Scholars' Mine Scholars' Mine

Doctoral Dissertations Student Theses and Dissertations

1970

Discrimination of signal and noise events on seismic recordings Discrimination of signal and noise events on seismic recordings

by linear threshold estimation theory by linear threshold estimation theory

David Nuse Peacock

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Part of the Engineering Commons, and the Geophysics and Seismology Commons

Department: Geosciences and Geological and Petroleum Engineering Department: Geosciences and Geological and Petroleum Engineering

Recommended Citation Recommended Citation Peacock, David Nuse, "Discrimination of signal and noise events on seismic recordings by linear threshold estimation theory" (1970). Doctoral Dissertations. 2157. https://scholarsmine.mst.edu/doctoral_dissertations/2157

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DISCRIMINATION OF SIGNAL AND NOISE EVENTS

ON SEISMIC RECORDINGS BY LINEAR

THRESHOLD ESTIMATION THEORY

by

DAVID NUSE PEACOCK, 1943-

A DISSERTATION

Presented t o the Faculty of the Graduate School of the

UNIVERSITY OF MISSOURI - ROLLA

In Partial Fulfillment of the Requirements for the Degree

DOCTOR OF PHILOSOPHY

in

GEOPHYSICAL ENGINEERING

1970

T2366 c.1 101 pages

193936

ABSTRACT

The object of this study is the investigation of a

linear threshold element technique for identifying surface

multiples on a single seismic trace. Traces of seismic

events were generated which contained primaries, surface

multiples, and various levels of Gaussian random noise.

Since it was necessary to separate the events as much as

possible, the traces were subjected to pulse-compression

deconvolution processing prior to LTE analysis. Mean

frequency, peak frequency, amplitude spectrum variance,

periodicity, and polarity were employed as pattern param­

eters. A set of weights was found that would maximize

the moment of inertia of the S line distribution of the

patterns subject to the constraint that the sum of the

squared values of the weights was minimized. It is shown

that the problem of the maximization of the moment of

inertia reduces to the solution of a simple eigenvalue

problem. Furthermore, the optimum set of weights is the

eigenvector corresponding to the largest eigenvalue of a

matrix proportional to the autocovariance matrix of the

pattern vectors. The classes of patterns representing

primaries and multiples on traces with high signal-to-noise

ratios were clustered and separ~ted, making identification

by inspection a simple procedure. Clustering and separation

of classes on traces with low signal-to-noise ratios was

less than optimum.

ACKNOWLEDGEMENTS

The author wishes to express his appreciation first

of all to his wife and parents for their patience and

understanding during the course of this investigation.

He is indebted to Dr. Hughes M. Zenor, his dissertation

supervisor for his encouragement and support. Special

thanks are due to Dr. Gerald B. Rupert and to Dr. John

C. Robinson for their critical review of the manuscript.

The author's gratitude is also extended to Dr. Robinson

for providing subroutine TPLITZ used in this research

i i i

and to Dr. Frank J. Kern for his constructive suggestions

and helpful insight into some of the facets of this research.

ABSTRACT .

ACKNOWLEDGEMENTS

LIST OF FIGURES

LIST OF TABLES .

I. INTRODUCTION

TABLE OF CONTENTS

II. SEISMIC TRACE SYNTHESIZATION AND PROCESSING

TECHNIQUES .

A. Reflectivity Spikes

B. Wavelet Generation

C. Primary Generation

D. Multiple Generation .

E. Noise Generation

F. Processing

1. Autocorrelation Function

2. Power Spectral Estimate

3. Wavelet Reconstruction .

4. Inverse Filter Construction and

Deconvolution

III. PATTERN RECOGNITION ANALYSIS OF SYNTHETIC

SEISMOGRAMS .

A. Parameters Investigated

B. Determination of Weighting Vector

C. Analysis and Discussion

IV. SUMMARY AND CONCLUSIONS .

V. APPENDICES

lV

Page

ll

iii

vi

viii

1

5

5

8

9

10

12

17

17

18

25

25

36

37

42

47

58

61

A. Mathematics Relative to Development of

Deconvolution Operator

B. Subroutine to Compute Deconvolution

Operator

C. Linear Threshold Element Data

D. Linear Threshold Element Details

VI. BIBLIOGRAPHY

VII. VITA

\'

68

69

82

87

LIST OF FIGURES

Figure

1. Hypothetical reflectivity series

2. Symmetric Ricker wavelet

3. Reverberation model

4. Trace of primaries and multiples

5. Trace of primaries, multiples and random

noise. P = 10

6. Trace of primaries, multiples and random

noise. P = 5

7. Trace of primaries, multiples and random

noise. P = 2

8. Autocorrelation function of synthetic seismic

trace. P =oo.

9. Power spectral estimate of synthetic seismic

trace. P =oo.

10. Statistically derived wavelet,-y(t).P=oo.

11. Deconvolution operator for synthetic seismic

trace. P = oo .

12. Deconvolution operator for synthetic seismic

trace. P = 10

13. Deconvolution operator for synthetic se1sm1c

trace. P = 5

14. Deconvolution operator for synthetic seismic

trace. P 2

15. Deconvolved synthetic seismic trace. P =oo

16. Deconvolved synthetic seismic trace. P = 10

Vl

Page

5

9

10

11

14

15

16

19

24

24

27

28

29

30

32

33

17. Deconvolved synthetic seismic trace. P

18. Deconvolved synthetic seismic trace. P

5

2

19. Partial amplitude spectrum of event no. 9 from

\' l

:)4

35

figure 15. t 0 = 1118 ms . :;9

20. S line distribution produced with unit weights.

p = 00 •

21. S line distribution produced with unit weights.

p = 10

22. S line distribution produced with unit weights.

p = 5 .

23. S line distribution produced with unit weights.

p = 2 •

24. S line distribution produced with eigenvector

corresponding to A x P = 00 rna •

25. S line distribution produced with eigenvector

corresponding to A x P = 10 rna .

26. S line distribution produced with eigenvector

corresponding to A P = 5 max.

27. S line distribution produced with eigenvector

corresponding to Amax. P = 2

48

49

so

51

53

54

55

56

LIST OF TABLES

Table

I. Times, polarities and amplitudes of primary

spikes .

II. Pattern vector coding for periodicity

component

III. Pattern vector coding for polarity component

IV. Eigenvalues and eigenvectors of the matrix B

V. Times of the central extremum, t 0 , for events

1n the reference set .

VI. Times of the central extremum, t 0 , and pattern

vectors for events of interest .

VI I l

Page

6

41

41

44

70

74

I. INTRODUCTION

Exploration geophysicists and other interested scientists

are constantly searching for new and better methods of ob­

taining and analyzing reflection seismic data. Furthermore,

the presence of a multitude of anomalous waveforms on seismic

records, such as surface waves, dispersed energy, multiples,

ghosts, diffractions etc. necessitates continual research

with the purpose of more fully identifying, understanding,

and controlling these forms of important seismic "noise".

The reader is referred to Olhovich (1964) for a comprehensive

outline of the major noise problems in reflection seismology.

The purpose of this research is to investigate the appli­

cations of pattern recognition analysis techniques to the

problem of identifying multiply reflected events (multiples)

in reflection seismic prospecting, a problem which is con­

sidered to be one of the most common and dominant of those

encountered by exploration geophysicists.

The recognition that multiple events were geophysically

possible, and, in fact, were quite common in occurrence, Has

first made by Leet (1937), who noticed their presence and

observed their basic characteristics on records from a well

which had been shot for velocity. The identification of

multiples as such was first made by Johnson (1943), who pro­

posed a specific geologic model which was necessary and suf­

ficient to explain their existence on seismograms from certain

regions of California.

problem were Ellsworth

Early investigators of the multiple

(19 4 8) ~ ,Johnson ( 19 4 8) , S 1 oat ( 19 4 8) ,

and Dix (1948), among others.

The concepts of statistical communication theory and

generalized harmonic analysis were first developed shortly

before, and during, World War II, and, subsequent to declass­

ification, were published by Wiener (1949). Other precursors

in this area included Bartlett (1948, 1950) and !\icc (1944).

During the early 1950's, the application of statistical com­

munication theory to the problems of reflection seismology

was led by Wadsworth et al (1953), and Jones and Morrison

(1954). The efforts of these and others, however, were

hindered by the unavailability of high speed data processing

equipment with large storage capacity. With the advent of such

equipment and supporting digital recording systems in the

early 1960's, a veritable technological revolution overcame

the industry. On an economical and practical basis, investi­

gators now had the wherewithal to attack the nagging seismic

problems, multiples included, with the techniques of statisti­

cal communication theory and allied concepts.

During the past few years a wealth of literature has

appeared concerning investigations which make usc of high

speed data processing techniques. Among the principal authors

are Robinson (1966, l967a; and Treitel 1964), Treitel (and

Robinson 1966; et al 1967), Schneider (et al 1964, et al

1965), Embree (et al 1963), Anstey (1960, 1964), and Anstey

and Newman (1966).

The field of pattern recognition developed from a late

1950's interest in "artificial intelligence" and "thinking

machines". In this respect~ Rosenblatt (1962) is recognized

as the father of the science of pattern recognition due to

his exhaustive study of a brain model~ the "perceptron".

Other early investigators included Sebestyen (1962), and

Marill and Green (1960). Almost from the outset, the science

was developed along two dissimilar lines. A geometrical or

"deterministic" approach was held to by Rosenblatt and others,

while Sebestyen and others conducted investigations based on

studies of the statistical properties of the images or pat­

terns to be recognized. Recently a trend has developed

toward unification of these two approaches. Techniques have

been developed whereby the n features of all patterns may be

weighted with an optimum (in some sense) set of statistically-

derived feature weights. This brings about clustering of

similar patterns within the n-dimensional pattern space so

that simple thresholds may be used to isolate each cluster.

See~ e.g., Fischler et al (1962) and Minsky and Papert (1969).

A technique of this nature has been applied with some success

in the field of diagnostic medicine, e.g., by Stark et al

(1962)~ among others.

The application of pattern recognition analysis tech­

niques to geophysical problems consists of one paper by

Mathieu and Rice (1969)~ who, with limited success, used the

4

statistical approach to differentiate between events on a

synthetic seismogram corresponding to shale and similar ones

corresponding to sandstone.

Chapter II of this study begins with a description of

the generation of a synthetic seismic trace consisting of

primaries and surface multiples. After addition of various

levels of random noise, this trace is subjected to a series

of mathematical processes leading to a pulse compression

deconvolved trace, which then provides the input for the

pattern recognition analysis. Chapter III consists of the

description of the linear threshold element technique ap­

plied in this problem and the discriminatory results

obtained through its use. Chapter IV includes a discussion

of the results and recommendations for further research.

Appendix A contains the mathematical details of the devel­

opment of the deQonvolution operator while Appendix B is a

listing of the Fortran subroutine used to obtain the operator.

Appendix C lists the input pattern vectors for the various

noise levels studied while Appendix D consists of the theoret­

ical development of the multiple identification scheme.

