Fitting Cluster Data as an Alternative to Simple Kriging for Estimation of Spatial Heterogeneity

Simple kriging is a powerful estimation tool in the hydrogeological field; it is not only unbiased, but also an exact estimator. Kriging requries computation or fitting of a variogram model and our proposed method starts with the very same process. After estimating a mean, variance, and length scale of the target spatial variable field, our method of estimating spatial heterogeneity utilizes clustering algorithms and an indicator function, rather than an inverted covariance matrix and vector of coefficients. Similar to an inversion modeling approach, by using the variogram data, a training set of random spatial fields are generated. Using only the information available to the kriging algorithm - measurements at specific locations - the training set and measured data are clustered. The cluster containing the true field data and training random fields are retained and used to fit a probability distribution. Our study compared the clustering approach to simple kriging approach with the assumption of an underlying Gaussian spatial process; mean squared error was reduced at every estimation location in the field by clustering relative to simple kriging. Additionally, the use of clustering algorithms can identify nonlinear relationships within the training data not captured by two-point covariance statistical models.