Kriging assumptions

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,... 

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. 

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 ... 

Kriging is based on the assumption that the variables between two known points follow a stochastic process, which is characterized by a variogram model. 

While the Kriged image plots are useful for thinking about how the chemical patterns might be distributed across space, we would like to point out that here we assume that the chemical concentrations are spatially dependent and that they vary stochastically across space. We make this assumption based on our knowledge of the micro-soilscapes of the plaza and patio, which are highly... 

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 ... 

Perform the optimization of the flow-sheet substituting the selected units by their metamodel. If we are dealing with a constrained problem in which the constraints are calculated through a metamodel, the errors introduced in the model could produce infeasibilities. One way to deal with this problem consists of simply consider that the problem is possibly feasible if the infeasibilities for each equation are inside the errors estimated by the kriging. In the refinement stage, as errors decrease, it is possible to confirm or not this assumption. 

In noisy systems it is not possible to verify if the gradient of the metamodel matches the gradient of the true function. In this case the stopping criteria is based on the assumption that if in two successive major iterations (at least one contraction must be performed) the optimal solution is the same, we would expect that the gradients also match the true gradients . This is only a heuristic based on the observation that, as the domain reduces, the accuracy of the kriging increases, and also the accuracy of the derivative information extracted from the kriging. 

Journel(7-9) provides a means for generating the desired mappings in situations such as the example site which are free from any undue assumptions regarding the distributional form of concentrations within blocks. This approach simply transforms the data into the sample cumulative probability distribution. This nonparametric geostatistical approach is sometimes called "indicator kriging."... 

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