II. SEISMIC TRACE SY0JTII1:SIZJ\TJON

AND PROCESSING TECII:'< I QULS

A. Reflectivity Spikes

The presence of energy reflecting horizons within the

crust of the earth can be represented mathematically for

seismic purposes by a time series of Dirac delta functions,

symbolized by 8 (t), which are commonly called "spikes".

The reflection coefficients, A-, of the set of horizons l

can be used to provide the amplitude and polarity charac-

teristics of the delta functions in order that the series

may be written as

5

f(t) == 2: 1\.o(t-T-) l l

(l) i

where for practicality only a finite quantity of time shifts

T- need be considered. l

A pictorial representation of a por-

tion of a hypothetical reflectivity series is as illustrated

1n figure 1. A6

Al A" J\4

l ;\C) J\11

C' L

01 1 I I I 1~10 I J 1 I I 'l'

A3 A12

As J\7 As

Figure 1. Hypothetical reflectivity series.

S:ince velocity generally increases with depth, the negative

spikes represent density-velocity inversions of the form

P- lv- 1< P- v-l+ l+ 1 l for normally incident longitudinal waves.

Table I. Times, polarities and amplitudes of primary spikes.

Time (ms.) Pol. Amp. Time (ms.) Pol. Amp. Time (ms.) Pol. Amp.

401 - .03841 2905 - .68638 2224 - .47185

1344 + . 1.5025 3918 - .69232 859 + . 77559

946 + .37073 1985 + .93882 1311 - .22791

1088 + .29316 2581 + .84390 2848 - .32324

247 - .40823 1032 - .46248 668 - .27220

3571 + .55618 3209 + .06620 762 + .03901

600 + . 74273 593 - .12976 1588 - .81425

726 - .20237 700 + .51423 2830 + . 75661

1933 + .34547 453 - .49192 1315 - .19867

1135 + . 40 46 0 3710 - .16022 3003 + . 79427

3926 + .41330 1052 + .57671 119 + .05785

3148 - .36088 3443 - .07415 1455 + .69384

809 + .95472 1118 + .50896 3225 + .54827

1715 - .63386 3790 + . 58515 3238 + .44751

1165 + .88296 3835 - .44876 1569 + . 79462

1878 + . 27076 3247 - .12200 615 + .70387

1568 + .21093 1270 + .51522 1247 + . 77538

2351 + .24331 1265 + .94982 2589 - .07734

86 2 + .68093 1662 + .10023 2207 + .84673

2 756 - . 71122 1537 + .49920 1790 + .63957 0'

Table I continued.

Time (ms.) Pol. Amp. Time (ms.) Pol. Amp. Time (ms.) Pol. Amp.

1856 - .44326 390 + .20431 3103 + .30712

3021 + .81310 3275 + .09300 3300 - . 70184

2745 - . 78237 3841 - .87038 3494 + . 72692

1759 - .94451 2356 - .80140 1746 + .51115 1935 + .22898 2655 + . 82295 1628 + .49932

15 76 - .23409 3730 + .38142 2899 - .38840 1983 + .63072 2490 + .00243 213 - .48489

2066 + .94430 1201 - .67417 3590 - . 73720

1979 - .86556 2791 - .42892 3655 - .38234

1682 + .76428 1635 + . 71887 3270 + .61458

3296 - .08094 3009 - .38142 302 + .23245

2783 + .38984 334 + .00146 3924 + . 31151

2067 + .16511 3196 + .02303 2038 + . 21188

1839 + .22663 3556 + .68587 1307 + .63902

232 + .5027o 1064 + . 21127 3955 + .85962

174 + .16121 1248 .69112 887 + .65790

3911 + .14326 30 70 + .80048 1801 + .00862

564 + . 72876 1606 .17496 1 79 7 + .58295

2315 - .02289 568 + . 71029 3135 + .b4554

131 + .06584 2270 + .87932 22 71 + .21346

8

For this particular problem a set of 120 spikes with

random amplitudes was considered to represent a realizable

series of primary reflection coefficients under the re-

striction that the polarity distribution consisted of 80

positive and 40 negative. The total number of spikes and

their polarity distribution are arbitrary; however, fnr

velocity generally increasing with depth, there should be

more positive spikes than negative. The time of occurrence

in milliseconds of each spike was arbitrarily selected

from a tabulation of random numbers \vithin the total time

length (4.0 seconds) of the trace. The times of occurrence,

polarities, and amplitudes of the spikes are listed in

table I.

B. Wavelet Generation

A symmetric wavelet as developed by Ricker (1945) and

of the form

RW (t) ( 2)

was computed where t is time and b 1s the wavelet breadth,

measured between the two maxima. This wavelet has a minimum

of -1 at t = 0, and maxima of 2 exp(-3/2), (approx. 0.45),

at t = + b/2. Ricker's original wavelet was normalized hy

4/~ to yield equation (2). For computational purposes,

the realistic value of 10 ms. was chosen for b and the \vavelet

was truncated at 30 ms. on hoth sides of its central axis of

symmetry. An illustration of a \vavc1ct of this form appears

in figure 2.

Figure 2. Symmetric Ricker >Vavelet.

C. Primary Generation

For continuous functions the general form of the convo-

lution process is X

1./J ( t ) =! h ( T ) g ( t - T ) d T -00

or equivalently, 'XJ

1./J(t) =/ g(T)h(t-T)dT (3)

-X

where t is a time shift. Considering h(t) as a "filter",

the output \j_J(t) consists of a linear combination of the

values of the input g(t) at all instants of time.

Lmploying the notation of \\.ainstein and Zubakov, (1962),

10

the discrete form of the convolution process is represented

as

N l/J ( t) T~l r ( T ) R W ( t - T ) (4)

where variables t and T are integers. By convolving a dis-

crete version of the modified Ricker wavelet of equation

(2) (sampled at 1 ms. intervals) with the set of primary

reflection coefficients, a synthetic seismic trace of pri-

mary events was obtained.

D. Multiple Generation

A strong event occurring at 0.564 sec. was then arbi-

trarily selected as the generating horizon of a series of

surface multiples. A reverberation model of the form useu

by the geophysical industry and illustrated in figure 3

was employed, where R represents the amplitude of the event

whose peak occurs at 0.564 sec.

0 0.564 sec.

R2

3 R

Figure 3. Reverberation model.

The alternation of polarity signifies a phase reversal at

each reflection of the multiple from the surface. The con-

volution of this multiple reflectivity series with another

"VA -0 - ~ " A 1\. Af'\AI A AA ...... AI\ I I t v vv •v V I 9F I vV I \( vr .. r~ v

0.0 0 .4 0 .8

I It A A A I • • "' A 1\ 0 - • ... f1 0 A 1\ A I - AA A II. I\ I I J\ A ~ ... J\l\ A ... ~ V y w•vv lJ'f y vv vy 0- • vv vvv-o.8 1.2 1.6

'f Jo.A-A 1 ~.A •u •. A.- *' ~a, ,.bf> 1 AAI\, AAA 1 A 1 • yl/fi' 4jtV ff 9HV V '7 V~ 9F v r -v VV 1.6 2 . 0 2.4

I I " 1\ I " A I A A ... " 0 Y'/1. J I A A. f4 " " • • V" A 'V y y V\(v lf"' • .. -v 1('{ 1 o v vir 2.4 2 .8 3.2

~- ~~....._ A • I 1\.A I J\1"'\AA I A • .,. AA *" 411+.. .111\ I r lj \IV I 1( 1V14V 44P 1( rq • yr v 3 . 2 3 . 6 4.0

Time (sec . )

Figure 4 . Trace of primaries and multiples .

......

......

1 2

modified symmetric Ricker wavelet of the same form as that

in equation (2) yielded a synthetic trace of multiple events.

However, more energy in relatively lower frequency bands was

incorporated into the trace by allowing the breadtl1 of the

wavelet to be 15 ms. and subsequently increasing the total

length of the wavelet to 80 ms.

The composite trace of primaries and multiples as shown

1n figure 4 was constructed by superimposing the two original

traces.

E. Noise Generation

To better simulate a seismic trace, random noise traces

were generated and added to the composite.

If the samples of a stationary noise time ser1es are

uncorrelated, i.e., statistically independent, it is shown

by 11ancock and Wintz (1966) that the covariance matrix of

the noise may be written as

(S)

where o 2 ]s the variance of the no]se samples and l lS the

identity matrix. The average signal-to-noise power ratio 1s

defined as

p (6)

Accordingly, several traces of Caussian random no1se h'cre

constructed using the IBM subroutine CAUSS (1967h).

trace had a mean value of zero and the variance was preset

l:l

so as to yield values of P equal to 10, 5, and 2. For this

study, the signal s was the trace of both the primary and

multiple events synthesized above, so that,more precisely,

equation (6) becomes

primary power + coherent noise power

incoherent noise power

A value of P equal to 2 is considered to be a physically

realizable minimum (Kern, 1970), and for this reason, higher

levels of random noise power were not considered.

The individual noise traces were then subjected to a

five-point moving average routine described by

i+2

ni = j~- 2 O.Znj (7)

where n. is the value of the noise trace at time sample 1. l

This "smoothing" process effectively filtered the noise

traces so as to make the frequency spectrum of each one

more nearly coincident with the frequency spectrum of the

trace of primaries and multiples.

Subsequently, each noise trace was superimposed on

the trace s(t) to yield the output traces shown as g(t)

1n figures 5, 6, and 7.

The traces g(t) arc synthetic representations of

field traces of a reflection seismic record which have

t • • 1\,11. -- .. . I A • 1\1\._1\1'\A... .... AA&It A/\.'4 • •v .. 9P , vv lrVf v• r- lJ

0 . 0 0 .4 0 . 8

1 A 1\. A A 0 1 .... ~ 1\ A 1\ • * fl - 1\ 1\. f\ ~ 11./l A A 1\ ... 1 A II. ~ A ,A A A . ~ '{T v ... v, v V \f y w: ~y v- , . y y yv~

0 . 8 1.2 1.6

V' AA-A ""-6 •• A ~-11 -· •• ~~ .-1111 ' "~\ Mil '_A I yirvv1\v ff 1441PV 'ij "f v v' -r 1JV

. 4

. • A I\. • 1\ I\ t A " • 1\ V' A J.. ~ A ft. t. A 1\. .... • VA A 'V r 1 VOV' V"' • - • " vr y v 41 vV' 2 . 4 2 .8 3 . 2

1\- A• /1. I -.. 1 0 AI\ 1 Jll,....AA lc: 1\ .... y 0 J\'t _,A AA AI\ \F V vv - y yyVv 41 V¥ V 4Py Vy v I

3. 2 3 . 6 4. 0

Time (s ec . )

Figu re 5 . Trace of primarie~ , mul tiples and rand om noise . P = 10 .

f-' ~

' - .. - A ., A - - + - .. A • ' A "= A 1'\ Jl. .... -Jt. A ~- A A .. - v ., .. y ~ "" ll vyy v«e> r- -, 0.0 0 . 4 0 . 8

' A A A A. -.-- - ~.A A. .. • A - -A A. A .._AA A ~A ... - - • .A A ,.~~~.. ~A A .... .._ fT- y -~'<1 v VYV w a. rr ~ 0. f y r 0. 8 1.2 1.6

'f .u- A,.. A "'f" .,-A-~ r· A, .,--''- ., ""'!. """· .,. J.. ... tvw1'Y'--F '¥ 1 ~, r ' , ... 1 .6 2 .0 2.4

... • fA. • AA. • AI ~A .,A . ..A ~- ,A.,_ fA. ., • .,. -A 'Y · T T •vrr - .., f¥ T • • ~ .. 2 . 4 2 . 8 3 . 2

~- .... ., 1\ ' - -' - AA • -~ .. A - ~ A .... ,. - .. ., ..A - v AA. :4 v-y vV* = f yy'V'\F «JF 4N ''*' 't Vf • • 1 u

3.6 3.2 Time ( se c . )

4 . 0

Fi gure 6. Trace o f p r imaries , mult ip les & r a ndom noise . P = 5 .

