Spatial Variability - Mineral Resource Estimation - page 112

Question 3: Fit the directional variograms with a licit var-

iogram model and comment on your results.
Comment on aspects of the variogram that are
uncertain.

found in typical continuous variable distribu-
tions. Calculate the nine indicator variograms
corresponding to the deciles of the Au grade
distribution.Thesevariogramsshouldbeit
with smoothly varying parameters. Fit the
variograms with, say, two spherical vario-
gram structures and plot the parameters as a
function of threshold.

6.6.4 Part Four: Cross Variograms

Question 1: Extend Part Three to direct and cross vario-
grams of the normal scores of Cu and Mo,
three directional variograms of the normal
scores of Mo, and three directional cross var-
iograms between the normal scores transforms
of Cu and Mu. Plot the correct sill on the cross
variograms and comment on the results.
Question 2: The most dificult aspect of using multiple
variables and cross variograms is itting a
model of coregionalization. The only practi-
cal model of coregionalization is the linear
model of coregionalization (LMC). Recall the
LMC and the constraints that it imposes on
variogram modeling.

References

Armstrong M (1984) Improving the estimation and modeling of the
variogram. In: Verly G et al (ed) Geostatistics for natural resources
characterization, part I, Reidel, Dordrecht, pp 1-19
Babakhani M, Deutsch CV (2012) Standardized pairwise relative var-
iogram as a robust estimator of spatial structure. Unpublished CCG
Research Paper 2012-310
Barnes, RJ (1991) The variogram sill and the sample variance. Math
Geol 23(4):673-678
Christakos G (1984). On the problem of permissible covariance and
variogram models. Water Resour Res 20(2):251
Clark I (1986) The art of cross-validation in geostatistical applications.
In: Proceeding of the 19th APCOM, pp 211-220
Clark I, Harper WV (2000) Practical geostatistics. Ecosse North Amer-
ica, Columbus, p 340
Cressie N (1985) Fitting variogram models by weighted least squares.
Math Geol 17(5):563-586
Cressie N (1991) Statistics for spatial data. Wiley, New York, p 900.
Reprinted 1993
David M (1977) Geostatistical ore reserve estimation. Elsevier,
Amsterdam
Davis BM (1987) Uses and abuses of cross-validation in geostatistics.
Math Geol 17(5):563-586
Deutsch CV, Journel AG (1997) GSLIB: geostatistical software library
and user's guide, 2nd ed. Oxford University Press, New York, p 369
Goovaerts P (1997) Geostatistics for natural resources evaluation.
Oxford University Press, New York, p 483
Gringarten E, Deutsch CV (1999) Methodology for variogram inter-
pretation and modelling for improved reservoir characterization. In:
Paper SPE 56654, presented at the SPE annual technical conference
and exhibition held in Houston, 3-6 October
Isaaks EH (1999) SAGE2001 User's Manual, Software license and
documentation. http://www.isaaks.com
Isaaks E, Srivastava RM (1988) Spatial continuity measures for proba-
bilistic and deterministic geostatistics. Math Geol 20(4):313-341
Isaaks EH, Srivastava RM (1989). An introduction to applied geostatis-
tics. Oxford University Press, New York, p 561.
Journel A (1988) New distance measures: the route towards truly non-
Gaussian geostatistics. Math Geol 20(4):459-475
Journel AG (1987) Geostatistics for the environmental sciences: An
introduction. U.S. Environmental Protection Agency, Environmental
Monitoring Systems Laboratory
Journel AG, Huijbregts ChJ (1978) Mining geostatistics. Academic,
New York
Myers DE (1991) Pseudo-cross variograms, positive definiteness and
cokriging. Math Geol 23:805-816
Omre H (1984) The variogram and its estimation. In: Verly G, David
M, Journel AG, Marechal A (eds) Geostatistics for natural resource
characterization: NATO ASI series C, v. 122—Part I. Reidel Publica-
tion. Co., Dordrecht, pp 107-125
Reed M, Simon B (1975) Methods of modern mathematical physics,
vol II. Academic Press, New York

Question 3: Fit an LMC to the variograms you calculated

in Question One. Document the procedure
youfollowedandshowtheinalnineexperi-
mentalvariogramswiththeittedmodel.

6.6.5 Part Five: Indicator Variograms

for Continuous Data

Consider the same 3-D data as in the previous two parts for
indicator variograms.
Question 1: Find or recalculate/remodel the normal scores
variogram of Cu. Consider directional vario-
grams.
Question 2: Use the bigaus program from GSLIB to
calculate the 0.1, 0.25 and 0.9 quantile indi-
cator variograms using the normal scores var-
iogram model. The 0.1 and 0.9 quantile var-
iograms will be identical since the Gaussian
distribution is symmetric.
Question 3: Calculate the experimental indicator vario-
grams corresponding to the 0.1, 0.25 and 0.9
quantiles and plot with the results of Question
2. Comment on any differences. Pay particu-
lar attention to the 0.1 and 0.9 quantile indi-
cator variograms and any asymmetry in the
experimental variograms.
Question 4: Three indicator variograms are suitable for
checking bivariate Gaussianity; however,
nine indicator variograms are more reason-
able to discretize the range of variability