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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.