1-' (J1

Ia n ~litO .... ~A .,A 0 7 _ ~ rt rt .... en A 0007 • A I\__ Aft .a ... - __ ..Alt.. •-•- A.A. 4 0 .. v• *'VI i.<$1 v a u.v " =9fi li~Vf o:cr ,.- =use y' 0.0 0.4 0.8

., AAA,.._ ............. rt ~....~>.A a • A , -A.A. _A ~AA A._ AA ,. e n =• ...... & 0 -"A ...AAA --,_, ovy-r ~ v· • V v y'V y«Uq?" ... ~T T 4 r· ,~,v,.

0.8 1.2 1 . 6

'f __ .u.A""_A "tf"··A-· ·- JA. . .,....Aa.-- .. -- A..,. MA ..... A.- .. v fY ... ,.,. ,-v . t'l -y- -·T ""f lr-

1.6 2.0 2.4

.. - • A A- - AA • AA ~ /L_, A .~ .A -.. .A.,_. fA. ~ ..... .,. .. & ...._A ~ CO 41 0 r OW' I U W41Jyy yV'-" 40.... - b ¥1¥ 17 yn; ~

2.4 2.8 3.2

it( ..... A ., A. aft~ •• fA ... --·A.- A .... ,. --A- .A~ ..... AA.. ._. ., • ..,. u GV. =r 0 f ·-yyvv v..; 4<> .. --r 1?F. vyr r =

3 . 2 3 . 6

Time (sec.)

Figure 7. Trace of primaries, multiples and random noise. P = 2.

4.0 1-' 0\

been recorded with no frequency filtering (i.e., wideband

recorded), and comprise the input to the processing schemes

which follow. For processing and analysis purposes, only

the form of the input wavelet was assumed known.

F. Processing

In order to isolate each event as much as possible

each trace g(t) was deconvolved with an inverse filter

constructed from the statistical information contained

within the trace itself. A description follows of the

various processes necessarY to calculate and apply an

1nverse filter or deconvolution operator. The mathemat-

ical development of the operator appears in Appendix A.

In order to construct a deconvolution operator for a

17

seismic trace, one must know the expression for the original

wavelet. which, when convolved with a reflectivity function,

yields the seismic trace. The wavelet is determined, accord-

ing to Robinson (1967b), by taking the inverse complex

Fourier transform of the square root of an estimate of the

power spectrum of the trace. This estimate may be obtained

by complex Fourier transforming the smoothed autocorrelation

function of the trace.

1. Autocorrelation Function

The autocorrelation of a continuous function h 1s

defined as 00

¢hh(T) = ~ h(t)h(t+T)dt (8)

-00

] 8

Using the discrete form of equation (8), represented by

the sampled data autocorrelation functions (ACF) of the

input traces were obtained where again the notation lS

that of Wainstein and Zubakov (1962) with t and T being

integer valued and the overbar signifying average value.

Because the ACF is an even function only ¢ (T) for T > 0 gg -

was calculated. Since the input traces had a sampling

interval of 1 ms., the value of ¢gg(T) was computed at 1 ms.

intervals, while, for practicality, the ACF's were truncated

at T = 260. The function ¢gg(T) for 0 < T < 260 with P = oo

(i.e., no random noise) is shown as figure 8. The ACF's for

P = 10, 5, and 2 are not shown since they appear much the

same as that in figure 8.

2. Power Spectral Estimate

If h(t) is considered to be the limiting case of a

periodic waveform whose period has approached infinity,

and if h(t) possesses finite energy such that

00 12 _[ lh(t) dt < M (10)

then the complex Fourier transform of h(t) is defined as

00

H(f) = J h(t) exp (-2njft)dt (11) -00

19

150

120

90

60

30

-30 Lag (ms.)

-60

Figure 8. Autocorrelation function -90

of synthetic seismic trace. p = 00

where f is frequency and j is the unit complex quantity.

The original time function, h(t), may be obtained from

its frequency domain representation, H(f), through the

inverse complex Fourier transform given by

H (t) 1 -z:;y--

00

~ H(f) exp (Znjft)df -00

H(f) is a complex quantity and hence may be uniquely

specified by its amplitude spectrum, IH(f)l' and its

phase spectrum,

e (f) -1 tan

Im rH (f)] Re [H (f)]

20

(12)

(13)

where Re and Im signify real and imaginary parts, respec-

tively.

For this investigation, the discrete Fourier transform

of the empirical autocorrelation function is given by

n-1

rgg (f) = T=~+l cj> ( T ) e Xp ( - 2 TI j f T ) gg (14)

considering ¢ (T) to be one cycle of a periodic function. gg

The frequency function f is called the Schuster periodo-

tram and is an asymptotically unbiased estimate of the

power spectrum of the trace g(t). "Unbiased estimate" is

interpreted to mean that the estimate, considered to be

a random variable, will have an expected value "close" to

the true value of the power spectrum. However, according

to Robinson (1967b) and others, the periodogram is not a

consistent estimate of the amount of energy present in

any given band of frequencies, i.e., it is not a consist-

21

ent estimate of the power spectral density. The definition

of "consistency" is that as given by Meyer (1965).

states that p is a consistent estimate of p if

lim Prob n-+ 00

where p 1s an estimate based on a sample x 1 ,

This

(15)

the parameter p. Bartlett (1948) states that the variance

of the periodogram at any given frequency does not approach

zero as the sample size, n, approaches infinity, and hence,

I~g(f) is inconsistent. A consistency theorem and discussion

regarding the limit of the variance as n approaches infinity

is presented in the above cited work by Meyer.

There are at least two ways to obtain a consistent

power spectral estimate. One way is to weight or "smooth"

the ACF in the time domain with a lag window function or

smoothing function and then take the Fourier transform of

the result to obtain the required estimate. Another way is

to compute the periodogram, and then through convolution,

perform a smoothing operation in the frequency domain to

arrive at the estimate. Since the two methods are mathe-

matically equivalent, time domain smoothing was chosen for

computational purposes.

22

Smoothing functions are, in general, quite well-known

and a wealth of literature describes their characteristics

and mathematical properties. Among the leading investigators

are Bartlett (1948,1950), Zaremba (1967), and Parzen (1967).

An easily understood exposition of the basic theory is that

authored by Blackman and Tukey (1959).

For reasons which follow, the particular smoothing

function chosen was the Bartlett or Fejer window, first

proposed by Bartlett (1948). This is defined as

I t I I t I wCt) = 1 - < t

' m t m

I t I (16)

0 > t m

Equation (16) represents an even triangular function,

the slope of whose sides is determined by tm' the truncation

point, and for which w(O) = 1. It has been shown by Parzen

(1967) and others that the location of the truncation point

affects both the resolution and the bias of the spectral

estimate. The t for this problem was chosen to be 64, m

that is, at the 64th lag of the ACF. The Bartlett window

is a member of a class of smoothing functions which, upon

application, always yield non-negative spectral estimates.

Other smoothing functions such as the Hamming do not neces­

sarily yield non-negative estimates. See Blackman and Tukey

(1959). It can be shown that the non-negative property

depends upon the non-negativity of the spectral window,

i.e., of the Fourier transform of the lag window. The

Bartlett lag window has as its Fourier transform

23

W(f) = (sin nftm)

2 (17)

Tift m

Thus, smoothing the empirical ACF with a Bartlett window

not only yields an unbiased, consistent, and therefore

much less erratic, spectral estimate upon Fourier trans­

formation, but also, as a "lagnappe", prevents the occur-

renee of any fallacious negative power.

Following the procedure outlined above, the smoothed

empirical autocorrelation functions weresubjected to a

discrete complex Fourier transformation to yield estimates

of the power spectrum for each signal-to-noise ratio

investigated. The fast Fourier transform (FFT) algorithm

as developed by Cooley and Tukey (1965) was utilized. The

details regarding its implementation and operation are

presented by Cooley et al (1967), Cochran et al (1967),

Welch (1967), Robinson (1968), and others. The actual

FFT subroutine used was that as published by Robinson

(196 7b). It was decided to use a 256 point FFT since this

would give a resolution in the frequency domain of approx-

imately 3.91 cycles per sec. The estimate of the power

spectrum for P = oo is shown in figure 9. Minor differences

appeared in the estimates for P = 10, 5, and 2.

1200

900

1-< <l>600 :s: 0

0..

300

1

(!.) 'lj

;:::$ 0 +->

·rl r-i p.. E

<::r::

-1

406£ 806£ 1206£ 1606£ 2006£

Frequency 6£ = 3.91 Hz.

Figure 9. Power spectral estimate

of synthetic seismic trace. P =oo.

40 80 120 160

Time (ms.)

Figure 10. Statistically derived

wavelet, - y(t). P = oo •

200

24

2406£

240

3. Wavelet Reconstruction

Through the use of the inverse FFT, the inverse

Fourier transform of the negative square root of each

power spectral estimate yielded the time function wavelet

-y (t). The wavelet obtained for P = oo is shown as figure

25

10. No significant variations were noted for the other

values of P. This wavelet is a time function derived from

the statistics of the seismic trace, and actually is an

"average" in some respect of the two distinct wavelets

originally utilized to synthesize the trace. The negative

root was taken since Ricker (1945) has shown that a sym­

metric wavelet of the form represented by equation (2)

contains only negative Fourier cosine components. This

implies that it has a constant phase spectrum of TI.

4. Inverse Filter Construction and Deconvolution

Know~ng the wavelet as a time function one may then

construct a deconvolution operator which theoretically will

yield spikes when convolved with the seismic trace. Rice

(1962) says heuristically that one wishes to "perform the

inverse of the reflection process". Mathematically one

seeks a solution to the matrix equation

cpA = y (18)

where y is the negative of the normalized wavelet obtained

from the preceeding analysis, cp is its normalized autocor-

relation matrix and A is unknown. The manipulation of y

26

implies that its phase is everywhere zero and follows the

theory of Rice (1962). In equation (18) the symmetric

matrix¢ is of the Toeplitz form, i.e., all elements along

any particular diagonal are the same. This property greatly

reduces both the required computer storage space and the

computations necessary to obtain a solution.

