Advertisement

Mathematical Geosciences

, Volume 46, Issue 7, pp 841–868

# Revisiting Multi-Gaussian Kriging with the Nataf Transformation or the Bayes’ Rule for the Estimation of Spatial Distributions

Article
• 290 Downloads

## Abstract

A multivariate probability transformation between random variables, known as the Nataf transformation, is shown to be the appropriate transformation for multi-Gaussian kriging. It assumes a diagonal Jacobian matrix for the transformation of the random variables between the original space and the Gaussian space. This allows writing the probability transformation between the local conditional probability density function in the original space and the local conditional Gaussian probability density function in the Gaussian space as a ratio equal to the ratio of their respective marginal distributions. Under stationarity, the marginal distribution in the original space is modeled from the data histogram. The stationary marginal standard Gaussian distribution is obtained from the normal scores of the data and the local conditional Gaussian distribution is modeled from the kriging mean and kriging variance of the normal scores of the data. The equality of ratios of distributions has the same form as the Bayes’ rule and the assumption of stationarity of the data histogram can be re-interpreted as the gathering of the prior distribution. Multi-Gaussian kriging can be re-interpreted as an updating of the data histogram by a Gaussian likelihood. The Bayes’ rule allows for an even more general interpretation of spatial estimation in terms of equality for the ratio of the conditional distribution over the marginal distribution in the original data uncertainty space with the same ratio for a model of uncertainty with a distribution that can be modeled using the mean and variance from direct kriging of the original data values. It is based on the principle of conservation of probability ratio and no transformation is required. The local conditional distribution has a variance that is data dependent. When used in sequential simulation mode, it reproduces histogram and variogram of the data, thus providing a new approach for direct simulation in the original value space.

## Keywords

Simple kriging Nataf transformation Multi-Gaussian  Conditional probability Prior Likelihood Stationarity Bayes’rule  Direct simulation

## References

1. Armstrong M (1998) Basic linear geostatistics. Springer, Berlin, p 153Google Scholar
2. Bailey TC, Gatrell AC (1995) Interactive spatial data analysis. Prentice Hall, Englewood Cliffs, p 413Google Scholar
3. Bourgault G (1997a) Spatial declustering weights. Math Geol 29(2):277–290
4. Bourgault G (1997b) Using non-Gaussian distribution in geostatistical simulations. Math Geol 29(3): 315–334Google Scholar
5. Caers J (2000) Adding local accuracy to direct sequential simulation. Math Geol 32(7):815–850
6. Chiles JP, Delfiner P (1999) Geostatistics modeling spatial uncertainty. Wiley series in probability and statistics. Wiley, New York p 695Google Scholar
7. Deutsch CV (1989) DECLUS: a Fortran 77 program for determining optimal spatial declustering weights. Comput Geosci 15(3):325–332
8. Deutsch CV (1996) Constrained modeling of histograms and cross plots with simulated annealing. Technometrics 38(3):266–274
9. Deutsch CV, Journel AG (1998) GSLIB Geostatistical software library and user’s guide, 2nd edn. Oxford University Press, Oxford, p 369Google Scholar
10. Der Kiureghian A, Liu PL (1986) Structural reliability under incomplete probability information. J Eng Mech 112(1):85–104
11. Emery X (2005) Simple and ordinary multigaussian kriging for estimating recoverable reserves. Math Geol 37(3):295–319
12. Goovaerts P (1997) Geostatistics for natural resources evaluation. Oxford University Press, Oxford, p 483Google Scholar
13. Hoel PG (1971) Introduction to mathematical statistics. 4th edn. Wiley, New York, p 409Google Scholar
14. Isaaks EH, Srivastava RM (1989) An introduction to applied geostatistics. Oxford University Press, Oxford, p 561Google Scholar
15. Journel AG, Huijbregts CJ (1978) Mining Geostatistics. Academic Press, London, p 600Google Scholar
16. Journel AG (1994) Modelling uncertainty: some conceptual thoughts. In: Dimitrakopoulos R (ed) Proceedings of geostatsitics for the next century. Kluwer Academic, Dordrecht, pp 30–43Google Scholar
17. Journel AG (2002) Combining knowledge from diverse sources: an alternative to traditional data independence hypotheses. Math Geol 34(5):573–596
18. Leuangthong O (2004) The promises and pitfalls of direct simulation. In: Leuangthong O, Deutsch CV (eds) Proceedings of geostatsitics Banff. Springer, Berlin, pp 305–314Google Scholar
19. Matheron G (1971) The theory of regionalized variables and its applications. Issue 5 of Les Cahiers du Centre de morphologie mathématique de Fontainebleau, École national supérieure des mines de Paris. p 211Google Scholar
20. Nataf A (1962) Détermination des distributions dont les marges sont données. Comptes rendus de l’Académie des Sciences, vol 225, France, pp 42–43Google Scholar
21. Soares A (2001) Direct sequential simulation and cosimulation. Math Geol 33(8):911–926
22. Tran TT, Deutsch CV, Xie Y (2001) Direct geostatistical simulation with multiscale well, seismic, and production data. SPE Annual Technical Conference and Exhibition, New Orleans, SPE Paper 71323Google Scholar
23. Turlach BA (1993) Bandwidth selection in kernel density estimation: a review. Discussion paper 9317. Institut de Statistique, Voie du Roman Pays, B-1348, Université catholique Louvain-la-NeuveGoogle Scholar
24. Verly G (1983) The multiGaussian approach and its applications to the estimation of local reserves. Math Geol 15(2):259–286
25. Xu W, Journel A (1995) Histogram and scattergram smoothing using convex quadratic programming. Math Geol 27(1):83–103

## Copyright information

© International Association for Mathematical Geosciences 2014

## Authors and Affiliations

1. 1.CalgaryCanada

## Personalised recommendations

### Citearticle 