. Under these conditions, Fu (1976) has noted that

pattern recognition can be considered as consisting of two subproblems.

1. What measurements should be taken from the input

patterns? Usually the decision of what to measure is

rather subjective and also depends on practical

considerations such as the availability of information or

the cost of measurements.

2. The second is the problem of classification based

on the measurements taken from the selected features.

A simplified block diagram of a pattern recognition system

is shown in Fig. 1.1.

Input Pat tern

Feature extraction

Class i-fication Dec is ion

Feature Measurements

Fig. 1.1: A pattern recognition system (After, Fu 1976)

3

There are two categories in pattern recognition classification -

supervised and unsupervised.

The former depends on prior information on the nature of classes

to be identified and establishes a classification of the samples based

on this information. Discriminant analysis or characteristic

analysis belongs to this category.

The unsupervised classifications are used when no prior

information is available on the nature of classes. Principle

component analysis or any cluster analysis techniques are in this

category.

These categories have been distinguished by Howarth (1973b)

as pattern classification and pattern analysis, respectively.

Applications of pattern recognition techniques to the Earth

Sciences and problems arising in their practical applications have

been fully discussed by Howarth (1973b, 1983).

The main objectives of the research involved in this thesis

are to investigate by computer methods any correlation between different

groups of data sets, and thereby to extract any features of geological

significance such as any potential mineralization or geological

variation.

This kind of computer-based analysis of a multivariate data

set is particularly useful because firstly computer analysis with more

data will provide unbiased and possibly more reliable results and also

subtle differences between data sets which are not very easy to

distinguish by other means may be readily distinguished in a reasonable

manner so that the result may provide insight into the geology.

Secondly full utilization of Landsat MSS data with its abundance and

repeatability of data would provide a cheap and fast way of data

analysis for preliminary reconnaissance surveying, particularly

4

where the area is not easily accessible and where any geological

or geochemical data are not readily available. Thirdly, the criteria

found in the chosen study area could be extrapolated to other areas

with similar geological conditions.

The research is involved with the integration of various

geophysical, geological and geochemical data bases with Landsat MSS

data for the chosen Bodmin Moor area of Cornwall, S.W. England, using

pattern recognition techniques as mentioned previously.

To achieve the research objectives, it is essential after

selecting good relevant data, to carefully make the digitization,

preprocessing (noise filtering) and data validation which are

essential to the data analysis.

Digitization of data was achieved by using either computer

interpolation techniques (gravity and geochemical data) or manually

(magnetic). For Landsat MSS data, an areal average method was used for

obtaining initial data for analysis and then the computer interpolation

technique was applied in order to project the data onto the same grid

points as used in the other methods.

In the noise filtering process, Landsat MSS and geochemical

data were filtered by a 3 by 3 spatial moving average method, and for

particular use in geophysical data, a 9 by 9 filter operator was

designed by using the concept of low-pass filtering.

The most important factors in data qualification, stationarity

and normality of data were also evaluated.

Various data sets were subject to analysis individually prior

to the multivariate data analysis using pattern recognition.

For geophysical data, various filter operators have been

designed for analysis to extract information due to regional or local

geological features. This includes vertical and horizontal

5

derivatives, upward and downward continuations, low- and high-pass

filtering and reduction to the pole of the magnetic data. Further

with the formula derived, pseudogravity filter operator for

2-dimensional data were then designed for data processing. This is of

great use to effectively analyse the coherence between gravity and

magnetic data, and thereby to extract spectral and further subsurface

structural features.

For Landsat MSS data, image enhancing techniques such as

contrast stretching using the Gaussian probability density function,

and ratioing techniques have been exercised and their colour composites

have been made for analysis.

The geochemical data were subject to level slicing by

concentration scale to examine the regional distribution of each

element and probability plots were applied to define threshold values

of anomalies and thus delineate spatial distribution of those anomalous oire&S.

Similarity analysis has been made of the data by using trend

surface analysis, analyses of correlation coefficients and similarity

map, and coherence analysis.

The information in this thesis will be presented in the

following sequence.

In Chapter 2, the general geomorphology and soil types,

regional geology and mineralization of the study area have been

reviewed. This is followed by data preparation and qualification in

Chapter 3. Chapter 4 deals with feature extraction of individual

data sets including geophysical, Landsat MSS and geochemical data.

The trend surface analysis is then described in Chapter 5 and

further similarity analysis in Chapter 6.

Finally various supervised and unsupervised pattern recognition

techniques have been applied for classification of geological signi-

fiance in Chapter 7.

6

CHAPTER TWO

GEOLOGICAL SETTING

2.1 Geomorphology and Soil Types

Cornwall can be conveniently divided into two major physio-

graphic upland units which are related to major rock types, the

Granites and the Slates. Extensive, gently sloping (influve) areas

lying from below 90m in Land1 s End to over 300m to the North Cornwall

characterize the land over Devonian and Carboniferous slates.

The granite upland rises from less than 200m in theLand's End area

to over 400m at Rough Tor on Bodmin Moor.

The climate is typical of southwestern Britain. Though very

mild onLand1s End and near the coast, conditions become more

climatically severe with increasing rainfall as the high Moor lands

are approached. Average annual rainfall ranges from over 1800mm

on the higher parts of the Bodmin Moor to below 1000mm on the coast

and along the lower parts of Land1sEnd.

The soils in the region are formed either directly in-situ

from local rocks, or by downslope movement of already weathered material

produced previously. Staines (1976, 1979) gives detailed description

of the soils in the region.

Much of the Devonian slate outcrop is dominated by soils belonging

to the Highweck series, of fine loamy typical brown earths in slaty

drift with depressions occupied by wetter soils. Close to the

granite contact are brown podzolic soils in the south but stagno-

podzolic soils in the north. On the volcanic outcrops in the north

brown podzolic soils dominate with humic gley soils in depressions.

The Upper Carboniferous outcrops in the north are occupied by stagno-

7

podzolic and surface-water gley soils.

In contrast, most of the higher, flatter ridges of the granitic

outcrop are occupied by stagnohumic gley soils, while the flanks are

occupied by stagnopodzols. Brown podzolic soils occur on the lower

formed granite land. On parts of Godolphin granite the granitic

head contains a significant proportion of the slate debris within it.

Depression sites such as broad valleys and valley heads are

occupied by humic gley and peat soils.

2.2 Geology

In the following description the regional geology of Cornwall

is abstracted mainly from Edmonds et al. (1975) and a detailed geology

of the Bodmin Moor area is reviewed from relevant Memoirs of the

Geological Survey of Great Britain Sheets and other published

references.

2.2.1 The Geology of Cornwall

The regional geological map of Cornwall is shown in Fig. 2.1.

The Pre-Cambrian igneous complex of the Lizard, which comprises

serpentinites, gabbro,hornblende schists and the Kennack gneiss and

granites, is known to contain the oldest rocks in this region. The next

oldest rocks are relatively small remanents of Ordovician quartzite

and Silurian limestone at Meneage and Nare Head. However, the region

is mainly dominated by intensely contorted thrust Devonian and

Carboniferous rocks (Killas) which are intruded by granite masses and

overlain in parts by relatively undisturbed Permian and later deposits.

Fig. 2.1 Geology of Cornwall (South-west England)

9

(a) Devonian and Carboniferous Rocks

The south-west peninsula of England is mainly composed of a broad

synclinal structure trending east-west. Devonian sediments occur

to the north and south, and the central area of the syncline is

occupied largely by Carboniferous rocks (Dearman, 1971).

The Devonian is divided into three groups; the Lower, the

Middle, and the Upper Devonian. It is mainly composed of slates, silt-

stones, conglomerate and some calcareous beds which vary in colour

from purple to green and grey. The Carboniferous rocks of south-west

England are divided into two series; a lower marine series of

evenly bedded shales and cherts with thin limestone, and an upper

series of a sequence of shales with turbidite sandstones and

siltstones (Reid, et al., 1911).

The Lower Carboniferous rocks occur in several tectonic slices

separated by thrusts and faults.

Extensive sheets of spillitic lava, tuffs and agglomerate from

volcanicity during the Lower Carboniferous times occur in northern

Cornwall and South Devon.

(b) Gran ite and Minor Intrusions

The oldest rocks of south-west England, apart from those of the

Lizard, are dark green dykes and sills called greenstones and they

include dolerite, gabbros and other basic igneous rocks. However,

the most widespread igneous rocks in Cornwall and Devon are granitic,

and they outcrop as cupolas from a single buried Airorican batholith.

The granites of south-west England occur as five great bosses

which trend WSW from Exeter to Land's End, and form the large topographic

features of Dartmoor, Bodmin Moor, St. Austell Moor, the Carnmenellis

and Land's End masses. Small bosses lie near the larger ones forming

10

Kit Hill and Hingston Down between Dartmoor and Bodmin Moor, Castle-an

Dinos and Belowda Beacon, north of St. Austell Moor and around

Carnmenellis Carn Brea, C a m Marth and Godolphin. Isolated from

Cornwall are the granite boss of the Scilly Isles and the underwater

HaigFras lying about 60 miles N.N.W. of the Scillies which may be part

of the same batholith displaced by faulting (Edmonds, et al., 1975).

Exposed contacts between granite and country rock (locally known as

killas) are always sharp, but often irregular and frequently complicated

by apophyses from the granites, or granite dykes cutting across the

contacts.

The Bouguer anomaly map of Bott, Day and Masson Smith (1958)

indicates that in general the granite contacts are steeper on the

south than on the north side. This is generally confirmed by the

aureoles (Edmonds, et al., 1975) and by gravity modelling of the granite

masses and drilling conducted by the Institute of Geological Science

(IGS) (Beer, Burley and Tombs (1975, Tombs 1977). Further confirmation

can be found in the 2-dimensional modelling conducted by Al-rawi (1980).

The granite is characterized by rugged tors dominating areas

of moorland or rough pasture. The rocks consist generally of quartz,

perthitic orthoclase crystals, plagioclase and brown mica (biotite).

Secondary white mica (muscovite) is locally present and consists of

tourmaline, zircon and apatite (Edmonds, et al., 1975).

(c) Mineralization

The mineralization in south-west England is widely recognised to

be related generally to the intrusion of the Amorican granites, while

later orogenic movements produced further mineralization. The

intrusion of quartz-porphyry, aplite and thin pegmatite dykes is also

associated with the late stages of granite emplacement. Hosking

11

(1949) suggested that the mineralization always occurs in close proximi

to the granite, but Dines (1956) noted that the relationship between

the mineral deposits to the granite bosses is not so close, based on

the study of lode distributions around the granite bosses.

A general fracture pattern, which eventually produced the

channels for the mineralizing fluids, probably developed at the

beginning of folding and granite intrusion. Later relaxation of the

pressure and the shrinkage caused by the cooling granite reactivated

the fractures. Local variations of stresses established distinct

local fracture patterns differing from the regional system.

The mineral content of the lodes varies both laterally and with

depth. Rich sections occur at changes of strike, intersections, and in

the steeply inclined sections. The lowest mineralization zone contains

dominant amounts of tin, while above it, is the copper rich zone, which

in turn is replaced upwards by a still more extensive lead rich zone.

Other elements present are zinc, arsenic, antimony, iron and manganese,

together with smaller amounts of tungsten, cobalt, nickel, uranium,

baryte and fluospar.

The gangue minerals associated with the ore-bodies in Cornwall

are mainly adularia, fluorite, white mica, tourmaline, chlorite,

hematite and quartz. Secondary alteration above the watertable

is widespread throughout the region. The upper parts of the lodes

consist mainly of gangue minerals with iron and manganese oxides,

forming the 'gossan1 or the so-called 'iron-hat' of the miner (Edmonds,

et al. , 1975).

There are currently five operating metal mines in S.W.England

and a number of developing prospects: Geever (Land's End), South Crofty

(Redruth), Wheal prosper and Mulberry (Lanivet), Wheal Concord

(Blackwater), and Hernerdon (Plymouth). All produce tin as the major

12

product with tungsten, copper, zinc, silver and gold as significant

associated products (Thorne, 1981).

(d) The Structure

During the Ordovician times the area was covered by a shallow

warm sea, deepening to the north towards the central Welsh trough.

The shallow sea became more extensive in Silurian times and stretched

from the Bristol Channel across eastern England.

Towards the end of the Silurian period the Caledonian Orogeny

uplifted the land to the north of the present peninsula and largely

produced a north-easterly structural trend. Later, the Amorican

orogeny encroached and produced easterly trending folds.

A Caledonoid zone in S.W. England is truncated south of a

line from Perranporth through Pentewan and the rest of the peninsula

is covered by the Amorican zone.

The Amorican orogeny, of late Carboniferous and early Permian

times, was characterized by north-south compressive forces which have

produced an east-west structural trend. Towards the end of

compression the Cornubian granite magmas, rising from great depths,

approached to within some hundreds of metres of the surface. Strata

were arched up over a number of cupolas and in places, as above the

Bodmin Moor granite, slices of pre-folded rock slid radially outwards

from the rising intrusion. Some of this low angle faulting took

place by reactivation along earlier thrust planes. As the granite

crystallised, and pressure relaxed, easterly trending normal faults

developed. North-westerly wrench faults may have been initiated by

earlier compressive forces, but many were reactivated in Tertiary times.

In general, the folds trend easterly but modification might have taken

place by Tertiary wrench faulting (Dearman,1964).

K E Y

HP Tertiary

New Red Rocks

Granite

Major fault lone

Major synclinal a m

Major anticlinal a m

Line ol confrontation between south-lacing and north-lacing lolds

Zone of intense detormation

Utiriltinil I'oini

Bude H'anmtt At<ntlit

Ciimhcuk/ Hitscv Ih'iicli• BosctsUe

FOLDS F A N N I N G O V E R S O U T H W A R D S I N T O

R E C U M B E N T F L E X U R E S P A R T L Y O B S C U R E C B Y \ L A T E Z I G Z A G F O L D S

LParfsfov

Tintsge I

^ T T e a r h . ' ' " ' R E C U M B E N t

V S O U T H - F A C I N G y

A v " " " (sr f- S & K ? & &

Portnrow ' o ^ ** T R U R O \ J P e n t e w a n

-f*c/ng f o l o s \ >

^Portnsdlcr

Doilnmn I'oinl Shirt Point

30 40

30

50

40

T~ 60 70

50 Miles

80 Kilometres

l.izonl I'oini

Fig. 2.2 The structural geology of S.W. England (After Edmonds, et al., 1975)

14

Open upright folds north of Bude fan out northwards

into northward-overturned folds and southwards into southward-

overturned folds. The latter become recumbent and isoclinial

towards Padstow, south of which the primary recumbent folds face

north.

The large low angle Rusey Fault marks a major tectonic break.

To the south of a line from Rusey Beach through Tremaine, deformation

is increased. Folding south of this belt is also recumbent but

south-facing with axes trending around E.N.E. and this pattern extends

to a line roughly from Padstow through the southern ends of the

Bodmin Moor and Dartmoor granites. At this line occurs a major

confirmation of south-facing folds to the north and north-facing folds to

the south within the St. Minver synclinorium. Possibly the two

opposing sets of folds are separated by a thrust or fault.

Immediately south of Padstow the folds are recumbent,

trending ENE, but south-dipping fold axial planes around Newquay, and

in a belt which extends eastwards, steepen southwards. A major

anticlinal axis, displaced locally by faults, runs eastwards from

Tremaine, north of Newquay, to Dartmouth.

The northern boundary of the Gamscatho Beds at Perranporth,

and all along the Perranporth-Pentewan line, was probably a slide on

which major north-facing recumbent folds moved in NNW directions

(Sanderson, 1971).

The Mylor Beds and Gamscatho Beds between Perranporth and

Penzance show recumbent first-phase folds trending ENE and facing north

or north-west, second-phase upright folds and third-phase flat-lying

folds.

The metamorphic rocks of the Lizard represent altered early

Devonian and older rocks. Movements at the end of Devonian times,

15

suggested by radiometric dating, may point to the thrusting north-

eastwards of a Lizard nappe.

The general structure of the region is shown in Fig. 2.2.

A general view of the structure is described by Dearman (1971)

and Edmonds et al. (1975) and a good description on structural

zonings of the Variscan fold in the region is given by Sanderson

and Dearman (1973). Geotectonic views of the Cornubian batholith

are mentioned by Bromley (1976) and Badham (1976), and Paleomagnetic

reconstruction for the upper paleozoic and the tectonic evolution

of the Variscan orogeny are given by Tarling (1979).

2.2.2 The Geology of the Bodmin Moor Area

The study area is in the British National Grid Square SX

and contained within British National Grid northing lines 9000-5190

and easting lines 20000-23810.

Fig. 2.3 shows a more detailed map of the area, centred on the

Bodmin Moor granite and surrounded by Devonian and Carboniferous sediments

with scattered minor complex intrusions. A part of the St. Austell

granite is also included in the south-west corner'of the area.

(a) Devonian Rocks

The oldest sediments in this region are of Lower Devonian

age; these are grouped into three categories; Dartmouth slates, Meadfoot

group, and Staddon grits. They consist mainly of slates, sandstones,

shales, conglomerates and some calcareous beds.

The Dartmouth slates are often siliceous and contain bands

and lenses of hard grit of quartzite. The rocks vary in colour from

purple to green and grey. In the Dartmouth slate the distinction

zz

Crackington Formation.

Crackington Form.* Lower C.

Lower Carboniferous.

Upper Devonian.

Upper and Middle D.

Middle Devonian.

Lower Devonian.

Lavas.

Elvan or quartz porphyry.

Diabase.(D).

Granite.

Picrite.

) CARBONIFEROUS

•DEVONIAN

/IGNEOUS

v.

>. '/Ay . -X' '.-y,^

Fig. 2.3 Geology of the Bodmin Moor area.

17

between cleavage and bedding is often obscure partly due to intense

thrusting causing the middle limbs of recumbent folds to be severed.

The Meadfoot group is distinguished by the presence of grey slates

with few fossils in their upper portion (Ussher, et al., 1907).

Surrounding the St. Austell granite, the Meadfoot group of slates contains

lenticules and beds of grit, often hard, compact, and evenly bedded

(Ussher,et al., 1909).

In the area between Bodmin and Grogley Downs the Staddon group

is encountered and the group consists of a number of thin bands

of grit or grey-wackes , hard and compact if fresh, dispersed

through a much thicker mass of killas of variable textures (Reid,et al,

1910).

To the north of the Staddon grit ridge in the Bodmin area are

the grey Middle Devonian slates, in which the only variations consist

of a few thin beds of limestone, some silty bands, and a few

intrusive sills (Ussher, et al., 1909). In the region between

the coast and Wadebridge, the slates are always grey and show well

the striking flatness of the cleavage planes. East of Wadebridge

the narrow Middle Devonian outcrop shows the usual EW strikes

but near the area where the granite outcrops the slate is disturbed

probably with much folding and faulting (Reid, et al., 1910).

The Upper Devonian rocks, unlike the Middle Devonian, are

very variable. In the lower part there are grey slates but higher

in the succession there occur purple and green slates, which are in

turn replaced by silty slates with thin bands of grit. The Upper

Devonian rocks are characterized by the presence of various belts

of contemporaneous lava and volcanic tuff (Reid, et al., 1910).

This group extends from near Tintagel Head, the north-west corner

of the study area passes north of Bodmin Moor granite to occupy a

18

large area to the East of the Bodmin Moor.

(b) Carboniferous Rocks

In the Boscastle district rocks termed Yeolmbridge Formation

assigned to the Lower Carboniferous, crop out in a belt trending

ESE from the coast near Boscastle. These rocks comprise mainly

fine-grianed, grey sericitic slates with scattered brown earthy

calcareous lenses and there appears to be a gradational downward

passage into possibly Upper Devonian slates. The slates are

commonly rust-spotted after pyrite, and aggregations of sericitic

and brown micaceous minerals show elongation due to tectonic stresses

(Mckeown, et al., 1973).

The Upper Carboniferous rocks are divided into two groups;

the Crackington Formation and the Bude Formation. However, the

description here is restricted to the Crackington Formation since

the Bude formation is outside the study area. The Crackington

Formation consists of dark grey cleaved shales with interbedded

siltstones, thin turbidite sandstones and a few thicker greyish

green sandstone units. To the east of Bodmin Moor there is a

transitional zone from the Lower Carboniferous to'Crackington

Formation (Mckeown, et al., 1973).

(c) Devonian Volcanic Rocks and Greenstones

There are various indications of contemporaneous volcanic

activity during Upper Devonian times. The volcanic rocks are splites,

schalsteins and tuffs. They are for the most part grey-green in colour.

The lavas are mainly tuffs with some vesicular, highly sheared and

occasionally porphyritic agglomerates.

19

Greenstones occur as sills or dykes, mainly intruded

into the lower beds of the Upper Devonian (Mckeown, et al. 1973, Reid

et al. 1910).

In the northern and eastern parts of the Bodmin Moor there

are many igneous rocks breaking through and penetrating the Upper

Devonian slates and often altering them to spicosites. These rocks

are mostly basic and ultrabasic. Several exposures of picrite are

flanked by long, wide, strap-shaped sills and possibly dykes of

diabase; partly contact-altered by the granite (Reid, et al., 1911).

(d) Granites and Later Intrusive Rocks

Except in the western and northern margins of the granite

moor, where the contacts are always marked by a steep slope, there is

generally no sharp ri se at the iunction with the slates. On the north

western side of the granite junction with the killas is a normal

intrusive one but towards the northern end it appears often to be a

fault. Over large areas the granite, whether fresh or decomposed,

shows little original variation either in composition or texture. In

hand specimen the granite is seen to be composed of orthoclase in

crystals about an inch long, quartz, in abundant grains of fair size,

a somewhat subordinate amount of plagioclase, and white and brown mica.

The Bodmin Moor granite is further linked to the other Cornish

granites by the occurrence within it of several good-sized

masses of finer granite, which are clearly intrusive. Schorl may be

an essential constituent of the intrusions as a whole. The

characteristic feature of this finer intrusive, is the almost complete

absence of the coarse grains of quartz, so abundent in the normal

rock (Reid, et al., 1910).

20

(e) Elvans

Numerous elvans or quartz-porphyry dykes, are associated with

the Bodmin Moor granite and they are most abundant toward the southern

and eastern end of the great intrusion, with a few extensions

westward toward Padstow and Wadebridge.

The elvans are slightly younger than the granite and the

edges of the elvans are always composed of fine-grained, felsic material,

and the centre is always coarser in texture than the edges.

The alteration appears to be connected to the infilling of

the tin-veins and is in all cases very pale in colour and almost

white. Two types of alteration can be recognised in the elvans

in this areas as in other districts of Cornwall; the first is

essentially the same as that which affects the granite when it is

converted into greisen. In this process the feldspar of the original

rock is replaced by aggregates of white mica and quartz. The second

type is tourmalinization (Reid, et al., 1910).

(f) Aureole of Thermo-metamorphism Surrounding the Granites

The Bodmin Moor granite is surrounded by an aureole of

thermometamorphosed rocks like those of other Cornish granites.

The surface width is approximately 600 to 1000 metres around the

Bodmin Moor.

The altered rocks are silvery-grey, or almost white but as the

proportion of chloritic material increases, the altered rocks become

darker and they also contain much more finely divided iron-ore

than those that are rich in white mica.

The unaltered killas is composed largely of minerals which contain

much water of hydration. The first effect of the granite intrusion

was to set free this water in the form of superheated steam, which

21

resulted in the metamorphic rocks within the aureole. The meta-

morphosed killas has almost always a foliated structure.

The eastern portion of the aureole of metamorphism presents

several points of special importance. In the first place, the rocks

in the eastern side of the granite were less affected by dynamic

action than those in the west, and in consequence they were less

schistose when metamorphosed by the intrusion. Further the junction

of the granite with the killas is often a fault, and the fault

trends, roughtly, east-west, the same as the mineral lodes (Reid,

et al., 1910).

The St. Austell granite and the smaller outlying granitic stocks

are also surrounded by well-marked aureole of contact meta-

morphism or pneumatolytisis (Ussher, et al., 1909).

(g) Mineralization

The principal mining areas in the area of study are situated

in the metamorphic aureoles of the great granite bosses as in other

areas of Cornwall.

In the Bodmin Moor area, the most important part of the area

for its mineral production has been the south-eastern part of the

granite. East-west striking lodes rich in tin and copper have been

extensively worked.

They also line up with similar striking lodes in the Kit Hill-,

which are mainly of tin and copper associated with lead, arsenic and

silver, etc. Most mines are rich in pyrites.

The oxides of iron also occur in the form of gossans,

particularly between St. Austell and Wadebridge to the west of the study

area, most iron lodes trend north-south and the lodes mainly

consist of red and brown hematites.

22

The deposits of manganese are usually associated with oxides

of iron; they appear to be connected with cherts in the Carboniferous

series (Reid, et al., 1911, Mckeown, et al., 1973).

Important lead zones have also been extensively worked at

Menheniod and Hendsfoot, and antimony was worked near Duloe and

Pillaton in the south of Bodmin Moor granite (Ussher, 1907).

Recently, some industries are more actively conducting some

exploration in the south-eastern margin of Bodmin Moor granite and near

the Kithill where a large number of east-west trending sulphide lodes

occur (Prest, 1982).

In St. Austell granite, the principal mining area is situated

in the metamorphic aureole on the southside of the granite. The

principal deposits occur in the killas and for the most parts are

associated with quartz-porphyry dykes (elvans). The lodes of

St. Austell and of the whole region in the north are mainly tin-bearing.

There also occur mineralized lodes younger than the main tin and

copper lodes. Some of these lodes contain minerals which are also

characteristic of tin-lodes, such as arsenic, copper, and small

quantities of uranium, cobalt, and nickel ores. These younger lodes

have been formed in fissures crosscutting the tin'or copper lodes,

and these younger lodes have a strike direction between north and

northwest, similar to the cross-courses.

St. Austell district is particularly important as a producer

of remarkably white and pure china clay, which is an alteration product

of the feldspathic constituents of the granite. While the eastern

part of the granite is comparatively free from this kind of

decomposition, the central and western portions show a widespread and

very complete kaolinization (Ussher, et al., 1909).

23

CHAPTER THREE

DATA PREPARATION AND QUALIFICATION

3.1 General Considerations in Data Preparation and Qualification

Bendat et al (1971) and Bath (1974) have described in

detail the data preparation procedures for digital processing of

data.

Data preparation is considered as one of the most arduous

tasks to be faced in any efforts to any data processing. This is largely

done by two steps; digitization and preprocessing.

Digitization is the process of creating machine readable

values for digital data processing, the sampling of which is usually

performed at equally spaced intervals. The problem is then

to determine an appropriate sampling interval,AS. On the one hand,

sampling at points which are too close together will yield correlated

and highly redundant data and thus unnecessarily increase the

labour and cost of calculations, whereas on the other hand sampling

at points which are too far apart will lead to confusion between high

and low frequency components in the original data', and lose the

resolution. The latter problem is known as 'aliasing'. In the space

domain, if the sampling rate is 1/AS samples /m where AS is the

spacing in meters, the maximum frequency we can pick up is f = 1/2AS.

Here f is so-called the Nyquist frequency or folding frequency.

Two practical methods exist for handling this aliasing problem

(Bendat et al, 1971). The first is to choose AS sufficiently small

so that it is physically unreasonable for data to exist above the

associated cutoff frequency f .

Generally it is a good rule to choose f to be one and a

24

half to two times greater than the maximum anticipated frequency

(Bendat et al. 1971, Agarwal 1968). The second method is to filter

the data prior to processing so that information above a maximum

frequency of interest is no longer contained in the filtered data.

The second method is preferred over the first in order to

save computing time and thus costs. However, the combined case

should be always considered between information of interest and

computing time.

After the data has been digitized and filtered, prior

to actual analysis, certain operations must be performed in order to

qualify the data, since the correct procedures for analysing

random data, as well as interpreting the analysed results, are

strongly influenced by certain characteristics which may or may not

be exhibited by the data.

The most important of these basic characteristics involved

with the study are the stationarity and the normality of the data.

Stationarity should be analysed because the analysis procedures

required for non-stationary data are generally more complicated

than those suitable for stationary data. The validity of an assumption

that the data have a Gaussian probability density' function should be

investigated as the normality assumption is vital to parametric

analysis of the data.

3.2 Interpolation and Extrapolation

An account of the interpolation technique used in the

gridding is described.

Serious problems with edge effects are usually encountered

when data processing either in space or in frequency domains. Thus,

25

an appropriate technique should be applied in order to reduce edge

effects which might lead to spurious results, in case of analysis in

frequency domain, or loss of information around edges by stripping

off half of the filter window around the border when convolving in

space domain. This can be done by applying an appropriate extra-

polation technique.

Extrapolation techniques both in space and in frequency

domain have been developed.

3.2.1 Interpolation

There are a variety of interpolation schemes, differing not

only on the principles but also sometimes on the properties of data.

In any case, the most important features are firstly to calculate

values as near as possible to the real values and secondly to introduce

minimum noise in the digitization process which can affect the computed

spectrum.

The gridding program applied in this study was originally

written by the USGS (Program W9322) and it utilizes the inverse-square

method. That is, the weighting factor applied is'inversely proportional

to the square of the distance. Mathematically, for a particular grid

point (i,j), the computed grid value Z(i,j) is

n 2

Z X(k) * l/dZ(k) Z ( i , j ) = — (3.1)

kS L l/d2(k)

t h

where x^k) t'"ie ^ originally measured value which is separated by

d(k) from the grid point (i,j). If the distance is in the grid unit,

the range of d(k) for a particular point (i,j) is

26

(i-1) <_ d(k) £ (i+1) in row

(j-1) _< d(k) £ (j + 1) in column

For those map points which are coincident to the data points,

the values themselves are assigned to the grid point. For missing points

which might occur particularly where the originally measured data th are sparsely distributed, either k local points or the 4-point

th boundary average method is applied. In the K local averaging method

for a missing point searches are made for calculated local grid

points by increasing the range of search-distance radially, and if

any calculated values are encountered, the average of those grid

values is taken for that missing point.

The 4-point boundary average method involves averaging

values at the nearest 4 points for the missing point value.

In this study, the latter method has been applied to obtaining

missing points in the gridding procedure because it is computationally

a little more efficient.

If the originally measured data is fairly uniformly

distributed, this is a very effective method particularly in potential

field data as the decay pattern of the field is generally inversely

proportional to the square of the distance. However, it has been

noted that if the data is sparse, it may introduce some high frequency

noise. This effect will be demonstrated in Section 3.3.

3.2.2 Extrapolation

(a) Extrapolation in Space Domain

In the space domain, information at edges can be lost up to

27

half of a filter window around the border of a map when convolving.

This can be avoided by generating data at ends of a profile or

around the border of a map. However, the problem with an extrapolation

is that in this case the information at edges is distorted so that

it might induce some spurious results.

A very reasonable way of extrapolation has been developed to

minimize the distortion, so that the information at edges might be

as useful as others (or at least no significant distortions at edges

in worst case ).

The technique uses the geometrical property of an exponential

function and the local background. This method is computationally

efficient.

(b) Extrapolation in Frequency Domain

The fundamental approach for reducing edge effects in

frequency domain is different from that of space domain, because of

differences in assumption of data structure. In the frequency domain

the data is assumed to be periodically continuous while in the space

domain it is not necessarily so.

The edge effects in the frequency domain-are due to the

truncation of data which results from violation of assumptions (Bath,

1974). This truncation of data introduces the so-called Gibb's

phenomenon where some unwanted oscillations are introduced into the data

in the processing. This phenomenon is larger at the edges of the data,

so that the processed results could be ambiguous at edges. Although

it is not presented here, the author has experienced serious edge

effects when Gaussian filtering was applied to the gravity data to

reduce the high frequency noise in the frequency domain. Another

problem is to avoid introducing any high frequency noise due to

28

overshooting when extrapolating, particularly with a short data

length.

The most common method may be the cosine tapering (Bath,

1974). Computationally a little more laborious, but one of the most

effective methods has been developed using Ku's (1977) algorithm

of simplified cubic spline method used in his interpolation scheme.

The so-called 'mean slope' method is advantageous compared to the

conventional cubic spline methods in that it attenuates overshooting

by interpolation so that high frequencies may not be introduced in

extrapolation even with very short data length.

Both extrapolation schemes either in space or in frequency

domains are described in detail in Appendix A.

3.3 Digitization of Two-dimensional Data

Considering the area being covered around the Bodmin Moor

and peculiarities of the research, a sampling interval of

300m would be required to make a 128 by 128 grid array and thus the 2

total area covered is about 1500Km . It is bounded by the British

National Grid coordinates 5190N - 9000N in the north-south and

20000E - 23810E in the east-west directions as mentioned in Section

2 . 2 . 2 .

Digitization was carried out either manually by projecting

a grid on the field contour maps and taking readings at regular interval

(magnetic), or automatically by using a computer program with irregular

data recorded on the magnetic tapes (gravity and geochemical data).

Landsat MSS data for the area were acquired by averaging for a certain

area and gridding using the computer program. Details of the method of

gridding each data set will be described in the next section.

29

Actually the first method is considered as the least

efficient way of performing conversion. This is usually done

in cases where no other alternatives exist. However, its accuracy,

when carefully done, compares favourably with other more sophisticated

methods. For example, the raw gravity in Fig. 3.2(a) obtained

by using the gridding program contains more high frequency noise

through the data than the raw magnetic map in Fig. 3.2(b) obtained

manually.

Besides noise being introduced in the digitizing process,

particularly when the gridding program is used for digitization, noise

also may be introduced in data acquisition procedures due to

geological, field disturbances, instrumental or cultural effects

and it may also be caused in data reduction procedures.

Aliasing was not examined by analysing the frequency

response of a profile but visually inspecting the contoured maps.

Since there is virtually no loss of resolution, as shown in

comparison between two sets of gravity maps in Fig. 3.1 with 128 by 128

grid data, and Fig. 3.2(a) with 64 by 64 grid data (which has been

taken for every second value in every second row), the data were

further reduced to a set of 64 by 64, to give a sampling interval of

600 meters. The advantage of reduced data size, is that the smaller

set is much more efficient to handle, particularly in multivariate data

analysis and yet it is still large enough for regional analysis of

multivariate data sets.

The noise in the data was reduced by applying appropriate

filtering techniques. This filtering aspect will be described in

the following sections.

30

3N*r(20oooe,qooo>4)

0 io 20 wm 1 I 1 ' S c a t e

Fig. 3.1 Contour map of Raw gravity data in gravity units. 128 by 128 array size.

31

(20QftOE,qoQON)

(a) Gravity

I 1 1 1 L

120000E ,<1ooon)

Fig. 3.2 Contour maps of potential field data (a) Gravity (b) Magnetics. 64 by 64 array size.

32

3.3.1 Digitization of Geophysical Data

(a) Gravity Data

The field data for the Bouguer gravity anomaly map was obtained,

reduced and compiled by the Institute of Geological Sciences (IGS).

Tombs (1977) has described the data reduction procedures in some detail. 3

A density of 2.67gr/cm was used for the reduction and the

Bouguer anomalies were calculated against the International Gravity

Formula 1967 and referred to the National Gravity Reference NET 1973.

For topography ranging within a radius of 50km (Hammer zone, A to M) 3

the data were terrain-corrected for an average density of 2.7gr/cm .

The terrain-corrected Bouguer anomaly values expressed in ten

times a gravity unit were dumped on the magnetic tape and gridded at

an interval of 300m. The density of field observations over the

Bodmin Moor area that was gridded is about 70 data points per 100 sq.km.

(b) Magnetic Data

The total magnetic field contour map was published by the

IGS in 1958 at 1/25,000 or 1/50,000 scales over the on-shore areas

of south-west England. The contour interval of this aeromagnetic map

is 10 gammas. This map was compiled from an airborne survey conducted

in 1957 by Hunting Survey Limited, under the contract to the Geological

Survey of Great Britain (now part of IGS) and the United Kingdom Atomic

Energy Resource Authority. The aeromagnetic survey was flown with a

flight line separation of 2km or closer, and with tie lines spaced

at about 10km. The mean terrain clearance was approximately 150m.

In the reduction of data , diurnal corrections were applied

for the data and then the backgrounds based on the IGRF (International

Geomagnetic Reference Field) was removed.

33

The contoured maps, at a scale of 1/25,000, or 1/50,000

where 1/25,000 scale of maps were not available, were obtained from

the IGS and digitized manually at every 300 meters grid intervals.

Some differences were observed between the two scales of maps, but

these were not significant and showed some consistencies, so that

reasonable adjustments could be made manually.

3.3.2 Digitization of Landsat MSS Data

The MSS in Landsat 1 and 2 is a four band scanner with six sensors

in each band. It operates by receiving the Earth's reflected solar

radiation in the spectral region from 0.5 to l.lym. The spectral

range of each band for the MSS used in the Landsat 1 and 2 are

Band 4 0.5 - 0.6)Jm

Band 5 0.6 - 0.7lJm

Band 6 0.7 - 0.8]Jm

Band 7 0.8 - 1.1pm

At the nominal orbital altitude and attitude, the MSS scans cross-

track swaths of 185km in width, simultaneously imaging six scan lines

across in each of the four spectral bands. The scan lines which are

projected on ground are approximately 185km long and in a west-

to-east direction. The arrangement of scanning is shown in

Fig. 3.3.

34

orrica

Fig. 3.3: MSS Scanning Arrangements (After European Space Agency Earthnet programme 1979).

The data obtained for this study was already radiometrically and

geometrically corrected and converted to a usable binary form on

Computer Compatible Tape (CCT), with the radiance values ranging

from 0 to 255 in all four bands. Each pixel of the data represents , r. ..m _,»m . . , approximately 57 x 79 in its area! coverage.

Further details on the Landsat MSS data acquisition procedure

are described in European Space Agency Earthnet programme (1979),

Siege I et al. (1980) and others.

The CCT tape for the south-west England produced at FUSINO

receiving station in Italy was purchased from the Remote Sensing Unit

(RSU) of the Space Department at the Royal Aircraft Establishment,

Farnborough. The CCT tape data was imaged by Landsat 1 on 24,4,1975.

35

The track number and frame number are 220 and 25 respectively, and

the centre location of latitude and longitude is (50.15N, 5.10W) in

decimal degrees of latitude and longitude.

The data over the Bodmin Moor area was obtained as follows. . lines pixels The data obtained was averaged for a point value by 8 x 10

which were then recorded on the DISC with X(E-W) and Y(N-S) coordinates

converted into the British National Grid Unit with respect to a

provisional origin. In order to project the Landsat data on the User's

grid", the coordinates were rescaled using a few reference points

on the map and readjusted against the gridding program mentioned

previously (Section 3.2.1) in order to obtain coincident data

sets on the Bodmin Moor area. Full detailed accounts are given in

Appendix B.

3.3.3 Digitization of Geochemical Data

The original geochemical data were obtained from the Wolfson

Geochemical Atlas, which was initially compiled by the Applied Geo-

chemistry Research Group at Imperial College for regional geochemical

analysis of stream sediment samples through Englahd and Wales. This

was to delineate potentially mineralized districts and also to provide

fundamental geochemical information related to the regional geology

(Webb, et al. 1978).

Digitization of the geochemical data was done by using the

gridding program to interpolate the data onto the User's grid

coordinates. Eight elements, Arsenic (As), Copper (Cu), Gallium (Ga),

" User's grid system on this thesis was devised by the author in order to standardise data from various sources for the purposes of this study. The origin is at (9000N, 20,000E) BNG.

36

Lithium (Li), Nickel (Ni), Lead (Pb), Tin (Sn) and Zinc (Zn), have

been chosen for the analysis since these elements are closely related

to local mineralization and geology (Howarth: Hale, personal

communication). The density of measured data over the area is about

40 samples per 100 sq.km.

3.4 Noise Evaluation

3.4.1 Estimates of Noise Contributions

The power density function yields information on the amplitude

of the dominant harmonics of which the data is composed (Agarwal, 1968).

This enables one to analyse noise levels in the data by using the

power spectrum of the data.

A study of power spectra was carried out for the raw and

filtered data sets using a two-dimensional Fast Fourier Transform

(FFT) program. For easy comparisons, the two-dimensional power spectra

were represented as one-dimensional spectra by averaging radially.

Fig. 3.4(a) to 3.4(n) show one-dimensional representation of power

spectra of the raw and filtered data. The extrapolation as mentioned

in 3.2.2(b) was applied prior to applying FFT for 'the analysis.

On the whole, the raw data show much higher power spectra

at shorter wavelengths than those of filtered ones. The raw gravity

data shows quite significant noise levels in Fig.3.4(a). This is

partly due to very near surface density variations, of little interest

to this study, and possibly occasional erorrs in the original

measurements or data reduction process. The gridding process may

also introduce some high frequency noise as mentioned before. The

contoured map of raw data in Fig. 3.2(a) clearly shows high frequency

effects. The energy spectrum of raw magnetic data in Fig. 3.4(b) also

represent those of raw data and dotted lines are of filtered data. (Nyquist frequencies indicated by t)

Fig. 3.4 continued

V.« i*. !• rfi 4. m •'.«« •'.«• i'm Ml Out OCT Cicus/tod Klin (1) Pb

(j) Li

i

(m) Sn

s

(k) Ni

39

shows some high energy levels at shorter wavelengths.

In the Landsat data the power spectra of raw data at shorter

wavelength are quite high compared to those of longer wavelengths,

which might depict the characteristic reflectivity of surface or

near surface features in the data.

The geochemical data have varying amounts of noise as

clearly shown in one-dimensional power spectra from Fig. 3.4(g) to

3.4(n).

Gilman et al. (1962) has noted in the study of power spectra

for time series of atmospheric and solar indices that the spectral

characteristics is subject to a general suppression of relative variance

at higher frequencies and consequent inflation at low frequencies, which

is called 'red noise'. Problems with discrete Fourier transforms

in estimating power spectra are also given by Cordell and Grauch (1982)

in their comparative study with integral Fourier transform. Thus, the

spectral representation may somehow be distorted. However, this

phenomenon is not considered to be serious in this study and has been

disregarded in the qualitative interpretation.

From the analysis of one-dimensional power spectra, it

can be concluded that the maximum powers are concentrated at longer

wavelengths, whereas at shorter wavelengths there are some spectral

highs which may be due to noise either of small scale geological

features of little interest to this study or digitization, etc.

3.4.2 Noise Filtering

Prior to the actual analysis of data, any high frequency

noise which may cause erroneous results should ideally be removed.

However, practically such a complete removal of unwanted noise is not

40

possible. Thus, some optimal procedures are applied to reduce the

noise, while preserving the signal.

Bee Bednar (1982) has noted in seismic data filtering that

weighting average is most effective when the additive noise and

signal component occupy different portion of the frequency spectrum.

Thus, for geophysical data, a 9 by 9 filter operator has been

designed using the principle of low-pass filtering as described in

Appendix C-5. This filter passes any frequencies lower than one and

a half of the Nyquist frequency. The filter operator so designed

is illustrated in Table 3.1.

TABLE 3.1: Noise filter coefficients (fourth quadrant)

0.05500 .04755 .02981 .01174 .00102 0.04755 .04109 .02568 .01006 .00087 0.02981 .02568 .01591 .00162 .00051 0.01174 .01006 .00612 .00227 .00018 0.00102 .00087 .00051 .00018 .00001

For Landsat MSS and geochemical data, a 3 by 3 box-car window

smoothing was applied to reduce the random noise in the data.

The calculated values might contain noise averaged over nine points,

but as assumed the error is random, the calculated errors tend to be

much smaller than the individual error. The effect of this procedure

through a frequency approach in a one-dimensional case is given by

Jenkins and Watts (1969, p.50).

The application of different noise filtering techniques is due

to their different physical properties, and perhaps due to the objective

of the study and the user's experience.

To analyse the noise levels of filtered data, a two-dimensional

power spectrum of data sets have been evaluated. The two-dimensional

power spectra of filtered data are shown in Fig. 3.5(a) to 3.5(n).

41

I

L

(a) Gravity

2

Mf QuCfcC 11$ .t ICIU'OOI" i 600" ' (b) Magnetic

Fig. 3.5 Two-dimensional power spectra of the filtered data in log scale (a) and (b) show gravity and magnetics, (c) to (f) are of Landsat MSS four bands, (g) to (m) are of eight geochemical elements of As, Cu, Ga, Li, Ni, Pb, Sn and Zn.

42 Fig. 3.5 continued

t n .

0 A

< ; J - <3 <

MfOtlCWCKS .IICKS/0«I« I»:i«»ail600ni (e) MSS Band 6

(g) As rxoucuciti .CTCifi/MT* mKivmitaoni

TO- : »

vjVi ^ • ^ u ° TO ii - o i

^ TO W

Er ' - TO

K-t:

• •taufxcifs .cicits/ooio i*it«von6ocmi

FXOuCkCltS .C»Cll5/0«1» imi«»«ll(00HI (f) MSS Band 7

MtOUtNCICI .CTCLCt/MT* INTIIVNll900A1 (h) Cu

'•lOutkCICS -CICKS/DO!" |NI(«vai I 600" I

(i) Ga (j) Li

Fig. 3.5 continued

IMMXIII ,CIClCt/0«t» |«t(a«*lllOWII (k) Ni

43

MfOuCXKI .C»CIIS/D««» l»»l«»KIMO"l

(1) Pb

(m) Sn (n) Zn

44

In this case extrapolation was not applied because it was unlikely

to significantly alter the general features of the spectrum. The

comparative noise levels between the raw and filtered data can be seen

more clearly in one-dimensional representation of two-dimensional

power spectra in Fig. 3.4(a) to 3.4(n).

All the two-dimensional power spectra show that the maximum

powers are concentrated at the centre and more or less circularly

symmetrically diminish to the high frequencies, except a few data

sets particularly including gravity and magnetics which have some

elongations to axial lines. Clearly noise levels have been

significantly reduced after the Nyquist frequency as indicated by

arrows in one-dimensional power spectra of filtered data.

Typical examples of filtered gravity and magnetic data are shown

in contoured maps of Fig. 4.1(a) and (b). Compared to the original

maps in Fig. 3.2(a) and (b), respectively, the filtered map eliminated

all high frequency noise and show smoother patterns than the original

maps (See Chapter 4).

3.5 Data Qualification

As mentioned in Section 3.1, two basic data qualifications

- stationarity and normality - have to be done for their validity in using

appropriate data processing, so that improvements for appropriate

statistical analysis can be achieved.

3.5.1 Stationarity

The stationarity of sampled random data depends upon the

physics of the phenomenon producing the data. Almost all physical

processes are more or less non-stationary. The degree of stationarity

45

may be a matter of gridding scale (Bath, 1974). In general we may

consider phenomenon with slow variation as stationary or quasi-

stationary, while it is customary to consider a rapidly varying

phenomenon such as seismic p-wave as transient and non-stationary.

There may be various evaluation schemes of stationarity

ranging from visual inspect of the data to detailed statistical tests

of appropriate data parameters.

Bendat et al (1971) has described an assumption of statinarity

can often be determined by a simple non-parametric test of sample

mean square values computed from the data. Mazzarella and Cesere (1980)

has written a program called MARKOV using Bendat's algorithm

for analysis of stationarity of a time series.

The mean square value MSQ is defined as,

MSQ = MEAN2 + DEVT2 (3.2)

where MEAN is the mean value and DEVT is the standard deviation

of the sample. Then the process of the test of stationarity of a random

sample in the program is summerized as follows:

1. Divide the sample in equal intervals (termed NA) of width (MS) where the data'in each interval may be considered independent.

2. Compute a mean square value (MSQ) for each interval and align these sample values in time sequence.

3. Test the sequence of mean square values for the presence of underlying trends or variations other than those due to expected sampling variation, using a nonparameteric test such as the RUNS test.

The Markov process was used by Palumbo and Mazzarella (1980) in a study of rainfall statistical prediction schemes of the atmosphere.

In this study, MARKOV has been modified to be used for

46

analysis of stationarity of the data line by line. Three different

kinds of tests have been conducted. The first is a test using

the filtered data itself and the second and third ones are tests

after regional trends have been removed. The two regional removal

techniques are 'Spencer's method of smoothing' and 'Double

Exponential filtering' (Davis, 1973, pp.226). The formulae used

are as follows:

1. Spencer's method of smoothing

? i = 350 ( 6 0 Yi + " ' W ^ i - l ' + 4 7 ( yi +2 + yi-2> +

33(yi+3+yi.3) • :8(yi+4yi_4) • fity^i-s' -

" 5 < yi+7 + yi-7 ) ~ 5 ( yi +8 + yi-8 ) " 3 ( yi + 9+ yi-9 ) " (yi+10+yi-10):>

(3.3)

2. Double exponential filtering

Y. = 0.3ly. + 0.16(y. +y. ,) + 0.08(y. +y. J + I J i l+l i-l i+2 i-2 0.04(y. +y. 0) + 0.02(y. ,+y. .) (3.4) J i+3 i-3 i+4 i-4

The result of the analysis is given in Table 3.2.

Tests for the original four Landsat MSS bands show that

about 30% of total lines are stationary at 0.05 per cent level,

but the rest of the data sets show none or very low rate of stationarity.

It is interesting to note that although the gravity and magnetic

data are usually considered to be at least quasi-stationary, none

of the lines satisfy the requirement of stationarity in the original

data. Geochemical data show also very low rate of stationarity in the

original data. These might be due to the regional trend of the data.

Thus, two kinds of regional removal have been applied further to

examine the stationarity; Spencer's method of smoothing and the

Double exponential filtering as mentioned before.

47

When Spencer's method of regional removal is applied, all

of the data sets generally increase the rate of stationarity with

maximum rate 75 per cent in the Landsat MSS band 6. In case of

the use of the Double exponential filter they show even higher

increase in their rates with maximum 100 per cent in the Landsat

MSS band 4. There are a few exceptions with geophysical data when

the rates are decreased compared to those of Spencer's method.

TABLE 3.2: Test result of stationarity. Row by row tests so that each variable consists of 64 test data profiles. Each variable was subjected to three different ways of testing: original, regional trend removed by Spencer's method and regional trend removed by the Double exponential filter (DEF). recorded numbers are those lines which satisfy the stationarity at 0.05 level of significance.

Regional trend Regional trend Variables original removed by removed by REMARKS

Spencer DEF

Gravity 0 36 16 number of Magnetic 0 12 10 degrees of MSS Band 4 18 25 64 freedom of MSS Band 5 23 27 61 the RUN test MSS Band 6 15 45 60 for these MSS Band 7 16 43 59 data = 15 As 7 12 30 Cu 0 16 26 Fe 6 29 50 Ga 3 33 45 Li 2 29 42 Ni 1 1 30 45 Pb 3 6 20 Sn 5 1 1 17 Zn 3 6 35

48

This may indicate that an appropriate regional removal should

be applied prior to exercising the test of stationarity. Howarth

has noted that it is possible to analyse the randomness of the data

by statistical tests of the trend surfaces. This account will be

described in Chapter 5.

3.5.3 Normality

The parametric statistical analysis assumes that the

distribution of samples is normal. Yet the results of an investigation

will be subject to considerable uncertainty if the samples are

drawn from a population whose characteristics are incompletely

understood (Howarth et al. 1979). Thus test of normality is essential

for appropriate statistical analyses.

Perhaps the most obvious way to test stationary random

data for normality is to measure the probability density function of

the data and compare it to the theoretical normal distribution. If

the sampling distribution of the probability density estimate is

known, various statistical tests for normality can be performed.

However, a knowledge of the sampling distribution of probability

density measurements requires frequency information for the data

which may be difficult to obtain in practical cases. In this case,

a nonparametric test is desirable.

One of the most convenient nonparametric tests for normality

is the chi-square goodness-of-fit test. Details of parametric

statistical tests of normality can be found in numerous statistical

references, for example, Davis (1973), Bendat et al. (1971) etc.

and various nonparametric tests are described by Siegel (1956) and

Bendat et al. (op. cit).

In this study, visual inspection of histogram of the

49

samples has been performed qualitatively to analyse the normality

of the data.

Histograms of various filtered data sets are shbwn in

Fig. 7.1 in Chapter 7. Histograms of four Landsat MSS data are

somewhat similar to patterns of normal distribution, except for long

tails toward low and high ends of data. The gravity data show the

typical pattern of bimodal distribution and the magnetic and all of

the geochemical data are positively skewed to some extent. As

expected, not only the geophysical and geochemical data but also the

four Landsat MSS data have been rejected at 5 percent of

significant level with nonparametric chi-square test of the normality.

Thus it is necessary to transform the data to near normal

distribution for approriate data analyses as otherwise the

analysed results will suffer from loss of reliability. The

transformation techniques will be described in Chapter 7. The

histograms of frequency distribution of filtered data are shown in

Fig. 7.1 in Section 7.2.

3.6 Discussions

Data preparation including digitization and noise filtering

may be one of the most difficult tasks in the data processing,

particularly when several different types of data are being

considered. Various different filtering techniques have to be

applied partly due to the different physical properties of the data

and depending on purpose of the study. This is because the noise

contents in the data are usually unknown except in a very few cases

where the input signal is correctly known. The concept of noise

is also partly subjective to the study involved depending on the type

50

of study.

In this study, the raw gravity and magnetic data were subject

to filtering with a specially designed filter operator in order to

reduce the noise, and all the other data were subject to box-car

smoothing filtering using a 3 by 3 window function.

Comparative plots of one-dimensional power spectra of the

raw and filtered data show significant reduction of energy at

shorter wavelengths, and thus random noise in the data has been

removed largely by filtering.

To increase the reliability of the result, it is essential

to perform certain operations in order to qualify the data for

appropriate data processing. These include tests of stationarity

of data for correct procedure of analysis and further normality

of data for an appropriate statistical analysis.

The filtered Landsat MSS show generally some degree of

stationarity, but the filtered geophysical and geochemical data show

none or a very low degree of stationarity. However, after removing

the regional trends by Spencer's method of smoothing or double exponential

filtering, the degrees of stationarity increase rapidly. Thus in

general it can be concluded that all the data are'at least quasi-station-

ary and also often stationary.

Except for Landsat MSS data, all the other data show significant

skewness, so that all the data including Landsat data were subject to

transformation of data to at least near-normality using the power

transform method for all the data except for gravity where an

arcsine transform method was applied.

51

CHAPTER FOUR

FEATURE EXTRACTION

4.1 Introduction

Feature extraction is the process of summarizing all of the

information and reducing it to a smaller, more manageable set, thus

eliminating as much of the redundant information as possible, while

retaining as much of the essential information as is required. There

are various techniques depending on the nature and properties of the

data set and the aim of study.

