PSEUDO-DEPTH SLICING

Wavelength filtering has been used for many years as a method for approximately separating the effects of shallower sources from those of deeper ones (Fuller, 1967). However, in designing appropriate filters there are a number of difficult issues to address, such as the transition wavelengths, rolloff characteristics between bands, and even the number of bands to use. Although it is often possible to obtain useful results by experimenting with these parameters, a systematic approach which yields near-optimal results under a wide range of circumstances is clearly desirable.

Pseudo-depth slicing offers such a systematic method for designing filters. The underlying framework for depth slicing is that of Wiener filtering (Wiener, 1949), which is widely used in seismic data analysis. Wiener filter design assumes that the signal (in this case, the gravity or magnetic map) can be regarded as the sum of two or more uncorrelated random processes. Under this assumption, the technique gives a unique way of separating the various components of the data which is optimal in a linear least-squares sense.

The practical problem then reduces to one of identifying in the data the uncorrelated components, which in a practical sense means finding a model for the power spectrum of these components. Such a model, which with various refinements has proved to be very successful in capturing the essential features of aeromagnetic data, was first introduced by Spector and Grant (1970). Spector and Grant's model assumes that the sources lie in a small number of vertically separated horizons, and that the magnetization distribution on each horizon is in some sense random. The latter assumption is, of course, somewhat fictitious, but turns out usually to be good enough to produce a useful model for the observed power spectrum.

Under Spector and Grant's assumptions the observed power spectrum is a sum of exponentials in spatial frequency. The logarithm of the power spectrum is thus approximately a series of linear segments, whose slopes correspond to the depths of the horizons. Inspection of the logarithmic power spectrum allows a determination of the number of horizons present; fitting a straight line to each segment yields a model which can be used to construct a set of Wiener filters that optimally extract the signal from the various horizons.

Figure 1: Reduced-to-pole magnetic intensity for a portion of the Upper Michigan Peninsula. Contour interval: 5 nT.

To illustrate how the technique can be used to extract information from a data set, we use a small portion of the aeromagnetic data from the Upper Michigan Peninsula. Figure 1 shows the reduced-to-pole total magnetic intensity over the sample area, contoured at 5 nT. Three features are evident: a broad anomaly with a high in the western part of the data and a low in the east; superimposed higher-frequency features which do not form any easily discernible pattern; and a linear feature cutting the southwest corner, which turns out to be a small part of a very large dike.

Figure 2: Radially averaged logarithmic power spectrum

Figure 3: Filter responses

Figure 2 shows the fit to the radially averaged power spectrum for the data in Figure 1. Note the logarithmic scale for the power. Four approximately linear segments can be discerned; the last of these is approximately horizontal, corresponding to white noise. Figure 3 shows the corresponding four filters. The output from the highest-frequency filter was discarded, as it indeed appeared to be simply noise.

Figure 4: Long-wavelength component of the reduced-to-pole magnetic data. Contour interval: 5 nT.

Figure 4 shows the low-frequency (long-wavelength) component of the data obtained using the first Wiener filter. Only the smoothly varying component, certainly due to basement structure or lithology or both, remains in this data set.

Figure 5: Intermediate-wavelength component of the reduced-to-pole magnetic intensity. Contour interval: 0.1 nT.

Figure 5 shows the intermediate-wavelength component. Note that the contour interval is now 0.1 nT. The dominant component is the dike in the southwestern corner of the data.

Figure 6: Short-wavelength component of the reduced-to-pole magnetic intensity. Contour interval: 0.1 nT.

Figure 6 shows the short-wavelength component of the data, again contoured at 1 nT. Some signal from the dike is visible in this component, although at an amplitude much reduced from Figure 5. The remainder of the image is a complex pattern of NNW-SSE and SW-NE trending anomalies. These trends reflect the major drainage patterns in the area, which has been glaciated during several periods. Some anomalies follow modern drainages; others almost certainly show paleochannels.

Thus, each component of the depth slicing reflects a distinct geological component of the data. The separation between components is never perfect, as evidenced by the presence of the dike anomaly in both the intermediate- and short-wavelength components. Geologically, this probably reflects the fact that the dike is a feature extending from the near surface to a considerable depth. Nevertheless, the data are much more readily interpretable than the raw reduced-to-pole map; at least three distinct types of features can be identified, one of them (the drainage anomalies) not easily discernible in the original data. Under conditions which favor the type of model used, comparable results can be obtained from other data and the technique has proved to be a versatile tool in both qualitative and quantitative analysis.

REFERENCES

Fuller, B.D., 1967, Two-dimensional frequency analysis and design of grid operators, in Hansen, D.A., Heinrichs, W.E., Holmer, R.C., MacDougall, R.E., Rogers, G.R., Sumner, J.S., and Ward, S.H., 1967, Society of Exploration Geophysicists' Mining Geophysics, Volume II, p. 658-708.

Spector, A., and Grant, F.S., 1970, Statistical models for interpreting aeromagnetic data: Geophysics, v. 35, p. 293-302.

Wiener, N., 1949, Time Series: The M.I.T. Press, 163p.