value at each grid point was then subtracted from the true value obtained via 

 equation (30) and the result was machine contoured at a contour interval of 

 0.5 mgl . This residual contour chart is shown in Figure 7. Figure 8 is a 

 histogram of the model-1 errors with the standard deviation and range noted. 



In order to compare the accuracy of the spline interpolation to that 

 which could be achieved through the use of a two-dimensional least-squares 

 polynomial surface, a twelfth-degree surface was fitted to the simulated survey 

 data obtained from model 1 . The contoured field values generated by this 

 surface are shown in Figure 9. Comparison of this surface with the true field 

 values in Figure 4 indicates that the least-squares surface reproduces the 

 general shape of the low-frequency components but does not depict the higher 

 frequency features. As stated previously, the least-squares surface can be made 

 to fit the high frequency components by reducing the size of the area over which 

 the surface is computed but this is not considered to be an appropriate approach 

 to interpolation when adequate survey data is available. 



By overlaying Figure 7 with the contour chart of the model field, it is 

 readily apparent that the errors in the interpolated values are correlated with 

 those regions of the model field which contain a large amount of energy at the 

 higher frequencies. This correlation is interpreted as arising from the fact that 

 the two-dimensional FFT is essentially a least-squares operation which produces 

 an estimate of the average amplitude spectrum over the entire region in which 

 the computation is carried out. This effect, coupled with the obvious non- 

 stationarity (in space) of the model field, caused the amplitude spectrum to 

 flatten out at an unrealistically low frequency (0.17 cycles/data interval). 

 In turn, this led to an estimate of track spacing and sampling interval which was 

 too large to adequately define the higher frequencies. In order to determine the 

 effect of a shorter sampling interval, a second model, designated model 2, was 

 constructed with track spacing and direction similar to model 1 but with a 

 sample interval of 1 nm. The histogram of the errors resulting from the 



24 