It is generally accepted (Robinson, 1969) that an

operator length of approximately twice the effective wave-

let length yields sufficiently reliable results upon de-

convolution. Rice (1962) has hypothesized that deconvolution

efficiency is directly proportional to the length of the

operator; however, the accumulation of computer round-off

error nullifies this hypothesis. A 49 point solution to

equation (18) was obtained through the utilization of

subroutine TPLITZ, a listing of which appears in Appendix B.

The deconvolution operators for the four signal-to-noise

ratios are shown in figures 11, 12, 13, and 14.

Efforts were made to reduce the amount of high frequency

side lobes present in the deconvolution operator a(t) by

prewhitening the amplitude spectrum of the derived wavelet.

¢ (0) was increased by 10, 20, and 50 per cent with the yy

results that all three experiments yielded more smoothly

varying filters whose side lobes were reduced in both number

and amplitude. However, upon convolving these operators with

the seismic traces according to

27

15

12

9

6

3 (j)

"0 ;:j .j....l .,...; ..-1 0 p... s ~ 50

Lag (ms.)

-3

-6

-9

Figure 11. Deconvolution operator

for synthetic seismic trace. p = 00 •

28

15

12

9

6

3 Q)

'"0 ;::J

+-1 .,...; .--1 ~ 0 s

-< 50

Lag (ms.)

-3

-6

-9

Figure 12. Deconvolution operator

for synthetic seismic trace. p = 10.

29

15

12

9

6

3

(J)

'"d ;:l +-' . ...,

0 rl 50 0... s Lag (ms.) <r:

-3

-6

-9

Figure 13. Deconvolution operator

for synthetic seismic trace. p = 5.

30

15

12

9

6

3

C)

'""0 ;:J -1-l

·..-1 0 r-f 0.. 50 s

<r: Lag (ms.)

-3

-6

-9

Figure 14. Deconvolution operator

for synthetic seismic trace. P = 2.

31

N

d(t) = L g(-r)a(t--r) (19) T=l

an unacceptable lack of resolution existed in d(t) and for

this reason the raw ACF of the wavelet was employed for the

solution of equation (18) and is the operator a(t) in

equation (19). This yielded the output traces shown in

figures 15, 16, 17, and 18 for values of P equal to oo ,

10, 5, and 2 respectively.

I I • A yA ' s • j I _,a tz Yt . ' . 1 • .. MJt • 4/Ji lll 0.0 0.4 0 . 8

1 #A ~~~ -. I •• - ., I "' .... • tt .,. . ..~ I -- r'" v f l -1" t t • T t , 0.8 1.2 1.6

Ill.,.. ~A ,1 .. wA W I r J... -'"'- I yA #Wh I 1 I • f I • T J ww-1.6~0 2.4

I I .... ' ... ' u ... 1 A Al . _.A.~ 1 " ,.. J t 1 ..,.., IT " 2.4 2.8 3. 2

... ·-· y ' . ' .., .,_ J, .. ,. y . .,, 4 ' ·-t ' ff )f" aw 1 -.JU •1 3.2 3.6 4. 0

Time (sec.)

Figure 15 . Deconvolved synthet i c seismic trace . P = oo.

~ N

I -·-.. I ·~··"'...,.,...I .... 1 W. I ...... '¥'\sial .J. .... • .I "'i-1"('••+ 't.,. U II ,, .. 0 - 0 0 .4 0.8'

0 . 8 1.2 1.6

~ ~ -A.-"fr I .,.,...,,, • ,. ... ~~~~ ,.,,'r I. I I ~ S: 't"l"'t :fo ;p .......

1 - 6 2 .0 2.4

l •. \ * "* fl ~ I -r I*' ,.,.. I ...M,.~ ..,.... .. ...,.,.... .... tr II.,. "'' v~ I ..,J 2 . 4 2.8 3 .2

"""'"'f":o"yt"':A II ol lloa ""' ""IIIII ............. I """ >IQII •• ··~· ~ • .. ""f• I ,.,. ~ ~' .,. ... ' - VJO "( 1 f" -· \lA! .. ..

3.2 3. 6 4 . 0

Time (sec.)

Figure 16 .De convo l ve d synthetic seismic trace . P = 10.

~

VI

I ............... •v••fl'to ..... I·IJt~ ,.,., ... (t .. .Y."ff'""'"' .,~. f .... 0.0 0.4 0.8

..., .. ,...,.~.,.. ,.,.,~ ... ·""'~ .... ,., ........... Jot~ .... ,J. ... .. 1 . 6 2.0 2. 4

.......... , nr• ... ..,. 0 .J ...... 'V' • • ., ..... ..,.,. .. .,, +r ",.."' •'f' .. .,J 2.4 2.8 3.2

.................... ~·· .. ~ . ...,..,,..,. , ... ,.,. .. 3.6 4.0

Time (sec.)

Figure 17. Deconvolved synthetic seismic trace. P = S.

~ ~

I ............ ~~, .... ,.,._,._,..At4M• '*•'~1"~,...,.,,.., .. 0.0 0 . 4 0.8

0 . 8 1.2 1.6

~·~~.-~, •'t"-t-·~··--.... ~~· ... ··.Jr... 1.6 2 . 0 2.4

•••rt * l1le ..... ..-.~1 rtcJ.J .. ~~ .. ...,J..,a.low: .. ,t .. tY•W: ,..,J 2 . 4 2. 8 3. 2

3 . 2 3.6 4.0

Time (sec.)

Figure 18. Deconvolved synthetic seismic trace . P = 2.

VI V1

III. PATTERN RECOGNITION ANALYSIS

OF SYNTHETIC SEISMOGRAMS

36

The following analysis may be termed a linear thresh­

old element (LTE) classificatory scheme. A linear thresh­

old element can be described as a mathematical device ,

which, when subjected to a pattern of n inputs, forms a

linear combination of the inputs in a manner which results

in a single output. The output is then compared to a thresh­

old with the result that the input pattern is classified

into one of two classes depending on whether the output is

less or greater than the threshold.

An early investigator of LTE techniques was Fisher

(1938) who used weighted sums of measurements of properties

to differentiate between species of Irises. More recently,

Mattson and Dammann (1965) used LTE's to determine and

code subclasses in an n-dimensional pattern space.

In the discussion that follows, the term "pattern" will

be considered to refer to one of the 38 seismic events of

interest selected from each deconvolved trace of figure 15,

16, 17, or 18. A set of properties is used to characterize

each pattern, while the numbers associated with these proper­

ties make up the pattern vector. Thus, if n properties of

each seismic event are measured and represented with n real

number (x1

, x2

, ... ,xn), these numbers then constitute a

pattern vector in an n-dimensional space. The events of

interest for the four signal-to-noise ratios are listed in

table VI in Appendix C.

A. Parameters Investigated

37

The parameters used to describe a pattern in this study

are peak frequency, mean frequency, amplitude spectrum vari­

ance, periodicity, and polarity. A discussion of each param­

eter follows.

It is observed that weathered materials near the surface

of the earth attenuate high frequency energy. Because a

surface multiple travels two or more times through the

weathered zone while a primary traverses the weathered layer

only once, it is to be expected that surface multiples con­

tain relatively more low frequency energy than primaries.

Epinat'eva and Ivanova (1959) utilized this fact as a basis

for constructing low-pass filters to eliminate certain types

of multiples and Mateker (1965) givPs an example to substan­

tiate the property of anoma1ous attenuation of high frequency

energy by the weathered zone.

One of the pattern parameters used was peak frequency,

that is, the frequency at which the maximum value of the

amplitude spectrum occurs. It was expected that primary

events would have higher peak frequencies than multiple

events. This parameter was measured from the amplitude

spectrum obtained for each event, the details of which will

be subsequently discussed.

38

The mean or average frequency was employed because it

was expected that primary events would exhibit higher means

than multiple events. The mean frequency for each pattern

was also obtained from its amplitude spectrum. The final

frequency parameter used in this study was the sample vari­

ance (with respect to frequency) of the amplitude spectrum

of each event. The relationship n

LAiCfi-:oz

'2 i=l q =--------- (20)

was used to calculate this parameter. In this expression

Ai and fi represent the amplitude and frequency respectively

of spectrum sample i, tis the mean frequency, and n is the

number of amplitude spectrum samples.

The amplitude spectrum for each pattern was obtained

through a 256 point FFT of the 11 points (11 ms.) considered

to constitute an individual event. The 11 points consisted

of the central extremum, the 5 points immediately preceding,

and the 5 points immediately following the extremum.

The pattern parameters of mean and peak frequency are

considered to be statistically independent since it seems

reasonable to assume that the value of one parameter in no

way influences the value of the other. Figure 19 shows the

amplitude spectrum of a typical pattern selected from

figure 15.

8

6

Q) '1j

;:::j .j.J

•M 4 r-1 .

~ -:r:

2

2011f 4011£ 6011f 8011£

Frequency 11£ = 3.91 Hz.

Figure 19. Partial amplitude spectrum of event no. 9 from figure 15. t 0 = 1118 ms.

Vl 1.0

40

Another parameter used in this study was the "periodic­

ity" of the trace. A group of 49 events was selected from

the first 2.0 seconds of the trace to serve as a reference

set to which the 38 patterns of interest were compared for

time periodicity. This search and identify operation can

be thought of as follows: Given an event in the reference

set whose central extremum occurs at time t 0 , is there an

event of interest (from the original 38) occurring at time

t = K t0

± o (21)

where K = 2, 3, ,n, and o = 0,1,2,3,4, or 5 ms? In

this equation, inclusion of an error term, o, allows for

the possible shift of the central extremum due to construc­

tive or destructive interference from neighboring events.

If a pattern of interest was found to satisfy these criteria,

this fact was coded with respect to the value of o. The

codes are listed in table II. A code of zero was employed

to denote nonsatisfaction of the above criteria. It was

arbitrarily assumed that the patterns corresponding to pri­

maries would have lower codes than those corresponding to

multiples. The reference events for the four signal-to-noise

ratios are listed in table V in Appendix C.

The final parameter included in each pattern vector

was "polarity". The extremum of each event exhibiting a

non-zero periodicity code was examined and coded for phase

Table II. Patt.e.rn vector coding for

periodicity component.

I o I (ms.)

0

1

2

3

4

5

K

odd

odd

even

even

Table III. Pattern vector coding for

polarity component.

Polarities of Reference Event and

Event of Interest

opposite

alike

alike

opposite

code

10

6

4

3

2

1

code

10

1

10

1

41

inversion according to the criteria listed in table III· ,

patterns having zero periodicity codes necessarily were

assigned zero polarity codes. Thus primaries were gen­

erally represented by lower polarity codes than multiples.

The pattern vectors for the four signal-to-noise

ratios are listed in table VI in Appendix C.

