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Popular  Methods  in  the  Light  of  Modern  Geostatistics  233


            The following sections present analytical results and numerical  applications
        in  which  various  sorts  of  geostatistical  kriging  are compared  to  BME-based
        techniques,  and  by  means  of which  the  BME theory  is amply  confirmed.



         Kriging    Estimators

        As we saw above, the BME theory  preserves the referents of earlier geostatistical
        results which  are derived  as its  limiting  cases.  This  powerful  feature of  BME
        is demonstrated  in this section  by showing that  it  can easily reproduce  certain
        well-known  kriging  results.  Of  course,  BME  can  lead  to  improved  maps  in
        situations  in  which  additional  sources  of  knowledge  become available  (higher
        order  moments,  multiple-point  statistics,  scientific  theories,  etc.).

        Simple   and ordinary    kriging

        We start with some basic results that  are valid  in the  restrictive case in which
        our  knowledge consists of  hard data  and  low-order  space/time  statistical  mo-
         ments.

         PROPOSITION    12.2:  When  the  general  knowledge  is limited  to  the
        mean  and  (centered)  covariance  and  the  specificatory  knowledge  in-
        cludes  only  hard  data,  the  BMEmode estimate is






        where,  as before,  x, denotes  the  mean  value and c it}  is the  z'fc-th element
        of  the  inverse  (centered)  covariance  matrix  c~^ ap.  Equation  12.5  coin-
        cides with  the  simple  kriging estimate.
        Proof:  Under  the  conditions  described in the  proposition,  the  BME  equation
        reduces  to



        or





        where                     and                                After
        derivation,  Equation  12.7  reduces to  Equation  12.5.
            Working  along  the  same  lines,  the  following  proposition  can  easily  be
        proven.
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