Page 220 - Modern Spatiotemporal Geostatistics
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Single-Point Analytical  Formulations           201

            The following proposition  deals with the situation in which the space/time
        variability  is expressed in terms  of  non-centered covariances.


        PROPOSITION     10.2:  The  specificatory  knowledge  is  as  described  in
        Proposition  10.1.  The  general  knowledge  consists of  the  non-centered
        ordinary  covariance  in  space/time.  The  BMEmode  estimate  x.k  is  the
        solution  of the  equation






                     are the ik-ib and fc-th elements, respectively, of the inverse
        matrix C^ , where



        is  the  matrix  of  the  spatiotemporal  non-centered  covariances  (7^  be-
                          and PJ  (i, j  =  1, ..., m, k);  and the parameter
        tween all points p t
        is of  the  form  of  Equation  10.15  with




         Proof:  Working  along the  lines  of  the  proof  of  Corollary  10.1  above, we find
        that  Equation  7.10  (p.  137)  can be written  as






        where                   is the ij-th element of the inverse matrix
        9ij(Xi,Xj)  = XiXj  (i,j  = l,...,m and  k),  a






         Equation  10.19  can  be simplified  as follows:





        which—taking into account Equations  10.15  and 10.18—can  be written  in the
        form  of Equation  10.16.

             In certain  practical  applications the  spatiotemporal  variogram can be cal-
        culated  more  efficiently  than  certain  other  second-order statistical  moments.
         In such cases the following  proposition  is useful, for  it  provides the  appropriate
         BME  formulation  in terms  of variogram  functions.
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