In this chapter, various feature extraction techniques have

been applied to sets of data individually including potential field

data, four Landsat MSS data, and eight geochemical elements of As,

Cu, Ga, Li, Ni, Pb, Sn, and Zn.

In geophysical data processing for gravity and magnetics, a

number of filtering techniques including derivatives, continuations,

high- and low-pass filtering, reduction to the pole, and pseudogravity

methods have been applied to extract any regional and local features

of geological interest.

Landsat MSS data were subject to image enhancement by contrast

stretching using the Gaussian probability density function and further

ratioing techniques in order to enhance features of the images.

For geochemical data, level slicing in concentration levels

(i.e. the slicing levels are determined by an exponential function; see

Section 4.4 for more details) was applied to examine the general

distribution patterns of each element. Probability analysis was also

made to determine threshold values to the geochemical halcfs for Cu, Pb,

Zn, and Sn.

Performing analysis of individual data sets is particularly important

52

prior to multivariate data analysis because firstly it extracts

certain types of anomalies and secondly properties of individual

data sets can be understood, so that the analysis and interpretation

of the multivariate data may become much easier.

4.2 Geophysical Data Processing

4.2.1 Filter Operators for Analysis of Potential Field Data

One of the most important problems in the interpretation of

potential field is the separation of the data giving information

characterizing different geological structures. This process is

called 'filtering', the procedure of which is to separate anomalies

of particular wavelength from the original data, so that the anomaly

features can be accentuated.

Various mathods were utilized to perform this separation

either in space or in the frequency domain, for example, vertical

derivatives, upward and downward continutations, lowpass and highpass

filtering, reduction to the pole, etc.

Analysis in the frequency domain involves the Fourier transform

of the data and removes those elements in the frequency domain corres-

ponding to unwanted effects, and applies an inverse Fourier transform

to obtain the resulting map in the space domain.

Alternatively a fully equivalent operation of convolution

filtering can be used to achieve the same objective without implementing

the Fourier analysis. The digital convolution of the filter with the

map is as follows

53

a-l b-l Z(x,y) = Z Z X(x - x,y - A) F(x,A) (4.1)

x=0 y=0

where X(x,y) is the input data, F(x,X) the lagged filter and Z(x,y)

is the filtered output. a and b are the filter windows in x and y

directions, respectively, and the filter windows are in most cases

equal.

Since the Fast Fourier Transform (FFT) was introduced (for

example, Cooley and Tukey 1965, Robinson 1967 etc), the frequency

analysis was preferred to the spatial analysis due to its computational

efficiency with readily available transfer functions. However, it has

some difficulty in computation of large data sets due to its requirement

of the double memory space. Further difficulty arises from well-known

'Gibb's phenomenon' which tends to introduce severe oscillations which

are particualrly high at edges of data as mentioned in Section 3.2.

Those problems in the frequency analysis can be avoided by

convolution in the space domain. However, convolution has two dis-

advantages; firstly much computational time is required and secondly

there occurs a loss of information at the margin of the data set when

convolving. Since most of the filter operators have the circularly

symmetrical property (i.e. in a discrete square grid the cirularly

symmetry is equivalent to 8-folded symmetry) except in a few cases

such as the reduction to the pole and pseudogravity, the computational

time with convolution can be considerably reduced by folding about the

orthogonal axes, so that for a moderate size of filter operator the

computation may be much less than that of frequency analysis. Comparative

numbers of multiplications involved between ordinary convolution, 8-

folded convolution, double Fourier transform (DFT), and FFT are given

in Table 4.1.

54

Table 4.1: Comparison of the number of multiplications required between ordinary convolution, 8-folded convolution, discrete Fourier Transform and Fast Fourier Transform.

filter size

Ordinary Convolution

8-folded Convolution

DFT FFT Remarks

5 x 5

7 x 7

9 x 9

11 x 11

13 x 13

15 x 15

17 x 17

21 x 21

25 x n'

49 x N

81 x N

121 x N'

169 x N'

225 x N'

289 x N'

441 x N'

6 x N'

10 x N'

15 x N

21 x N'

28 x N'

36 x N'

45 x N'

55 x N

.. Data 4N log_N 2 size N x N

2

In case of 64 x 64 data set, FFT requires at least 49 N

multiplications (this number includes multiplications with transfer

function and the inverse FFT) which is even greater than 8-folded

convolution with 17 x 17 filter size.

The other disadvantage in convolution, strip-off effect at edges,

can be solved if an appropriate extrapolation technique such as the

exponential extrapolation as mentioned in Section 3.2 is applied.

Therefore, in many cases such as vertical derivatives, continua-

tions, pass filters etc., spatial analysis is preferable to the

frequency analysis computationally, and in its usefulness of the processed

results if the filter operators are properly designed.

As noted by Mufti (1972% it is possible to design relatively

small operators which will give results practically equivalent to those

from much larger operators but with a greatly reduced number of numerical

operation. Thus, analysis in the space domain by convolution can be

55

used as it is computationally efficient and effective.

4.2.2 Review of Filter Operators

(a) Derivatives

The second vertical derivative filtering is often used to

enhance the character of maps from the potential field surveys. The

method has the property of amplifying small-scale fluctuations relative

to broad-scale anomalies, so that the resultant map may indicate

certain geological features of small scale not easily identifiable

on the original map. The theoretical second derivative is unsuitable

for use with practical data because of noise in the data. The presence

of noise can lead to divergence of the expressions for differentiation

of the map. This fact has led to the awareness that useful expressions

are with the theoretical equations with an appropriate weighting function.

Various different operators have been proposed by many authors for

computation of second vertical derivatives: The early works include

those by Peter (1949), Henderson and Zietz (1949), Elkins (1951),

Rosenbach (1953) and Nettleton (1954) etc. and later by Henderson (1960)

Paul (1961), Agarwal and Lai (1971), Agarwal and Lai (1972) etc.

Recently Gupta et al (1982) has successfully applied so called optimum

second vertical derivatives designed by analysing the power spectrum

of the gravity data in geological mapping and mineral exploration .

Frequency analysis of the filter operators (Me sko 1965, 1966, Danes and

Oncley 1962, Darby and Davies 1967, Zurflueh 1967, Fuller 1967,

Kanasewich and Agarwal 1970, Agarwal and Lai 1971 etc.) provides an

insight to understand the nature and purpose of differentiation when

applying these operators upon gridded data.

56

It may be clear from these analyses that there is no uniquely

preferable operator in practice. The choice of operators depends on the

depth or size of the structure of interst, and the probable random

noise present in the data. Agarwal and Lai (1971) have shown that

although the filter operator by Rosenbach (1953) is much nearer to the

theoretical response than that of Elkins (1951), it is not to be preferred

to the latter in some practical applications where the data contain high

frequency noise. This is because the Rosenbach filter operator tends

to emphasize the noise or smaller features of no interest and thus lead

to difficulties in the interpretation.

Agarwal and Lai (1972) suggested empirically that the maximum

frequency content of the anomaly is inversely proportional to the depth

of the structure. Mathematically, they represented this by expression:-

Maximum frequency * Depth = Constant

It is therefore obvious that in cases where we are interested in deeper

structures a lowpass filter such as Elkins (1951), or Agarwal and Lai

(1972) where a = 0.23 (see appendix C.8), would yield good results,

whereas such filters may not have much application for shallow structures.

(b) Upward and Downward Continuations

The process of potential field continuation above or below the

original datum can be used to map subsurface geological structures by

interpreting the continued field. <

Upward continuation is used in potential field data interpretation

as a filter to remove the effect of shallow sources relative to those

of basement (steenland and Brod 1960, Nettleton and Cannon 1962,

Kanasewich and Agarwal 1970).

57

Downward continuation can be used to sharpen the anomalous

sources and to separate sources with overlapping effects. It has much

the same general effect in localizing anomaly boundaries and trends

as a second derivatives or residual calculations (Nettleton, 1976).

Upward continuation is a straight forward operation, since the

surfaces are in field-free space. In downward continutation it is

more unstable because of the inherent uncertainty in the location and

size of the structures represented by the field at the datum plane

because we must assume that there are no sources between the levels

over which the continuation is being made.

Roy (1966, 1967) and Rudman, et al (1971) have examined using

model studies the reliability of the downward continuation to determine

the maximum possible depth and to outline the variation in the shape of

homogeneous sources at various depths. If continuation is carried out

to depths greater than the source, the continued field will begin to

oscillate: this oscillation could be a criterion of depth of the source.

The half maximum value at the oscillation level may be used as an

outline of the top of the source.

Peters (1949) and Henderson and Zietz (1949a) have designed

sets of coefficients for computation of the continued field. Henderson

(1960) has reviewed the effectiveness of various coefficient sets,

including comparison of actually measured and continued fields.

Fundamentals of continuation fields in the frequency domain

have been described by Henderson (1966), Nagi (1967) and Roy (1967) etc.

The computation of continued fields in the frequency domain can be

achieved by multiplying the Fourier transformed data with the transfer t 2 x TO z (u + u ) , , . . . , function, e , and applying an appropriate weight Wt in the

case of downward continuation. If Z is positive, downward continuation

58

is computed, while if Z is negative, upward continuation is computed.

In the above formula u and u are angular frequencies in x and y

directions, respectively. In case of upward continuation no weigth

is applied since the continued field is much smoother than the

original field (Fuller, 1967).

(c) Lowpass and Highpass Filtering

Two-dimensional wavelength filtering is a means of separating

anomalies of different wavelength from each other, so that regional

trends, or other local features that are not easily detectable on the

original map can be emphasized. Separation by filtering is based on

the assumption that the near surface, presumably higher wave number

components of the field, can be isolated from the lower wavenumber

trends caused by deeper sources.

The general theory for the design of two-dimensional filters

to be applied to regular grid data has been given by Darby and Davies

(1967), Fuller (1967) and Zurflueh (1967).

The basic procedure applied in these papers is to specify in

digital form the desired wavenumber response of th'e filter (lowpass,

highpass or bandpass). Filter coefficients to be applied in the space

domain are determined by taking the inverse Fourier transform of the

wavenumber response.

Baranov (1975) has applied the function sin (ttx/Q)/ttx in order

to design the lowpass filter operator in the space domain. Since the

convolution of any function F(x) with the above function has the effect

of replacing the function F(x) with an arbitrary spectrum by a function

f(x) whose spectrum does not exceed tt/Q, various lowpass filter operators

can be designed by varying Q, so that desired lowpass filtering maps

are obtained for interpretation.

59

Alternatively, one can also carry out the filtering operation

in the frequency domain and then obtain the inverse Fourier transform

of the resultant spectrum.

Polynomial surface fittings are also applied to problems which

can be analysed with two-dimensional wavelength filters. However,

there are several advantages in using wavenumber filters as pointed out

by Zurflueh (1967). First pseudo-anomalies are a severe problem with

polynomial surfaces, especially if the data have a large dynamic range.

Filters are more reliable in this respect. Secondly the goodness of

fit of the polynomial varies in different parts of a map and the fit

becomes unreliable at the edges. Wavelength filters yield uniform

results over the whole extent of a map. Thirdly, filter programs can

process larger amounts of data in one computer operation than is possible

with polynomial programs.

(d) Reduction to the Pole

Reduction to the pole method is a mapping procedure helpful

in the interpretation of magnetic anomalies. Its purpose is to convert

data which have been recorded in the inclined earth's magnetic field

to what the data would have looked like if the magnetic field had been

vertical for the causative body, thereby reducing the distortion in the

pattern of magnetic anomalies resulting from the dependence on the angle

of magnetic inclination. Thus the map of data reduced to the pole may

be interpreted more easily (Baranov and Nandy 1964, Bhattacharrya 1965,

Kanasewich and Agarwal 1970, Nettleton 1976).

Some examples of filter oporators for computation of the reduced

to the pole are given by Baranov and Nandy (1964) and Baranov (1975).

Ervin (1976) has developed an algorithm for the reduction to the

pole using a Fast Fourier Series by considering the inducing magnetic

60

field only. Bhattacharrya (1965) has described the filtering procedure

in the frequency domain, taking into account both the induced and

remanent components of magnetization. Kanasewich and Agarwal (1970)

has modified Bhattacharrya's method in order to take advantage of FFT.

Spector (1975) has also shown the transfer function of the reduction to

thepolewhich can be implemented with FFT.

Obviously, in computational effort, analysis of reduction to the

pole in the frequency domain may be preferred. However, as mentioned in

Section 3.2, it is preferable to analyse in the space domain in order to

avoid difficulties in the frequency analysis with requirement of large

computer memory space and also spurious oscillations in the processed

map by FFT.

(e) Pseudogravity

Baranov (1957) first introduced the pseudogravity formula using

thepoisson's relation of practical application to gravity and magnetic

data interpretation.

The pseudogravity is a process to convert the magnetic field

into a gravity field. The transform involved is the elimination of the

distortion due to the oblique angularity of the normal magnetic field,

so that the resulting anomalies will be located on the vertical line

above the disturbing magnetized bodies and do not depend on the inclination

of the magnetic field, nor on the direction of the magnetization. Thus,

the interpretation and all the subsequent computations become very simple.

However, as was noted by Affleck (Baranov, (1957)), the pseudo-

gravity anomalies represent the effects of the magnetic rocks only, while

the observed gravity anomalies represent the combined effects of all

rocks, magnetic and non-magnetic. Thus Aina (1979) has pointed out in

61

his review of the method that the full capabilities of Poisson's

equations in combined gravity and magnetic data interpretation have not

been fully realized, and he has also suggested that further development

should be made for special cases of situations whereby anomalies are

caused by arbitrary distribution of magnetization and density in irregular

shaped bodies.

The application of this method was made for the Fourier transform

approach by Kanasewich and Argarwal (1970), for the Matrix method by

Bott and Ingles (1972), and for the Hilbert transform method by Shuey

(1972).

Spector (1975) has also given the transfer function of the pseudo-

gravity which can be implemented into FFT in the frequency domain. So

far in the literature no description has been given of attempts at

interpretation of potential field data by analysis with filter operators

in the space domain on a regular grid.

4.2.3 Theoretical Background of Filter Operators

Baranov (1975) has described details of potential fields and

derived some filter operators such as vertical derivatives, continuations,

reduction to the pole and so on.

The theoretical background of potential fields is briefly

described and further developments to pseudogravity and first horizontal

derivatives for the magnetic field are described.

The Laplace equation holds for potential fields,

V2u = 0 (4.2)

It can be solved in the cartesian coordinates as,

62

, v -iax-iBy-yz u(x,y,z) = e 3 (4.3)

2 2 2 where y = a + 3 (4.3.1)

A more generalized solution of Eq (4.2) is given by Baranov

by multiplying by a constant as

u(x,y,z) = 4/

u(0)B) e-iax~^y-yz d a d e (4.4)

- 7 T

where u(a,3) is the Fourier transform of the function

u(x,y,0) = u(x,y), that is

u(a,3) = ^ ^ ( x . y ) dxdy (4.5)

Filter operators such as continuations, reduction to the pole, first

vertical derivative, lowpass filter and first horizontal derivative

for potential fields have been computed for this study using Baranov's

method derived from Eq(4.4). Pseudogravity and first horizontal deri-

vative operator for magnetic data have also been designed in this study

from the derived equations. The second vertical derivative operator

is readily available from many publications as detailed in Section

4.2.2a. In this study the second vertical derivative operator has been

computed by using Agarwal's method (1972).

For all filter operators so designed, Hann's shortening

operator (Blackman and Tukey, 1958) used by Fuller (1967) were applied

for the edge correction in order to prevent a sharp cut-off at the

edges by truncation of the operators, and finally a weighted normalization

designed by the author has been applied (i.e. the sum of filter

63

operators is equal to 1 except those for derivative operators in which

case it is equal to zero).

The formula for shortening operator is

S(k,n) =

+ cos 7r(k2 + nV/(x 2 + y V for | k| £ x

|n| < y

0 for |k| > x

|n| > y (4.6)

The newly developed normalization procedure is

W(k,n) = W(k,n,d) + |w(k,n,d)| (4.7)

S S where AA = 1 - £. E, W(k,n,d) in cases of continuations, pass filters, k=I n=l reduction to the pole and pseudogravity operators and AA = -

W(k,n,d) in cases of derivative operators , AB = I I |w(k ,n,d)| and k— i. n— i

S the filter size.

A comparative study of the shortening operators by Agarwal and

Singh (1977) has shown that Martin's shortening operator is best for

normalizing sets of filter coefficients. In many cases, however, the

shortening operator combining Hann's shortening and weighted

normalization seems to be much preferred to Marten's method of

shortening in the author's experience. This is because the shortening

operator designed by the author can achieve both sharpening at the

edges and normalization by weighting rather than the normalization

only produced by Martin's method which often encounters problems with

sudden edge cutoffs.

Detailed calculations of filter operators including the

64

filter coefficients used in the study are given in Appendix C and

the programs for computing filter operators are listed in Microfiche 2

in rearfolder.

4.2.2 The Description of GEOPAK Program

Spatial filtering is an areal analysis technique that can be

applied to any form of contourable geological information. The process

involves digitizing the input data, designing a filter and convolving

the filter with the data.

GEOPAK has been written initially by ROBINSON (1971) in order

to utilize spatial filtering techniques. This is an integrated

package of FORTRAN IV subroutines whose main features are those of the

convolution and transformation of the convolved data into alphanumeric

character sets to print out the map data on the lineprint output.

The program has been modified in order to enhance the

efficiency and effectiveness of the computing by adapting several

significant subroutines. These include

1. 8-folded convolution

2. extrapolation of the data

3. gray-scale line-print output, etc.

(1) 8-folded convolution; The convolution of the data in the space

domain is the operational Fourier transform of multiplication. The

process thus requires the multiplication of the corresponding map data

with a filter and adding them all to produce a resultant data set

representing the value at the centre of the filter window and this

process goes through the map until an entire new map is produced.

Spatial filtering permits a large map to be handled compared to the

analysis in the frequency domain.

65

The convolution is thus the main routine of the program, which

largely dictates the computing time. The initial program computes the

convolution by multiplying every filter coefficient with the corresponding

map data and adding the all resultants together, so that the computing

is rather slow.

Since most of the filter operators have the circularly symmetrical

property (this is equivalent to 8-folded symmetry in the gridded system),

it is possible to reduce the number of multiplications significantly

by folding about symmetrical axes. The comparative numbers of

multiplication are shown in Table 4.1.

(2) Extrapolation; One of the most serious problems in spatial

filtering with convolution is the loss of information when convolving

the data with a filter operator. The strip-off of data by convolution

is as much as a half of the filter window around the border of the map.

To avoid this loss of information at the edges of the data, a reliable

extrapolation technique was designed using the exponential function

(see Section 3.2 and Appendix A).

(3) The gray-scale line-print output; This program subroutine

initially written by Blenkenship (Gonzalez and Wintz, 1977) has been

modified to print out the map by darkness depending on the amplitude

of the values. Several slicing techniques including equal-interval,

log-level, power level and absorption level slicings are employed in

the routine in order to be suitable for a variety of data sets which

show different patterns of frequency distribution. For example, geo-

physical data are commonly applied with equal-level slicing, while

geochemical data may be adequately treated with power-level or

absorption level slicing.

The flow chart of the GEOPAK program used and its descriptions

are illustrated in Appendix D and the program list is on Micro-iche 1

66

in rearfolder.

4.2.5 Qualitative Interpretation of Potential Field Data

(a) Description of the filtered gravity and magnetic maps

The filtered maps of gravity and magnetic data are shown in

Fig. 4.1 (a) and (b), respectively. In Fig. 4.1 (a), the main features

of the Bouguer anomaly map are that large negative anomalies occur in

the centre and southwest corner of the map which correspond to the Bodmin

Moor and a part of the St. Austell granites. Further a small patch of

negative anomaly appears in the centre of the eastern margin, the locality

of which corresponds to the Kithill granite. All these negative anomalies

are attributed to the low density of the granitic masses and they do

appear to be connected in the subsurface with the general WSW-ENE trend,

as noted by many authors,for example, Bott, et al 1958.

Apart from the granite bosses, the Bouguer gravity anomaly is

gradually increasing toward the sediments surrounding the granites, and

the southern flank of the anomaly over the Bodmin Moor granite usually

shows steeper gradients than the northern part.

Another interesting feature is the EW trending high anomaly in

the southern border of the map. Similar features also appear in the

northeast and northwest corner of the map. These features are difficult

to interpret uniquely from the gravity data alone. However, considering

the gravity data with the magnetic data may indicate that the northern

and southern gravity anomalies are due to different causes; in the

south the anomaly might be due to the limit of the granite batholith

and a deepening of the sedimentary basin but in the north it might be

due to uprising of the basement with only a local thickening of the

sediments as a contributing factor in the distortion of the contours

67

(a) Bouguer gravity anomaly (in gravity units)

O to ao Km • * * * * Scale

Fig. 4.1v Contour maps of the fittered potential fields of the Bodmin area with a cutoff wave length of 800 m.

68

in the northeast of the map.

Some irregularities in the contour map, particularly around

the Bodmin Moor granite may be due to differences in subsurface relief

and density of overburdens possibly partly controlled by fracture

patterns.

The magnetic map in Fig. 4.1 (b) shows strong positive and

negative anomalies trending eastwest or WNW-ESE in the northeast

quadrant of the map. The anomalies trending WNW-ESE correspond

generally to areas in which basic intrusive trending in this direction

are widely distributed. The strong negative anomaly north of the

main granite is coupled with a high positive anomaly to the north

of the negative althou^iit is not included in the map area.

Al-rawi (1980) has noted by bandpass filtering of the regional

data that the large positive anomaly can be attributed to deeply buried

basic rocks and he suggests that the direction of the remanence

vector in the basic rocks is southerly dipping. Thus, the negative

anomaly might be partly complementary to the positive anomaly.

Elsewhere, there appear no particular features but the general

tendency is the gradual increase from south to north with a largely

east-west trend. The magnetic map shows generally no relation to

the granite bosses, but may reflect largely the trend of crystalline

or metamorphosed basement under the main granite, which may be getting

deeper to the south.

Areas of some distortion and steep gradient in the northeast

may be related to some local magnetic features as well as some structural

features such as folds and faults etc.

(b) First-and second-derivative maps

The gravity and magnetic derivative maps including first

69

vertical, second vertical, and first horizontal derivatives have been

calculated with the appropriate filter operators designed (see Appendix

C),by convolving them with data from the Bodmin Moor area using the

GEOPAK program package as mentioned before.

Fig. 4.2 (a) to (c) show the first, second vertical derivative

and first horizontal derivative maps of the gravity fields, respectively.

The first and second derivative maps show broadly similar features;

small patches of positive anomalies are distributed around the sediments

surrounding the granites, and some highs also occur on the granite area.

However, the first horizontal derivative map in Fig. 4.2 (c) shows

almost the mirror image of the vertical derivative maps. The positive

anomalies in the first horizontal derivative map are generally around

the boundary of the granites where the steepest gradients occur, whereas

the positive anomalies in the first and second derivative maps appear

to be outside the granite outcrop. The reason may be that the first

horizontal derivatives pick up maximum gradients of any anomalies,

while vertical derivatives outline local anomalies or edges of anomalous

bodies. Particularly anomalous areas of vertical derivatives might

correspond to areas of inhomogeneities in the shallow subsurface which

might be related to local structures such as small granitic subsurface

intrusions. These kinds of subsurface intrusions could be related to

local mineralizations.

On the other hand, all the magnetic derivative maps from Fig.

4.3 (a) to (c) show that the general anomaly patterns are very similar

to each other, except in the magnitude of anomalies. The greater

amplitude in the second vertical derivative map (Fig. 4.4 (b)) may

indicate that the successive vertical derivatives accentuate the

relative effect of the shallower features such as the near surface

70

liOoooE, Rooo N) • —

M

(a) First vertical derivative map \o 2o Km

Scale lao&ooe, qoooN)

(b) Second vertical derivative map

Fig. A.2 Derivative maps of the gravity data. (a) First vertical derivative, (b) Second vertical derivative and (c) First horizontal derivative map.

71

continued from Fig. 4.2

0 § ip £.0 Km Scale

72

0 10 oo Km 1 I I I L

Fig. 4.3 Derivative maps of (a) First vertical derivative and (c)

the magnetic data. derivative, (b) Second vertical First horizontal derivative map.

73

continued from Fig. 4.3

0 io 20 Km \ i i • • (20ODOE, qoooN.) Scale

N 4

(c) First horizontal derivative map

74

basic intrusions the same as those in the gravity maps.

Except in the north-eastern part of the map, there appear no

significant features. The strong positive anomaly trends in a WNW-ESE

direction and there are associated negative anomalies to the south of the

positive ones. Other patches of anomalies trend approximately east-

west in the middle of the eastern margin of the map. All these anomalies

have been shifted very little from the original magnetic map.

Elsewhere, weak anomaly patches bearing an east-west trend

appear on either side of the main anomaly in the north-west and north-

east corner of the map.

All of these anomalies correspond to those areas where basic

rocks have been geologically mapped, so that these anomalies would

appear to be caused by the shallow basic rocks which have a high magnetic

susceptibility contrast with the surrounding rocks. In both gravity and

magnetic derivatives, numerous interruption zones can be found in the

maps. They are often found to be associated with fractures, faults,

and other geological lineaments.

One must be cautious in applying derivative filtering that

the resultant maps of the derivative filtering do not necessarily

contain high and low anomalies which have structural significance but

may be the result of algebraic properties, particularly as might be

produced in the gravity derivative maps.

(c) Upward and downward continuation maps

The gravity and magnetic data in the Bodmin Moor area were

continued upward to 1200 metres and downward to 600 metres being

multiples of grid interval (600 m) .

The upward continuation map of the gravity field in Fig. 4.4

(a) shows smoothed anomaly features associated with the granites. The

75

o 10 20 Km

. 4.4 Upward and downward continuation maps of the gravity data. (a) upward (h = 2), (b) downward (h = 1) continuation maps, h is in terms of grid spacing (600 m). Contour interval 25 gravity unit.

76

£o Km

Fig. 4.5 Upward arid downward continuation maps of the magnetic data. (a) upward (h = 2) and (b) downward (h = 1) continuation maps, contour interval 25 gammas.

77

(a) Gravity |0 20 Km

S c a l e (IOOOOE^OOON)

V / (b) Magnetics

Fig. 4.6 Upward continuation maps of the gravity and magnetic data (h = 3). (a) Gravity (b) Magnetic.

78

contribution of filtering in the upward continuations is not very \

significant partly because in the filtering process of the raw data

only the high frequency components have been significantly reduced.

On the other hand, the downward continuation map in Fig. 4.4

(b) shows that this operation has sharpened the local inhomogeneities.

In Fig. 4.5 (a) the upward continued magnetic map also shows smoothed

anomalies, attenuating largely small anomaly features trending WNW-

ESE and showing the broad E-W anomaly trend associated largely with

basement structures.

The strong negative anomaly north of the Bodmin Moor granite

appears to indicate a magnetic body extending to depth. The downward

continuation map in Fig. 4.5 (b) certainly shows a sharpening of small

anomaly features and localizes them compared to the filtered map in

Fig. 4.1 (b).

This may be comparable to filtering features of the derivative

maps which also have the same general effect in localizing individual

anomalies.

A further upward continued map to 1800 metres (h = 3) in Fig.

4.6 (a) and (b) shows smoother features than the map at 1200 metres

(h = 2). Particularly the magnetic anomaly features trending NW-SE in

the northeast margin of the granite have nearly completely disappeared

in the filtered map to 1800 metres (h = 3). These upward continued

maps clearly indicate the causative basic rocks extend to a limited

depth only.

(d) Lowpass and highpass filtered maps

The lowpass filtered map has been calculated from the filter

operator designed to pass frequencies lower than 0.3 cycles/600 m.

This is equivalent to a cutoff wavelength of approximately 2000 m. The

79

(a) Lowpass filtered map o 11> ao Km t 1— J 1 1

Scale

Fig. 4.7 Lowpass and highpass filtered maps of the gravity data. (a) lowpass filtered and (b) highpass filtered maps. Cutoff wavelength is 2 km.

80

(a) Lowpass filtered map 0 >o 20 Kffl 1 I ! ! l

s c a l e

Fig. 4.8 Lowpass and highpass filtered maps of the magnetic data (a) lowpass filtered and (b) highpass filtered maps. Cutoff wavelength is 2 km.

81

highpass filtered map is conveniently called that because it is the

residuals of the lowpass filtering, and it must represent effectively

a high frequency range of the data as described in Section 4.2.2.

Fig. 4.7 (a) and (b) illustrates the lowpass and highpass

filtered maps of the gravity data, respectively. The lowpass filtered

map shows the regional trend, eliminating the small local features,

whereas the highpass filtered map delineates local anomalies associated

with shallow geological structures.

The highpass filtered map is much the same as those of vertical

derivatives (see Fig. 4.2 (a) and (b)) and its patterns are strikingly

inverse to those of the horizontal derivative map in Fig. 4.2 (c).

The lowpass filtered map of the magnetic data in Fig. 4.8 (a)

shows the broad features to be largely east-west trending, which

reflect the deep subsurface structural features possibly associated

with the American earth movement.

The highpass filtered map in Fig. 4.8 (a) shows much the same

patterns as the vertical derivative maps, emphasing and isolating

local anomalies more clearly. Henderson and Zietz (1949) and

Swartz (1954) have also described the similarity between residuals and

derivatives. The positive anomaly trending WNW-ESE appears to be due

to the effect associated with the shallow basic igneous rocks since

this anomaly is nearly attenuated in the lowpass filtered map, as

also described in the upward continuation map (see Fig. 4.6). However

the strong negative anomaly abruptly interrupted in the centre of the

northern margin of the map appears to be caused by a body extending

from near the surface to great depth as partly confirmed in the

upward continuation map.

(e) Reduction to the pole map

82

Fig. 4.9 shows the reduction to the pole map of the total

magnetic intensity field. This has been calculated by convolving

the magnetic data with the filter operator designed (see Appendix

C.2), by assuming that the magnetization vector of the total field

is the same as that of the inducing field (Inclination = 68°,

Declination = 10°W). Thus if the magnetic anomalies are mainly caused

by only induced magnetization the reduction to the pole map will produce

a symmetrical anomal located above the causative body.

The reduction to the pole map is very similar to the filtered

map (Fig. 4.1 (b)), but a careful examination shows a slight shift of

the anomaly patterns in the former map toward the north. This may

be expected if we consider the steep inclination and a shallow

declination applied in the design of the filter operator. According

to Cornwell (1967), mean values of 188° in declination, -13°in

inclination were calculated for the Exter lava which is of the

permian age. Although the ages may be different between the two lavas,

considering large difference from the present dipole field direction

and the high Q value (>1), the magnetization vector of the total field

for the areas of volcanic lavas might be much different from the current

inducing field vector. Al-rawi (1980) indicated this in his regional

study for the strong positive anomaly coupled with negative anomaly

in the north of the granite, so that those strong positive or negative

anomalies may not be correctly represented by the transformations.

(f) Pseudogravity map

Further transformation has been made to convert the magnetic

data into gravity. Fig. 4.10 illustrates the pseudogravity map

calculated from the 21 by 21 filter operator. The anomaly patterns

[•20ooo&,qooou)

o to 30 Km I 1 ! ! I

Scale

Pseudo-gravity map of the magnetic data

84

are much the same as the reduction to the pole map except the anomalies

are shifted very little toward the north. The smoothed feature of Fig.

4.10 may be due to the filter size (21 by 21), whose coefficients are

not alternating but this phenomenon does not arise in the case of

the reduction to the pole because they are of alternating signs.

4.3 Landsat MSS Data Processing

4.3.1 Introduction

Remote sensing plays an important role in geological exploration

because it provides a quick, economic, overview particularly in arid

and semi-arid areas where the terrain is inaccessible. Geological

applications of remote sensing are well described in numerous references,

for example, by Siegal et al (1980), Sabins (1978), Smith (1977), etc.

Though aerial photographs were recorded by Daguerre as early as

1839 (Reeves,et al,1975), the concept of the Earth Resources Technology

Satellite (ERTS; later Landsat) was developed in the late 1960's by

joint NASA/USGS studies, stimulated by the proven geological value of

orbital photographs from the Mercury and Gemini flights.

Landsat 1 was launched in 1972 and began returning imagery

every 18 days from both imaging systems: the Return Beam Vidicon (RBV)

television system and the Multi-Spectral Scanner (MSS). Land 2 and

3 (c) were launched in 1975 and in 1978, respectively. The images have

been used for a wide variety of applications. Landsat 4(d) is now

operational, after being successfully launched on schedule on July 16

1982.

In the beginning of the geological applications of Landsat images

85

it was a simple small-scale comparison with the small

geological map (Lowman, 1969). Structural sketch maps made from the

colour-composite have in many cases shown lineaments not included in

the geological map. The detection of many large unknown structures

gave indication of potential value of Landsat imagery for the study of

regional structures.

In this study, geological applications of remote sensing data

are two fold: regional lithologic mapping and mineral exploration.

Lithologic mapping from images can often be accomplished by

analysis and interpretation of the spectral and spatial information

within the images. The spectral and spatial information available

for lithologic identification is expressed by landform development,

drainage pattern and density, vegetation differences, and spectral

reflectivity, all integrated in the context of climatic effects (Abrams,

1980).

Contrast enhancement is one of the most widely used image

processing techniques for lithologic mapping. It is the process of

redistributing frequencies in an image to maximize the contrast.of

areas of interest. Details on various contrast st-retching methods

are described by Taranik (1978), Siegal and Gillespie (1980) and

others.

Drake (1975) and Houston et at (in Freden et al, 1974)

have shown that colour-composite mapping using bands 4, 5 and 7

made it possible to make many stratigraphic subdivisions.

Thillaigovindarajan et al (1979) have also noted that using the 2

I S Additive Viewer the false-colour composite was found to be the

best for interpretation of linear and circular features.

Rationing is another common processing technique for lithologic

mapping. This is a method of enhancing minor differences between

86

materials by defining the slope of the spectral curve between two

bands. Ratio images also tend to reduce the effects due to topography

and to emphasize the changes in brightness values in materials (Chavez,

1975).

However, at the same time, ratio images accentuate noise, making

interpretation more difficult. In addition, dissimilar materials

having similar spectral slopes but different albedos, which are easily

separable in standard images, may become inseparable in ratio images

(Siegal et al 1980). Ratio images have been used in many geological

investigations for lithologic mapping by Blodget et al (1975) and others.

Rowan et al (197 6) has noted that the R4/5, 5/6, and 6/7 composite

was found to provide the greatest amount of information for discriminating

rocks.

Newton (1974, 1981) has noted by comparative study of several

different types of aerial photography for geological interpretation

that colour and infrared photographs are only marginally better for

study of superficial deposits and even under relatively unfavourable

conditions, normal black-and-white photography can be of great value,

particularly in the early stages of mapping an unknown area.

However, in many cases, colour photographs provide greatly

increased information content over black-and-white images since the

human eye can discriminate many more shades of colour than it can

tones of gray.

Thorough study of visible and near-infrared spectra of minerals

and rocks are mentioned by Hunt et al (1971 a, b, c, d, e, 1972, 1973,

1974 (a) and (b), 1977) .

The application of Landsat imagery to exploration for minerals

and hydrocarbons is one of the most difficult to study because of the

proprietary nature of the information. Correlation of lineament

87

studies with field mapping, aeromagnetic and gravity surveys,

geochemical sampling and relations to known ore deposits have been

described by Raines (1976, 1977), Salas (1977), and Rowan et al

(1981). Economic applications have been summarized by Vincent

(1977) and many others.

For mineral exploration, there are three promising methods

of study: the regional study of linear features, analysis of areas

of alteration through the use of multispectral data, and the

application of geobotanical techniques.

The regional features in Landsat images may generally

indicate structural regions, representing the surface expression

of faults, fold, lithologic contacts, or other geological discontinuities.

Many authors (for example, Rowan et al 1975, Correa 1975, Baker 1977,

Carter and Rowan 1977, Mercanti 1977, and Rowan et al 1979) have noted

that the study of linear features has often led to the discovery of

ores. This may be because ore deposits are generally related to some

type of deformation of the lithosphere and u.ost theories of ore formation

and concentration comprise tectonic and deformational concepts (Siegal et al

1980). Fernandez, et al (1979) and others have noted that of the many Landsat

features the circular ones are most likely to be correlated with the

presence of mineral deposits.

Regarding the analysis of alteration zones, gossans may be

valuable indicators of mineral deposits that are concealed beneath

the weathered surface, although not all gossans are associated with

ore bodies. The colours of gossans contrast with those of adjacent

country rocks, but most gossans are relatively small so that they

are difficult to detect on Landsat images. Hydrothermal alteration

zones are more areally extensive, so that it is one of the most

useful indirect approaches to exploration for ore bodies that is

based on studies of gaugue minerals associated with ore deposits.

88

Rowan et al (1976, 1977) and Carter and Rowan (1979) have

noted that ratioing of MSS bands have proven to be the most

effective method to detect hydrothermal alteration.

Mineral concentrations at depth often accompanied by

secondary surface indicators, many of which (e.g. tonal anomalies

associated with geochemical or geobotanical stress-gossans or

alteration aureoles) can often be detected by computer-assisted

multispectral image analysis (Baker, 1977). Lyon (1975) has

also suggested that it may be possible to detect vegetation

anomalies associated with porphyry copper deposits by examining

spectral ratios on a pixel by pixed basis.

The principle applications of geobotanical techniques for

mineral exploration in remote sensing are largely based on three

factors;

1. observation of the distributions of indicator plants,

2. vegetation density changes, especially bare spots,

and 3. morphological changes in plants.

Detection by indicator plants with any remote sensing

technique may be difficult due to the small areal extent of most

patches of such species. The general environment of mineral

deposits are often unfavorable for vegetation growth so that grassy

clearings or stunted plant growth may result. Such clearings are

probably the most widely used geobotanical indicator in prospecting

(Lag and Bolvikan,1974). Some morpholgical and physiological

changes in plants, and toxicity symtoms due to concentration of

some elements may be detectable by remote sensing techniques due to

different reflectance caused by changing colours or changes in growth

89

form.

Gausman (1977) has described in some detail the properties

of the spectral reflectance of a leaf. However, the spectral

reflectance and emittance properties of a plant are much more

variable and more complex than that of a leaf. Colwell (1974)

has mentioned factors on the leaf reflectance properties to be

important when considering a vegetation canopy.

However, though geobotanical symtoms related to a mineral

deposit are in general obvious, there are subtle effects that can

only be seen on the Landsat image by the trained eye.

The unique advantages and characteristics of Landsat MSS

are its synoptic and repetitive character of space-acquired imagery.

The computer compatible digital format of the MSS images is further

advantageous compared to the photoimagery since the MSS images can

utilize its dynamic range of brightness values for further enhancement.

Ligget et al (1974) has shown that in reconnaissance

exploration for mineral, groundwater, and geothermal resources, and

in the study of geological hazards, Landsat MSS may provide cost

savings of about 10 to 1 compared with conventional methods.

Computer processing of Landsat MSS data for geologic applications

is well described by Gillespie (Siegal and Gillespie, 1980) and

Taranik (1978) etc.

4.3.2 Basic Principles in Landsat MSS Data Processing

Two aspects of the image processing were applied in this study.

Firstly, the regional study over the whole Cornwall area was conducted

for regional geological mapping such as lithologic mapping, detection

90

of linear features or any alteration zones which might be related

to any mineralization. This is based partly on the data produced 2 . . .

by using I S Additive Viewer set up at the Remote Sensing Unit m

Imperial College. Sampling over Cornwall was taken at every fourth

pixel in every third line since this is approximately a square

grid image and also fits within the limitation of the TV screen of

the system which holds a maximum of 512 x 512 array size at any one

time.

A false-colour composite of bands 4, 5 and 7 in blue, green

and red light, respectively, was made for the interpretation. Colour

-ratio composite techniques usually offer an efficient means for

combining stretched band ratio images for discrimination of rock types.

Discrimination is increased not only because information from several

ratio images is combined, but also the human eye is capable of

discriminating much better in colours than in shades of gray.

Since we can have 6 ratioed data from four spectral bands,

a total of 20 colour ratio combinations is possible if we are using

the colour additive viewer.

Rowan et al (1976) have noted that the most useful combination

for discriminating the main rock types and altered areas in exposed

regions, was the colour subtractive view of MSS R4/5, 5/6, and 6/7 in

cyan, yellow and magenta, respectively. However, facilities using

the diazo process to produce the colour subtractive view of composite

images were not available to the author. Therefore, an optimum

combination for geologic analysis of the study area was determined

using the colour additive viewer with the following ratio image

combinatiore: R5/4, R6/5 and R7/6 in blue, green and red, respectively.

All were subjected to histogram normalization.

To extract any linear features, individual black-and-white images

91

were produced on a pixel by pixel basis using a ULCC (University

of London Computer Centre) 64 level gray scale picture processing

package, PICPAC routine. In the gray level slicing process, a

contrast stretching technique was designed using a probability

density function of Gaussian distribution with a mean of zero and

a variance of 1.

The mathematical expression for the probability density

function of the Gaussian distribution with a mean of zero and a

variance of 1 is:

PDF = f(x) = - 00 < x < 00 (4.8) /2tt

Detailed procedures of the level slicing technique and program used

are given in Appendix D.

The second aspect is the processing of the digitized images

over the Bodmin Moor area to examine the regional spectral features

of the data. Again, in this study, a false-colour composite of MSS

bands 4, 5, and 7, and a colour composite of band ratios 5/4, 6/5 2

and 7/6 were enhanced by histogram normalization in the I S system

to produce colour additive pictures.

In order to form meaningful images for ratios between

Landsat bands, it is usually necessary to subtract a constant bias

value from each band (Chaven 1975 ) due

to direct scatter of sublight from the atmosphere into the sensor.

It depends mainly on the wavelength of the scattered light. Usually

11 for band 4, 5 for band 5 and 3 for band 6 are subtracted from

the recorded values. No atmospheric noise is considered for band 7.

However, in this study, this aspect was not considered.

Instead, the following procedures have been performed in the ratioing

92

since relative significance between two methods can be preserved

similarly.

To make the ratioing exact, the least square regression

line was calculated between two bands as follows:

y = a Q + a 1 x (4.9)

where y is the numerator and x the denominator in ratioing.a^ is the

intercept value at x = 0, and a^ the slope.

The calculated coefficients between two bands are given

in Table 4.2.

Table 4.2 Computed coefficients of linear trend between two variables

numerator denominator 0 b^(intercept values in x-axis)

Band 5

Band 6

Band 7

Band 6

Band 7

Band 7

Band 4

Band 4

Band 4

Band 5

Band 5

Band 6

-27.1280

5.1274

1.2360

41.4188

34.2470

-22.6880

1.7689

2.5926

2.4650

1.5808

1.5526

1.1648

-26.2012

-22.0578

In order to calculate the ratios, the intercept values (S q) were first

subtracted from the numerator (y) and then divided by the relative

denominator (x), that is

z i = " a o ) / x i ( 4 . 9 . 1 )

However, for ratio data of R6/5 and R7/5, the intercept values of

93

x-axis (b^) were subtracted from the denominators rather than subtracting

a^ from the numerators. In this way, the ratio can be performed in

the first quadrant with the least square linear regression line passing

through the origin, and the significance of the relative data are

preserved in this manner since the data are being standardized prior

to any multivariate data analysis.

4.3.3 Feature Extraction and Interpretation

Either black and white images of individual bands, colour-

composites of the original Cornwall region and digitized data over the

Bodmin Moor area, or their ratios, produced by photographic process

have been subjected to detailed interpretation. The photographic

processing of the images caused loss of some resolution.

Band 5 is by far the best for geological mapping and band

7 is optimum for recognizing contacts between land and water, vegetation

differences, and probably differences in soil moistures.

(a) Black and White MSS images of Cornwall

Gray scale images of the Cornwall area are enhanced by contrast

stretching with Gaussian distribution and are shown in Fig. 4.11 (a)

to (d). Frequency distributions of the original four bands are shown

in Fig. 4.12(a) and (d). From the histograms shown, spectral features

of Land in the higher range are well separated from those of water in

the lower range on bands 6 and 7; but the features is not so distinct

on bands 4 and 5. Particularly the spectral distribution pattern of

land in the near-infrared bands (bands 6 and 7) shows a near-normal

dis tribution.

Although distinctive geological features may not be seen

(a) Band 4

(b) Band 5

Fig.4.11 Contrast stretched black-and-white Band images over Cornwall. a) Band 4, b) Band 5, c) Band 6 , d) Band 7 and e) Lineaments stidied from Band 5.

Fig.4-11.continued.

Band 6

Band 7

96

Fig. 4.11. continued

(e) Lineaments studied from black-and-white MSS Band 5 images.

97

63 127 191 255

(a) Band 4

il G3 127 191 255

(b) Band 5

TOMiliTO G3 127

(c) Band 6 191 255

,l\llUuHHUl.... 63 127

(d) Band 7 191 2 5 5

Fig.4.12 Frequency distributions of black-and-white MSS Band images over Cornwall, a) Band 4,b) Band 5, c) Band 6 and d) Band 7.

98

in the images, they generally show some regional features which are

related to the regional geology.

Bands 4 and 5 in Fig. 4.11 (a) and (b), respectively, show

particularly high reflectance over the high Moor lands. Although

this feature does not appear in the Land's End and Carnmenellis granite

areas, careful investigation makes it possible to outline the granite

boundaries throughout the peninsula in all four bands.

Although it is not so clear, probably due to the photographic

process, dark colours in the coast areas and some river channels in

the northwest may be due to high spectral features of sanddunes. Some

areas along the southeast coast line are also dark, which is attributed

to high reflectance by cultural features and possibly sanddunes. As

noted by Davis, et al (1981), the high reflectance of sanddunes is due

to the lack of iron staining and. other dark minerals in the visible

bands. Although mineralized areas usually appear to be bright

(Rowan, et al 1976) it is not so apparent in the study area. This

may be due to the almost complete coverage of vegetation which may not

always show distinct reflectance properties from other features in the

black and white images. Also further detailed distinction between

different rock types is not possible because the vegetation may mask

subtle differences of the reflectances between the various rock types.

Numerous linear features are apparent through the area. Some

of them may be drainage patterns but many may represent geological

structures such as fractures, faults, or curved features representing

folds. Some of these lineaments correspond to some known major faults

and other could not be correlated with any of the geological features

in the existing geological map.

Between the images, band 5 is by far the best in the study of

99

lineaments and the next is band 7 in the study area. All interpreted

linear features are summarized in Fig. 4.11 (e).

The conspicuous northwest trending regional linear features

around the Bodmin Moor granite area marked well-known wrench-faults.

These lineaments are manifested in the space image by the tonal

differences in vegetation and alignment of subsequent stream segments

and topographic forms etc.

The contact line between Caledonian and Amorican zones is

also marked between Perranporth in the west and Pentowan in the east.

Lines of confrontation or oppositely facing folds may also be seen

in the images as in the north of Padstow.

Other features which are interesting to note are the parabolic

features cut by lineaments in areas between Bodmin Moor and Dartmoor

granites. These features appear to be in the south of the recumbent

fold belt on the structural geology map (see Fig. 2.2) but these may

be parts of many recumbent folds in the area.

The lineaments which cannot be identified with any of the

geologic features in the geologic map are possibly related to deeply

buried concealed structures.

In any case, such detailed remote sensing studies which also

involved correlation of geophysical data interpretation and also

checking these lineaments from the standpoint of their correspondence .

to concealed structures will be very important because the structural

control is very often the main factor in the location of mineral

deposits as described in Section 4.3.1.

Many lineaments in the area may be related to local mineral-

izations. Correlation with possible mineralization can also be

supported by available geochemical maps (Webb et al 1978) in the area.

100

However, the enhanced images do not provide any clear

indication of the presence of the mineral deposits. Thus, in all

cases in the study area, spectral tonal variations found to exist

in the Landsat imagery of heavily vegetated regions requires

detailed ground examination to determine whether they are the result

of geologic, simple botanical or other factors such as any cultural

features.

(b) False-colour composite of Cornwall

Fig. 4.13 is the false-colour composite of bands 4, 5 and 7

using blue, green and red filters, respectively. The map shows the

major physiographic and structural features actually visible on

it and a few well-known features such as the wrench faults around

the Bodmin Moor are also included. Many lineaments, expressing

faults or folds are obvious.

On the other hand, the Landsat images show albedo differences,

generally resulting from vegetation patterns, which may be of some

geologic value by directing attention to areas or features with

distinctive vegetation patterns.

In this colour composite, vegetation appears to be red

because of the high reflectivity of vigorous vegetation in MSS band 7

compared with MSS bands 4 and 5 (Rowan et al 1976).

In the study area, the colour composite may be most useful

for discrimination of vegetated areas. The darkest red, and therefore

the densest vegetation, occurs in areas surrounding the Bodmin Moor

granite. Some red tinges are apparent throughout the whole peninsula.

Particularly, Carnmenellis, Land's End and some part of Lizard complex

show uniform distribution of vegetation. Colour composites appear

to offer a little improvement over the stretched MSS images for

101

discriminating geological features. High moor lands of St. Austell,

Bodmin Moor and Dartmoor granites are white to greenish white in the

image, but this feature does not appear on the low lying Land's End

and Carnmenellis granite areas. Sanddunes appear to be white in the

NW and SE coasts and along some river channels. The greenish feature

in the middle of the Lizard complex may be due to sparse vegetation

or stunted growth in serpentine rocks. Cultural features in the municipal

areas are blue as in St. Austell and Plymouth because of the high

reflectivities of those features in the visible bands.

Although most of the metamorphic aureoles appear to be light

brown to reddish, the colour composite may not offer a reliable means

for detecting altered areas in the study area where the surface is

almost completely covered by cultivated vegetation. These do not show

distinct spectral features from others in the study area.

(c) Colour-ratio composite of Cornwall

The colour-ratio composite shown in Fig. 4.14 was analysed by

comparing with the regional geological map. The colour-ratio composite

image virtually offer little improvement over the false-colour composite

for discriminating any rock types.

Areas of high Moor granite and sanddune are blue, while the

cultural features appear to be dark blue. Another blue feature in the

Lizard complex corresponds largely to areas of granite and gneiss.

Vigorously vegetated areas appear to be bright orange in the ratio

composite. One interesting point to note is that surrounding the

Bodmin Moor and St. Austell granites and over Land's End and Carnmenellis

granites are marked dark orange distinctive from other vegetated areas

having bright orange colour. It is not so clear whether this feature

is directly related to reflective features from the altered areas or

102

Fig.4.13 False colour-composite of MSS Band 4,5 and 7 in blue,green and red,respectively.

Fig. 4. 14 Colour-composite of MSS Band ratios R5/4,R6/5, and R7/6 in blue,green and red,respectively.

103

not since the comparative colour density differences between altered

areas and vigorously vegetated areas are still too subtle to be clearly

differentiated. Areas of serpentine rocks are more apparent in the

ratio composite than in the false-colour composite.

Many linear features are apparent in both false-colour

composite and band ratio composite images. These kinds of composites

may be sometime advantageous in studying regional lineaments since they

remove small local variations. However, it will not be pursued further

here since it has been already described in the previous section.

(d) Description of surface maps for the Bodmin Moor area

Surface maps in the study area have been plotted for two

comparative bands in order to increase perspective views: bands 4 and

7.

In Fig. 4.15 (a) the high reflective feature of MSS band 4

is apparent over the high Moorland areas. Typical highs arise in many

parts of the St. Austell granite and a few in the Bodmin Moor granite.

These may be due to the high reflectivity of clay minerals in open-pit

mining areas in the visible spectral bands. However, the high

reflectance features do not appear in band 7 as shown in Fig. 4.15

(b). Instead, the reflectance features in these areas are lower

than the vegetated surroundings. The sudden drops in the NW, SE and

a little in the SW corner of the map reflect the low reflectivity of

the offshore water. Although details on every feature are not described

here, the general feature is a sort of reflection of morphological

features rather than any geological ones.

In general, on land areas in band 4, high reflective features

occur in high or dry areas while low ones occur in wet and thickly

vegetated areas.

104

I 1.641

(a) MSS BAND 4

( 1 . 64 1

Fig. 4.15 Surface maps of MSS bands 4 and 7 over Bodmin Moor area, Cornwal1. The geometrical coordinate (1,1) is oriented as the northwest corner.

105

I 9.411

I 34 . 4 11

MSS BAND 7 OVER THE BODMIN MOOR GRANITE

Fig. 4.16 Surface maps of MSS Bands 4 and 7 over Bodmin Moor granite. The geometrical coordinate (9.20) represents the northwest part of the map.

106

On the contrary, in band 7, thick and vigorous vegetation

shows highest reflective features while dry areas and water show low

reflective features.

To examine more detailed reflective features over the Bodmin

Moor granite area, the data was partitioned for the area as shown in

Fig. 4.16 (a) and (b).

The high reflectance in band 4 is evident, the peaks of which

largely correspond to high tors in the area and toward the boundary the

spectral feature is flattened. On the other hand, the reflectance

features in band 7 of Fig. 4.16 (b) are reversed to those in band 4.

The high reflective features in band 4 appear to be low in band 7, and

the flat feature in band 4 disappear in band 7 and it rather shows some

irregular highs. These may mean the spectral feature in band 7 may be

little affected by the topographic elevation. Instead, it may be

influenced by vegetation conditions and surface and near-surface

moisture contents, judging from the fact that the area is almost

covered by vegetation which varies according to elevation.

The high moor lands consist of dry and brown vegetation

overgrazed by cattle, while lower lands are mainly composed of vigorous

vegetation and/or high moisture content in the soils particularly in

lowland depressions.

(e) False-colour composite of Bodmin Moor area

Spatially averaged data in Fig. 4.17 (a) to (d) show regional

reflective features over the Bodmin Moor area. Bands 4 and 5 are closely

similar in spectral features as in Fig. 4.17 (a) and (b).

Regionally the highest reflectance features occur in the high

lands of Bodmin Moor and St. Austell granites. Additional highs occur

sporadically throughout the study area, which may correspond to areas

107

( a ) b a n d 4 ( b ) b a n d 5

->p Km sea l e

( c ) b a n d 7

y J

f P i t

H ( d ) c o m p o s i t e

F i g . 4 . 1 7 R e p r e s e n t a t i o n o f MSS B a n d s 4 , 5 a n d 7 b y n o r m a l s l i c i n g o v e r t h e B o d m i n M o o r a r e a a n d t h e i r f a l s e c o l o u r - c o m p o s i t e ( d ) ( a ) B a n d 4 , ( b ) B a n d 5 a n d ( c ) B a n d 7 .

*'Overlay' in the rear folder is approximately the regional geology of the Bodmin Moor area.

