Changes in dispersion variance consequent upon inaccurately modelled semi-variograms

Mathematics and Computers North-Holland

in Simulation

30 (1988) 11-16

CHANGES IN DISPERSION VARIANCE MODELLED SEMI-VARIOGRAMS

CONSEQUENT

UPON INACCURATELY

Peter I. BROOKER Department

of Geology and Geophysics, University of Adelaide, Adelaide, SA SOOO,Australia

The semi-variogram function which describes the spatial variation of samples as a function of their distance apart is an intermediary in geostatistical calculations. It is used in the calculation of several variances: the estimation variance which quantifies the accuracy of block estimates and the dispersion variance which measures the variability of blocks in the deposit. However the semi-variogram must be accurately modelled if these variances are to be well-known. The changes in dispersion variance calculated here when a spherical semi-variogram is estimated inaccurately show the need for care in its estimation, especially in the selection of the correct nugget effect.

1.

INTRODUCTION Previous papers (1,2) which the author presented at SSA conferences in 1982 and 1984 dealt

with the generation of data sets which may be used as simulated models of orebodies.

By "min-

ing" such computer models the mining engineer is able to discover how the deposit will behave under various operating conditions. He can therefore design preferred extraction, stockpile and blending practice.

The educator is able to demonstrate the best estimation procedures to use

in evaluating an orebody by working with such models and this was discussed in SSA-80 (3). Important geostatistical measures, the estimation variance, measuring the accuracy of block estimates, and the dispersion variance, used to predict the variability of mining units of a certain size within larger units, were introduced in that paper. The simulation model generated to mirror the orebody must have the statistical characteristics of the deposit studied.

In particular, the semi-variogram, measuring the correlation be-

tween neighbouring samples as a function of their distance apart, is required (along with the average value and variance of the samples). and reveals two types of variability.

This function is estimated from the sample data

At small distances the so-called nugget effect, due to

errors of measurement as well as physical nuggets of mineralisation, is observed, whilst for increasing distances the function increases in value (spatial variation). For it to be of use in geostatistical calculations the function must be modelled as a positive definite function and the division between nugget effect and the spatial variation can be critical in the successful implementation of geostatistical methods.

037%4754/88/$3.50

0 1988, IMACS/Elsevier

Science Publishers

B.V. (North-Holland)

12

P. I. hooker

In an earlier function.

paper

Brooker

This was followed

variance

if

The importance

of

model chosen

for

in dispersion square,

for

correctly

used form for

variance,

for

determining

the semi-variogram, blocks

of

the

size

parameters

and from a half

is

statistic,

apportions

spherical

the block

side

is

nugget

for

zero

in this

variance, chosen

in a deposit effect

to a wide variety length

demonstrated

when the

and spatial

model,

of the model to vary between

to ten times

again effect

square

in the relative

appropriate

this

in the estimation

dispersion

nugget

one unit

modelling

estimated.

the semi-variogram

improperly

Semi-variograms

the true

the changes

is poorly

in the important

square

in accurately

(5) which investigated

to 25%, 50% and 99% error

and plotted.

ed by allowing

the difficulty

the semi-variogram

to the variation

corresponding

calculated

(4) demonstrated

the semi-variogram

The most widely

variance calculated from modelled semi-variograms

by a study

the model chosen

paper by reference

effect

/ Dispersion

variation.

and the changes

twenty units and range are

of orebodies and two for

are discuss-

relative

nugget

the range.

DISPERSIONVARIANCEAND THE SEMI-VARIOGRAM

2.

Samples taken taken

more uniform.

sporadic

of tons

of the mining unit

ing the spatial of blocks

where g(v,v)

of

This

= g(V,V)

the average

in value

Journel

even though they be

hand mining block

a few kilograms change

are averaged

(6,

values

are much

out when considered

in variability

and Huijbregts

volume V can be written

according

to the

the function

p.67)

show that

describ-

the variance

as

- g(v,v) value

of the semi-variogram

g(h)

as each end of the vector

sweeps out the volume v.

Knowledge of the dispersion ential

that

design

stockpiles.

pendent

dissimilar

in terms of the semi-variogram,

of samples.

v in a larger

represents

the engineer

the plant.

quite

On the other

blocks.

can be quantified

var(v/V) h independently

samples

of the mining

variability

of size

are frequently

zone of the deposit.

