Page 159 - Modern Spatiotemporal Geostatistics
P. 159
140 Modern Spatiotemporal Geostatistics — Chapter 7
Estimating the value of the variable at p k is equivalent to solving the
BMEmode of Equation 7.6, which in this case reduces to
where the integration is over the range [xiiX:;] °f tne s°ft datum. After the
evaluation of the partial derivatives and the substitution of the c^'s from
Equation 7.13, Equation 7.14 gives
The desired BMEmode estimate Xk is obtained as the solution to Equation
As was mentioned in previous chapters, BME formalism is very general,
and one has considerable freedom in choosing covariance, variogram, and gen-
eralized covariance models. Homogeneous/stationary or nonhomogeneous/
nonstationary, separable or nonseparable, etc. models can be used depending
on one's understanding of the basic features of the problem (see also "Spa-
tiotemporal Covariance and Variogram Models" in Chapter 11, p. 224). The
next numerical example is concerned with the effect of incorporating skewness
into the mapping calculations.
EXAMPLE 7.3: Based upon the spatiotemporal hard data configuration of Fig-
ure 7.1, x^(Pfe)"rea''zat'ons (^ = 1> • • • > 1000) were generated assuming the
same statistics as in Example 12.7 (p. 238). For each one of these realizations,
the BMEmode estimate was obtained at pointpk, assuming various
skewness values (i.e., 0,2,3, and 3.5) as prior knowledge. Then, the corre-
sponding estimation errors
were calculated. The pdf's of these estimation errors are plotted in Figure 7.2.
The error pdf's change considerably as the skewness values vary, thus showing
the importance in estimation accuracy of the skewness values incorporated by
BME.
Furthermore, Bogaert et al. (1999) have suggested a computational tech-
nique that incorporates into BME analysis information available about the non-
Gaussian shape of the univariate pdf of the random field. This technique in-
volves a suitable transformation of the posterior pdf to the Gaussian space.
Then, the pdf is back-transformed to the original space using a smoothed es-
timate of the transformation.