108

with less vegetated or dry surface. Sanddunes near the coast lines

or cultural features also appear to give high reflectance.

Low reflectance in the visible bands may be attributed to

spectral features of waterlogged or intensely vegetated lands.

On the contrary, band 7 generally reflects the opposite

features to either band 4 or 5, except perhaps areas with water and

wet land. The high reflective features of high moor lands show lower

reflectance than the surroundings in band 7. Areas of high reflectance

largely correspond to highly vegetated areas. Low reflectance

in band 7 is generally attributed to water, sparse or dry vegetation

and features of the larger towns, while high reflectance may be due

to vegetation, the density and vividness of which is proportional to

its magnitude of reflectance.

The colour composite in Fig. 4.17 (d) shows that high tors

and open-pit mining areas are white and high moor lands with dry vege-

tation, cultural or dry lower land areas appear to be light blue,

and areas of intense vegetation show reddish. Dark colour represents

areas of moisture.

Many of the altered areas in the aureoles' of the granites in

the south are greenish to orange, but much of the altered zones in the

north to northeast of the Bodmin Moor granite are reddish.

It is uncertain whether these features are due to spectral

features of altered areas or not in the study area because they may be

obscured by vegetation cover although vegetation in altered areas often

shows certain distinct spectral features which can be distinguished from

others.

(f) Colour-ratio composite of the Bodmin Moor area

The stretched colour-ratio composite of R5/4, R6/5 and R7/6

109

assigned by blue, green and red, respectively, are shown in Fig. 4.18

(a) to (d). Geologically, although almost no improvement is offered

by the ratio data compared to the false-colour composite of the original

baids some relative reflective features between bands can be seen in

the ratio composite.

In Fig. 4.17 (a), the high land of Bodmin Moor granite are

bright, but St. Austell granite and two spots in the Bodmin Moor granite:

one in the northwest and the other in the southern margin, appear to

be dark. This indicates that though dry vegetated lands have relatively

higher reflectance on band 5 than band 4, the reflectivity of the open-

pit mining regions is relatively higher in band 4 than band 5 spectral

regions.

In the lowlands of sediments dark colour may be due to

relatively low reflectance of band 5 due to strong chlorophyll absorption

at near 0.68 ym spectrum. The offshore area in the northwest corner

of the map is dark but the offshore area in the south appears to be

white. Though the reason is not clear, considering shallow and maybe

dry stream channels flow in the northwest while more active water

channels flow in the southern coast through populated areas, it might

be due to differences in reflective features of contaminated water by

suspended sediments in the south from un- or less-contaminated water

in the northwest, since there are strong absorptions in blue and green

bands from the suspended sediments in the water as noted by Specht et

al (1973). This feature does not appear in the MSS ratio R6/5 and

R7/6. Most of the drainage system on land areas may also appear to

be bright in the band ratio R5/4 probably due to suspended sediments

by strong current or contamination.

In Fig. 4.18 (b), the general features of R6/5 are that water

and granitic moor lands are dark while areas of intense vegetation

(c) R7/6 (d) Ratio composite

4 . 1 8 C o l o u r r e p r e s e n t a t i o n o f MSS B a n d r a t i o R 5 / 4 , R 6 / 5 a n d R 7 / 6 i n b l u e , g r e e n a n d r e d r e s p e c t i v e l y a n d t h e i r c o l o u r - c o m p o s i t e ( d ) . ( a ) R 5 / 4 , ( b ) R 6 / 5 , ( c ) R 7 / 6 .

0 Io jiQ KlT\ 1 1 1 1 i Sea I e

111

appear to be bright green, the degree of which may be proportional to

its brightness and density.

Band ratio R7/6 in Fig. 4.18(c) shows that townships and

high moorlands are dark while water and vegetated areas are bright.

Nowhere do the images show any direct relation of the

distinct leflective features of alteration zones or any mineralization

though the colour composite of the ratioed data shows some indication

of alteration around the metamorphic aureoles surrounding the granites.

The colour-ratio composite in Fig. 4.18 (d) shows a more

dynamic range of reflectance features. The high land of Bodmin Moor

granite and maybe dry killas are blue but the high moor land of open-

pit mining and township areas appear to be dark.

The offshore, water and densely vegetated areas are reddish

to brown. The greenish to yellowish colours may be due to different

density and brightness of vegetation, the order of which is from dense

and bright to less dense and less bright.

4.4 Geochemical Data Processing

Geochemical methods of prospecting have been proven in practice

with a number of mineral discoveries. Almost invariably, geochemical

methods have been used in conjunction with other geological or geophysical

exploration methods (Rose, et al, 1979).

The stream sediment sample values obtained from the study from

the Wolfson Atlas project (see Section 3.3.3) approximate to a composite

sample of those materials derived from the catchment area upstream

from the sample sites, so that in appropriate circumstances the patterns

of metal distribution in the rocks and/or soils may be reflected to a

degree in corresponding variations in the composition of the stream

1 12

sediment except in areas of contamination (Webb, et al.,1978).

In this section the geochemical data for the stream sediment

samples were analysed to investigate the regional distribution patterns

of each element in relation to geology, and possibly delineate those

areas with potential mineralization prior to the multivariate regional

geological mapping as a tool to assist detection of possible mineralized

areas.

Geochemical elements usually have bimodal or polymodal concentration

distributions. When potential mineralization is of concern,the sample

distribution may be treated simply as bimodal; background and anomaly.

However, the background and anomaly populations commonly overlap, so

that completely satisfactory discrimination of background and anomaly

samples is not possible.

There are various ways of separating anomalous samples from

background, depending on the amount of data, purpose of the study and

economic consequences of the selection, etc. (Rose,et aL, 1979).

Most cases are involved with threshold selection to define

anomalous samples from backgournd. Others may include the extraction

of a local anomaly by subtracting regional values 'from the sample

values or using a number of sophiscated multiple statistical treatments

such as discriminant analysis or factor analysis.

Webb et al (1978) and Mancey (1980) have noted that colour

subtractive views of percentile slicing of three geochemical elements

in cyan, yellow and magenta illustrate the regional geology most

effectively.

In this study, threshold methods (including representation by

the absorption-scale level slicing method to evaluate the general

distribution patterns of the eight elements (As, Cu, Ga, Li, Ni, Pb,

Sn, and Zn), and probability plots) have been applied further to evaluate

1 13

the data as a means of assessing the potential for mineralization.

4.4.1. Principles of Regional Geochemical Data Processing

(a) Analysis of the regional distribution patterns of geochemical

elements

To analyse the regional distribution patterns in stream

sediments of As, Cu, Ga, Li, Ni, Pb, Sn and Zn, the data were sliced

using absorption levels.

The absorption level slicing technique takes the form:

x = aeby (4.10)

where y is the slicing level value and x is the sample value, a is the

minimum of the logarithmic data values and b the difference between

the maximum and minimum logarithmic values divided by the number of

gray levels (GN).

In another expression Eq. (4.10) becomes

y = ( l n X ~ a) * GN (4.11) b'

where b = ^r, so that the gray level of each sample value is determined G N

directly from the above equation, or the gray level threshold values

for the data are determined first and the level values are applied to

the data for selected levels by iterative method.

This kind of slicing may be adequate for application to

the geochemical elements since the frequency distributions of most

geochemical elements are approximately log-normal. Thus, the distribution

pattern can be histogram-normalized which may give maximum discriminatory

114

power of the regional distribution.

(b) Detection of geochemical haloes using the probability plot

The probability plot has been effectively used as a means of

separating the anomalous samples from the background.

The method makes use of probability plot paper devised by

Hazen (1913). The cumulative percentage scale of this paper is

specially arranged, so that when the ordered concentration values

for any normally distributed population are plotted, the points all

fall close to a straight line. If a bimodal or polymodal distribution

are mixed with distributions which are themselves normally distributed ,

they will, when plotted, give a curve which is the resultant of two

or more straight lines. There is a point, or points, of inflexion

on this sigmoidal curve where the direction of the curve changes.

These points of inflexion indicate that there are two or more populations

involved.

The plotting procedures are as follows:

When the number of samples is small, the data are arranged in

ascending order of magnitude, each individual sample is plotted in

such a way that there are equal percentage intervals between each

sample. In general, if there are N samples, the first in the sequence

is plotted on the 'probability' line whose value is 50/N % and the

succeeding (N-l) samples at equal intervals of 100/N %. Further

details are described by Sinclair (1976).

These procedures were implemented in the interactive graphic

program called GIRAF initially written by Steven Earle, Department of

Geology, Imperial College. The mean and standard deviation of each

sub-population are calculated statistically. The Kolmogorov-Smirnov

statistical test of normality of the sample is also implemented in

115

the program.

4.4.2 Regional Distribution of Geochemical Elements

The geochemical maps sliced by absorption levels in Fig. 4.19 show the regional distribution patterns in stream sediments. Slicing levels, with their relative symbols, are listed in Table 4.2.

In addition, means and standard deviations were calculated for lithologic units (including Upper and Lower Carboniferous rocks, Upper, Middle and Lower Devonian sediments and granite areas) in order to provide information in the lithologic distribution of the elements. The statistics for anomalous zones defined from the probability plot by threshold values were separated for more reasonable lithologic estimation of each element. The calculated statistics are given in Table 4.3.

Considering the data as a whole, there is a broad similarity in the distribution of groups of elements although there are some variations in details.

Principal composite patterns of the regional geochemical distribution are as follows.

1. Generally most of metallic sulphide elements such as As, Cu, Pb, Zn and Ni tend to be low on the granite compared to the surrounding country rocks. These typical lows could be due to kaolinization (Hale, pers. comm.). These elements show also typical low values in the Upper Carboniferous rocks (except Ni), and in the lower Devonian sediments except in areas of contamination by drainage patterns.

Ga, Li and Sn are largely higher on the granite. They also appear to be high in the middle Devonian near the granite, which may

N MOOR AREA

117

be due to the secondary enrichment by drainage systems. Extensive

lateral distribution of high tin patterns in the eastern part of the

St. Austell granite has in part been related to reworking during the

pliocene marine transgression. This explanation is deduced from the

absence of supporting arsenic and/or copper values (Webb et al 1978).

As, Cu, Ni and Zn are uniformly low on the granites while

Pb shows some high on the granite.

2. As, Cu, Ni, Pb and Zn generally tend to concentrate in

a number of areas around the granitic aureoles. Again, there are

marked differences within the overall pattern in distribution and

metal concentration.

Peak values for As and Cu occur east, southeast and south

of Bodmin Moor and near the Kithill granites. Pb and Zn show

maximum concentration in southeast and northwest of the main granite,

and near the Kithill granite. Zn is concentrated east of the St.

Austell granite and shows moderate highs east and south of the Bodmin

Moor granite.

Except in areas of drainage contamination and possibly old

mining activities and smelting (particularly in the southeastern part

of the study area and east of the St. Austell granite), the distribution

of the elements associated with mineralization is in broad accord with

mineral zoning.

3. Ga and Li are characterized by their low distribution

patterns in the country rocks, particularly Carboniferous and Lower

Devonian sediments, but they show patterns in the Middle Devonian

rocks:as high as the granite areas. This may be attributable to

redistribution of those components by particularly abundant drainage

systems south and west of the main granite.

Sn peaks occur in most of the mineralized areas east and

118

south of the Bodmin Moor granite and over the St. Austell granite. As

Hosking et al (1965) pointed out in the study of sediments of the

rivers to the south of the St. Austell granite mass, the wide dispersion

of the high tin values may be derived from the slate rather than from

particular mineralized zones or veins.

Table 4.2 Slicing-level values for 8 geochemical elements used in colour plotting in Fig. 4.19.

element

colour level

slicing-level element

colour level As Cu Ga Li Ni Pb Sn Zi

1 Dark Blue 10-2 30-7 6-8 4-6 5-8 1-3 1-3 11-20

2 Blue 2-4 7-15 8-11 6-10 8-13 3-7 3-10 20-40

3 Cyan 4-6 15-33 11-15 10-15 13-21 7-16 10-30 40-75

4 Green 6-11 33-73 15-20 15-23 21-33 16-42 30-100 75-150

5 Yellow 11-21 73-160 20-27 23-36 33-53 42-100 100-300 150-250

6 Orange 21-39 160-360 27-37 36-56 53-86 100-270 300-900 250-500

7 - Lemon 39-71 360-800 37-50 56-87 86-D8 270-690 900-3000 500-1000

8 Red 71- 800- 50- 87- 138- 690- 3000- 1000-

units are ppm

4.4.3 Probability Analysis

Fig. 4.20 (a) to (h) shows the probability plots of the

regional geochemical elements. The plotted lines show assymmetrically

placed sigmoidal curves. There is a point of inflexion where the

direction of the curves change, as indicated by arrows. This point

suggests that there are two population involved; a population of small

concentrations, and mixed with them a small population of large

119

Table 4.3: Means and standard deviations of 8 geochemical elements from the Bodmin Moor area in lithologic units. Threshold values of geochemical anomalies were determined from the probability analysis in Section 4.4.3.

Upper C 510

Lower C 69

Upper D 1000

Middle D 672

Lower D 896

Crani te 713

Anomalous area

* 23.1 10.7 10.0 6.5 8.1 65.1

As** 3.71 7.59 7.18 7.69 5.58 6.0 21.11

1.1- 21 .0 7.3-35.0 1.5-36.9 1.7-36. 1.1-32.3 37.2-131.0

510 43 872 610 894 707 224

44.6 199.0 90.7 58.5 73.8 44.6 961.7

Cu 15.96 128.41 157.23 56.69 94.48 64.11 283.52

24.7-169 .5 38.3-55.86 14.6-571.8 21.9-560.6 12.5-546.1 3.6-565.2 576.2-1763

510 69 876 651 888 701' 165

16.0 16.4 20.1 22.3 14.7 23.4 44.5

Ca 2.34 3.40 2.95 2.16 3.84 3.22 9.47

9.6-22.5 9.4-23.6 10.8-27.7 13.6-28.3 6.8-31.0 13.7-30.8 31.2-68.4

510 69 1000 672 839 636 134

9.2 10.7 12.2 14.2 10.3 17.9 44.6

L i 2.04 2.38 2.82 2.74- 2.49 4.36 25.31

4.9-19.5 5.6-15.4 4.3-22.4 9.1-23.7 5.3-24.6 7.3-29.9 25.0-135.8

510 69 1000 672 829 462 318

62.2 68.3 60.2 61.7 55.1 26.2 117.9

Ni 12.47 10.07 13.47 16.53 15.70 14.84 31.14

31.1-91.0 44.2-87.2 13.9-91.8 11.7-88.8 5.7-89.2 6.3-64.3 92.5-221.3

507 53 969 667 885 , 713 66

45.6 171.3 105.1 96.0 78.8 45.2 668.5

Pb 34.38 89.18 92.32 73.01 66.29 33.57 327.67

16.3-282 .6 29.7-373.6 17.2-388.0 26.0-382.5 2.0-368.7 2.0-325.1 390.0-1750

510 60 877 653 895 709 156

24.9 882.4 212.8 464.6 453.6 408.9 3725.9

Sn 64.20 456.82 369.11 493.34 458.20 446.28 1425.20

0-937.9 46.7-1806.5 0-1815.9 5.3-1822.2 5.2-1793.1 3.9-1828.9 1836.7-8880.:

510 52 869 518 662 524 725

157.8 312.0 214.4 246.4 223.3 111.3 796.7

Zn 60.59 87.06 84.43 100.79 85.21 70.50 255.02

51.1-524 .0 173.8-511.2 61.5-524.6 63.4-528.9 11.5-529.8 11.3-518.7 537.9-1905.6

510 55 933 617 870 692 183

* Means minimum and Maximum

* * Standar deviat ions • • number of s neiples

un i t s are ppm

120

'10 70 30 40 50 SO 70 60 90 9S 99 99 C U M U L A T I V E P E R C E N T

-i 1 1 1—r

I 7 5 10 70 10 40 50 60 70 60 90 95 9e 99 C U M U L A T I V E PERCENT

77 !7 IC 57 I.C '7 60 90 95 96 99 C U M U L A T I V E P E R C E N T

1 7 5 10 75 JO 4C 50 6C >0 80 90 95 96 99 C U M U L A T I V E P E R C E N T

Fig.4-20 Probability plots of 8 geochemical elements: As,Cu,Ga,Li,Ni,Pb,Sn and Zn.

121

Fig.4-20 continued.

C U M U L A T I V E P E R C E N T C U M U L A T I V E PERCENT

C U M U L A T I V E P E R C E N T C U M U L A T I V E P E R C E N T

122

concentrations. The two straight lines represent an approximation to

these two populations. The actual values for these two sub-populations

are plotted as dots along the straight lines. The straight lines are

actually fitted from the dot points by multiplying the percentages for

the smallest individuals with the inverse of its proportion to the total.

For example, if the small sub-population is 95% of the total, then the

inverse '100/95' is multiplied by the percentages of concentrations

involved. Likewise, for the largest concentrations '100/5' is multi-

plied to the percentages of concentrations from the larger sub-population.

The dots along the sigmoidal curve are the resultant estimate of the

two straight sub-populations. The Kolmogorov-Smirnov non-parametric test

was applied here as a test of goodness-of-fit of the estimate curves to

the observed data.

The statistical results calculated are shown in Table 4.4. The

total means and even the means of sub-population 1 are much higher than

the world mean crustal concetrations except for Ga, Li, and Ni. Ga, Li

and Ni are lower in total means and means of sub-population 1, but the

means of sub-population 2 of Ga and Ni are higher than those from the

world statistics.

However, in the case of Ni, the anomalous values were encountered

in the lower range, which may represent a local geological feature that

the granites are low in nickel.

In all cases, Komogorov-Smirnov tests show that the estimated

values are not significantly different from the observed samples at a

confidence level of 90% as shown in Table 4.4.

To further analyses the spatial distributions of the anomalous

samples to the background, colour pictures have been produced for the

main sulphide elements (Cu, Pb, Sn and Zn) which are closely related

to local mineralization. This was done by using the Colour Additive

Table 4.4 Statistical results of probability analysis

. Sample . _ , . , sub-pop 1 sub-pop 2 K-S tesl Variable r Std.Dev. Threshold 77 .. — — - _r / — . nr.„ Mean . Mean Std. Percent Mean Std. Percent at 90%

Dev. Dev.

As 11 15.6 3.7 8.6 7.2 94.86 54.9 19.6 5.14 A 0.804

Cu 110 206.8 575 93.2 66.9 96.19 790 238 3.81 A 2.079

Ga 20 6.7 31 18.6 4.5 96.61 36.1 2.9 3.39 A 0.612

Li 15 11.9 25 13.8 4.8 95.58 40.8 12.0 4.42 A 0.989

Ni 55 23.3 28 18.1 7.1 16.15 60.0 14.1 83.85 A 0.225

Pb 110 152.3 390 107 71 96.61 478 150 3.39 A 1.706

Sn 920 1496.3 1830 349 325 83.64 3600 1250 16.36 A 1.313

Zn 230 167.8 533 206 95 95.73 671 138 4.27 A 0.732

A: not significantly different at the significance level of 90%

K-S: Kolmogorov-Smirnov statistics

units are ppm

124

Viewer at the Remote Sensing Unit in Imperial College. Threshold values

estimated from the probability plots are shown in Thble 4.4.

Arsenic has been excluded from this analysis because As may not

be a good pathfinder as noted by Shouls et al (1968) since it is

inconsistently associated with other elements in mineralization.

The patterns were produced with five levels as follows:

1. blue: background (in the probability plot, the background

is treated by the sample values below the point of arrow).

2. light green: less than -1 SD (Standard Deviation) of

sub-population 2.

3. yellow : between -1 SD and 0 SD.

4. orange : between 0 SD and +1 SD.

5. red : greater than 1 SD.

As shown in Fig. 4.21, the pictures indicate generally well defined

anomaly patterns which might be related to local mineralization.

Most copper concentrations occur in the east and southeast

of the Bodmin Moor granite and a small patch near the Kithill granite.

They are largely associated with anomalous tin concentration as well.

These may occur associated with local copper-tin mineralization. Part

of the concentration in the southeast of the Bodmin Moor granite may

have contaminated drainage.

Sn also shows an extension of its anomaly further to the

south of the main granite, on the St. Austell granite and in its eastern

part.

Pb and Zn are largely assocaited with each other southeast and

northwest of Bodmin Moor, and near the Kithill granite areas. A strong

Zn anomaly occurring in the eastern part of St. Austell granite may be

due to a combination of mineralization with some contamination of the

drainage sediments from local smelting.

125

N

M b \ * • -

i (a) Cu (b) Pb

(c) Sn (d) Zn

10 20 Km

Scdi\e

F i g . 4 . 2 1 S p a t i a l d i s t r i b u t i o n o f g e o c h e m i c a l h a l o e s o f C u , P b , S n a n d Z n .

126

4.5 Conclusions

A variety of data processing techniques chosen with reference

to the properties of the data, have been applied to the geophysical,

Landsat MSS and geochemical data, to extract regional and local

features of geological significance.

In geophysics, regional features such as magnetic basement features

and regional gravity fluctuations were defined by regional analysis

including low-pass filtering and upward continuation. Local anomaly

features of magnetic sources and density variations such as tertiary

undulation or subsurface granite cusps which could be related to local

mineralization were also enhanced by horizontal and vertical derivatives

and highpass filtering in order to visualize those features more clearly.

In Remote Sensing, contrast stretching and ratioing techniques,

and their colour-composites applied to Landsat MSS data from Cornwall

and the Bodmin Moor area enabled the extraction of various regional

features showing lithologic or structural patterns. In particular,

many known lineaments were confirmed and further linear features

(which are not on the geological map and are probably geologically

significant) were detected from the enhanced images.

In addition,level slicing for geochemical data by the absorption

scale employed in this study shows the regional distribution patterns

of geochemical elements in conjunction with geological units. Further,

probability plots were effectively applied to determine threshold

values and thereby spatial distribution of geochemically anomalous zones

has been effectively delineated and these may be related to local

mineralizations.

127

CHAPTER FIVE

TREND SURFACE ANALYSIS

5.1 INTRODUCTION

Trend surface analysis has been widely used in the Earth

Sciences for recognition and measurement of trends that can be

represented by lines or surfaces.

It is a multiple regression technique used to fit

statistical equations or models to geographically distributed

data in order to define trends on a map surface. A trend analysis

illustrate regional,and residuals from the trend surface, and this

type of analysis may provide valuable geological information about linear

structures and related tectonic features.

Henkel (1968) has reviewed somfe of the geophysical trend

analysis and noted that apart from their use for examination of

linear features in the data they can be employed for various

analytical procedures. Geophysical anomalies with certain trends

can be distinguished as anomaly patterns using gridding operators

with specific properties such as the directional filter as described

by Fuller (1967). Trend surface analysis can also indicate

regional effects and detect whether these have been removed from

the residual anomalies.

Agarwal (1968) applied an empirical method using cross

correlation coefficients in order to trace the trend due to various

geological factors. There are numerous applications of the trend

surface analysis in geological studies. Excellent review on

problems arising in the trend surface analysis and its application

is given by Howarth (1983).

128

Polynomials are perhaps the most widely used as can be seen

in most of the statistical references. Lower order trend surfaces

may depict the regional or large-scale features and their residuals

may reflect the local or small-scale geological features.

Other approximations can also be used for trend surface

analysis such as orthogonal polynomials (the Fourier and Chebyshev

series, etc.) or the moving average method, etc.

Comparison of advantages and disadvantages of different

methods cannot be directly evaluated because of the different purposes

for which they are employed. Criteria for selecting the most

appropriate method for a trend surface depends upon the geological

objectives. Some comparison between polynomial, moving averages, and

the Double Fourier Series methods for various analysis are given by

Davis (1973). Some details on the trend surface analysis with

polynomial fitting and double Fourier series can be found in

Harbaugh and Merriam (1968).

In my study, first- and third-degree trend surfaces have

been applied for analysis. This is because it can be anticipated

that a basic pattern of similarity between variables should lie in

the lower degree terms.

5.2 Mathematical Procedures

The general n-degree polynomial for two independent

variables, often geographic coordinates, is as follows:

129

Z = b. + ( b x + b Y) + (b X 2 + b.XY + b Y 2) + 0 1 2 3 4 5 + (b X° + b Xn~^Y + .... + b. Y n) (5.1) k k+1 k+n

where k = n(n+l)/2 and n is the degree of the polynomial and

b^ (i=0,1,2,...,k+n) the coefficients to be calculated.

In this study, first and third degree trend surfaces have been

analysed by solving equations of polynomials by means of the least

square criterion. The expressions for the least square fit

of the first- and third-degree polynomials are given in Appendix E.

A statistical test can be performed to indicate the reliability

of the fit of the calculated values to the original data. Usually

the "goodness-o'f-fit" depends on the degree or order of the

calculation.

The most commonly used measure*of reliability of the fitted

surfaces is the sum of squares test which has been widely used,

for example, by Merriam and Sneath (1966), and Howarth (1967), etc.

Howarth (1967) calculated the percentage sum of squares accounted

for by first-, second-, and third-degree trend surfaces fitted to

random data in order to measure the reliability in extracting regional

trends, and noted that if the sum of squares test produces values

that fall below 6.0, 12.0, and 16.2 per cent for the first-, second-,

and third-degree trend surfaces respectively, the distribution of

data points is not significantly different from random at the 0.05 level.

Nordiffe (1969) derived improved values for critical % sums of

squares. If F^ for level of significance Ot, r^ = K-l, r2=n-k-2,

where K is the number of coefficients in trend surface equation, n the

number of data points, then

130

F (k-1) % 'explained' = x ,00 (5.2)

a

e.g. for n = 500 minimum % accounted for by random trends will be

F = 1.8 and 4.3% at a = 0.01, and F = 1.2% and 3.4% at a = 0.05 a ' a for linear and cubic polynomials, respectively.

In this study, the sum of squares has been calculated to

indicate the goodness-of-fit of the trend surfaces as follows.

R _ SSR 2 SST

n n Z X" . - ( Z X

regression regression )2/n

n n Z XZ, - ( Z X . ,r/n . , observed . , observed i=l i=l

(5.3)

where n is the number of data.

Further the F-test value can also be obtained by

F = [ SSR/m SSD/(n-m-1) (5.4)

where m and (n-m-1) are degrees of freedom of regression and

deviation from polynomial regression, respectively. m is one

less than the number of coefficients for the surface being used.

SSR and SSD are the sums of squares due to regression and deviation,

respectively. SST is the total variation, that is,

SST = SSR + SSD (5.5)

The F-test provides a measure of the random effect of the

regression.

131

Further details of the statistics employed can be found in

Davis (1973). The program in his reference (p.332) has been

modified by the author to apply to regularly gridded data sets for

this study.

5.3 Application to the Bodmin Moor Area Data

First- and third-degree trend surface analysis have been

applied to the gravity and magnetic data, four Landsat MSS bands,

and eight geochemical elements in the Bodmin Moor area. The are of

geochemical elements^ As, Cu, Ga, Li, Ni, Pb, Sn and Zn.

The first-degree maps of the analysed data are shown in

Fig.5.1(a) to (n), and the third-degree maps are illustrated in

Fig. 5.2(a) to (n). The residual maps of the third-degree polynomial

were contoured and shown in Fig. 5.3(a') to (n).

The trend surface coefficients are tabulated in the Table 5.1

(a) for the first-degree and (b) for the third-degree. Statistical

data pertinent to application of analysis of variance to particular

trend-surface analysis have been calculated. These data include (a) sums

of squares apportioned among first- and third-degree regression

components, (b) sums of squares associated with deviations of regression

from the original components, and the number of degrees of freedom

associated with regressions and deviations. From these data, the

goodness-o-f-fit (R^) and F ratio is calculated as shown in Table 5.2,

and further the F ratio values are used to determine the significance

level, expressed as a percentage by reference to Tables of F.

132

5.3.1 Geophysical data

(a) Gravity data

The first-degree trend surface fitted to the gravity data reveals

a gradual increase toward the east-northeast direction perpendicular

to the north-northwest linear direction in Fig. 5.1(a).

The generally low fit of the first-degree trend surface, of

14.6 per cent of total sum of squares (Table 5.2) indicates that the

bulk of variation of the gravity is not properly accounted for by the

first-degree trend surface.

A third-degree trend surface fitted to the same data in

Fig. 5.2(a) resulted in a significantly improved fit, accounting for

81.6 percent of total sum of squares. A dominant feature is the

major trend of low gravity values connecting the Bodmin Moor and

St. Austell granite bosses and this trend turns nearly E-W in the

eastern part of the study area showing clearly the trend in low values

of the Cornubian granite batholith (see Section 4.2.5). Outside

of this gravity trend, the trend surface increases toward the

Upper Paleozoic sediments.

(b) Magnetic data

The first-degree trend surface of the magnetic data in

Fig. 5.1(b) shows nearly E-W trends with a gradual increase from

south to north. Quite a high fit of the first-degree trend surface,

of 68.8% of the total sum of squares may indicate that the linear

trend surface may depict the regional trend of the magnetic source

in this area. This trend which is different from that of the

gravity data corresponds well to the Amorican structural trend.

The third-degree trend surface shows some improvement in

fitting, of 79.5%. The general trend is very similar in features

133

(c) MSS Band 4 (d) MSS Band 5

Fig. 5.1: Comparison of trend surfaces of degree 1 for 14 variables in the Bodmin Moor area. The variables are of (a) Gravity, (b) Magnetic, (c) MSS Band 4, (d) MSS Band 5, (3) MSS Band 6 (f) MSS Band 7, (g) As, (h) Cu, (i) Ga, (j) Li, (k) Ni (1) Pb, (m) Sn and (n) Zn. The values on the maps increase in order of (3 2 1 & A B C).

