minesight for modelers.pdf

5/22/2018 MineSight for Modelers.pdf

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Introduction to Geostatistics

Objective: To make you familiar with thebasic concepts of statistics, and thegeostatistical tools available to solveproblems in geology and mining of an oredeposit

Classical Statistics

Sample values are realizations of a randomvariable

Samples are considered independent

Relative positions of the samples areignored

Does not make use of the spatialcorrelation of samples

Geostatistics

Sample values are realizations of randomfunctions

Samples are considered spatiallycorrelated

Value of a sample is a function of itsposition in the mineralization of the deposit

Relative position of the samples is takenunder consideration.

Topics

Basic Statistics

Data Analysis and Display

Analysis of Spatial Continuity (variogram)

Basic Statistics

Statistics

Geostatistics

Universe

Sampling Unit

Support

Population

Random Variable

Definitions

Statistics

The body of principles and methods fordealing with numerical data

Encompasses all operations from collectionand analysis of the data to the

interpretation of the results


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Geostatistics

Throughout this workbook, geostatistics will

refer only to the statistical methods and tools

used in ore reserve analysis

Universe

The source of all possible data (for example,

an ore deposit can be defined as the

universe; sometimes a universe may not

have well defined boundaries)

Sampling Unit

Part of the universe on which a measurement

is made (can be a core sample, channel

sample, a grab sample etc.; one must specify

the sampling unit when making statements

about a universe)

Support

Characteristics of the sampling unit

Refers to the size, shape and orientation ofthe sample (for example, drillhole coresamples will not have the same support asblasthole samples)

Population

Like universe, population refers to the totalcategory under consideration

It is possible to have different populationswithin the same universe (for example,

population of drillhole grades versuspopulation of blasthole grades; samplingunit and support must be specified)

Random Variable

A variable whose values are randomly

generated according to a probabilistic

mechanism (for example, the outcome of a

coin toss, or the grade of a core sample in adiamond drill hole)


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Frequency Distribution

Probability Density Function (pdf)

Discrete:

1. f(xi) 0 for xiR (R is the domain)2. f(xi) = 1

Continuous:

1.f(x) 02.f(x)dx = 1

Frequency Distr ibution

Cumulative Density Function (cdf)

Proportion of the population below a certain

value:

F(x) = P(Xx)1. 0F(x) 1 for all x2. F(x) is non decreasing

3. F(-)=0 and F()=1

Example

Assume the following population of

measurements:

1, 7, 1, 3, 2, 3, 11, 1, 7, 5

PD F

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5 6 7 8 9 10 11

CDF

0

0.1

0.2

0.3

0.40.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5 6 7 8 9 10 11

Descriptive Measures

Measures of location:

Mean

Median

Mode Min, Max

Quartiles

Percentiles


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Mean

m = 1/n xi

i=1,...,n

Arithmetic average of the data values

Mean

What is the mean of the example population:

1, 7, 1, 3, 2, 3, 11, 1, 7, 5

m =?

Mean

m= (1+ 7+ 1+ 3+ 2+ 3+ 11+ 1+ 7+ 5)/10=

= 41/10=

= 4.1

Mean

What is the mean if we remove highest

value?

Mean

m= (1+ 7+ 1+ 3+ 2+ 3+ 1+ 7+ 5)/9=

= 30/9=

= 3.33

Median

M = x(n+1)/2 if n is odd

M = [x n/2+x(n/2)+1]/2 if n is even

Midpoint of the data values if they are sortedin increasing order


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Median

What is the median of example population?

M=?

Median

Sort data in increasing order:

1, 1, 1, 2, 3, 3, 5, 7, 7 ,11

M = 3

Other

Mode

Minimum

Maximum

Quartiles

Deciles

Percentiles

Quantiles

Mode

The value that occurs most frequently

In our example:

Mode=?

Mode

1, 1, 1, 2, 3, 3, 5, 7, 7 ,11

Mode = 1

Quartiles

Split data in quarters

Q1 = 1st quartile

Q3 = 3rd

quartile

In example:

Q1=?

Q3=?


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Quartiles

1, 1, 1, 2, 3, 3, 5, 7, 7 ,11

Q1= 1

Q3= 6

Deciles, Percentiles,Quantiles

1, 1, 1, 2, 3, 3, 5, 7, 7 ,11

D1= 1

D3= 1

D9= 7

Mode on the PDF

Mode (also min)Mode (also min)

MaxMax

Mean on the PDF

Mean(=4.1)Mean(=4.1)

Median on the CDF Descriptive Measures

Measures of spread:

Variance

Standard Deviation

Interquartile Range


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Variance

S2 = 1/(n-1) (xi-m)2 i=1,...,n

Sensitive to outlier high values

Never negative

Variance

Example:1, 1, 1, 2, 3, 3, 5, 7, 7 ,11

M=4.1

S2= 1/9 {(1-4.1)2+ (1-4.1)2+ (1-4.1)2+ (2-4.1)2+ (3-4.1)2+

(3-4.1)2+ (5-4.1)2+ (7-4.1)2+ (7-4.1)2+ (11-4.1)2 } =

= 1/9 (9.61+ 9.61+ 9.61+ 4.41+ 1.21+ 1.21+ 0.81+ 8.41+

8.41+ 47.61) =

= 100.9/9 =

= 11.21

Variance

Remove high value:1, 1, 1, 2, 3, 3, 5, 7, 7

M=3.33

S2= 1/8 {(1-3.33)2+ (1-3.33)2+ (1-3.33)2+ (2-3.33)2+

(3-3.33)2+ (3-3.33)2+ (5-3.33)2+ (7-3.33)2+

(7-3.33)2 =

= 1/8 (5.43+ 5.43+ 5.43+1.769+ 0.109+ 0.109+ 2.789+

13.469+ 13.469) =

= 48/8 =

= 6

Standard Deviation

s = s2

Has the same units as the variable

Never negative

Standard Deviation

Example:

S2= 11.21

S = 3.348

S2 = 6

S =2.445

Interquartile Range

IQR = Q3 - Q1

Not used in mining very often


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Descriptive Measures

Measures of shape:

Skewness

Peakedness (kurtosis)

Coefficient of Variation

Skewness

Skewness = [1/n (xi-m)3] / s3

Third moment about the mean divided bythe cube of the std. dev.

Positive - tail to the right

Negative - tail to the left

Skewness

Example:

1, 1, 1, 2, 3, 3, 5, 7, 7 ,11

M=4.1

Sk= 1/10 {(1-4.1)3+ (1-4.1)3+ (1-4.1)3+ (2-4.1)3+

(3-4.1)3+ (3-4.1)3+ (5-4.1)3+ (7-4.1)3+

(7-4.1)3+ (11-4.1)3 } =

= 1/10 (-29.79-29.79-29.79-8.82-1.33 1.33+ 0.73+

24.39+ 24.39+328.51) =

= 277.2/10 =

= 27.72

Skewness

Remove high value:

1, 1, 1, 2, 3, 3, 5, 7, 7

M=3.3

Sk= 1/10 {(1-3.3)3+ (1-3.3)3+ (1-3.3)3+ (2-3.3)3+

(3-3.3)3+ (3-3.3)3+ (5-3.3)3+ (7-3.3)3+

(7-3.3)3 } =

= 1/10 (-12.17- 12.17- 12.17- 2.2- 0.03- 0.03+ 4.91+

50.65+ 50.65) =

= 67.44/9 =

= 7.49

Positive Skewness Peakedness

Peakedness = [1/n (xi-m)4] / s4

Fourth moment about the mean divided bythe fourth power of the std. dev.

