Abstract
Simulation of multigaussian stochastic fields can be made after a Karhunen-Loéve expansion of a given covariance function. This method is also called simulation by Empirical Orthogonal Functions. The simulations are made by drawing stochastic coefficients from a random generator. These numbers are multiplied with eigenfunctions and eigenvalues derived from the predefined covariance model. The number of eigenfunctions necessary to reproduce the stochastic process within a predefined variance error, turns out to be a cardinal question. Some ordinary analytical covariance functions are used to evaluate how quickly the series of eigenfunctions can be truncated. This analysis demonstrates extremely quick convergence to 99.5% of total variance for the 2nd order exponential (‘gaussian’) covariance function, while the opposite is true for the 1st order exponential covariance function. Due to these convergence characteristics, the Karhunen-Loéve method is most suitable for simulating smooth fields with ‘gaussian’ shaped covariance functions. Practical applications of Karhunen-Loéve simulations can be improved by spatial interpolation of the eigenfunctions. In this paper, we suggest interpolation by kriging and limits for reproduction of the predefined covariance functions are evaluated.
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References
Braud, I.; Obled, C. 1991: On the use of Empirical Orthogonal Function (EOF) analysis in the simulation of random fields. Stochastic Hydrol. Hydraul. 5, 125–134
Creutin, J.D.; Obled, C. 1982: Objective analysis and mapping techniques for rainfall fields: An objective comparison. Water Resour. Res. 18(2), 413–431
Christakos, G. 1992: Random Field Models in Earth Sciences, Academic Press, Inc. U.S.A. ISBN 0-12-174230-X
Dagan, G. 1989: Flow and Transport in Porous Formations. Springer-Verlag, Berlin. ISBN 3-540-51602-6
Davenport, W.B.; Root, W.L. 1958: An Introduction to the Theory of Random Signals and Noise, McGraw-Hill, New York
Davis, R.E. 1976: Predictability of sea surface temperature and sea level pressure anomalies over the North Pacific ocean. J. Phys. Ocean. 6(3), 249–266
Fortius, M.I. 1973; Statisticheski ortogonalnye funktsii stohasticheskogo processa v zamknutom intervale. (Statistical orthogonal functions of a stochastic process on a closed interval) Fizika Atmosfery i Okeana Vol IX(l), (in Russian)
Fortius, M.I. 1975; Statisticheski ortogonalnye funktsii stohasticheskogo polia, opredelennye dlia zamknutogo prostranstva. (Statistical orthogonal functions of a stochastic field, determined for a closed space) Fizika Atmosfery i Okeana Vol XI(11), (in Russian)
Gomez-Hernandez, J.J.; Srivastava, R.M. 1990: ISIM3D: An ANSI-C three-dimensional multiple indicator conditional simulation program. Com. and Geosci. 16(4), 395–440
Gottschalk, L. 1993: Correlation and covariance of runoff. Stochastic Hydrol. Hydraul. 7, 85–101
Hisdal, H.; Tveito, O.E. 1992: Generation of runoff series at ungauged locations using empirical orthogonal functions in combination with kriging. Stochastic Hydrol. Hydraul. 6, 255–269
Hisdal, H.; Tveito, O.E. 1993: Extension of runoff series using empirical orthogonal functions. Hydrol. Sci. J. 38(1), 33–49
Holmström, I. 1963: On the method for parametric representation of the state of the atmosphere. Tellus, XV, 127–149
Holmström, I. 1969: Extrapolation of meteorological data. Serie Meteorologi Nr 22, Sveriges Meteorologiska och Hydrologiska Institut, Stockholm, Sweden
Holmström, I. 1970: Analysis of time series by means of empirical orthogonal functions. Tellus, XXII, 638–647
Holmström, I.; Stokes, J. 1978: Statistical Forecasting of Sea Level Changes in the Baltic, SMHI Rapporter, Meteorologi och klimatologi, Nr RMK 9, Sveriges Meteorologiska och Hydrologiska Institut, Norrköping, Sweden
Johansen, G. 1993: Simulation algorithms for gaussian related random functions. M.Sc. Thesis in Mathematical Statistics, NTH, University of Trondheim, Norway
Journel, A.G. 1983: Nonparametric Estimation of Spatial Distributions. Math. Geol., 15(3), 445–468
Journel, A.G.; Alabert, F. G. 1990: New method for reservoir mapping, J. Petr. Tech., Febr., 221–218
Mantoglou, A. 1987: Digital simulation of multivariate two- and three-dimensional stochastic processes with a spectral turning bands method. Math. Geol., 19(2), 129–149
Mantoglou, A; Wilson, J.L 1982: The turning bands method for simulation of random fields using line generation by spectral method. Water Resour. Res. 18(5), 1379–1394
Matheron, G. 1973: The intrinsic random functions and their applications. Adv. Appl. Probab. 5, 439–468
Obled, C.; Creutin, J.D. 1986: Some developments in the use of empirical orthogonal functions for mapping meteorological fields. J. Climate Appl. Meteor. 25(9), 1189–1204
Omre, H.; Sölna, K; Tjelmeland, H. 1991: Simulation of random functions on large lattices. SAND/04/1991 Norwegian Computing Center, Norway
Press, W.H.; Teukolsky, S.A.; Vetterling, W.T., Flannery, B.P. 1992: Numerical Recipes in C, The art of Scientific Computing. 2nd edition, Cambridge University Press, New York
Rubin, Y.; Journel, A.G. 1991: Simulation on non-gaussian space random functions for modeling transport in groundwater. Water Resour. Res. 27(7), 1711–1721
Storch, H.v.; Hannoschöck, G. 1985: Statistical aspects of estimated principal vectors (eofs) based on small sample sizes. J. Clim. Appl. Meteror., 24, 716–724
Tompson, A.F.B.; Ababou, R.; Gelhar, L.W. 1989: Implementation of the three-dimensional turning bands random field generator. Water Resour. Res. 25(10), 2227–2243
Wilkinson, J.H..; Reinsch, C. 1971: Linear Algebra, vol. II of Handbook for Automatic Computation. Springer Verlag, New York
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Kitterrød, NO., Gottschalk, L. Simulation of normal distributed smooth fields by Karhunen-Loéve expansion in combination with kriging. Stochastic Hydrol Hydraul 11, 459–482 (1997). https://doi.org/10.1007/BF02428429
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DOI: https://doi.org/10.1007/BF02428429