Fig. 5.1 continued

~~~~.

(e) MSS Band 6

(g) As

Ii) Ga

"~W""J:~ .- . . - . = : .::::.. : )'

:§l 1'1 ~ :i' =- 'I ::;: ;··S ;;;;. 'Ill - ··n == .'I!! ~ 11':1: ". , = ··.t~: ~ .= ~;.,tt: ~ - .. ~. = .'''.: ,,. = ::,'B:~ =- ::.;::'1:~ ._" ~.

I I II illl

.....• I

il . I

I .!

(f) MSS Band 7

(h) eu

.:::::: : . -.­. -: ==-:.=" .­.­.­.­.­.­.­.-:=" .' .'

(j) Li

...... ~ .... =

-::

(m) Sn (n) Zn

136

to the first-degree ones, showing the general E-W trend. A broad

basin-like feature appears in the south of the map which may represent

a local basin of non-magnetic rocks. The location of this basin

is nearly coincident geographically with the central part of the

Trevone basin. The general increase toward the north may reflect

a shallower magnetic basement in a northern direction.

5.3.2 Landsat MSS data

The first-degree trend surfaces of all Landsat MSS bands

reveal a generally common NNW-SSE linear trend. Trends of MSS bands

4 and 5 increase towards SW and are the reverse of the trends of the gravity

data, which is largely attributed to the high reflectance over the

granite masses compared with the relatively low reflectance in

the surrounding sediments.

However, in MSS bands 6 and 7, the direction of increase of

the trends is opposite and almost corresponds to those of the gravity.

This may be due to the lack of dominant features over the granite

relative to higher reflectance of the green vegetations in the

surrounding sediments toward the infrared spectral bands as described

in Section 4.3.3.

The extremely low fit of the first-degree trend surfaces of

all bands (0.0315 ~ 0.0525) indicates that the linear trend may not be

reliable for analysis of linearity of the data. The third-degree

trend surfaces in the MSS bands in Fig. 5.2(c) to (f) have resulted

in a much improved fit for the various data sets ranging from 27.3

to 46.5% of the total.

The values of the third-degree trend surfaces of the four bands

are the reverse of those of the gravity data. Low value gravity trends

corresponds to high value MSS data trends. The higher trends

137

fdr .piiiiiiiiiiiiih!,-

III' 4

^ i l i l i l " 1 ? t* M H U U n t M * .

(b) Magnetic

(c) MSS Band 4 (d) MSS Band 5

Fig. 5.2: Comparison of trend surfaces of degree 3 for 14 variables in the Bodmin Moor area. The variables are of (a) Gravity, (b) Magnetic, (c) MSS Band 4, (d) MSS Band 5, (e) MSS Band 6, (f) MSS Band 7, (g) As; (h) Cu; (i )Ga ; (j ) Li ; (k ) Ni ; (1) Pb; (m) Sn and (n) Zn.

The value on the maps increase in order of (3 2 1 & A B C)

Fig. 5.2 continued

a m i i w i K a H f f i i i

4444***** nllilllll!:

(e) MSS Band 6 (f) MSS Band 7

N

» . i t u i i u n : : 1!!!. i:::..: 11 • 1111:':

(g) As (h) Cu

(i) Ga (j) Li

Fig. 5.2 continued 1 39

(m) Sn (n) Zn

i

140

Table 5.1 Trend surface coefficients of degree 1 and degree 3

\ \ e n i : 1" Ts

v a r i u b l e ^ r ^ ^ b 0 b l b 2

G r a v i l y - 5 1 1657 1 . 7 7 2 5 - 1 1428 M a g n e t i c 39 7 1 8 5 - 0 5 7 3 - 3 0 8 4 2 Band 4 33 0 8 1 1 - 0364 0 0 1 4 2 Band 5 30 5 4 7 6 - 0454 0 0 3 2 0 Band 6 87 0 4 4 3 0 1447 - 0 8 4 0 Band 7 78 4 3 7 1 0 1940 - 1 1 5 3 As 3 1989 0 2 0 1 7 0 0 5 0 0 Cu - 2 6 7 8 2 1 2 8 1 9 6 1 2 8 0 0 Ga 25 3184 - 1544 - 0 0 8 8 Li 16 . 7 5 9 9 - 1810 0 1246 Ni 55 8134 0 1375 - 1687 Pb 5 8 . 2 3 9 5 1 0 0 6 3 0 5 3 0 5 Sn 5 0 2 6269 - 1 8 4414 31 2 4 7 9 Zn 1 4 5 . 4777 0 3 9 4 5 2 0 6 4 5

(b) Degree 3

b o b l b 2 b 3 b 4 b 5 b 6 b_ b s b 9 v a r i a b l e ^ ^ G r a v i t y 274 . 3 9 8 1 - 1 6 . 3 2 5 8 - 1 3 . 6 1 4 6 0 . 4156 0 . 0 7 4 9 0 . 1 4 0 8 - . 0 0 2 2 - . 0 0 4 2 0 . 0 0 4 3 - . 0 0 0 6 Magnet i c 60 . 6937 - 3 . 2 1 1 1 - 2 . 6 0 9 6 0 . 1123 - . 0505 - . 0 3 2 9 - . 0 0 0 7 - . 0 0 0 7 0 . 0 0 1 0 0 . 0 0 0 6 Band 4 25 . 6 6 8 0 0 . 3744 0 . 2 8 9 3 - . 0 0 4 5 - . 0 1 1 0 - . 0 0 2 3 - . 0 0 0 0 0 . 0 0 0 2 - . 0 0 0 0 0 . 0 0 0 0 Band 5 11 . 7776 1 . 0 2 4 9 0 . 8 4 3 3 - . 0 1 5 7 - . 0 2 4 5 - . 0 1 0 2 0 . 0 0 0 0 0 . 0 0 0 3 - . 0 0 0 0 0 . 0 0 0 1 Band 6 34 . 3 8 6 6 3 . 1 4 2 9 1 . 1 0 0 0 - . 0 4 9 1 - . 0 5 3 7 0 . 0 2 6 4 0 . 0 0 0 2 0 . 0 0 0 4 0 . 0 0 0 1 - . 0 0 0 5 Band 7 18 . 9 5 7 0 3 . 5 5 9 6 1 1 1 4 9 - . 0 5 5 7 - . 0 5 7 6 0 . 0 3 4 8 0 . 0 0 0 3 0 . 0 0 0 4 0 . 0 0 0 1 - . 0 0 0 6 As 18 . 5 8 7 6 0 . 0 0 7 5 - 2 . 0 5 0 6 - . 0 1 4 5 0 . 0 4 6 0 0 . 0 6 6 0 0 . 0 0 0 1 0 . 0 0 0 1 - . 0 0 0 7 - . 0 0 0 6 Cu 346 . 0 5 1 6 - 2 8 8 7 9 7 - 2 2 2 9 9 5 0 . 8 7 7 1 0 . 5 6 3 3 0 . 6 6 0 5 - . 0 0 8 7 - . 0 0 0 3 - . 0 0 7 4 - . 0 0 5 7 Ga 20 . 0 5 0 5 0 1036 0 0 0 4 5 0 . 0 0 2 2 - . 0 0 5 0 0 . 0 0 2 1 - . 0 0 0 1 0 . 0004 - . 0 0 0 4 0 . 0 0 0 1 Li 1 . 3 3 6 9 0 . 3 0 6 6 0 . 7 6 7 4 0 . 0 1 2 5 - . 0 2 5 4 - . 0 1 2 3 - . 0 0 0 3 0 . 0 0 0 7 - . 0 0 0 5 0 . 0 0 0 3 Ni 99 . 3 7 4 6 - 2 . 5 1 5 8 - 2 3 1 0 2 0 . 0 3 5 6 0 . 0 4 9 3 0 . 0 5 3 3 0 . 0 0 0 0 - . 0 0 0 7 0 . 0 0 0 1 - . 0 0 0 6 Pb 558 . 8 3 3 6 - 2 9 . 1 1 8 8 - 2 4 . 5 8 5 7 0 . 5 4 8 7 0 . 6 4 8 7 0 . 5 1 4 5 - . 0 0 3 8 - . 0 0 2 9 - . 0 0 4 3 - . 0 0 4 3 Sn - 2 7 9 . 2 5 4 2 48 7 9 0 5 - 8 2 . 7 7 6 4 - 1 1 5 8 5 0 . 0 8 2 7 5 . 7 0 2 4 0 . 0 0 5 2 0 . 0 2 5 4 - . 0 4 8 6 - . 0 5 0 8 Zn 368 . 4 6 0 8 - 3 . 9 4 0 1 - 1 6 . 4 9 0 4 - . 2 5 6 8 0 . 5 5 2 8 0 . 4 0 3 2 0 . 0 0 4 2 - . 0 0 2 7 - . 0 0 5 1 - . 0 0 2 9

141

Table j.2 Summary of test values of goodness-of-fit and F-test

TOs- degree

variable^^

first degree third degree TOs- degree

variable^^ goodness-of-fit F-test goodness-of-fit F-test

Gravity 14.65 351.4 81.61 2015.2

Magnetic 68.80 4513.0 79.51 1761.9

Band 4' 5.25 113.5 27.35 170.9

Band 5 3.15 66.6 30.75 201.6

Band 6 3.53 74.9 45.78 383.3

Band 7 ' 4.66 100.1 46.51 394.8

As 6.07 132.3 29.03 185.7

Cu 7.65 169.6 23.34 138.3

Ga 18.36 460.2 51.44 480.9

Li 11.66 270.1 42.73 338.7

Ni 3.57 75.7 15.8 85.2

Pb 1.90 39.7 28.64 182.2

Sn 20.07 513.9 40.82 313.1

Zn 5.36 115.8 14.40 76.4

Those underlined satisfy random effect at significance level =0.05 according to the criteria by Nordiffe(1969).

142

in MSS data connect the Bodmin Moor and St. Austell granite areas.

Bands 4 and 5 show their maximum values over the St. Austell

granite with general high trends in the NE, which decrease towards

the NW, NE and SE boundaries of the map. However, although the

general high trends in bands 6 and 7 are similar to those of bands

4 and 5, the typical highest peak in the St. Austell area disappears

in bands 6 and 7, and the high trend extends toward the NE border

of the map and decreases towards NW and south of the map. This is

again due to relatively low reflectance of bands 6 and 7 over

sparsely vegetated high Moor lands and relatively high reflectance

of well developed vegetation in the low lands associated with the

sediments. The sudden decrease at the NW and SE boundaries along

the coast lines is certainly attributed to the relatively low

reflectance of water compared to land areas in all bands.

5.3.3 Geochemical data

Again, the first-degree trend surfaces fitted to the eight

geochemical elements show the broad regional trend of each data set.

As, Cu and Pb elements in Fig. 5.1(g), (h) and (1) are similar to

each other and generally increase towards the SE. This may indicate

that these three elements may be regionally associated with each other

throughout the area. The high common occurrence of these elements in

the mineralized zones in the east of Bodmin Moor, and to a lesser

extent near the east of St. Austell granite, might be the main contributing

factor to the close correlation between the three trends.

Ga in Fig. 5.1(i) increases toward the west with a linear

trend of nearly N-S, and Li in Fig. 5.1(j) has a NE trend with an

increasing gradient to the NW. The general linear trends of Ga

143

and Li may be due to their high occurrence in the granite and to some

degree the neighbouring sediments as mentioned in Section 4.4.2.

Ni and Sn in Fig. 5.1(k) and (m), respectively, appear

to be similar in their NW-SE regional trends, but their values are

increasing in opposite directions. Ni increases to NE but Sn

shows its increase to SW. Zn has a linear direction of ENE-WSW

and increase toward the SE of the map.

The low reliability of these trends is indicated by generally

low fits of their linear trend surfaces of the data, ranging from

1.9% in Pb to 20.0% in Sn.

The third-degree trend surfaces show an improvement when

fitted to the same data with much higher fits of the regression

to the total sum of squares ranging from 14.4% in Zn to 51.4% in Ga.

Third-degree trend surface of the eight elements are shown in

Fig.5.2(g) to (n). Again, As, Cu and Pb in Fig. 5.2(g), (h) and (1),

respectively, show similar regional trends to each other. They

increase towards the eastern boundary and NW corner of the map, with

a saddle ridge of low values passing NNE to WSW in the west of the map.

The high trends represent those areas of most mineralization in

the east of the Bodmin Moor granite extending to the Kithill granite,

and also in the northwest part of the map.

Li and Ga appear to be similar in their third-degree trends,

which are largely moderately high over the Bodmin Moor granite

and the highest trend appears on the St. Austell granite in the

southwest corner of the map.

The regional trend apparent on the map of Ni in Fig. 5.2(k)

shows a broad depression with a low trend over the Bodmin Moor granite

and particularly low values over the St. Austell granite areas.

Surrounding the granites the trend increases toward the map boundary

144

in the east, northeast and northwest.

Sn in Fig. 5.2(m) shows a highest trend on the St. Austell

granite and the high trend extend to ENE, generally following the

southern margin of the Bodmin Moor granite. It decreases to the north

and SE corner of the map. This is largely related to the high content

of Sn over the St. Austell granite area with its extensions to

the southern margin of the Bodmin Moor granite and eastwards to the

Kithill granite.

Zn in Fig. 5.2(n) appears with a high trend extending NE

in the south of the map which joins the eastern boundary of the

St. Austell granite and the east of the Bodmin Moor granite area where

a number of zones of sulphide mineralization occur. Another high

appears in the northwestern part of the map, which is largely attributed

to Pb-Zn mineralization to the east of Wadebridge.

5.3.4 Residuals from the third-degree trend surfaces

The spatial distribution of third-degree trend surface

residuals is shown in Fig. 5.3(a) to (n) in order to extract local

isolated features which might have significance in geology or local

mineralization. Furthermore, the residual maps may provide

information on associations in detailed variations between variables.

a. Geophysical data (Gravity and Magnetic)

Clusters of gravity residual values in Fig. 5.3(a) reflect

local variations in the original data. The negative clusters that

show in the middle and SW corner of the map depict the trend of low

gravity values over the granite masses, and positive trends occur

outside the granite margins.

145

(200ooE,qooaN)

(a) Gravity (b) Magnetic

(c) MSS Band 4 (d) MSS Band 5

Fig. 5.3 Contoured Residual Maps of the Third degree polynomial: (a) Gravity, (b) Magnetic, (c) MSS Band 4, (d) MSS Band 5, (e) MSS Band 6, (f) MSS Band 7, (g) As, (h) Cu, (i) Ga, (j) Li (k) Ni, (1) Pb, (m) Sn and (n) Zn.

. »P . Km Sea l <8

Fig. 5.1 continued

(e) MSS Band 4

(g) As (h) Cu

(i) Ga (j) Li

147

Fig.4-20 continued.

148

This may be due to the steep gradient of the Bouguer anomaly

along the margin of the granites due to the steep intrusive contacts

of the granites with the surrounding sediments. Those dense

basic intrusives around the granite margin might also contribute to

emphasize these features. Sharp decreases towards the edges of the maps,

particularly in NW and NE, may be due to unreliable edge effect typical

in the polynomial fittings as described in Section 4.2.2c. However,

the E-W basin-like trend in the southern margin of the map may be of

some structural significance. This has been partially described

in Section 4.2.5 and will be described further in Section 6.3.3.

The magnetic residuals in Fig. 5.3(b) are similar in their

features to the vertical derivatives and highpass filtering maps

(see Section 4.2.5b) except in some margins of the map. It illustrates

well the isolation of local anomalies as described in the vertical

derivative maps. The high reliability' of this result is indicated by

its high 'goodness-to-fit' of the third-degree trend surface to the

data (See Table (5.2.).

b. Landsat MSS data

The third-degree residuals of MSS bands 4 and 5 are shown

in Fig. 5.3(c) and (d), respectively. The positive clusters that

aggregate in the middle and southwest corner of the map both represent

high reflectance over the high granite Moor areas. They are largely

attributed to dry and brownish vegetation over high elevations possibly

due to cattle overgrazing and partly due to the open-pit mining of

Kaolin as described in Section 4.3.3. These aspects were observed

on a field trip through the area in April 1982.

149

In the flat and low lands of the map, the low reflectance

may be caused by cultivated green vegetation toward the end of

April when the image was taken, particularly in band 5 which has

a strong absorption around 0.68]Jm of spectral value.

Dr Hawkes of I.G.S. who accompanied me on the field trip

considers it is unlikely that there has been much change in land use

between April 1975 when the Landsat data were imaged and 1982 when

the field data were collected.

On the other hand, MSS bands 6 and 7 are similar to each

other in their residuals in Fig. 5.3(e) and (f), but different from

those of bands 4 and 5. There is a lack of high reflectance in the

residuals over the granite Moor areas as in bands 4 and 5, and they

show rather negative values over the granite areas while they show

some positive values outside the granite, particularly near the coast-

line. Strong negtive reflectivity of the off-shore areas are distinct

in Bands 6 and 7 and show the clear feature of the coastline. This

feature is not so sharp in Bands 4 and 5.

The reflective feature over the Kithill granite area is not

similar to those of other granite areas in all bands. It may be

due to different vegetation coverage in areas lower than high Moor

areas as described in Section 4.3.3. The vegetation over the Kithill

granite might have consisted of a green agricultural cover which is

similar to the vegetation in the cultivated lower flat areas.

c. Geochemical data

Residuals of the geochemical elements from Fig. 5.3(g)

to (n) may show more direct relationship to local mineralization

and geological features than any other variables. In elements with

common sulphide minerals high positive residuals occur around the

150

margins of granite masses. Particularly high residuals are distinct

in the east to southeast borders of the Bodmin Moor granite and

this high trend extends further to the south, following some drainage

patterns. The extension of these high residuals to the Kithill granite

area is also characteristic of most elements with common sulphide

minerals such as As, Cu, Pb, Zn and Sn. Around the south to

southwest border of the Bodmin Moore granite and further towards the

eastern border of the St. Austell granite where there had been some

mining activities for sulphides in the past, high As, Cu, and Zn

residuals similarly occur. Zn also shows some drainage pattern

in the southeast of the St. Austell granite which might also be

partly due to contamination by old mining and smelting. Between

Wadebridge and the Bodmin Moor granite in the northwestern part of the

map, high positive residuals of Pb and Zn occur. This may be related

to local Pb-Zn mineralizations. Some anomalous residual of As

also occur in this area.

A small patch of positive Zn residuals is also shown in the

north of the granite where many basic volcanic rocks occur.

In Fig. 5.3(m), positive Sn residuals occur mainly near the

southern and southeast margins of the Bodmin Moor granite and over the

St. Austell granite and its eastern part.

Ni appears as largely high residuals around the metamorphic

aureoles of the granites and as typically low residuals over the granite

masses. A typical high residual in the southeast of the St. Austell

granite may be due to both effects, mineralization and contamination

by mining activities, etc.

The elements, Ga and Li in Fig. 5.3(i) and (j), respectively,

show in general, positive residuals in the granitic areas and from

the Bodmin Moor granite it extends further to the northeast of the

151

map which might indicate a contribution from local volcanic rocks

in the area.

5.4 Discussion

Regional trends of the data over the Bodmin Moor area were

extracted by analysis of trend surfaces of first- and third-degree, and

local irregularities or isolated anomalies were calculated by

subtraction of the computed third-degree trend surfaces from the

original data.

Some statistics were computed to examine the 1goodness-of-fit'

of the computed regression to the original data, and further random

effects of the data were assessed using the calculated F-test values.

The degree of randomness of the data has, to a certain extent, the inverse

relationship with the degree of 1goodness-of-fit1.

At the 0.1% level of significance (a = 0.001), test statistics

of a F-distribution with degrees of freedom (df)* (2,4093) and

(9,4086) for the total number of 4096 data values have critical values

of F ~ 6.9 and ~3.0, respectively. The computed test values of all

fourteen variables for both cases of degrees 1 and 3 fall well within

the critical region, that means the computed test values are all much

greater than the critical values, so that the regression of data

are fitted well for first- and third-degree polynomials.

*F for [(k-1), (n-k-2)]df where k = no. of coefficients in equation

of trend surface and n = no. of data points.

152

In a different way of expression for assessing the randomness

of data, the regression effect is significantly different from the

random effect. Anything with % sums of squares accounted for >1%

is definitely'significant1 in a statistical sense, but will still not

necessarily be geologically very meaningful.

Although the comparative study of similarity among different

data may not be so reliable in the trend surface analysis, it

provides at least some insight into the basic patterns of similarity

between variables.

To facilitate the comparison, a diagram was made of the

lienar directions of the data, with indication of the predominant

directions of various trends in Fig. 5.A.

Apart from the magnetic data and Zn, all the linear trends

can be effectively divided into two groups.

Group A consists of gravity da'ta, four Landsat MSS bands, Li,

Ni and Sn, and trends largely in a northwest to southeast direction.

Among those in this group, MSS bands 4 and 5 Li and Sn show the same

increasing direction of the trends to SW, while gravity, Ni, MSS bands

6 and 7 increase to NE.

The similarities and dissimilarities of the regional trends

of the variables in this group may be understood by the fact that those

variables commonly show most dominant features over the main granite

area.

Group B consists of As, Cu, Pb and Ga, and trends in the range

between North and NE. Sulphide related elements As, Cu, and Pb are close

to each other and show their increases to SE but Ga increases opposite

to the others to NW.

153

Fig. 5.4 Linear directions of the data and predominant directions of the linear trends. Values in the parenthesis are 'goodness of fit'. Higher values are more 'important'.

154

Significant geological features in Group B are evident.

There is an anomalously high common occurrence of As,

Cu, and Pb in the Middle and Upper Devonian and Lower Carboniferous

rocks with the maximum values in the mineralized zones, particularly

in the eastern flank and southeastern part of the Bodmin Moor granite,

which indicates that these elements are most likely associated in

these areas. There is low occurrence of these elements within the

granite itself and in the Lower Devonian and Upper Carboniferous

rocks.

By contrast, Ga shows its high concentration within the

granites and surrounding superficial sediments where streams drain

from the granite, while Ga has a low concentration in the rest of

the sedimentary rocks.

The residuals derived from the third-degree trend surfaces

illustrate localized anomaly patterns which might be closely

related to specific mineral concentrations.

The author's study has shown that analysis of the third-degree

trend surfaces of the variable is a more reliable means of assessing

the regional trends and local anomaly patterns than the other methods

of trend surface analysis considered.

155

CHAPTER SIX

SIMILARITY ANALYSIS

6.1 INTRODUCTION

There are basically three different types of map comparison,

overall similarity, spatial similarity and spectral (or

structural) similarity. These can be applied to geological data,

similar types of maps or different types'of maps.

In this chapter, these three methods of similarity analysis

have been employed to investigate the inter-relationships between

map variables.

Correlation coefficients were calculated for overall similari

analysis, and for spatial similarity the similarity map of Davis

(1973, p.393-407) was used between gravity and pseudogravity, and

between some geochemical elements. Coherence analysis was used for

analysis of spectral similarities and also for the structural

similarities between gravity and pseudogravity transform of magnetic

data.

6.2 Principles of the procedure

6.2.1 Overall Similarity

The Pearson product-moment correlation coefficient has been

calculated in making comparisons between variables. The calculated

coefficients result in a measure of statistical overall correspondenc

between two variables with no consideration of the sample location at

all.

When the correlation coefficients are estimated from the m

156

sample data, the computed formula is as follows,

n £ (X -X.)(X. -X.)

Sii k=l l k 1 J k J

= — (6.1) ij S S n _ n _ 1 3 [ Z (X., -X.) Z (X., -X.)

k=1 lk 1

k=1 Jk J

i = 1,2,...,m

j = 1,2,...,m

where S. and S. are the variances of variables X. and X. respectively 1 J . 1 J and S.. the covariance between the two variables, n is the ij number of samples in each variable.

Because of variability of correlation estimates, it is

desirable to verify that a non-zero value of the sample correlation

coefficients reflects any existence of statistically significant

correlation between variables. This is accomplished by testing the

hypothesis that the population correlation coefficient p^j = 0.

The acceptance region for a test of the hypothesis of zero correlation

given by Bendat et al (1971) is,

/—t" i+R.. ^El l n _ i l < |Z« | (6.2)

ij 2

where Z is the standardized normal variable and n the number

of samples. Values outside the above interval would constitute

evidence of statistical correlation at the a level of significance.

For example, at the significance level of Ot = 0.01, the

acceptance region of the hypothesis is I R — I £ 0.0404 (6.2.1)

Thus, if the absolute value of a computed correlation coefficient

is greater than 0.0404 , the hypothesis is rejected at the significance

level of CL = 0.01, so that the two variables can be said to have

statistical correlations.

157

The overall similarity analysis is very effective to see

concisely the degree of association between variables since it

gives a single number between -1 and 1. As noted by Howarth (1979),

although it may be seriously affected by several factors such as

erroneous data values, scaling of the data and so on, the

Pearson correlation matrix will give much useful information on

the structure of interelement relationships for little computational

effort.

However, it' does not provide information on spatial

significance, so that it might be difficult to interpret it in relation

the geology. There are also many other ways of overall similarity

measurement such as distance coefficient or cosine theta coefficient,

etc. Some detailed accounts of these overall similarity measures

are described by Harbaugh and Merriam (1968), Davis (1973) and Sokal

and Sneath (1963).

6.2.2 Spatial Similarity

A possible way of comparison of two maps with relation to

the spatial distribution is the so-called 'similarity map1 (Davis,

1973).

This is similar to matching coefficients or cross-association

coefficients as described by Harbaugh and Merriam (1968), and

Z-trend maps by Robinson and Merriam (1971), in which the data are

not necessarily required to be numerical and the results are commonly

reduced to two categories.

Instead, a similarity map is a numerical application of

those techniques and there are no particular limits on the number of

categories.

158

This is a very effective way of spatial comparison between

different sets of the same type of variables, since it is more complex

if two different maps have been mapped separately for comparison

(Davis, 1973).

This technique can be equally applied to two different types

of variables since the problem inherent in different maps based

on different variables can be avoided if the two original maps are

converted to standardized form so that both variables can be

represented by common dimensionless digits.

The standardized form is

X.-X Z. = (6.3) l s

where Z^ and X^ are the standardized and measured data at point i,

respectively, and X and s the mean and standard deviation of the

variable.

Because the map variables are standardized, it is not really

necessary to compute a correlation to obtain a measure of similarity

between the two maps. Instead, the surfaces can be multiplied

together. If both surfaces deviate from the mean in the same direction,

their product will be positive, while if they deviate in opposite

directions their product will be negative.

Care must be taken in that the high correlation does not

always imply the same geological significance since both negative

or positive ones with the same magnitude always produce the same

results.

Although it may provide information of some structural or

other geological significance, as noted by Davis (1973), this method

may not give direct insight into structural features themselves.

Rather, it provides spatial distributions of similarities between

map variables which may provide further qualitative information of

159

structural or other geological significance, if the original maps

are consulted in its interpretation.

6.2.3 Coherency Analysis

Coherency analysis has been widely used for analysis of

spectral or structural similarity in the potential fields (Agarwal,

1968; Kanasewich and Agarwal 1970, QEB Inc. Lakewood Co. 1978 etc.).

This provides an excellent measure of similarities between waveforms when

random noise is absent (QEB Inc. Lakewood Co. 1978). The mathematical

formulation is as follows:

Consider two spatial functions f(x,y) and g(x,y) where x and

y are an orthogonal coordinate system, and f and g are map data

sampled at discrete spatial intervals Ax and Ay. Letting

and represent the power spectrum of f and g respectively,

and P_ (io ,03 ) the cross-power spectrum between f and g where 0) and f g x y x

U) are angular frequencies in x and y directions, the coherence

between f and g are as follows.

|Pf (o> >|2

Coh( f, g)= x y , r- (6.3) P.(0) ,03 ) P (03 ,0) ) f x y g x y

Theoretically, the above coherence function would be equal to

1, independent of frequency, while this is not so in practical applications

due to windowing and smoothing effects (Bath, 1974).

To calculate Coh(f,g), the power spectra of both f and g,

together with the crosspower spectra between f and g have to be

determined. These are calculated by a Fast Fourier transform

algorithm (as mentioned in Section 4).

The Fourier transform of a function f defined over the

160

x-y plane, is

^ -id) x — icu y F(u>x,u> ) = J J f(x,y)e X e y dxdy (6.5)

—CO —CO

In practice, we deal with a discrete sample function over

a finite areal extent.

For the discrete case, Equation (6.5) becomes

N-l M-l -27Ti(^ + JLL) F(u,v) = Z Z f(k,n)e AXAY (6.6)

k=0 n=0

where AX and AY are the sample intervals in the X and Y directions

respectively, and N and M represent the total number of samples in X

and Y. u and v are the frequency indices, and k and n the location

indices in the x and y directions respectively.

The power spectrum P^ is obtained by

P (u,v) = |F(u,v)|2 (6.7)

and the cross-power spectrum P^ by

P. (u,v) = F"(u,v) G(u,v) (6.8) f >g

where * indicates the complex conjugate function. G(u,v) is the

Fourier transform of function g.

Thus, for the practical discrete case Equation (6.4) can

be written as

161

Coh(f,g) =

u +Au v +Av

u=u v=v F*(u,v)G(u,v)

u +Au v +Av u +Au v +Av Z Z |F(u,v)|2 Z Z |G(u,v)|2

u=u v=v u=u v=v

(6.9)

where F and G are the corresponding Fourier transforms of f and g,

and Au and Av are the bandwidths over u and v respectively. Thus

Equation (6.9) is an estimator for the coherence over a band centred

at u = uj + Au/2 and v = v^ + Av/2

A plot of coherence coefficients as a function of wavenumbers

will illustrate those spectral regions which have degrees of association

between f and g. If the coherence is very low for a particular band of

wavenumbers, f and g are said to be incoherent in that wavenumber band.

If f and g are, respectively, the Bouguer gravity field and a

pseudogravitational transform of the magnetic field, the incoherence

may indicate that the magnetic and gravity data do not represent the

same source body. Or, it may also indicate that significant

remanent magnetization is present. Random noise may also contribute,

to a certain degree, to reduce the coherence. On the other hand,

if the coherency is very high for a particular band of wavenumbers, then

the causative bodies of magnetic and gravity may come from the same source

unit. Thus, Equation (6.4) is of great value in interpreting

the subsurface structures of an area. The main point in utilizing

Equation (6.4) is that it is assumed that the rock unit is both

anomalously dense and magnetic, and the direction of the rock

magnetization is uniform throughout the area and also the same as

that of the present day inducing field, or alternatively, we know already

162

the precise direction of the magnetic vector.

Where considerable remanent magnetization is present,

and where the direction of the remanent magnetization is unknown,

then the coherence between the gravity and transformed magnetic fields

will be low even if the causative body is the same.

A programe 'COHAN' was written by the author using Equation

(6.4) and Bartlett windowing was implemented in the smoothing process

for the coherence analysis and further for bandpass filtering

'BPFILT' was also written to delineate structural features in space

domain for those spectral regions applied for the analysis.

6.3 Applications of Similarity Analysis

6.3.1 Overall Similarity

The Pearson-product moment correlation coefficients were

calculated by using Equation (6.1) for the data over the Bodmin Moor

area. The calculated results are shown in Table 6.1. The coefficients

show the existence of inter-relationships between variables and their

relative strengths.

Taking the significance level of OL = 0.05 as the critical value

then lower values are an indication of non-association and the range

of the correlation coefficients (R) falling below 0.0306 are those

of non-association as noted in the previous Section. Thus, the results

have been classified into five groups according to the magnitude of

the coefficients.

a.very strong association |R| > .9

b.strong association .6 < |R| < .9

c.significant association .3 < |R| < .6

163

d. weak association .0306 _< |R) < .3

e. non-association |R| < .0306

Statistical results of inter-relationship are discussed on the basis

of their attitudes and patterns of the regional distribution as

described in Chapter 4 and Chapter 5.

(a) Very strong association

Only Landsat data fall in this group. The coefficients

between bands 4 and 5, between bands 6 and 7 show above 0.96 of R

which may indicate very similar reflectances of the near-surface

features between those two visible spectral bands and between the near-

infrared spectral bands as described in Section 4.3.3. Although it

is not shown here, bands 6 and 7 with band ratios R6/4, R7/4, R6/5 and

R7/5 are also very strongly associated. None of the ratio data show

the strong association with band 4 or 5.

(b) Strong association

As-Cu: A strong correlation between the two elements

may be due to their common occurrence in the soil particularly

associated with mineralized zone in the study area contributes to high

increase in the correlation coefficient between these two elements.

Ga-Li: The common high occurrence of these elements in

the granite and local stream draining from the granite near its

margin together with their low occurrence in the sediments, cause

the strong association between these two elements.

164

(c) Significant association

Gravity-Magnetic, B4, B5, Ga, Li, Ni, and Sn: The gravity

data shows some positive correlation with magnetic and nickel.

Particularly the positive correlation with nickel can be seen from the

fact that high nickel occurrence appears in the aureoles of the

granite and typical low occurrence within the granite itself.

Negative correlations with bands 4 and 5 are expected due

to high reflectances of those visible bands over the high Moor lands.

Ga, Li and Sn elements are also negatively correlated with the

gravity due to their high occurrences on or near the granites and

low occurrences toward the sediments.

Magnetic-Sn: They show a moderately negative correlation

between these two data sets. Generally this may be due to the high

occurrence of tin in the south where the magnetic trend decreases

gradually, and the low occurrence of tin in the north where in

general there is a regional high magnetic trend.

B4,B5,-B6,B7, Ga, Li, and Ni: Bands 4 and 5 show positively

moderate correlations with bands 6 and 7 as expected.

Positive correlation with Ga and Li are due to their common

high values over the granites and low values towards the surrounding

sediments.

Negative correlation with nickel may be due to the absence

of nickel in the granites and its concentration in the surrounding

aureoles possibly in the form of pentlandite and pyrrhotite.

As, Cu-Pb, Sn and Zn: their moderate positive association

can be explained partly by their common occurrences in the soils

and also their association in mineralization and surrounding anomalous

zones.

165

Ga,Li - Ni,Sn: Ga and Li show moderately negative correlations

with Ni due to their inverse relationships in the granites and

surrounding metamorphic aure oles, and positive correlations with Sn

are due to their high occurrence in the granites and their margins

and low occurrence in the sediments.

Zn-Ni, Pb: Zn shows moderate correlations with Ni and Pb.

Particularly the significant association with Pb may be in part

due to their common association in mineral occurrences.

(d) Non-association: The data sets below are statistically

non-correlated at 95% confidence level in the area.

Gravity - Zn Magnetic - B6, B7, Ga B5 - As B6 - Ga, Li, Ni, Sn, Zn B7 - Ni, Sn Cu - Ga

To display complex relationships among variables more

effectively, a cluster algorithm 'CLUSTER1 (Davis, 1973, p.467)

was used to construct a dendrogram of the correlation matrix.

CLUSTER uses weighted pair-group average clustering. The

method operates on an N x N similarity matrix. The algorithm pairs

those two individuals, i and j say, which have the highest similarity

and replaces columns (and rows) i and j by a single column with

arithmetic 'average' similarity coefficients. The process is

then repeated by pairing individuals and clusters of previously combined

individuals.

Details of the weighted pair-group method can be referred

to in Davis (1973), Sokal and Sneath (1963) and Parks (1966). The

process of averaging together members of a cluster and treating

TABLE 6.1: Correlation Matrix (untransformed data)

Gravity Magnetic B4 B5 B6 B7 As Cu

Gravity 1.0000

Magnetic .3914 1.0000

B4 -.3873 -.0956 1.0000

B5 -.4480 -.1627 .9679 1.0000

B6 -.2319 .0111 .5410 .5645 1.0000

B7 -.2039 .0253 .4424 .4680 .9911 1.0000

As -.2763 -.2275 -.0663 -.0254 .1458 .1586 1.0000

Cu -.1248 -.2151 -.1126 -.0930 .0375 .0493 .7554 1.0000

Ga -.5481 -.0043 .3917 .3572 .0211 -.0314 .0336 -.0187

Li -.5800 -.1853 .5125 .4857 .0031 -.0625 -.0599 -.0775

Ni .4533 .1876 -.3651 -.3728 -.0306 .0067 .0971 +.0696

Pb .1671 -.0864 -.1993 -.2240 -.1802 -.1754 .4009 .4207

Sn -.5222 -.3479 -.0974 -.0900 .0269 .0265 .3413 .3575

Zn .0158 -.2394 -.1578 -.1624 .0224 .0440 .4519 .3864

Ga Li Ni Pb Sn Zn

0000 8250 1.0000

3237 -.5024 1.0000

0729 -.2078 .2049

3294 .3545 -.2276

0830 -.1484 .3695

1.0000

-.0685 1.0000

.4838 .2010 1.0000

• 1762 •3306 •4850 .6394 • .7938 .9482 .0990 .2534 .4078 .5622 .7166 .8710 1.0254

Gray

Ga

Li

Ni

Mag.

Sn

As

Cu

Pb

Zn

B4

B5

B6

B7

Fig. 6.1

.0990 .2534 .4078 .5622 .7166 .8710 1.0254 .1762 .3306 .4850 .6394 .7938 .9482

Dendrogram - clustering with absolute correlation coefficients (untransformed data) Values along X-axis are similarities.

.5641

.8250

.4332

.2726

.3479

• 1841

• 7554 • 4150 .4838 .1333 • 9679 .5040 .9911

CTN

168

them as a single new number introduces distortion into the dendrogram

and this distortion becomes increasingly apparent as successive

levels of clusters are averaged together.

However, this is an efficient way to represent inter-

relationships between variables in a cogent tree-structural manner.

The analysed results are shown in the dendrogram of Figure 6.1.

Absolute values of all correlation coefficinets were considered in

the analysis since the similarity and dissimilarity are different

in verbal sense. If similarity values above about 0.25 are taken

to be significant in grouping, there are three groups apparent in

the dendrogram.

Group 1 consists of gravity, Ga, Li, Ni, Sn and magnetic

in their association with descending order of correlation.

Group 2 consists of mainly sulphide metallic elements

including As, Cu, Pb and Zn.

Group 3 are those from all MSS bands.

The groupings of similarities are somehow different from

those of linear trends in Section 6.4. This is because significance

of magnitude of all individual values, are taken into account in the

similarity analysis with correlation coefficients while only regional

aspects of the sampled data contribute to the trend surface

analysis which ignores detailed variations of individual values.

Mancey (1980) has noted that the inter-relationships can

be found better with the transformed data rather than analysis with

the untransformed data. Analysis made with the transformed data

in this study has shown similar results with the untransformed data.

However, certain differences have been found in clustering results

between these two. Grouping at the same level of similarity in the

TABLE 6.2: Correlation Matrix (transformed data)

Gravity Magnetic B4 B5 B6 B7 AS

Gravity 1.0000

Magnetic .3246 1.0000

B4 -.3929 -.0517 1.0000

B5 -.4528 -.1204 .9689 1.0000

B6 -.0431 .1239 .4291 .4142 1.0000

B7 .0497 .1567 .2029 .1810 .9563 1.0000

As -.2947 -.2071 -.0520 .0075 .0977 .0949 1.0000

Cu .0865 -.1472 -.2720 -.2825 -.0257 .0328 .6248

Ga -.5554 .1220 .3366 .3159 -.0263 -.1235 .1413.

Li -.6820 -.1260 .3592 .3850 -.0694 -.1738 .1109

Ni .4803 .1543 -.3401 -.3536 .0368 .1215 .1757

Pb .1587 -.1603 -.2954 -.2978 -.0881 -.0433 .5346

Sn -.6720 -.6240 .1207 .16059 -.0987 -.1353 .3154

Zn .1603 -.2363 -.2480 -.2486 .0060 .0674 .4943

Cu Ga Li Ni Pb Sn Zn

0000 0737

1622

3556

5611

2891

5718

0000 7834

2862

1421

,2656

,2080

0000 4338

2905

,4695

,2511

1.0000

.2897

-.2345

.5471

1.0000

.0598 1.0000

.5589 .2142 1.0000

o vo

• 1751 • 4759 • 6263 • 7767 • 9271 • 0999 • 2503 • 4007 .5511 .7015

I I • I I

• 8519 1.0023 Gray

. — Sn

.— Ga

Li

,— Ni

— B4

— B5

— As

Cu

— Pb

Zn

Mag

B6

B7

•0999 .2503 .4007 .5511 .7015 .8519 1.0023 •1751 .3255 .4759 .6263 .7767 .9271

• 6720

• 4931

• 7834 • 3587 .3315 • 9689

• 2367

• 6248 • 5404 • 5589 • 1721

• 1333

• 9563

o

Fig.- 6.2 Dendrogram - clustering with absolute correlation coefficients (transformed data) • Values along X-axis are similarities.

171

transformed data (R = .25), B4 and B5 are clustered into group 1

and B6 and B7 are independently clustered as a group 3. Magnetics

shows no association at all to any of the groups. Clustering

of sulphide metallic elements remains same as Group 2 in the untrans-

formed data.

Qualitatively, the result of transformed data seems to be

better in its clustering in this study. Correlation coefficients

for the transformed data and dendrogram of clustering result are

given in Table 6.2 and Fig. 6.2 respectively.

6.3.2 Similarity map

The application of 'similarity map' analysis was made to

following data; between gravity and pseudogravity, As and Cu, Cu and Sn,

and Pb and Zn.

Tov avoid overweighting by outliers of data, the data were

reset to ±3.291 when the standardized data were greater than

3.291 in absolute values. This ensures the data fall in the

confidence level of 99.9%

Maps of the surface-product of two standardized data are given

in Fig. 6.3(a) - (d). Areas of high coincidence of two surfaces are

clearly shown, as are areas where they depart markedly from one another.

In Fig. 6.2(a) the similarity map between gravity and pseudogravity

shows high coincidence in the northeastern part of the map which

might depict dense magnetic bodies. In the northwest there is

moderately high correspondence. Along the sub-surface trend of the

Cornubian batholith there also occur a moderately positive correlations.

Low correlations or negative correlations occur in the northern

part of the granite and toward the southeast of the granite masses.

172

(2£0OPE, qooo H)

N

(a) Gravity and pseudogravity

Scale

M

3.0 Km

(b) Cu and Sn

Fig. 6.3 Similarity maps. Hatched areas represent negative values whereas blank areas are positive.

173

F i g . 6 .3 cont inued

(c) Pb and Zn 3 io ao 1 i » • —•

S e a l e

174

These may mean that either gravity and magnetic anomalies come from

different sources, or remanent magnetization may exist so that the

transformation of the magnetic field into gravity may not be complete.

Particularly in the area where very high negative magnetic anomaly

appears, the low negative correspondence appears to be due to the

latter as already discussed in Section 4.2.5e.

High correspondences between Cu and Sn in Fig. 6.3(b) largely

occurs east and southeast of the Bodmin Moor granite and Kithill

granite, and the northeast margin of the St. Austell granite,

and negative correlations occur in the southern margin of the Bodmin

granite and east of the St. Austell granite where largely high Sn occurs

along the granite margins while copper values are low. Another

negative trends occurs in the southeast of the map where largely high

Cu occurs partly due to mineralization and possibly contamination by

mining activities and smelting.

Elsewhere, there appears largely low positive correspondence

throughout the map. In Fig. 6.3(c) high association of Pb and Zn

appears in three areas: one in the northwest of the granite, another

to the southeast of the granite and the third in the Kithill granite

area. Strong negative correlations occur in the northwest of the main

granite and in the southwest of the study area. The first one coincides

with high Pb mineralization and the last two areas correspond with

high Zn mineralization and possible areas of contamination.

Elsewhere, except along the southern and eastern part of

the Bodmin granite and northwest of the map, where largely low negative

association occur, Pb and Zn are largely positively associated,which

means that they are positively correlated as shown in correlation

coefficients in Tables 6.1 and 6.2, respectively, for the untrans-

formed and transformed data.

175

In Fig. 6.3(d), the inter-relationships between As and Cu are

largely positive except a few areas of weak negative association.

The strong positive association which appears in the southeast of the

Bodmin granite and . Jn the Kithill granite may reflect those

highly mineralized or partly contaminated area.

The low correlation on the southern margin of the granite is

noteworthy because the copper and tin mineralization occur in this

area but the arsenic shows low values in this region which may imply

that Arsenic should not be used as a pathfinder for mineralization

as it is inconsistently associated with other elements in mineralization

as noted by Shouls et al (1968).

6.3.3 Coherence Analysis

A contour map of coherence coefficients calculated by

Equation (6.9) for the Bodmin area is plotted in Fig. 6.4 as a function

of spatial frequency. Interpreted regional subdivisions are also

illustrated in the map.

A bandwidth of Q = 2, was applied for smoothing to compute

the power and cross-power estimates together with the Bartlett windowing.

Coherence contours show discrete regions of high and low

coherence on the map. There are four distinct regions labelled A, B,

C, and D.

Region D has been ignored since the spatial region of D may be

insignificant in terms of geological results.

To interpret the significance of the spectral regions A, B, and

C, two dimensional filters were designed which pass or reject

gravity and/or magnetic data within the frequency limits defined by

these regions. These filters were then applied to the data in order

176

Fig. 6.4 Coherence vs. Frequency - Bodmin Moor Area

177

to produce filtered maps of various coherent and incoherent regions.

Digital filters were designed in the two-dimensional

frequency domain by assigning a weighting value of 1.0 to

frequencies within the desired region of the coherence plot, while

a value of 0 was assigned to frequencies outside the desired

region.

A cosine taporing method was applied to the frequency weights

for the transition zone between the pass and reject bands, in order

to reduce spurious oscillation due to the sharp truncation o'f data.

The filter function was then multiplied by the Fourier transform of

the gravity and magnetic data, and the resulting product was inversely

transformed to yield a filtered version of the original data. The

filtered maps are shown in Fig. 6.5 to 6.7.

a. Region A

The coherence of this region is not very high, but it is

presented here to illustrate spatial features for those low frequency

regions with a moderate magnitude of the coherence.

In Fig. 6.5(a) the gravity map was bandpass filtered to show

the general low trend of the gravity cupolas interconnected in the

subsurface and high ridges along the northern margin which may

correspond to the ridge between the two basins of Bude and Trevone.

This also largely corresponds to the E-W trending magnetic high

to the north of the granite.

The inverse association between gravity and magnetic data

in the southern part of the map may be due to the following reasons

as partly described in Section 4.2.5(a).

It indicates that the causative anomalies of magnetic and

gravity fields do not come from the same source bodies; the gravity

178

( 2 0 0 o o E , q o o o N )

M

(a) Gravity Coherence Map CtooooEjloeoN)

(b) Magnetic Coherence Map o to so Km , • »

s cai e Fig. 6.5 Bandpass filtered maps of coherence region A

179

high of the E-W trend may be due to regional undulations of the main

granite and thickening of the sedimentary basin, or the limit of lateral

extension of the main granite underneath, while the magnetic anomaly

may come from magnetic basement of high susceptibility under the

batholith or sediment basin. The area may correspond

geographically to the central part of the Trevone basin.

b. Region B

The moderately high coherence in'this spectral region may be due

to the general E-W structural trends in the north of the Bodmin Moor

granite as shown in the bandpass filtered gravity and magnetic maps

in Fig. 6.6(a) and (b), respectively. Although small fluctuations

occur in the filtered gravity map in this spectral region, the general

high and low gravity trends corresponds to those of the magnetic data

in the north of the Bodmin Moor granite.

The curved shape with alternating signs in this region may

be due to several slices of rocks which have different magnetic

properties. This may be due to shallow basic rocks intruded into

zones of stratigraphical weakness which formed during the intrusion

of the granite bosses and were therefore aligned along the margin of

the granites.

Freshney and Taylor (1971) have noted that slices of Lower

Carboniferous rocks are inserted into Upper Devonian rocks as far

south as Tintagel. They have also noted that rocks in a zone of

isoclinal folds below the Rusey Thrust show the effect of low-grade

metamorphism with development of fine chlorite, particularly

in Devonian and fine pyrite and pyrrhotite spots. Thus the magnetic

patterns may be partly due to stratigraphic effects of Lower Carboni-

ferous and Upper Devonian rocks.

180

(lOOooE, qoooW?

M

(a) Gravity Coherence Map

(2dOOD*,qooou)

(b) Magnetic Coherence Map

o to 2.0 W -Jl

S c a l e

Fig. 6.6 Bandpass filtered maps of coherence region B

181

The middle and southern parts of the map do not show

distinctive features. These areas may not be correlated

significantly as shown in Similarity map in Fig. (6.3(a)).

c. Region C

The incoherent gravity and magnetic maps of Region C

of the coherence plot were obtained by applying the filtering mentioned

above. Fig. 6.7(a) and (b) show the results of bandpass filtered

gravity and magnetic maps, respectively.

A number of intermediate and short wavelength highs and lows

trending generally N-S are apparent. In Fig. 6.6(a) the alternating

low and high N-S trends through the map area may indicate

the subsurface undulation of the granite and sedimentary rocks

e.g. local thickening of the superficial deposits, probably local

sub-regional tectonic faulting contributes somehow to those fluctuations

and some discontinuities in the area.

On the other hand, the magnetic map of incoherence Region C in

Fig. 6.5(b) shows local small scale anomalies apparently different

from those of gravity and trending largely N-S in the north of the

map where a number of basic volcanic rocks are intruded into the

Devonian sediments.

6.4 Discussion

Analysis of various similarity measures shows general inter-

associations between map variables.

The correlation coefficients provide a measure of overall

similarity in a simple number which might be geologically informative.

They may also be of further value in multivariate data analysis.

182

(a) Gravity Coherence Map

(b) Magnetic Coherence Map P • . ^o Hm

S c a l e Fig. 6.7 Bandpass filtered maps of incoherent region C

183

For example, the extremely high correlation coefficients, such

as between Bands 4 and 5 or Bands 6 and 7, may be redundant in the

multivariate data analysis, so that one of the two variables may be

reasonably removed in the multivariate data analysis, in order

to reduce the number of data sets for efficient and effective

computation. Further consideration can be given to those variables

which are not correlated with any other variables at all, since these

variables may occur completely independently from others.

This aspect will be further discussed in Chapter 7.

The similarity map clearly shows spatial correspondence

between variables, so that associated mineralized zones or

any geologically significant aspects such as structural information or

geological provinces, could be analysed by interpreting those maps

with original data.

Coherence analysis between the' gravity and a pseudogravity

transform of the magnetic data is very useful to analyse spectral

correlations between them. Furthermore, bandpass filters designed

from the coherence map for particular spectral regions, such as high

coherence or incoherence have been effectively used to extract structural

features for those spectral regions, so that causative anomalies

for both fields have been more clearly identified. In particular,

stratigraphic features in the northern part of the study area may

be due to shallow basic intrusions occurring along the stratigraphic

beddings or zones of structural weakness developed during the

Amorican orogeny. Stratigraphic overlappings between Upper Devonian

and Lower Carboniferous sediments may have contributed to these

effects to a lesser extent due to their differences in susceptibility.

184

Many known and unknown linear features could be delineated

by bandpass filtering designed from the coherence analysis between the

gravity and pseudogravity transform of the magnetic data in the

study area because they can be emphasized by the application of

filtering for particular spectral region.

185

CHAPTER SEVEN

CLASSIFICATION AND IDENTIFICATION OF MULTIVARIATE DATA

7.1 INTRODUCTION

Four types of pattern recognition techniques have been

used in order to examine their applicability to various combinations

of multivariate data sets with the purpose of defining geological

provinces, or areas of possible mineralization.

The methods include Factor Analysis (FA), Cluster Analysis,

empirical discriminant analysis and characteristic analysis. As

already mentioned in Chapter 1, the first two methods require no

pre-determined or established training set for the analysis and they

attempt 'natural' grouping of the multivariate data sets using

certain similarity or dissimilarity criteria, (i.e. unsupervised

learning). On the other hand, the last two types belong to supervised

learning which requires certain prior known training sets in order to

classify remaining unknown samples. The procedures of each classi-

fication method, together with reviews of references, will be

described rn the following sections.

Parametric statistical analysis assumes that the frequency

distributions of multivariate data are normal, otherwise results

will be less confident. Thus, the data were transformed to near-

normality by using the power transform technique (Howarth and Earle,

1979) or an arc sine method where it is appropriate, and comparison

was made between the results from untransformed and transformed data

sets.

Details on the transformation techniques applied will be

described in the next section.

186

In addition, one of the most

data analysis - the selection of the

pattern recognition analysis is also

difficult tasks in multivariate

best sets of variables for

described.

7.2 Transformation of Data

Parametric statistical analysis assumes that the data is

normally distributed. In a real situation most of the data are

skewed to some extent, so that they do not conform to the normal

distribution.

As shown in Fig. 7.1(a) to (t), histograms of various

original data sets clearly show some degree of skewness, and particular

geochemical elements such as As, Cu and Pb are highly positively

skewed. Many transformation methods have been proposed for reducing

this skewness.

Various transformation techniques are described by Hoyle (1973)

Mancey and Howarth (1980) have shown that the power transform

suggested by Box and Cox is one of the most effective techniques for

de-skewing data. In this study, the power transform was applied

to all the data sets except gravity, in which an arc sine method