Describes the degree to which the curvetends to be pointed or peaked

Higher values when the curve is peaked

Usefulness is limited


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CV = s/m

No units

Can be used to compare relative dispersionof values among different distributions

CV > 1 indicates high variability


In our example:

CV = 3.348/4.1 =0.817

Remove high value:

CV = 2.445/3.33=0.743

Normal Distribution

f(x) = 1 / (s 2) exp [-1/2 ((x-m)/s)2] symmetric, bell-shaped

68% of the values are within one std. dev.

95% of the values are within two std. dev.

Normal Dis tr ibution curve

Std. normal distribution

mean = 0 and s = 1

standardize any variable using:

z = (x-m) / s

Normal Dist ribution Tables

The cumulative distribution function F(x) isnot easily computed for the normaldistribution.

Extensive tables have been prepared to

simplify calculation

Most statistics books include tables for thestd. normal distribution


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Example of cdf (normal)

Find the proportion of sample values above 0.5 cutoff in a

normal population that has m =0.3, and s = 0.2Solution:

First, transform the cutoff, x0 , to unit normal.

z = (x0 - m) / s = (0.5 -0.3) / 0.2 = 1

Next, find the value of F(z) for z = 1. The value of F(1) = 0.8413from Table

Calculate the proportion of sample values above 0.5 cutoff,

P(x > 0.5), as follows:

P(x > 0.5) = 1 - P(x 0.5) = 1 - F(1) = 1 -0.8413 = 0.16 Therefore, 16% of the samples in the population are > 0.5

Lognormal Distribution

Logarithm of a random variable has a normal

distribution

f(x) = 1 / (x 2 ) e -u for x > 0, > 0where

u= (ln x - ) 2 / 22

= mean of logarithms= variance of logarithms

Conversion Formulas

Conversion formulas between the normaland lognormal distributions:

Lognormal to normal:

= exp (+2 /2)

2 = 2 [exp(2) - 1]Normal to lognormal:

= log -2 /2 2 = log [1 + (2 / 2)]

Lognormal Dis tr ibution Curve

Three-Parameter LN Distribution

Logarithm of a random variable plus a

constant, ln (x+c) is normally distributed

Constant c can be estimated by:c = (M2 - q1 q2 ) / (q1 + q2 + 2M)

Bivariate Distr ibution

Joint distribution of outcomes from tworandom variables X and Y:

F(x,y) = Prob {Xx, and Yy} In practice, it is estimated by the proportion

of pairs of data values jointly below therespective threshold values x, y.


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Statis tical analysis

To organize, understand, and/or describedata

To check for errors

To condense information

To uniformly exchange information

Error Checking

Avoid zero for defining missing values

Check for typographical errors

Sort data; examine extreme values

Plot sections and plan maps for coordinateerrors

Locate extreme values on map; Isolated?Trend?

Data Analysis and Display Tools

Frequency Distributions

Histograms

Cumulative Frequency Tables

Probability plots

Scatter Plots

Q-Q plots

Data Analysis and Display Tools

Correlation

Correlation Coefficient

Linear Regression

Data Location Maps

Contour Maps

Symbol Maps

Moving Window Statistics

Proportional Effect

Histograms

Visual picture of data and how they aredistributed

Bimodal distributions show up easily

Outlier high grades

Variability

# CUM. UPPER

FREQ. FREQ LIMIT 0 20 40 60 80 100

- -- - - - -- - - - - -- - +. .. . .. . .. +. . . .. . .. . +. . . .. . .. . +. . . .. . .. . + . .. . .. . ..+

86 .093 .100 +*****. +

34 .130 .200 +** . +

48 .182 .300 +*** . +

73 .261 .400 +**** . +

86 .354 .500 +***** . +

80 .440 .600 +**** . +

84 .531 .700 +***** . +

74 .611 .800 +**** . +

70 .686 .900 +**** . +

60 .751 1.000 +*** . +

43 .798 1 .100 +** . +

28 .828 1 .200 +** . +

29 .859 1 .300 +** . +

31 .893 1 .400 +** .+

25 .920 1.500 +* .+

19 .941 1.600 +* .

16 .958 1.700 +* .

8 . 9 66 1 .8 00 + .

9 . 9 76 1 .9 00 + .

3 . 9 79 2 .0 00 + .

6 . 9 86 2 .1 00 + .

4 . 9 90 2 .2 00 + .

1 . 9 91 2 .3 00 + .

3 . 9 95 2 .4 00 + .

3 . 9 98 2 .5 00 + .

1 . 9 99 2 .6 00 + .

0 . 9 99 2 .7 00 + .

0 . 9 99 3 .5 00 + .

0 . 9 99 3 .6 00 + .

0 . 9 99 3 .7 00 + .

1 1.000 3.800 + .

- - -- - - -- - - - -- - + . . . .. . . ..+ . . .. . .. . .+ . .. . .. . ..+ . . .. . .. . .+. . . .. . .. . +

925 1.000 0 20

Histogram in text file


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Histogram Plot Histograms with skewed data

Data values may not give a singleinformative histogram

One histogram may show the entire spread

of data, but another one may be required to

show details of small values.

Histograms with skewed data Cumulative Frequency TablesCUTOFF SAMPLES PERCENT MEAN C.V.

CU ABOVE AB OVE ABOVE

.000 2399.00 100.00 .5129 .8782

.200 1717.00 71.57 .6858 .6133

.400 1240.00 51.69 .8365 .4809

.600 840.00 35.01 1.0025 .3889

.800 522.00 21.76 1.1917 .3229

1.000 310.00 12.92 1.4012 .2663

1.200 205.00 8.55 1.5682 .2266

1.400 133.00 5.54 1.7165 .2106

1.600 72.00 3.00 1.9206 .2002

1.800 35.00 1.46 2.1697 .1966

2.000 21.00 .88 2.3614 .1947

2.200 11.00 .46 2.6118 .2006

2.400 6.00 .25 2.8667 .2134

2.600 2.00 .08 3.6550 .0174

2.800 2.00 .08 3.6550 .0174

3.000 2.00 .08 3.6550 .0174

3.200 2.00 .08 3.6550 .0174

3.400 2.00 .08 3.6550 .0174

3.600 2.00 .08 3.6550 .0174

Min. data value = .0000

Max. data value = 3.7000

Std. Deviation = .450

C.V. = Coeff. of variation = Standard deviation / mean

2399 Intervals used out of 2412

Probability Plots

Shows if distribution is normal or lognormal

Presence of multiple populations

Proportion of outlier high grades

Probability Plot


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Scatter Plots

Simply an x-y graph of the data

It shows how well two variables are related

Unusual data pairs show up

For skewed distributions, two scatter plots

may be required to show both details near

origin and overall relationship.

Scatter Plot

Linear Regression

y = ax + b

a = slope, b = constant of the line

a = r (y/x) b = my - amx

Linear Regression

Different ranges of data may be described

adequately by different regressions

Cu


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Q-Q plot Covariance

Covxy= 1/n (xi-mx)(yi-my) i=1,...,n

Where

mx = mean of x values and

my = mean of y values

High Positive Covariance

x-mx0

y-my0

mx

my

Covariance Near Zero

Large Negative Covariance Covariance

It is affected by the magnitude of the data

Values:

Multiply x and y values by C, then

covariance increases be C2

.


