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The GLOBEC Kriging Software Package - EasyKrig3.0, May 1, 2004 Copyright (c) 1998, 2001, 2004, property of Dezhang Chu and Woods Hole Oceanographic Institution. All Rights Reserved. Kriging is a technique that provides the Best Linear Unbiased Estimator of the unknown fields (Journel and Huijbregts, 1978; Kitanidis, 1997). It is a local estimator that can provide the interpolation and extrapolation of the originally sparsely sampled data that are assumed to be reasonably characterized by the Intrinsic Statistical Model (ISM). An ISM does not require the quantity of interest to be stationary, i.e. its mean and standard deviation are independent of position, but rather that its covariance function depends on the separation of two data points only, i.e. E[(z(x) - m)(z(x') - m) ] = C(h), (1) where m is the mean of z(x) and C(h) is the covariance function with lag h, with h being the distance between two samples x and x': h = || x - x' ||. (2) Another way to characterize an ISM is to use a semi-variogram, gamma(h) = 0.5* E[ (z(x) - z(x') )^2]. (3) The relation between the covariance function and the semi-variogram is gamma(h) = C(0) - C(h). (4) The kriging method is to find a local estimate of the quantity at a specified location, x(L). This estimate is a weighted average of the N adjacent observations: z(x(L)) = sum( lambda(i) z(x(i)), (5) where i is from 1 to N, and x(L) are the coordinates of an arbitrary point whose value is what we want to estimate. The weighting coefficients lammbda(i) can be determined based on the minimum estimation variance criterion: See Eq.(6) in Description.doc file (6) subject to the normalization condition. sum(lambda(i)) = 1, (7) where i is from 1 to N. Note that we don't know the exact value at , but we are trying to find a predicted value that provides the minimum estimation variance. The resultant kriging equation can be expressed as See Eq.(8) in Description.doc file (8) where mu is the Lagrangian coefficient. In addition, we have replaced the covariance function with the normalized covariance function [normalized by C(0)]. Equivalently, by using Eq. (4), the kriging equation can also be expressed in terms of the semi-variogram as See Eq.(9) in Description.doc file (9) where we have used normalized semi-variogram, i.e., semi-variogram normalized by C(0) as we did in deriving Eq. (8). Having obtained the weighting coefficients (lambda_beta) and the Lagrangian coefficient (mu) by solving either Eq. (8) or Eq. (9), the kriging variance, Eq. (6), can be expressed as: See Eq.(8) in Description.doc file (10) The above equations are the basis of the Easykrig software package. References. Deutsch, C. V and A. G. Journel, 1992. GSLIB: Geostatistical Software Library and User's Guide. Oxford University Press, Oxford, 340 p. Journel, A.G. & C.J. Huijbregts, 1992. Mining Geostatistics. Academic Press, New York, 600 p. Kitanidi, P.K. 1997. Introduction to Geostatistics. Applications in hydrogeology. Cambridge University Press. 249 p. Marcotte, D. 1991. Cokriging with MATLAB. Computers & Geosciences. 17(9): 1265-1280.