Influence of parameter estimation uncertainty in Kriging: Part 2 - Test and case study applications

The theoretical approach introduced in Part 1 is applied to a numerical exa mple and to the case of yearly average precipitation estimation over the Ve neto Region in Italy. The proposed methodology was used to assess the effec ts of parameter estimation uncertainty on Kriging estimates and on their es timated error variance. The Maximum Likelihood (ML) estimator proposed in P art 1. was applied to the zero mean deviations from yearly average precipit ation over the Veneto Region in Italy, obtained after the elimination of a non-linear drift with elevation. Three different semi-variogram models were used, namely the exponential. the Gaussian and the modified spherical, and the relevant biases as well as the increases in variance have been assesse d. A numerical example was also conducted to demonstrate how the procedure leads to unbiased estimates of the random functions. One hundred sets of 82 observations were generated by means of the exponential model on the basis of the parameter values identified for the Veneto Region rainfall problem and taken as characterising the true underlining process. The values of par ameter and the consequent cross-validation errors. were estimated from each sample. The cross-validation errors were first computed in the classical w ay and then corrected with the procedure derived in Part 1. Both sets, orig inal and corrected, were then tested, by means of the Likelihood ratio test . against the null hypothesis of deriving from a zero mean process with unk nown covariance. The results of the experiment clearly show the effectivene ss of the proposed approach.