Kriging random functions

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 ) ... 

To use what is termed simple kriging, only the assumption that the random function is intrinsic needs to be made. The problem with this assumption is that the expected value of the phenomena of interest is rarely a constant. For example, the expected concentration of lead in the soil around a smelter would decrease as the distance from the smelter increased. If this decrease (or trend) is gradual enough, it is often assumed that within a limited neighborhood the random function has a "local stationarity" and then simple kriging is used, since generally only the observations within the limited neighborhood are used in the estimation process. 

To use what is termed universal kriging, it is assumed that Z(2 ) is an intrinsic random function of order k. But the problem of identifying the drift and the semi-variogram when they are both unknown is still present. However, Matheron (11) defined a family of functions called the generalized covariance, K(h). and the variance of the generalized increment of order k can be defined in terms of K(h ). That is. 

Simple kriging is actually a subset of universal kriging since the assumption that Z(2 ) is an intrinsic random function of order 0 is the same as the assumption that ZCjc) is intrinsic. Additionally, when l x) is intrinsic, the generalized covariance and the semi-variogram are related as follows ... 

In the previous section, an overview of the kriging assumptions was given. When these assumptions are accepted, a kriging system of linear equations can be developed. Whether the random function,... 

The theoretical basis of the kriging method is the so-called intrinsic hypothesis. A random function Z(x) accomplishes this hypothesis if two assumptions are fulfilled ... 

Simple Kriging Consider the problem of interpolating a random function. That is, given M sample values ipi,i = 1,..., M, at each of M spatial points Xj, estimate the value of the function, 99 at the point x. 

Application of Kriging to the discretization of random fields has been intrcxiuced by Li and Der Kiureghian (1993). The random field tx(x, co) is approximated by a linear function of M nodal values a,(ca) = a(x co)... 

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