Page 105 - Modern Spatiotemporal Geostatistics

86 Modern Spatiotemporal Geostatistics — Chapter 3

where I = (I mh+i,..., I m) is the domain of \ aoft\ k (ui) are the lower (upper)
bounds of the intervals /».

EXAMPLE 3.15: Other forms of commonly used soft data have a probabilistic
character, such as

is the cumulative distribution function (cdf) obtained from ,5; and
where F s

where the function h(-) represents an empirical chart, a model, a justified
belief, etc. (for instance, a hydrologist may be uncertain about the accuracy of
individual hydraulic conductivities obtained at two different wells, but he/she
can justifiably assign probability values to their differences). Equation 3.32 may
be derived as a special case of Equation 3.33 for a properly selected cdf (i.e.,
the Xi values lie with probability one within known intervals Ii). Soft data can
also be expressed in terms of interval probabilities, i.e.,

EXAMPLE 3.16: In some cases the knowledge base ,5 rnay involve probabilistic
logic relationships, such as

where the symbols "A" and "—>" denote conjunction and material conditional,
respectively (see also the discussion of conditional probability and the proba-
bility of conditionals in Chapter 4, beginning on p. 98).

EXAMPLE 3.17: There are cases of practical importance in which soft data are
available not only at the data points p aoft but also at the estimation point p k
itself. Such a case is one in which the specificatory knowledge has the form

See also Chapter 6.
In practice, there exist several methods for encoding the soft probability
functions mentioned above (F s, p s, etc.). Some of these methods depend on
the physical situation considered and the background of the experts involved.