was applied for transformation of the data.

The generalized power transform is:

z = (xA-l)/A , X K 0 1 x > 0 (7.1)

In A , A = 0 J

where z is the set of transformed data, x the set of original data

and A the power coefficient.

Howarth and Earle (1979) have implemented various optimization

criteria in the computer program for computation of A in order to

187

either reduce the skewness to zero, or jointly reduce the skewness

to zero and kurtosis to three. Mancey and Howarth (1980) have also

shown its applicability to an even larger data set- by making a

subset of the data for computation of X. Mancey (1980) has also

discussed in detail the power transformation method.

In this study, a subset of data for estimation of X has

been made by taking values at every second column in every second row.

The mean X of computational results obtained using three

kinds of optimization technique (Dunlap and Duffy, Skewness/Kurtosis 2/1

and maximum likelihood schemes) are shown in Table 7.1

Table 7.1 Mean X values estimated using the Dunlap and Duffy,skewness/kurtosis 2/1, and maximum likelihood schemes

Variables mean X Variables mean X

gravity 1 , .2301 BR 7/6 9, .5013 magnetic 0, .2126 As .2543 Band 4 .3003 Cu .3677 Band 5 0. .5933 Ga .0687 Band 6 7. .1788 Li -1 , .077 Band 7 4, .8857 Ni .8893 BR 5/4 7. .4403 Pb . 1445 BR 6/4 5. .3950 Sn 0. .0611 BR 7/4 3. .8829 Zn 0. .0176 BR 6/5 4. .6374 Gravity SVD 1. .0482 BR 7/5 3. ,3534 Magnetic SVD 1 . . 1906

The transformation of the data sets have been performed using

mean X values obtained from three different optimization techniques,

mentioned above. This may be most reasonable method (Howarth, perc.

comm.) and also the X values calculated from the three methods are

usually close to each other.

188

Histograms of the power transformed data except for the gravity

data are shown in Fig. 7.2(b) to (t). Generally the X transform has

regularized the data. Particularly the data having approximately a

unimodal distribution have been improved significantly (which means

that even in the bimodal, the population of the anomalous pattern

is relatively small compared to the background population - these

are the most common cases in geochemical elements).

This could be beneficial in parametric statistical analysis.

The Landsat MSS bands and their ratios (Fig. 7.2(c) to (1)) have

been reasonably de-skewed by the power transform. However, though

in all cases the overall asymmetries have been de-skewed clearly,

certain data have not been much improved in terms of normality compared

to the original data.

This phenomenon is particularly significant with the

geophysical data including gravity and magnetic which have patterns

of strong bimodal distribution. Some geochemical elements (eg, Ni)

are also not much improved. These data show strong bimodal distribution

patterns.

It appears to the writer that in the case of univariate data

the power transform is most effective, while in the case of typical

bimodal or polymodal data sets it improves its overall skewness

but the effectiveness is decreased with the degree of polymodal

distribution.

Thus, the arc sine method has been used for the gravity of

typical bimodal distribution as follows.

-1 (X.-X . Z. = sin l min (7.3) i (X -X . max min

where X . and X m m i max are the minimum and maximum of sample values,

respectively.

u - n V n - i n . * -m.m ti.m nt.ii n..n m..i .^TS'TT^^riS^^R^^^^^^^'Ei:!. ij.'.T rt.it

a ) GRAVITY b ) MAGNETIC c ) MSS BAND 4

11.11 ll.ti 11.11 m.n m.m <!.•• 41..i h.m m!|< 71.11 m.'ll im.'m uitu tiltn d ) MSS BRNO 5 ) mss BAND 5

n . n <i.i> it .M

f ) MSS BAND 7

: . n I.M i. i i i . n i . i i

g ) M S S 3R5/4 • ••! •••< l-ll l.m 1.11 i.m i.m <.|l c.x l.ll l.ll h) MSS 3R5/4 i) MSS 3R7/4

Li :.»• i.ii t.ii i.it i.,, i.ii i. ii :.i« .i.n o.ii i.»I i.m i.ii <.io i.m i.ii i.II

j ) m5s br6/5 k) mss 3R7/5 1 )mSS BR7/6

Fig.7.1 Histograms of the filtered data.

190

Fig.7.1 continued.

to ii.m m.m o.'n 57T77 m ) 0 S ELEMENT

P ) LI ELEMENT

s) SN ELEMENT

11.11 11.11 im.m im.o iii.n in.n m.n nt.it iiu.h UJ i.m ii.ii to.ti ti.it ii.m <•.:<

N ) CU ELEMENT O ) GFI ELRMRN"

».i ' I.ti n.tt tt.tr M.n ti.il It.I I .»7i* iiJti tiTti u.'m H. 'n uJL • i.ii iii.ii -...11 im.ii itl.m <|,.l« in .01 111.11

q ) N f F ELEMENT r ) pR ELEMENT

-tn.it iii.m kii.ii mii.ii imi.m imi.m m i l .1,1.11 ,,:„ in.m mi.hut.m 1,77m t )ZN ELEMENT

191

v = — r ^ — * —i i r- - - i 1 •f.tt - I . * H . ' l -t.tt ••»• I.M l.tl t.M l.lt I H t.M I I . . I I I .M lf.lt f.Of |.M f. l l I . I t » . » 1.11 ?•>

a) GRRV1TY AS b)MRGNETJC c) *SS BOND 4

i.t? t.tt «.«» it.M i ? .u m.m it.K n H i i i i i i i i u i i m i i M n t m i n i M i i u i i i i i f n M i m i m . i l c i : i V i n » i f i i M i i » i M i n i t < i i i n r n n i i i t » i i i t i i . . i « . i . i

d ) MS5 SKND 5 e ) ^SS BOND 6 f ) » S S 3RND 7

I.M .... .... .... I . n „ . „ „..c „.i. „ . , , , „ . „ .„. ' „ V , . n i , . „.',, , . ' „ „;..

g ) M S S BR5/4- h ) MS5 BR6/4 ^ ^ M<55 bri/4

? ' : . ' ! « . « f. l . l l ..t.lt I I I . I I M l . I I IM I . l l i"..U 11.11 .11.11 111.11 11 f I .M 1.11.1. 1111..I t - l l t.ft t.M I . . I t.ll 0.1

j ) HS5 5R6/5 k)MSS KR7/5 1 ) MSS HR7/6

Fig 7 2 Histograms of ^transformed data except for the gravity data(a) in which case the arc sine transform was applied.

192

Fig.7.2 continued,

•Vii i.n i.n I .n t.41 I.u i.n *.«« i.k i.»» i.m in »." ».•• " :.»i '•,t *•** m ) AS ELEMENT

:••>« i.ii ».»« i p ) L ! E L E M E N T

S ) S N E L E M E N T

N ) CU ELEMENT O ) G A E L E M E N T

!M mi Mt c7»» '>••» <••>* M . i i t i . t * •».«; •»»•»»

UkL wl >:•< t.'n i-ii Tm

Q ) N I E L E M E N T r) P B E L E M E N T

JiiL I M it"

LU '„ .!„ .!„ .!.. •.«•r.. T ) Z N E L E M E N T

193

From the characteristic of the sine curve it is clear that the

middle part of the data is compressed while both ends of the

distribution are stretched, so that typical bimodal distribution

could be transformed near to normality as shown in Fig. 7.2(a).

As Prelat (1977) has noted, this transform also prevents the variance

from being a function of the mean as occurs in a binomial distribution.

7.3 Selectioft of Variables in Multivariate Data Analysis

Selection of an appropriate group of variables is one of the

most difficult tasks in multivariate data analysis in order to

effectively analyse the data for geological exploration.

If the number of samples is kept small, the collection may not

be representative of the truth, and if the number of variables is

small, the variation due to some of the geological processes might not

be represented. Therefore, statistical and mathematical approaches

that require the use of automated procedures on computers should be

applied to a large set of samples analysed for a large number of

variables.

However, more data will not always guarantee better results

and we also have to face the problem of computer capacity to

accommodate such a large number of variables from large samples, so

that an optimum number of variables should be chosen in order to be

efficient in computation and effectively solve the geological

problems.

Castillo-Muhoz (1973) has shown in the study of the overall

recognition success rate for the testing set with a linear discriminant

fucntion that the maximum recognition success rate for samples of

known type presented to the classifier as 'unknown1 (the testing set)

194

is attained (for his data) with particular combinations of either 6, 7,

or 8 variables. Thus, the use of a certain limited number of variables

can be more successful in its analysed results and efficient

in computation than a larger number of variables.

There have been some propositions for the selection of variables

as for example in Schultz and Goggans (1961) and Mucciardi et al.

(1971) etc.

In this study, two methods were used in the selection: trial by

intuition and experience, and principal component analysis.

The trial method may be entirely user-dependent although

the selection should be performed based on geological reasoning.

Knowledge of the characteristics of each variable and geological

information are essential to be effective in the selection. Analysis

of inter-relationships between variables such as correlation coefficients

may also provide information in order to reduce the number of variables

to a smaller size as described in Section 6.3.

Principal component analysis is a well-known procedure in

reducing the dimension to a smaller size. As will be described in the

following sections, this is a linear transformation of variables

which are uncorrelated with each other. Each principal component

score calculated by transformation, would constitute a certain

proportion of the total variance (represented in its eigenvalue),

so that those scores for which variance becomes an insignificant

proportion of the total variance, could be removed from the further

analysis with minimum loss of information since these insignificant

scores probably consist of noise contributions as noted by Mancey (1980)

and Siegal (1980), etc. Thus, the reduced numbers of component

scores chosen can be used further in the cluster analysis or any dis-

criminant analysis. In addition, if any training set is well

established in the study area, then, a method called 'BAKWRD'

195

developed by Howarth (1973a) would be very effective and efficient

in the selection of variables. This is a sequential backward method

in order to find the best set of variables for effective discrimination.

It is suboptional in the sense that the overall best combination out

of all possible combinations may not be found.

As noted by Howarth (1973a) exhaustive examination of all

possible combinations is not practicable for more than about 7

variables. In most cases, setting-up an appropriate training set itself

is difficult in practice, so that this method has not been used in the

selection scheme of variable sets in this study.

In areas of known geology and mineralization, trial by

ISODATA (a cluster analysis program which will be described later

in this Chapter) could be very effective in the selection. As mentioned

above, the practical use of this method for thorough examination is

limited by the number of possible combinations of variables

{m(m-l)/2} where m is the total number of variables to be analysed.

Where any prior information is not available, the most

reasonable way of selection may be to apply principal components

analysis in order to reduce to a smaller manageable size of scores

which may contain most of the information in the multivariate data.

There appear to be no objective methods of determining what

combinations of the data will give the best result in relation to

geology or mineralization. Thus, the complication of variable selection

schemes leads the potential user to an individual optimum method

either by intuition or a statistical method.

Pattern recognition techniques, therefore, have been used in

this study to cluster three different kinds of untransformed and

transformed data chosen by trial using information available (such as:

correlation coefficients; examination of eigenvector matrix; and

196

comparison of the results from ISODATA analysis to the known geological

information) and two untransformed and transformed data sets

derived from the principal components analysis.

The three sets of data are as follows:

1. 6 Landsat MSS data set: B4, B5, B7, R5/4, R6/5 and R7/6.

2. 8 geochemical element set: As, Cu, Ga, Li, Ni, Pb, Sn and

Zn.

3. 9 mixed variable set: Gravity, Magnetic, B5, R5/4, R7/6, Cu,

Ni, Sn and Zn.

The two data sets computed from the PCA are the first 8 PCA scores

derived from the following 16 variables: Gravity, Magnetic, B4, B5,

B7, R5/4, R6/5, R7/6, As, Cu, Ga, Li, Ni, Pb, Sn and Zn.

PCA is calculated so that the first component scores have the

largest variance of the total, the second component scores have the

second largest variance of the total, and so on. Thus, the first few

component scores contain most of the information of the multivariate

data set, while the component scores towards the end are most likely to

contain the noise contribution of the data.

Therefore, by choosing the first few component scores and

ignoring the rest of the component scores, we represent all information

in it effectively for further easier manipulation of the data.

Joreskog et al. (1976) have noted by an empirical test that the Scree

Test proposed by Cattell (1966) has been found to be the most useful.

In this Scree Test, if the magnitude of the eigenvalues are

plotted against the successive component number, there is a point on the

slope of this plot where an exponential decay portion in the beginning

and the consistent linear portion of the later components come together.

This point divides the exponential part in which the major information

is contained (and thus these scores are to be retained for further

197

20

<u 10 3 .—i CO > C Q) Ml •H W

\ * \ Exponential \ portion

\ . Linear portion

V — .

principal components number 11 12 3 16

20

s? w (1) 3 i CO > C 0) M •H W

10.

\ ^ Exponential \ portion

\ Linear portion

Figure 7.3:

I 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 principal components number

Cattell's Scree Test for determining the correct number of principal components calculated from 16 variables: Gravity, Magnetics, B4, B5, B7, R5/4, R6/5, R7/6, As, Cu, Ga, Li, Ni, Pb, Sn and Zn.

198

analysis) and the latter, linear, part that contains noise which can

be ignored in the further analysis. The 8 PCA scores chosen are

based on the criteria as shown in fig. 7.3.

As noted by Davis (1973), since the discriminant analysis is so

clo sely related to multiple regression, most of the procedures

for selecting the most effective set of predictors can also be used

to find the most effective set of discriminators. In this sense, the

'Ridge regression' technique presented by Howarth in the recent

Departmental Seminar (November 1982)seems to be very effective since

the aim of this technique is to find best predictor elements which

are stable and have high discriminating power. However, this technique

was not attempted, in this study, because of shortage of time.

Mathematical procedures for ridge regression are described by Jones

(1972) and Turner (1980).

7.4 Unsupervised Classification

7.4.1 Factor analysis

(a) Factor analysis procedures

Usually the multivariate data show intercorrelation to some

extent between variables so that the axes of the probability density

function are not statistically orthogonal.

Factor analysis (FA) or principal components analysis (PCA)

is a multivariate statistical technique that generates new variables whose

probability density function have orthogonal axes (i.e. uncorrelated

variables). Although the two techniques have much in common and

are often referred to in the literature as the same thing, there are

fundamental differences between them as described by Howarth

and Sinding-La'rsen (Howarth, 1983, Chapter 6). In PCA each

component is determined so as to maximize the common

199

variance (C ) that is common to all the variables,whereas in FA a

given number of 'factors' (less than the number of variables) are defined

so as to account maximally for the common intercorrelation between

the variables. PCA is thus variance-oriented whereas FA is

correlation-oriented.

The mathematical bases of the PCA and FA are described in many

references,e.g. Cooley and Lohnes (1971), Davis (1973) and Chien (1978),

etc. Comprehensive descriptions of the two methods are also given by

Howarth (1983).

When we are considering heterogeneous data sets prior to any

actual computation, we have to normalize the data set by standardization

in order to give each variable equal weighting. The standardization

procedure is given in Equation (6.3) for a single variable. In

PCA, the correlations between variables which are represented by the

covariance matrix have to be computed. • When the correlation

coefficients R. . are estimated from the sample data, the computational ij formula for the case of m variables is given in Equation (6.1) of

Section 6.2.

To generate independent variables when R ^ is not zero, one must

find a transform that transforms matrix R to R' so that for all i \ j,

R L j = 0. To do this, one must compute the eigenvalues of R. The

eigenvalues are actually the variances for each factor of the transformed

scores associated with R' and the eigenvalues are the solutions

to the characteristic polynomial of R. The characteristic equation

of R is a polynomial of degree m obtained by expanding the determinant

200

C =

Rn~xi R 21

31

ml

R 12

R22" X2

"32

m2

13

R23

R33~ X3

1m 2m 3m

R - • • • R -X mJ mm m

= 0

(7.4)

Here the eigenvalues X^ are the solutions to the equations. With

the known eigenvalues, a column matrix of the eigenvectors V can be

determined as follows:

V RV = R1 (7.5)

where V is the transpose of V and R1 is the matrix having the

eigenvalues (X.,X.,X0,...,X ) as diagonal elements. I 2 J m

The matrix formed from the column vectors (normalized to

unit length) defines the linear transformation that is applied to

the m-variables to produce the principal component scores.

The eigenvalues are ranked in descending order^, so that the

first transformed map will have the largest variance, and the next

will have the second largest variance and so on.

Some of the components scores may be contributing little to

the overall variance, and fewer principal components, say P variables,

than m may adequately represent the data. The P derived variables

that may retain as much of the variance of the original data as

possible are the first principal components.

In FA, the correlations among the m variables are assumed to be

accountable by a model in which the variance for any variable is

201

distributed between a number (p) of common factors and anumber

of 'unique factors', the remaining being attributed to random error

(Howarth, 1983). Thus,the factor model X^ may be expressed as

P X. = Z z. f + e. (7.6) l . lr r i r= 1

t h where fris the r common factor, p the specified number of factors

and z. the factor loading of X. on factor f and e. the random lr l r i

variation unique to the factor associated with X^.

In psychometrics, most problems are concerned with identifying

the common factors. However, in geological factor study, factor scores may

be more important because it retains geometrical information with the

significance of analysed results.

The variance for each variable has been reduced to unity by

normalisation. Therefore, the sum of squares of the factor loading P 2 of the common factors Z (Jl) provides a measure of the degree of

i= 1

fit of the samples to the common factors. This is referred to as

'communalities'.

The magnitude of the communalities is dependent upon the

number of factors that are retained. For this reason, care must be

taken in choosing the number of factors.

In the true factor analysis model, it is assumed that the

number of common factors is related to a number of known underlying

causative influences. The number of these is postulated beforehand

and used to define the factor solution. In geological work this is

not usually possible and the PCA solution, truncated to K(<m) components

is usually taken as the initial solution. This may be 'improved'

upon, in the sense that a more easily interpretable solution is obtained,

by subsequent (orthogonal or oblique) rotation to a final set of

202

variables, generally referred to in the geological literature as 'factors'.

The Scree test for analysis of information content for the

components has been described in the previous section. However, it is

not practically possible to display effectively all information for

analysis if the number of components containing information is larger.

Some workers recommend retaining all factors which have eigen-

values greater than one. Another approach may be to use only two or

three factors, because this is the maximum number that can conveniently

be displayed as scatter diagrams. Particularly when colour-composite

plotting techniques are used, choosing three factor scores is the most

effective in order to represent the resultant data by means of colour

additive or colour subtracting views as noted byHowarthet al ( 1976).

Any number larger than this increases the dimensionality of the

problem.

To ease the interpretation problems of the factor scores, a

further rotation of the factor can be applied. In this study, the

Varimax scheme of Kaiser has been used, taking the first three

principal components as the starting point.

The Varimax criterion involves maximization of the variance

of the loadings on the factors by obliquely rotating the principal

components axes but still preserving orthonormality between components

(Trochimczyk and Chayes, 1978) and thereby adjust the factor loadings

so that they are either near ±1 or near zero. Consequently for each

factor there will be a few significantly high loadings and many

insignificant loadings and the interpretation in terms of the original

variables is more easily apparent.

Factor analysis based on PCA in this way has certain important

properties which are essential in multivariate data analysis as noted

by Howarth (Pers. Comm.). First, it is repeatable ; second, data

203

compression can be achieved effectively; and third, interpretation of the

resulting map may be simplified.

In this study, the FACTOR program described in Davis (1973,

p519) has been significantly modified to apply it to the regularly

gridded multivariate data sets. The analysis has been made on a line

by line basis, keeping individual data files separate in order to

facilitate easy manipulation - this means that the multi-data files

individually recorded on a magnetic tape can be copied on the disc

when the job is run.' A line-by-line basis analysis could reduce the

dimension of the multivariate data set significantly so that the Central

Memory space is considerably reduced to a manageable size for the job.

The modified program is listed on Microfiche 3.

One restriction is that since the maximum number of files

allowed in a program run is limited by 50, the individual data files

may be limited to about 15 files in a run, so that for more than 15

individual data files the program might have to be modified.

204

(b) Applications of the factor analysis

PCA was first performed on the normalized variance-covariance

matrices (i.e.: correlation matrix) of a total of 8 data sets from

untransformed and transformed 6 Landsat MSS, 10 Landsat MSS, 8

geochemical elements and 9-mixed variables, the statistical results

being given in Table 7.2 to 7.5. Standardized data were reset to

±3.291, if the standardized data were larger than 3.291 in absolute

value, in order to avoid overweighting by extreme data values.

In all cases, the first threfe principal components were taken

for further Varimax rotation to yield the FA scores. The reason

for this is that not only do the first three components represent most

of the total variation in this study, but it is also most convenient 2 . . . .

to apply to the 'I S1 colour additive viewer at the Remote Sensing Unit

in the College to produce a colour-composite picture. However,

there is inevitably some loss of information judging from the Scree

Test criterion as shown in Fig. 7.4, in using only 3 components.

Statistical results, given in Table 7.2 to 7.5, show percentage

contributions of each component to the total variance and their

cumulative percentage.

The three factor scores retained from FA were stretched from 0 to 255 and then subjected to histogram normalization for effective

2 viewing on the I S viewer, as described in Section 4.3.1. The resultant

maps are shown in Fig. 7.3 to 7.10. In the maps, white and/or bright

colours represent high values and dark colours represent low values of

untransformed (U)

30

20

10

transformed (T)

1 2 3 4 5 6 7 1 2 3 4 5 6 7 principal component numbers

(c) 8-geochemical element sets

3

2C

1G (U)

30

20

10 (T)

50

40

30

20

10

60

50

40

30

20

(U)

50

40

30

20

101

(T)

> \ 2 3 4 5 6 2 3 4 5 6 (a) 6 Landsat MSS variable sets

50

40

30

20

(T)

3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 10/1 2 3 4 5 6 7 (b)loLandsat MSS variable sets

N) o Cn

8 9 10 1 2 3 4 5 6 7 8 9 1 (d) 9-mixed variable sets

Fig. 7.4 Cattell's Scree test for determining the correct number of principal components of four different variable sets: 6MSS data, 10 MSS data, 8-geochemical data and 9-mixed variable data sets

206

the component. Detailed discussions on the resulting map will be

described in the following sections.

(b.l) Six Landsat MSS data

Fig. 7.5(a) to (d) and 7.6(a) to (d) are the results of FA for

the untransformed and transformed 6 Landsat data sets, respectively.

From the statistical results in Table 7.2, the first three

factor scores of the FA constitute about 95% of the total variance

in both untransformed and transformed data sets and thus the first

three factor scores represent effectively most of the total variance,

though there still occurs some loss of information according to

the Scree Test criterion as shown in Fig. 7.4(a).

(b.1.1) Untransformed data

The first factor score is very highly correlated with brightness"

in the visible bands 4 and 5, and band ratio R7/6, and to a lesser

extent, with band 7. This factor is responsible for 51.1% of the total

variance. It is apparent that this factor is a measure of brightness

in the visible bands. High negative correlation with R7/6 may indicate

that this ratio may reflect inversely similar features to those of bands

4 and 5 as shown in Fig. 4.18(c) in chapter 4 and further the

similarity may be seen from high negative correlation coefficients

(-.7493 and -.7310) with bands 4 and 5, respectively.

Thus bright areas may represent high Moorlands, or dry or

township areas while dark zones may indicate those areas with wet or

thick vegetation. Sanddunes in the northwest and southeast coast lines

show up as bright areas.

"'brightness1 referred in this section means the intensity of the value.

207

The second factor score appears to be correlated with

brightness in the band 7 and band ratio R6/5, and to a lesser extent,

with band 4 and 5. Considering high correlations between R6/5 and two

infrared bands 6 and 7 (0.9390 and 0.9656, respectively), this

factor is a measure of brightness in the infrared bands.

Thus, bright areas may correspond to areas of green vegetation,

while dark zones may represent areas of water or dry vegetation.

Township areas and sanddunes are also dark in the image.

The third factor score is very highly 'correlated with band

ratio R5/4 and to a lesser extent, with band 5, and thus the factor score

map may be comparable to the R5/4 picture as shown in Fig. 4.18(a) in

Chapter 4. Therefore, high values which produce bright areas of this factor

are represented by dry vegetation or contaminated waters while dark

spots indicate areas of bare soils (or kaolin), green vegetation or

uncontaminated water.

The composite picture in Fig. 7.5(d) shows effectively the

combination of the three factors and is responsible for 95.9%

of the total variance. The general features are more detailed than the

false-colour composite in Fig. 4.17 in Chapter 4.

The Bodmin Moor granite area appears to be blue except for

two spots in the northwest and middle south which are greenish yellow.

St. Austell granite area also appears to be greenish-yellow. This

feature represents areas of open-pit mining for kaolin in this region.

Township areas and sanddunes along the southeast and northwest coastlines,

and dry and sparsely vegetated lowlands, appear to be green. Red tints

represent thick vegetation.. Water or water-logged areas appear to be

black and blue.

Reddish orange to magenta occur around the granite margin

where most of the mineralization in this region is located. However,

direct relationships between those two features are not so distinctive

208

( c ) F a c t o r 2 3 0 4 - , % - ( d ) C o m p o s i t e 9 6 . 0 %

F i g . 7 . 5 : FA o f t h e u n t r a n s f o r m e d 6 - M S S v a r i a b l e s e t : MSS b a n d s 4 , 5 , 7 , R 5 / 4 , R 6 / 5 a n d R 7 / 6

6 t 'Q ao Km

sea 1 e

209

( b ) F a c t o r 2 3 2 . 9 %

( c ) F a c t o r 3 1 4 . 9 % ( d ) C o m p o s i t e 9 4 . 8 %

3.0 K m

Sea 1 e

F i g u r e 7 . 6 FA o f t h e t r a n s f o r m e d 6 MSS v a r i a b l e s e t : MSS b a n d s 4 , 5 , 7 , R 5 / 4 , R 6 / 5 a n d R 7 / 6 .

210

Table 7.2 Statistical results of Factor Analysis for 6-landsat MSS data sets:B4,B5,B7,R5/4,R6/5 and R7/6.

1. Untrasformed data, a) Eigenvalues with their associated percentages.

Factor Eigenvalue Percentage variance

Cumulative per-centage variance

1 3.0645 51.07 51.07 2 1.8261 30.45 81.51 3 0.8674 14.46 95.97 4 0.2349 3.92 99.88 5 0.0045 0.07 99.95 6 0.0026 0.04 100

b) Rotated factor matrix ^factor variancte^ 1 2 3

B4 0.9532 0.2342 0.0196 B5 0.9207 0.2279 0.2596 B7 0.2259 0.9596 0.1606 R5/4 0.1220 0.1423 0.9817 R6/5 -.0425 0.9842 0.0668 R7/6 -.9057 0.1855 -.0282

2. Transformed data, a) Eigenvalues with their associated percentages.

Factor Eigenvalue Percentage variance

Cumulative per-centage variance

1 2.8181 46.97, 46.97 2 1.9746 32.91 70.88 3 0.8930 14.88 94.76 4 0.2616 4.36 99.12 5 0.0493 0.82 99.94 6 0.0034 0.06 100

b) Rotated factor matrix *^\factor variabTe^

1 2 3

B4 0.9730 0.1096 0.0497 B5 0.9386 0.0846 0.2806 B7 0.0993 0.9819 0.0317 R5/4 0.1693 -.0312 0.9838 R6/5 -.1265 0.9670 -.0689 R7/6 -.8754 0.2455 -.0523

211

in the study area.

(b.1.2) Transformed data

Transformed data in Fig. 7.6(a) to (d) are almost the same

as those of the untransformed data in Fig. 7.5(a) to (d),

respectively. The first three factors represent 94.9% of the total

variance.

It can be expected from the fact that though the transformation

of the data changes values completely different from the original

ones, the frequency distributions of the transformed data are very

similar to the untransformed data (see Section 7.2.2, Fig. 7.1 and

7.2).

Comparison of the statistical results of eigenvalues and rotated

factor matrices in Table 7.2(a) and (b> clearly indicates the

similarities between the two analysis.

In order to illustrate how effectively the chosen six variable

sets represent the multivariate Landsat MSS data, FA has been applied

to the total 10 variables including four MSS bands and the whole combination

of 6 ratio data from the four bands.

The results of the computed factor scores for the untransformed

and transformed data sets are given in Fig. 7.7(a) to (d) and

Fig. 7.8(a) to (d), and the statistical results are summarized in

Table 7.3.

The resultant features are strikingly similar to each other.

The first three factors scores represent 97.3 and 95.7% of

the total variances for the untransformed and transformed data sets,

respectively.

Therefore, the chosen 6 variable sets represent the total

Landsat MSS data effectively and they are thus computationally

212

( a ) F a c t o r 1 6 1 . 9 % ( b ) F a c t o r 2 2 6 . 5 %

( c ) F a c t o r 3 ( d ) C o m p o s i t e 9 7 . 3 %

10 _i i — i 1. ao Km

scale

F i g u r e 7 . 7 : FA o f t h e u n t r a n s f o r m e d 1 0 - M S S v a r i a b l e s e t : MSS b a n d s 4 , 5 , 6 , 7 , R 5 / 4 , R 6 / 4 , R 7 / 4 , R 6 / 5 , R 7 / 5 a n d R 7 / 6

213

( c ) F a c t o r 3 9 . 1 % ( d ) C o m p o s i t e 9 3 . 7 %

( a ) F a c t o r 1 3 6 . 7 % ( b ) F a c t o r 2 2 9 . 9 %

F i g u r e 7 . 8 FA o f t h e t r a n s f o r m e d 1 0 - M S S v a r i a b l e s e t : M S S b a n d s 4 , 3 , 6 , 7 , R 5 / 4 , R 6 / 4 , R 7 / 4 , R 6 / 5 , R 7 / 5 a n d R 7 / 6

o io io Km J 1 ! S i

Scale

214 Table 7.3 Statistical results of Factor Analysis for 10-Landsat MSS data sets:B4,B5,B6,B7,R5/4,6/4,7/4, 6/5,7/5,and 7/6.

1. Untransformed data. a) Eigenvalues with their associated percentages.

Factor Eigenvalue Percentage variance Cumulativ centage v 1 6.1899 61.90 61.90 2 2.6488 26.49 88.39 3 0.8903 8.90 97.29 4 0.2512 2.51 99.80 5 0.0138 0.14 99.94 6 0.0028 0.03 99.97 7 0.0020 0.02 99.99 8 0.0011 0.01 99.999 9 0.0001 0.00 99.999

10 0.0000 0.00 100

LrSSnce

b) Rotate4 factor matrix ^-nLactor varialDle--. 1 2 3

B4 B5 B6 B7 R5/4 R6/4 R7/4 R6/5 R7/5 R7/6

0.2050 0.2088 0.9195 0.9580 0.1782 0.9854 0.9897 0.9947 0.9965 0.1757

-.9602 -.9316 -.3640 -.2469 -.1404 -.0120 0.0011 0.0259 0.0291 0,8961

-.0032 0.2372 0.1200 0.1215 0.9735 0.1442 0.1442 0.0304 0.0497 -.0182

Transformed data. Eigenvalues with their associated percentages. Factor Eigenvalue Percentag varianc e Cumulative per-e centage variance

1 2 3 4 5 6 7 8 9 10

5.67 2.99 0.91 0.28 0.12 0.01 0.01 0.00 0.00 0.00

56.74 29.87 9.08 2.85 1.20 0.11 0.10 0.03 0.01 0.00

56.74 86.61 95.69 98.54 99.74 99.85 99.95 99.98 99.99 100

Rotated factor matrix. --—doctor variaBT^- 1 2 3

B4 B5 B6 B7 R5/4 R6/4 R7/4 R6/5 R7/5 R7/6

0.0799 0.0623 0.8972 0.9717 -.0131 0.9812 0.9846 0.9788 0.9816 0.2358

-.9764 -.9470 -.3750 -.1272 -.1911 0.1302 0.1374 0.1156 0.1286 0.8648

0.0221 0.2547 0.0436 0.0068 0.9808 0.0308 0.0356 -.0813 -.0525 -.0444

215

efficient by reducing the dimension of multivariate data analysis

by removing those variables with high correlations as redundancy.

(b.2) Eight geochemical element sets

The results of FA for the untransformed and transformed eight

geochemical element sets are shown in Fig. 7.9(a) to (d) and

7.10(a) to (d), respectively, and statistical results are given in

Table 7.4.

The eigenvalue of each factor in the table shows its relative

magnitude to the total variance. This relative significance is

more clearly shown in the percentage representation of each variance

to the total. In the case of the untransformed data, the

contribution of the first three factor scores to the total variance

are 34.0%, 29.2%, and 11.3%, and thus the first three factor

scores represent 74% of the total variance.

On the other hand, in the case of the transformed data the

contribution of the first factor to the total variance increases to 38.7%,

but those of the second and third factors decrease slightly to

28.4% and 9.7% compared to those of the untransformed data.- Thus,

the overall contribution of the first three factors to the total

variance is an increase by 2.2% in case of the transformed data.

This could imply that the results of the transformed data set would

contain more information at least for the first three FA scores.

(b.2.1) Untransformed data

The characteristics of the first three factor scores which have

the largest portion of the total variance are as follows.

Factor 1 is very highly correlated with As, Cu and Sn and to a

lesser extent, with Zn and Pb. This is interpreted as old mining

regions in the study area. Thus, the factor may indicate Sn haloes

216

( c ) F a c t o r 3 1 1 . 3 % ( d ) C o m p o s i t e 7 4 . 6 %

F i g u r e 7 . 9 FA o f t h e u n t r a n s f o r m e d 8 - g e o c h e m i c a l e l e m e n t s e t : A s , C u , G a , L i , N i , P b , S n , a n d Z n

0 io 30 Km 1 I ! i 1 s^aie

217

Table 7.4 Statistical results of Factor Analysis for 8-geochemical element sets; As,Cu,Ga,Li,Ni,Pb,Sn and Zn.

1, Untransformed data. a) Eigenvalues with their associated percentages. Factor Eigenvalue Percentage Cumulative per-

variance centage variance 1 2.7227 34.03 34.03 2 2.3383 29.23 63.26 3 0.9047 11.31 74.57

0.7707 9.63 84.20 5 0.5333 6.67 90.87 6 0.3602 4.50 95:37 7 0.2343 2.93 98.30 8 0.1358 1.70 100

b) 'Rotated factor matrix. ^\£actor 1 2 3 variably As 0.8163 -.0365 0.3531 Cu 0.8481 -.0863 0.2879 Ga O.OI76 0.9422 0.0150 Li 0.0263 O.936O -.1834 Ni -.1804 -.4422 0.6039 Pb 0.2060 -.0436 0.7746 Sn O.6859 O.378O -.1705 Zn 0.3179 -.0178 0.7643

2. Transformed data, a) Eigenvalues with their associated percentages. Factor Eigenvalue Percentage Cumulative per-

variance centage variance 1 3.0957 38.70 38.70 2 2.2751 28.44 67.14 3 0.7753 9.69 76.83 4 O.6723 8.40 85.23 5 0.4423 5.53 90.76 6 0.3420 4.28 95.03 7 0.2467 3.08 93.12 8 0.1506 1.88 100

b) Rotated factor matrix. ^^actor 3 ^^actor 1 2 3 variabit^^ As 0.8008 0.2279 -.1359 Cu 0.8436 -.0848 -.1029 Ga -.0438 0.9578 -.0684 Li -.1126 0.8619 -.3701 Ni 0.5031 -.2010 0.6731 Pb 0.76 2 -.1096 0.0761 Sn 0.3263 0.2224 -.8167 Zn 0.8200 -.1893 0.0941

218

mainly associated with copper and arsenic most probably related

to mineralization in this region.

The brightest colour appears to be in the eastern flank

and southeast of the main granite. Relatively high concentrations

also occur near the Kithill granite, southern margin of the

Bodmin Moor granite and northeast of the St. Austell granite.

Most of metamorphic aureole zones appear to be dark, which

may be due to weakly negative association of Ni to component 1 and

general low association of other elements. Weak association of Pb

and Zn elements to the component resulted in dark colours for areas

associated only with Pb-Zn haloes.

Factor 2 shows its strongest association with Ga and Li and

to a lesser extent, with Sn positively but with Ni negatively. Thus,

this factor may reflect areas of granites and associated kaolization.

Prominent highs occur in the St. Austell and Bodmin Moor granite

and along its southern margins where drainage patterns flow down from

the granite inland to the surrounding sediments.

Factor 3 is mainly associated with Ni, Pb, and Zn, and to a

lesser extent, with As and Cu. Therefore, this factor may reflect

areas of Pb-Zn and Ni haloes in the study area, which are not necessarily

directly related to the local mineralization since contamination by old

mining activities, etc. might have occurred around the region.

The brightest areas are near the Kithill grnaite, northwest

and southeast of the main grtanite, and the eastern part of

the St. Austell granite, and to a lesser extent, along the metamorphic

aureoles of the Bodmin Moor granite. The granitic areas are typically

low in this component scores.

The composite picture in Fig. 7.9(d) representing 74.6%

of the total variance shows the combination of the three factors to

represent in an effective manner most of the information of the

219

data set.

Though the geological provinces are not clearly detailed, the main

granitic areas and different types and intensity of mineralized

zones are well defined. Upper and Middle Devonian sediments appear

to be the same, largely greenish colour. These rock units have been

differentiated from Carboniferous and Lower Devonian sediments which

mark the similar feature of dark colour.

Magenta and bluish colours outline the granite areas. Reddish

lemon colours in the eastern flank of the Bodmin Moor 'and north of the

St. Austell granite mark areas of tin mineralization associated with

copper and arsenic. Orange colours in the southeast of Bodmin Moor

and near the Kithill granites mark again areas of tin haloes associated

with most of the metallic sulphide elements but they are weaker than the

previous ones and sometimes contamination by drainage systems is

apparent. Magenta in the south of the main granite represents the tin

high associated with high As and Cu.

Bluish green in the south, southeast and northwest of the main

granite is mainly associated with Pb-Zn haloes probably related to the local

mineralization in this area. The widespread red colour in the east

of the St. Austell granite may be due to high Sn which may be attributable

to either contamination by drainage in this region or derived by local

sediments of shale rich in tin.

(b.2.2) Transformed data

Fig. 7.10(a) to (d) shows the characteristic features of the

first three factors and their composite obtained from the transformed

8-geochemical elements of As, Cu, Ga, Li, Ni, Pb, Sn and Zn. Compared

to the untransformed pictures, some distinct features are evident in the

transformed data though the general features are similar to each other.

220

( c ) F a c t o r 3 9 . 7 % ( d ) C o m p o s i t e 7 6 . 8 %

F i g u r e 7 . 1 0 FA o f t h e t r a n s f o r m e d 8 - g e o c h e m i c a 1 e l e m e n t s e t : A s , C u , G a , L i , N i , P b , Sn a n d Z n o io oo km j i i i i

S c a l e

221

Factor 1 is highly correlated with As, Cu, Pb and Zn, and to

a lesser extent, with Sn and Ni. This factor generally represents

patterns related to the local sulphide mineralization and

contamination by drainage or smelting. Thus, this factor is an

indication of geochemical haloes of sulphide elements around the

granites.

The brightest and thus the highest values appear in the south-

east of Bodmin Moor and around the Kithill granite. Slightly less

dominant values occur around the aureoles of the granites: east,

south, northwest and north of the Bodmin Moor granite and another

one northeast of the St. Austell granite.

Factor 2 is very similar to that of the untransformed data.

It is strongly correlated with Ga and Li and marginally correlated

with Sn, As and Ni, so that granitic features and their flanks are

dominant in the picture.

Factor 3 shows its strong association with Sn, and to a lesser

extent, Li in a negative manner and further with Ni in a positive

manner, so that Sn concentrations in and near the granites appear

to be dark while granitic aureoles where Ni enrichment occurs, are

bright in the picture.

The composite picture in Fig. 7.10(d) shows combined geological

features better compared to the untransformed data, while localization

of anomaly patterns are more diffuse than the untransformed data.

Carboniferous sediments have been clearly differentiated

from the lower Devonian rocks which have similar features to the

Carboniferous rock units in the untransformed data. It may be difficult

to say that the Upper Devonian rocks are markedly different from

the Middle Devonian but they show somewhat different features to the

transformed data.

222

Granitic areas are blue to black. The Bodmin Moor granite

has been delineated very well but some confusion still arises with

the St. Austell granite mainly due to heavy alteration by secondary

enrichment.

The anomalous bulge on the southern part of the Bodmin Moor

granite in both untransformed and transformed data sets may be due to

the secondary enrichment by drainage flowing down from the granite upland

to the lower sedimentary areas in this region.

Mineralized zones are a bit diffuse compared to the untransformed

data but they are regionally comparable with them.

The main geochemical haloes of sulphide elements are marked by

red to magenta, but Pb-Zn haloes in the northwest of the map appears

to be yellowish white. Areas of nickel enrichment are bright

green.

(b.3) Nine-mixed variable sets

Fig. 7.11(a) to (d) and 7.12(a) to (d) show the results of FA

for the untransformed and transformed 9-mixed variable sets of Gravity,

Magnetic, B5, R5/4, R7/6, Cu, Ni, Sn and Zn, respectively. Statistical

results are given in Table 7.5.

For the first three factors, the untransformed and transformed

data sets show 65 and 71% of the total variance, respectively, so

that the transformed data shows higher percentage of the total variance

(by 6%) th an the untransformed data. This again indicates that the

result obtained from the transformed data set is at least statistically

more significant than that from the untransformed data set as in the

analysis of 8-geochemical element sets. Detailed descriptions for

each factor follow in the next section.

223

(b.3.1) Untransformed data

The general features of the three main FA factors for the

untransformed data set in Fig. 7.11 are as follows.

Factor 1 is highly associated with Landsat MSS data, i.e.

B5, R5/4 in a negative manner and R7/6 in a positive manner, and to

a lesser extent with gravity and magnetics, so that Factor 1 could be

a measure of brightness of Landsat MSS data. Thus, though there are

some variations in detail, the general features are a reflection

of band 5 as shown in Fig. 4.17 in Chapter 4.

Water or water-logged areas and densely vegetated areas are

bright while dry and brown high Moorland, dry lower land with less

vegetation or township areas are dark blue. Sanddunes in the northwest

and southeast coast lines appear to be dark.

Factor 2 is mainly positively associated with Zn, Cu, Ni and, to

a lesser extent, with Sn positively but with magnetic data negatively, so

that this factor may define the geochemical haloes. Granitic areas appear

to be low in this factor. Negative correlation of the magnetic data in

this component may indicate that most of geochemical haloes in the

study area appear to be in areas of negative magnetic anomalies.

Factor 3 is mainly correlated with gravity and Ni, and to a

lesser extent, with magnetic in a positive manner but Sn in a negative

manner. Weak negative correlations occur also with Cu and B5.

Dark features mark the Cornubian batholith trend, while the

bright colour are mainly in the Lower Carboniferous and Upper Devonian

sedimentary areas. Thus bright colours may represent areas of Ni

concentration associated with high Bouguer gravity anomalies and

possibly high magnetic values.

The colour-composite of the three factors in Fig. 7.11(d),

representing 65% of the total information, show largely the regional

224

( c ) F a c t o r 3 1 2 . 6 % ( d ) C o m p o s i t e 6 5 . 2 %

F i g u r e 7 . 1 1 FA o f t h e u n t r a n s f o r m e d 9 - m i x e d v a r i a b l e s e t : G r a v i t y , M a g n e t i c , B 5 , R 5 / 4 , R 7 / 6 , C u , N i , S n a n d Z n

0 io 56 Km 1 I i S I S e a l e

225 Table 7*5 Statist ical results of Factor Analysis for 9-mixed. variable

sets: Gravity fMagnetic,B5»R5A»R7/6fCufNifSn and Zn.

1. Untransformed data. a) Eigenvalues-with their associated percentages.

Factor Eigenvalues Percentage variance

Cumulative per-centage variance

1 2.7527 30.58 30.58 2 1.9800 22.00 52.58 3 1.1361 12.62 65.21 4 O.947O 10.52 75.73 5 0.6755 7.51 83.24 6 0.5828 6.48 89.71 7 0.4274 ^.75 94.46 8 0.3133 3.48 97.9^ 9 0.1852 2.06 100

b) Rotated factor matrix. "^\factor 1 2 3

Gravity O.3532 -.0128 0.7757 Magnetic 0.2977 -.4246 0.4889 B5 -.8709 -.1915 -.2056 R5A -.6367 0.1845 -.0863 R7/6 0.7735 0.1945 0.0380 Cu 0.1000 0.6487 -.3101 Ni 0.1928 0.5184 0.6570 Sh 0.1019 O.2721 -.8021 Zn 0.0436 0.8458 O.O519

2. Transformed data.

a) Eigenvalues with their associated percentages.

Factor Eigenvalue Percentage variance

Cumulative per-centage variance

1 3.0404 33.78 33.78 2 2.1957 24.40 58.18 3 1.2101 13.^5 71.63 4 0.8240 9.16 80.78 5 0.6187 6.87 87.66 6 0.4456 4.95 92.61 7 O.3O8O 3.42 96.03 8 0.2084 2.32 98.34 9 0.1490 1.66 100

b) Rotated factor matrix. —factor variable—- 1 2 3 Gravity 0.7531 -.3181 -.28 56 Magnetic 0.7445 0.1949 -.1122 B5 -.1368 0.2509 0.8848 R5A -.2416 0.0195 0.5796 R7/6 0.0011 -.0952 -.8637 Cu -.2437 -.73^6 -.1990 Ni 0.4056 -.7667 -.1087 Sn -.9215 -.1267 O.O733 Zn -.1435 -.8823 -.0692

226

geological features.

The granitic areas are marked by dark and blue colours except

areas of contamination by drainage pattern and old mining. In the

northern part of the Bodmin Moor granite where a part of the granite

is marked by a greenish hue different from other granitic areas, may

be due to the secondary geological environment. This area largely

corresponds to a depression, so that geochemical elements can be

enriched due to a poor drainage system.

Most of the Upper Devonian appears to be similar to the

Carboniferous rock unit in the north.

The Middle Devonian may not be defined very well and is marked

largely by a greenish and dark orange. The Lower Devonian generally

appears to be dark due to the generally low content of most geochemical

elements used in the study. Blue features in or near the granites

may be related to the drainage patterns. Offshore areas are marked by

blue colours except coastal areas of contaminated streams drainage to

the sea, particularly in the southeast corner of the study area.

Mineralized zones are again indicated by varying degrees of

colour from red to yellow, and occur in the east and southern margin

of the Bodmin Moor, around the Kithill and St. Austell granite areas.

(b.3.2) Transformed data

The results of FA for the transformed 9-mixed data set are

shown in Fig. 7.12(a) to (d). Though the information in the three main

factors is equivalent to the untransformed data set, marked differences

are that factor 1 in the untransformed data corresponds to factor 3

in the transformed data set in a negative mode, factor 3 in the

untransformed data corresponds to factor 1 in the transformed data,

and factor 2 in both data sets are of the opposite signs.

227

PCA MXD9 (B R C : T)

0 1 0 2 0 Km 1 1 1 1 1 Sea I e

F i g u r e 7 . 1 2 FA o f t h e t r a n s f o r m e d 9 - m i x e d v a r i a b l e s e t : G r a v i t y , M a g n e t i c , B 5 , R 5 / 4 , R 7 / 6 , C u , N i , Sn a n d Z n

m e a n s t h e v a l u e s a r e r e v e r s e d .

228

Their information contents to the total variances are also different

(see Table 7.5). However, for easy comparison, the pictures of

each factor for both data sets have been made in the same fashion.

Regionally transformed pictures are similar to the corresponding

pictures in the untransformed data though the percentaged eigenvalues

significantly different compared to the total (for example, factor

1 in the untransformed data is 30.5%, while the corresponding factor 3

in the transformed data is 13.5% of the total variance). However,

there are marked differences in details between the two corresponding

pictures. Since quantitative comparison is not possible and the

regional aspects of both pictures are similar to each other, further

description will be omitted except for one point.

The geochemical haloes defined in the transformed data show again

a more diffuse pattern than the untransformed data, and the transformed

9-mixed data set for geological mapping is marginally better than the

untransformed 9-mixed data set in this study. This may be partly

due to histogram normalization applied for all FA scores in colour 2

processing using the 'I S' colour additive viewer.

However, in case of the 8-geochemical data sets, geological

mapping with the transformed data set is much better than the untrans-

formed data.

7.4.2 Cluster Analysis

(a) Cluster analysis procedures

Cluster analysis has been widely used by Numerical Taxonomists

as a classification tecnique and is increasingly employed in the field

of geological sciences, especially for geochemical applications

(Rhodes 1969, Obial 1970, Obial and James 1973, Castillo-Munoz 1973,

229

Crisp 1974, and Mancey 1980, etc).

This is a statistical treatment of data that is able to elaborate

on elemental and sample associations. The purpose of cluster analysis is

to divide the set of m-variables into subsets or clusters, those

objects in the same cluster being close or similar in some sense, and those

in different clusters being distant or dissimilar.

Cormack (1971) has defined three different types of clustering

procedure as

1. Hierarchical classification - the classes are them-selves grouped in a repetitive process at different levels to form a dendrogram.

2. Partitioning - the classes are mutually exclusive, thus forming the subset of the original data.

3. Clumping - the classes may overlap, the classes are regarded as different types of class.

Clustering is considered to be -partitioning the data set

into 'natural' and 'homogeneous' groups and finding the 'most

representative elements'. The 'natural' concept of clustering is that

there should be parts of the space in which points are very dense,

separated by parts with low density (Carmichael, 1968). Two most

important aspects must be taken into consideration in order to satisfy

this condition, namely, the choice of similarity measure and the way

of grouping of samples.

Several measures of similarity have been described in the

literature. They fall into two main categories: those using numerical

data such as the correlation coefficient, distance coefficient, or

cosine data coefficient; and those using non-numerical data coded into

two- or multi-scale forms, such as matching and association coefficients.

Details of computational procedures for these measures are given by

Sokal and Sneath (1963), Harbaugh and Merriam (1968), Fu (1976), etc.

230

The most widely used coefficients are the distance coefficients.

The commonest measure in the distance coefficients is the Euclidean

distance d.. which is expressed as

where X., and X., denote the Kth variable measured an object i and j, lk jk

respectively.

As expected, a low distance indicates the two objects are .

similar, whereas a large distance indicates dissimilarity.

The Euclidean distance measure has the property of giving

extra weight to outlying values of a variable. This is partly

overcome by scaling (Cormack, 1971). Several other different forms

of distance measure: are summarized by Cormack (1971) in his excellent

review on clustering methods.

In this study, the data for the analysis have beeen transformed

in order to give each variable equal weighting by standardization

and absolute values larger than 3.291 reset to 3.291 in order to

avoid extra weight to outliers, so that the data lies within 99.9%

confidence level.

Similarity measures with properties like correlation coefficients

are often attempted. The mathematical formula to calculate the

correlation coefficient generally used for clustering is given by

Equation 6.1 in Chapter 6. Some arguments against using it are when

circular features occur, because it can give R.. < R., when variables i j lk

E. and E. are more similar than E. and E, (Eades, 1965). Ball (1965) l j i k

also describes that the covariance matrix suffers from being sensitive

to noise and to the distribution of the data set. Sokal and Sneath

(1963) note that in the large data sets used in computing the

(7.6)

231

correlation coefficients reduce the sensitivity to a directional

property.

As also noted by Marriott (1971) in his examination of the

classification method, the method minimizing the determinant of the

within-group dispersion matrix is scale-dependent, and it may be

rather sensitive to the inclusion of highly correlated variables, and

the existence of genuine multimodality can be masked by the inclusion

of irrelevant variables, especially if the modes are ellipsoidal rather

than spherical.

Most techniques for grouping have developed as algorithms

without formal basis. A formal approach would set up a criterion

to be optimized over the set of partitions of the data set.

Cormack (1971) has summarized three types of groupings of samples:

(i) agglomerative: successive groupings of the individual sample into groups

(ii) divisive: partitioning of the data set into smaller groups

(iii) clustering: finding partitions in the data set with properties approximating to some desired criterion.

Agglomerative methods have been widely used in geochemical

applications as well as in Numerical Taxonomy (Crisp 1974, Sneath

and Sokal (1973). The divisive method has not been much used because

this type of process suffers from inherent difficulties if early

misclassifications are used for later subdivision (Crisp, 1974).

Gower (1967) has suggested that considerable distortion may be

caused by the dendrograms presented in the grouping process.

In agglomerative methods, there are three kinds of this

technique: a weighted pair group average method (WPGM), an

unweighted pair group average method (UPGM), and a single linkage

232

method (SLM), are in common use as criteria of entry of the samples

into a cluster. Obial and James (1973) suggest UPGM is most suitable

