Fast and exact simulation of stationary Gaussian processes through circulant embedding of the covariance matrix

SIAM Journal on Scientific Computing

Volumn 18, Issue 4, 1997, Pages 1088-1107

  Dietrich, C R ^a   Newsam, G N ^b  

^a GRIFFITH UNIVERSITY   (Australia)

^b DEFENCE SCIENCE AND TECHNOLOGY ORGANISATION   (Australia)

Author keywords

Geostatistics; Monte Carlo simulations; Random fields; Time series

Indexed keywords

COMPUTER SIMULATION; CORRELATION METHODS; FAST FOURIER TRANSFORMS; MATRIX ALGEBRA; MONTE CARLO METHODS; RANDOM PROCESSES; STATISTICAL METHODS; TIME SERIES ANALYSIS;

CIRCULANT EMBEDDING; CORRELATION FUNCTIONS; COVARIANCE MATRIX; GEOSTATISTICS; STATIONARY GAUSSIAN PROCESSES; TOEPLITZ CORRELATION MATRIX;

GEOLOGY;

EID: 0031173547     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/S1064827592240555     Document Type: Article

References (35)

1. M. ABRAMOWITZ AND I. A. STEGUN, Handbook of Mathematical Functions, Dover, New York, 1970.
2. S. BARNETT, Matrices, Methods and Applications, in Oxford Applied Mathematics and Computing Sciences Series, Clarendon Press, Oxford, 1990.
3. T.C. BLACK AND D.L. FREYBERG, Simulation of one-dimensional correlated fields using a matrix-factorization moving average approach, Math. Geol., 22 (1990), pp. 39-62.
4. L. BORGMAN, M. TAHERI, AND R. HAGAN, Three-dimensional frequency-domain simulations of geological variables, in Geostatistics for Natural Resource Characterization, Part 2, G. Verly, M. David, A. G. Journel, and A. Marechal, eds., Reidel, Dordrecht, the Netherlands, 1984, pp. 517-541.
5. D.C. CHAMPENEY, A Handbook of Fourier Theorems, Cambridge University Press, London, 1987.
6. G. CHRISTAKOS, The space transformation in the simulation of multi-dimensional random fields, Math. Comput. Simulation, 29 (1987), pp. 313-319.
7. M.W. DAVIS, Production of conditional simulations via the LU triangular decomposition of the covariance matrix, Math. Geol., 19 (1987), pp. 91-98.
8. A. DEMBO, C.L. MALLOWS, AND L.A. SHEPP, Embedding nonnegative definite Toeplitz matrices in nonnegative definite circulant matrices, with application to covariance estimation, IEEE Trans. Inform. Theory, IT-35 (1989), pp. 1206-1212.
9. C.R. DIETRICH, Computationally efficient Cholesky factorization of a covariance matrix with block Toeplitz structure, J. Statist. Comput. Simulation, 45 (1993), pp. 203-218.
10. C.R. DIETRICH AND G.N. NEWSAM, A fast and exact method for multidimensional Gaussian stochastic simulations, Water Resour. Res., 29 (1993), pp. 2861-2869.
11. C. R. DIETRICH AND G. N. NEWSAM, Fast and exact method for multidimensional Gaussian stochastic simulations: extension to conditional simulations, Water Resour. Res., 32 (1996), pp. 1643-1652.
12. C. R. DIETRICH, A simple and efficient space domain implementation of the turning bands method, Water Resour. Res., 31 (1995), pp. 147-156.
13. F. MA, M. S. WEI, AND W.H. MILLS, Correlation structuring and the statistical analysis of steady-state groundwater flow, SIAM J. Sci. Statist. Comput., 8 (1987), pp. 848-867.
14. I. S. GRADSHTEYN AND I. M. RYZHIK, Table of Integrals, Series, and Products, Academic Press, New York, 1979.
15. A.G. JOURNEL AND C.T. HUIJBREGTS, Mining Geostatistics, Academic Press, New York, 1978.