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Covariance

C=20.975

C = 2097.5

Correlation

Three scenarios between two variables:

Positively correlated

Negatively correlated

Uncorrelated


r = Covxy/ xy

r = 1, straight line, positive slope

r = -1, straight line, negative slope

r = 0, no correlation

May be affected by a few outliers

It removes the dependence on the

magnitude of the data values.


= 0.99


= -0.03


= -0.97


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It measures linear dependence

= -0.08

Data Location Map

Contour Maps Symbol Maps

Each grid location is represented by asymbol that denotes the class to which thevalue belongs

Designed for the line printer

Usually not to scale

Moving Window Statistics

Divide area into several local areas ofsame size

Calculate statistics for each smaller area

Useful to investigate anomalies in meanand variance

Proportional Effect

Mean and variability are both constant

Mean is constant, variability changes

Mean changes, variability is constant

Both mean and variability change


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Proportional Effect Plot


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Spatial continuity

H- scatter plots

Plot the value at each sample location

versus the value at a nearby location

Spatial continuity

A series of h-scatter plots for several

separation distances can show how the

spatial continuity decays with increasing

distance.

You can further summarize spatial

continuity by calculating some index of the

strength of the relationship seen in each

h-scatter plot.

Spatial continuity Moment of inertia

For a scatter plot that is roughly symmetric

about the line x=y, the moment of inertia

about this line can serve as a useful index of

the strength of the relationship.

= moment of inertia about x=y= average squared distance from x=y

=1/n [1/2 (xi-yi)2]

=1/2n (xi-yi)2

Moment of inertia

X

(X-Y)/2

(X,Y)

X-Y

Y

Variogram

Measures spatial correlation betweensamples

(h) = 1 / 2n [Z(xi) - Z(xi+h)]2

Semi-variogram will be referred asvariogram for convenience


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Variogram

Function of distance

Vector

Depends on distance and direction

Variogram parameters

Range

Sill

Nugget Effect

TYPE: NORMAL TRANSFORMATION: NONE VARIABLE: CU

FROM TO PAIRS DISTANCE DRIFT V(H) MEAN1 0 - 50 2666 30.9 .2536E-01 .1030E+00 .7488E+00

2 50 - 100 9734 79.2 -.8209E-03 .2186E+00 .8056E+003 100 - 150 23036 126.4 -.2306E-01 .2490E+00 .7981E+004 150 - 200 32117 175.6 -.1505E-01 .2560E+00 .7652E+005 200 - 250 45989 225.4 .2757E-02 .2601E+00 .7419E+006 250 - 300 47351 275.1 -.2589E-01 .2508E+00 .7286E+00

7 300 - 350 51794 324.7 -.2560E-01 .2505E+00 .7417E+008 350 - 400 46522 373.8 -.3154E-01 .2434E+00 .7270E+00

9 400 - 450 40313 424.5 -.4489E-01 .2448E+00 .7113E+0010 450 - 500 32113 473.7 -.4401E-01 .2270E+00 .6915E+00

.2790E+00 +

.2647E+00 + X

.2504E+00 + XX XX XX

.2361E+00 +

.2218E+00 + X X

.2075E+00 +

.1932E+00 +

.1789E+00 +

.1645E+00 +

.1502E+00 +

.1359E+00 +

.1216E+00 +

.1073E+00 + X

.9300E-01 +

.7869E-01 +

.6439E-01 +

.5008E-01 +

.3577E-01 +

.2146E-01 +

.7154E-02 +- - - - - - - + - - - - - - - - - +

250. 500

Sample variogram output Data for computation

Computation 1

For the first step (h=15), there are 4 pairs:1. x1 and x2 , or .14 and .282. x2 and x3 , or .28 and .193. x3 and x4 , or .19 and .104. x4 and x5 , or .10 and .09Therefore, for h=15, we get(15)=1/(2*4)[(x1-x2)2+(x2-x3)2+(x3-x4)2+(x4-x5)2 ]= 1/8 [ (.14-.28)2 + (.28-.19)2 + (.19-.10)2 + (.10-.09)2]= 0.125 [(-.14)2 + (.09)2 + (.09)2 + (.01)2 ]= 0.125 ( .0196 + .0081 + .0081 + .0001 )= 0.125 ( .0359 )(15) = 0.00448

Computation 2

For the second step (h=30), there are 3 pairs:1. x1 and x3 , or .14 and .192. x2 and x4 , or .28 and .103. x3 and x5 , or .19 and .09Therefore, for h=30, we get

(30) = 1/(2*3) [(x1-x3)2 + (x2-x4) 2 + (x3-x5)2 ]= 1/6 [(.14-.19)2 + (.28-.10)2 + (.19-.09)2 ]= 0.16667 [(-.05)2 + (.18)2 + (.10)2 ]= 0.16667 ( .0025 + .0324 + .0100 )= 0.16667 ( .0449 )

(30) = 0.00748


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Computation 3

For the third step (h=45), there are 2 pairs:1. x1 and x4 , or .14 and .102. x2 and x5 , or .28 and .09Therefore, for h=45, we get(45) = 1/(2*2) [(x1-x4 )2 + (x2-x5)2]= 1/4 [(.14-.10)2 + (.28-.09)2 ]= 0.25 [(.04)2 + (.19)2 ]= 0.25 ( .0016 + .0361 )= 0.25 ( .0377 )(45) = 0.00942

Computation 4

For the fourth step (h=60), there is only one pair:

x1 and x5 . The values for this pair are .14 and .09,respectively. Therefore, for h=60, we get(60) = 1/(2*1) (x1 - x5 ) 2

= (.14-.09)2

= 0.5 (.05)2

= 0.5 ( .0025 )(60) = 0.00125

If we take another step (h=75), we see that there areno more pairs. Therefore, the variogram calculationstops at h=60.

Class Size

Three possible options:

Lag distance = 50

0-50, 51-100, 101-151 etc..

Lag = 50, tolerance = 25

0-75, 75-125, 125-175 etc..

Lag = 50, strict tolerance = 25

0-25, 25-75, 75-125 etc..

Windows and Band Widths

Fitting a Theoretical Model

Draw the variance as the sill (c + c0 )

Project the first few points to the y-axis. This is anestimate of the nugget (c0 ).

Project the same line until it intercepts the sill. Thisdistance is two thirds of the range for sphericalmodel.

Using the estimates of range, sill, nugget and theequation of the mathematical model underconsideration, calculate a few points and see if thecurve fits the sample variogram.

If necessary, modify the parameters and repeat StepFour to obtain a better fit.

Variogram models

Spherical

Linear

Exponential

Gaussian Hole-Effect


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Variogram Models Variogram Models

Sample Variogram Plot Types of Anisotropy

Geometric

same sill and nugget, different ranges

Zonal

same nugget and range, different sills

Anisotropy Modeling Anisotropy

Geometric


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Modeling Anisotropy

Zonal

Modeling Geometrical Ani sot ropy: a recipe

Calculate variograms in different directions

Keeping nugget and sill the same, fit one-dimensional models to the samplevariograms in all directions

Make a rose diagram of ranges and find thedirection of the longest range

If diagram looks like a circle, no anisotropy.If diagram looks like an ellipse, there isanisotropy. Use ellipse pattern in searchparameters.

