2-D and 3-D Trend Analysis - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 236

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b c

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Fig. 7.1 Top of Arbuckle Formation in central Kansas. Contours are in feet below sea level (After
Davis 1973 ). Geographic co-ordinates are in arbitrary units (area shown measures 320 km on a
side.) (a) Contour map of original data; (b) Linear trend surface; (c) Residuals from linear trend
surface (Source: Agterberg 1984 , Fig. 1)

number of observation points ( n ) satisfies the condition n m .If n ¼ m , a special

type of surface would be fitted with the property T k ¼ Y k and no residuals. In most

applications, n is several times larger than the number of coefficients ( m ).

7.1.1 Top of Arbuckle Formation Example

A typical example of trend surface analysis is shown in Fig. 7.1 (after Davis 1973 ).
Figure 7.1a gives the contours (in feet below sea level) of the top of Ordovician rocks
(Arbuckle Formation) in central Kansas. The linear trend surface is shown in Fig. 7.1b
and the corresponding residuals in Fig. 7.1c . The sum of the values R k contoured in
Fig. 7.1c and the values T k (Fig. 7.1b ) gives the original pattern (Fig. 7.1a ).