Simulation of normal distributed smooth fields by Karhunen-Loéve expansion in combination with kriging |
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| Authors: | Nils-Otto Kitterrød Lars Gottschalk |
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| Institution: | (1) Norwegian Water Resources and Energy Administration, Hydrology Department, Box 5091, Maj. 0301 Oslo, Norway;(2) Department of Geophysics, University of Oslo, Blindern, Box 1022, 0315 Oslo, Norway |
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| 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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| Keywords: | Karhunen-Loéve expansion Empirical Orthogonal Functions stochastic simulation gaussian fields analytical covariance functions eigenfunctions kriging |
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