Rose diagram

0o90o

135o

Length of axes correspond to variogram ranges

45o

Variogram Contours

Nested Structures Variogram types

Normal

Relative

Logarithmic

Covariance Function Correlograms

Indicator Variograms

Cross Variograms


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Relative Variogram

R (h) = (h) / [m(h) + c]2

c is a constant parameter used in the case of a

three- parameter lognormal distribution.

Pairwise Relative Variogram:

PR (h) = 1/(2n) [(vi -vj ) 2/((vi +vj )/2)2 ]

vi and vj are the values of a pair of samples atlocations i and j, respectively.

Logarithmic Variogram

Variogram using the logarithms of the datainstead of the raw data

y = ln x or

y = ln (x + c) for 3-parameter lognormal

Reduces or eliminates the impact ofextreme data values on the variogramstructure

T ransformation from Logs

To transform log parameters back to normal values:

1. Ranges stay the same

2. Estimate the logarithmic mean () and variance (2). Usethe sill of the logarithmic variogram as the estimate of 2

3. Calculate the mean, () and the variance (2 ) of thenormal data:

= exp ( + 2/2) 2 = 2 [exp (2 ) -1]

4. Set the sill of the normal variogram = the variance ( 2 )5. Compute c (sill-nugget) and c0 (nugget) of the normal

variogram:

c = 2 [exp (clog ) - 1]

c0

= sill - c

Covariance Function Variograms

C(h) = 1/N [vi vj - m-h . m+h ] v1 ,...,vn are the data values

m-h is the mean of all the data valueswhose locations are -h away from someother data location.

m+h is the mean of all the data valueswhose locations are +h away from someother data location.

(h) = C(0) - C(h)

Correlograms

(h) = C(h) / ( -h . +h )-h is the standard deviation of all the data

values whose locations are -h away fromsome other data location:

2-h = 1/N (vi2 - m2-h )+h is the standard deviation of all the data

values whose locations are +h away fromsome other data location:

2+h = 1/N (vj 2 - m 2+h )

Indicator Variogram

1, if z(x) < zci(x;zc) ={

0, otherwise

where:x is location,zc is a specified cutoff value,z(x) is the value at location x.


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Cross Variograms

CR (h) = 1/2n [u(xi)-u(xi+h)]2 * [v(xi)-v(xi+h)]2

Used to describe cross-continuity betweentwo variables

Necessary for co-kriging and probabilitykriging

Cross Validation

Predicts a known data point using aninterpolation plan

Only the surrounding data points are usedto estimate this point, while leaving the datapoint out.

Other names: Point validation, jack-knifing

Cross Validation

The least amount of average estimationerror

Either the variance of the errors or theweighted square error (or variance) isclosest to the average kriging variance.

The weighted square error (WSE) is givenby the following equation:

WSE = [(1/i 2) (ei)2 ] / (1/i2)

Cross Validation Report

Variable : CU

ACTUAL KRIGING DIFF

Mean = 0.6991 0.7037 -0.0045

Std. Dev. = 0.5043 0.3870 0.2869

Minimum = 0.0000 0.0200 -0.9400

Maximum = 3.7000 2.1000 2.2100

Skewness = 1.0641 0.5634 1.3559

Peakedness= 2.0532 -0.0214 7.0010

Ave. kriging variance = 0.3890

Weighted square error = 0.0815


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The Necessi ty of Modeling

Suppose we have the data set below

It provides virtually no information about theentire profile

Deterministic Models

Depend on:

Context of data Outside information (not contained in data)

Probabilist ic Models

The variables of interest in earth sciencedata are typically the end result of vastnumber of processes whose complexinteractions cannot be describedquantitatively.

Probabilistic random function modelsrecognize this uncertainty and provide toolsfor estimating values at unknown locationsonce some assumptions about the statistical

characteristics of the phenomenon aremade.

Probabilistic Models

In a probabilistic model, available sampledata are viewed as the result of a randomprocess.

Data are not generated by a randomprocess; rather, their complexity appears asrandom behavior

Random Variables

A random variable is a variable whose

values are randomly generated according to

some probabilistic mechanism.

The result of throwing a die is a random

variable. There are 6 equally probable

values of this random variable: 1,2,3,4,5,6

Functions of Random Variables

Since the outcomes of a R.V. are numerical

values, we can define another random variable by

performing mathematical operations on the

outcome of a random variable.

Example: if D is the variable defined as the result

of throwing a die, 2D can be the variable defined

as the result of throwing the die and doubling the

result.


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Parameters of a Random Variable

The set of outcomes and their corresponding

probabilities is sometimes referred to as the

probability distribution of a random variable.

These probability distributions have

parameters that can be summarized.

Example: Min, Max etc


The complete distribution can not be

determined from the knowledge of only a few

parameters.

Two random variables may have the same

mean and variance but their distributions

may be different.


The parameters can not be calculated by

observing the outcomes of a random variable.

From a sequence of observed outcomes all we can

calculate is sample statistics based on that set of data.

Different set of data will produce different statistics.

As the number of outcomes increases, the sample

statistics becomes more similar to their model parameters.

In practice, we assume that the parameters of our random

variable are the same as the sample statistics.


The two most commonly parameters used in

probabilistic approaches to estimation are the

mean or expected value of the random

variable and its variance.

Expected value

Expected value of a random variable is itsmean or average outcome.

= E(x)

E(x) refers to expectation:

E(x) = - x f(x) dx

where f(x) is the probability density functionof the random variable x.

Variance of a Random Variable

The variance of a random variable is theexpected squared difference from the meanof the random variable.

2 = E (x-)2 = - (x-)2 f(x) dx

Std. dev. is


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Expected value

Example:

Define R.V. L=outcome of throwing two dice

and taking the larger of the two values.

What is the expected value of L?

E(L)=1/36 (1)+3/36 (2)+5/36 (3)

+7/36(4)+9/36 (5)+11/36 (6) =

= 4.47

Joint Random Variables

Random variables may also be generated in

pairs according to some probabilistic

mechanism; the outcome of one of the

variables may influence the outcome of the

other.

Covariance

The dependence between two randomvariables is described by covariance

Cov(x1 ,x2) = E {[x1 - E(x2)] [x2 - E(x2)]}

= E(x1 x2) - [E(x1)] [E(x2)]

Independence

Random variables are consideredindependent if the joint probability densityfunction satisfies:

p(x1 ,x2 ,...,xn) = p(x1) p(x2) ... p(xn)

i.e., probability of two event happening isthe product of each events probability

Expectation and variance

Properties:

C is a constant, then E(Cx) = C E(x)

If x1 , x2 , ..., xn have finite expectation, then

E(x1 +x2 ...+xn ) = E(x1) + E(x2) + ... + E(xn)

If C is a constant, then Var(Cx) = C2 Var(x)

If x1 , x2 , ..., xn are independent, then

Var(x1 +x2 ...+xn) = Var(x1)+Var(x2)+...+Var(xn)

Var(x+y) = Var(x) + Var(y) + 2 Cov(x,y)

Weighted Linear Combinations of Random Variables

Estimate is an outcome of a random variable

that is created by a weighted linear

combination of other random variables.

Expected value and Variance (same

definition as before)


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Random Functions

R.F. is a set of random variables that have

some spatial locations and whose

dependence on each other is specified by

some probabilistic mechanism.