B. Determination of Weighting Vector

Consider the mathematical representation of an LTE

given by

42

(22)

where xik

i ;:::: 1,

i :::: 1, ,n is the kth pattern vector and w. l

,n is a weighting vector. Each pattern vector

1s thus reduced to a single number Sk. A set of weights is

to be found such that the values of S for one class of

patterns will be markedly different from the S values of

the other class. This difference can be observed by con-

structing a histogram of S values which will be referred

to as an S line distribution.

It is shown in Appendix D that an optimum set of weights

1s one which will maximize the moment of inertia of the S

line distribution given by

p

M:::: L ( 2 3)

k;::::l

43

subject to the constraint that

n 2 L: w.

l = min. (2 4)

i=l

In equation (23) S is the mean S value and p is the total

population of patterns. Equation (23) is expanded and

rearranged to yield

n n

M = L 2: i=l j=l

w.b .. w. l lJ J (25)

where the b .. are elements of a matrix B which is proportion­lJ

al to the sample covariance matrix of the pattern vectors

, ... ,xnk), k = 1, ... 'p. It is further shown

that the set of weights is the eigenvector corresponding

to the largest eigenvalue of the matrix B.

Each signal-to-noise ratio was analyzed twice: once

using all the pattern parameters previously mentioned and

once using all except the parameter of amplitude spectrum

variance. Accordingly, the matrix B with elements

b ... = lJ

p

I: k=l

(x.k-x.) (x.k-x.) l l J J

(26)

was computed for each value of P. The IBM subroutine EIGEN

(1967a) was used to compute the eigenvalues and their cor­

responding eigenvectors which are listed in table IV. After

calculating the eigenvalues of B, a normalized version of

the eigenvector corresponding to the largest eigenvalue was

44

Table IV. Eigenvalues and eigenvectorst of the matrix B.

Eigenvalues +- 53951. (f) 12501. 654.3 161.7

!-< 0 .378 .922 .078 -.001 +J u .923 (j) -.383 .053 .011 > s:: -.052 -.025 .557 .829 (j)

bO -.060 .,...; -.046 .825 -.559

~

p =00 Variance not included.

Eigenvalues

56814. 35796. 1196.1 604.9 161.1 +-

(f) .238 .637 .678 .278 -.019 !-< 0 .911 .105 -.394 -.060 .021 .;..l

u -.047 -.029 -.156 (j) . 542 .824

> s:: -.052 -.398 -.278 . 7 7 4 -.566 (j)

bO .,...; -.329 . 762 -.533 -.164 .015 ~

p = 00. Variance included.

Eigenvalues +- 64710. 18123. 966.0 144.7 (f)

!-< 0 -.063 .997 -.043 - . 0 2 9 .;..l

u .997 .065 .042 .008 (j)

> s:: -.032 .046 (j)

.534 .844 bO

.019 .843 -.536 .,...; -.033 ~

p = 10. Variance not included.

t Eigenvectors are listed in the order: mean frequency, peak frequency, periodicity, polarity, amplitude spectrum variance.

45

Table .IV. continued.

Eigenvalues

9496 7. 29630. 4030.2 865.9 141.9 +-

IJ) -.181 .663 .723 .070 -.012 l-< 0

. 75 8 ~ .565 -.327 -.017 -.001 u <l) -.035 .025 -.067 .513 .855 ~ !=: -.032 .007 -.105 <l) . 848 -.518 0.0

·r-l -.625 .490 -.596 -.113 -.019 ~

p = 10. Variance included.

Eigenvalues

+-IJ) 60099. 34206. 890.2 146.4 l-< 0 -.095 .994 -.047 -.009 ~

u <l) .995 .096 .032 -.008 ~ !=: -.014 .032 .554 .832 <l)

0.0 •r-l -.034 .031 .831 -.555 ~

p = 5 . Variance not included.

Eigenvalues

78074. 37840. 12354. 877.1 139.4 +-

IJ) -.222 .859 -.460 -.036 -.002 l-< 0 .807 .427 .407 .020 -.016 ~ u .543 <l) -.020 .028 .016 .839 ~ !=: -.034 .018 -.014 .838 -.544 <l) 0.0

·r-l -.546 .281 .788 -.028 -.020 ~

p = 5 . Variance included.

46

Table IV continued.

Eigenvalues

+- 70 58 7. 24872. 1019.8 160.4 Vl H 0 -.014 .999 -.040 -.029 +J u .999 .016 .022 .015 (])

> s:: -.024 (])

.045 .497 . 866 OfJ -.012 .020 . 86 7 -.498 •r-i

lJ-4

p = 2. Variance not included.

Eigenvalues

+- 96306. 37550. 6149.0 918.0 158.0 Vl

. 716 .676 .039 -.018 H -.170 0 -!-} .790 .510 -.340 -.025 .008 u (])

-.043 .479 > -.030 .029 .876 s:: (]) -.018 .019 OfJ

-.088 .871 -.482 •r-i

-.646 -.096 -.015 lJ.4 -.588 .476

p = 2. Variance included.

entered in equation (22) and an S value for each pattern

obtained. These were plotted as points on an s line and

presented in figures 24 through 27, forming the basis for

the pattern recognition analysis.

C. Analysis and Discussion

The measured values of the frequency parameters and

47

was

are

the assigned codes of the polarity and periodicity parameters

are such that the multiple events should have lower S values

than primaries. Thus, identification of separate clusters

of primaries and multiples should be possible upon visual

examination of the S line distributions provided.

In figures 20 through 27, circles are used to denote

primaries and triangles are used to denote multiples.

Figures 20 through 23 illustrate, for comparison, the

results for four signal-to-noise ratios using non-optimum

(unit) weighting vectors. The two classes of patterns,

primaries and multiples are not readily separable. This lS

true even under the ideal signal-to-noise ratio ofoofor the

case where all five parameters were employed.

Figure 24a shows very good clustering of the multiples

and sufficient separation of the two classes so as to permit

identification by inspection. For the same signal-to-noise

ratio, employment of all five parameters results in acceptable

clustering of the multiples but a scattering of anomalous

primaries between the two clusters of dissimilar events

(figure 24b). In figure 2Sa it is observed that two primaries

d - multiple 0

- primary

-1.0 -0.6

5

4 (!)

3 '-HU 0~

(!)

2 IJ) 1-< .6. (!)$-<

1 s ;:::l ~ ·rl u

E--<U

-0.2 0.0 0.2

Normalized S

(a) Variance not included.

-1.0 -0.6

6

5

-0.2 0.0 0.2

Normalized S

(b) Variance included.

48

00

000::0::00 oo

a:x:oo~ooo

0.6 1.0

0

a:o

a:o

0 o:ocoo 0

0.6 1.0

Figure 20. S line distribution produced

with unit weights. P = oo •

,6. - multiple

o - primary

-1.0 -0.6

5

4 (])

lH t)

3 0 s:: (]) 0

2 Vl !-< (]) !-< 0 s ;:::s

1 •r-i t)

E-< t)

0

-0. 2 0.0 0.2

Normalized S

(a) Variance not included.

6

5

4 (])

lH t)

3 0 s::

(]) Vl !-<

2 (]) !-< s ;:::s

1 •r-i t)

E-<U 0

0

-1.0 -0.6 -0.2 0.0 0.2

Normalized S

(b) Variance included.

49

00000

0

0.6 1.0

0

0

0

0 GIXX> 0

0 D,ax/so d:J.

<X> ax>~

0.6 1.0

Figure 21. S line distribution produced

with unit weights. P = 10.

~- multiple

o - primary

-1.0 -0.6

-1.0 -0.6

5

4

3

2

1

Q) '-HU 0 .::

Q) til!--< Q) !--< ~ ;::l

.,...; u E-<U

0

-0.2 0.0 0.2

Normalized S

0

co

0 0:00

o .C,.o~

oo.C.oco~

0.6

(a) Variance not included.

5

4

3

2

1

Q) '-HU 0 .::

Q) til !--< Q) !--< ~ ;::l

.,...; u E-< u

0

-0.2 0.0 0.2

Normalized S

(b) Variance included.

ooo

0 a:tn> (])

0 cn:tXlXtX>

0 O~(J)

0.6

Figure 22. S line distribution produced

with unit weights. P = 5.

50

1.0

1.0

a- mu1 tip1e

o - primary

-1. 0 -0.6

5

4

3

2

1

Q) 4-IU 0 s::

Q) til 1-< Q) 1-< s ::I

·rl u E--< u

-0.2 0.0 0.2

Normalized S

(a) Variance not included.

5

4 Q)

3 4-IU OS::

Q)

2 tllH Q)H

1 S::l

•rl u t--<U

-1.0 -0.6 -0.2 0.0 0.2

Normalized S

(b) Variance included.

51

0 co 000

0 f). 0/:S;D 0 ClD

0.6 1.0

0 0

0.6 1.0

Figure 23. S line distribution produced

with unit weights. P = 2.

52

exhibit S values in the same range as four of the multiples

and that the S line distribution appears to show three rather

distinct clusters of patterns. Figure 25b shows more intra­

cluster dispersion than does figure 25a. The S line distri­

bution for the four parameter case with P = 5 mistakenly

clusters two primaries as multiples, while in the five param­

eter case intracluster dispersion again appears unacceptable.

Figure 27a and 27b indicate the deleterious effects of the

high level of random noise. Figure 27a shows an equal

number of primaries and multiples in the same cluster as

well as two multiples surrounded by a scattering of primaries.

Figure 27b shows practically no separation of the two classes

as well as poor clustering of the primary events.

In general, less intracluster dispersion and more

intercluster dispersion were obtained by not utilizing the

variance of the amplitude spectrum as a pattern descriptor.

Both of these characteristics would seem to be desirable

for good clustering and separation.

Several attempts were made to find combinations of

two or three of the pattern parameters which would result

in good clustering of the pattern classes when plotted on

an S line. Limited success was attained by using peak

frequency and mean frequency, but only for the trace which

had a signal-to-noise ratio of 00 •

In this case the S line

6.- multiple

o- primary

lH 0

U)

(J)

s •r-i

(J)

u r::::: (J) !-; !-; ;::j u

5 0

4 00

3 £::. 00:0

2 £::. 0 ooxa:oo E--< u 1 ~ 0 0::0 0 0 ()XO)OCO)J ro

-1. 0 -0.6 -0.2 0.0 0.2 0.6 1.0

Normalized S

(a) Variance not included.

~i 0

(J) CX)

lH u 0 r:::::

(J) 0 Oa:x:>

U) !-; (J) !-; A 0 00 OCXX) s ;::j I

•.-1 u ltiO. A 0 00 0 E--<U

0 0 CXXlO CXXXXX> 0 0

0

-1.0 -0.6 -0. 2 0.0 0. 2 0.6 1.0

Normalized S

(b) Variance included.

Figure 24. S line distribution produced with

eigenvector corresponding to A P = oo • max.

