Page 132 - Modern Spatiotemporal Geostatistics

Mathematical Formulation of the BME Method 1 1 3

which together with the normalization equation

must be solved with respect to (J.Q, p, a,x(t a), & — l,—,m,k and A = 1, 2.
In this formulation, while the mean X(t) may not be known explicitly, it can
be considered to be known implicitly as the solution of Equation 5.17. For
illustrative purposes, let us consider the simple case in which a = k so that
Xa = Xk and fj, a,\(t a) = n\(tk), A = 1, 2. After some analytical manipula-
tions, Equations 5.20 and 5.21 give

where fJ>\(0), A = 1, 2 are the initial conditions. These conditions are related to
the mean and the variance of X(t) at t = 0, namely and
By substituting Equation 5.22 into Equation 5.21, the
/ZQ is found. The set of equations in Equation 3.18 (p. 79) can also be included
in the analysis in a similar fashion. Indeed, the 9^-operator is given in this case
by

which is substituted into Equation 3.18, leading to

or