Stochastic Modeling and Geostatistics. AAPG Computer Applications in Geology. Volume 2. Edited by Jeffrey M.Yarus and Richard L.Chambers. In press. Modified weighted least squares semivariogram and covariance model fitting algorithm. Alexander Gribov 1 , Konstantin Krivoruchko 1 , Jay M. Ver Hoef 2 1 Environmental Systems Research Institute, 380 New York Street, Redlands, CA 92373-8100 2 Alaska Department of Fish and Game, 1300 College Road, Fairbanks, AK 99701 Introduction Broadly speaking, geostatistics can be divided into two parts – 1) modeling the semivariogram or covariance, and 2) kriging. We can further distinguish three steps that are often used (often, but not always, see for example Barry and Ver Hoef 1996, Diggle 1998) for modeling the semivariogram or covariance: 1. Compute the empirical semivariogram or covariance, 2. Choose a model among the family of valid semivariograms or covariances, and 3. Estimate the semivariogram or covariance by fitting the valid model to the empirical semivariogram or covariance. Once the semivariogram or covariance has been estimated, it can be used in one of the forms of kriging (e.g., ordinary kriging, simple kriging, universal kriging, indicator kriging, etc.) This article proposes some new methods for both computing empirical semivariograms and covariances (step number 1) and fitting semivariogram and covariance models to the empirical data (step number 3). We will describe the algorithm for fitting semivariogram and covariance models that is used in the Geostatistical Analyst extension to GIS ArcInfo/ArcView 8.1 (Krivoruchko, 1999). One of the goals of the software is to obtain good default parameters for semivariogram and covariance models that allow novice users to create a prediction (kriging) map. Thus, it is necessary to find an algorithm that is reliable for a wide range of data types and conditions. Empirical semivariogram estimation Let us first consider the empirical semivariogram. Let z(s p ) be a value observed at the pth location s p , where s p = (x p , y p ) is the vector containing the x- and y-spatial coordinates. Define the lag h as the vector from point s 1 to point s 2 as h = s 2 - s 1 . The semivariogram cloud is defined as half of squared differences, γ pq =0.5⋅[z(s p ) - z(s q )] 2 for all possible pairwise lags between s p and s q ,