53

~- multiple

o - primary

-1.0 -0.6

5

4 (j) 4-lu

03 0 ~

(j) rn H

.&2 (!) H 00 s ;:j

L::.a> 1 ·r-1 u oo.c:..oo E-<U

0

-0.2 0.0 0.2

Normalized S

(a) Variance not included.

5 (!)

4-1 u 0 ~

0 4 (!) (f) H

3 (j) H

0 s ;:j •r-1 u

0 0 <XDcO E-< u 0

-1.0 -0.6 -0.2 0.0 0.2

Normalized S

(b) Variance included.

0

0

CX)

000:0 0

0 «XXX> cr:xxDCX> CX)

0.6

0.6

Figure 25. S line distribution produced with

eigenvector corresponding to A P = 10. max.

54

0

1.0

1.0

L1- multiple 0

- primary 5

4

3

<!) 4-iU 0 I=!

0

0

0 2

/':::LX:s) 1

<!) (f) !-< <!) !-< s ::s

•rl u E-U 000~ oco 000(11)])000 0

0

-1.0 -0.6 -0.2 0.0 0.2 0.6 1.0

Normalized S

(a) Variance not included.

54-i (])

u 0 I=!

0 4 <!) U) !-<

0 0 0 3 <!) !-< s ::s ·rl u

0 0 0 0 0 00 00 E-U 0

AO L» 0 0 0 0 OX> cxx:x:m:o 0 0

-1.0 -0.6 -0.2 0.0 0.2 0.6 1.0

Normalized S

(b) Variance included.

Figure 26. S line distribution produced with

eigenvector corresponding to A P = 5. max.

55

~- multiple

o- primary Q) 4-1U 0 s::

Q) Ul 1-<

3 a.> 1-< E ~

2 •r-1 u E-<U

0

56

0

0 o oa:o

oo o o o cr:. oLP o o o:x:o o cxx:o oo o

-1.0

-1. 0

-0.6 -0.2 0.0 0.2

Normalized S

(a) Variance not included.

0

00 000~ 00 co

-0.6

0

0 0

0 o a:oo

Q)

5 4-1 u 0 s::

Q) 4 Ul 1-<

a.> 1-<

3 E ~ •r-1 u

2 E-< g o o o a:x:ooooo o o

-0.2 0.0 0.2

Normalized S

(b) Variance included.

0.6

0.6

Figure 27. S line distribution produced with

eigenvector corresponding to A P = 2. max.

1.0

1.0

showed not two, but three clusters of patterns, viz., one

cluster of multiples and two clusters of primaries. For

57

the traces with values of P equal to 10, 5, and 2, no success

with a reduced number of pattern parameters could be attained.

Because there are over 120 combinations of the five parameters

for the four signal-to-noise ratios it was impractical to

investigate all such combinations. However, it is felt that,

in general, acceptable results are not obtainable with three

or less parameters.

In figures 24 through 27 a threshold has not been in­

dicated since no a priori information regarding threshold

location is known. If a priori S line distribution infor-

mation for a given signal-to-noise ratio is provided from

previous analyses, it would seem feasible to specify the

probabilities of the occurrence of Type I and Type II errors

and thereby select a threshold before analyzing a trace of

seismic data. If such a priori information has not been

provided one would not be able to select a threshold in

accordance with any preset statistical confidence level.

Instead, one could only assume that S line clustering and

intercluster separation is indicative of the existence of

two classes of patterns.

ss

IV. SUMMARY AND CONCLUSIONS

The object of this study was the investigation of a

linear threshold element technique for identifying surface

multiples on a single seismic trace. Traces of seismic

events were generated which contained primaries, surface

multiples, and various levels of Gaussian random noise.

The traces were subjected to pulse-compression deconvolution

processing in preparation for LTE analysis. The pattern

parameters employed were mean frequency, peak frequency,

amplitude spectrum variance, periodicity and polarity. A

set of weights was found that would maximize the moment of

inertia of the S line distribution of the patterns subject

to the constraint that the sum of the squared values of

the weights was minimized. The classes of patterns repre-

senting primaries and multiples on traces with high signal­

to-noise ratios were clustered and separated, making identi-

fication by inspection a simple procedure. Clustering and

separation of classes on traces with low signal-to-noise

ratios was less than optimum.

The results of figures 24 through 27 generally in­

dicate that an LTE classificatory scheme provides a usable

tool for identification of surface multiples on single

seismic traces. The figures indicate that the "performance"

of the technique deteriorates with increased random noise.

The rather poor results obtained with the LTE technique for low

59

signal-to-noise ratios is attributed to the relatively poor

performance of the deconvolution operator. It is felt that

a matched filter such as that described by Wainstein and

Zubakov (1962) would provide better resolution of individ­

ual events on traces with low signal-to-noise ratios.

Treitel and Robinson (1969) discuss the advantages of using

matched filters for deconvolving low signal-to-noise ratio

seismic traces.

It appears from this study that the "proper" selection

of pattern parameters effectively governs the power of the

technique. This limitation is of the same form as that for

pattern recognition techniques applied to other problems.

The question arises: "What properties does one need to

measure in order to specify a pattern?" This question must

remain largely unanswerable since the answers are most prob-

ably data dependent. The fact that virtually no success

could be attained with a reduced number of pattern parameters

implies that two dissimilar clusters of the input patterns

do not exist in the reduced pattern space. This does not

rule out the possibility that two dissimilar clusters of

another set of input patterns might exist in some two or

three dimensional pattern space.

If the process of frequency dependent attenuation is

accepted, it would seem more correct to employ a time-varying

attenuation process rather than the time-invariant process

60

used in this study. With time-invariant frequency attenua-

tion, the variance of the amplitude spectrum is felt to be,

at best, a pattern descriptor of minor importance. There

does not appear to be any physical reason to suggest that

the variance of the amplitude spectrum of a primary should

be different from that of a multiple. If however, a time­

varying attenuation process were assumed for the multiples,

one would generally expect their amplitude spectra to be

different in shape from the spectra of the primaries. This

difference in shape might be evident not only in the variance,

but also in higher-ordered statistics of the amplitude spec-

trum. The simplest time-varying attenuation process might

well be an exponential of the form exp(-aft) where f and t

are frequency and time, respectively, and a is a constant.

More physical realism could be obtained by considering

a seismic trace to contain non-surface-generated as well as

surface-generated multiples which have geometrically simpler

propagation paths.

On more general terms, the concepts of pattern recog­

nition analysis should be applied to other problems of re­

flection seismology, measuring space variations in addition

to time variations. Some examples of such problems might be

the identification of diffractions generated from faults and

the general problem of automatic reflection picking. Initial

efforts on the latter subject have been made independently by

Hileman et al (1968) and other research workers.

V. APPENDICES

APPENDIX A

MATHEMATICS RELATIVE TO DEVELOPMENT

OF DECONVOLUTION OPERATOR

61

For continuous functions, the autocorrelation of an

input function, h, is defined by

00

¢hh(T) ~~ h(t)h(t+T)dt. -00

(A-1)

Changing the variable of integration with the substitutions

dx == dt,

~ _:1-_ 00 ' +oo -

one obtains 00

¢hh(T) :::: J( h(X-T)h(x)dX. -~

This may be written as 00 J h(:x)h[x+(-T)jdx == ¢hh(-T)

-00

which implies that

(A-2)

(A- 3)

(A- 4)

(A- 5)

so that the autocorrelation of a continuous function is an

even function.

Using the notation of Wainstein and Zubakov (1962), the

discrete autocorrelation is

¢gg(T) ~ g(t)g(t+T),

where the variables t and T are integers.

(A-6)

62

Since the process g is assumed to be time stationary, the

expected value required in equation (A-6) is independent

of time t so that

g(t)g(t+T) = g(O)g(T) (A- 7)

and

g(t)g(t+T) = g(-T)g(O) (A- 8)

which implies that

cjJ ( T) = </J (-T) (A- 9) gg gg

for discrete functions.

In order to obtain equation C 1~, first consider a

rectangular pluse defined by

u(t) = ~ c:m 2t2 :m) = 0 elsewhere

The autocorrelation of u(t) is given by

!¥-IT! <Puu(T) =J( u(t)u(t+T)dt

m 2

which integrates to

(-t <T<t ) m-- m

= 0 elsewhere.

(A -10)

(A-ll)

(A-12)

63

The autocorrelation of u(t) equals the autoconvolution of

u(t) since u(t) is symmetric. The Fourier transform of u(t)

lS

2

=Ff-m [

-j'ITft j'ITft J e m -e m

-j2'ITf

_rr sin 'ITftm =l~ 'IT£ Fm [si:f::tm J

(A- 13)

(A- 14)

Since convolution in the time domain corresponds to multi-

plication in the frequency domain, the Bartlett lag window

of equation (A-12) has as its Fourier transform the square

of the function in equation (A-14). That is,

W(f) = tm[sin 'ITftm ]2 'ITftm (A-15)

Equation (A-15) is identical to equation (17) in Chapter II.

An interesting property of the Fourier transform pair

given by equations (16) and (17) is that smoothing an

empirical autocorrelation function with equation (16) results

in non-negative spectral estimates. Consider the autocor-

relation function of a seismic trace g(t) represented by

00

¢(1:) =I g(t)g(t+T)dt (A-16) -00

¢ may be represented in a Fourier series as

64

7T

¢ ( T) = 2; f e j n T J ¢ ( x) e - j n x dx -do -7T

(A-17)

Since it is physically impossible to perform the summation

for doubly infinite values of n, the partial sum of this

series may be written as

N 7T

<P ( T ) "' 2; -~ e j n T I ¢ ( x) e - j nx dx

which, according to Wiener (1949), may be expressed as

7T

1 j sin [x (N+~)] -- ¢(x+T) 21T X

-7T sin Cz) dx .

This represents a weighted average of ¢(x) such that

7T

_1_/ sin [ x (N+~)] = 1 21T • (X)

-7T Sln z

(A-18)

(A-19)

(A- 2 0)

which implies that the total weight of all points is unity.

However, the weights can be negative as well as positive or

zero. Therefore, the Fourier coefficients, which are

actually lines in the periodogram of ¢, may become negative

by considering only a finite portion of the autocorrelation.

Since power is inherently positive the periodogram is not

considered to be a good estimate of the power spectrum.

Now consider the Cesaro partial sum of equation (A-17):

(A- 21)

The Cesaro method of partial summation is discussed in

Olmsted (1961) and other texts on advanced calculus.

65

Equation (A-21) can be written as

rr Nx 1 J sin2 2 2TIN ¢(x+T)

2 X

-rr sin 2 dx (A-22)

again according to Wiener (1949).

weighted average of <P(x) such that

This represents another

1 2TIN

7T Nx

f sin2 2 . 2 X

Sln Z -7T

dx = 1 , (A- 2 3)

which is of the same form as equation (A-20) except that the

weighting is non-negative. This implies that the estimate of

the Fourier coefficients of <P will be non-negative.