for geochemical data. Details of comparison of the three methods

are described by Obial and James (1973), and full descriptions of these

methods may be found in Sneath and Sokal (1973).

Some restrictions in choosing the grouping technique are

imposed by the nature of the data and its scientific purpose in classi-

fying. Cormack (1971) has stated that the single linkage is the only

hierarchical clustering method which satisfies both the stability

requirement and certain other sample requirements about the way in which

a classification should represent the mutual dissimilarities of

objects. However, Jardine (Cormack, 1971) argues that it is nevertheless

a method of classification which may produce inhomogeneous clusters

what Cormack referred to as 'the chain effect'.

Several dozen different methods of classification have been

developed so far. Overall reviews are given by Cormack (Op. Cit.).

Comparative reviews on some different clustering techniques are

given by Dubes and Jain (1976) and Gower (1967). Practical problems

in cluster analysis are described by Marriott (1971). Many clustering

techniques have been introduced to solve the problems of pattern

analysis and pattern classification.

Sammon (1970) has introduced on-line interactive graphic systems.

The dynamic clustering method developed by Diday (1973) is a non-

hierarchical classification procedure which can help the geologists in

exploration work.

In this study, a clustering method called 'ISODATA' initially

developed by Ball and Hall (1966) was used for classification.

The version used for this study has been modified from

the original ISODATA algorithm by David Wolf of the Stanford Research

233

Institute, and adopted for use in the Imperial College Computer

Centre CDC 6400 Computer by Richard Howarth (Crisp, 19V4). A

simplified flowchart is given in Fig. 7.13.

The ISODATA method is one of the most famous of the square-error

clustering methods (dubes and Jain, 1976). This method allows the

comparison of individual samples with a series of k cluster group means,

which are tentatively computed based on the distance threshold, from the

overall mean, and thus the samples are allocated to the nearest

cluster centre, and the means are recomputed for the new clusters.

This process is iteratively applied until the means represent the clusters

with a specified error-of-fit. Refinement of the basic process

allows for clusters to be joined or split depending on local conditions

and user's thresholds.

The repartitioning of the samples about the means is based

on the maximum variance criterion such -that the sum of square distances

of the features from their cluster centres is minimized. Thus,

for a fixed number of clusters the program minimizes the total squared

error, E

2 K 2 E = z e, (7.7)

k=l k

where e. 2 = Z Z (X. .-C, . )T(X. .-C, .) (7.7.1) k icr kJ kJ j=i i£ck i=l,2,...,n

and C is the kth cluster centre where C = (C C ,...,C n and K K K I KZ Km

m are numbers of measurements and variables, respectively. The performance of the algorithm is highly dependent upon

a set of control parameters.

234

Fig.7.13 Flowchart for ISODATA (After Dubes and Jain,1976)

235

The initial clustering is controlled by a user supplied

threshold, the "spherical factor". A new cluster centre is formed if

the Euclidean distance of a sample from the overall mean is greater

than the spherical factor multiplied by the overall standard

deviation or root mean square average distance between clusters.

Iteration of the basic process allows small clusters to be

discarded, and clusters to be fused into larger clusters

(lumped) or broken down into small clusters (split) as required by

the user. For each iteration, the user control options are the

maximum number of clusters that can be lumped in iteration (NCLST), the

threshold distance for lumping or splitting (THETA C), the number of

discards (THETA N) and an option whether lumping or splitting is

performed in the iteration.

Thus the algorithm consists of a basic partitioning process

that is iteratively used with the initial, lumping or splitting

routines to describe the data.

Since the ISODATA computes the similarity by means of the

Euclidean distance of each sample to the cluster centres, the

algorithm does not suffer from the distortions inherent in other

multivariate techniques based on the computation of a similarity

measure of a covariance matrix. It is also capable of handling larger

data sets and yet allows the detail of each data point to be considered,

a feature which is lost when a similarity matrix is used for the

analysis. The maximum likelihood estimation of the means of a mixture

of Gaussian distributions makes the technique a very powerful tool

in the analysis of mixed populations (Crisp, 1974).

The basic output of ISODATA program is a listing of cluster

centres, a table of distances between them, the squared-error for each

cluster, a nd the membership of each cluster. Several other statistics

236

are also computed.

Further detailed description on the method can be found in

references by Ball and Hall (1966) and Crisp (1974).

The central processing time used for the analysis is linearly

proportional to the number of measurements, the number of variables

and the number of iterations. CPU time for 10 iterations with 5

partitions of data per iteration in the analysis of a 9-variable set,

each of which consists of 1024 measurement values, was typically 280 seconds

on Cyber 174.

(b) Testing the ISODATA program

Crisp (1974) has tested the program using a standard set of

well-known multivariate data which consists of 150 samples with four

variables containing typically 3 species of iris. One of the three

species (category 1) is distinct from the other two species, which

are moderately well sorted into clusters but are not linearly separable.

He has found from the test that the clustering is most sensitive to

changes in the spherical factor. He has also noted that the distinct

category 1 is perfectly identified both at low and high spherical

factors, and the threshold for forming 50 clusters is shown where

the error is minimized with the iris test data set. The remainder

of the controls had very little effect on the basic clusterings.

In this study, in order to understand how the user parameters

affect the clustering in practical application of large data sets

four actual data sets (untransformed and transformed 8-geochemical

elements and 9-mixed variables) were used to test the algorithm for

the two most important cases.

237

1. Test of ISODATA by varying the spherical factor with the rest

of the control parameters fixed.

2. Test of ISODATA by varying the Euclidean distance threshold

with the rest of the control parameters fixed.

In both cases, 10 to 12 iterations were made with a fixed number of

partitions per iteration (5) and the maximum number of "lumpings" per

iteration (9). The numbers of discarded samples were kept the same in

all cases as follows: In the first 3 iterations no discards were

applied, in the next 3 iterations 5 discards were used, and for the

rest of the iterations 10 discards were used in the analysis. By

gradually increasing the discards in the following iterations, we may

effectively discard small clusters while keeping the maximum number

of discards around 5% or less to the total. In this way, it may

be more efficient computationally in practical cases where a larger number

of measurement values are dealt with and it may be more effective

in the analysed result in practical cases if we ignore fugitive

clusters, which have insignificant proportion of the total measurements,

since we can usually expect that those discards are from either

abnormally high or low values. An example of the input controls for

the run of the program is shown in Table 7.6

Table 7.6 An example of the input parameters for ISODATA

80.10 5102412 554 CLUSTER ANALYSIS OF 8 GEOCHEM.VARI.(TRN) (11F10.4)

00000000000 9 3.0 0LUMP 9 3.0 0LUMP 9 3.0 0LUMP 9 3.0 5LUMP 9 3.0 5LUMP 9 3.0 5LUMP 9 3.010LUMP 9 3.010LUMP 9 3.010LUMP 9 3.010LUMP 9 3.010LUMP

238

The results of the first case where the spherical factor was varied,

are shown in Table 7.7(a) and (b), 7.8(a) and (b), and

7.9(a) to (d) for the untransformed and transformed 8-geochemical

elements, 9-mixed variable sets and 8-PCA score sets respectively.

The results of the second case where the Euclidean distance

threshold was varied for the untransformed 8-geochemical elements and

transformed 9-mixed variable sets are shown in Table 7.10(a) and

(b), respectively.

From the analysis, it was noted that the spherical factor is

almost the only one sensitive to the results of clustering if numbers of

pattern classes and discards are kept the same.

The criterion for the assessment of the results in this study

is the error percentage of the clustering with consideration of the

numbers of pattern classes and discards.

From Tables 7.7 to 7.9, the minimum error occurs where the

clustering constitutes around 40 pattern classes in this case, and it

is gradually increased with increasing the number of iterations. Thus,

partitioning the data around 40 cluster centres in this case may be

the very distinct natural clusters to be identified, based on the

criterion of error of fit. However, this may not be so meaningful

in regional assessment of geological applications since 40 classes in

this study still make complications in interpretation. The most useful

one for the more meaningful interpretation in the study, was found

to be when the number of clusters is about 8 to 12.

For both untransformed and transformed 8-geochemical elements,

the best optimum spherical factor is 0.4 and the next is 0.6, but

in case of 9-mixed variable set the best optimum spherical factor

is 0.6 followed by 0.8. This phenomenon is varied in case of 8-PCA

score sets. It appears that the optimum spherical factor may vary

Table 7.7 Test results of ISODATA of 8-geochemical data sets "by varying the spherical factor with the rest of the input parameters constant(thetac=3.)

a) Untransformed 5-geochcmlc.nl data.

srtuf. iteration

0.1 0 . 2 0.1 0 . 6 0.8 1.0

1 * * * 5 0 / 0 15.83

50/0 15.51

50/0 11.13

5 0 / 0 11.13

50/0 11.19

50/0 15.16

2 16/0 13.50

13/0 13.80

16/0 10.73

16/0 11.88

11/0 12.61

31/0 15.90

3 ' "3/0 12.55

10/0 12.90

11/0 11.16

12/0 13.58

12/0 12.51

26/0 18.06

1 38/0 12.11

31/10 13.03

35/8 12.08

21/50 11.15

18/61 11.59

18/12 22.17

5 31/10 12.9"

30/10 13.70

31/8 13.36

17/50 16.26

11/61 13.19

11/12 25.86

6 31/10 13.39

26/10 11.18

29/8 13.91

11/50 19.79

10/61 21.16

11/12 28.99

7 21/16 13.18

Zl/16 15.87

20/56 13.21

8/77 23.57

5/71 30.81

6/25 36.01

8 19/16 11.91

18/16 16.88

16/56 15.71

5/77 32.32

1/71 37.72

5/25 39.71

9 16/16 16.72

16/16 I 8 . 5 6

12/56 19.8"

1/77 37.99

3/71 12.17

1/25 H . 3 5

10 13/"6 19.99

13/16 2 0 . 7 0

10/56 22.33

3/77 12.13

2/71 52.29

1/25 H.32

11 10/16 21.37

10/16 26.81

8/56 26.03

2/77 51.25

1/71 72.06

1/25 11.32

12 8/16 27.61

8/16 30.66

6/56 36.38

1/77 70.79

1/71 72.06

1/25 I I . 3 2

b) Transformed 8-geochcmical data.

sph.f. iteration

0 . 1 0 . 2 0 . 1 0 . 6 0 . 8 1 . 0

+ * * * 50/0

2 2 . 6 1 50/0 22.53

5 0 / 0 15.77

5 0 / 0 13.72

1 1 / 0 13.79

2 1 / 0 2 0 . H

2 1 6 / 0 1 7 . 7 3

1 6 / 0 20.3?

1 2 / 0 1 5 . 1 0

1 2 / 0 1 1 . 5 2

33/0 15.11

17/0 25.31

n 1 1 / 0 1 6 . 5 1

1 0 / 0 1 8 . 1 1

37/0 15.93

37/0 15.19

31/0 1 8 . 1 6

13/0 29.51

A- 31.11 1 6 . 8 6

2 8 / 2 0 17.69

3 0 / 1 2 1 6 . 6 5

27/8 1 8 . 0 0

2 0 / 1 6 2 0 . 9 1

9/0 35.68

c J 2 8 / 1 1 17.37

23/20 1 8 . 9 6

2 5 / 1 2 1 8 . 1 0

2 2 / 8 19.93

15/16 21.19

6 / 0 13.83

6 2 1 / 1 1 2 0 . 2 6

1 9 / 2 0 2 0 . 5 2

2 2 / 1 2 19.56

17/8 2 3 . 1 2

1 2 / 1 6 28.05

5/0 5 1 . 1 8

7 16/23 23.17

11/20 2 1 . 2 3 •

15/19 23.79

1 2 / 8 27.31

7/22 37.21

1/0 5 5 . 1 0

8 12/23 2 7 . 0 2

1 0 / 2 0 3 0 . 2 9

11/19 28.97

9/8 32.36

5/22 13.27

3 / 0 7 3 . 2 2

c 8/23 35.83

7 / 2 0 37.81

7/19 38.89

6 / 8 1 3 . 0 2

1 / 2 2 5 0 . 6 1

2 / 0 7 6 . 2 2

10 6/23 41.36

6 / 2 0 1 1 . 8 6

5/19 19.OI

3/8 50.31

3/22 6 9 . 2 2

2 / 0 7 6 . 2 1

11 5/23 4 8 . 5 8

5/2C 1 8 . 1 2

4/19 53.60

1 / 8 51.67

3 / 2 2 6 9 . 2 2

2 / 0 7 6 . 2 1

12 1/23 5 l . 1 i

1 / 2 0 52.03

3/19 7 0 . 6 6

3/8 72.33

3/22 6 9 . 2 2

2 / 0 7 6 . 2 1

ho to VO

* number of patterns/number of discards

* * error percentage

Table 7.8 Test results of ISODATA of 9-mixed variable sets by varying the spherical factor with the rest of input parameters constant (thetac=3.)•

a) Untransformed 9-mixed variable sot. b) Transformed 9-mixed variable set.

sph.f.

iteration 0.1 0.2 0.4 0.6 0.8 1.0 sph.f.

iteration 0.1 0.2 0.4 0.6 0.8 1.0

1 * * *

50/0 23.96

50/0 21.19

50/0 1 6 . 7 0

•50/0 14.34

50/0 13.07

33/0 17.47

1 * * * 50/0 23.94

50/0 25.27

50/0 20.60

5 0 / 0 16.14

50/0 14.80

31/0 19.40

n / 46/0 20.48

45/0 18.58

4 5 / 0 1 6 . 0 6

45/0 13.97

46/0 13.55

28/0 19.85

46/0 21.64

48/0 22.89

45/O 18.72

4 5 / 0 16,04

43/O 15.73

26/0 21.04

3 41/0 19.56

<K)/0 13.12

'K)/0 15.57

41/0 14 .79

40/0 15.75

25/0 2 1 . 9 8

3 41/0 20.90

45/0 22.01

40/0 18.46

40/0 16.87

39/0 16.51

20/0 24.16

4 27/25 19.70

31/20 18.15

29/23 15.93

2e/29 15.74

25/21 18.15

16/12 23.18

4 25/36 20.79

27/35 20.67

26/33 19.01

32/19 17.27

30/14 17.70

15/7 26.91

5 ?3/25 2 0 . 2 3

29/20 18.31

25/23 17.50

24/29 17.73

22/21 19.48

14/12 25.30 _ .

5 22/36 21.49

23/35 2 0 . e 6

23/33 19.79

26/19 18.46

26/14 19.16

12/7 .30.44

6 2 0 / 2 5 2 0 . 9 6

26/20 18.75

20/23 19.44

19/29 20.78

16/21 2 3 . 1 8

11/12 30.98

6 18/36 23.22

21/35 2 1 . 2 9

19/33 2 2 . 0 9

22/19 21.12

21/14 22.04

8/7 36.78

7 15/y* 22.65

21/35 19.72

14/51 2 1 . 5 2

15/48 21.20

11/29 27.44

9/20 33.25

7 13/46 25.35

1 6 / 4 4 2 3 . 2 3

16/33 24.08

14/46 24.22

17/24 23.27

6/7 46.56

8 1 2 / 3 4 2 5 . 9 5

18/35 2 0 . 8 9

1 2 / 5 1 24.68

13/48 23.57

9/29 32.74

3/20 42.66

8 10/46 30,22

1 2 / 4 4 27.46

12/33 28.89

10/<*6 28.23

14/24 26.91

6/7 46.55

9 1 0 / 3 4 31.64

15/35 2 2 . 9 4

9/51 31.22

10/48 20.17

8/29 42.26

6/20 45.71

9 6/46 33.97

9/44 32.35

9/33 33.33

8/46 32.99

11/24 28.87

6/7 46.55

10 9 / 3 4 34.18 13/35 25.05

8/51 40.47

9/48 31.44

6/29 51.50

5/20 48.85

10 6/46 43.67

7/44 40.24

8/33 37.36

7/46 38.19

8/24 34.e2

6/7 46.55

11 8/34 43.14

9/35 32.86

7/51 43.41

8/43 40.97

5/29 48.20

4/20 51.57

11 5/46 48.80

5/44 49.21

7/33 40.20

- 6/24 44.44

6/7 46.55

12 7/34 41.46

8/35 35.61

6/51 39.49

7/48 •43.91

4/29 59.72

3/20 59.88

12 5/46 46.04

5/44 46.31

6/33 44.76

- 5/24 48.89

6/7 46.55

* number of patterns/number of discards

* * error percentage

Table 7.9 Test results of ISODATA of 8-PGA score sets by varying the spherical factor with keeping the rest of input parameters constant*

a) Untransformed 8-PCA score set(thetac-3.;

sph.f.

iteration 0.2 0.4 0.6 0.8

1 * 50/0 50/0 50/0 50/0 * * 20.18 18.37 16.46 15.95

2 45/0 4 3 / 0 44/0 't4/0 18.32 17.12 16.84 16.70

3 41/0 40/0 39/0 39/0 17.58 17.30 18.02 17.60

4 32/24 31/19 28/14 28/22 17.75 17.89 19.42 19.53

5 27/24 28/19 24/14 23/22 19.21 18.58 21.24 22.86

6 23/24 24/19 18/14 20/22 21.33 20.67 25.99 24.85

7 15/44 l9/?8 16/14 12/59 24.74 24.17 27.27 30.76

e 13/44 16/28 14/14 9/59 26.86 2 6 . 5 6 30.82 38.74

9 10/44 14/28 11/14 8/59 36.86 28.39 37.90 46.37

10 8/44 12/28 9/14 7/59 45.17 36.73 48.84 51.70

11 7/44 9/28 8/14 6/59 49-55 43-57 53.42 59.17

12 6/44 8/28 7/14 5/59 54.58 50.38 57.87 6 8 . 1 3

b) Transformed 8-PCA score 3et(thetac-3.) sph.f.

iteration 0.2 0.4 0.6 0.8

1 * * * 49/0 24.33

49/0 25.15

50/0 20.85

50/0 18.15

2 45/0 2 2 . 2 3

46/0 23.20

4 7 / O 20.21

44/0 19.10

3 41/0 22.15

4 3 / O 22.06

4 3 / O 20.84

38/0 20.46

4 31/18 2 2 . 6 7

31/23 2 2 . 2 7

35/13 21.53

28/14 23.51

5 27/18 23.57

27/23 23.77

31/13 22.61

24/14 26.04

6 24/18 24.87

22/23 26.91

26/13 24.96

18/14 31.97

7 19/28 28.06

16/31" 30.00

19/21 28.77

12/23 38.89

8 16/28 31.71

13/31 34.08

15/21 33.29

10/23 44.27

9 12/28 40.52

10/31 41.89

12/21 39-57

8/23 55.89

10 10/28 43.00

8/31 49.41

8/21 54.92

6 / 2 3 6 3 . 2 8

11 7/28 57.31

6/31 58.97

7/21 57.48

5/23 70.02

12 6/28 60.23

5/31 67.41

6/21 59.13

4 / 2 3 73.49

* * *

number of patterns/number of discards error percentage

Table 7.9 continued.

c) Untransformed 8-PCA score set(thetac-5«) sph.f.

iteration 0.2 0.4 0.6 0.8

1 * 5 0 / 0 50/0 50/0 50/0 * * 20.18 18.37 16.46 15.95

2 4 5 / 0 43/0 44/0 44/0 18.30 17.12 16.84 16.70

3 41/0 40/0 39/0 39/0 17.58 17.30 18.02 17.60

4 32/24 31/19 28/14 28/22 17.75 17.89 19.42 19.53

5 27/24 28/19 24/14 23/22 19.21 18.58 21.24 22.86

6 23/24 24/19 18/14 20/22 21.33 20.67 25.99 24.85

7 15/44 19/28 16/14 12/59 24.74 24.17 27.27 30.76

6 13/44 16/28 14/14 9/59 26.86 2 6 . 5 6 30.82 38.74

9 10/44 14/28 11/14 6/59 36.86 28.39 37.90 53.06

10 8/44 12/28 8/14 4/59 45.17 36.73 49.86 64.44

11 6/44 8/28 6/14 3/59 55.69 51.55 60.74 72.54

12 3/44 5/28 3/14 2/59 74.09 66.41 79.15 79.70

* number of patterns/number of discards

* * error percentage

d) Transformed 8-PCA score set (thetac-5.)

sph.f.

iteration 0.2 0.4 0.6 0.8

1 * * * 49/0 24.33

49/0 25.15

50/0 20.85

50/0 18.15

2 4 5 / 0 2 2 . 2 3

46/0 23.20

47/0 20.21

44/0 19.10

3 41/0 22.15

43/0 22.06

43/0 20.84

38/0 20.46-

4 31/18 22.67

31/23 22.27

35/13 21.53

28/14 23.51

5 27/18 23.15

27/23 23.77

31/13 22.61

24/14 26.04

6 24/18 24.87

22/23 26.91

26/13 24.96

18/14 31.97

7 19/28 28.06

16/31 30.00

19/21 28.77

12/23 38.89

8 16/28 31.71

13/31 3 4 . O 8

15/21 33.29

10/23 44.27

9 12/28 40.52

10/31 41.89

12/21 39.57

8/23 55.89

10 10/28 43.OO

8/31 49.41

8/21 54.92

5/23 64.67

11 7/28 57.31

6/31 58.97

5/21 65.07

3/23 78.22

12 4/28 70.34

3/31 76.48

3/21 78.57

2/23 86.41

Table 7.10 Test results of ISODATA by varying the criterion(thetac) with the rest of the constant.

Euclidean distance input parameters

a) Untransformed 8-geochemical data (spherical factor=0.4)

thetac 0.1 0.5 1.0 3.0 5.0 8.0 iteration

0.5 5.0

1 * 5 0 / 0 5 0 / 0 5 0 / 0 50/0 50/0 50/0 * * 11.13 11.13 11.13 11.13 11.13 11.13

2 , 5 0 / 0 5 0 / 0 46/0 46/0 46/0 46/0 10.'47 10.47 10.73 10.73 10.73 10.73

3 5 0 / 0 5 0 / 0 41/0 41/0 41/0 41/0 10.39 10.39 11.46 11.46 11.46 11.46

4 48/8 48/8 35/8 35/8 35/8 35/8 10.29 10.29 12.08 12.08 12.08 12.08

5 48/8 48/8 33/8 31/8 31/8 31/8 10.29 10.29 12.81 13.36 13.36 13.36

6 48/8 48/8 32/8 29/8 29/8 29/8 10.29 10.29 12.88 13.94 13.94 13.94

7 37/99 37/99 25/56 20/56 20/56 20/56 8.10 8.10 11.66 13.24 13.24 13.24

8 37/99 37/99 2 5/56 16/56 1 6 / 5 6 16/56 8.10 8.10 11.58 15.71 15.71 15.71

9 37/99 37/99 25/56 12/56 12/56 12/56 8.10 8.10 11.54 19.84 19.84 19.84

10 - - 24/56 10/56 10/56 10/56 1 1 . 6 9 22.83 22.83 22.83

11 _ - 2 3 / 5 6 8 / 5 6 8/56 8/56 11.93 26.08 26.08 26.08

12 - - 23/56 6/56 5/56 5/56 11.91 36.38 40.01 40.01

b) Transformed 9-mixed variable set (spherical factor-0.6) thetac

Iteration 0.5 1 . 0 3.0 5.0 8.0 11.0 12.0

1 * * * 50/0 50/0 16.14 16.14

5 0 / 0 16.14

50/0 16.14

50/0 16.14

50/0 16.14

5O/O 16.14

2 50/0 15.41

4 9 / 0 15.48

45/0 16.04

45/0 16.04

45/0 16.04

W o 16.04

45/O 16.04

3 50/0 15.19

49/0 15.26

W o 16.87 w°

16.87

40/0 16.87 w°

16.87

40/0 16.87

u 44/21 14.89

4 3 / 2 2 14.94

32/19 17.27

3 2 / 1 9 17.27

32/19 17.27

32/19 17.27

32/19 17.27

5 44/21 14.89

43/22 14.94

23/19 19.46

28/19 18.46

28/19 18.46

28/19 18.46

28/19 18.46

6 44/21 14.89

43/22 14.94

22/19 21.12

22/19 21.12

22/19 21.12

22/19 21.12

22/19 21.12

7 33/108 13.81

32/109 13.85

l'i/46 24.22

14/46 24.32

14/46 24.32

14/46 24.32

14/46 24.32

8 33/103 13.81

3 2 / 1 0 9 13.85

10/46 22.23

10/46 28.23

10/46 28.23

10/46 28.23

10/46 28.23

9 33/108 13.81

32/109 13.35

3/46 32.99

8/46 32.99

8/46 32.99

8/46 32.99

8/46 32.99

10 33/103 13.61

3 2 / 1 0 9 13.85

7/46 - 33.19

5/46 44.70

5/46 44.70 44.70

5/46 44.70 ho -O U>

* number of patterns/ number of discards * * error percentage

244

vto determine-the oft'-«num sf her? cal faetor farevery M a from data set to data setj so that it might be necessary/whenever a

different variable set is applied for the analysis, but it is noted in

this study that both transformed cases remain the same as the

untransformed cases.

The Euclidean distance threshold may not affect the clustering

results provided the number of pattern classes is kept the same and

the number of discards remains same. However, the merging process

is considerably slower if the distance criterion is small.

In some cases of low Euclidean distance threshold, it may

never converge to form a small number of clusters, which may

be meaningful for interpretation. This has to be avoided in practical

computations. This phenomenon is shown in Table 7.9 where & equals

to 0.1, 0.5 and 1.0.

Otherwise, above a certain value (in this study ^ = 3), the

reuslts are virtually identical as far 'as the numbers of pattern

classes and discards remain the same.

It is also significant to note that the performance of

clustering is remarkably stable though sometimes there may be a small

improvement by iteration as indicated by vertical dark lines for the

same number of clusters with the same number of discards in the Tables.

Thus, the most efficient and effective procedures of the application

of the algorithm drawn from the analysis may be as follows.

1. Determine the optimum number of partition per

iteration (3-5) and the maximum number of clusters per iteration (in

this study it was 9).

2. Determine the discards in each iteration; the total number

of discards may be kept around 5% of the total measurements or less.

This can be done as follows. In the first few iterations, no

discards may be applied and the number of discards is increased

245

gradually in the following iterations.

3. Find the Euclidean distance threshold large enough to

form meaningful clusters by a computationally reasonable number of

iterations (in case of this study $ = 3 or larger).

4. Find the optimum spherical factor by varying the value from

0.1 to about 1.0 with the rest of the user's parameters fixed.

Thorough examination of all parameters was impracticable with the

test data sets used in the study because of the computation cost

and time. This is also partly because those parameters (the number '

of partitions per iteration and the maximum number of lumpings per

iteration) only make effects marginal to the final results, as also

noted by Crisp (1974). However, further analysis for varying degrees

of variable sets may be useful and thus provide means of efficient and

effective use of the sophisticated ISODATA program.

(c) Applications of cluster analysis

In this study the clusters designation by letters are different

for all the untransformed and transformed data sets.

(c.l) 8-geochemical element sets

The results of both untransformed and transformed 8-geochemical

element sets are shown in Fig. 7.14(a) and (b), respectively.

Though Crisp (1974) has noted that the untransformed data shows

much faster convergence to a smaller number of clusters and allows

high rate of discarding than the transformed data, this is not unique

in the case of this study. From Tables 7.7(a) and (b) to 7.9(a) and

(d) for the test results of ISODATA obtained by varying the spherical

factor and keeping the rest of the input control options constant, it

appears that the convergence rates of the untransformed data are

246

not always faster than the transformed data, and even the number of

discards of the untransformed data are not uniquely high in the case

of untransformed data with the same initial condition. Thus,

apart from the initial condition of the program run, not only are

their convergence rates but also the discards may be dependent upon

the .d Ts puersior\ patterns-: vn<thfo-'ettch var? able between variables

so that no uniqueness may be drawn for general cases.

The clusters produced show a marked consistency to changes in

spherical factors, as was also noted by Crisp (1974).

The means and standard deviations of 12 classes produced by

the run with iteration 9 for the untransformed data set is shown in

Table 7.11(a) and those of 10 clusters with the same iteration for the

transformed data set is shown in Table 7.11(b).

The clustering results may not be so meaningful with relation

to known geology as in the FA. This may be expected because (1) the

stream sediment sample values themselves are the resultant of regional

redistribution of upstream geology, and (2) the sampling process also

introduces smoothing effects with adjacent values,

and (3) the noise filtering process will merge the data to further distort

the geochemical composition on the spot.

However, bearing in mind the secondary environments together

with those factors mentioned above, they are similar to results obtained

from other methods, and thus indicate a consistency in the results.

Some regional characteristics are evident in relation to the

known geology or geochemical haloes in both untransformed and trans-

formed data sets, which might be enough to justify the use of the

method in classification.

The Bodmin Moor granitic area is well defined except

the northern part of the granite body but there are some confusions

247

(a) untransformed data

— l i t h o l o g i c a l boundary

Figure 7.14 Results of clustering by ISODATA for 8-geochemical element sets: As, Cu, Ga, Li, Ni, Pb, Sn and Zn

Table 7.11 Means and standard deviations of clustering for the 8-geochemical data sets.

a) Untransformed data (iteration 9,12 patterns,discards 56) b) Transformed data (iteration 9, 10 patterns, discards 3*0 variable

cluster no As Cu Ca Li Ni Pb Sn Zn no of

samples variable

cluster no. As Cu Ca Li Ni Pb Sn Zn no. of

samples A * * * -.46

. 2 1 -.31

. 1 2 -•94

.39 -.49

.14 0.01

.45 -.31

.28 -.48

.26 -.34

.31 246 A * * *

0.11 .61

0.06 .55

0.69 .43

0.99 .52

-1.30 .48

-.39 .60

1.10 .42

-.79 .46

64

3 - . 2 1 .34

-.28 .13

0.10 .33 - . 2 2

.24 O .63

.51 -.10

.60 -.44

.34 -.14

.3" 355 B 0 . 2 0

.49 0.41

.41 0.21

.43 -.22

.38 0.67

.47 1.30

.50 -.61

.44 0.59

.72 130

C O .23 .36

0.90 .72

-.83 .44

-M .12

0.26 .48

0.85 .44

-.33 .23

0.64 .35

35 C -1.10 .84

-.39 .44

-.65 .61

-1.00 . 5 4

0.34 .44

-.96 .39

-1.30 .50

-.53 .37

123

0 -.03 .20

-.25 .11

0.Z7 .16

-.13 .10

O .92 .44

0.55 .53

-.42 .27

2.10 .70

28 D -.06 .69

-.39 .28

0.30 .39

0 . 2 5 . 5 2

0.52 .61

-.45 .45

-.39 .71

-.12 -.38

201

2 -.50 .05

-.30 .08

1.50 .92

1.50 .72

-1.40 .73

-.53 .03

2.10 .40

2.90 .65

13 E -1.40 .56

-.38 .44

2.80 .51

2.20 .29

-2.00 .40

-2.00 .75

1.10 .51

-2.00 .88

22

F -.28 .30

-.36 •23

0.68 .47

0.70 . 6 0

-1.60 .44

-.39 .19

-.16 .47

-.84 .13

130 F 1.30 .53

1.10 .85

0.37 .40

0.44 .40

0.16 .73

0.34 .55

1.10 .40

0.67 .44

132

G 0.39 . 6 9

0.32 •55

0.02 .36

-.26 . 2 1

0.14 .34

2.70 .52

-.47 . 1 8

0.73 .40

19 G 1.50 .53

1.70 •3.6

-.58 .80

-.53 .70

0.61 . 6 9

1.70 .61

0.67 .43

1.50 .64

65

H 0.10 .53

-.02 .45

-.02 .59

-.05 • 35

-.46 .72

-.39 .20

2.20 .71

-.07 .57

72 H -.51 .61

-.17 .58

-1.30 .61

-.82 .47

0 . 0 3 .54

0.09 .55

0.19 .65

0.08 .39

147

I -.57 ..07

-.33 .03

3.00 .53

3.00 .4 7

-1.80 .39

-.16 .03

1.50 1.10

-.94 .26

24 I -.64 .31

-.08 .52

1.50 .73

1.70 .34

-.61 1.50

-.91 .28

1.50 .08

2 . 1 0 . 6 9

19

•J 1.20 .53

1.60 .97

0.12 .29

-.05 .13

0.34 .57

0.17 .37

0.74 .54

0.46 .40

21 J -.32

.67 -1.60

.79 0.86

.44 1.30

.46 -1.80

• 35 -.39

.53 0.19

.'•5 -1.50

.40 87

X 3.30 .07

2.00 .77

-.80 .30

-.46 . 1 1

0.71 .41

2.00 .54

0.64 .74

3.00 .32

12

. 1 3.20 .21

3.20 .16

0.39 .20

0.15 .09

-.70 .55

0.1b .20

0 . 9 3 .26

0.26 .24

" l3

ho -f> 00

* Arithnatic moans in uni ts of standard deviation from the sample means. * * Standard deviation

249

in the St. Austell granite. This may be due to effects of heavy

secondary enrichment which occurred throughout the area either by

drainage or old mining activities as mentioned previously.

As in the FA, the anomalous bulge to the south of the Bodmin Moor

granite and the indentation to the north of the Bodmin Moor granite

are evident in the clustering.

The untransformed data in Fig. 7.14(a) shows a broad

distinction between lithologic units. The general features of the

clustering are that Carboniferous and Lower Devonian rocks are

assigned as the same cluster A, Upper and Middle Devonian rocks are

clustered into a group assigned to as cluster B. Bodmin Moor granite

is largely well defined as a cluster (F) but the St. Austell

granite area is defined as clusters E, F, I and H.

Clusters D, E, G, J, K and L largely represent areas of high

geochemical haloes of metallic elements. Cluster D is marked by high Zn,

cluster E is characterized by markedly high Sn and Zn, and cluster G

represents areas of high Pb and moderately high Zn. Cluster K shows a

prominent high in most metallic sulphide elements including As, Cu, Pb

and Zn, which in this study area might be most closely associated

with areas of high metallic concentration near the Kithill granite.

Cluster L shows exceptionally high values only with As and Cu and

a moderate high Sn, and cluster J is characterized by moderately high

values in As and Cu. Cluster H represents areas of high Sn content,

and cluster C shows a moderately high value in most sulphide elements,

which may be due to contamination by drainage, etc. Discards are

mostly related to anomalous areas of either very high or very low values

in the geochemical elements.

In the transformed data, the lithologic units are generally better

defined with the same number of classes compared to the untransformed

250

data. The Carboniferous rocks (cluster C) have been differentiated

from the Lower Devonian rocks (cluster H) although some mis-

classifications still arise probably due to similar geological environments

between them. Marked lows in Ga, Li and Sn contents are characteristics

of the cluster C and H, and most of the metallic elements are prominently

low in these clusters as well.

The Bodmin Moor granite area is also differentiated into two

distinct groups : Cluster J of low Sn content and cluster A of high

Sn content. In the St. Austell granite area more localization of

pattern classes have been produced. Clusters A, E, and I are characterized

by high Ga and Li but are low in most metallic elements except cluster

I with dominant high Zn. Cluster F shows marked high in As and Cu,

and to a lesser extent, in Pb and Zn, so that this cluster is the most

representative of high sulphide elements in the St. Austell granite

in this study area.

Upper and Middle Devonian rocks are still smeared into a group

on the map but there occur two distinct groups: Cluster B is moderately

high in most sulphide elements but cluster D is generally much lower

in these components than the statistical means in the study area

(See Table 4.2) except nickel. Thus, cluster B is more likely associated

with some sort of mineralization in these rock units than cluster D.

Various differential clusters of geochemical haloes in

the untransformed data are collectively classified as cluster F and G of

much broader features in the transformed data than the untransformed

data. Cluster G shows marked highs in As, Cu, Pb, and

Zn while cluster F shows marked high in As and Cu but moderate high in

Pb and Zn contents. Discards in the transformed data are only related

to the very low contents of geochemical elements.

From the characteristic features of the maps, the regional

251

geology may be better defined and delineated by the transformed data,

while areas of geochemical haloes of metallic elements are characterized

by more diffuse patterns in the transformed than the untransformed

data.

(c.2) 9-mixed variable sets

The results of clustering from both untransformed and transformed

9-mixed variable sets are shown in Fig. 7.15(a) and (b), respectively.

The statistical means and standard deviations' of the clusters

for the untransformed and transformed data are also illustrated

in Table 7.12(a) and (b), respectively.

Generally the results show, regional geological features,

as well as mineralization patterns in both cases, although outlining

pattern classes are different from those of 8-geochemical element

sets in Fig. 7.14(a) and (b). Again, in the analysis the transformed data

is better in defining the regional geology but it shows a more diffuse

pattern in delineating areas of mineralized zones compared to the

untransformed data.

The Bodmin Moor granite is outlined by clusters F and H in

the untransformed map and clusters F and J in the transformed map.

Again, the anomalous bulge to the south of the main granite is evident

in both data sets. Depression in the northern part of the granite has

been differentiated from the granite in the transformed data but not

into the same cluster as the surrounding sediment as in the 8-geochemical

element sets. Of particular interest is the cluster F in the untrans-

formed data and J in the transformed data which largely correspond to

the Bodmin Moor granite. These characteristics might be due to the

dominant features of the MSS bands and gravity data over the granite area.

The St. Austell granite area is also clustered as different from the

252

lithological boundary

Figure 7.15 Results of clustering by ISODATA for 9-mixed variable sets: Gravity, Magnetic, B5, R5/4, R7/6, Cu, Ni, Sn and Zn

Table 7.12 Means and standard deviations of clustering for the 9-mixed variable sets.

aj Untransformed data ( iteration 9, 10 patterns, discards 18 ) b) Transformed data ( iteration 6j 10 patterns« discards 16 )

variable Cravity Magnetic B5 R5/1 R7/6 Cu Ni Sn Zn no. of

variable

cluster no« Gravity Magnetic »5 R5/1 R7/6 Cvl Hi En Zn no. of

samples cluster no. samples

variable

cluster no« no. of samples

A * * * 0 . 1 3

.69 -.52

.52 0 . 0 2

.37 0 . 2 8

.58 0 . 0 0

. 6 3 -.27

. 2 0 0 . 1 2

.59 -.31

. 1 1 -.05

.31 330

A * * * -.23

O ? - 0 2

'65 ".it

0 0 0 . 2 3

<59 0.21

<51 -.15

<36 0 . 6 2

<55 0.2S

.60 0 . 1 6

.67 197

-2.60 -.62 -.16 B 1.10 1.50 - 2 . 9 0 -2.60 2 . 6 0 0.71 0.73 -.97 0 . 1 1 22

3 1 . 2 0 1 . 5 0 -2.60 - 3 . 0 0 1 . 9 0 -.09 0.73 -.62 -.16 22 . 0 2 .21 . 0 7 .25 . 6 0 . 0 0 2 .001 .001 .001 22

.02 . 2 8 .05 . 2 1 .38 .001 .001 . 0 0 0 .001 0 . 9 3

.61

.05 .38 C 0 . 9 3

.61 1.10 -.26 -.51 0 . 3 2 -.19 0.10 -1.20 - . 3 1 207

C 0.97 1.60 -.32 -.13 0.39 -.31 O . 3 8 -.62 -.39 201 0 . 9 3

.61 .19 .27 .51 . 5 5 .10 • 51 .18 . 5 1 207

.51 .59 .25 .51 . 1 6 .08 .51 . 0 2 .33 D 0.82 -1.20 -.06

. 5 1 .51 .59 .25 .51 .51 .33 D 0.82 -1.20 - 2 . 3 0 -.06 1.80 0.21 -.18 -.16 0 . 1 7 130

D 1.10 -1.10 -1.90 - . 0 1 1 . 1 0 0 . 0 3 - . 2 1 -.55 -.19 36 • 39 .17 .05 .55 .85 .87 •51 .70 . 7 1 130

.16 .18 .83 .53 .83 .58 .37 .07 • 35 E 1.20 .17

-1.20 .16

- 2 . 3 0 - . 0 7 1.70 O . 2 3 -.17 -.17 0 . 1 6 31 E 0.26 -.17 0 . 0 5 0 . 0 0 - . 0 2 0.35 0.75 -.23 1.60 33

1.20 .17

-1.20 .16 .86 .51 .06 .86 .52 .69 . 7 2

31

.60 .62 .32 .11 .56 .76 .83 .15 .77 F -1.20 .18

-.35 -.05 0.06 0.21 -.25 -1.30 0.93 - . 9 1 97 F -1.20 0.01 1.60 1 . 1 0 -1.10 - . 1 2 -1 . 3 0 -.13 -.81 88

-1.20 .18 .31 ..31 . 6 0 • 53 •70 .56 .53 .61

97

.36 .33 .81 . 6 3 . 6 1 .05 .79 .20 . 2 1 C -1.60 -.55 -1.80 -2.10 -.39 -1.80 1.20 -1.80 22 C -1.10 -.71 0.33 - . 9 3 -.29 -.29 - 1 . 1 0 2.20 2.90 12 .25 .10 .80 • 51 .65 .19 .11 .17 .82

22

M .06 .59 . 5 6 .37 .08 .70 • 37 .65 H -.31 O . I 9 1.30 1.80 - 1 . 0 0 - . 1 6 O . 1 7 -.73 -.23 • 51

1 1 r H -1.10 -.15 -.06 0.16 0.32 -.16 -.93 1.30 -.39 111

• 51 . 1 0 .03 .78 . 5 1 .38 . 1 1 • 56 -.23

• 51 1 1

M .21 .32 . 5 6 . 1 8 .31 .83 1.20 .53 I -.13 -.66 - . 0 3 0.16 0 . 0 9 1.50 0.21 1 . 0 0 1.10 162. 2.10 -7.70 -1.70 2.20 -.85 13

•73 .37 .36 •59 . 5 5 .65 .71 . 1 1 .67 162.

I -1.50 -.59 2.10 - 1 . 9 0 -7.70 -.35 -1.70 2.20 -.85 13 •59 . 5 5 .65 .71 .67

.18 .07 .80 .62 .72 .06 . 1 0 .50 .18 J - 1 . 1 0 0.002 1.30 1.30 -1.10 -1.70 -1.80 0.12 -1.50 . 1 6

66

-.63 .23

2.80 .65

17 .26 .21 .71 .71 . 6 3 .88 .39 . 1 2

-1.50 . 1 6 ro

J -.71 .52

-.63 .23

- . 1 1 .32

0.18 .17

0.25 . 1 1

2.80 .65

0.12 .37

1.20 .89

0.97 1.10

17 . 6 3 .39 . 1 2

-1.50 . 1 6 on

u>

* Arithmatic means in units of standard deviation from the sample meana. * * standard deviations.

254

Bodmin Moor granite, showing different mean intensity of variables,

as in the 8-geochemical element sets. In both granite areas, tin

rich zones are differentiated from the other granite areas, as a group

H in the untransformed data and group F and I in the transformed

data.

Upper Devonian and Carboniferous rocks are generally grouped

into a cluster C except in the area east of the main granite in both

untransformed and transformed data sets. The reason for this exception

is not clear in the east of the main granite area unless it is due to

the solifluxion process or secondary effect mainly due to basic

rocks distributed in this area.

The Middle and Lower Devonian rocks are assigned as a cluster

A in the untransformed data but they are largely classified as

different clusters in the transformed data.

A seemingly odd cluster boundary between the Middle and Lower

Devonian, and a large cluster C associated with Carboniferous and

a large part of the Upper Devonian sediments in the north, might be due

to the dominant physical properties of the geophysical data which may

reflect the subsurface features which are irrelevant to the surface

geologic distinction, e.g. a dense or magnetic rock at depth.

Areas of main geochemical haloes have been grouped as two

different clusters, J and E, in the untransformed data but these are

•largely defined as a cluster I in the transformed data. In the untransformed

data, group J represents areas of main sulphide mineralized zones while

group E is largely coincident with areas of Pb-Zn anomalies.

The regional outlining of offshore areas is evident in both

untransformed and transformed data in the northwest and southeast corners

of the map.

Clustering of 9-mixed data sets seem to be much more stable than

255

those of 8-geochemical data sets though some odd outlining of geological

boundaries are recognized. This account will be further discussed

in the training set analysis in the following section.

Again in this analysis, the transformed data may be better

in defining the regional geological features while anomalous areas

may be more localized by the untransformed data.

(c.3) 8-PCA score sets

The multivariate data sets of 8-PCA scores derived from

16 variables have been further applied to analyses by the clustering

method and EDF (Empirical Discriminant Function). The proportion

of information for each component is already shown in Fig. 7.3

in Section 7.3. The eigenvector matrices of 8-principal components

for the untransformed and transformed data are illustrated in Table

7.13(a) and (b), respectively.

The analysed results of clustering for the untransformed and

transformed data sets are shown in Fig. 7.16(a) and (b), and

statistical results of clusterings are given in Table 7.14(a) and (b),

respectively.

The regional features of clustering are largely comparable to the

9-mixed variable sets in Fig. 7.15(a) and (b). However, detailed definition

of lithologic classification might be much better in the 8-PCA

data sets.

The Bodmin Moor granite is much better outlined in the 8-PCA

data sets than any other multivariate data sets. Lower Devonian

sediments may also be better defined in the 8TPCA data sets than any

other variable sets.

Between untransformed data sets from 9 -mixed and 8-PCA data the

8-PCA score data set is also better in defining anomalous areas but this

256

Table 7.13 Eigenvectors of the first 8 principal components of PCA with 16 variables; Gravity, Magnetic, B4, B5,B7,R5/4,R6/5,R7/6, As,Cu, Ga, Li, Ni, Pb, Sn and Zn.

(a) Untransformed data. l c. 3 6 5 6 7 fi

1 . 1 4 6 1 . 2 0 3 0 . 0 9 3 7 . 2 9 7 6 . 0 1 7 2 - . 0 6 1 1 - . 2 3 8 3 - . 0 6 7 5 ; . 1 1 1 1 . 1 2 0 0 . 1 2 0 1 . 5 2 5 1 - . 3 7 8 0 . 2 7 6 6 - . 2 6 7 6 1 - . < . 0 > 3 . 1 6 6 0 . 3 2 3 5 . 0 3 9 6 . 0 1 6 6 - . 1 9 5 6 - . 0 2 1 0 <i - . < > 1 2 0 . o m . 1 9 9 6 . 2 6 5 5 - . 1 0 3 6 - . 0 1 0 6 - . 0 3 6 7 - . 0 6 6 0 •> - . 1 4 7 0 . 5 5 6 7 - . 0 7 7 6 . 2 1 3 6 . 0 2 8 8 - . 0 5 0 7 . 1 7 7 5 r- - • 1 4 7 7 - . 1 2 0 2 . 2 5 2 0 - . 1 6 6 1 - . 5 5 5 6 - . 0 7 3 1 . 5 5 6 8 - . 1 5 5 9 7 - . n ? T , - . 1 0 6 ( 1 . 5 6 6 8 - . 1 5 7 0 . 2 8 2 5 . 0 5 3 6 - . 0 7 1 0 . 1 r? 2 4

n . 3 l ' . o - . 0 1 0 : 1 . 0 1 0 2 - . 6 9 2 3 . 1 6 0 2 . 0 1 2 3 . 1 6 9 1 . 2 5 7 2

c . 0 0 3 ' . - . 0 0 7 1 - . 0 1 7 5 . 0 5 0 3 . 0 6 8 6 - . 3 2 6 3 . 1 0 9 7 - . 1 7 6 9 1 0 . 0 3 0 6 - . < • 7 2 3 - . 0 0 0 2 . 0 0 6 1 . 0 6 0 0 - . 6 3 8 7 - . 1 2 8 2 - . 2 2 6 2 1 1 - . 1 1 1 0 - . 0 0 7 7 - . 2 6 2 6 - . 0 0 6 3 . 6 0 8 7 . 0 6 6 1 . 6 1 8 0 . 0 3 5 9 1 ? - . 3 7 0 7 • O 3 6 4 - . 2 7 6 6 - . 0 6 0 5 . 2 1 5 5 . 1 1 8 1 . 1 6 5 6 . 0 8 3 9 1 3 . 2 " 3 0 - . U R 0 6 . 1 6 9 7 . 2 2 3 0 . 1 2 3 9 . 6 6 9 1 . 2 5 5 2 - . 5 2 6 0 1 6 . 1 0 3 2 - . 2 8 6 6 - . 1 6 9 7 . 6 5 1 2 . 0 0 8 9 - . 1 1 6 6 . 1 6 1 3 . 5 6 3 0 I f . - . 1 - » 1 6 -.2991 - . 2 0 3 1 - . 3 2 2 6 . 1 6 9 3 . 1 5 0 2 - . 6 1 2 8 - . 3 1 6 7 ' 6 . 1 1 0 6 - . 3 9 0 - . a 2 9 7 . 2 3 6 9 . 0 5 3 6 . 5 3 5 2 . 0 6 0 0 . IOCJ

(b) Transformed data.

i 3 6 5 6 7 e 1 . 3 4 0 0 . 2 A 1 1 - . 0 1 7 9 - . 2 7 6 5 . 1 2 5 9 - . 0 7 7 0 - . 0 1 7 0 . 0 5 3 6 5 . 0 C 2 H . 3 2 2 5 . 0 0 7 6 . 1 7 2 9 . 5 2 6 0 . 2 9 9 9 - . C 3 0 2 - . 2 5 6 3 3 - . 3 5 1 0 . 0 5 4 4 . 3 1 7 1 - . 2 6 9 5 . 1 6 7 6 - . 2 2 3 5 - . 0 3 6 6 • 0C 1 5 t - . 3 A <3 0 . 0 1 0 9 . 3 1 6 0 - . 2 8 1 3 . 0 5 1 6 - . 0 3 1 0 . 0 0 1 3 -.02ce «. . 0 7 1 0 . 0 7 2 5 . 6 1 8 6 . 2 3 6 2 - . 1 3 0 3 * - . 0 6 3 6 - . 0 8 9 1 . 1 6 2 6 f -. PM - . 1 1 0 8 . 1 2 0 6 - . 2 29e - . 3 2 1 8 . 7 6 6 1 . 1 3 9 8 - . 0 3 5 2 7 . 1 M 0 . 0 6 6 5 . 5 6 7 8 . 3 1 9 3 - . 1 3 0 3 - . 0 7 3 3 . 0 0 7 1 . 0 3 9 3 E . 3 0 2 3 . 0 0 1 7 - . 1 1 2 7 . 6 5 2 1 - . 2 6 9 0 . 1 6 6 6 . 0 1 5 3 . 0 6 5 7 c . 0 3 0 0 - . 5 0 6 . 1 7 6 0 . 0 2 6 9 . 1 6 8 0 . 3 1 0 7 - . 2 1 1 1 - . 1 9 6 6 ; r . 1 < m - . 3 9 7 6 . 0 7 0 2 . 0 0 5 1 . 1 9 8 0 - . 1 3 8 7 - . 1 6 6 0 - . 5 9 7 3 1 1 - . 2"60 - . 0 9 5 2 - . 0 6 9 6 . 3 6 3 8 . 6 6 0 1 . 0 6 5 6 . 1 5 3 8 . 1 6 6 0 1 ? - • ? 7 0 - . 1 0 0 0 - . 1 0 1 6 . 3 2 1 6 . 1 8 1 6 . 0 6 9 6 . 1 5 8 6 . 2 6 6 0 1 ? . 2 O 5 0 - . 0 7 6 1 . 1 3 1 5 - . 1 6 1 1 . 2 1 6 5 . 0 6 0 8 . 6 9 6 2 . 0 1 3 6 1 4 . 2 9 6 6 - . 3 6 3 7 - . 0 0 7 2 - . 1 5 2 7 . 2 0 6 1 . 0 7 5 5 - . 6 6 8 1 . 5 6 6 0 1 5 - . 1 " ' , 1 - . 6 0 0 6 - . 0 6 6 3 . 1 6 6 6 - . 2 7 8 7 - . 2 9 3 9 . 1 7 9 6 - . 1 7 6 0 : > . 2 2 6 0 - . 3 6 6 0 . 1 1 6 6 - . 1 6 0 6 . 0 7 8 1 - . 1 2 8 6 . 3 3 5 8 • 2 7 C 5

257

— - — lithological boundary

(b) transformed data

Figure 7.16 Results of clustering by ISODATA for 8-PCA score sets

Table 7.14 Means and standard deviations

a) Untransformed data ( iteration 10, 12 patterns, discards 28, error 3 6 . 7 ) variable

cluster no. F1 F2 F3 F4 F5 F6 F7 F8 no. of

samples

A * -.42 -.79 -.31 -1.40 0 . 0 7 0.33 -.97 -.49 87 * * .36 .61 .28 .51 .38 .64 •97 .91

B 2.40 1.20 -3.30 0.52 1.30 - . 6 7 0.46 0.19 22 .09 .04 .00 .20 .11 .16 .33 .20

C 0.57 0.74 0.75 0.14 1.10 -.57 0 . 1 7 -.35 211 .32 .32 .30 .48 .48 .49 .71 .61

D 0.15 0.01 0.43 0.02 -.53 0.70 -.08 0.01 333 .29 .43 .30 . 6 9 .64 .58 .99 .71

E -1.20 0.41 -.04 -.35 -.66 -.65 0.85 0.64 148 .68 .30 .37 1.00 .61 .33 .57 1.10

F 0.39 -1.10 0,20 1 . 6 0 0.01 O . 5 8 0 . 5 8 2.00 39... • .24 .71 .19 .46 .71 .91 .62 .88

G -.44 -2.70 -.34 -.75 0.35 - 2 . 3 0 -.46 -1.10 27 .24 .42 .22 .61 .47 .66 .64 .57

H 0 . 3 1 -2,20 0.03 0.77 -.11 0.08 0.13 -.41 49 .31 .85 .29 • 85 .56 .67 . 6 9 .91

I 1.80 0.68 -2.30 -.51 -2.20 -.32 -.24 -.22 32 .48 .53 .69 .81 .50 .31 .82 .69

J -.79 - . 6 9 -1.80 -.48 1.10 2.90 -.05 -.38 16 .82 .47 .88 .92 .70 .50 1.30 1.50

K -2.10 O . 3 8 -1.70 1.50 0.41 -.68 0 . 9 6 13 .68 .42 .52 .68 .34 .70 .50 .77

L -2.80 0.72 -1.80 1.60 1.20 0.58 -2.10 -.43 19 .81 .44 .46 .63 .76 • 63 .77 •58

* Arithmatic means in un i t s of standard dev ia t ion from the sample means. Standard dev ia t ions .

clustering for the 8-PGA score sets.

b) Transformed data ( iteration 9, 12 patterns, discards 28, error 40.5 ) variable

cluster no. F1 F2 F3 F4 F5 F6 F7 F8 no. of

samples

A * * * -.16

.43 -1.20

.57 0.14

.48 0.67

.70 -.33

.51 0.01

.58 -.24

.36 -1.10

.70 130

B 1.90 . 1 0

0.08 .02

-3.20 .07

1.20 .24

2.00 .16

-.07 .36

-.67 .05

0.61 . 1 1

22

C 0.43 .41

-.86 .77

0.75 -.11 .75

0.39 .64

-.04 .51

-.02 .99

0.81 .79

180

D 0.39 .47

0.79 .79

0 . 3 4 . 6 0

0.30 .64

0.24 .65

0.29 .67

0.61 .77

-.31 .65

264

E -1.80 .59

0.27 .69

-.04 .28

1.30 .56

0.05 .53

-1.60 .52

0.03 -.95 .66

15

F -1.50 .69

0.13 .35

0.15 .49

-.89 .65

0.23 •53

1.80 .57

0 . 0 0 .76

-.20 . 6 0

68

G O .29 .y*

0.18 .65.

-.00 .58

-1.20 .78

-1.20 .67

-.76 .63

-.08 .65

0 . 2 9 .86

141

H - . 9 3 . 4 9

-.61 .41

-.81 .59

.29

.74 0.46

.65 -1.90

.61 1.70

.83 0 . 7 6

.72 22

I 0.13 .71

O . 2 3 .47

-.42 • 93

-1.10 .66

2 . 1 0 .72

-.44 .76

-.85 .42

-.04 .46

27

J -3.00 .38

0.34 .56

-.58 .23

0 . 0 9 .49

1 . 6 0 .56

-3.00 .33

0.06 .17

-1.00 . 6 0

18

X 1.40 .36

-.16 .73

-2.90 .47

- . 6 9 •55

-1 . 3 0 .64

0.39 .71

-.55 .32

-.28 .83

30

I -1.40 .67

0.41 .40

-.45 .36

0.97 .76

-.61 .57

O .56 .63

-1,00 .49

1.00 . 9 1

79

259

may be not always so in the case of transformed data sets although

lithologic classifications are generally better in the case of the

transformed 8-PCA score data.

However, though more definitive information could be drawn

from the data sets obtained by the 8-PCA component scores than any

other data sets chosen largely by intuition, there are certain

disadvantages in using PCA component score sets for futher classification

analysis since it is very difficult to interpret the pattern classes

in terms of geologic features of individual data sets, for example,

physical, chemical or spectral.

7.5 Supervised Classification

7.5.1 Discriminant analysis

(a) Discriminant analysis procedure's

Discriminant analysis has been applied to a wide variety of

research and problems of prediction as a statistical technique for

examining differences between two or more groups of measurements with

respect to several variables simultaneously.

In Earth Sciences particularly in geochemistry, discriminant

analysis is one of the most widely used multivariate procedures

because it is a powerful statistical tool and it provides an

additional link between univariate and multivariate statistics

(Davis, 1973).

It is a classification tool for use when prior information is

available on the nature of classes which we are interested in

distinguishing between. A training set consisting of a typical example

of these classes is used to find a decision rule which discriminates

well between these classes on the basis of the observed measurements, and

260

this rule is then applied to classify the remainder of the data.

The simplest of the discriminant function is a Linear

Discriminant Function (LDF). It transforms an original set of

measurements on a sample into a single discriminant score. That

score represents the sample's position along a line defined by the

LDF. The basis of the LDF is that of a search for a linear surface

in the hyperplane that effectively separates the classes. In the case

of two groups based on the presence or absence of ore mineralization, LDF

thus consists of finding a transform which gives a maximum ratio of the

between groups variance to the within-groups variance.

If we regard our two groups as consisting of two clusters

of points in multivariate space, we must search for the one orientation

along which the two clusters have the greatest separation, while

simultaneously each cluster has the least inflation. Fig. 7.17 shows

how the group A and B are adequately separated by the LDF.

A good description on the LDF is given by Harbaugh and Merriam

(1968), Rhodes (1969) and Davis (1973).

Howarth (1973) has suggested that the LDF may not be capable

of separating classes that are nonlinear. He used the Empirical

Discriminant Function (EDF) as an alternative to predict both the

stratigraphic affinity and occurrence of mineralization.

This method is the exponential form of the polynomial

method of Specht (1967). It is based on the nonparametric estimation

of a probability density function for each category to be classfied,

so that the Bayes decision rule may be implemented. A smoothing function

is applied so that a density function is estimated from the training

samples.

The effect of smoothing by varying the smoothing parameter Q

is shown in Fig. 7.18. As O increases, the five distinct modes of the

261

Figure 7.17 Two overlapped bivariate distributions showing the effective classification by projecting onto the discriminant function line (After Davis, 1973)

262

Xo= |oo...o. ..»•[

Figure 7.18 Interpolated one-dimensional probability density function for set of five training samples with increasing values of smoothing parameter O. (After Howarth, 1971a)

263

training samples are gradually smoothed out. A detailed account of the

mathematics involved is given by Specht (1967), Howarth (1971a, 1971b,

1973a) and Castillo-Muhoz (1973).

Ideally, the classification should be performed in such ways

that the classes for the classification are mutually exclusive

and exhaustive, and the variable set chosen should allow the perfect

assignment of unknown data to one of the classes. Hence the choice

of the variable set becomes an important consideration as already

discussed previously. Howarth (1973b) has reviewed the various

methods available, and he finds that the stepwise feature selection

techniques provide the best overall success rates in the classification.

As already described in Section 7.3, these techniques (BAKWRD)

consist of an iteration of the classification with the variable

causing the greatest improvement in the overall success rate being

added to or deleted from the variable sret. The problems of testing

the classifier performance is the large number of possible combinations

of the variate set, thati,s,{m(m-1 )/2} where m is the total number of

variables to be analysed. This point has been discussed already in

Section 7.3 and Howarth (1973b) has noted that the optimum method is

training with successive eliminations (also known as

11 jackknife"). These procedures are implemented in the EDF package

program developed by Howarth (1973a) and used by Castillo-Munoz (1973).

The mathematical procedures of EDF described by Howarth (1973a)

are given in Appendix G.

In this study, the EDF has been used to classify those areas

most closely related to the training set chosen by means of geological

and geochemical information and results of the cluster analysis. Further

details on the training set selection schemes will be described in

the next section.

264

(b) Selection of the initial condition

The control parameters required by the EDF computer program

(Howarth, 1971b) are as follows: (1) the number of variables to be

considered, (2) the number of pattern clusters into which the data

are to be grouped, (3) the a priori probability of the occurrence

of each of the classes specified, (4) the smoothing parameter

for the estimation of the probability density function for each variable

in each class, (5) whether the data are to be left untransformed or

log-transformed prior to exercising the discriminant analysis,

(6) a threshold value that an unassigned sample is to be assigned to in

one of the given categories, and (7) the data matrix coordinates

of one or more rectangular training areas for each class.

The program will then classify all the samples on the basis of

relative probability that an unassigned sample resembles each of the

defined classes. If the absolute probability that an input sample belongs

to one of the given categories falls below the threshold value, the

sample is then assigned to the 'unknown' class (Howarth, 1971b).

Since the input data are from different sources, provisions for log

transformation (base 10) in the program were not used. Instead, the input

data, have been provided from either untransformed or

data transformed by the power transformation technique or arc sine

transformation technique (see Section 7.2). There are options in

the program for assigning different a priori probabilities for the

occurrence of each class, or for giving a different penalty for making

an incorrect classification and these were arbitrarily made equal for

all classes, since it is not possible to determine them prior to the

analysis.

Above all, the most important initial step is the training set

selection and choice of the smoothing parameter(cr). The training set

265

samples are to be chosen in such ways that an artificial grouping of

the results should be avoided as far as possible.

Thus the training areas for the classes of the multivariate

data sets were selected by considering the results of the cluster

analysis by ISODATA, and the geological map was used to select

lithologically homogeneous geologic formations as far as possible.

Geochemical homogeneity was also taken into account by referring to the

regional distribution patterns of geochemical elements produced as

in Fig. 4.18 in Section 4.4. In particular, clustering results by

ISODATA played an important role since when only the lithologic

homogeneity was considered the EDF results were dubious.

In this study, seven training classes for the classification

were selected to represent the major lithological classes, including

a class for the potential mineralization. Further, a training class

representing the offshore area was also- selected for the 9-mixed

variable sets and 8-PCA score sets. Upper and Lower Carboniferous sediments

were grouped together into a class since they are generally intercollated

with each other by folding so that it might be difficult to

separate them in the regional sense. Upper and Middle Devonian strata

generally show geochemical homogeneity but they were separated as two

different groups in order to see what might be expected to be

produced. The Lower Devonian was assigned as a class though in some

occasions it shows geochemical homogeneity with Carboniferous rock

units and Middle and Upper Devonian rocks.

The Bodmin Moor and St. Austell granites are grouped as

different clusters because they are somehow different in geochemical

composition, and this was confirmed in the cluster analysis by

ISODATA which produced the granites as different clusters in the higher

number of pattern classes, though they were merged into one cluster

266

in the lower number of pattern classes. Selection of a class for

mineralized zones has been made by examining spatial information of

geochemical haloes of metallic sulphide elements as described in

Section 4.4 and the results of ISODATA analysis. Part of the

Crackington Formation in the east central part of the study area

(transitional zone) and basic intrusives in the northeast of the Bodmin

Moor granite were ignored since regionally these minor groups could be

merged into the neighbouring major groups.

However carefully they are chosen, in practical applications the

training sets selected by the investigator will always be too small

to reliably be extrapolated to new data sets (Nagy, 1968,

Howarth 1971a).

After having selected an appropriate training set in the study

area, the success of the method depends on a correct choice of the