The highly

over the hundreds size

from an orebody

from one particular

It will

variance

is fundamental

know the order

Similarly

the mill

of

be seen from the above

on the semi-variogram

the variation

feed

limits equation

it

is necessary

is

estimated

that

in the operation in block

values

must be set to allow that this

since

of

so that

should

is

ess-

he can properly

smooth functioning

the dispersion

function

It

a mine.

variance

be as well

of

is de-

known as is

possible. The semi-variogram, of data Y (x+h) difference function

g(h),

and Y(x),

separated

between the data at this g(h)

is built

up.

g(h) g(h) The parameters co + c.

This

from the sample data by taking

by displacement distance.

As the distance

then is modelled,

= Co + C(l.S(h/a)

frequently

- O.S(h/a)**3)

half

account

between the pairs by the

so-called

. . ..h<

of all

the average

pairs squared

is varied spherical

a

. . . . haa

= Co + C

to be identified

h, and calculating

are the range,

a, the nugget

effect,

Co, and the sill,

a form:

13

P. I. Brooker / Dispersion variance calculated from modelled semi-variograms Probably termine ing is

the most important

is the nugget greater

model chosen cussed

here

will study

of variation, pair 3.

effect.

the spatial

be a pure nugget as they

which deals

the sill

of parameters

is

effect.

effect with

fixed

grid

aspect

is too

coarse

The consequences

isotropic

of

parameters

(certainly

if

to de-

the grid

may not be observed such misassignment

spac-

and the

will

be dis-

variance.

situations

at an arbitrary

of these

of the variation

the dispersion

in which there

value,

(a, e) where a is the range

lo.,

and e is

is no preferred

Then every model is

the relative

nugget

direction

specified

effect,

by a

e = Co/C.

DISPERSION VARIANCECALCULATIONS Only one geometrical

unit

are considered

situation

inside

Base Dispersion

The dispersion functions

of

0.5 units

considered

Clark

each point

when considering

(7).

of

is

calculated

Models

for

length

blocks

twenty units.

for

percentage

change,

nugget

value

spectrum

effect

of side

length

one

Such a two dimensional

as a contour

relative

to the correct

at that

point)

0.5

1.

the auxiliary

to 10 units

value,

of

With these

in the parameters

variance

point

using

from 0 to 2 in steps

map in Figure

to changes

at that

of models

from 0.5 units

varies

variance

of the base map and the dispersion value

a large

which the range varies

are plotted

of dispersion

to the correct

side

Square mining

a bench of an open pit.

which the relative

compared with the original (relative

deposit

here.

Variance

and the results

A fixed

is considered

a square

variance

and for

base the robustness ated.

situation

is appropriate

3.1

of

If the sampling

than the range)

in so far

In this

and at the same time most difficult

(a,e)

in steps 0.2

values

are as a

may be investig-

is made in one parameter

recalculated.

The recalculated

and the percentage

change

for

value

in dispersion

is

variance

is determined. 1c!. (32

5.0

I 1

5

>..

. 6.

+

f

I

.

.

.

.

*

*

.

.

.

.

0

5.0 range Figure

1. Dispersion

variance

as a function

of range and relative

nugget

effect.

Sill

0

equals

10

14

P.I. Brooker

The percentage

changes

The extent

ent ly.

and 3 for

of these

particular

points

/ Dispersion

variance calculated from modelled semi-variograms

in the parameters variations (a,e)

are *25%,

250% and *99% and are

on the semi-variogram

model is

applied

illustrated

independ-

in Figures

of the base map.

4

5

6

a

7

9

10

distance

Figure

2.

Changes in semi-variogram 0

12 ’

1 I

2 I

when range

3 I

4 I

changes.

5 1

6 I

Base (a,e)

7 I

8 I

= (5,0.4)

9 I

10 12

-8

Figure

3.

Changes in semi-variogram

when relative

nugget

effect

changes.

Base (a,e)

=

(831)

2

P. I. Brooker

3.2

Changes in Dispersion

In this nugget

case

effect

dispersion

so that

variance

is

variance

less

ranges

Figure

against

extreme for

than 25% for

ranges

the variation

in dispnrsi.on

4 is not a contour range.

