4.5geostatistics

UNSW Mining Engineering

INTRODUCTORY GEOSTATISTICS

Point Estimation Techniques; Ordinary Kriging

(The following notes are largely based on An Introduction to Applied Geostatistics, 1989, byEdward H. Isaaks and R. Mohan Srivastava and the illustrations are copied from thattextbook.)

Resources are generally estimated on the basis of three dimensionally oriented sample data.The sample data are used to interpolate values at regularly distributed points lacking sampledata. The regularly distributed, estimated values are then used to estimate average values forblocks and sub-blocks occupying the resource volume. For tabular ore bodies, coal seamsand the like, the data distribution may be effectively two dimensional. The polygonal,triangulation, local sample mean and inverse distance methods of estimating point values willbe illustrated by a two dimensional case where a single point estimate is made, based onsample values from 7 points around that point. The techniques involved and the strengthsand weaknesses of these four methods are illustrated by the following tables and figures, 1 to5.

Ordinary kriging is also used to estimate the same point value. Tables and figures, 6 to 9,illustrate the technique and demonstrate that kriging is much more computationally intensivethan the non-geostatistical methods. The ordinary kriging estimate is as inaccurate as theothers because the available sample data is inadequate to permit accurate estimation by anymethod. All of the techniques are designed to produce unbiased estimates by ensuring that theweights applied to each of the known samples sum to 1. Ordinary kriging is designed to domore, to be the "best, linear, unbiased estimator". Again, the weights sum to 1 but, inaddition, they are calculated to minimise the variance of the errors. Since the true values areunknown, the errors can only be determined against the random function model selected asbest representing the real distribution of sample values. Minimising the variance of the errorshas the unfortunate side effect of excessively smoothing ordinary kriging estimates.

The effects of bias and the spread of variance about the mean error are illustrated on figure10. Globally unbiased and conditionally biased and conditionally unbiased estimation errorsare illustrated on figure 11.

Geostatisticians normally calculate the variogram values at all the available lags for thesample data and select a positive definite (capable of a unique solution) random functionmodel (or nested set of models) which best fits the spatial continuity revealed by thatexperimental variogram. The covariance is then used for solving the ordinary kriging system.In working the single point estimation, only the covariance function was used. This ignoredthe possibility of anisotropy in the distribution of values. In many cases, the spatialcontinuity functions will be different in different directions. Such anisotropy is usuallycaptured in three directions in the kriging system. For a tabular ore body, the geologicaldescription of those three directions would be along strike, down dip and at right angles to thedistribution of the ore body. The selected random function model can only be used for pointestimations within the domain for which the model approximates the true distribution of datavalues. The domains may be defined by variography but they usually reflect geologicalcontrols of mineralisation.

Block estimates are based on averaging the point estimates from regularly distributed pointswithin the block. Discretisation, determining the number of point estimates required, aims toidentify the number above which more point estimates do not significantly alter the estimate.As the number of point estimates increases exponentially, the lowest acceptable number cansave a great deal of computation time. Average block grade estimates can be calculated

MFPPE Introductory Geostatistics

UNSW Mining Engineering

directly from the kriging equations, providing they are modified to take account of thecoordinates of the discretising points. This makes for more complex kriging equations butreduces computation substantially.

Ordinary kriging can be a good estimator of block grade and a better estimator of globalgrade. It is sensitive to extremes of grade; smoothing can spread high grades into low gradeareas and vice versa. Estimating grades of small blocks from widely spaced data can result inlarge estimation errors. The aim of minimising error variance means that ordinary kriging isan optimisation which may not suit the needs of determining mining block grades in order toseparate ore from waste.

Alternatives to ordinary kriging include the non-linear kriging methods: Indicator Kriging,Log-normal Kriging, Disjunctive Kriging, Uniform Conditioning and Multi-GaussianKriging. These are better able to deal with the problems of extreme sample values. Thekriging estimates produced by these methods can be interpreted directly as a conditionalprobability or as the mean of a conditional distribution. Indicator Kriging (including MultiIndicator Kriging) is generally the best of the non-linear kriging methods (personal opinion).The estimates it produces are conditional probabilities.