smoothing parameter ( a ) . Since it is td some extent, data-dependent

(Howarth, 1971a), an optimum value must be determined prior to

performing the classification with EDF. Alternatively, the smoothing

option with successive increments is desirable to classify with the

data successively and compare the results with known geologic

information. However, analysis in this way may not be so desirable compu-

tationally and in its effectiveness since much effort has to be expended

in order to calculate for varying degrees of the smoothing parameter

and it also requires a complete knowledge of geology in the study

area for correct assessment of classification results obtained by

varying smoothing parameters.

Howarth (1973a) has presented a computer program BAKWRD

initially written for feature selection schemes. The program uses a

method of evaluating the probable classification success of the

training set originally proposed by Kanal and Chandrasekaran (1968).

267

In this study, BAKWRD was used to obtain a classification

performance table showing the true and assigned classifications of the

training set samples.

Fig. 7.19 shows for varying smoothing parameters the correct

performance rates of each class of the training set samples chosen

from untransformed and transformed 8-geochemical, 9-mixed variable sets

and 8-PCA score sets.

From the figures, the correct classification rates are high when

the smoothing parameter is between 0.1 and 2.0, but the rates decrease

gradually towards higher smoothing parameters. If the smoothing parameter

is very low (a=0.01), there is no misclassification at all but many

are unclassified. This indicates that certain optimum smoothing

parameters must be applied in general for the data set used in this

study. The plotted values of the correct classification rates at

Q = 0.01 on the figure are unclassified rates rather than misclassified

rates.

From the analysis, the transformed data sets are more stable

and higher in correct classification rates than the untransformed

data in all cases. Comparing results between the three different variable

sets, the 8-PCA score set is the highest and 9-mixed variable set

is second. Training set samples from the transformed 8-PCA score

data set show 100% correct classification rate bewteen Q = 0.1 and

Q = 2.0, and even at O = 5.0 the performance rate is over 90%. Except

in the untransformed 8-geochemical element set, they show 100%

correct classification rate between O = 0.1 and a = 1.0 and even the

untransformed 8-geochemical element is 100% between O = 0.1 and O = 0.5.

Assuming the training set samples chosen for the analysis are

adequate, optimum smoothing parameters for the data sets chosen for the

study lie between 0.1 and 0.5 (this tendency has been generally noticed

100 •*

90 J

80 b

70

60

50

40

30 .

20 .

10 .

untransformed 8-geochemical » transformed 8-geochemical + untransformed 9-mixed * transformed 9-mixed A untransformed 8-PCA n transformed 8-PCA

i i 1 0 smoothing parameter

Figure 7.19 Correct classification performance rates of the training set for the untransformed and transformed variable sets

r o 00

269

in the test with other trianing set samples). Thus the smoothing

parameters O = 0.2 or 0.5 were chosen as optimum values for the analysis

of EDF in this study. Indeed, comparison of the results of EDF

analysis with O = 0.2 and (7=0.5 shows marked resemblances between

each other and some differences noted are generally marginal.

The performance rates of each class in the training set samples

for the optimum smoothing parameters (CT = 0.2 and 0.5) analysed by

BAKWRD program are 100% for all data sets as shown in Fig. 7.18.

(c) Applications of EDF

(c.l) 8-geochemical element sets

The results of classification using the chosen 7 classes of the

training set are shown in Fig. 7.20(a) and (b) for the untransformed

and transformed data, respectively. Even with the small training set samples

(12.5% of the total samples) the lithologic units are largely well

classified in the analysis though there are some variations in

details. The classification reuslts are almost comparable to

the reuslts of clustering by ISODATA as shown in Fig. 7.14(a) and

(b). Classification of granitic areas, Carboniferous and Lower Devonian

rocks, Upper and Middle Devonian rocks in both analyses provide

general resemblances between Fig. 7.14 and 7.20.

The Bodmin Moor granite which is assigned as clusters E and F,

is probably the most successfully classified. This may be attributable

to the uniform regional lithologic features of the units and no serious

effect of secondary environment. However, the granite clusters are in

general spread over into the Devonian sediments, partiuclarly in

the south of the granite. An exception is in the northern part of the

granite where a portion of the granite area is assigned as Devonian

270

(a) untransformed data

— lithological boundary

t BY JINC COOHOlNOJtS (b) transformed data

Figure 7.20 Results of classification by EDF for the 8-geochemical sets: As, Cu, Ga, Li, Ni, Pb, Sn and Zn

271

or Carboniferous sediments. As these features have also been

recognized in results of the cluster analysis of the same data sets and

FA, the spreading to the south is attributed to the drianage system

flowing down from granitic uplands into Devonian lowlands

to the south, and the granite area assigned as Devonian or Carboniferous

sediments in the north corresponds to a poorly drained topographical

depression where thick oxide coating might have altered the chemical

characteristics of the soil.

Cluster F,'classified as the St. Austell granite in the southwest

corner of the study area, is also well represented except in a few

marginal areas where there might be distortion of data by secondary

effects such as drainage or mining activities and also by digitizing

and smoothing in the data preparation processes.

The Carboniferous rock unit has been generally well classified,

but the Upper and Middle Devonian rocks are intermixed together

which could be expected since these two rock units are similar in

geochemical composition. Parts of Lower Devonian rock (Cluster D)

have been assigned as cluster A of Carboniferous rock, which would indicate

there are some geochemical similarities between the two rock units.

Group G assigned as potential mineralization zones has been

differentiated in the east of the main granite and near Kithill and partly

in the southwestern part of the map between Bodmin Moor and St. Austell

granites.

Results of the classification for the transformed data in

Fig. 7.20(b) are similar in features to the untransformed data except

for the drainage pattern in the southeastern part of the map, and a slightly

more diffuse pattern in defining potential mineralized zones in

the east and south of the main granite and the northeast margin of

St. Austell grnaite, which are, to some extent, comparable to the

clu stering results of the same data in Fig. 7.14(b). Therefore, further

272

detailed descriptions will be omitted for the analysed results.

(c.2) 9-mixed variable sets

The results of classification by EDF analysis for the 9-mixed

variable sets in Fig. 7.21(a) and (b) generally show similar features

comparable to the results of clustering by ISODATA.

Though the training set samples chosen are small, EDF

analysis might be as stable as the sophisticated ISODATA program in

its performance of classification as long as the training set samples

are chosen carefully.

(c.3) 8-PCA score sets

The classification results of 8-PCA score sets by EDF are shown

in Fig. 7.22(a) and (b) for the untransformed and transformed data,

respectively.

Though regional features might be comparable to the clustering

results in Fig. 7.16, some distinctions in classification appear between

both results.

The Carboniferous and Lower Devonian sediments have been generally

differentiated from the Upper Devonian in the EDF analysis. In the

clustering analysis in Fig. 7.16, the Carboniferous and part of the

Upper Devonian rocks have been clustered as the same group, and Middle

and Lower Devonian sediments have been grouped into a single cluster.

The depression in the northern part of the Bodmin Moor

granite is again classified into the Devonian sediments, the feature

of which does not appear in the clustering analysis of the same data sets

as in Fig. 7.16.

There also occur some differences in classification of

273

(a) untransformed data

lithological boundary

M

*Voo 17.00 14.00 10.00 71.00 tostinc cooooinoies

(b) transformed data

Figure 7.21 Results of classification by EDF for the 9-mixed variable sets: Gravity, Magnetics, B5, R5/4, R7/6, Cu, Ni, Sn, Zn

274

lithological boundary

Figure 7.22 Results of classification by EDF for the 8-PCA score sets

275

anomalous units though their localities appear to be generally in the

same region.

Again, in these analyses, the transformed data set shows

more diffuse patterns in defining mineralized zones although the litho-

logical definitions have been classified in a similar way.

As in the cluster analysis, significant misclassifications also

arise in classifying different sedimentary rock units in the EDF analysis,

which could mean that they are, to a certain extent, geologically

similar and those' facies might be similar physically, chemically

or spectrally unless much distortion occurred in the preprocessing

of the data.

7.5.2 Characteristic Analysis (CHARAN)

The applications of characteristic analysis in this study is

mainly introductory in its application to the data. This is mainly due

to the limited time for the research.

(a) Characteristic analysis procedures

Characteristic analysis is a multivariate technique that has

been used to attempt the regional assessment of exploration targets

for a variety of types of deposit (Botbol, 1970, 1971, Botbol, et al.

1977, 1978, Sinding-Larsen et al. 1981, McCammon et al. 1981). It was

originally developed as a method for integrating regionalized multi-

variate data in geology, geochemistry and geophysics (McCammon et al.

1981).

There are basically three concepts involved with the method; data

transformation, 'favourable' model formulation and regional cell

evaluation.

276

The CHARAN program initially written by Botbol, Sinding-Larsen

and co-workers has been modified to use in this study.

1. Data transformation

The initial step in the CHARAN program is to transform the

data into binary form by assigning the value of 1 meaning favourable

or the value of 0, meaning unfavourable or unevaluated.

The manner in which this transform is performed depends on the

nature of the exploration model and the nature of data. For geochemical

data, favourability may be defined by local anomalies calculated

from second derivative surfaces, highpass filtering and so on.

Reduction of local anomalies by the second derivative method is

illustrated in Fig. 7.23.

Where a cell or location has a negative second derivative for

a particular variable, it is labelled '!' and is of interest because

the values within the cell are higher than the values in the neighbouring

cells. The 'O's represent all other data which may have no potential

values of interest. For those variables with known negative anomaly

representation, simple reversal of binary notation may be applied.

For geophysical data, on the other hand, favourability may be

determined on the basis of regional gradients, local increases or decreases,

and recognition of any special features such as any lineaments, etc.

For geological map data, the presence of a particular rock type

or any structural linear features could be a criterion to be considered.

This type of binary representation yields maps that indicate only

those areas that have values of major interest in exploration.

The latest version of the characteristic analysis program

facilitates the dynamic range of data values, by coding the input in

ternary form (1, 0, and -1) (McCammon et al. 1981). Here the value of

. l . l . l . t . o . o . o . o . l . t . l . l . l . t . o . l . t . t . t . o . o , ' . '

Figure 7.23 Hypothetical data profile showing areas above local inflection points (second derivative negative) labelled '1' and other locations labelled '0' (After Botbal et al. 1977) ^

278

1 means favourable as before, the value of 0 meaning unevaluated, and the

value of -1 meaning unfavourable. Adaption of logical combinations of

transformed variables make it most powerful to utilize the evaluation

of dynamic range of models.

2. 'Favourable' model formulation

After having generated a binary or ternary array for each

variable, the next step is to determine the relative weights of each

variable for region cell evaluation from a model.

The favourability of a given cell is defined as a weighted

linear combination of either the binary or ternary transformed variables,

as follows,

f = a.X + a X + ... + a X (7.8) 11 2 2 m m

where a^ and X , (i=l ,2, . . . ,m) represent the weights and m transformed

variables, respectively.

The weights, a^, in Equation (7.8) are determined by 'a product

matrix' defined by a selected model set of the transformed variables.

Mathematically,

(X'X)a = Xa (7.9)

where A is the largest eigenvalue of (X'X). X is the n x m matrix

of m variables for n selected cells that comprise the model.

The a^'s are the elements of the eigenvector a_ associated with

X and are scaled such that f in Equation (7.8) lies between 0 and 1

in case of binary input but -1 and 1 for ternary data set.

The model cells may be selected in areas of known geology or

deposits, depending on the nature of study, i.e. mineralized model or

lithologic model. Generally the model should be generalized in its appli-

279

cation not by the unique features of deposits or lithology in an area, but

by inclusion of regional cells that do not contain particular

deposits or lithology.

3. Region-cell evaluation

After the weights of the variables, which comprise a model,

have been calculated by Equation (7.9), the degree of association

for unknown region cells outside the model areas is thus calculated

by multiplying the binary or ternary vector that represents the

transformed variables of each region cell with the characteristic

vector of the model calculated as in Equation (7.8).

(b) Application of characteristic analysis

In this study, the original version of characteristic analysis

utilizing the binary form of the input variables was applied to test its

applicability to the data from the Bodmin Moor area.

Five sulphide geochemical elements of As, Cu, Pb, Sn and Zn were

selected for consideration and their model cells were chosen for use in

order to analyse for areas of potential mineralization.

Eight model cells were selected on the basis of known geochemical

haloes related to the known mineral lodes or veins in the eastern

part of the Bodmin Moor and near the Kithill granites. The model was

characterized with respect to the elements used. Table 7.15 shows the

characteristic weights for the five elements derived from the product

matrix of the model. Pb is the most strongly weighted component

of the model vector, followed by As and Sn. The ratio of the eigenvalue

(the characteristic root in the table) and the total number of I's in

the model cells indicates the degree of anomaly overlap.

Low overlap means low dependence of cells within the model, which

280

reflects the combination of cells and/or variables which does not

produce a diagnostic result, while high overlap indicates that the

model chosen shows a strong dependence adequate enough to produce the

diagnostic results.

In the case of this study, the model has a high degree of

similarity (about 73%).

The results of the analysis with this model are shown

in Fig. 7.24. Five classes were chosen to illustrate the degrees

of association between the region cells and the model.

There occur five most prominent non-model anomalies on the

map. All occur as individual mineral belts(?) which may or may not

be genetically related, but they are largely aggregated with moderate

degrees of association around them.

However, none of the prominent anomalies are associated with the

known mineral lodes or veins. This may indicate that the manner

in which the data transformation is performed in the program may not

be adequate for this kind of regional analysis.

It seems to the author that in any future analysis with more

time available it might be advisable to examine if the data trans-

formation for geochemical elements should be performed by other methods

such as highpass filtering, etc.

Obviously full utilization of the new version of characteristic

analysis with logical combinations would facilitate its dynamic

range of application to the assessment of the regional geology or

mineralization.

281

Figure 7.24 Results of characteristic analysis for a variable set of As, Cu, Pb, Sn and Zn. Rectangulars are the selected model cells. Degrees of association is indicated by symbols in the increasing order as blank, ',-,+,0

282

Table 7.15 Characteristic weights of five elements (As, Cu, Pb, Sn and Zn) for the model selected as indicated by two rectangulars in Fig. 7.24

Element Weights Remarks

As .47 * = 17.49 Cu .42 ** = 24 Pb .55 Sn .47 Zn .27

* The characteristic root ** Total sum of l's in the model cells

7.6 Conclusions

Four kinds of supervised and unsupervised pattern recognition

techniques have been applied to various sets of the multivariate data

from the Bodmin Moor area. The usefulness of each kind of technique

has been shown in use for regional assessment of geology or potential

mineralization.

In all cases, the classification of the Bodmin Moor granite

is the most successful. The regional geological mapping of sedimentary

rock units is in general confused, but the data sets which are most

appropriate are the 8-geochemical ones followed by 8-PCA score sets.

This might be due to similarities between different sediments,

particularly near their boundaries which would be due to secondary effects

such as contamination or smoothing, etc. Alternatively, this might be

due to inappropriate combinations of the multivariate data (particularly

of gravity and magnetics) for the classification, though the gravity data

might contribute significantly in classifying the granitic areas. In

283

this study area, these potential fields usually represent subsurface

geological features which might be unrelated to the surface lithologic

features.

However, the training set test of BAKWRD for varying the

smoothing parameter (a) has shown that the 8-PCA score sets are

most stable, followed by the 9-mixed data sets and the last by the 8-

geochemical element sets. This indicates that an appropriate

combination of data sets having various properties, for example, physical,

chemical or spectral, would constitute more stable data sets and thus

draw better results. In addition, in all cases, the transformed data

are more stable than the untransformed data, indicating favourability

of the transformed data over the untransformed data in the application

of pattern recognition techniques.

Thus, in this study, regional lithologic mapping of the

transformed data is better than the untransformed data, but at the expense

of diffuse patterns for localizing mineralized zones, whereas the

untransformed data are better in defining potential mineralization which

largely corresponds to areas of known mineral lodes or veins.

Dicards occurred both at typically low and high values in both

untransformed and transformed data in this study although many workers

(for example, Crisp 1974, Castillo-Munoz 1973, and Mancey 1980, etc.)

have noted in the analysis with geochemical data sets that the discards

might occur usually at high anomalous values of geochemical elements.

From the analyses of the techniques, the following general

conclusions may be drawn.

1. The factor analysis method is by far the most capable

of handling as large a number of data measurements as may be required,

since the dimension of the program depends on the input data structure

which can be modified as required. However, the display of major

284

component scores and the choice of combination of the major components

are not immediately possible and are more complicated than other

classification methods. The interpretation may also be somehow

subjective.

2. The cluster analysis by ISODATA is, to some extent,

subjective because of various initial options which would affect the

results. However, this method is much more conclusive (at least

from the statistical point of view), since the error percentage

of the clustering results together with various statistics useful

for interpretation, are calculated though the error rate is much

dependent on the number of discards.

Immediate recognition of patterns from the line-printer output

would add•to its usefulness for rapid and effective analysis.

3. Both 'EDF and CHARAN' supervised classification

techniques applied are very subjective since the classification results are

entirely dependent upon the user's input controls and no unique results

can be drawn.

However, these classification methods would be most usefully

applied to areas where the training set for regional geology or

particular mineralization is well-known and thus this is used further to

classify unknown areas by comparison.

While there are complications in setting-up the user's initial

conditions, their simplicity and nonparametric nature could make them

most appropriate in solving complex geological problems arising in

pattern recognition.

285

4. Transformation of the data may be useful at least for

statistical requirements. However, in this study, unique advantages of

transformed data over the untransformed data are not significant,

though the regional geological mapping seems to be generally better

with the transformed data but at the expense of diffuse patterns of

mineralized zones.

Problems with the transformed data might lie in the fact that

transformation of individual variables to near-normality may not

guarantee that the data set drawn from the transformed data are truly

Gaussian, as noted by Mancey (1980).

However, tests of correct classification rate with the

training set samples indicate that the transformed data are more stable

than the untransformed data. The test results also show that 8-PCA

score sets are the most stable compared to the smoothing parameter Q,

followed by the 9-mixed variable sets, and the least stable one is

the 8-geochemical data set.

This would indicate that combining the geophysical, remote

sensing and geochemical data might constitute a better multivariate

data set than any set of data drawn from any single set of properties

(for example, physical, chemical or spectral) of the region of interest.

5. Though the usefulness of combining various data having

different properties has been noted from the study conducted, odd

lithologic outlining of the 9-mixed variable sets indicates that some

data sets such as gravity and magnetics may not be adequately applied

in the analysis, since they usually show subsurface geological features

which might be different from the surface or near-surface geological

features.

286

6. None of the pattern recognition techniques applied in

this study are to be uniquely favoured over other methods. They are

largely complementary to each other. Particularly when EDF is used

for classification, it is almost essential to perform the cluster

analysis or some other unsupervised classification techniques to ensure

that the training set for the EDF constitutes 'natural' groupings.

287

CHAPTER EIGHT

CONCLUSIONS AND RECOMMENDATIONS 8.1 Conclusions

Data preparation

Careful data preparation is essential in multivariate analysis;

otherwise, the results of analysis may be misleading. Data prepara-

tion included digitization, noise filtering and data qualification.

1.1 Digitization of data was achieved using either computer inter-

polation technique ( gravity and geochemical data ) or manually(magnetic).

For Landsat MSS data, an areal average method was used for obtaining

i n i t i a l data for regional analysis and then the computer interpolation

technique was applied in order to project the data onto the same grid

points as used in the other methods.

1.2 Spectral analysis of the raw data indicated that the power at

shorter wavelengths appears to be due to random noises. Thus, the

noise in the data has been reduced by either a specially designed

f i l t e r operator ( gravity and magnetics) or box-car smopthing f i l ter ing

using a 3 ty 3 window function ( Landsat MSS and geochemical data ).

Comparative plots of one-dimensional power spectra of the raw

and filtered data show significant reduction of energy at shorter wave-

lengths, and thus the random noise in the data has been considerably

reduced ( See Fig. )•

1.3 For data qualification, two important factors, stationarity

and normality, were evaluated for the f i l tered data •

The Landsat MSS data show some degree of stationarity, but

the geophysical and geochemical data show at bestveiytaudegrees of

stationa x i t y . However,", after removing the regional trends by

either Spencer's method or Double exponential f i l te r ing ( Davis,1973,

p226 ) f the degrees of stationarity increase rapidly, so that in general

I

288

i t can "be concluded that a l l the data are at least quasi-stationary

( See Table 3.2 ) .

Test of normality by visual inspection of histogram shows

significant skewness except Landsat MSS data, in which cases they

show, in general, symmetrical patterns.

Feature extraction

Prior to multivariate data analysis, feature extraction tech-

niques were needed, the type of technique being applied to each of the

14- data sets being determined by the physical, chemical or spectral

characteristic property. This enabled emphasis to be placed upon

regional or local features of geological significance or potential

mineralization.

2.1 In geophysics, regional features ,for example, due to magnetic

basement and/or regional gravity trends were defined by regional analysis

including lowpass f i l ter ing and upward continuation. Sources of

magnetic and gravity anomalies usu&hjdifferent from each other in this

study area. For example, the regional magnetic trend reflects base-

ment features while the regional gravity trend shows regional features

of the granite cupolas and adiacent sediment basins. Local anomaly

features of small scale magnetic and gravity sources ( e.g. perhaps

subsurface granite cusps ) were enhanced by vertical and horizontal

derivatives and highpass f i l ter ing in order to visualise those features

more clearly.

2.2 In remote sensing, feature enhancement techniques including

contrast stretching and ratioing methods, and derived colour-composites

applied to Landsat MSS data from Cornwall and the Bodmin Moor granite

area enabled the extraction of various regional features showing l i tho-

logical or stricturS-l patterns. I n particular, the granite cupolas

and many known lineaments ( e.g. wrench faults, the contact line between

289

Caledonian an ( l Amorican zones, and l ine of confrontation in the north

of Badstow, etc.) were confirmed and further linear features ( which

are not on the geological map hut are probably geologically s ignif icant)

were detected from the enhanced images.

This work for the study of lineaments shows that black-and-

white band 5 w as the best, followed by band 7> and for l i tholog ica l

mappings, the colour-composite of band ratios provide the best.

2.3 For geochemical data, level s l ic ing by concentration scale

employed shows the regional distribution patterns of geochemical elements

in relation to geology. Metallic sulphide elements including As,Cu,

Pb,Zn and Ni tend to be low in the granites compared to the surrounding

country rocks; this could be due to kaolization. These elements (

except Ni ) show also typical ly low values in the Upper Carboniferous

and Lower Devonian sediments except in areas of contamination by drainage,

etc. The metallic sulphide elements tend to concentrate in a number

of areas around the granitic aureoles.

Ga, Li, and Sn show high concentrations on the granite and i t s

margin, which may be due to the secondary enrichment by drainage systems.

They are characterized by their low values in the country rocks, parti*

cularly in the Carboniferous and Lower Devonian sediments.

Spatial distributions of the anomaly patterns of metallic

elements ( Cu, Pb, Sn and Zn ) were delineated by the probability analy-

s i s and these may be related to the local mineralizations. Sn haloes

appear in the east, southeast and along the southern border of the

Bodmin Moor granite and in the K i t h i l l granites. Also Sn haloes

appear in most of the St.Austell granite and in the northeast to east

of the St.Austell granite. The east and southeast of the Bodmin Moor

granite and the K i th i l l gimite appear to be associated with Cu haloes.

Pb and Zn haloes are generally associated in the southeast and

northwest of the Bodmin Moor granite and in the K i t h i l l granite. Zn

haloes occurring in the east of the St.Austell granite may be mainly due

290

to contamination by old mining, etc.

Trend surface analysis

Regional trends of the data sets over the Bodmin Moor area were

extracted by analysis of f i r s t - and third- degree trend surfaces, and

local irregularit ies or isolated anomalies were calculated from the

third-degree trend surface residuals.

3.1 The re l i ab i l i t i e s of the first-degree trend surfaces are generally

low in the analysis. However, two distinct groups can be found.

Group 1 consists of gravity, four Landsat MSS bands, Li, Ni and Sn; i t

trends approximately in a NW to SE direction. These variables common-

ly show most dominant features over the granite areas; bands k and Hi

and Sn show high occurrences while bands6 and 7> and Ni are typical ly

low on the granites.

Group 2 consists of As, Cu, Pb and Ga; i t trends between north

and NE. The signif icant geological features are high common occurrences

of As, Cu and Pb in the Upper and Middle Devonian rocks; the maximum

values are associated with the mineralized zones, particularly in the

east and southeast of the Bodmin Moor and in the K i th i l l granites.

There i s a low occurrence of these elements within the granite cupolas

and in the Lower Devonian and Upper Carboniferous rocks. Zn element

belongs to this category.

By contrast, Ga shows i t s high levels of occurrence within the

granites and surrounding superficial sediments where streams drain down

from the granites, while low concentrations occur in the rest of the

sediments.

3.2 The residuals derived from the third-degree trend surfaces

i l lustrate localized anomaly patterns which might be related to specific

lithology or mineral concentrations. However, detailed f ie ld checking

i s needed in order to identify these anomalies. This i s partly due

291

to low significances of the trend surface analysis, particularly near

the edge of the data.

3*3 In the analysis, the third-degree trend surfaces provide more

reliable means of assessing the regional trends and local anomaly patterns.

Similarity analysis

Three different types of similarity analyses were made to establish

inter-associationships between variables: correlation coefficients,

similarity map and coherence analysis.

4.1 The correlation coefficients which provide means of evaluating

overall similarity are computationally simple and perhaps geologically

informative (See Table 6.1 & 6.2 for the untransformed and transformed

data, respectively ). Hierachical clustering of the correlation

matrix worked better with the transformed data and the results show

distinct three groups. Group 1 consists of gravity, Sn, Ga, Li, Ni,

bands 4 and 5 their association with the descending order of correlation.

*£he common features of this group are dominant features over the granites.

Group 2 is mainly of metallic sulphide elements including As, Cu, Pb and

Zn. Group 3 is composed of bands 6 and 7 which show typically high

reflectance features over the vegetation cover.

The groupings of similarities are different from those of linear

trends. This is because only regional aspects of the sample data

contribute to the trend surface analysis which ignores detailed variations

of individual values.

4.2 The similarity maps show spatial correspondence between variables,

so that associated mineralized zones or any geologically significant

features such as structural information or geological provinces, could

be analysed.

A strong association between gravity and pseudogravity appears

in the northeastern part of the study area which may depict dense magnetic

291-1

sources. The Cornubian batholith trend shows a relatively low but

positive association.

High associations between Sn and Cu, and As and Cu, commonly

occur in the east and southeast of the Bodmin Moor and in the K i th i l l

granites. Another high spatial correlation between Sn and Cu appears

in the northeast of the St.Austell granite. Pb and Zn show their

high associations in the southeast and northwest of the Bodmin Moor and

in the K i th i l l granites. The local i t ies of these correspond to areas

of known mineral lodes except for some drainage patterns, particularly

in the southeast of the main granite. Strong negative associations

between Pb and Zn in the east of the St.Austell granite correspond to

high Zn haloes probably contaminated by old mining act iv it ies,etc.

4-.3 The coherence between gravity and pseudogravity transform of

the magnetic data was in general low. The remanent magnetization of

the magnetic source, bodies might contribute to the low coherence.

Bandpass f i l ter ings designed for particularspectral regions

such as high coherence or incoherence show structural features for those

spectral regions. In particular, stratigraphic features in the north-

ern part of the study area ( See Fig. 6.6 ), derived from the bandpass

f i l ter ing of the high coherence spectral region, might be due to shallow

dense basic rocks occurring along the stratigraphic beddings or zones of

weakness developed during the Amorican orogeny. Stratigraphic over-

lappings between Upper Devonian and Lower Carboniferous rocks might have

contributed to these effects to a lesser extent due to their differences

in magnetic susceptibi l i t ies.

Pattern recognition thechniques

Pattern recognition techniques provide much more powerful means

of extracting regional geology as well as anomaly patterns by combining

various data sets than any feature extraction techniques for individual

data set.

291-2

The resear-ch was concerned with classification mapping of regional'

multivariate data which were derived from geophysical, remote sensing

and geochemical data against the known geology. The study has shown

how various pattern recognition techniques (FA, ISODATA, EDF and CHARAN)

based on the multivariate data can be used to solve geological problems.

Multivariate analysis requires that the set of samples be suffi-

ciently large to be representative and this set of samples should be >

analysed for a large enough number of variables. Selection of the

ootimum number of variables to solve the problem being tackled is crucial.

In this study two methods of selection were used (a) trial by

intuition and experiences and (b) principal components analysis.

The first method utilized all available information on geology, geophysics,

geochemistry including correlation coefficients and the eigenvector matrix

and three different bands of untransformed and transformed data were used.

1. 6 Landsat MSS data set : E^, E5, B7, R6/5, R7/6

2. 3 geochemical element set : As, Cu, Ga, Li, Ni, Fb, Sn, Zn

3. 9 mixed variable set : gravity, magnetic, B5» R R 7 / 6 , Cu, Ni,

Sn, Zn

The second method utilized two untransformed and transformed data sets

from the first 8 PCA scores derived from the 16 variables, gravity,magnetic,

B^, B5, B7» R5A» R6/5» R7/6, As, Cu, Ga, Li, Ni, Pb, Sn, Zn

5.1 In this work, it was found that the regional lithological

mapping could be best achieved using transformed data, while untrans-

formed data were more suited to defining potential mineralization zones.

However, the Bodmin Moor granite rock unit was consistently identified

by all of the methods.

291-3

5.2 The boundaries of the sedimentary rock units in the regional

geological map are not clearly defined. The most appropriate data

set in this study are the transformed 8-geochemical one, followed by the

transformed 8-PCA score set. The transformed 8-geochemical data set

was in general better in defining the regional l ithological units,

particularly Carboniferous and Lower Devonian rocks. The Lower

Devonian rocks was similarly well-defined in the analysis of the trans-

formed 8-PCA data set.

The relatively poor definition of the sedimentary rock units

could be due to the mixture of transported material with local material

, or smoothing effect in the digitization,etc. Alternatively, this

might be due to inappropriate combinations of the multivariate data

(particularly of gravity and magnetics) for the classification, though

the gravity data might contribute significantly in classifying the granite

areas. In this study area, these potential f ields usually represent

subsurface geological features which might be irrelevant to the surface

l ithological features.

5.3 Comparative contribution of each variable to the multivariate

data was derived on the basis of eigenvalues(Fig.7.3) and eigenvectors

(Table 7.13) obtained by PCA of 16 data sets(gravity,magnetic,B^,B5,B7,

R5/^,R6/5,R7/6,As,Cu,Ga,Li,Ni,Fb,Sn and Zn). The overall contribution

of the f i r s t 8-PCA scores i s set arbitrary to be 1.

From Table 8.1, columns of "original* shows the nature and

magnitude of the contribution of each variable of the multivariate data

(i.e. positive or negative and its weight). Columns of "absolute"

illustrate the significance of the contribution by individual data sets

of the multivariate data (i.e. overall contribution of each variable to

the multivariate data set) and it is indicated by the descending "order".

291-4

Table 8.1

variably

untransformed data transformed data

variably * original * * absolute **order original absolute order Gravity 1.1092 .0670 6 .0905 .0631 8

Magnetic .1005 .0721 3 .1225 .0611 12

Band 4 -.0470 .0673 5 -.0548 .0 675 2

Band 5 -.0578 .0657 9 -.0635 .0615 11

Band 7 .0114 .0596 12 .0927 .0529 16

R5/4 -.0512 .0677 4 -.0429 .0617 9

H 6/5 .0327 .053^ 13 .1095 .0592 13 R7/6 .0452 .0657 9 .0786 .0588 14

As -.0614 .0423 16 -.0252 .0564 15 Cu -.0693 .0506 14 -.0186 .0634 7 Ga -.0436 .0656 11 -.0185 .0647 6

Li -.0729 .0659 8 -.0577 .0 675 2

Ni .0979 .0726 2 .0923 .0616 10 Pb .0378 .0 660 7 -.0039 .0654 5 Sn -.1162 .0728 1 -.1171 .0682 1

Zn .0458 .0459 15 .0119 .0669 4

* analysed results from the orig inal eigenvector matrix

* * analysed results from the absolute values of eigenvector matrix

* * * the order of overall contribution derived from the absolute matrix

In the untransformed data(in descending order),gravity,magnetic

and Ni are strong positive contributors, while Sn and to a lesser extent,

Id show strong negative contribution. In the transformed data ,

magnetic, R6/5, Band 7> Ni, gravity and R7/6 are strong positive

contributors, while again Sn and to a lesser extent, Li show strong

negative contribution.

In the analysis of the overall contribution by the absolute

eigenvector matrices, the most signif icant element i s Sn followed by

Ni, magnetic and R5/4, etc. and the least s ignif icant one i s As in the

untransformed data. In the transformed data, again Sn i s the most

s ignif icant element followed by Band 4, L» and so on. THe least

contributor i s Band 7«

291-5 However, the ratios of the least contributors to the highest

contributors are O.58 and O.78, in the untransformed and transformed

data, respectively. This indicates that even the least contributors

in this analysis are s t i l l s ignif icant in multivariate data analysis.

I t was also found that each of the pattern recognition techniques

had i t s own specific advantages in i t s method of application ( as

described in Section 7*6 ) • As a result, the techniques may be

regarded as complementary to each other. Particularly when EDF or any

supervised class i f icat ion technique i s used, i t i s almost essential to

perform the cluster analysis or any other unsupervised class if icat ion

methods to ensure that the training set for the supervised classif ication

constitutes 'natural' and 'homogeneous' groupings.

5.5 I t was indicated that combining geophysical, remote sensing and

geochemical data might constitute better multivariate data set than any

set of data drawn from any single set of properties,either • -physical,

chemical or spectral, of the region of interest, since the training set

test by BAKWRD for varying smoothing parameter (-6") has shown that 8-PGA

score sets are the most stable, followed by 9~roixed data sets and the

l a s t by 8-geochemical data sets. In addition, in a l l cases, the

transformed data are more stable than the untransformed data, indicating

favourability of the transformed data over the untransformed data in the

application of pattern reconition techniques.,

5.6 In the analysis of ISODATA and EDF, discards occurred both at

typical ly low and high values in both untransformed and transformed data.

5.7 I t i s important to note that the conclusions in this section

are tentative since the data used for some of the analyses were not as

good as orig inal ly expected.

<91-6

8.2 Summary of Recommendations for further work

Feature extraction

1.1 As pointed out in Section 4.2.2 , analysis of reduction to the pole and pseudogravity in the frequency domain is preferable to analysis in the space domain.

Regional analysis including lowpass filtering and upward conti-

nuation could also be more conveniently and effectively achieved by

analysis in the frequency domain if edge effects be avoided by an

appropriate extrapolation. This is because for regional study in the

space domain, it is needed to design a large filter operator for effective

analysis, which results in a large amount of computing time.

1.2 In geologically virgin areas, the most effective way of mineral

exploration seems to be using the abundant Landsat data ( particularly

Landsat D if it is commercially available ) for the preliminary regional

study to outline lithological boundaries, structural features, hydrothermal

alteration zones or any botanical symptoms such as clearing or any toxic

effects of vegetation by metal concentrations. This is followed by

ground checking to delimit interesting areas of economic potential for

further effective detailed survey, by either geophysically, geochemically

or combining them together with geological mapping. Thus, Landsat

MSS data would play a very important role in the future study, particularly

in remote areas where the geology is not well-known.

Pattern recognition techniques

2.1 What is optimum and effective methods in applying pattern recogni-

tion techniques for geologically unknown areas ?

(l) Probably the most important thing in the pattern recognition is

selection of variables to a smaller manageable size which can be

conveniently handled by various pattern reconition techniques.

There are two common ways of doing this.

291-7

The f i r s t method i s to study the correlation coefficients in order to

find interrelationships "between the variables. I f any strong

association i s found , then only one of the variables i s selected

and the others are removed as redundancy. Complete independent

variables( the correlations to others are very low ) may also be

removed since these variables may not be representative of the local

geologyfetc. Further analysis of eigen-values and eigenvector

matrix would indicate how: much each of the variables contributes to

the multivariate data set. I f needed, further reduction can be

made by removing those variables whose contribution to the major

components scores are low.

Alternatively, PCA can be applied to yield the principal components

scores, each of which constitutes a certain proportion of the total

variance ( represented in i t s eigenvalue ) . Those components

scores whose variance become insignif icant proportion of the total

variance, could be removed fromthe analysis with minimum loss of

information since these components scores probably consist of noise

contributions as noted by Mancey(l980),

(2) Analysing the frequency distribution of each variable :

The frequency distribution i s supposed to be normally distributed.

I f i t i s not normally distributed , then a transformation technique i s .

applied to transform the data near to- normal.

(3) Applying appropriate unsupervised class if icat ion techniques ( such

as clustering by ISODATA ) : I f the ISODATA package i s used, then

the data set i s tested for various options for optimum results ( as

described in detail in Section 7«4.2(b)) .

(4) Supervised class if icat ion techniques : I f any training set i s

obtained with the unsupervised class i f icat ion , further study with

the supervised class i f icat ion methods(e.g. EDF ) can be pursued for

comparison.

291-1694

2.2 The use of on-line interactive graphic systems to handle large

data sets could improve the efficiency and effectiveness of pattern

recognition analysis methods.

2.3 In pattern reconition studies, i t i s better to combine data sets

having different properties, (e.g. physical, chemical or spectral ), than

to use data sets derived from a single property. This i s because any

s ingle property may not always present enough c lass i f icat ion of the

regional geology or mineralization. For example, Landsat MSS data

used in this study did not provide information on potential mineralization.

2.4 One of the most important but d i f f i cu l t tasks in multivariate

data analysis i s to select an appropriate variable data set for the

best discriminating power as described before. Thus, the use of

more powerful means of setting-up the multivariate data set i s desirable.

The Ridge Regression" technique might be one of the available methods

which could be more fu l l y used.

2 . 5 I t should be noted that in some cases regional trends in the

data might have to be removed in the preparation procedures, since they

might only ref lect deep-seated basement features having l i t t l e relevance

to surface or near-surface geological targets. I n these instances,

the regional variations may res t r i c t the effective dynamic range of the

relevant data.

For th i s reason, shallow re s i s t i v i t y and radiometric data, which

ref lect only immediate surface or near-surface physical features, would

be preferable as geophysical data on which to carry out multivariate

analysis, i f geological mapping or perhaps evaluation of potential

mineralization i s required. In remote sensing, as Lyon (1976) and

Kahle et a l . (1979) pointed out, the thermal infrared bands (7-15Am)

may offer a better basis than the v i s ib le bands for geological studies

since many rock-forming minerals including s i l i ca tes , carbonates, and

sulphates, etc. show much stronger absorption peaks in the infrared

2.91-9

spectral range than in the v is ib le bands, where the distinctions are

not so well made. Therefore, the recently launched Landsat D(4),

which has their detection, would be expected to contribute greatly in

the future study of pattern recognition f ie lds .

The addition of the 1.6 /im and 2.2^um bands in i t would also

allow discrimination between unaltered and hydrothermally altered rocks

better than the visible bands as noted by Pbdwysocki et a l . (1979).

In addition, the HCMM (heat capacity mapping mission) data, covering

v is ib le and near-infrared wavelengths, 0.55 to 1.1 jxm. and the thermal

infrared, 10.5 to 12.5/Am, are also expected to be of geological value

in rock discrimination.

2 £ The latest version of the characteristic analysis (NCHARAN)

allows a more definitive means of data transformation and manipulation

by log ica l processes, this technique could be applied more widely in

the future to geological data studies.

2 y There i s no absolute guarantee that even though the individual

data set may be transformed to the near-normal distribution, the overall

multivariate data derived from such individual data sets would constitute

multivariate normal distribution patterns. I t might, therefore, be

worthwhile to apply transformation of the multivariate data set collect-

ively using suitable transformation techniques such as the power trans?-

form, after the data have been transformed into a single dimension or

made dimensionless by standardization.

2.8 Finally, i t i s recommended that this research be continued in a

variety of geological environments to provide more examples and definite

conclusions. This should be extended to allow the application of

pattern recognition techniques over areas of relatively unknown and

complex geology, possibly to locate new mineral deposits.

292

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315

APPENDIX A

NEW EXTRAPOLATION TECHNIQUES

A.1 Extrapolation in Frequency Domain Using Bicubic Spline Method

As mentioned in Section 3.2.2(b), in practical application of

frequency analysis, truncation of data to a limited number always intro-

duces some oscillations throughout the data, particularly at the

edges of area of data coverage. Therefore, some technique must be

applied to avoid the problem which otherwise would lead to spurious

results.

Extrapolation of data at both ends of a profile or around the

border of a map to be analysed is a method of reducing such edge

effects.

Sato (1954) first attempted to find a solution by adding a

certain number of zeros at ends of the original array. Recently Tsay

(1975) has considered two methods with greater success compared to

Sato's method. Tsay's first method is similar, at least in idea, to

Sato's except in adding an array of certain constants rather than

the zeros in Sato's method. Tsay's second method called Cosine Series

method uses the symmetry of data by adding an array T(kAX), k=-2n,

-2n+1,...,-1, to the left side of the original data to be continuous

and letting

T(-AX) = T(AX)

T(-2AX) = T(2AX)

T(-2nAX) = T(2nAX)

Tsay's first method can be very successful when the data at the edges

316

change smoothly around the background level as demonstrated by his

experiments with the upward continuation.

However, it seems to have difficulties if the data at edges

change rapidly with the sudden truncation often encountered in this

study, for example, the gravity data in the northern part of the

St. Austell granite which is in the middle of the western border of

the study area, and also the magnetic data with strong negative values

in the middle of the northern boundary area. The problem is the

violation of continuity which is required in Fourier analysis.

This may cause some oscillations in the data. In the case of Cosine

Series method, this leads to a large increase in the array size and

thus increases memory requirements and time for two-dimensional array, e.g.

100 * 100 input array will produce 200 * 200 array which can be

computationally prohibitive for practical purposes.

Since the frequency analysis assumes the data to be infinitely

continuous, we need some extrapolation of data in such a way that two

points at the edges should be coincident at zero level to minimize fictitious

oscillations and also no high frequency signal should be introduced by

extrapolation.

The most common way may be the cosine tapering. A more sophisti-

cated method can be the application of the bicubic spline method used

by Ku (1977) in his interpolation.

As shown in Fig. A.1, if we have a data profile with its length

L which has to be extrapolated by extending to lengths and at

both ends in order to make the length of data to the power of two,

which is often required for the computational convenience.

317

yl

Fig. A.l: The original profile and its extrapolations

Taking two values at each end and using the datum value (this is the

background which must be removed prior to any actual data processing,

so that the datum could be always zero), we can draw a profile which

is continuous after extrapolation as shown in Fig. A.2.

datum

yn-l n

Fig. A.2: Continuity of a profile after extrapolation

Following Ku, the simplified version of the cubic polynomial

f(x) = C_ + C. (x-x. ) + C (x-x. J 2 + C/x-x. ,)3 U I i-l L l-l 3 l-l X . , < X < X . 1-1 1 (A.l)

We can approximate the function, in order to derive extrapolating

values at edges.

318

Letting

SL(i-l) = yi-l i-2 x. - x. l-l i-2

SL(i) yi " yi-l x. - x . . 1 1-1

(A.2)

C (i) = tan(0.5(tan 1(SL(i-l)) + tan ](SL(i))))

for i = 2,3, ... ,n (A.3)

and C(1) = c (2)

C (n+1) = C (n).

The remaining coefficients C^ and C^ in Equation (A.l) can then

be obtained by the following relation:

' h 2 h 3 ^

2h. 3hT l I

r c 2u) i

C3(i)

y. - y. - h.C.(i) I l-l l 1 C^i+1) - C (i)

(A.4)

where h. = x. - x. i=l,2,...,n. l l l-l

Let us consider one edge of a profile

SL(1) = yn " yn-l x - x n n-1 V yn-l

if we express the extrapolation interval in terms of unit length,

but at the end of a profile, we force SL(2) = 0, in order to satisfy

the boundary condition at the end of extrapolation (the boundary value

and its slope are zero).

Hence, C^(x'), (^(x1) and C^(x') are readily obtained using

Equation (A.3), Equation (A.4) and the relation in Equation (A.2).

C Q can be obtained using the datum value. As mentioned above,

319

it should be zero in order to minimize fictitious oscillations

called Gibb's phenomenon in the frequency analysis.

A. 2 Extrapolation in Space Domain

As mentioned in Section 3.3.2, in order to avoid the loss of

information at the ends when convolving, it is necessary to extrapolate

the data at ends of a profile by one half of the filter window.

Since the harmonic field can be expressed usually by the

exponential function (see Equation 4.2), it might be reasonable to

consider that the adjacent values in potential fields have exponential

relationships. This justifies the use of the exponential function

for extrapolation. One further thing to be considered is that values

to be extrapolated should.be bounded to certain limits, for example,

maximum and minimum of the data, local highs and lows, or possibly some

portions of the differences of the last two values at the end of the

profile.

The mathematical procedure is as follows:

We may generalize the extrapolation function as

y = a*ebx + c (A. 5)

Rewriting Equation (A.5),

Y = a* eb x (A.6)

where Y = y - c.

Differentiating Equation (A.6),

Y' = ab*ebx = bY (A.7)

Equation (A.7) means that the slope of the function is equal

320

to the original value multiplied by a constant b. Thus the extra-

polation function has the property of the constant rate of slope

changing. This establishes the geometric property which is to be

applied to the limits of the function.

Let the difference at a particular two adjacent values be

equal to d=(y n_ ry n) and

then,

yn-l " yn =

yn ~ V H yn+l yn+2

= b*d

= b2*d (A.8)

' _ = ,N yn+N-l yn+N

where N is the number of points to be extrapolated.

Summing Equation (-A.8) at both sides and letting N tend to

infinity, then the sum S is

