Equivalence between Kriging and CPDF Methods for Conditional Simulation

Abstract

Currently the kriging and conditional probability density function (CPDF) methods are widely used in solving the conditional simulation problems involving stochastic processes and fields. For the fundamental understanding of these two methods, this paper considers their applications to the conditional simulation of a one-dimensional, univariate and stationary stochastic process or field. The major findings of this study are as follows. First, the two methods are completely equivalent if the stochastic process is Gaussian with a zero mean. Specifically, the best linear unbiased estimate (BLUE) and the kriging variance are identical to the corresponding conditional mean and variance, respectively. Second, when the kriging method is used, the conditional simulation of a nonzero mean stochastic process (with a known value of the mean) is not equivalent to the (nonzero) mean plus the conditional simulation of the zero mean stochastic process obtained by subtracting the nonzero mean from the original process. Third, it can be shown that the second moment of the process conditionally simulated with the help of the kriging method are not identical to the target second moment (a priori known statistics). Finally, the kriging method is not suitable for the conditional simulation of non-Gaussian stochastic processes if no other assumptions or conditions are made for the reasons indicated in the paper, although the estimation (BLUE) may still be performed, as claimed by its proponents.