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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