Conditional simulation endeavours to generate a set of simulated values on a specified spatialgrid where:

the histogram and the variogram of the simulated values are very similar to that of theconditioning data;

the simulated values are the same as the conditioning data values at data locations.

If several geostatistical populations are defined, the simulated values in each of thepopulations should have the above properties. A properly constructed ConditionalSimulation may be considered as one possible reality in that it honours the known propertiesof reality. It can provide a model of uncertainty for the grade of a mineable block which canbe integrated with a loss function for a particular mining operation to achieve optimal oreselection. Conditional simulation is not compatible with current pit optimisation software.

MFPPE Introductory Geostatistics 2

60 70 80

SampleNo. X Y V

Distancefrom

65E,137N1 225 61 139 477 4.52 437 63 140 696 3.63 367 64 129 227 8.14 52 68 128 646 9.55 259 71 140 606 6.76 436 73 141 791 8.97 366 75 128 783 13.5

140

130

An Introduction to Applied Geostatistics

Table 11.1 Distances to sample values in the vicinity of 65E,137N

Figure 11.1 The data configuration shown in this figure is used to illustrateseveral point estimation methods in the following sections. The goal is to estimatethe value of V at the point 65E,137N, located by the arrow, from the surroundingseven V data values.

791

140 -477

130

696 6

227 753645

(a)

140

130

60

70

80

(b)

60

70

80

254 An Introduction to Applied Geostatistics

Figure 11.4 (a) The polygons of influence and (b), Delaunay triangles for thedata configuration given in Figure 10.1.


Figure 11.2 A perspective view showing the discontinuities inherent in polygonalestimates.

696


AOIK

Figure 11.5 An example showing how three nearest data can be weighted bytriangular areas to form a point estimate. The data are located at the corners ofthe triangle. The data value at I is weighted by the triangle area AOJK, at J by

area AO1K, and at K by Aon.

— (22.5)(696) + (12.0)(227) + (9.5)(606)

= 548.7 ppm44

Table 11.2 Inverse distance weighting calculations for sample values in the vicinityof 65E,137N

SampleNo. X Y V

Distancefrom

65E,137N 1/di1 225 61 139 477 4.5 0.2222 0.20882 437 63 140 696 3.6 0.2778 0.26103 367 64 129 227 8.1 0.1235 0.11604 52 68 128 646 9.5 0.1053 0.09895 259 71 140 606 6.7 0.1493 0.14026 436 73 141 791 8.9 0.1124 0.10567 366 75 128 783 13.5 0.0741 0.0696

El/di = 1.0644

LocalSampik-

603.7


Table 11.3 The effect of the inverse distance exponent on the sample weights and

on the V estimate.

1/de

V p=0.2 p= 0.5 p = 1.0 p = 2.0 p = 5.0 p = 10.01 477 0.1564 0.1700 0.2088 0.2555 0.2324 0.01062 696 0.1635 0.1858 0.2610 0.3993 0.7093 0.98743 227 0.1390 0.1343 0.1160 0.0789 0.0123 <.00014 646 0.1347 0.1260 0.0989 0.0573 0.0055 <.00015 606 0.1444 0.1449 0.1402 0.1153 0.0318 0.00196 791 0.1364 0.1294 0.1056 0.0653 0.0077 <.00017 783 0.1255 0.1095 0.0696 0.0284 0.0010 <.0001it( in ppm.) 601 598 594 598 637 693

I ate- Va-1 ude, . x24.2

C

0 / 0C.:

(a) (b)7(h) C(h)

Co + Co + C

Figure 12.2 An example of an exponential variogram model (a) and an exponentialcovariance function (b).

