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Analytical Expressions of the Posterior Operator 133

Table 6.1. Examples of parameters and operators in Equation 6.17.

Soft Data
Equation B D 5f
3.32 I I Xao/t Eq. 6.1

3.33 1 I ^(X»y* Eq. 6.3
3.34 J R1 dFXC; h) J(0 Xso/t Eq. 6.8
3.35 Xso/t Eq. 6.9

COMMENT 6.2 : One should keep i n mind that th e posterior operator i n

Equation 6.17 i s a mathematical quantity representing ou r state o f knowl-

edge o f the physical phenomenon under study. Consequently, th e ^ -operator
has epistemic features that allow it to possess whatever properties the mod-

ern geo'statistician chooses based on available knowledge, the goals of the

study, etc. The only requirements are that the corresponding pdfmust satisfy

the mathematical conditions of a pdf and that the results of any calculations

we make with this pdf agree with the physical data, scientific laws, etc.
The above results certainly do not exhaust all possibilities regarding the
^-operators. These results deal with knowledge that can be expressed in terms
of the natural variable X. However, knowledge that depends on some other
variables Y\, Y%, etc. can also be studied by BME (e.g., Chapter 9). The
reader is encouraged to consider operators that best describe physical knowl-
edge about a scientific or engineering problem of his/her own interest and derive
mathematical expressions for the relevant posterior operator. Certain imple-
mentations of the BME formulas above could be computationally intensive,
especially when multiple integrations over pdf's are involved. However, contin-
uing progress in numerical approximation techniques (Monte Carlo, etc.), as
well as the fast-developing technology of workstations and parallel computation
is expected to handle such computational issues efficiently.
The derivation of the BME posterior pdf (Eq. 6.17) does not involve any
of the restrictive assumptions and approximations used by other geostatistical
methods, such as the multi-Gaussian and indicator approaches. Limitations of
the multi-Gaussian characterization of posterior probability distributions include
(Goovaerts, 1997): (a) strong assumptions are made about the multivariate
probability distribution, which usually cannot be checked in practice; (b) ex-
tremely large and low values are considered spatially uncorrelated, which is of-
ten an invalid assumption; (c) the corresponding variance is data-independent.
The indicator approach suffers from theoretical and practical limitations such
as (Olea, 1999): (i) unfeasible values for cdf may be obtained; (ii) the derived
cdf sometimes fail order-relationship requirements; (m) the large number of
indicator thresholds involved require heavy computational and inference efforts
(to reduce these efforts approximations are introduced, which may make things
worse); (iv) the approach can account for only a few cases of imprecise data;