Geodesic
Computationally efficient generation of Gaussian conditional simulations in geological modeling problems
Matematičeskoe modelirovanie, Tome 24 (2012) no. 11, pp. 83-96.

Voir la notice de l'article provenant de la source Math-Net.Ru

The generation 2D and 3D normally distributed random fields conditioned on well data are often required in reservoir modeling. Such fields can be obtained by using: unconditional simulation with kriging interpolation, sequential Gaussian simulation, Cholesky factorization of the covariance matrix. However, all methods have disadvantages. First and second one have a fallible correlation function of realizations. This disadvantage could cause incorrect values of oil flow rates. Cholesky factorization, which has the advantage of being general and exact, has high computational costs for geological modeling problems. In this work we use spectral representation of conditional process. It is shown that covariance of two arbitrary spectral components could be factorized into functions of corresponding harmonics. In this case the Cholesky decomposition could be considerably simplified. A feature of this approach is its accuracy and computational simplicity.
Keywords: stationary processes, Gaussian processes, spectral method, Cholesky decomposition, geological modeling.
Mots-clés : Fourier transform, covariance matrix 
@article{MM_2012_24_11_a6,
     author = {I. R. Minniakhmetov and A. H. Pergament},
     title = {Computationally efficient generation of {Gaussian} conditional simulations in geological modeling problems},
     journal = {Matemati\v{c}eskoe modelirovanie},
     pages = {83--96},
     publisher = {mathdoc},
     volume = {24},
     number = {11},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MM_2012_24_11_a6/}
}
TY  - JOUR
AU  - I. R. Minniakhmetov
AU  - A. H. Pergament
TI  - Computationally efficient generation of Gaussian conditional simulations in geological modeling problems
JO  - Matematičeskoe modelirovanie
PY  - 2012
SP  - 83
EP  - 96
VL  - 24
IS  - 11
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MM_2012_24_11_a6/
LA  - ru
ID  - MM_2012_24_11_a6
ER  - 
%0 Journal Article
%A I. R. Minniakhmetov
%A A. H. Pergament
%T Computationally efficient generation of Gaussian conditional simulations in geological modeling problems
%J Matematičeskoe modelirovanie
%D 2012
%P 83-96
%V 24
%N 11
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MM_2012_24_11_a6/
%G ru
%F MM_2012_24_11_a6
I. R. Minniakhmetov; A. H. Pergament. Computationally efficient generation of Gaussian conditional simulations in geological modeling problems. Matematičeskoe modelirovanie, Tome 24 (2012) no. 11, pp. 83-96. http://geodesic.mathdoc.fr/item/MM_2012_24_11_a6/

[1] Komarov S. G., Geofizicheskie metody issledovaniya skvazhin, Nedra, M., 1963

[2] Zakrevskii K. E., Geologicheskoe 3D modelirovanie, OOO “IPTs Maska”, M., 2009

[3] Buhman M., Powell M., “Radial basis function interpolation on an infinite regular grid”, Algorithms for approximation, v. II, Chapman Hall, London, 1990, 146–169 | MR

[4] Samarskii A. A., Gulin A. V., Chislennye metody, uchebnoe posobie, Nauka, M., 1989, 429 pp. | MR

[5] Gikhman I. I., Skorokhod A. V., Vvedenie v teoriyu sluchainykh protsessov, Nauka, M., 1977 | MR

[6] Lavrik D. A., Minniakhmetov I. R., Pergament A. Kh., “Regulyarizovannye algoritmy statisticheskogo otsenivaniya funktsii v zadachakh geologicheskogo modelirovaniya”, Matem. modelirovanie, 234 (2011), 23–40 | MR

[7] Cressie N., “The Origins of Kriging”, Mathematical Geology, 22 (1990), 239–252 | DOI | MR | Zbl

[8] Krige D., A statistical approach to some mine valuations and allied problems at the Witwatersrand, Master's thesis of the University of Witwatersrand, 1951

[9] Matheron G., “Principles of geostatistics”, Economic Geology, 58 (1963), 1246–1266 | DOI

[10] Bulinskii A. V., Shiryaev A. N., Teoriya sluchainykh protsessov, FIZMATLIT, M., 2005, 408 pp.

[11] Kramer G., Lidbetter M., Statsionarnye sluchainye protsessy, per. s angl., Mir, M., 1969

[12] Matheron G., “The intrinsic random functions and their applications”, Adv. Appl. Probability, 5 (1973), 439–468 | DOI | MR | Zbl

[13] Yaglom A., An Introduction to the Theory of Stationary Random Functions, Prentice-Hall, Englewood Cliffs, NJ, 1962 | MR | Zbl

[14] Emery X., “Simple and Ordinary Kriging Multigaussian Kriging for Estimating recoverable Reserves”, Mathematical Geology, 37:3 (2005), 295–319 | DOI | MR | Zbl

[15] Shinozuka M., Zhang R., “Equivalence Between Kriging And Cpdf Methods For Conditional Simulation”, J. Eng. Mech., 122 (1996) | DOI

[16] Deutsch C., Journel A., Gslib, Geostatistical software library and user's guide, Oxford University Press, Oxford, 1997

[17] Pergament A. Kh., Minniakhmetov I. R., Akhmetsafina A. R., “Sistema testov dlya algoritmov geologicheskogo modelirovaniya”, Vestnik TsKR Rosnedra, 5 (2009), 8–16

[18] Haugh M., The Monte Carlo framework, examples from finance and generating correlated random variables, IEOR E4703: Monte Carlo Simulation Course Notes, 2004

[19] Halbleib R., Voev V., “Forecasting Multivariate Volatility using the VARFIMA Model on Realized Covariance Cholesky Factors”, Journal of Economics and Statistics, 231:1 (2011), 134–152

[20] Wang J., Liu C., “Generating multivariate mixture of normal distributions using a modified Cholesky decomposition”, Winter Simulation Conference, 2006, 342 | DOI

[21] Minniakhmetov I. R., Stokhasticheskoe modelirovanie uslovnykh gaussovskikh protsessov, Preprint No 79, IPM im. M. V. Keldysha RAN, M., 2011

[22] Lidl R., Pilz G., Applied Abstract Algebra, Wiley, New York, 1999, 217–219 | MR