0 if IN = 051(h) = 1 Co + Ci(1 — exPCIP)) if I h i > 0

( Co + if = 0C(h)

CiexP(4111 ) if 1111> °Both of these functions, shown in Figure 12.2, can be described by

the following parameters:

• Co, commonly called the nugget effect, which provides a discon-tinuity at the origin.

• a, commonly called the range, which provides a distance beyondwhich the variogram or covariance value remains essentially con-stant.

• Co + Ci , commonly called the sill [31, which is the variogramvalue for very large distances, 7(00). It is also the covariancevalue for IN = 0, and the variance of our random variables, 52.

ss nte, Co

den Le‘an A) ,-0.-311,1

LL 4 h le- ;), 2 ,51inc-e

Table 12.2 A table of distances, from Figure 12.1, between all possible pairs ofthe seven data locations.

distanceLocation 0 1 2 3 4 5 6 7

0 0.00 4.47 3.61 8.06 9.49 6.71 8.94 13.451 4.47 0.00 2.24 10.44 13.04 10.05 12.17 17.802 3.61 2.24 0.00 11.05 13.00 8.00 10.05 16.973 8.06 10.04 11.05 0.00 4.12 13.04 15.00 11.054 9.49 13.04 13.00 4.12 0.00 12.37 13.93 7.005 6.71 10.05 8.00 13.04 12.37 0.00 2.24 12.656 8.94 12.17 10.05 15.00 13.93 2.24 0.00 13.157 13.45 17.80 16.97 11.05 7.00 12.65 13.15 0.00

0.050.060.361.220.220.19

10.001.00

1.001.001.001.001.001.001.000.00

010C20030C40050C60C70

1

The D matrix is

D=

The inverse of C is

2.613.390.890.581.340.680.181.00

the C matrix is;...,- A A

C12013 U14 U15 016 ("17 IA A A AL,22 ti23 td24 U25 026 027 1032 033 034 035 036 037 1

042 043 044 045 046 C47 1

052 053 054 055 056 057 1062 063 0_64 065 066 067 1072 073 C74 On urn 077 1

1 1 1 1 1 1 1 0

10.00 5.11 0.44 0.20 0.49 0.26

5.11 10.00 0.36 0.20 0.91 0.490.44 0.36 10.00 2.90 0.20 0.110.20 0.20 2.90 10.00 0.24 0.15

0.49 0.91 0.20 0.24- 10.00 5.11

0.26 0.49 0.11 0.15 5.11 10.00

0.05 0.06 0.36 1.22 0.22 0.19

1.00 1.00 1.00 1.00 1.00 1.00

011021031041051061Cr'

0.127 -0.077 -0.013 -0.009 -0.008 -0.009 -0.012 0.136-0.077 0.129 -0.010 -0.008 -0.015 -0.008 -0.011 0.121- 0.013 -0.010 0.098 -0.042 -0.010 -0.010 -0.014 0.156- 0.009 -0.008 -0.042 0.102 -0.009 -0.009 -0.024 0.139- 0.008 -0.015 -0.010 -0.009 0.130 -0.077 -0.012 0.118-0.009 -0.008 -0.010 -0.009 -0.077 0.126 -0.013 0.141-0.012 -0.011 -0.014 -0.024 -0.012 -0.013 0.085 0.188

0.136 0.121 0.156 0.139 0.118 0.141 0.188 -2.180

C-1=

9227(0.13) + 848 9783

(0.09) (0•9)

947°5)69B (0.32)(0.17) •

9. "(0.13)

Figure 12.3 The ordinary kriging weights for the seven samples using the isotropicexponential covariance model given in Equation 12.25. The sample value is givenimmediately to the right of the plus sign while the kriging weights are shown in

1 parenthesis.

