2-D and 3-D Trend Analysis - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 270

FEET

0 100 200 300 400

.01 − .23 − .41 − .37 − .16 − .01 − .23 − .52 1.00 .52 .23 − .01 − .16 − .37 − .41 − .23 .01

− .08 − .25 − .34 − .21 − .00 .20 .31 − .01 − .22 − .37 − .50 − .47 − .16 − .03

.61 .55 .30

− .26 − .35 − .16 − .05 .23 .28 .45 .48 .30 .04 − .15 − .38 − .42 − .48 − .17 − .03 .05

− .44 − .20 − .04 .25 .35 .46 .50 .31 .04 − .18 − .39 − .48 − .53 − .27 − .08 .05 .13

− .33 − .15 .15 .38 .47 .47 .38 .08 − .16 − .40 − .48 − .49 − .34 − .13 .09 .20 .13

.22 .24 .19

− .17 .01 .33 .45 .40 .30 .05 − .15 − .38 − .49 − .50 − .45 − .11 .09

Fig. 7.29 Two-dimensional autocorrelation function for data from Table 7.4 , Whalesback copper
deposit (Source: Agterberg 1974 , Fig. 61)

FEET

PERIOD 1600 800 533 400 320 267 229 200

0 0 1 1 0 5 5 1 0 1 5 5 0 1 1 0 0

1 0 1 2 0 1 1 0 0 6 18 16 5 0 0 0 0

0 0 0 1 1 1 1 0 1 11 29 28 12 2 0 1 0

1 0 0 0 0 0 0 0 2 10 23 27 15 3 0 1 2

1 1 0 1 1 1 0 0 2 4 8 13 11 3 0 0 1

1 1 0 1 2 1 1 0 1 1 1 3 5 3 1 1 1

Fig. 7.30 Part of two-dimensional power spectrum for data in Table 7.4 ; sharply defined peak
reflects zoning of copper percent-foot values shown in Fig. 7.28 (Source: Agterberg 1974 , Fig. 77)

and phase) are estimated simultaneously but then a nonlinear model should have
been developed and applied. The linear model can only be used when p / m and q / n
are assumed to be known. For each separate sine wave, R 2 was plotted as a
percentage value in Fig. 7.31a . The result can be regarded as a 2-D power spectrum
for irregularly spaced data. Since three coefficients were fitted for each sine wave
(including the constant term that is related to the mean value), each R 2 value could
be converted into an F -value at the upper tail of a theoretical F -distribution with
3 and 17 degrees of freedom. The F 0.95 - and F 0.99 -values correspond to 27-% and
37-% values for R 2 and these were contoured. There are two contoured peaks in
Fig. 7.31a and one of these corresponds to the peak in Fig. 7.30 . The contour map of
the sine-wave whose amplitude falls on the well-developed peak of Fig. 7.31a is
shown in Fig. 7.31b , and its profile for a line along the surface ( V 10,600) in

Fig. 7.31c . The 20 original observations were projected onto Fig. 7.31c .along lines
parallel

¼

to the contours. Obviously, the contours in Fig. 7.31b are an