Therefore, assuming that the above arguments will with-

stand the transition from continuous to discrete autocorrela-

tion functions, one may "smooth" the discrete autocorrelation

function of equation (A-6) with the Bartlett lag window of

equation (A-12) to obtain the Cesaro approximation to <P given

by

The

$ gg ( T) " ~ gg ( T) ( 1-1 ~~ ) 0

IT I< t -m

discrete Fourier tm

......

~(f) = 2: T=-t m

transform of equation (A-24)

( 1 _tjmTj) e-j2TifT <Pgg(T)

which will necessarily be non-negative.

(A-24)

is then

(A-25)

66

In the design of a deconvolution operator one seeks a

set of filter weights a(t) which, when convolved with the

seismic trace, will yield a set of spikes as approximations

of reflection coefficients. This problem may be thought of

in more detail as follows: Find a filter such that its

convolution with a given wavelet will compress the wavelet

as nearly as possible into a spike. In symbols, consider a

spike m(t+s) and a wavelet y(t). Find the filter a(T) such

that N

X (t) = L a(T)y(t-T) (A-26)

T=l

will approximate m(t+s) with the smallest error. Thus, the

error N

s = x(t)-m(t+s) = L a(T)y(t-T)-m(t+s) (A- 2 7)

T=l

is to be made as small as possible (in the mean square sense).

Utilizing the overbar to denote expected value,

-- 2 s2 = x 2 (t)-2x(t)m(t+s)+m (t+s)

N N

= m2 (t+s)+ L L: a(T)a(n)y(t-T)y(t-n) T=l n=l

N -ZT~ a(T)m(t+s)y(t-T) .

Since the auto- and crosscorrelation functions

m2 (t+s) = ¢rnrn(O),

y(t-T)y(t-n) = ¢YY(T-n),

(A- 28)

(A- 29)

(A-30)

(A-31)

and

m(t+s)y(t-T) = ¢ (T+s) ym

67

(A-32)

are evident, one may simplify equation (A-29) to obtain

N = ¢mm (0) + L

T=l N

N L a(T)a(n)¢YY(T-n) n=l

-2 L ·a(T)¢ (T+s). T=l ym

(A-33)

The requirement of minimization of the mean squared error is

now accomplished by imposing the restriction

= 0 T=l, ... ,N

on equation (A-33) to obtain

N L a(n)¢YY(T-n) = <~>ym(T+s)

n=l

(A-34)

(A-35)

forT= 1, ... ,N. Realizing that the crosscorrelation of a

spike with the input wavelet yields a reversed wavelet, one

may express equation (A-35) in matrix form as

<PA = y (A-36)

where in this case, the reversed wavelet is the same as the

original because the wavelet is symmetric. Equation (A-36)

is identical to equation 08) of Chapter II.

APPENDIX B

SUBROUTINE TO COMPUTE DECONVOLUTION OPERATOR

SUBROUTINE TPLITZ(JM,R,GAMMA,A) DIMENSION R(SO),C(90),GAMMA(SO),CC(90),A(SO),AA(90) DOUBLE PRECISION A,R,GAMMA,C,CC,AA A(1)=GAMMA(1)/R(1) CC (1)=R(2)/R(1) JK=JM-1 DO 20 M=1,JK I=M+1 IF (M- 1) 12 , 12 , 7

7 SUMA=O.O SUMB=O.O DO 9 K=2,M SUMA=SUMA+C(K-1)*R(K) KK=K-1 MM=M-KK

9 SUMB=SUMB+C(KK)*R(MM+1) CC(1)=(R(I)-SUMA)/(R(1)-SUMB) DO 10 K=2 ,M NN=M-K CC(K)=C(K-1)-CC(1)*C(NN+1)

10 CONTINUE 12 SUMD=O.O

SUMC=O.O DO 13 K=1,M NN=M-K SUMC=SUMC+A(K)*R(NN+2) SUMD=SUMD+CC(K)*R(NN+2)

13 CONTINUE AA(I)=(GAMMA(I)-SUMC)/(R(1)-SUMD) DO 16 K=1,M AA(K)=A(K)-CC(K)*AA(I) A (K) =AA (K) C (K) =CC (K)

16 CONTINUE A(I}=AA(I)

20 CONTINUE RETURN END

69

APPENDIX C

LINEAR THRESHOLD ELEMENT DATA

70

Table V. Times of the central extremum, to, for events in the reference set. p = oo.

Event to Event to Number (ms.) Number (ms.)

1 213 26 1249

2 232 27 1265

3 247 28 1270

4 302 29 1307

5 390 30 1315

6 453 31 1455

7 564 32 1537

8 568 33 1569

9 600 34 1588

10 615 35 1606

11 668 36 1628

12 700 37 1635

13 809 38 1662

14 860 39 1682

15 887 40 1693

16 946 41 1715

17 1032 42 1746

18 1052 43 1759

19 1088 44 1790

20 1118 45 1797

21 1128 46 1856

22 1135 47 1934

23 1165 48 19 79

24 1201 49 1984

25 1246

71

Table V continued. p = 10.

Event to Event to Number (ms.) Number (ms.)

1 214 26 1249

2 231 27 1265

3 246 28 12 71

4 301 29 1307

5 390 30 1315

6 453 31 1455

7 564 32 1537

8 567 33 1569

9 601 34 1588

10 615 35 1607

11 669 36 1628

12 700 37 1634

13 809 38 1662

14 860 39 1682

15 887 40 1691

16 945 41 1715

17 1032 42 1747

18 1052 43 1759

19 1088 44 1791

20 1118 45 1797

21 1128 46 1856

22 1136 47 1933

23 1165 48 19 79

24 1200 49 1984

25 1246

72

Table V continued. p ::: 5.

Event t Event t Number (m~.) Number cm2.)

1 214 26 1249

2 231 27 1265

3 246 28 12 71

4 301 29 130 7

5 390 30 1315

6 453 31 1455

7 564 32 1537

8 567 33 1569

9 601 34 1588

10 615 35 1604

11 669 36 1628

12 700 37 1634

13 809 38 1662

14 860 39 1682

15 887 40 1691

16 945 41 1715

17 1032 42 1747

18 1052 43 1759

19 1088 44 1791

20 1118 45 1797

21 1128 46 1856

22 1136 47 1933

23 1165 48 1979

24 1200 49 1984

25 1246

73

Table V continued. p = 2.

Event to Event to Number (ms.) Number (ms.)

1 214 26 1249

2 231 27 1265

3 246 28 1269

4 301 29 1307

5 399 30 1315

6 453 31 1455

7 564 32 1537

8 569 33 1569

9 610 34 1588

10 615 35 1607

11 669 36 1628

12 700 37 1634

13 809 38 1662

14 860 39 1682

15 886 40 1691

16 945 41 1715

17 1031 42 174 7

18 1052 43 1759

19 1088 44 1791

20 1118 45 179 7

21 1128 46 1856

22 1134 47 1933

23 1165 48 1979

24 1200 49 1984

25 1246

Table VI-A. Times of the central extremum, to, and pattern vectors for events of interest. p = 00

Event to Mean Peak Perio- Pol- Var-Number (ms.) Freq. Freq. dicity arity iance 1 213 106.6 105.8 0 0 34.1 2 232 103.9 98.0 0 0 32.4 3 247 106.2 105.4 0 0 34.7 4 453 109.6 108.9 0 0 36.0 5 564 121. 8 43.3 0 0 94.5 6 568 123.6 43.3 0 0 9 7. 2 7 809 109.6 108.6 0 0 36.0 8 860 108.5 66.8 0 0 80.0 9 1118 106.2 93.7 0 0 54.0

*10 1128 89.0 8. 5 10 10 88. 7 11 1135 153.1 99.6 6 1 104.4 12 1201 109.3 108.6 6 10 35.4 13 1246 130. 5 128.5 0 0 31. 7 14 1249 145.0 125.0 0 0 59.6 15 1265 142.3 51.9 0 0 89.6 16 1682 122.9 114.0 0 0 41.4

*17 1693 76.9 10.0 6 10 58.3 18 1715 103.6 101.9 1 10 32. 5 19 1934 107.6 94.1 0 0 54.4

'-l * Denotes a multiple event. .j::,.

Table VI-A continued.

Event to Mean Peak Perio- Pol- Var-Number (ms.) Freq. Freq. dicity arity iance 20 19 79 164.3 101.9 3 1 106.9 21 1984 144.0 98.0 0 0 97.1 22 2207 107.3 107.8 0 0 34.4 23 2224 105.8 98.8 6 10 35.3

*24 2256 78.2 9.8 10 10 77.4 25 2270 105.3 101.9 10 1 3 7. 9 26 2581 108.4 106.2 6 10 35.4 27 2655 109.6 107.4 0 0 36.0 28 2745 117.9 110.1 0 0 41. 7 29 2756 119.1 109.7 0 0 43.1 30 2783 103.8 45.7 6 1 41.9 31 2791 103.4 81.2 0 0 38.1

*32 2819 93.9 5.6 6 10 79.4 33 2830 113.8 112.5 0 0 36.3 34 2848 101.9 94.1 0 0 36.0 35 3003 137.9 105.0 3 10 88.5 36 3008 164.4 81.2 0 0 116.7 37 3021 110.9 106.2 6 1 39.3

*38 3384 57. 7 15.8 10 10 26.2 * Denotes a multiple event

'-1 (Jl

Table VI-B. Times of the central extremum, to, and pattern vectors for events of interest. p = 10.

Event to Mean Peak Perio- Pol- Var-Number (ms.) Freq. Freq. dicity arity iance

1 214 152.5 82.9 0 0 130.9 2 231 179.5 102.7 0 0 117.1 3 246 203.3 90.2 0 0 159.7 4 453 165.0 119.9 0 0 105.6 5 564 155.6 48.4 0 0 161.0 6 567 158.0 46.1 0 0 129.9 7 809 135.3 106.6 0 0 87.0 8 860 101.3 39.4 2 10 68.0 9 1118 151.6 0.0 0 0 138.9

*10 1128 144.6 0.0 10 10 175.7 11 1136 185.9 106.6 4 1 147.1 12 1200 163.5 120.7 4 10 104.9 13 1246 16 7. 3 135.5 0 0 90.9 14 1249 166.2 111. 7 0 0 86.2 15 1265 142.7 59. 7 0 0 89.2 16 1682 189.4 113.6 0 0 114.3

*17 1691 179.6 53.5 6 10 180.3 18 1715 145.4 110.5 1 10 86.9 19 1933 142.1 51.2 0 0 121.6

'-J * Denotes a multiple event. 0\

Table VI-B continued.

