2-D and 3-D Trend Analysis - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 257

Fig. 7.18 Autocorrelation coefficients computed for residuals. Source: quadratic trend surfaces
(see text for method used) computed with profiles ( broken lines ) across theoretical

two-dimensional model used for kriging (Source: Agterberg 1984 , Fig. 10)

zones. On the other hand, Derry ( 1960 ) assumed separate, subparallel channels. It
will be seen that the results obtained by our trend + signal + noise model fit in with
Derry's hypothesis and not with Bain's hypothesis.
The following procedure was followed to obtain a 2-D autocorrelation function for
residuals from the trend shown in Fig. 7.17 . By using polar coordinates, distances
between pairs of boreholes were grouped according to (1) distance (1-km spacing) and
(2) direction of connecting line (45 wide segments). This gave domains of different
sizes. The autocorrelation coefficients for pairs of values grouped according to this
method are shown in Fig. 7.18 . These values were assigned to the centers of gravity of
their domains. A 2-D quadratic exponential (Gaussian) function with superimposed
noise component (nugget effect) was fitted by trend surface analysis of logarithmically
positive autocorrelation coefficients obtained from the grouped data. Both the input
pattern of autocorrelation coefficients and the fitted function are symmetrical with
respect to the origin where the observed autocorrelation coefficient is equal to
1 because it includes the noise component. A profile through the origin across the
quadratic trend surface fitted to logarithmically transformed autocorrelation function
is a parabola with its maximum at the origin. Each parabola becomes a Gaussian curve
when the antilog is taken. Four profiles across the fitted function are shown in
Fig. 7.18 . The maximum of the fitted function fell at 0.325 indicating that only
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½ % of the variance of the residuals is explained by the signal versus 67 ½ %by