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

Abstract

The theoretical approach introduced in Part 1 is applied to a numerical example and to the case of yearly average precipitation estimation over the Veneto Region in Italy. The proposed methodology was used to assess the effects of parameter estimation uncertainty on Kriging estimates and on their estimated error variance. The Maximum Likelihood (ML) estimator proposed in Part 1, was applied to the zero mean deviations from yearly average precipitation 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 assessed. 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 parameter and the consequent cross-validation errors, were estimated from each sample. The cross-validation errors were first computed in the classical way and then corrected with the procedure derived in Part 1. Both sets, original and corrected, were then tested, by means of the Likelihood ratio test, against the null hypothesis of deriving from a zero mean process with unknown covariance. The results of the experiment clearly show the effectiveness of the proposed approach.

Influence of parameter estimation uncertainty in Kriging: Part 1 - Theoretical Development

“…The entity and the importance of the bias and of the increase in variance will be elaborated in Part 2 by means of a case study based upon the average yearly precipitation over the Veneto Region in Italy (Todini et al, 2001). Although, for the sake of simplicity, the development of the methodology in this paper does not consider errors in data, it can be expanded easily to take them into account.…”

“…The application of the technique to real data with this independence hypotheses showed that the assumption of independent errors may involve significant distortions in the estimation of the parameters particularly for certain types of variograms (Todini et al, 2001).…”

Stochastic rainfall interpolation and downscaling

“…Obviously, this introduces additional uncertainties related to the many processes involved in the rainfall-runoff transformation and the modelling scheme used, and makes the validation of downscaling models only affordable in an indirect and therefore scarcely reliable way. To overcome this difficulty, standard synthetic tests based on Monte Carlo approaches should be set up and, most important, agreed upon by the hydrological and meteorological communities, to test objectively the different approaches in terms of expectation and reduction in uncertainty (Todini, 2001). Nonetheless, the investigation of detailed case studies is of great relevance and must be encouraged.…”

“…As an alternative to full Bayesian approaches, in view of these difficulties, a new technique is presented by Todini (2001) and demonstrated for the interpolation of the rainfall field in a region (Todini et al, 2001).…”

“…However, a sharp distinction cannot be made since interpolation and downscaling can both be incorporated in one single approach, e.g. in order to exploit jointly the information content of both remotely sensed and rain gauge data (Fiorucci et al, 2001;Todini, 2001).…”