Parameters of RF

The set of realizations of a random functionand their corresponding probabilities areoften referred as the probabilitydistribution

Like the histograms of sample values,these probability distributions haveparameters that summarize them

Random Functions

Parameters commonly used to summarize

the behavior of the random function:

Expected value

Variance

Covariance

Correlogram

Variogram

Reality vs Model

Reality:

sample values

summary statistics

Model:

possible outcomes with correspondingprobabilities of occurrence

parameters

It is important to recognize the distinction

between a model and the reality

Linear Estimators

all estimation methods involve weightedlinear combinations:

estimate = z* = wi z(xi) i = 1,...,n

The questions:

What are the weights, wi ?

What are the values, z(xi) ?

Desirable Properties

Desirable properties of an estimator:

Average error = 0 (unbiased)

E (Z - Z * ) = 0

where Z * is the estimate and Z is the true valueof the random variable

Error variance (spread of errors) is small

Var (Z - Z * ) = E (Z - Z * )2 = small

Robust

Question:

How to calculate the weights so that they satisfythe required properties?


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Random Process Assumptions

Strong stationarity

Second order stationarity

Intrinsic hypothesis

Stationarity

The independence of univariate and bivatiate

probability laws from the location x is referred

as stationarity.

(They may depend on separation distance h)

Strong Stationarity

In order for a random function Z(x) to meet the

strong stationarity requirement, the following

properties must be satisfied:

E[Z(x)] = m, m = finite and independent of x

No gradual increase or decrease in grade for some

specified direction (no drift).

Var[Z(x)]= 2 , 2 = finite and independent of xConstant parameter value of the underlying

density functions.

Second Order Stationarity

E[Z(x)] = m, m = finite and independent of x

E[Z(x+h). Z(x)] - m2 = C(h) = finite and independent of x

For each pair of random variables Z(x+h) and Z(x), the

covariance exists and depends only on the separation

distance h.

The covariance does not depend on the particular

location x within the deposit.

The stationarity of covariance implies the stationarity of

the variance as well as the variogram.

Under this assumption, the relationship between the

variogram and the covariogram is:

(h) = C(0) - C(h) = Var[Z(x)] - C(h)

Intrinsic Hypothesis

The intrinsic hypothesis of order zero:E[Z(x)] = m, m = finite and independent ofxE[Z(x+h)- Z(x)]2 = 2(h) = finite andindependent of x (variogram function)

We assume no drift , and the existenceand the stationarity of the variogram only.If condition of no drift in a deposit cannotbe satisfied, the intrinsic hypothesis oforder one is invoked.

Intrinsic Hypothesis

Intrinsic hypothesis of order one:E[Z(x+h)-Z(x)]=m(h)=finite and independent of x

E[Z(x+h)-Z(x)]2=2(h) = finite and independent of x The difference in the mean must be finite,

independent of the support point x, and dependonly on the separation distance h.

In performing local estimation using ordinarykriging, the intrinsic hypothesis of order zero is

invoked. Universal kriging may be employed underthe first order hypothesis.


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Ensuring Unbiasedness

Estimated value: Z* = i Z(xi)Estimated error: R* = Z*-Zo = iZ(xi) - ZoAverage error: r = 1/n R*Set expected value of average error to zero:

E{r} = E{1/n R*} = 1/n E{R*} = 0To guarantee that E{r} = 0, make E{R*} = 0

E{R*} = E{i Z(xi) - Zo}= iE{Z(xi)} - E{Zo}

Using the strong stationarity requirement:

E{Z(xi)} = E{Zo} = E{Z}

Therefore,

E{R*} =iE{Z} - E{Z} = 0 =>(i -1) E{Z} = 0 => i -1 = 0 => i =1

Ensuring Unbiasedness

Sum of weights, wi = 1 Two limitations:

The average error is not guaranteed to bezero, only the expected value

The result is valid only if the linearcombination belongs to the same statisticalpopulation

Estimation methods

Traditional:

Polygonal

Triangulation

Inverse distance

Geostatistical:

Kriging

Polygonal

Assigns all weight to nearest sample.

Advantages:

Easy to understand

Easy to calculate manually

Fast

Declustered global histogram

Disadvantages:

Discontinuous local estimates

Edge effect

No anisotropy

No error estimation

Triangulation

Weight at each triangle is proportional to the

area of the opposite sub triangle.

Advantages:

Easy to understand and calculate manually

Fast

Disadvantages: Not unique solution

Only three samples receive weights

Extrapolation?

3d?

No anisotropy

No error control

Inverse Distance

Each sample weight is inversely proportional tothe distance between the sample and the pointbeing estimated:

z* = [ (1/dip) z(xi ) ] / (1/ dip) i = 1,...,n

wherez* is the estimate of the grade of a block or a point,

z(xi) refers to sample grade,

p is an arbitrary exponent,

and n is the number of samples


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Inverse Distance

If p tends to zero =>local mean sample

If p tends to => nearest neighbor method(polygonal)

Traditionally, p = 2

Inverse Distance

Advantages:

Easy to understand Easy to implement

Flexible in adapting weights to differentestimation problems

Can be customized

Disadvantages:

Susceptible to data clustering

p?

No anisotropy

No error control

Ordinary kriging

Ordinary kriging is an estimator designedprimarily for the estimation of block gradesas a linear combination of available datain or near the block, such that estimate isunbiased and has minimum variance.

Definition:

Ordinary kriging

B.L.U.E. for best linear unbiasedestimator.

Linear because its estimates are weightedlinear combinations of available data

Unbiased since the sum of the weightsadds up to 1Best because it aims at minimizing thevariance of errors.

Kr iging Est imator

z* = wi z(xi ) i = 1,...,nwhere

z* is the estimate of the grade of a block ora point,

z(xi) refers to sample grade,

wi is the corresponding weight assigned toz(xi),

and n is the number of samples.

Kr iging Estimator

Desirable Properties:

Minimize 2 = F (w1, w2, w3,,wn) r = average error = 0 (unbiased)

wi = 1


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Error variance

Using R. F. model, you can express the error

variance as a function of R.F. parameters:2R=

2z + (ij Ci,j ) - 2 i C i,o

where

2z is the sample varianceCi,j is the covariance between samples

Ci,o is the covariance between samples and

location of estimation.

See Isaaks and Srivastava pg 281-284

Error variance

2R= 2z + (ij Ci,j ) - 2 i C i,o

Error increases as variance of dataincreases

Error variance increases as data becomemore redundant

Error variance decreases as data arecloser to the location of estimation

Ordinary Kriging

Minimize error

2R= 2z + (ij Ci,j ) - 2 i C i,o

i = 1 Use Lagrange method (Isaaks and

Srivastava, pg 284-285).

Result:

Ci,o = (i Ci,j) + i = 1

Kriging System (point)

Previous equation in matrix form:

Point Kriging (cont.)

Matrix C consists of the covariance values Cijbetween the random variables Vi and Vj at the

sample locations.

Vector D consists of the covariance values Ci0between the random variables Vi at the samplelocations and the random variable V0 at the

location where an estimate is needed.

Vector consists of the kriging weights and theLagrange multiplier.

Kriging System (block)


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Block Kriging (cont.)

In point kriging, the covariance matrix D

consists of point-to-point covariances. In blockkriging, it consists of block-to-point covariances. Covariance values CiA no longer a point-to-pointcovariance like Ci0 , but the average covariancebetween a particular sample and all of the pointswithin A:CiA = 1/A CijIn practice, the A is discretized using a number ofpoints in x, y and z directions to approximate CiA .