16. T. KAILATH, A theorem of I. Schur and its impact on modern signal processing, in Schur Methods in Operator Theory and Signal Processing, I, I. Gohberg, ed., Birkhauser-Verlag, Basel, 1986, pp. 9-30.
17. P.R. KING AND P. J. SMITH, Generation of correlated properties in heterogeneous porous media, Math. Geol., 20 (1988), pp. 863-877.
18. H. LEV-ARI AND T. KAILATH, Triangular factorization of structured matrices, in I. Schur Methods in Operator Theory and Signal Processing, I. Gohberg, ed., Birkhauser-Verlag, Basel, 1986, pp. 301-324.
19. A. MANTOGLOU AND J.L. WILSON, The turning bands method for simulation of random fields using line generation by a spectral method, Water Resour. Res., 18 (1982), pp. 1379-1394.
20. K.V. MARDIA AND A. J. WATKINS, On the multimodality of the likelihood in the spatial linear model, Biometrika, 76 (1989), pp. 289-295.
21. G.N. NEWSAM, On the asymptotic distribution of the eigenvalues of discretizations of a compact operator, in Proc. Center for Mathematical Analysis, Vol. 17: Special Program on Inverse Problems, Australian National University, Canberra, Australia, 1988.
22. G.N. NEWSAM AND C.R. DIETRICH, Bounds on the size of nonnegative definite circulant embeddings of positive definite Toeplitz matrices, IEEE Trans. Inform. Theory, 40 (1994), pp. 1218-1220.
23. B.S. RAMARAO, A. M. LAVENUE, G. DE MARSILY, AND M. G. MARIETTA, Pilot point methodology for automated calibration of an ensemble of conditionally simulated transmissivity fields, 1. Theory and computational experiments, Water Resour. Res., 31 (1995), pp. 475-493.
24. J.B. READE, On the sharpness of Weyl's estimates for eigenvalues of smooth kernels, SIAM J. Math. Anal., 19 (1988), pp. 627-631.
25. M. SHINOZUKA AND C.M. JAN, Digital simulation of random processes and its applications, J. Sound Vibrations, 25 (1972), pp. 111-128.
26. G. SPOSITO, W. A. JURY, AND V.K. GUPTA, Fundamental problems in the stochastic convection-dispersion model of solute transport in aquifers and field soils, Water Resour. Res., 22 (1986), pp. 77-88.
27. A.F.B. TOMPSON, R. ABABOU, AND L.W. GELHAR, Implementation of the three-dimensional turning bands field generator, Water Resour. Res., 25 (1989), pp. 2227-2243.
28. J.J. WARNES, A sensitivity analysis for universal kriging, Math. Geol., 18 (1986), pp. 653-676.
29. M. WEGENER, C. R. DIETRICH, AND G. N. NEWSAM, Generation of non-Gaussian sea surfaces for IR scene simulations, Infrared Technology XXI, SPIE's 1995 International Symposium on Optical Science, Engineering and Instrumentation, San Diego, CA, 1995.
30. P. WHITTLE, On stationary processes in the plane, Biometrika, 41 (1954), pp. 434-449.
31. J.W. WOODS, Two-dimensional discrete Markovian fields, IEEE Trans. Inform. Theory, 18 (1972), pp. 101-109.
32. d, Report CMA-SR06-93, Centre for Mathematics and Its Applications, Australian National University, Canberra, Australia, 1993.
33. A.M. YAGLOM, Correlation Theory of Stationary and Related Random Functions, Springer-Verlag, New York, 1987.
34. D.L. ZIMMERMAN, Computationally exploitable structure of covariance matrices and generalized covariance matrices in spatial models, J. Statist. Comput. Simulation, 32 (1989), pp. 1-15.
35. A. ZYGMUND, Trigonometric Series, Cambridge University Press, London, 1959.