The set of weights that will provide unbiased estimates with a minimumestimation variance is calculated by multiplying C- 1 by D:

W1 0.173W2 0.318V,3 0.129

w= V,4

W5= C-1 • D 0.086

0.151W6 0.057/177 0.086

IL. 0.907

Figure 12.3 shows the sample values along with their correspondingweights. The resulting estimate is

:60 = E 717iVi

i=1

(0.173)(477) + (0.318)(696) + (0.129)(227) + (0.086)(646) +

(0.151)(606) + (0.057)(791) + (0.086)(783)

= 592.7 ppm

(1 c (A. v 8 2 4- .2)

m

M 0

MIN

n•••n•••

1

_j_ 131a0

n•n


10.

0

0Residual

Residual

Figure 11.7 Skewed distribution of estimation error. The error distribution in (a)is positively skewed in contrast to the more or less symmetric distribution shown i(b).

m -0Mt,. 0

kResidual Residual

Figure 11.8 The error distribution in (a) shows a greater spread or variance aboutthe mean error, than (b).

-hi

m <0 -m>0

.•11•••n•

--F0 0 0

Residual Re idual Residual

Figure 11.6 Biased hypothetical distributions of error. In (a) the distribution ofestimation errors indicates a negative bias, in (b), a positive bias, and in (c), nobias.

35

-35 0


+IF + *

35 —

+ +

++ +++++

+ + + +

++ + 41. ++ + -FF ++4+ ÷ 4.

+ +2- + * ++ +

-+ +

-+ + +++ + . + Ilt+t + + +

-it - . . t -+ * -F +14-

0 . .;-4.1: Vitt4 .+ 4 .

+ + 4++ + ++ + + + +++

* +if+ +++ + + +

+ +

• -35 90 0 90

Estimated Values Estimated Values

Figure 11.10 In (a) the estimation error or residuals are globally unbiased, how-ever they are conditionally biased. For some ranges of estimates the average of theresiduals will not be equal to 0. In (b) the estimates are globally unbiased as wellas conditionally unbiased; for any range of estimates, the positive residuals balancethe negative residuals.

ANISOTROPY

POINT ESTIMATION EXAMPLES ASSUMED ISOTROPY

UNCOMMON IN MOST MINERALISED SYSTEMS

ANISOTROPY USUALLY REVEALED BY VARIOGRAPHY IN 3DIRECTIONS, WITH THE GEOLOGICAL CONNOTATION BEING:

ALONG STRIKE, DOWN DIP, ACROSS LAYERING.OTHER POSSIBILITIES EXIST

DOMAINS

AREA OR VOLUME WHERE SELECTED RANDOM FUNCTIONMODEL APPLIES. AGAIN, DOMAINS USUALLY REFLECT

GEOLOGY.

BLOCK KRIGING

BASED ON AVERAGE OF REGULARLY DISTRIBUTED POINTESTIMATES. CAN BE COMPUTED DIRECTLY.

DISCRETISATION

ORDINARY KRIGING

CAN BE A GOOD ESTIMATOR OF BLOCK GRADE, BETTER OFGLOBAL ESTIMATE.

SENSITIVE TO EXTREME VALUES - SMOOTHING - OPTIMISINGPOOR ESTIMATOR OF SMALL BLOCK GRADES FROM WIDE

SPACED DATA

ALTERNATIVES TO OK

NON-LINEAR KRIGING METHODS - INDICATOR ANDMULTI-INDICATOR K., LOG-NORMAL K., DISJUNCTIVE K.,

UNIFORM CONDITIONING, MULTI-GAUSSIAN K.HANDLE EXTREME VALUES BETTER. PRODUCE ESTIMATESINTERPRETED AS CONDITIONAL PROBABILITIES OR MEANS

OF CONDITIONAL DISTRIBUTIONS.

CONDITIONAL SIMULATION

HISTOGRAM AND VARIOGRAM OF SIMULATED VALUESSIMILAR TO CONDITIONING DATA;

SIMULATED VALUES SAME AS CONDITIONING DATA ATDATA LOCATIONS.

C.S. REPRESENTS A POSSIBLE REALITYCAN BE USED TO ACHIEVE OPTIMAL ORE SELECTIONNOT COMPATIBLE WITH CURRENT PIT OPTIMISATION

SOFTWARE.

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