S = lim(y - y M ) = d(l+b+b2 + ... + b N) (A.9) yn-l •'n+N

rt-*»

Since the extrapolated function has to be asymptotic to

certain limits, the sum of Equation (A.9) should be a constant. To

satisfy this condition, |b| should be less than 1.

Therefore, S = d -rXr- (A. 10) 1 -b

Letting S = d + rd, which means the maximum amplitude of

extrapolation is some multiple (r) of d, then

d + rd = d -b

b = (A.11) 1+r

321

Accordingly, for different values of r, the extrapolation

values are readily determined.

We evaluate Equation (A.11) further in practical cases.

(i) If r = 1, then we obtain b = y from Equation (A.11). This

means that if we extrapolate the function with a maximum change the same

as the amplitude d, the rate of slope changing should be y, so that every

point to be extrapolated changes half of the difference of the

previous two points in the equal grid system.

(ii) If r = 0, then b = 0 which means no slope changing. This

is equivalent to Tsay's first method which adopts the extrapolation

by adding a certain constant at the ends.

(iii) If r = oo, then b = 1, which means a constant slope. As

mentioned above, in order for the extrapolation function to be bounded,

b should be less than 1, .so that this case is unacceptable.

From the above examples, as long as the curve does not reverse

the trend of the end two values, b must be bounded from 0 to less than

1, while r varies from 0 to a certain constant.

Therefore, by examining the data set the value r may be

reasonably determined for practical applications.

In the#author's experience, r should be less than or equal

to 2 and if the differences between the last two values at the end of

a profile are large, then r = 0.5, might be reasonable. In general

case, it is more reasonable to apply r = 1.

Direct use of an exponential function for extrapolation has

some complications in practical computing due to its property of

divergence at one end. This complication could be removed if we apply

the geometrical property of the exponential fucntion as mentioned

above, so that the computation can be very effective.

322

APPENDIX B

SAMPLING OF LANSAT MSS DATA FOR THE BODMIN MOOR AREA

The orientation of Lansat frame does not coincide to the

UTM (Universal Transverse Mercator) grid which has been used for

the British National Grid (BNG)). Fig. B.1 is a diagram showing the

relationship between the Landsat frame orientation and UTM grid

systems. In the figure, the angle -9- includes the convergence of

the meridians (i.e. the angle at the frame centre between true north

and grid north) and satellite nominal heading.

GN

O x

0 image frame centre GN direction of UTM grid north OX direction of across-track scan motion OY direction of along-track satellite motion

Fig. B.1: Frame orientation

The orientation of frame for the CCT tape image of the Bodmin Moor

area is 0.2800308466 in radians (= 16 degrees) with respect to the

UTM grid.

There is no unique way of sampling of Landsat data in the user's

323

grid system correctly, so that some approximation has to be made

by the trial and error method.

In this study, original data large enough to cover the study

area for gridding have been obtained by averaging 8 by 10 pixels.

At the same time, x and y coordinates with respect to a provisional

origin are recorded in the user's grid system by approximating each

pixel size 57 by 79 square metres and finally rescaling the coordinate

values using the distance between some reference points so as to

apply the gridding package mentioned in Chapter 3.1 for the study area.

The detailed gridding procedures are as follows.

(1) Find the approximate location of the study area in the CCT tape

and outline an area large enough to cover the study area. This may

be done by trials using the grey scale plotting program package

GEOPAK (See Chapter 4).

(2) Take a provisional origin 0'(0,0) at the top left corner

and perform sampling by averaging 8 by 10 pixels in order to approximate

an area of 600 by 600 square metres. At the same time, x and y

coordinate values of each sample are computed with respect to the

origin in the orthogonal axes of the BNG and recorded. This has been

done as follows.

Assuming a pixel size of 57 and 79 in metres in E-W and N-S

directions respectively, then the grid in a profile between adjacent

sample points is Ax = 57* 1 Ocos(-90 and Ay = 57*10sin(-90 , so that the

coordinates of x and y are shifted by Ax and Ay for the immediate next

sampling point in the profile. For the next profile, the origin is moved

by (-79*8 sin(&), 79*8cos($0) and for the rest of the data in

the profile it moves (Ax,Ay) with respect to the previous point as

before, and this procedure is continued for the rest of the data.

324

(3) Print the data with the gray scale map on the line printer

particularly for MSS band 7 since this band shows most clearly shore

lines etc. due to its relative high reflectance of land to water

From the gray scale map and topographic map, find a few control

points by approximation, so that the coordinates of the control points

can be estimated in the gray scale map, in order to calculate the

distances of x and y coordinates separately between two control

points for scaling. The origin of the user's grid system may also

be estimated (X ,Y ).

(4) Compute the scale factor since the distance AB may not be

same in both grid systems. For example, from Fig. B2 assuming the

distance AB is approximated 1 1 . and 1 „, 1 _ in x and y directions xl y 1 x2 y2

both user's grid and gray scale systems respectively, the scale factor

in the x direction is S. = 1 ,/l Likewise the scale factor in fx xl x2 the y direction is S. = 1 ,/l fy yl y2 t

(5) These are used in the gridding program; subtract xq»Y q from

recorded x and y coordinates and scale the coordinate values by

multiplying the scale factors so as to apply the gridding package for

the data of the Bodmin Moor area.

325

Although it may not represent those values in the user's grid

coordinates exactly, the regional Landsat MSS data for a partiuclar

area can be obtained reasonably by the procedures mentioned above.

The correct projection of the grid point values is mainly dependent

upon the accuracy of the control points chosen in the both grid

systems for scaling and estimation of the grid origin 0(X ,Y ).

326

APPENDIX C

CALCULATION OF FILTER OPERATORS

Baranov (1975) has described most of the filter functions

except for the pseudogravity and the horizontal derivative of the

magnetic data which have been derived by the author as part of his research,

The second vertical derivative operator has been computed by

Agarwal's method (1972). Using these filter functions all filter

coefficients have been computed for the analysis of the potential

fields.

C. 1 Calculation of continuations

From Equation (4.5) if we let the x and y coordinates in

discrete form in terms of the grid interval be k and n respectively,

the spectral function can be written as

U = H U(k,n)e k n

kai + n8i ( C . l )

Then, the downward or upward continued at (x,y) to the altitude z

can be expressed, from Equation (4.4-), as

U(x,y,z) = g Z U(k,n) — ]— n 4tt

e _ Y Z + ( k - x ) a i + ( n - y ) e i . d a d 8

-tt (C.2)

Letting x = y = 0, then

U(0,0,z) = £ Z U(k,n) — ^ 4tt

e-YZ+kai+n8idad3 (C. 3)

where U(k,n) = U(k,n,0).

We then have U(0,0,z) = £ Z U(k,n) C(k,n)

where

327

C(k,n) = 4tt"

e-Yz+kai+n8i d a d 3 (C.4)

-tt

Hence C(k,n) represents the filter function of continuations.

If the sign of z is positive, an upward continuation coefficient

is computed, while if the sign is negative, a downward operator is

computed.

C.2 Calculation of reduction to the pole

From harmonic function of Equation (4.4-), the total magnetic

field is expressed as

T(x,y,z) = -X ff T(a,3) e-Yz-xai.-y2i dadf3 (Q 5 ) 4tt J.

where T(a,3) = H T(k,n) e k a i + n 8 1 (C.5.1) k n

The expression for the field at point (x,y,z) reduced to the pole in

the northern hemisphere can be derived from the potential field

Equation (4.4-) by integrating the total magnetic field in the upward

directions of the magnetization vector and inducing field as follows,

Tn(x,y,z) = 1

4tt T(a,S)e-zy-xai-Si £ dadB (C.6) AB

where A = A o Y + A.a. + A„3-3 li 2 I B = V-y + V a. + Vi. 3 1 l 2 l

and a n d 3 r e d-"-rect:'-on c o s i n e s of ^ (unit vector

of inducing field and V (magnetization vector) respectively.

Substituting T*(a,3) by the discrete form of Equation (C.5.1)

and letting x = y = z = 0, then

328

T0(0) = Z T(k,n) n 4tt'

kai+ngi y , J 0 e -j—- dad 3 AB (C.7)

Z T(k,n)C(k,n)

Thus, the filter coefficient C(k,n) = 1 kai+n8i y 477

-it < a,3 < it

AB dad 3

(C. 8)

C.3 Calculation of first vertical derivative

From harmonic function of Equation (4.4-), we obtain the field as

g(X,Y,Z) = 4tt

where g(a,3) £ Z g(k,n)e k n

S(o.B) e-yz-xai-y6i dad0

kai+n3i

(C. 9)

(C.9)'

The vertical derivative is then

ll = _L 4tt

g(a,3)e-yz"xai-yeiY dad3 (C.10)

Replacing g(a,3) in discrete form and letting x = y = z = 0,

then

If - E g s(k'n) 7 2 4tt

kai+n3i , JQ e y dad3

Z g(k,n) C(k,n) n

where C(k,n) = , 2 4tt .. kai+n3i , ,0 e y d a d3

(C.ll)

(C.12)

-7t < a, 3 < 7t

329

C.4 Calculation of horizontal derivative in gravity

In Equation (C.9) if we integrate over z from zero to infinity,

the result is a new representation for the potential;

U(x,y,z) = - 1 4tt'

g(a,3) e ' y - dadB (C.13)

Thus the horizontal derivatives in x and y directions are

9u 9x

9u 9y

4tt

4tt'

f n v -Yz-xai-y3i ia , | g(a,3) e ' — dadS

g(as3)iYz-xCli-y3i dad3

(C.14)

The derivatives at origin (x=y=z=0) are

9u 9 x J 0 4TT2

' 3u 1 9y 0 . 2 4TT

|(a,3) y dad3

g(a,3) y dad3

(C.15)

Therefore,

9u 9x

' 9u 9y

Z g(k,n) C (k,n) n x

Z g(k,n) C (k,n) n y

(C.16)

where

Cx(k,n) = 4TT

C (k,n) = y • 2 4tt

ekai+n3i ai Y

kai+n3i 3i ,.fl e — dad 3 Y

(C.17)

330

C.5 Calculation of lowpass filters

Baranov (1975) has described that the convolution expressed

by the formula

f ( x) = F(x) (C. 18) 7TX

has the effect of replacing the function F(x) with an arbitrary

spectrum by a function f(x) whose spectrum does not exceed tt/Q in

absolute value.

The discrete form of the 2-dimensional case f(x,y) at a single

point x = y = 0 is

£(0.0) - 2 2 F(kq,nq) sin,(l,kq/Q) * " ° < W Q > . kTT nTT k n

= Z Z F(kq,nq) * C(k,n) (C.19) k n

where the lowpass filter coefficient

C(k,n) = sin(fTkq/Q) « ,in(imq/Q) ( , 5 TTk nn

and q is the grid spacing.

In the study, Q = lOq has been chosen for calculation of the

lowpass filter operator. This is equivalent to cut-off frequency

about 0.3 cycles/data interval (600m). For highpass filtering, no

calculation of the filter operator has been made. Instead, the residuals

of lowpass filtering have been computed to represent the highpass

filtering because the residual should preserve all spectra higher than

tt/Q in the data.

331

C.6 Derivation of pseudogravity

The general expressions of the gravimetric potential and

magnetic potential are

U = G f a - dv (C.21) Jv y

V = J JV(—)dv (C.22) v ^

respectively, where G is the gravitational constant, O the density

contrast and J the magnetization vector.

From the above two equations, we can write the following

relation:

-JVU = GQV (C.23)

Letting v the unit vector .of magnetization and |j|= Ga, Equation (C.23)

then becomes

-v VU = V. (C.24)

If T^ is defined as a component of the magnetic field in a

direction ofunit vector X

TX(M) = - X VV(M) = - ^ (C. 25)

Combining Equation (C.24) and Equation (C.25)

332

Fig. C.l Geometry of the Magnetic Field

In the meantime, the gravity field g^ is

3u 80 ~ 9z (C.27)

Differentiating Equation (C.27) with respect to X and V, and

Equation (C.26) with respect to z and equating both, then

d2g 0 dXdV

8T; IT (C.28)

From the harmonic function of Equation (C.5),

V M ) - - T 4tt

T(a,3)e" Y z- x a i- y 3 i dadB (C.29)

If we assume the directions of vectors X and V to be

downward as in common in the northern hemisphere, Equation (C.28) must

333

be integrated in directions opposite to the vectors since the half

lines in Fig. (C.l) do not cut any magnetic sources.

It will be defined by the references s and t as

OQ = OM + MQ = OM + Xs + Vt (C.30)

If (x,y,z) are the coordinates of M, those of the variable points

Q will be

x + X s + Vjt, y + X^s + v2t, z + X^s + v3t (C.31)

Therefore, T T

!o(x,y;z) - / " / " A J T f(«.6)e- y z- x a i- y 6 i -As-Bt 4tt 0 0 -tt

where A = X.a. + X0B. + X_y li 2 I 3 B = v.a. + V-6. + v y Ii 2 i 3

TdadBldsdt

(C.32)

(C.32.1)

Since |e ^Sds = J- and 0 0

Equation (C.32) becomes

- uu f -Bt 1 e dt -

g0(x,y,z) = — 2 47T

T(a )6)e- Y Z" x a i- y 6 ii dadg A D

(C.33)

The spectral function T(a,8) is given in Equation (C.5.1). Therefore,

the pseudogravity at x = y = z = 0 is

g0(0) = £ Z T(k,n) 1 k n 4tt'

kai + n3i Y „ ,o a I d a d 3

= g Z T(k,n) C(k,n) (C.34)

where the pseudogravity filter operator

1 C(k,n) = 4tt'

kai + n3i Y .jo AB d0tdB

- I T < a, 3 < TT (C.35)

334

Practical calculation of pseudogravity

The .pseudogravity filter operator using Equation (C.35) can be

easily obtained by applying the same notation as Baranov (1975)

used in computation of the reduction to the pole operator. „/ r\ \ kOti + n(3i Y Letting F(a,3) = e , then mtegrati AB on

Equation (C.35) becomes

1 C(k,n) = 4tt

F(a,B) + F(-a,B) + F(a,-B) + F(-a,-3)]dad3

0 < a,3 < it (C.36)

This shows that the function C(k,n) is real.

Letting again G(a,3) = Re[(F(a,3) + F(a,-B)

C(k,n) = 1 27T

G(a,3) dad3

(C.37)

(C.38)

Referring Fig. C.2 /rr jx

6 ty

C(k,n) = 1 2tt

2tt

2TT

da 0 0 rTT r 3

d3 0 0

tt >a da

0 0

G(a,3)d3

G(a,B)da

[G(a,3)+G(3,a)]d3

-7T IT a

(C.39) Fig. C.2: Reduction of the region of integration to a triangle

Letting 3 = Otu, the variable u then varies from 0 to 1 and

Equation (C.39) becomes

335

/•TT

C(k,n) = 2TT

z-1 ada

0 0 1 /-TT

du 2TT

[G(a,au)+G(au,a)]du

[G(a,au)+G(au,a)]ada (C.40)

In Equation (C.37) the two terms differ only in the sign of u,

thus Equation (C.40) can be written as /•l /-7T

C(k,n) = 1 2tt

du

0 0 (k+nu)a.

F(a,au) = e

Re[F(a,au)+F(au,a)]ada

i /1+u2 P1 Qli a D11D12

(C.41)

(C.42)

F(au,a) = e ( k u + n ) a i / H u 2 P2 Q2i

a D D 21 22

(C.43)

where

12

A32(l-ru2) + U 1 + X 2 u ) 2

2 2 2 V 3 (1+u ) + (v +\>2u)

A3v3 (1+U2) - (A^A

X1 D21 D 22

[(A]+A2u)V3 + (V]+V2)A3] /1+U' A32(1+U2) + (Aju+ A2)2

2 2 2 V3 (1+u ) + (v]u+v2) A3V3(1+U2) -(A1U+A2)(V1U+V2) [(AiU+A2)v3 + (vlU+v2)A3] /l+u'

(C.43)'

In each of Equation (C.42) and Equation (C.43) appears an

exponential factor e xai with T = T.J = k + nu in Equation (C.42)

T = T 2 = ku + n in Equation (C.43) /•TT

Then 2TT

xai da = cf(-r) + i sf(x) (C.44)

336

where Cf(x) =

and Sf(t) =

sin TTT 2tt2T l-cos TTT 2TT 2T

(C.44•0

when TTT < 0.001, Cf(x) (1 — 2TT 6

Sf ( T ) = ± (X - J L . ) ( C . 4 4 . 2 )

where X = TTT.

C.7 First horizontal derivative of the magnetic field

As mentioned before (see Equation (C.25)), the magnetic field,

is related to the potential field according to the following relationship,

X - - « (C.45)

taken in a given direction A.

Finding the partial derivatives in x and y directions we have

magnetic field components

T = - — x 8x

T y 9y

(C.46)

are Thus, the horizontal derivatives in each direction x and y

m i _ _ 92V x V 1

dx , . (C.46.1) T • -

337

As shown in (C.6), Equation (C.45) leads to the solution:

oo V = J T(x+A|t, y+X2t, z+X3t)dt (C.47)

where K x + A ^ , y+A2t, z+X3t) = — -4TT

n\ -Yz-xai-y(3i -At, ,n T(a,3)e ' y dad3

(C.47.1)

and A is the same conventional defined in (C.32.1)

the spatial function T(a,3) is

T(a,3) = Z T(k,n)e n kai+n3i (C.48)

Substituting Equation (C.48) in Equation (C.47.1)and integrating

over t, then we obtain expressions for the coefficients to calculate

the derivatives at origin ( x = y = z = 0 ) .

VOc.n) 4TT

T '(k,n) = Y 4TT

T(k.n) e k a i + n e i a 2 dad3/A

T(k,n) a 2 d a d 3 / A

(C.49)

-TT < a , 3 < tt

Kanasewich and Agarwal (1970) has shown the horizontal

gradient along any declination D to be

T'(H) = T ' sinD + T 'cos D x y

Thus, the horizontal derivative is

1

(£.50)

T'(H) = 4tt'

T(k,n) e k a i + n 8 1 ( ai sinD+3icosD)dad3/A

Therefore, T'(H) = Z Z T(k,n) C(k,n) k n

(C.51)

(C.52)

338

where C(k,n) = ekai+nBi (iasinD+iBcosD)dadB/A (C.53) 4tt'

-7t < a , b < it .

Practical calculation of the horizontal derivative

Again, using the same notation as before, the practical

computation of the filter operator can be carried out without

difficulties. kai+nBi 2 Letting C (k,n) = — X t _ JL

and

4tt J.

1 C (k,n) = _ y / 2 J 4tt

a /A dadB

kai+nBi n2

(C.54)

B /A dadB (C.55)

Let us consider Cx(k,n). First the region of integration can be

reduced to the square 0 < a, B < 7t.

•kai+nBi a^ Letting F(a,B) = e

Whence the integral C (k,n) becomes

C (k,n) = -hr x . 2 4tt [F(a,B)+F(xx,-B)+F(-a,B)+F(a,-B)JdadB

0 < a,B < 7T.

Since the above function is real, we will set again

G(a,B) = Re[F(a,3)+F(a,-B)] (C.56)

Thus, the integration Equation (C.54) is

Cx(k,n) = 2tt

G(a,B)dadB (C.57)

The small square region of integration will be divided into two

triangles by the bisector B = a, letting 3 = au.

339

We can derive the integration as follows: rl\ /-I

Cx(k,n) = - L 2tt J

ada [G(a,au)+G(au,a)]du

2TT'

0 0 RL /-TT

du

0 0

[G(a,au)[G(au,a)]ada

If we integrate from u = -1 to 1, we get, by referring

Equation (C.57)

and

,.1 C (k,n) = — ^

2tt du Re[F(a,au)+F(au,oO jada

-1

2 \ (k+nu)ai (-(X +X u)i + X0(/l+u )a F(a,au) = e 1 2 3

(X /l+u + (Xj+X u) (k+nu)aif . v e (P-Q]i)a

D 1 1

(C.58)

(C.59)

2 2 2 where D^ = X^ (l+u ) + (X^+X^)

P = X3/l+u2

Q1 = (X1+X2u)

In the same way,

F(au,a) =e (ku+n)ai a (X3/l+u + X1ui+X2i)

(C.60)

(ku+n) = (P-Q2i)a

22

2 2 2 where D 2 2 = X3 (l+u ) + (X^u+X^

P = X3 /l+u2

Q2 = A]U + X2

340

In Equation (C.59) and Equation (C.60), appears an exponential TOti

factor e with T = k+nu in Equation (C.59)

T = t 2= ku+n in Equation (C.60)

The integration over a in Equation (C.58) is then reduced aTT

1 2,2

TOti ? e a da = C f ( x ) + i S f ( x )

2 3 2 where Cf(x) = sinTTx/(2x) + costtx/(ttx ) - s in(7TT)/(T TT )

2 3 2 S f ( x ) = - COSTTX/2 + sinTTX/ (x TT) + (cosTTX-1 ) / ( X TT )

Thus Cx(k,n) = J $(u)du (C.61) -1

P C f ( x ) + Q ] S f ( x ) P 2 C f ( x ) + Q 2 S f ( x ) where $(u) = +

11 22

likewise, referring to Equation (C.56) Cx(k,n) = C^(n,k)

C (k,n) = y -1

where

$1(u)du (C.62)

R.u 2 Cf(x)+u 2 S .S f (x ) R_u 2 C f ( x ) + S.u 2 S f ( x ) V » > - , „ •

= u $(u)

Finally,

C(k,n) = C (k,n) sinD + C (k,n)cosD (C.63) x y

341

C.8 Second vertical derivative

The basic computational principle of the second vertical

derivative applied in this study starts from the 2-dimensional

Fourier transform expressed as follows. oo

G(u,v) = |J g(x,y)e-2,,i(ku+nv)dkdn (C.64) —CO

Grau (1966) has shown that if G(u,v) is circularly symmetrical,

Equation (C.64) can be written as oo

G(R) = 27T J rg(r) JQ(27TrR)dr (C.65) 0

where u = R cosOt

v = R sina

x = r cos3

y = r sin3

and OL and 3 are angles of radial vectors with respect to positive u

and x axes in frequency and space domains, respectively. J ^ ^ h t R ) is the

Bessel function of zero order defined as

R 2TT „ . „ T fo I 27TirR COSU , f„ rr\ J (2nrR) = — — e du (C.66) o 2TT J

0 Agarwal and Lai (1972) have used this expression to design second

vertical derivative filter operators using the above notation. The

discrete form of Equation (C.65) as shown by Agarwal and Lai

is M

G(p) = 4y V w(r ) J Cpr ) (c.67) 2 (__, n 0 K n S r=0

where s is the grid spacing.

342

In case of second vertical derivative, G(p) is

G(p) = p2 exp(-Ap2) (C.68 )

2 2

where p = /u +v is the radial frequency, and u and v angular

frequencies in x and y directions, respectively. X is the

weighting factor.

Agarwal and Lai (op.cit.) have computed for the case of

X = 0.1, 0.15, 0.20 and 0.23 using Rational Approximation (Agarwal and

Lai, 1971). As described by Agarwal, the application of appropriate

X depends on the feature of interest and noise levels in the data.

For example, if X is smaller, it might be good for low noise level

of data to extract smaller features, while if X is larger it can be

applied to the data with some noise level for larger features.

Most of derived second vertical derivative filter operators

are bound between X = 0.1 and 0.3. The filter operator of Rosenbach

(1953) might be equivalent to that of X = 0.1 and the filter operator

of Elkin (1951) to that of X = 0.23.

The author has calculated the derivative coefficients

for X = 0.1 to X = 0.23. Both coefficients by Agarwal et al and the

author are listed in the Table C.l

Small differences in both (i) and (ii) in Table C.l may be

due to different method of calculation of the Bessel function of the first

kind (Jq(x))given in Equation (C.65) The author has used NAG Library

subroutine in the Imperial College collection and Agarwal and Lai has

used the Rational Approximation Method. Details of the Rational

Approximation Method can be found in Agarwal and Lai (1971).

Although the NAG routine is not listed here, the general

mathematical approximation is described.

Table C.l: Weights for the different ring averages in various second vertical derivatives.

(i) Derived wei ghts

radii R n A = = 0.05 0

Smoothing Parameter .10 0.15 0.20 0 .23

0 +4 .81396 +2. 72876 +1.68768 +1.12599 +0. 90856 S -6 .19426 -1 . 27348 +0.45395 +0.99468 + 1 . 09571

s/2 + 1 .86564 -1 . 64542 -2.15992 -1.87427 -1 . 64024 s/4 -1 .00698 +0. 44528 +0.27892 +0.02197 -0. 04013 S/5 +0 .55962 -0. 27584 -0.28208 -0.27377 -0. 31599 S/8 -0 .05443 +0. 02908 +0.02698 +0.00960 -0. 00216 s/ io +0 .01040 -0. 00568 -0.00552 -0.00419 -0. 00575

(ii) Agarwal and Lai's

radii R n A = = 0.05 0

Smoothing Parameter .10 0.15 0.20 0 .23

0 +4. .81777 +2 .72648 +1.68540 + 1 .12441 +0 .90749 S -6. .21198 -1 .26288 +0.46459 + 1.00206 + 1 .10074

S/2 + 1 . .891 18 -1 .66072 -2.17527 -1.88491 -1 .64754 S/4 -1 . .03145 +0 .46357 +0.29728 +0.03473 -0 .03128 S/5 +0. .58084 -0 .29126 -0.29486 -0.28267 -0 .32221 S/8 -0. .05735 +0 .03085 +0.02875 +0.01084 -0 .00126 s/ io +0. .01100 -0 .00604 -0.00589 -0.00446 -0 .00594

344

*NAG subroutine for approximation of the Bessel function of the

first kind Jn(x)•

The routine is based on the three Chebyshev expansions;

since jq (-x ) = j^Cx), so the approximation need only consider x > 0

(1) for 0 < x < 8

r ' 2

J-(x) = ) a T (t) , with t = 2(|-) - 1 (C.69 ) 0 L. r r 8 r=0

(2) for x > 8

j (x) = / T O / P ( x) cos(x - y) - Q(x) sin(x - y) 0 / n x ^ O 4 4 (C.70 ) i

where Pn(x) = ) b T (t) 0 L, r r r=0 2

with t = 2(-) - 1 x and Q (x) = - V c T (t) o x (_ r r

r=0

(3) for x near zero, Jq(x) ^ L (C.71 )

345 Table C.2 Filter Operators.

First Vertical Derivative - . 0 0 0 1 0 - . 0 0 0 2 6 - . 0 0 0 6 3 - . 0 0 0 4 0 . 0 0 5 4 6 - . 0 0 0 4 0 - . 0 0 0 6 3 - . 0 0 0 2 6 . 0 0 0 1 0 - . 0 0 0 2 6 - . 0 0 0 8 4 - . 0 0 1 6 4 - . 0 0 4 6 1 - . 0 2 5 0 3 - . 0 0 4 6 1 - . 0 0 1 6 4 - . 0 0 0 8 4 - . 0 0 0 2 6 - . 0 0 0 6 3 - . 0 0 1 6 4 - . 0 0 5 4 8 - . 0 0 7 9 1 . 0 3 7 8 3 - . 0 0 7 9 1 - . 0 0 5 4 8 - . 0 0 1 6 4 - . 0 0 0 6 3 - . 0 0 0 4 0 - . 0 0 4 6 1 - . 0 0 7 9 1 - . 0 7 7 8 1 - . 4 5 0 5 6 - . 0 7 7 8 1 - . 0 0 7 9 1 - . 0 0 4 6 1 - . 0 0 0 4 0

. 0 0 5 4 6 - . 0 2 5 0 3 . 0 3 7 8 3 - . 4 5 0 5 6 2 . 1 8 9 7 5 - . 4 5 0 5 6 . 0 3 7 8 3 - . 0 2 5 0 3 . 0 0 5 4 6 - . 0 0 0 4 0 - . 0 0 4 6 1 - . 0 0 7 9 1 - . 0 7 7 8 1 - . 4 5 0 5 6 - . 0 7 7 8 1 - . 0 0 7 9 1 - . 0 0 4 6 1 - . 0 0 0 4 0 - . 0 0 0 6 3 - . 0 0 1 6 4 - . 0 0 5 4 8 - . 0 0 7 9 1 . 0 3 7 8 3 - . 0 0 7 9 1 - . 0 0 5 4 8 - . 0 0 1 6 4 - . 0 0 0 6 3 - . 0 0 0 2 6 - . 0 0 0 8 4 - . 0 0 1 6 4 - . 0 0 4 6 1 - . 0 2 5 0 3 - . 0 0 4 6 1 - . 0 0 1 6 4 - . 0 0 0 8 4 - . 0 0 0 2 6 - . 0 0 0 1 0 - . 0 0 0 2 6 - . 0 0 0 6 3 - . 0 0 0 4 0 . 0 0 5 4 6 - . 0 0 0 4 0 - . 0 0 0 6 3 - . 0 0 0 2 6 . 0 0 0 1 0

Second vertical derivative (<<=0 .1; normalized) 0 0 - . 0 0 1 1 9 0 - . 0 0 1 1 9 0 0 0 . 0 2 1 6 3 - . 0 9 4 4 3 . 4 7 8 2 3 - . 0 9 4 4 3 . 0 2 1 6 3 0 - . 0 0 1 1 9 - . 0 9 4 4 3 - . 7 1 9 7 7 - . 6 0 3 6 1 - . 7 1 9 7 7 - . 0 9 4 4 3 - . 0 0 1 1 9 0 . 4 7 8 2 3 - . 6 0 3 6 1 4 . 0 5 9 0 5 - . 6 0 3 6 1 . 4 7 8 2 3 0 - . 0 0 1 1 9 - . 0 9 4 4 3 - . 7 1 9 7 7 - . 6 0 3 6 1 - . 7 1 9 7 7 - . 0 9 4 4 3 - . 0 0 1 1 9 0 . 0 2 1 6 3 - . 0 9 4 4 3 . 4 7 8 2 3 - . 0 9 4 4 3 . 0 2 1 6 3 0 0 0 - . 0 0 1 1 9 0 - . 0 0 1 1 9 0 0

First horizontal derivative(gravity) - . 0 0 0 1 7 - . 0 0 0 4 1 - . 0 0 0 8 6 - . 0 0 0 4 6 . 0 4 2 0 5 - . 0 0 0 4 6 - . 0 0 0 8 6 - . 0 0 0 4 1 - . 0 0 0 1 7 - . 0 0 0 4 1 - . 0 0 1 1 4 - . 0 0 1 8 6 - . 0 0 4 2 0 - . 0 1 7 0 7 - . 0 0 4 2 0 - . 0 0 1 8 6 - . 0 0 1 1 4 - . 0 0 0 4 1 - . 0 0 0 8 6 - . 0 0 1 8 6 - . 0 0 4 9 8 - . 0 0 5 4 0 . 1 4 5 5 4 - . 0 0 5 4 0 - . 0 0 4 9 8 - . 0 0 1 8 6 - . 0 0 0 8 6 - . 0 0 0 4 6 - . 0 0 4 2 0 - . 0 0 5 4 0 - . 0 3 5 3 9 . 1 0 2 4 6 - . 0 3 5 3 9 - . 0 0 5 4 0 - . 0 0 4 2 0 - . 0 0 0 4 6

. 0 4 2 0 5 - . 0 1 7 0 7 . 1 4 5 5 4 - . 1 0 2 4 6 0 - . 1 0 2 4 6 . 1 4 5 5 4 - . 0 1 7 0 7 . 0 4 2 0 5 - . 0 0 0 4 6 - . 0 0 4 2 0 - . 0 0 5 4 0 - . 0 3 5 3 9 . 1 0 2 4 6 - . 0 3 5 3 9 - . 0 0 5 4 0 - . 0 0 4 2 0 - . 0 0 0 4 6 - . 0 0 0 8 6 - . 0 0 1 8 6 - . 0 0 4 9 8 - . 0 0 5 4 0 . 1 4 5 5 4 - . 0 0 5 4 0 - . 0 0 4 9 8 - . 0 0 1 8 6 - . 0 0 0 8 6 - . 0 0 0 4 1 - . 0 0 1 1 4 - . 0 0 1 8 6 - . 0 0 4 2 0 - , . 0 1 7 0 7 - . 0 0 4 2 0 - . 0 0 1 8 6 - . 0 0 1 1 4 - . 0 0 0 4 1 - . 0 0 0 1 7 - . 0 0 0 4 1 - . 0 0 0 8 6 - . 0 0 0 4 6 . 0 4 2 0 5 - . 0 0 0 4 6 - . 0 0 0 8 6 - . 0 0 0 4 1 - . 0 0 0 1 7

First horizontal derivative(magnetic;D=350* 1=68° ) . 0 0 1 0 8 , .00035 - . 0 0 0 5 2 - . 0 0 3 8 1 - . 0 0 4 5 4 - . 0 0 5 0 1 - . 0 0 2 0 7 - . 0 0 0 3 8 . 0 0 1 3 3 . 0 0 0 0 9 ,V0663 - . 0 2 4 8 0 - . 0 1 0 3 2 - . 0 0 0 6 7 . 0 0 0 9 9 . 0 1 2 5 2 . 0 0 8 4 0 . 0 0 0 6 0 - . 0 0 3 3 3 . 0 1 4 1 4 . 0 2 6 9 8 - . 0 0 2 1 4 - . 0 6 7 0 1 - . 0 3 3 3 2 . 0 3 5 9 5 - . 0 1 0 1 1 - . 0 0 2 0 4 . 0 1 4 7 2 - . 0 3 8 3 4 . 0 1 8 4 4 . 0 8 4 8 5 - . 1 3 1 5 7 . 1 1 9 0 8 . 0 2 8 0 2 - . 0 4 5 7 1 . 0 1 4 0 1 - . 0 1 9 2 4 . 0 1 5 5 4 - . 0 8 4 6 8 - . 1 2 3 6 8 . 5 3 9 9 1 - . 2 0 2 2 1 - . 1 1 4 5 4 . 0 1 2 8 9 - . 0 2 1 0 7 . 0 1 4 5 1 - . 0 4 0 3 9 . 0 1 9 5 3 . 0 8 3 1 0 - . 1 2 1 4 5 . 1 0 6 0 1 . 0 3 4 1 6 - . 0 4 7 8 2 . 0 1 4 5 3 - . 0 0 2 6 5 . 0 0 8 1 5 . 0 2 8 8 6 - . 0 0 7 6 5 - . 0 6 3 2 2 - . 0 3 1 3 5 . 0 3 0 9 2 - . 0 1 6 5 4 - . 0 0 2 0 8 . 0 0 0 0 3 . 0 0 7 1 0 - . 0 2 2 0 9 - . 0 0 9 1 3 - . 0 0 0 2 8 . 0 0 1 5 4 . 0 1 6 2 4 . 0 0 7 3 8 . 0 0 0 7 9 . 0 0 1 1 6 . 0 0 0 1 1 - . 0 0 0 2 4 - . 0 0 4 2 3 - . 0 0 4 3 2 - . 0 0 4 4 1 - . 0 0 2 5 8 - . 0 0 0 3 0 . 0 0 1 1 8

Lowpass filter (Q=10) . 0 0 0 5 6 . 0 0 1 4 0 . 0 0 2 5 1 . 0 0 3 6 2 . 0 0 4 4 4 . 0 0 4 7 4 . 0 0 4 4 4 . 0 0 3 6 2 . 0 0 2 5 1 . 0 0 1 4 0 . 0 0 0 5 6 . 0 0 2 5 1 . 0 0 1 4 0 . 0 0 2 9 5 . 0 0 4 8 4 . 0 0 6 6 7 . 0 0 8 0 1 . 0 0 8 4 9 . 0 0 8 0 1 . 0 0 6 6 7 . 0 0 4 8 4 . 0 0 2 9 5 . 0 0 1 4 0 . 0 0 2 5 1 . 0 0 4 8 4 . 0 0 7 5 9 . 0 1 0 2 0 . 0 1 2 0 8 . 0 1 2 7 7 . 0 1 2 0 8 . 0 1 0 2 0 . 0 0 7 5 9 . 00484 . 0 0 2 5 1 . 0 0 3 6 2 . 0 0 6 6 7 . 0 1 0 2 0 . 0 1 3 5 2 . 0 1 5 8 9 . 0 1 6 7 5 . 0 1 5 8 9 . 0 1 3 5 2 . 0 1 0 2 0 . 0 0 6 6 7 . 0 0 3 6 2 , 00444 . 0 0 8 0 1 . 0 1 2 0 8 . 0 1 5 0 9 . 0 1 8 5 9 . 0 1 9 5 8 . 0 1 0 5 9 . 0 1 5 8 9 . 0 1 2 0 8 ,00801 . 0 0 4 4 4 , 00474 . 0 0 8 4 9 . 0 1 2 7 7 . 0 1 6 7 5 . 0 1 9 5 8 . 0 2 0 6 0 . 0 1 9 5 8 . 0 1 6 7 5 . O l 2 7 7 ,00849 . 0 0 4 7 4 . O l 2 7 7 ,00444 . 0 0 8 0 1 . 0 1 2 0 8 . 0 1 5 8 9 . 0 1 8 5 9 . 0 1 9 5 8 . 0 1 8 5 9 . 0 1 5 8 9 . 0 1 2 0 8 00801 . 0 0 4 4 4 0 0 3 6 2 . 0 0 6 6 7 . 0 1 0 2 0 . 0 1 3 5 2 . 0 1 5 8 9 . 0 1 6 7 5 . 0 1 5 8 9 . 0 1 3 5 2 . 0 1 0 2 0 0 0 6 6 7 . 0 0 3 6 2 00251 . 0 0 4 8 4 . 0 0 7 5 9 . 0 1 0 2 0 . 0 1 2 0 8 . 0 1 2 7 7 . 0 1 2 0 8 . 0 1 0 2 0 . 0 0 7 5 9 0 0 4 8 4 . 0 0 2 5 1 0 0 1 4 0 . 0 0 2 9 5 . 0 0 4 8 4 . 0 0 6 6 7 . 0 0 8 0 1 . 0 0 8 4 9 . 0 0 8 0 1 . 0 0 6 6 7 . 0 0 4 8 4 0 0 2 9 5 . 0 0 1 4 0 0 0 0 5 6 . 0 0 1 4 0 . 0 0 2 5 1 . 0 0 3 6 2 . 0 0 4 4 4 . 0 0 4 7 4 . 0 0 4 4 4 . 0 0 3 6 2 . 0 0 2 5 1 0 0 1 4 0 . 0 0 0 5 6

346

Upward continuation (h=2) 00010 . 0 0029 . 0 0058 . 00096 . 00129 . 00144 . 00129 . 0 0096 . 00058 00029 . 00010 000 29 . 00071 . 00142 . 00239 . 00332 . 00371 . 00332 . 00239 . 0 0142 00071 . 00029 00050 . 00142 . 00298 . 00540 . 00817 . 00952 . 00017 . 00540 . 0 0298 001 42 . 00050 00 096 . 0 0239 . 00540 . 01107 . 01901 . 02344 . 01901 . 0 1 1 0 7 . 00540 00239 . 0 0096 00 I 29 . 0 0332 . 0 0817 .01901 . 03916 . 05310 . 03916 . 01901 . 00817 00332 . 0 0129 001 44 . 00371 . 00952 . 0234 4 . 05318 .07611 . 05318 . 02344 . 00952 00371 . 00144 00129 . 00332 . 0 0817 .01901 . 03916 . 05318 . 0 3916 . 01901 . 00817 00332 . 0 0129 00096 . 0 0239 . 00540 . 01107 . 01901 . 02344 . 01901 . 0 1 1 0 7 . 00540 00239 . 0 0096 00050 . 00142 . 0 0298 . 00540 . 00817 . 00952 . 00817 . 0 0540 . 0 0 2 9 8 00142 . 00058 00029 . 00071 . 0 0142 . 00239 . 00332 . 00371 . 00332 . 0 0 2 3 9 . 0 0142 00071 . 0 0029 00010 . 0 0029 . 0 0058 . 00096 . 00129 . 00144 . 00129 . 0 0 0 9 6 . 0 0 0 5 8 00029 . 00010

Upward continuation (h=3) 00007 . 00020 . 00039 . 00065 . 0 0092 . 0 0114 . 00122 . 0 0114 . 00092 00065 . 00039 . 00020 . 00007 00020 . 00045 . 00086 . 00140 . 00201 . 00252 . 00272 . 00252 . 00201 00140 . 00086 . 00045 . 00020 00039 . 00086 . 00162 . 00271 . 0 0403 . 0 0 5 1 9 . 00567 . 0 0 5 1 9 . 00403 00271 . 00162 . 00086 . 00039 00065 . 00140 . 00271 . 00477 . 0 0 7 4 9 . 0 1 0 1 5 . 01132 . 0 1 0 1 5 . 00749 00477 . 00271 . 00140 . 00065 00092 . 00201 . 00403 . 0 0749 . 0 1 2 6 7 . 0 1 8 3 6 . 02106 . 0 1 8 3 6 . 0 1267 00749 . 00403 . 00201 . 00092 00114 . 00252 . 00519 . 0 1015 . 0 1 8 3 6 . 02851 . 03373 . 02851 . 01836 01015 . 00519 . 00252 . 00114 00122 . 00272 . 0 0567 . 01132 . 0 2106 . 0 3373 . 04050 . 03373 . 02106 01132 . 00567 . 00272 . 00122 00114 . 00252 . 00519 . 01015 . 0 1836 . 02851 . 03373 . 02851 . 01836 01015 . 00519 . 00252 . 00114 00092 . 00201 . 00403 . 00749 . 0 1 2 6 7 . 0 1836 . 02106 . 0 1836 . 01267 00749 . 00403 . 00201 . 00092 00065 . 00140 . 00271 . 0 0477 . 0 0 7 4 9 . 0 1015 . 01132 . 0 1015 . 00749 00477 . 00271 . 00140 . 00065 00039 . 00086 . 00162 . 00271 . 0 0403 . 0 0 5 1 9 . 00567 . 0 0519 . 0 0403 00271 . 00162 . 00086 . 00039 00020 . 00045 . 00086 . 00140 . 00201 . 0 0252 . 00272 . 00252 . 00201 00140 . 00086 . 00045 . 00020

00007 . 00020 . 00039 . 0 0065 . 0 0092 . 0 0114 . 00122 . 00114 . 00092 00065 . 00039 . 00020 . 0 0007

Downward continuation (h=-l) . 00107 .00444 . 01252 . 05022 . 24866 . 05022 . 01252 . 00444 . 00107

.00444 ,01179 ,03900 . 12843 ,68208 .12843 ,03900 , 01179 ,00444

,01252 .03900 .09982 .37977 .76083-.37977 . 09982 . 03900 . 01252

,05022 ,24866 ,05022 . 01252 - . 0 0 4 4 4 . 0 0107

. 12843 - , , 68208 . 12843 - . 0 3 9 0 0 . 0 1179 - . 0 0 4 4 4

.37977 1. , 76083 . 37977 . 09982 - . 0 3 9 0 0 . 01252

1, . 20809- 5 < . 63189 1, . 20809 - . 3 7 9 7 7 . 1 2843 - . 0 5 0 2 2

•5, . 631891 5 < .59474-•5 < . 63189 1 . 7 6 0 8 3 - . 6 8 2 0 8 . 24866

1, . 20809- 5 i , 63189 1 . 2 0809 - . 3 7 9 7 7 . 1 2843 - . 0 5 0 2 2

. 37977 1, . 76083 - . 3 7977 . 09982 - . 0 3 9 0 0 . 01252

. 12843 _ , , 68208 . 1 2843 - . 0 3 9 0 0 . 0 1179 - . 0 0 4 4 4

- . 05022 . 24866 - . 0 5022 . 01252 - . 0 0 4 4 4 . 00107

347

Reduction to the pole(21*21:D=350°, 1=68° )

. 0 0 0 0 1 _ . 0 0 0 0 1 _ . 0 0 0 0 4 . 0 0 0 0 4 . 0 0 0 0 9 - . 0 0 0 0 7 - . 0 0 0 1 7 - . 0 0 0 0 9 - . 0 0 0 2 5 . 0 0 0 0 4 - , . 0 0 1 1 5 - . 0 0 0 7 0 . 0 0 0 0 2 - . 0 0 0 1 1 . 0 0 0 0 5 - . 0 0 0 0 1 . 0 0 0 0 6 . 0 0 0 0 3

. 0 0 0 0 4 . 0 0 0 0 2 . 0 0 0 0 1 . 0 0 0 0 1 - - . 0 0 0 0 4 - . 0 0 0 0 5 - . 0 0 0 1 1 - . 0 0 0 1 0 - . 0 0 0 2 1 . 0 0 0 1 5 - . 0 0 0 3 3 . 0 0 0 1 2

. 0 0 0 5 5 . 0 0 1 3 3 . 0 0 0 7 3 - . 0 0 0 2 6 . 0 0 0 0 8 - . 0 0 0 0 5 . 0 0 0 1 2 . 0 0 0 0 5 . 0 0 0 1 1

. 0 0 0 0 6 . 0 0 0 0 6 . 0 0 0 0 3

. 0 0 0 0 4 - . 0 0 0 0 5 - . 0 0 0 1 2 - . 0 0 0 1 1 - . 0 0 0 2 5 - . 0 0 0 2 0 - . 0 0 0 4 2 - . 0 0 0 2 6 - . 0 0 0 6 3 - . 0 0 0 0 1 - . 0 0 2 4 9 - . 0 0 1 5 4 . 0 0 0 0 6 - . 0 0 0 1 8 . 0 0 0 2 0 . 0 0 0 0 8 . 0 0 0 2 4 . 0 0 0 1 4

. 0 0 0 1 7 . 0 0 0 0 9 . 0 0 0 0 7 . 0 0 0 0 4 - . 0 0 0 1 1 - . 0 0 0 1 1 - . 0 0 0 2 6 - . 0 0 0 2 3 - . 0 0 0 4 9 . 0 0 0 3 7 - . 0 0 0 7 9 - . 0 0 0 3 8 - . 0 0 1 2 6 . 0 0 2 4 4 . 0 0 1 4 4 - . 0 0 0 4 7 . 0 0 0 2 9 . 0 0 0 0 7 . 0 0 0 4 4 . 0 0 0 2 7 . 0 0 0 3 7

. 0 0 0 2 2 . 0 0 0 2 1 . 0 0 0 1 0 - . 0 0 0 1 0 - . 0 0 0 0 9 - . 0 0 0 2 5 - . 0 0 0 2 3 - . 0 0 0 5 2 - . 0 0 0 4 4 - . 0 0 0 9 4 - . 0 0 0 6 7 - . 0 0 1 4 7 - . 0 0 0 2 6 - . 0 0 4 8 8 - . 0 0 3 0 8 . 0 0 0 2 5 - . 0 0 0 0 8 . 0 0 0 7 5 . 0 0 0 4 9 . 0 0 0 7 5 . 0 0 0 4 7

. 0 0 0 4 8 . 0 0 0 2 7 . 0 0 0 2 2 - . 0 0 0 0 7 - . 0 0 0 2 2 - . 0 0 0 1 9 - . 0 0 0 5 0 . 0 0 0 4 4 - . 0 0 1 0 0 - . 0 0 0 8 4 - . 0 0 1 8 0 - . 0 0 1 1 4 - . 0 0 2 9 8 . 0 0 3 9 7 . 0 0 2 7 7 - . 0 0 0 5 5 . 0 0 1 2 2 . 0 0 0 8 3 . 0 0 1 4 4 . 0 0 0 9 4 . 0 0 1 0 0

. 0 0 0 5 8 . 0 0 0 5 1 . 0 0 0 2 6 - . 0 0 0 1 8 - . 0 0 0 1 4 - . 0 0 0 4 4 - . 0 0 0 3 6 - . 0 0 0 9 6 - . 0 0 0 8 2 - . 0 0 1 9 5 - . 0 0 1 6 2 . 0 0 3 6 2 - . 0 0 1 4 0 - . 0 0 9 8 3 - . 0 0 6 1 1 . 0 0 1 5 8 . 0 0 1 3 1 . 0 0 2 8 1 . 0 0 1 8 7 . 0 0 2 0 2 . 0 0 1 18

. 0 0 1 0 7 . 0 0 0 5 6 . 0 0 0 4 6 - . 0 0 0 0 8 - . 0 0 0 3 7 - . 0 0 0 2 3 - . 0 0 0 8 5 . 0 0 0 6 3 - . 0 0 1 8 5 - . 0 0 1 5 7 - . 0 0 4 0 9 . 0 0 3 4 5 - . 0 0 8 6 9 . 0 0 5 3 0 . 0 0 7 1 0 . 0 0 2 0 2 . 0 0 5 9 4 . 0 0 3 9 3 . 0 0 4 1 5 . 0 0 2 3 3 . 0 0 2 1 4 . 0 0 1 0 9 . 0 0 0 9 6 . 0 0 0 4 3 . 0 0 0 3 2 - . 0 0 0 0 8 - . 0 0 0 7 6 - . 0 0 0 3 0 - . 0 0 1 6 8 - . 0 0 0 9 9 - . 0 0 3 8 6 - . 0 0 3 2 5 - . 0 1 0 3 9 - . 0 0 8 1 6 - . 0 2 7 0 0 - . 0 1 0 4 4 . 0 1 4 7 5 . 0 0 9 3 4 . 0 0 9 3 3 . 0 0 4 5 6 . 0 0 4 2 9 . 0 0 1 9 2 . 0 0 1 8 9 . 0 0 0 7 5 . 0 0 0 7 6

. 0 0 0 3 5 - . 0 0 1 0 9 . 0 0 0 6 5 - . 0 0 2 3 3 . 0 0 0 9 2 - . 0 0 4 8 8 . 0 0 0 6 4 - . 0 1 1 9 8 - . 0 0 4 1 6 . 0 5 4 2 3 - . 0 1 0 9 4 . 0 8 6 4 1 . 0 3 0 5 1 . 0 2 5 8 1 . 0 0 7 5 7 . 0 0 9 7 0 . 0 0 2 1 3 . 0 0 4 2 4

. 0 0 0 5 7 . 0 0 1 8 5 . 0 0 0 1 1 . 0 0 5 6 8 - . 0 0 8 6 3 . 0 1 1 8 5 - . 0 1 7 1 3 . 0 2 1 9 7 - . 0 3 2 2 9 . 0 3 9 8 4 - . 0 6 6 4 8 . 0 8 1 2 4 - . 2 5 7 5 1 . 9 4 7 4 4 . 4 0 1 8 5 - . 0 4 8 5 8 . 0 8 7 0 6 - . 0 3 1 5 5 . 0 3 9 1 4 - . 0 1 8 6 7 . 0 1 9 9 9 - . 0 1 0 3 9 . 0 0 9 8 5 - . 0 0 5 0 7 - . 0 0 0 1 9 - . 0 0 0 3 1 - . 0 0 0 4 9 - . 0 0 0 7 9 - . 0 0 1 1 8 - . 0 0 2 0 6 . 0 0 3 1 6 - . 0 0 6 5 3 - . 0 1 1 9 8 - . 0 3 8 8 5 - . 1 0 8 3 4 . 0 0 2 0 4 . 0 2 1 4 2 . 0 1 1 7 6 . 0 0 8 7 3 . 0 0 4 8 3 . 0 0 3 6 2 . 0 0 2 0 4

. 0 0 1 5 4 . 0 0 0 8 4 . 0 0 0 6 0 - . 0 0 0 1 2 - . 0 0 0 3 6 - . 0 0 0 3 4 - . 0 0 0 8 6 - . 0 0 0 8 9 - . 0 0 2 0 0 - . 0 0 2 3 9 - . 0 0 5 1 3 . 0 0 7 2 5 - . 0 1 4 1 1 . 0 0 5 1 2 - . 0 0 3 1 2 . 0 0 0 0 0 . 0 0 4 9 1 . 0 0 3 4 9 . 0 0 3 4 9 . 0 0 2 0 4 . 0 0 1 8 0

. 0 0 0 9 7 . 0 0 0 8 2 . 0 0 0 3 9 - . 0 0 0 1 9 - . 0 0 0 1 9 - . 0 0 0 4 8 - . 0 0 0 4 8 . 0 0 1 0 9 - . 0 0 1 1 7 - . 0 0 2 4 2 - . 0 0 2 7 8 . 0 0 5 3 6 . 0 0 5 5 5 - . 0 1 9 3 3 - . 0 0 8 3 8 - . 0 0 0 6 1 - . 0 0 0 0 4 . 0 0 1 8 3 . 0 0 1 2 5 . 0 0 1 5 3 . 0 0 0 8 9 . 0 0 0 8 6 . 0 0 0 4 5 . 0 0 0 3 9 - . 0 0 0 1 0 ... . 0 0 0 2 5 - . 0 0 0 2 6 - . 0 0 0 5 9 . 0 0 0 6 1 - . 0 0 1 2 5 - . 0 0 1 3 0 - . 0 0 2 4 8 - . 0 0 2 3 8 . 0 0 4 1 9 . 0 0 5 2 3 . 0 0 0 1 3 . 0 0 2 2 5 . 0 0 0 0 2 - . 0 0 0 0 3 . 0 0 0 8 1 . 0 0 0 5 1 . 0 0 0 A) . 0 0 0 3 9 . 0 0 0 4 0 . 0 0 0 2 0 . 0 0 0 1 2 - . 0 0 0 1 3 - . 0 0 0 3 1 - . 0 0 0 3 2 . 0 0 0 6 6 . 0 0 0 6 6 . 0 0 1 2 7 - . 0 0 1 1 9 . 0 0 2 1 6 . 0 0 1 4 4 . 0 0 8 1 2 - . 0 0 4 0 1 . 0 0 0 6 9 , 0 0 0 9 3 . 0 0 0 0 7 - . 0 0 0 0 2 . 0 0 0 3 9 . 0 0 0 2 2 , 0 0 0 3 2 . 0 0 0 1 7 . 0 0 0 1 7 . 0 0 0 0 6 . 0 0 0 1 5 - . 0 0 0 1 6 . 0 0 0 3 4 . 0 0 0 3 4 . 0 0 0 6 7 - . 0 0 0 6 2 . 0 0 1 1 5 . 0 0 0 8 5 . 0 0 1 7 3 . 0 0 3 3 1 . 0 0 0 6 7 , 0 0 1 2 0 , 0 0 0 2 8 . 0 0 0 4 1 . 0 0 0 0 7 . 0 0 0 0 1 . 0 0 0 1 9 . 0 0 0 0 9 . 0 0 0 1 4 . 0 0 0 0 6 . 0 0 0 0 6 . 0 0 0 0 7 - . 0 0 0 1 6 . 0 0 0 1 7 . 0 0 0 3 5 . 0 0 0 3 2 . 0 0 0 6 1 . 0 0 0 4 9 . 0 0 0 9 5

- , , 0 0 0 4 3 . 0 0 4 0 0 - . 0 0 2 0 3 , 0 0 0 3 0 , 0 0 0 5 8 - . 0 0 0 1 2 - . 0 0 0 1 9 . 0 0 0 0 5 . 0 0 0 0 1 , 00008 . 0 0 0 0 3 . 0 0 0 0 5 , 0 0 0 0 3 . 0 0 0 0 7 - . 0 0 0 0 8 , 0 0 0 1 7 , 0 0 0 1 6 • . 0 0 0 3 2 . 0 0 0 2 7 - . 00051 . 0 0 0 3 1 , 00077 , 0 0 1 9 0 . 0 0 0 5 4 , 0 0 0 5 7 , 0 0 0 1 5 . 0 0 0 2 7 . 0 0 0 0 4 . .'10008 . 000'>3 , 00000 . 0 0 0 0 3 . 0 0 0 0 1 , 0 0 0 0 2 , 0 0 0 0 3 - . 0 0 0 0 7 , 0 0 0 0 7 , 0 0 0 1 5 - . 0 0 0 1 4 - . 0 0 0 2 7 - . 0 0 0 1 9 . 0 0 0 4 1 , 00012 , 0 0 1 9 3 - . 0 0 0 9 8 ,00011 , 0 0 0 2 8 . 0 0 0 0 7 . 0 0 0 1 2 . 00001 . 0 0 0 0 3 00001 , 0 0 0 0 0 . 0 0 0 0 1 ,00001 , 0 0 0 0 2 - . 0 0 0 0 3 , 0 0 0 0 6 , 0 0 0 0 6 ~ . 0 0 0 1 3 . 0 0 0 1 0 . 0 0 0 2 1 . o o o t o

- . 0 0 0 3 3 ,00094 . 0 0 0 3 1 ,00024 , 0 0 0 0 6 - . 0 0 0 1 3 . 0 0 0 0 3 . 0 0 0 0 5 . 0 0 0 0 0 00001 ,00000 . 0 0 0 0 0

348

Pseudogravity (21*21: D=350°, 1=68° ) . 0 0002 . 00004 - . 00000 . 00012 - . 0 0 0 0 0 .00024 - . 00000 . 00039 - . 0 0 0 0 0

. 00054 . 00053 . 00060 - . 0 0 0 0 0 . 0 0063 - . 00000 . 00051 - . 00000 . 0 0 0 3 ?

. 00000 . 00013 . 00006

. 00004 . 00007 . 00012 . 00018 . 00026 . 00035 . 00045 . 00056 . 0 0 0 6 8

. 00077 . 00096 . 00100 . 00097 . 0 0097 . 00090 . 00081 . 00067 . 0 0 0 5 3

. 00038 . 00025 . 00014 - . 0 0000 . 00011 . 00018 . 00026 - . 0 0 0 0 0 . 00048 - . 00000 . 00078 - . 0 0 0 0 0

. 00111 . 00112 . 00128 - . 0 0 0 0 0 . 0 0140 - . 0 0 0 0 0 . 00120 - . 0 0 0 0 0 . 00081

. 00061 .00041 - . 00000

. 00010 . 00016 . 00024 . 00034 . 0 0048 . 00062 . 00081 .00101 . 0 0 1 2 6

. 00148 . 00190 . 00202 . 00200 . 00202 . 00190 . 00172 . 00145 . 0 0 1 1 8

. 00089 . 00062 . 00040

. 00000 . 00021 - . 00000 . 00044 . 0 0059 . 00079 - . 0 0 0 0 0 . 00131 - . 0 0 0 0 0

. 00201 . 00210 . 00249 - . 0 0 0 0 0 . 00281 - . 0 0 0 0 0 . 00237 .00201 .00161 - . 00000 . 00087 - . 00000

. 0 0017 . 00025 . 00037 . 00052 . 00071 .00094 . 00125 . 00160 . 0 0210

. 00260 . 00356 . 00390 . 00393 . 0 0396 . 00365 . 00322 . 00267 . 0 0 2 1 3

. 00160 . 00115 . 00077 - . 00000 . 00030 - . 00000 . 00061 - . 0 0 0 0 0 .00111 . 00145 . 00197 - . 0 0 0 0 0

. 00350 . 00388 .00491 - . 0 0 0 0 0 . 0 0548 . 00499 . 00422 - . 0 0 0 0 0 . 0 0 2 6 8

. 00000 . 00143 - . 00000

. 00022 . 00033 . 00049 . 00065 . 00092 . 00121 . 00169 . 00224 . 0 0 3 2 5

. 00449 . 00718 . 00840 . 00829 . 00780 . 00660 . 00542 . 00423 . 0 0 3 2 5

. 00239 . 00171 .00114

. 00000 . 00037 . 00000 . 00073 - . 0 0 0 0 0 . 00136 - . 0 0 0 0 0 . 00262 . 0 0371

. 00625 . 00846 . 01225 . 01304 . 0 1 0 6 9 - . 0 0 0 0 0 . 00650 - . 0 0 0 0 0 . 0 0 3 6 9

. 00000 . 00190 - . 00000

. 00029 . 00031 . 00061 . 00062 . 00116 . 00116 . 00217 . 00234 . 00481

. 00730 . 02527 . 03028 . 01929 . 0 1436 . 00967 . 0 0757 . 00526 . 00411

. 00280 . 00209 . 00132

. 00050 . 00001 . 00105 . 00003 . 0 0196 . 00008 . 00362 . 00030 . 0 0 7 8 3

.00294 . 08180 . 05639 . 01821 . 0 1777 . 00817 . 00891 . 00438 . 0 0 4 7 9

. 00233 . 00243 . 00111

. 00026 . 00034 . 00055 . 00067 . 00104 . 00122 . 00192 . 00237 . 0 0 4 0 9

. 00653 . 01676 . 02215 . 01636 . 01234 . 00888 . 00684 . 00498 . 0 0 3 8 0

. 00270 . 0 0196 . 00129 - . 00000 . 00034 - . 0 0000 . 0 0067 - . 0 0 0 0 0 . 00122 - . 0 0 0 0 0 . 00225 . 0 0 3 2 3

. 00489 . 00760 . 00968 . 00990 . 00864 - .00000 . 00560 - . 0 0 0 0 0 . 0 0330

. 00000 . 00174 . 00000

.00021 . 00031 . 00044 . 00061 . 00082 . 00110 . 00144 . 00194 . 0 0 2 6 0

. 00358 . 0 0457 . 00577 . 00625 . 0 0593 . 00524 . 0 0 4 3 8 . 00354 . 0 0 2 7 5

. 00207 . 00149 . 00101 - . 00000 . 00027 - . 00000 . 0 0053 - . 0 0 0 0 0 . 00095 . 00125 . 00160 . 0 0000

. 00265 . 00351 . 00394 - . 0 0 0 0 0 . 00414 . 00380 . 00332 - . 0 0 0 0 0 . 0 0 2 2 0

. 00000 . 00121 . 00000

. 00015 . 00023 . 00033 . 00046 . 00061 . 00080 . 0 0103 . 00132 . 0 0 1 6 3

. 00203 . 00224 . 00266 . 0 0293 . 00292 . 00276 . 00246 . 00209 . 0 0169

. 00130 . 00094 .00064 - . 00000 . 00018 . 00000 . 0 0037 . 00050 . 00065 - . 0 0 0 0 0 . 00104 - . 0 0 0 0 0

.00151 . 00190 .00202 - . 0 0 0 0 0 . 0 0209 - . 0 0 0 0 0 . 00180 . 00154 . 0 0 1 2 6

.00000 . 00070 - , , 00000 *

. 00008 . 00014 .00021 . 00029 . 0 0039 . 00052 . 00065 .00081 . 0 0 0 9 7

.00115 . 00119 ,00137 . 0 0149 . 0 0 1 4 8 . 00142 . 00128 . 00110 . 0 0090 ,00069 . 00049 ,00032 .00000 . 00010 .00015 . 00022 - . 0 0 0 0 0 . 00039 - . 0 0 0 0 0 . 00060 - . 0 0 0 0 0 .00083 .00101 ,00105 - . 0 0 0 0 0 . 0 0105 . 00000 . 00089 . 00000 . 0 0 0 6 ? .00046 . 00032 ,00000 ,00003 . 00006 .00010 . 0 0015 . 00021 . 00028 . 00035 .0004 4 . 000V, 1 ,00060 . 00060 ,00068 . 00073 . 00071 . 00067 . 00060 . 00050 . 0 0040 ,00029 . 00019 ,00011 ,00001 . 00003 ,00000 . 00010 . 00000 . 00019 - . 0 0 0 0 0 . 00030 . 0 0 0 0 0 .00040 . 00048 ,0004V - . 0 0 0 0 0 . 0 0 0 4 7 . 00000 . 00038 .OOOOO . 00024 ,00000 . 00010 ,00004

349

APPENDIX D GEOPAK Program

Fig. D.l Flow chart of the GEOPAK

350

Descriptions of GEOPAK

1. IN; 5 Alphanumeric characters corresponding one of

those in the right of Fig. D.l.

2. ARGS: Argument numbers, consisting maximum 5 data.

3. Subroutine : APLOT

a routine to change the map data into alphanumeric data

sets. L = ARGS(l) output file recorded in alphanumeric

characters.

4. Subroutine CARDIN:

a routine to read in the data input

ARGS(1) = input file.

Three more input cards are required for the numbers of row and

column and title.

5. Subroutine CPLOT:

Contour plotting routine

ARGS(l) = 0 Read in options from input card otherwise, compute from the ARGS

ARGS(2) = number of contours

ARGS(3) = the increment of X multiplied by 100

ARGS(4) = the increment of Y multiplied by 100.

6. Subroutine CONVOL:

8-folded convolution routine with ARGS(l) = output file and ARGS(2) \ 2

Subroutine CONALL:

ordinary convolution routine with ARGS(l) = output file and ARGS(2) = 2.

351

7. Subroutine EXPOL:

a routine to extrapolate the data around the border of a map

ARGS(1) = the amplitude of maximum extrapolation in terms of differences at edge two points

8. FILT 6:

a routine to read in the filter coefficients

ARGS(l) = filter coefficient input file number

ARGS(2) = filter size

9. Subroutine GETNUM:

a routine to read in a data from a processing file

ARGS(l) = the number of first row to read

ARGS(2) = the number of last row to read

ARGS(3) = the number of first column to read

ARGS(4) = the number of last column to read

ARGS(5) = input file number

10. Subroutine GPLOT:

a gray scale plotting routine

ARGS(l) = 1 = equal-integral slicing 2 = power slicing 3 = log-slicing 4 = Exponential slicing

ARGS(2) = 0 An option to print higher values darker

ARGS(3) \ 0 An option to print lower values darker

11. HALT9: Stop execution to end

12 LIMIO: a routine to read in level values and corresponding character sets to change map data into alphanumeric characters

13: Subroutine PRYNT:

a line-print output routine

ARGS(1) = numbers of output printing

352

Subroutine PUTNUM:

a routine to write the map data in a file. The usage of ARGS is same as subroutine GETNUM

SIZE 3:

a routine to read in numbers of rows and columns

ARGS(l) = numbers of rows

ARGS(2) = numbers of columns

Subroutine SIMI:

a routine to compute the similarity map between two maps.

This routine requires to read two input data.

ARGS(I) = 0 no standardization of data input otherwise standardize the data

ARGS(2) = output file number

SYMB 6: to read in a character set to change the map data into alphanumeric data

ZMNX7: a routine to find the maximum and minimum of the data

Subroutine NORM:

A routine to reset a data into a certain range

ARGS(l) = lower limit of the range

ARGS(2) = upper limit of the range

ARGS(3) = ouput file number

ARGS(4) = 2 Apply A power transform and reset - otherwise just reset

ARGS(5) = A value multiplied by 100, in case of applying

A transform of the data

353

APPENDIX E

CONTRAST STRETCHING USING THE PROBABILITY DENSITY FUNCTION OF GAUSSIAN DISTRIBUTION FOR PICPAC GRAY SCALE PLOTTING

The Landsat MSS data are recorded in the range from 0 to

255 and the PICPAC plotting package in ULCC utilizes a maximum

of 64 gray levels, so that the Landsat data shculd be rescaled to

lie between 0 and 63 by using image enhancing techniques such as

contrast stretching.

In the procedures, a probability function of Gaussian distribution

has been applied in image enhancing in order to plot with at ULCC

gray scale mapping package called PICPAC.

Detailed procedures are as follows:

(1) Evaluate the frequency statistics of each band from 0 to 255

and establish its cumulative frequency distribution.

(2) Calculate 64 levels of theoretical Gaussian frequency

statistics with mean zero and variance 1 using Equation 4 in Section 4.

Assign 2.5% of total frequencies at each end value and establish

theoretical cumulative frequency statistics.

(3) Compute 64 gray level values either by fitting 2-points

linear interpolation or by fitting a Chebyshev polynomial of second

order for 3 points partial interpolation as shown in the figure (

(Anderson & Houseman, 1942).

354

Fig. E.l

*DECK HISTOGRAM STRETCHING PROGRAM HSTST (INPUT »OUTPUT»TAPE5=INPUT ..TAPE6=OUTPUT 1 .TAPE11.TAPE12.TAPE13.TAPE14)

C C A PACKAGE OF FORTRAN 10 PROGRAM TO CONDUCT THE GRAY-LEVEL C SLICING FOR EITHER EQUAL FREQUENCY DISTRIBUTION QR NORMAL C DISTRIBUTION FUNCTION WITH MEAN ZERO AND VARIANCE 1. C C THE PROGRAM INCLUDES TWO METHODS J 2-POINT LINEAR INTER-C POLATION AND PARTIAL INTERPOLATION WITH SECOND-ORDER C CHEBYSHEV POLYNOMIAL. IN CASE OF SOLVING THE CHEDY-C SHEV POLYNOMIAL . NEWTON-RAPHSON ITERATION TECHNIQUE C IS APPLIED . C C INPUT CONTROL CARDS i C IOPT(l ) = 0 STOP C = 1 ANALYSIS OF EQUAL FREQUENCY DISTRIBUTION C = 2 NORMAL DISTRIBUTION C IOPT< 2) = 1 FOR SOLVING BY 2-POINT LINEAR INTERPOLATION C = 2 FOR SOLVING BY CHEBYSHEV POLYNOMIAL C IOPT(3) = LU INPUT FILE NUMBER C C N NO. OF INPUT GRAY LEVELS C NL NO. OF OUTPUT GRAY LEVFI.S c DIMENSION X (65)» Y(65).Z(65 >»F(65). FT(65) DIMENSION FF<258).ZIN(256).IOPT(3) DATA SUMF/O.0/ DATA N f NL/256 » 64/ c READ (5.5) <IOPT(I> . 1 = 1.3) 5 FORMAT(312) IA = IOPT(I) IF(IA.LE.0 ) STOP IB = IOPT(2) LU = IOPT < 3) NL1 = NL+1 PI2 = SQRT(2.*3.141592) C c rf.ad in frequenty data

rewind i i) READ < LI). I'V, (?TN< I . J-.-i . n . 10 FORMAT(1 OLA.0) r C '"QMPUT F THE CUMULiVIIVF E RED''I ni. v lUNi'MON Of- . 1 NPU 1 C

ff <1) = 0.0 do 15 1 = 1.n

15 ff(i+1 ) = ff(i)+zin(i ) ff(n + 2 > ^ ff(ne1 ) C sum = ff(nf1) C go to (100.200). ia

c r c0mpu1f the fqi'ai distribution function 0

10a 'qntinuf a l a hi im / el. dat (nl"'

C CONHMPUTE THE THEORETICAL CUMULATIVE EQUAL DISTRIBUTION C FUNCTION

DO 105 1 = 1 . NL. 1 105 F ( I ) ••= AL

FT(1 ) = 0 . 0 DO 110 1=2.NL1

110 F T ( I ) = F T ( I - l ) + F<I ) FT(NLl) = SUM GO TO ( 1 2 0 . 1 3 0 ) . I D

C COMPUTE GRAY LEVELS BY USING 2-POINT LINEAR INTERPOLATION 120 CONTINUE

CALL LINT2P(Y » FF.FT.NL » N) GO TO 300

C COMPUTE GRAY LEVELS BY USING CHEBYSHEV POLYNOMIAL 130 CONTINUE

CALL CHED2D(Y.FF.FT.N.NL) GO TO 300

C COMPUTE THE NORMAL DISTRIBUTION FUNCTION C 200 CONTINUE

DX = "J . 92/FLOAT (NL-2 ) XX = - 1 . 9 6 + DX/2 . DO 220 1=3 .NI.. F ( I ) = E X P ( - X X * X X / 2 . ) / P I 2 SUMF = SUMFEF(I) XX = XX I DX

220 CONTINUE C C ACCEPT 2 . 5 PERCENT OF TOTAL FREQUENCY FOR EACH EDGE LEVEL C

CC = 5 . * S U M F / 1 9 0 . F(1) = 0.0 F ( 2 ) = CC F(NL1) = CC SUMF = SUMF 1-2 . *CC

C COMPUTE THEORETICAL FREQUENCIES AT EVERY LEVEL DO 225 1=2.NL1

225 F ( I ) = F(I>*SUM/SUMF C C COMPUTE THE THEORETICAL CUMULATIVE FREQUENCY FUNCTION

F T ( 1 ) ' 0 . 0 DO 230 I>2.NL1

230 FT ( T ) = L ) ( I 1 ) F F ( I ) (I GO TO ( 2 4 0 . 2 5 0 ) . I P

C COMPUTE GRAY LEVELS BY USING 2-POINT INTERPOLATION 240 CONTINUE

CALI... I... I. N I 21•' i T . FF . F T « NL. . N ) GO TO 300 ^

C C COMPUTE GRAY LEVELS BY USING CHEBYSHEV POLYNOMIAL ^

250 CONTINUE CALL CHED2D(Y.EF.FT.N.NI.)

300 CONTINUE C GENERATE VALUES FOR X-AXIS FOR GRAPHS

DO 310 1=1.NL1 310 X( l> = 1 -1

WRITE(6 .320) 320 FORMAT -3X. •ORDER' ? 5X- " LEVEL " . 10X. "FREQUENCY" )

DO 325 1=1»NLI IT = 1-2

325 WFv'I TE ( 6 i 330 ) I I i Y ( I ) . F ( I ) 330 F0RMAT(5X»I3 .F0 .1rHXfFlO.n

HO 340 1 = 1 »NL1 IY = I N T ( Y ( I > + 0 . 5 )

340 Y ( I ) = IY c. C COMPUTE NO. OF FREQUENCIES AI EACH LEVEL

DO 350 1 = 1»NL1 350 Z <I) = 0 . 0

DO 360 1=1»NL 11 = Y ( I ) +1 12 = Y ( I +1 ) DO 355 J =11r12

355 Z ( I+1 ) = Z C1 + 1) + Z I N ( J ) 360 CONTINUE

CALL GRAFt C(X » F » NL1) CALL GRAF IC < X . Z • Nl. I ) END SIJDROUTINE LIN T 2F" ( Y r F F' r F f t Ml. t N ) DIMENSION Y<65)»FT(65)»FF(25G^ C J = 2 Y(1) = 0.0 DO 70 1 = 1»NL 10 CONTINUE i f t j . g t . n ) go ro go I F ( F T ( I > . E O . F F ( J ) ) GO TO 50 IF (FT ( I > . GT . FF( J - 1 ) . AND . F'T < I ) . I.T. F'F ( J > > GO TO 60 I F ( F T ( I ) . L T . F F ( J - l ) ) GO TO 40 J = J+l GO TO 10

40 J = J - l GO TO 10

50 Y ( I ) = FLOAT(J-l) J = J + l GO TO 70

60 A = F F ( J ) - F F ( J - l ) B = FF(J ) -A*FLOAT(J) Y ( I ) = ( F T ( I ) - B ) / A - 1 . J = J+l

70 CONTINUE 80 CONTINUE

Y(NIF1) = 255 . RETURN END SUBROUUNF CHflLMH V.FF.I l .N.NI i DIMENSION ( v .<>5 ) , I F i 25G > » F T « 65 > » E1< 3 ) r E2 < 3 > DATA £ 1 / 1 . , 0 . » 1 .0 . DATA E2/1 . 0 » - 2 . r 1 . <

Y(1 ) = 0.0 DO 70 I =2 ? Nt CON TINUL l F ( J . G T . N ) GO TO GO I F ( F T ( I ) , L T . F F ( J - l ) ) GO TO 45 I F ( F T ( I ) . E O . F F ( J - l ) ) GO TO 40 IF ( F" T ( I > . G1 .F F ( J - 1 ) . AND . FT ( T ) . I I . FT (..(>> GO TO .1 = JF 1 i.O 10 (5 Y( n = JL0AT< J-l >

J = JT1 GO TO 70

45 J = J - l GO TO 35

50 CONTINUE k - J - i

SM =• 0 . 0 SE1 = 0 . 0 SE2 = 0 . 0 DO 60 L = 1f 3 I.L. - N+L-l SM = SM + FFU.L) SFI1 = SF. 1 +FF ( LL ) *E 1 ( L ) SE2 = SE2+FF(LL)*E2(L>

60 CONTINUE AO = SM/3. Al SE1/2. A2 - S E 2 / 6 . D = FLOATYK > C = FT( 1 )

65 CONTINUE BJ = B-FLUAIhJ) H = ((.: • ( AO + AI *BJ+A2#3. #BJ*B.J-2 . *A?> >/<Al#B+A2*6 . *B.J> B = D + H IF ( U . o r . 0 . I ) GO TO 65 Y ( I > = B - l . J = J+l

70 CONTINUE 80 CONTINUE

Y(N L +1 ) ; 2 5 5 . RETURN END

U) Ui ON

357

APPENDIX F

TREND SURFACE ANALYSIS OF FIRST- AND THIRD-DEGREE POLYNOMIALS

F.l Least square fit for a flat plane using first-degree polynomial

As the simplest case of trend surface analysis, the first-degree

trend surface which is equivalent to a flat plane, is represented by

Z = b + b X + b2Y (F.l)

where b^, b^, b^ are the unknown coefficients to be determined, and

X and Y are the values for x- and y-coordinates, respectively.

By applying the least square solution for the equation, and

solving for the unknown coefficients, we can obtain the mathematical

expression of the trend surface.

The mathematical procedure is as follows: n 2

S = I (b +b X. + b Y.- Z.) (F.2) . , O l i 2 1 i i= 1 where n is the number of data values. If S is to be minimized, it is

^ 9 S 9S 9S re necessary that -ttt— = = -^r— = 0. (r .3) 0 3bl 2

Differentiating (F.3) against the unknown coefficients, b^, b^ and

b^, and setting equal to zero, we then obtain the three normal

equations to find the solution.

bQn + b] X + b2 Y = Z

bQ X + b] X 2 + b2 XY = XZ

bQ Y + b] XY + b2 Y 2 = YZ J

(F . 4)

In matrix form

358

n ZX ZY ZX 2 EXZ ZXY ZY ZXY 2 ZY

> ' zz "

b , = ZXZ

y , 2YZ ,

(F.5)

In more simple matrix algebra terms

A] [B] = [c; (F.6)

where [A] is the XY matrix, [B] the column b^b^b2 vector and [C]

the column vector containing Z.

Equation (E.6) can be solved in the usual fashion as

[B] = [A] ^C] (F.7)

where [A] ^ is the inverse of the matrix [A]

F.2 Least square fit using third-degree polynomial

The third-degree polynomial is

Z 3 = bn + (b X+b Y) + (b X2+b,XY+b Y 2) + (b,X3+b X2+b_XY2+b_Y3) 0 1 2 3 4 5 6 7 8 9 (F.8)

By the same manner as in the first-degree polynomial, solving for

the coefficients using the least square criteria, we obtain the normal

equation as follows:

f Zn ZX ZY . . EYJ 1 ZX 2 ZX ZXY . . 3 . . ZXY ZY ZZXY 2 ZY . . 4 . . ZY

3 ^ ZY ZXY3 ZY4 . . . . ZY6 ,

r ZZ

ZXZ

ZYZ

3 ZY Z

(F.9)

359

In matrix algebra terms;

[A] [B] = [C] (F.10)

The solution is, as before, by multiplying the inverse of matrix A

[B] = [A]"1 [C] . (F.ll)

360

APPENDIX G

EMPIRICAL DISCRIMINANT FUNCTION (EDF)

Let us assume that for each class , j = l,2,...,k, we

have measurements on a m-feature vector X = {X ,X , . . . ,X }, and I 2 m

that the a priori probabilities h , j = l,2,...,k, of occurrence

of each class are known. Let the multivariate probability density

function for the jth category be f(X), that is, the probability

that X belongs to category j. These functions may be of any form

provided that they are everywhere non-negative integrable, and that

their integrals over all space equal unity. The classifier must

perform the classification on the basis of this information with a

minimum of misrecognition.

Defining a decision function d(X), where d(X) = d^ means that

X is assigned to , let e^ be the loss (penalty) incurred if

d(X) = d., when X is a member of fi.. It is assumed that the loss J i

is zero for a correct decision. The problem is now to choose a

decision criterion such that the average loss over all class is

minimized. Fu (1968) has shown that under the optimal rule, in the

sense of minimizing the average loss, Z I L ( X ) is smaller than

under any other decision rule. Using a symmetrical loss function

d(X) = d.; e. = 0 l I d(X) = d., j \ i; e. = const. J i

when X is a member of , then the Bayes decision rule is to assign

X to the category for which h e f (X) is a minimum. r r r

361

While it may be possible to estimate the a priori probabilities

and loss-function values to the correct order of magnitude for each

category., it may be impossible to know the probability density

func t ions.

Again following Specht, we will estimate the probability

densities for each category as a sum of exponentials based on the

set of training samples, which have a positive probability of occurrence,

and we will assume that samples not in the training set but near

a given sample point (in m space) will have about the same

probability of occurrence as the training samples.

Assuming that the estimated probability density function

for a category is smooth and continuous, and that the first partial

derivatives are small, Specht proposed that an interpolation

function g(X,X.) be found such that 1

f(X) = - Z g(X,X.) m . 1 l

where m is the number of training patterns available and g(X,X^) is

the contribution of the ith training patterns to the estimated density.

If it is assumed that each training pattern contributes independently

to the overall density distribution, and that g(X,X^) is a function

of the Euclidean distance of X from the ith pattern point in m

space, following Specht we write

-(X-X.)'(X-X.) i l g(X,X.) = (2TT)m/2am

exp

where a is a 'smoothing parameter1. The estimated

denstiy function for the ft.th category is then

362

k f0 (X) = L — . t" V e

(2TT)mA2a2 k A, 1= 1

where X .. is the ith training pattern from category ft.

The probability density function (for each category of the

training set) obtained with the increase of the smoothing parameter

may in some cases overlap other functions along the concentration

scale (Fig.l). Extrapolation of the data can take place in both

directions along the concentration scale. If a relatively large

smoothing parameter has to be used in order to extrapolate satisfactorily

between data points, points on the ends of the different populations

(e.g ft and ft , Fig. 1) could be assigned to rather different categories.

f r f ) f A ( X )

2 Concentration .scole

Figure 1 Extrapolation effect of two unimodal probability density functions for categories ft^ and ft^ with increase of CJ(q <Q_).

MICROFICHE 1

GEOPAK

MICROFICHE 2

COHAN and BPFILT

MICROFICHE 3

FACTOR

Overlay showing geology and drainage

C:Carboniferous ;UD:Upper Devonian MD:Middle Devonian LD:Lower Devonian BG:Bodmin Moor Granite SG:St.Austell Granite i

Overlay showing geology and drainage