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Multipoint Analytical Formulations 221

PROPOSITION 11.4: Assume that hard data are given at points p t
(i = 1, 2, . . . , nih) and soft data of the probabilistic type (Eq. 3.33) at
points p i (i = mh + 1, . . . , m). General knowledge is the (centered)
ordinary covariance. The posterior pdf is

where

The BME posterior pdf s (Eqs. 11.14 and 11.15) may be interpreted as a
natural synthesis of the two knowledge bases, i.e., the general (covariance)
and the specificatory (soft data). In the case of single-point analysis (p = 1),
important parameters in spatiotemporal mapping are the conditional mean

and the variance

where and
and the A is defined as in Proposition

COMMENT 11.1 : Th e computational BM E formulations presented above

account fo r various specificatory knowledge bases fe.g. , combinations o f

hard and soft data), thus leading to a non-Gaussian posterior pdf, in gen-
eral. Furthermore, the BME estimates are nonlinear. When only hard
data ar e used, i.e. , m/ j = m, th e multiple integral and th e subscript s i s

dropped in Equations 11.14-11.17. In this case, the mean and the vari-

ance of the posterior pdf become, respectively and

These quantities coincide with the sim-

ple kriging (SK) estimate and its error variance, respectively. This result

shows that BME analysis can offer a unified framework which contains the

existing kriging methods as its limiting cases. More specifically, the im-

plication of the unified framework in numerical calculations is that when
(j = {mean and covariance} and S = {hard data}, BME will provide the

same estimates as the kriging techniques (for a detailed discussion see Chap-

ter 12). When higher order space/time moments, physical laws, soft data,

etc. ar e added, th e kriging methods cannot b e used. Th e BM E techniques,

however, can still be implemented, leading to useful results.