Page 254 - Modern Spatiotemporal Geostatistics
P. 254

Popular  Methods  in  the  Light  of  Modern Geostatistics  235

        showing  that  the  BME  and  SK  estimates  coincide  in  this  case  (which  is an
        expected  result).
                            Table  12.1.  The g a  functions.




                           Normalization  constraint

                            Variogram  constraints







        EXAMPLE   12.5:  Consider  the  case  of  Example  12.4  above,  where the  vari-
        ograms  7^  (i, j  = 1,  2,  k)  between the three  points  are spatially  isotropic/
        temporally stationary.  The well-known  MMSE estimate provided by space/time
        ordinary  kriging (OK)  is


        We will compare the estimate  (Eq.  12.12) with the one obtained by BME  using
        the  same  knowledge  base.  Since the  known  statistics are the  variograms,  the
        information  available concerns the differences               The
        statistical  constraints  are shown  in Table  12.1.  The  difference
        is  not  considered,  for  it  is a linear combination  of  the  previous two differences
        d lk  and  V2fe-  The  OC

        The  Lagrange multipliers //i  and  (12  are found  by solving the following system
        of  equations


        and



        The  solution  of  Equations  12.14  and  12.15  gives



        The  BME  equation  reduces  to



        the solution of which  is (note that due to isotropy/stationarity,


         Hence,  the  BME  estimate  (Eq.  12.18)  is the  same  as the  OK  estimate  (Eq.
         12.12), as was expected.
   249   250   251   252   253   254   255   256   257   258   259