Kriging Variance

2ok= CAA - [(i CiA) + ]

Data independent

Block Discretization

To be considered:

Range of influence of the variogram used in

kriging.

Size of the blocks with respect to this range.

Horizontal and vertical anisotropy ratios.

Advantages of kriging

Takes into account spatial continuitycharacteristics

Built-in declustering capability

Exact estimator

Calculates the kriging variance for eachblock

Robust

Disadvantages of kriging

computer required

prior variography required

more time consuming

smoothing effect

Assumptions

No drift is present in the data(Stationarity hypothesis)

Both variance and covariance exist and arefinite.

The mean grade of the deposit is unknown.


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Effect of scale Effect of shape

Nugget Effect Effect of range

Effect of Anisotropy Search Strategy

Define a search neighborhood within whicha specified number of samples is used

If anisotropy, use an ellipsoidal search

Orientation of this ellipse is important

If no anisotropy, search ellipse becomes acircle and the question of orientation is nolonger relevant


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Search Strategy Include at least a ring of drill holes with enough

samples around the blocks to be estimated

Dont extend the grades of the peripheral holes tothe undrilled areas too far

Increasing vertical search distance has moreimpact on number of samples available for a given

block, than increasing horizontal search distance (invertically oriented drillholes)

Limit the number of samples used from each

individual drillhole

Search strategy (cont.)

Octant or Quadrant Search Importance of kriging plan

An easily overlooked assumption in every estimate

is the fact the sample values used in the weighted

linear combination are somehow relevant, and that

they belong to the same group or population, as the

point being estimated. Deciding which samples are

relevant for the estimation of a particular point or a

block may be more important than the choice of an

estimation method.

Declustering

Clustering in high grade area:

Nave mean=(0+1+3+1+7+6+5+6+2+4+0+1)/12 = 3

Declustered mean=[(0+1+3+1+2+4+0+1) +(7+6+5+6)/4] /9 ==2

Declustering

Clustering in mean grade area:

Nave mean=(7+1+3+1+0+6+5+1+2+4+0+6)/12 = 3

Declustered mean=[(7+1+3+1+2+4+0+6) +(0+6+5+1)/4] /9 ==3


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Declustering

Clustering in low grade area:

Nave mean=(7+1+6+1+0+3+4+1+2+5+0+6)/12 = 3

Declustered mean=[(7+1+6+1+2+5+0+6) +(0+3+4+1)/4] /9 ==3.33

Declustering

Data with no correlation, do no need

declustering (pure nugget effect model)

If variogram model has a long range andlow nugget, you may need to decluster.

Declustering

Cell declustering

Polygonal

Cell Declustering

Each datum is weighted by the inverse ofthe number of data in the cell

Polygonal Declustered Global Mean

DGM = (wi . vi ) / wi i=1,...,n

where n is the number of samples, wi arethe declustering weights assigned to eachsample, and vi are the sample values. Thedenominator acts as a factor to standardizethe weights so that they add up to 1.


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Cross Validation

To check how well the estimationprocedure can be expected to perform.

Temporarily discard the sample value at a

particular location and then estimate the

value at that location using the remaining

values.

Cross validation

It may suggest improvements

It compares, does not determineparameters

Reveals weaknesses/shortcomings

Cross validation

Check:

Histogram of errors

Scatter plots of actual versus estimate

Cross validation

Remember:

All conclusions are based on observations

of errors at locations were we do not need

estimates.

We remove values that, after all, we are

going to use.

Quantifying Uncertainty

One approach:

Assume that the distribution of errors isNormal

Assume that the ordinary kriging estimate

provides the mean of the normaldistribution

Build 95 percent confidence intervals bytaking 2 standard deviations either of theOK estimate


Kriging Variance2ok= CAA - [(i CiA) + ]AdvantagesDoes not depend on dataIt can be calculated before sample data are

available (from previous/know variography)

Disadvantages

Does not depend on data

If proportional effect exists, previous assumptions

are not true


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Same Kriging Variance!!!


Other approach

Incorporate the grade in the error variance

calculation:

Relative Variance = Kriging Variance /Squareof Kriged Grade


Combined Variance = sqrt (local variance *kriging variance)

where local variance of the weighted average (2w ) is:

2w = w2i * (Z0- zi )2 i = 1, n (n>1)where

n is the number of data used,

wi are the weights corresponding to each datum,

Z0 is the block estimate,

and zi are the data values.


Relative Variability Index(RVI) =SQRT(Combined Variance) / Kriged Grade

Change of Support

N = 4M = 8.825

Change of Support

N = 16

M = 8.825


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Change of Support

>10N = 2 = 50%M = 11.15

Change of Support

>10

N = 5 = 31%

M =18.6

Change of Support

The mean above 0.0 cutoff does notchange with a change in support

The variance of block distributiondecreases with larger support

The shape of the distribution tends tobecome symmetrical as the supportincreases

Recovered quantities depend on block size

Affine Correction

Assumptions: The distribution of block or SMU grades hassame shape as the distribution of point orcomposite samples. The ratio of the variances, i.e., variance ofblock grades (or the SMU grades) over that ofpoint grades is non-conditional to surroundingdata used for estimation.

Kriges Relation

2p = 2b + 2 pb

2p = Dispersion variance of composites in thedeposit (sill)2b = Dispersion variance of blocks in the deposit

2 pb = Dispersion variance of points in blocks

This is the spatial complement to the partitioning ofvariances which simply says that the variance ofpoint values is equal to the variance of blockvalues plus the variance of points withinblocks.

Kriges Relation (contd)

Total 2 = between block 2 + within block 2

2p = calculated directly from the composite orblasthole data

2 pb = calculated by integrating the variogramover the block b

2b = calculated using the Kriges relation:2b = 2p -2 pb


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Kriges Relation (contd)

How to calculate 2 pb ?

Integrating the variogram over a block provides

variance of points within the block

2 pb = block = 1/n2 (hi,j)

Calculation of A.C.

K2 = 2b/ 2p 1

(from the variogram averaging):

K2 = [ (D,D) -(smu,smu) ] / (D,D)= 1 - [ (smu,smu) / (D,D) ] 1

Affine correction factor, K = K2 1

Affine Correction (cont.)

Use affine correction if:

(2p -2b) /

2p 30%

Affine correction of Variance

Indirect Lognormal method

Assumption: all distributions are lognormal;

the shape of distribution changes with changes in

variance.

Transform:

znew = az

b

old

a = Function of (m, new ,old ,CV)b = Function of (new,old,CV), see the notes

CV: coefficient of variation = old/ mold

Indirect Lognormal method

Disadvantage:

If the original distribution departs from log

normality, the new mean may require rescaling:

znew = (mold/mnew) zold


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Change of Support (other)

Hermite Polynomials:

Declustered composites are transformedinto a Gaussian distribution

Volume-variance correction is done on theGaussian distribution

Then this distribution is back transformedusing inverse Hermite Polynomials

Change of Support (other)

Conditional Simulation:

Simulate a realization of the composite (orblasthole) grades on a very closely spacedgrid (for example, 1x1)

Average simulated grades to obtainsimulated block grades

Change of Support (applications)

Design a search strategy:

Decluster composites/variogram

Define SMU units

Apply change of support from composites to SMU

Calculate the SMU GT curves.

Guess at a search scenario

Krige blocks => create GT curves

Compare GT curves of block estimates to GT

curves of SMUs

Adjust search scenario etc..

