Kriging techniques

Data analysis, using elementary statistics and Kriging techniques [23] in the construction of the geochemical maps enables an understanding of the heavy metal distribution and the origin of the regional pollution. 

Geostatistical techniques, such as variography and kriging, have been recently introduced into the environmental sciences (O Although kriging allows mapping of the pollution plume with qualification of the estimation variance, it falls short of providing a truly risk-qualified estimate of the spatial distribution of pollutants. 

Ideally, to characterize the spatial distribution of pollution, one would like to know at each location x within the site the probability distribution of the unknown concentration p(x). These distributions need to be conditional to the surrounding available information in terms of density, data configuration, and data values. Most traditional estimation techniques, including ordinary kriging, do not provide such probability distributions or "likelihood of the unknown values pC c). Utilization of these likelihood functions towards assessment of the spatial distribution of pollutants is presented first then a non-parametric method for deriving these likelihood functions is proposed. 

Probabilistic techniques of estimation provide some Insights Into the potential error of estimation. In the case of krlglng, the variable pCic) spread over the site A is first elevated to the status of a random function PC c). An estimator P (2c) is then built to minimize the estimation variance E [P(2c)-P (2c) ], defined as the expected squared error ( ). The krlglng process not only provides the estimated values pCiyc) from which a kriged map can be produced, but also the corresponding minimum estimation variances 0 (39 ) ... 

The weights Xf(z) are cut-off z-dependent, and are determined as solutions of a linear system ( ) In fact, the indicator estimators used for the case-study underlying Figures 3 were obtained by a- slightly more elaborate technique called probability kriging or PK (9, 10) ... 

Important issues in groundwater model validation are the estimation of the aquifer physical properties, the estimation of the pollutant diffusion and decay coefficient. The aquifer properties are obtained via flow model calibration (i.e., parameter estimation see Bear, 20), and by employing various mathematical techniques such as kriging. The other parameters are obtained by comparing model output (i.e., predicted concentrations) to field measurements a quite difficult task, because clear contaminant plume shapes do not always exist in real life. 

Kriging-based estimation techniques also represent an important toolset allowing the recognition of spatial distribution patterns... 

The technique of kriging uses a weighted moving average interpolation method [BUSCHE and BURDEN, 1991] in the sense of a straightforward linear regression [JOURNEL, 1988]. The weights, w, are determined on the basis of the experimental... 

Most metamodel techniques assume that the errors (e) are independent and identically distributed (with a normal distribution) f W(0,C7 ) Vx. However, the errors in the predicted values are usually not independent, but they are a function of x. Therefore, the kriging fitting approach is comprised of two parts a polynomial term and a departure from that polynomial ... 

Relationships within the 1 aetitiae-geniculata complex have also been investigated by using the correspondence analysis of allelic frequencies and the kriging linear interpolation (Matheron, 1969, 1 97Q). Thi.a. last.technique, widely employed in... 

A way to overcome this problem is to generate an approximation of complex analysis code that describes the process accurately, but at a much lower cost. Metamodels offer an approximation in that they provide a model of the model . Clarke et al. (2005) [58] suggested metamodelhng techniques, namely response surface methodology (RSM), radial basis function (RBF), kriging model and multivariate adaptive regression sphnes (MARS) as potentially useful approaches. Computer deterministic experiments have been addressed by Charles et al. (1996) [59], Simpson et al. (1998) [60], CappeUeri et al. (2002) [61] and Aguire et al. (2(X)7) [62],... 

Access to a full range of model approximation techniques such as polynomial, Kriging, or neural networks based response surfaces, sensitivity based Taylor series linearization, and variable complexity models. 

Commonly used surrogate modeling approaches include regression techniques (Haidar and Mahadevan 2000), polynomial chaos expansion (Najm 2009), kriging (Stein 1999), etc. Each of these methods uses different types of basis functions, and one may approximate G better than the other, and it is necessary to choose a suitable surrogate model based on the application of interest. 

Indicator Kriging This technique transforms the observations into integer valued quantities with the transformation... 

Failure of this property is not addressed in disjunctive kriging. For reasons of space, non-Gaussian spatial statistics are not discussed in this chapter. However see [56] or [97] for examples of non-Gaussian techniques. 

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