2-D and 3-D Trend Analysis - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 252

Fig. 7.13 Location of 200 wells in Kansas. Good ( 1964 ) fitted trend surfaces to elevation on top of
Arbuckle Group for points. (Linear trend surface was shown in Fig. 7.1 .) Observations have been
randomly divided into three samples. Solid circles are for control sample 1; open circles for control
sample 2; triangles for sample 3. Smaller circle represents area for estimating autocorrelation
function used for kriging (Source: Agterberg 1970 , Fig. 2)
The preceding experiments have been discussed in publications by statisticians
including those by Tukey ( 1970 ), Haining ( 1987 ) and Cressie ( 1991 ). In the 1950s
and 1960s, most earth scientists took a trend surface approach to their mapping
problems but later the extra advantages in taking a random-field approach (i.e.,
universal kriging) became clearly established. As pointed out by Cressie ( 1991 ,
p. 164) it can be said that comparison of results with or without the random-field
approach is easy because the trend-surface model is a special case of a random field
model. Consequently, when the spatial-covariance structure is known, universal
kriging generally gives more precise predictions than trend-surface analysis,
because universal kriging chooses optimal weights to be applied to the data.
However, as Watson ( 1971 ) has shown it is possible for trend-surface prediction
to be just as precise as universal kriging even when the residuals do not satisfy a
pure white noise model as would be required for analysis-of-variance applications
to decide on optimal degree. Cressie also points out that, in practice, there is a price
to pay for using universal kriging because: “One must obtain (efficient) estimators
of variogram parameters, whose effects on mean-squared prediction errors should
be assessed.” The following two sections contain further examples of universal
kriging. The objective of the modeling will be to separate the spatial variability into
three components: (1) regional trend; (2) localized “signals” based on the assump-
tion that the residuals from the regional trends are weakly stationary and possess the
same regional autocorrelation fun; and (3) strictly local white noise.