GT: grade tonnage curves


Reconciliation between BH model and

Exploration model:

Calculate GT curves of exploration model

Apply change of support from BH model toExploration model

Calculate the adjusted BH model GTcurves.

Compare GT curves of block estimates toGT of adjusted BH model estimates.

C. of S. for Ore Grade/Tonnage Estimation Equivalent Cutoff Calculation

(zp - m) / p = (zsmu - m) / smuzp = the equivalent cutoff grade to be appliedto the point (or composite) distributionm = mean of composite and SMU distribution

p = square root of composite dispersionvariancezsmu = the cutoff grade applied to the SMUm = mean of composite and SMU distributionsmu = square root of SMU dispersionvariance


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Equivalent Cutoff Calculation

zp = ( p/ smu ) zsmu + m [1 - ( p/ smu )]

The ratio p/ smu is basically the inverseof the affine correction factor K.

This ratio is 1.

Numeric Example

Let the mean of composites = 0.0445, and

the specified cutoff grade zsmu = 0.055

If the ratio p/ smu = 1.23, what is theequivalent cutoff grade?

zp=1.23 (0.055) + 0.0445 (1 - 1.23) =0.0574

Therefore, the equivalent cutoff grade to beapplied to the composite distribution is0.0574.

Equivalent Cutoff

if the specified cutoff grade is less than themean, the equivalent cutoff grade becomesless than the cutoff

if the specified cutoff grade is greater thanthe mean, the equivalent cutoff gradebecomes greater than the cutoff.


Other:

Almost required in MIK


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Simple Kriging

Z*sk = i [Z(xi ) - m] + m i = 1,...,n

Z*sk - estimate of the grade of a block or apointZ(xi ) - refers to sample gradei - corresponding simple kriging weightsassigned to Z(xi )n - number of samplesm = E{Z(x)} - location dependent expectedvalue of Z(x).

Cokriging

Suitable when the primary variable has not

been sampled sufficiently.Precision of the estimation may beimproved by considering the spatialcorrelations between the primary variableand a better-sampled variable.Example: extensive data from blastholesas the secondary variable - Widely spacedexploration data as the primary variable.

Cokriging

....................................... ..... ...........

[Cov{didi}] [Cov{dibj}] [1] [0] [ i] [Cov{x0di}]

....................................... ..... ...........

[Cov{dibj}] [Cov{bjbj}] [0] [1] [ j] [Cov{x0bj}]

....................................... x ..... = ...........

[ 1 ] [ 0 ] 0 0 d 1

....................................... ..... ...........

[ 0 ] [ 1 ] 0 0 b 0

....................................... ..... ...........

[Cov{didi}] = drillhole data (dhs) covariance matrix, i=1,n

[Cov{bjbj}] = blasthole data (bhs) covariance matrix, j=1,m

[Cov{dibj}] = cross-covariance matrix for dhs and bhs

[Cov{x0di}] = drillhole data to block covariances

[Cov{x0bj}] = blasthole data to block covariances

[ i] = Weights for drillhole data

[ j] = Weights for blasthole data

dand b= Lagrange multipliers

[Cov{didi}] = drillhole data (dhs) covariance matrix, i=1,n

[Cov{bjbj}] = blasthole data (bhs) covariance matrix, j=1,m

[Cov{dibj}] = cross-covariance matrix for dhs and bhs

[Cov{x0di}] = drillhole data to block covariances

[Cov{x0bj}] = blasthole data to block covariances

[ i] = Weights for drillhole data

[ j] = Weights for blasthole data

dand b= Lagrange multipliers

Cokriging-steps fo r Dr ill and B lasthole data

Regularize blasthole data into a specified block size.Block size could be the same as the size of the modelblocks to be valued, or a discreet sub-division of suchblocks. A new data base of average blasthole blockvalues is thus established.

Variogram analysis of drillhole data.

Variogram analysis of blasthole data.

Cross-variogram analysis between drill and blastholedata. Pair each drillhole value with all blasthole values.

Selection of search and interpolation parameters.

Cokriging.

Universal Kriging Outlier Restricted Kriging

Determine the outlier cutoff gradeAssign indicators to the composites based onthe cutoff grade

0 if the grade is below the cutoff

1 otherwiseUse OK with indicator variogram, or simplyuse IDS , or any other method to assign theprobability of a block to have grade above theoutlier cutoff.Modify Kriging matrix.


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ORK matrix Nearest Neighbor Kriging

Utilize nearest samples (assign more weight)

Non-Linear kriging methods

Indicator krigingProbability krigingLognormal krigingMulti-Gaussian krigingLognormal short-cutDisjunctive kriging

Parametric (assumptions about distributions)or non-parametric (distribution-free)

Why Non-Linear To overcome problems encountered withoutliers To provide better estimates than thoseprovided by linear methods To take advantage of the properties on non-normal distributions of data and therebyprovide more optimal estimates To provide answers to non-linear problems To provide estimates of distributions on ascale different from that of the data (the

change of support problem)

Indicator Kriging

Suppose that equal weighting of N given samples is used

to estimate the probability that the grade of ore at a

specified location is below a cutoff grade.

The proportion of N samples that are below this cutoff

grade can be taken as the probability that grade estimated is

below this cutoff grade.

Indicator kriging obtains a cumulative probability distribution

at a given location in a similar manner, except that it assigns

different weights to surrounding samples using the ordinary

Kriging technique to minimize the estimation variance.

Indicator Kriging

The basis of indicator kriging is the indicatorfunction:At each point x in the deposit, consider thefollowing indicator function of zc defined as:

1, if z(x) < zc

i(x;zc ) =0, otherwise



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Indicator Kriging

Examples:

Separate continuous variables into categories:

I(x) = 1 if k(x) 30, 0 if k(x) >30Characterize categorical variables and

differentiate types:

I(x) = 1 for heterozygote, 0 for homozygote

Indicator Kriging (applications)

Some drill holes have encountered a particular

horizon, some were not drilled deep enough, some

penetrated the horizon but the core or the log is

missing:

Use I(x) = 1 for drill hole assays above the horizon

and I(x) = 0 for assays below the horizon. Use

indicator kriging and calculate the probability of the

missing assays to be 1 or 0.


Some data may represent a spatial mixture of two

or more statistical populations (for example, clay

and sand.

Separate populations:

I(x) = 1 for clay, 0 for sand.

Then calculate the probability of an unsampled

location to be clay or sand.

Krige (local estimates) unsampled locations using

only data belonging to that population

Final estimate can be a weighted (by

probabilities) average of the local estimates.


Extreme values:

Separate population to 1 and 0 based on

outlier cutoff. Proceed then as though you

are dealing with two spatially mixed

populations.

Multiple Indicator Kriging

Same as indicator kriging but instead of one

cutoff, we use a series of cutoffs.

Multiple Indicator Kriging

THE INDICATOR FUNCTION:At each point x in the deposit, consider thefollowing indicator function of zc definedas:

1, if z(x) < zci(x;zc ) =0, otherwise



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Indicator Function at point x The (A;zc) function

(A;zc ) = 1/AA i(x;zc ) dx [0,1]

Proportion of Values z(x) zc within area A Local Recovery Functions

Tonnage point recovery factor in A:

t*(A;zc) = 1 -(A;zc)

Quantity of metal recovery factor in A:

q*(A;zc) = zc u d (A;u)

A discrete approximation of this integral is given by

q*(A;zc) = 1/2 (zj + zj-1) [*(A;zj) -*(A;zj-1) ] j=2,...,n

Local Recovery Functions

This approximation sums the product ofmedian cutoff grade and median (A;zc)proportion for each cutoff grade increment.The mean ore grade at cutoff zc gives the

mean block grade above the specified cutoffvalue.

