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Multipoint  Analytical  Formulations            223






        By substituting the  last  equation  into the  BME  equation  (Eq.  9.29),  we find








        which  can also  be written  as Equation  11.18.
            Several of  the  single-point  results of  Chapter 10 may be derived as special
        cases  of  the  analysis  of  the  present  chapter.  The  BME  equation,  Equation
        10.21  (p.  202),  e.g., is a special case of  Equation  11.18  above; etc.

        Other    Combinations

        Analytical  results  for  applications  involving  combinations  of  vectorial  and
        multipoint  scenarios can  also  be  derived.  For  illustration,  consider  the  fol-
        lowing example.
        EXAMPLE   11.1:  Assume that  the  specificatory  knowledge about  the  natural
        fields  X(p)  and Y(p)  are the hard data  and    as well as the soft
        data xsoft ar|d $'soft (°f any °ftne various forms considered in Chapter 3). The
        general  knowledge consists  of the  ordinary  auto- and cross-covariance matrices




        Under these circumstances, the  BME  estimates Xk e  will  be the solutions  of  the
        following set of  BME  equations












        where                     denotes the ij-th element of the inverse covari
        ance matrix,                         and



            At  this  point  we  have  in  our  possession  a  number  of  models which  are
        largely  analytically  lucid.  Certainly,  the  analytical  formulations  become more
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