It

effect

all

is

if

a plot

units

the percentage for

all

of the relative

of percentage

the spatial

aspect

(-99% change in range),

Otherwise

variance

are independent

map but rather

assigned

than four

in dispersion

variance

can be seen that

ranges.

greater

15

as the Range changes

changes

and a nugget

is

variance calculated from modelled semi-variograms

Variance

the percentage

is not recognised sion

/ Dispersion

change in

of the variation

the change in disper-

change in dispersion

the cases

considered.

can be extreme as the plot

for

variance

For smaller

99% change in range

shows. 0

1 I

200 '

4 I

3 I

2 I

6 I

5 I

1

8

I

I

10

9

200 'I

180160-

x 99% 0 50% A 25% .-25% Z-50% X-99%

140120-

fk

loo

2 .c cl

80-

b

40-

2$

20-

:: 8

60-

O -2o-4o-6O-8O-lOO-120

; 0

I 1

I 4

I 3

I 2

I 7

I 6

I 5

; -120 10

I 9

I 8

range Figure

4.

3.3

Percentage

change

Changes in Dispersion

Again the percentage so that effect iance 4.

Figure

5 is

and applies may occur

if

Variance

change

a plot for

in dispersion

all

in dispersion

of percentage ranges.

the parameters

variance

It

for

as the Relative variance

change

given is

of the second

variance

that

large

change in range.

changes.

independent

in dispersion

can be seen again

are incorrectly

percentage

Nugget Effect

against

changes

parameter

relative

in dispersion

nugget var-

estimated.

CONCLUSIONS The semi-variogram

is an artefact

in the calculation

of

important

geostatistical

variances.

P. I. Brooker

16 0

220

0.2 I

/ Dispersion

variance calculated from modelled semi-variograms

0.4 I

0.6 I

0.8 I

1 I

1.2 I

1.4 I

1.6 I

1.8 I

2 ' 220

I 0.4

I 0.6

I 0.8

I 1.0

I 1.2

I 1.4

I 1.6

I 1.8

2

200180160-

x 99% + 50% A 25% q-25% X-50% z-999

%1402 G * al 0 2 "1

120loo8060-

-60

i 0

I 0.2

; -60

relative nugget effect Figure 5. Percentagechange in dispersionvariance for given percentage change in relative nugget effect. One of these is the dispersionvariance which determinesthe variabilityof, say, the production of one day in that of a month or a year. Such measures are vital in correct design of plant operating conditions. Since the semi-variogramis necessary in the calculationof the dispersion variance it is importantthat it be estimatedas accuratelyas possible. It has been shown that the proper division of the semi-variograminto a random component or nugget effect and a spatial component is of great importance. If this is not done the dispersionvariance can be inaccurate and the consequencesfor plant operation can be important. Accurate assessmentof the semi-variogram requires a large enough suite of samples taken at small enough distances apart so that the correct nugget effect can be ascertained.

REFERENCES

(1) Brooker, P.I. and Paul, C, Numerical simulationof a two dimensionalorebody, Proceedings of the Fifth Conferenceof the SimulationSociety of Australia,(1982)133-136.

(2) Brooker, P.1, Simulationof spatially correlateddata in two dimensions,Mathematicsand Computers in Simulation27 (1985) 155-157. (3) Brooker, P-1, Using simulationmodels as an aid in teaching geostatistics,Proceedingsof the Fourth Conferenceof the SimulationSociety of Australia (1980) 126-131. (4) Brooker, PSI, Semi-variogramestimationusing a simulated deposit, Mining Engineering, Vol 35, No. 1, (1983) 37-42. (5) Brooker, P.I., A parametric study of robustnessof kriging variance as a function of range and relative nugget effect for a spherical semi-variogram,MathematicalGeology, Vol 18, No. 5, 1986, 477-488. (6) Journel, A.G. and Huijbregts,C.J., Mining Geostatistics,(AcademicPress, London, 1978) (7) Clark, I., Some auxiliary functions for the sphericalmodel of Geostatistics,Computers and Geosciences,Yol. 1, (1976) 255-263.