Mean ore grade at cutoff zc :m*(A;zc) = q*(A;zc) / t*(A;zc)

Est imation of (A;zc)

(A;zc) proportion of grades z(x) below cutoff zc within panelA. (unknown since i(x;zc) known at only a finite number ofpoints).

(A;zc) = 1/n i(xj ;zc) j=1,...,nor

(A;zc) = j i(xj ;zc) xj D j=1,...,N

where n is the number of samples in the panel A,N is the number of samples in search volume D,

j are the weights assigned to the samples,

j = 1, and usually N >> n.

Ordinary kriging is used to estimate (A;zc) from the indicatordata i(xj ;zc). We use a random function model for i(xj ;zc),which will be designated by I(xj ;zc).


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Indicator Variography

I(h;zc ) = 1/2 E [ I(x+h);zc ) - I(x;zc ) ]2

Median Indicator Variogram

m(h;zm ) = 1/2n [ I(xj+m+h);zm ) - I(xj;zm ) ]2

j=1,,n

Indicator variogram where cutoff corresponds

to median of data

Order Relations Advantages of MIK

It estimates the local recoverable reserveswithin each panel or block.

It provides an unbiased estimate of therecovered tonnage at any cutoff of interest.

It is non-parametric, i.e., no assumption isrequired concerning distribution of grades.

It can handle highly variable data.

It takes into account influence of neighboringdata and continuity of mineralization.

Disadvantages of MIK

It may be necessary to compute and fit avariogram for each cutoff.

Estimators for various cutoff values may not showthe expected order relations.

Mine planning and pit design using MIK resultscan be more complicated than conventionalmethods.

Correlation between indicator functions of variouscutoff values are not utilized. More informationbecomes available through the indicator crossvariograms and subsequent cokriging. These formthe basis of the Probability Kriging technique.

Change of Support

Function *(A;zc) and grade-tonnagerelationship for each block is based ondistribution point samples (composites).

Selective mining unit (SMU) volume is muchlarger than sample volume, therefore, onemust perform a volume-variance correction tothe initial grade-tonnage curve of each block.


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Affine Correction

Equation for affine correction of any

panel or block is given by*v (A;z) = * (A;zadj)wherezadj=adjusted cutoff grade = K(z - ma)+ma

Use affine correction if:

(2p -2b) /2 p 30%

Grade Zoning

Grade zoning is usually applied to controlthe extrapolation of grades into statisticallydifferent populations

Often grade zones or mineralizationenvelopes correspond to different geologicunits

Grade Zoning (contd)

Determine how the grade populations areseparated spatially

Is there a reasonably sharp discontinuitybetween the grades of the differentpopulations?

Or is there a larger transition zone betweenthe grades of the different populations?


Discontinuity between grade populations:


Transition zone between gradepopulations:


Discontinuity between the gradepopulations is best modeled using adeterministic model, i.e., digitized theoutlines

Transition zone between the gradepopulations is best modeled using aprobabilistic model, i.e., indicator kriging


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Characterizing the contact betweendifferent spatial populations:

Calculate the difference between theaverage grades within each population as afunction of distance from the contact:

Dzi = zi - z(-i)


If the average difference in grade Dzi vs

distance from the contact is more or lessconstant, then there is probably adiscontinuity between the differentpopulations :


If the average difference in grade Dzi vsdistance from the contact is small for smalldistances but increases with increasingdistance, then there is likely a transitionzone between the different populations:

Grade Zone Bias Check

Often mineralization envelopes lead tobiased ore reserve models. To check:

Interpolate using the nearest neighbor(polygonal) method)

Use the search parameters corresponding to

the model of spatial continuity

Disregard all grade zoning

Compare at 0.0 cutoff grade, the tons andgrade of the polygonal model to those of themineralization envelope model.


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model nuggetSill (without

nugget)Ranges

Directions(MEDS)

EXP 0.007 0.078 80/60/60 45/0/0


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DIST EAST NORTH ELEV VALUE DH

84.24 2628.40 5432.60 2435.0 0.6300 53

86.58 2628.40 5432.60 2450.0 0.3700 53

93.25 2628.40 5432.60 2465.0 0.3100 53

93.25 2628.40 5432.60 2405.0 0.4700 53

103.42 2628.40 5432.60 2390.0 0.6100 53

103.42 2628.40 5432.60 2480.0 0.4100 53

103.69 2728.30 5439.10 2480.0 0.7300 54

132.10 2618.80 5558.90 2435.0 0.0100 61

133.60 2618.80 5558.90 2420.0 0.0000 61

133.60 2618.80 5558.90 2450.0 0.0100 61

138.02 2618.80 5558.90 2465.0 0.0000 61

138.02 2618.80 5558.90 2405.0 0.0100 61

145.09 2618.80 5558.90 2390.0 0.0000 61

145.09 2618.80 5558.90 2480.0 0.0000 61

158.24 2829.60 5537.40 2435.0 0.3300 62

159.50 2829.60 5537.40 2420.0 0.2300 62


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77.99 2728.30 5439.10 2480.0 0.7300 54

84.20 2628.40 5432.60 2435.0 0.6300 53

85.52 2628.40 5432.60 2450.0 0.3700 53

89.38 2628.40 5432.60 2465.0 0.3100 53

89.38 2628.40 5432.60 2405.0 0.4700 53

95.47 2628.40 5432.60 2390.0 0.6100 53

95.47 2628.40 5432.60 2480.0 0.4100 53


84.20 2628.40 5432.60 2435.0 0.6300 53

85.52 2628.40 5432.60 2450.0 0.3700 53

89.38 2628.40 5432.60 2465.0 0.3100 53

89.38 2628.40 5432.60 2405.0 0.4700 53

95.47 2628.40 5432.60 2390.0 0.6100 53

95.47 2628.40 5432.60 2480.0 0.4100 53


84.20 2628.40 5432.60 2435.0 0.6300 53

85.52 2628.40 5432.60 2450.0 0.3700 53

89.38 2628.40 5432.60 2465.0 0.3100 53

89.38 2628.40 5432.60 2405.0 0.4700 53

95.47 2628.40 5432.60 2390.0 0.6100 53

95.47 2628.40 5432.60 2480.0 0.4100 53

99.08 2618.80 5558.90 2435.0 0.0100 61


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Cutoffs 0.42 0.56 0.68 0.78 0.88 0.98 1.12 1.30 1.46 1.66


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Cutoff MeanVariogram

typeNugget Sill-nugget Range

0.42 0.1907 3 0.05 0.20 75

0.56 0.4802 3 0.05 0.20 750.68 0.6132 3 0.05 0.20 63

0.78 0.7241 3 0.06 0.18 63

0.88 0.8232 3 0.06 0.17 57

0.98 0.9199 1 0.05 0.15 51

1.12 1.0279 1 0.03 0.13 41

1.30 1.2050 1 0.03 0.10 41

1.46 1.3727 1 0.03 0.06 41

1.66 1.5539 1 0.02 0.03 32


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minesight for modelers.pdf

Documents

results5222018 minesight

core sample

grab sample

possible data

orientation ofthe sample

numerical data

randomvariable samples

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