Event to Mean Peak Perio- Pol- Var-Number (ms.) Freq. Freq. dicity arity iance 20 19 79 165.7 102.3 0 0 111. 8 21 1984 150.3 98.0 0 0 104.3 22 2207 155.7 88.6 0 0 113.1 23 2224 158.6 101.5 0 0 129.7

*24 2255 169.5 0.0 6 10 195.5 25 2270 137.8 109.3 4 1 82.4 26 2581 145.8 110.1 6 10 102.5 27 2655 160.1 112.5 0 0 122.4 28 2746 163.6 101.5 0 0 123.0 29 2756 163.4 116.0 0 0 100.0 30 2782 16 7. 2 86. 7 10 1 124.1 31 2791 154.4 0.0 0 0 139.8

*32 2820 195.3 0.0 10 10 212.6 33 2831 172.6 108.9 2 10 138.0 34 2848 171.3 58.6 0 0 160.7 35 3003 161.1 107.8 10 10 139.2 36 3009 184.7 80.8 6 10 137.5 37 3021 147.7 108.9 0 0 98. 2

*38 3384 235.0 0.0 10 10 207.8

* Denotes a multiple event. '-.1 '-.1

Table VI-C. Times of the central extremum, to, and pattern vectors for events of interest. p = 5.

Event to Mean. Peak Perio- Pol- Var-Number (ms.) Freq. Freq. dicity arity iance

1 214 161.4 84.7 0 0 142.0 2 231 193.2 109.3 0 0 120.3 3 246 219.2 97.6 0 0 163.9 4 453 176.7 129.3 0 0 112.9 5 564 169.7 53.9 0 0 177.6 6 567. 171. 7 54.3 0 0 136.5 7 809 143.7 106.6 0 0 97.2 8 860 104.6 54.7 2 10 70.6 9 1118 162.7 0.0 0 0 151.0

*10 1128 163.2 0.0 10 10 193.0 11 1136 196.8 105.4 4 1 159.0 12 1200 176.9 125.0 4 10 113.7 13 1246 174.6 137.1 0 0 97.0 14 1249 173.8 120.3 0 0 91. 7 15 1265 150.9 65.2 0 0 99.8 16 1682 201.8 113.2 0 0 117.0

*17 1691 191.3 58.2 6 10 184.6 18 1715 154.0 115.2 3 1 90.9 19 1933 152.1 53.9 0 0 131.9 '-J

00 * Denotes a multiple event.

Table VI-C continued.

Event t Mean Peak Perio~ Pol- Var-Number (mg.) Freq. Freq. dicity arity iance

20 1979 166.8 108.2 0 0 112.3 21 1984 152.9 100.0 0 0 105.4 22 2208 166.3 80.8 0 0 125.2 23 2225 177.6 104.3 0 0 150.5

*24 2255 186.6 0.0 6 10 199.5 25 2270 157.7 115.2 4 1 101.5 26 2581 155.1 113.2 6 10 112.2 27 2655 172.4 116.8 0 0 134.6 28 2744 169.0 116.8 0 0 125.5 29 2756 176.0 119.9 0 0 110.0 30 2782 183.4 96.8 10 1 134.4 31 2791 166.6 0.0 0 0 153.3

*32 2 816 244.7 0.0 2 10 146.9 33 2831 178.0 105.0 2 10 149.1 34 2848 187.4 70.7 0 0 170.8 35 3003 168.7 105.8 10 10 149.5 36 3009 194.3 84. 7 6 10 144.0 37 3021 157.4 110.1 0 0 108.9

*38 3382 298.6 84.7 10 10 145.3

* Denotes a multiple event. '-.1 \.0

Table VI-D. Times of the central extremum, to, and pattern vectors for events of interest. P = 2.

Event to Mean Peak Pe rio- Pol- Var-Number (ms.) Freq. Freq. dicity arity iance

1 214 165.8 69.9 0 0 149.6 2 231 206.0 113.6 0 0 117.9 3 246 239.8 95.3 0 0 168.3 4 453 194.4 130.4 0 0 124.9 5 564 192.9 54.3 0 0 200.1 6 569 213.9 62.5 0 0 158.8 7 809 155.1 105.8 0 0 114.4 8 860 106.4 42.2 2 10 74.6 9 1118 174.6 0.0 0 0 168.3

*10 1128 187.3 0.0 10 10 211.1 11 1134 193.3 76.9 2 1 174.4 12 1200 193.3 132.0 2 10 121.5 13 1246 181.9 138.2 0 0 104.1 14 1249 180.8 117. 2 0 0 97.8 15 1265 162.0 62.9 0 0 117.9 16 1682 216.3 113.2 0 0 121.9

*17 1691 200.9 58.6 6 10 193.2 18 1715 160.0 116.4 1 10 87.5 19 1933 163.8 31. 2 0 0 147.2

00

* Denotes a multiple event. 0

Table VI-D continued.

Event to Mean Peak Perio- Pol- Var-Number (ms.) Freq. Freq. dicity arity iance 20 1979 168.6 105.4 0 0 117.6 21 1984 157.8 98.4 0 0 112.6 22 2208 178.5 0.0 0 0 138.5 23 2223 161.9 9 7. 6 0 0 119.3

*24 2255 201. 8 0.0 6 10 202.8 25 2270 163.8 113.2 4 1 105.3 26 2581 162.5 114.0 6 10 121.4 27 2654 191.7 119.5 2 10 153.0 28 2746 182.5 98.4 0 0 143.9 29 2756 190.3 117.9 0 0 129.3 30 2 782 206.4 94.1 10 1 145.3 31 2 791 178.5 0.0 4 1 171.4

*32 2820 214.9 0.0 10 10 199.8 33 2831 203.8 105.8 2 10 162.3 34 2848 202.7 28.1 3 1 184.1 35 3003 182.1 106.6 10 10 170.5 36 3009 20 7. 6 82.0 6 10 152.5 37 3021 169.6 109.3 0 0 120.9

*38 3382 249.4 70.3 10 10 182.4

* Denotes a multiple event. 00 1-1

APPENDIX D

LINEAR THRESHOLD ELEMENT DETAILS

A linear threshold element may be represented by

n

where x. 1 ' i = 1,

s = 2: i=l

,n is

x.w. 1 1

a pattern vector and w. 1

82

(D-1)

i = 1, ... ,n is a weighting vector such that the S value of

one class of patterns will be markedly different from the S

values of another class of patterns if, indeed, two classes

of patterns exist. Consider a total population of p patterns

and let the ith component of the kth pattern by represented

Then the S value for the kth pattern is given by

'

and the mean S value over the population is

p n

S = _1_ L L: x.kw. p k=l i=l 1 1 '

(D-2)

(D- 3)

which, upon interchanging the order of summation, may be

writ ten as

s =

or more simply,

n 2: i=l

p w.-

1- E 1 p k=l

n s = 2:

i=l x.w.

1 1

(D-4)

(D- 5)

where x. l

is the mean of the ith element of (x1

, 'X ) • n

83

If two separated classes exist in the n dimensional space,

the moment of inertia of the population will be a maximum

with respect to some axis within the space. Therefore, the

moment of inertia of the S line distribution, given by p

M L: k=l

(D-6)

will also become large if a weighting vector wi, i = 1, ... ,n

has forced the S values to be different for the two classes.

The value of M, however, can be made as large as desired

by allowing lw-1, i = 1, l

,n to increase without bound.

Hence, combining a constraint of the form

n 2 L: w.

i=l l min.

with equation (D-6) to yield n - 2 L (Sk-S)

k=l n A = L: i=l

w. l

2

(D- 7)

(D- 8)

results in a ratio, the maximization of which will tend to

maximize the moment of inertia while tending to minimize the

sum of the squares of w .. l

Before considering the maximization of A, an important

observation and some simplification of equation (D-8) are in

order. Utilizing equations (D-2) and (D-5), equation (D-6),

which is the numerator of equation (D-8), may be written as

p

M = L k=l [ ~ (x.k-x. )w. J . 1 l l l l=

84

(D-9)

Upon expansion, rearrangement of the summand, and inter­

changing the order of summation, one obtains

If

then

n M= L

i=l

n L: j=l

p

p w.w. L (x.k-x.)(x.k-x.)

l J k=l l l J J

L (x.k-x.)(x.k-x.) =b .. k= 1 l l J J l J

n n

M = L i=l

L: j=l

w. b .. w. l lJ J

(D-10)

(D-11)

(D-12)

Here the b .. are elements of a matrix which is proportional to lJ

the sample covariance matrix of the pattern vectors xk

(xlk' ... ,xnk), k = 1, ... ,p.

In the light of equation (D-12), ratio (D-8) may be

written as n L i=l

Differentiating equation

n

L: j=l n 2: i=l

(D-13)

b .. w.w. lJ l J

2 w· l

(D-13)

with respect to each w., and 1

equating the result to ·zero will yield the conditions required

to maximize A. Accordingly,

ClA

aw. 1

= 0 (D-14)

85

implies that

n n n 2: 2: b .. w. -A 2: w. = o i=l j=l lJ 1 i=l l

(D-15)

Equation (D-15) corresponds in matrix form to the relation

T (B-AI)W = 0 (D-16)

where I is the identity matrix. Equation (D-16) can have a

non-trivial solution only if

IB-AII = 0 (D-17)

Thus, the problem of maximizing equation (D-8) for A is

reduced to the problem of finding the largest eigenvalue

of the matrix B, methods for which, being well-known, will

not be discussed here. The set of weights w., i = 1, ]_

which will produce this maximum is the eigenvector corres-

ponding to A where the eigenvectors are related to the max.

eigenvalues by

'n'

(D-18)

The matrix B is real and symmetric and therefore the

eigenvalues of B are necessarily real. A proof of this can

be found in DeRusso et al (1966).

The eigenvector W corresponding to the eigenvalue Amax.

is the set of weights which maximizes the moment of inertia

of the S line distribution while minimizing the sum of the

squares of w .. l

These weights, when used in equation (D-1),

86

result in a variable S such that two classes of patterns in

an n dimensional space are displayed as two clusters on the

S line.

87

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93

VI I. VITA

David Nuse Peacock is the son of Walter H. Peacock and

Helen Nuse Peacock. Born March 16, 1943 in Washington, D.C.,

he received his primary education at St. Peter's School and

his secondary education at Point Pleasant Beach High School,

both in Point Pleasant Beach, New Jersey. He obtained his

B.S. in Geology from the University of Missouri School of

Mines and Metallurgy, Rolla, in 1964 and his M.S. in Geophys­

ical Engineering from the University of Missouri at Rolla in

1966. In September, 1966 he enrolled in the Mining and Petro­

leum Engineering Department of the University of Missouri at

Rolla as a Ph.D. candidate in Geophysical Engineering. During

his graduate studies he was employed for two summers by Geo­

physical Service, Inc. in New Orleans, La. and Dallas,Tex.

His studies were partly financed through a National Science

Foundation Traineeship and a Graduate Assistantship in the

Mining and Petroleum Engineering Department, where he was a

laboratory assistant in Geophysical Engineering. lle held

the position of Assistant Instructor in Mathematics at U.M.R.

for the academic year 1968-1969. He is a member of the Euro­

pean Association of Exploration Geophysicists, American Asso­

ciation for the Advancement of Science, New Jersey Academy of

Science, and is an associate member of the Society of Explor­

ation Geophysicists. In June, 1969 he married the former

Donna Ruth Koch of Rolla, Missouri.

1.93936