The paper addresses the representation and simulation of random fields using wavelet bases. The probabilistic description of the wavelet coefficients involved in the representation of the random field is discussed. It is shown that a broad class of random fields is amenable to a simplified representation. Further, it is shown that a judicious use of the local and multiscale structure of Daubechies wavelets leads to an efficient simulation algorithm. The synthesis of random field samples is based on a wavelet reconstruction algorithm which can be associated with a dynamic system in the scale domain. Implementation aspects are considered and simulation errors are estimated. Examples of simulating random fields encountered in engineering applications are discussed.
Issue Section:
Technical Papers
1.
Basseville
M.
Benveniste
A.
Chou
K. C.
Goldin
S. A.
Nikoukhah
R.
Willsky
A. S.
1992
a, “Modeling and Estimation of Multiresolution Stochastic Processes
,” IEEE Transaction on Information Theory
, Vol. 38
, No. 2
, pp. 766
–784
.2.
Basseville
M.
Benveniste
A.
Willsky
A. S.
1992
b, “Multiscale Autoregressive Processes, Part I: Schur-Levinson Parametrizations, Part II: Lattice Structure for Whitening and Modeling
,” IEEE Transactions on Signal Processing
, Vol. 40
, No. 8
, pp. 1915
–1953
.3.
Belkin
G.
1993
, “On the Representation of Operators in Bases of Compactly Supported Wavelets
,” SIAM Journal of Numerical Analysis
, Vol. 6
, No. 6
, pp. 1716
–1740
.4.
Belkin
G.
Coifman
R.
Rokhlin
V.
1991
, “Fast Wavelet Transform and Numerical Algorithms I
,” Communications on Pure and Applied Mathematics
, Vol. 44
, pp. 141
–183
.5.
Clippingdale, S., and Wilson, R., 1989, “Least-Square Image Estimation on a Multiresolution Pyramid,” IEEE International Conference on Acoustics, Speech and Signal Processing, Glasgow, pp. 1409–1412.
6.
Daubechies
I.
1988
, “Orthogonal Basis of Compactly Supported Wavelets
,” Communications on Pure and Applied Mathematics
, Vol. 41
, pp. 909
–996
.7.
Daubechies, I., 1992, Ten Lectures on Wavelets, Society for Industrial and Applied Mathematics, Philadelphia, PA.
8.
Dijkerman
R. W.
Mazumdar
R. R.
1994
, “Wavelet Representations of Stochastic Processes and Multiresolution Stochastic Models
,” IEEE Transactions on Signal Processing
, Vol. 42
, No. 7
, pp. 1640
–1652
.9.
Fenton
G. A.
Vanmarcke
E. H.
1990
, “Simulation of Random Fields via Local Average Subdivision
,” Journal of Engineering Mechanics
, Vol. 116
, No. 8
, pp. 1733
–1749
.10.
Fenton
G. A.
1994
, “Error Evaluation of Three Random-Field Generators
,” Journal of Engineering Mechanics
, Vol. 120
, No. 12
, pp. 2478
–2497
.11.
Flandrin
P.
1989
, “On the Spectrum of Fractional Brownian Motions
,” IEEE Transactions on Information Theory
, Vol. 35
, No. 1
, pp. 197
–199
.12.
Flandrin
P.
1992
, “Wavelet Analysis and Synthesis of Fractional Brownian Motion
,” IEEE Transactions on Information Theory
, Vol. 38
, No. 2
, pp. 910
–917
.13.
Fournier
A.
Fussel
D.
Carpenter
L.
1982
, “Computer Rendering of Stochastic Models
,” Communications of the ACM
, Vol. 25
, No. 6
, pp. 371
–384
.14.
Herley
C.
Vetterley
M.
1993
, “Wavelets and Recursive Filter Banks
,” IEEE Transactions on Signal Processing
, Vol. 41
, No. 8
, pp. 2536
–2556
.15.
Keshner
M. S.
1982
, “1/f Noise
,” Proceedings of IEEE
, Vol. 70
, pp. 212
–218
.16.
Lewis
J. P.
1987
, “Generalized Stochastic Subdivision
,” ACM Transactions on Graphics
, Vol. 6
, No. 3
, pp. 167
–190
.17.
Luettgen
M. R.
Karl
W. C.
Willsky
A. S.
1994
, “Efficient Multiscale Regularization with Application to the Computation of Optical Flow
,” IEEE Transactions on Image Processing
, Vol. 3
, No. 1
, pp. 41
–64
.18.
Mignolet
M. P.
Spanos
P. D.
1992
, “Simulation of Homogeneous Two-Dimensional Random Fields: Part I—AR and ARMA Models
,” ASME JOURNAL OF APPLIED MECHANICS
, Vol. 114
, pp. 260
–269
.19.
Newland, D. E., 1993, An Introduction to Random Vibrations, Spectral and Wavelet Analysis, Longman House, Harlow, U.K.
20.
Rubinstein, R. Y., 1981, Simulation and the Monte Carlo Method, John Wiley and Sons, New York.
21.
Samaras
E.
Shinozuka
M.
Tsurui
A.
1983
, “ARMA Representation of Random Processes
,” Journal of Engineering Mechanics
, Vol. 111
, No. 3
, pp. 449
–461
.22.
Shinozuka
M.
Jan
C-M.
1972
, “Digital Simulation of Random Processes and Its Applications
,” Journal of Sound and Vibration
, Vol. 25
, No. 1
, pp. 111
–128
.23.
Spanos
P. D.
Mignolet
M. P.
1992
, “Simulation of Homogeneous Two-Dimensional Random Fields: Part II—MA and ARMA Models
,” ASME JOURNAL OF APPLIED MECHANICS
, Vol. 114
, 270
–277
.24.
Tewfik
A. H.
Kim
1992
, “Correlation Structure of the Discrete Wavelet Coefficients of Fractional Brownian Motion
,” IEEE Transactions on Information Theory
, Vol. 38
, No. 2
, pp. 904
–909
.25.
Vanmarcke, E. H., 1983, Random Fields: Analysis and Synthesis, MIT Press, Cambridge, MA.
26.
Walter, G. G., 1994, Wavelets and Other Orthogonal Systems with Applications, Boca Raton, FL.
27.
Wornell
G. W.
1990
, “A Karhunen-Loeve-Like Expansion for 1/f Processes via Wavelets
,” IEEE Transactions on Information Theory
, Vol. 36
, No. 4
, pp. 859
–861
.28.
Wornell
G. W.
Oppenheim
A. V.
1992
, “Estimation of Fractal Signal from Noisy Measurements using Wavelets
,” IEEE Transactions on Signal Processing
, Vol. 40
, No. 3
, pp. 611
–623
.29.
Zeldin, B. A., 1995, Ph.D. Dissertation, Rice University, Houston, TX.
30.
Zeldin, B. A., and Spanos, P. D., 1995, “Random Field Simulation Using Wavelet Bases,” Proceedings of the Seventh International Conference on Application of Statistics and Probability in Civil Engineering, M. Lemare, J.-L. Favre, and A. Mebarki, eds., Balkema, Rotterdam, Paris, France, July 10-13, pp. 1275–1283.
This content is only available via PDF.
Copyright © 1996
by The American Society of Mechanical Engineers
You do not currently have access to this content.
