2-D and 3-D Trend Analysis - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 251

Fig. 7.12 Typical kriging
interpolation problem;
values are known at five
points. Problem is to
estimate value at point P 0
from values at points P 1 -P 5
(Source: Agterberg 1974 ,
Fig. 64)

P 5

P 4

P 3

P 0

P 2

P 1

7.2.1 Top of Arbuckle Formation Example

As a contribution to the discussion of which technique is better: trend surface
analysis or kriging, Agterberg ( 1970 ) performed the following experiment. A set
of 200 elevation data for the top of the Arbuckle Formation in Kansas (Fig. 7.13 )
was randomly divided into three samples: two “control” samples (No. 1 and No. 2)
each consisting of 75 data and a test sample (No. 3) of 50 data. Three different
techniques were applied to samples No. 1 and No. 2 and the results used to make
predictions of the elevations at the top of the Arbuckle Formation at the 50 points of
sample No. 3. These three techniques were: (1) Trend surface analysis; (2) Kriging,
and (3) Trend surface analysis plus kriging of residuals. Linear, quadratic, cubic
and quartic trend surfaces were fitted to the entire data set and the two smaller
control samples. Details of how 2-D autocorrelation functions were constructed for
kriging applied to residuals from these surfaces are given in Agterberg ( 1970 ).
Estimates of 2-D autocorrelation functions based on residuals from the linear,
quadratic, and cubic surfaces fitted to the entire data set are shown in Fig. 7.14 .
Results of the experiments involving the subsamples are given in Tables 7.2 and
7.3 . Analysis of variance to decide on the best trend surface cannot be used because the
residuals from the surfaces are autocorrelated. From the results shown in Table 7.2 it
can be concluded that the quadratic, cubic and quartic fits perform equally well. Note
the drop in percentage explained sumof squares fromquadratic (74%) to cubic (60%)
for predictions made by control sample 1. Table 7.3 contains results for sample 3 on
the basis of the control samples after subjecting the deviations from the fitted surfaces
to kriging. Kriging on its own, without trend surface analysis, is about as good as the
fitting of a quadratic trend surface. Because there are strong trends in the data,
universal kriging (of. Huijbregts and Matheron 1971 ) is preferable to simple or
ordinary kriging. Kriging of residuals improves the overall degree of fit in all
experiments.