On the least‐squares residual anomaly determination

This paper discusses an approach to determine the least‐squares optimum order of the regional surface which, when subtracted from the Bouguer gravity anomaly data, minimizes distortion of the residual component of the field. The least‐squares method was applied to theoretical composite gravity fields each consisting of a constant residual component (sphere or vertical cylinder) and a regional component of different order using successively increasing orders of polynomial regionals for residual determination. The overall similarity between each two successive residual maps was determined by computing the correlation factor between the mapped variables. Similarity between residual maps of the lowest orders, verified by good correlation, may generally be considered a criterion for determining the optimum order of the regional surface and consequently the least distorted residual component. The residual map of the lower order in this well‐correlated doublet is considered the most plausible one and may be used for gravity interpretation. This approach was successfully applied to the Bouguer gravity of Abu Roash dome, located west of Cairo in the Western Desert of Egypt.

Some aspects of regional‐residual separation of gravity anomalies in a Precambrian terrain

The Bouguer gravity field measured over two Archean greenstone belts of northwestern Ontario is analyzed using three different regional‐residual separation techniques. The purpose of the analysis is to obtain a residual map suitable for gravity modeling studies to help define the subsurface characteristics of the greenstone belts and associated granitic areas. The methods used to derive the regional and residual maps are spectral factorization, upward continuation, and graphical smoothing. The substantial differences in the three sets of maps emphasize the ambiguity and subjectivity of the separation process. Each method may yield nonunique results. For example, in the spectral factorization technique, the filter design is dictated by the clarity of the slope change between the short‐ and long‐wavelength features and, in the case of the upward continuation technique, by the choice of the continuation height. The graphical method is empirical clearly nonunique. The regional map obtained through graphical smoothing is the most satisfactory for the purpose stated since it bas been designed to have minimal contributions from the shallow and broad greenstone masses outcropping at the surface. These features are clearly visible in the spectrum‐based regional map and to a lesser extent in the upward‐continued regional map. All three types of residual maps follow the general outline of the geologic units and thus are probably equally useful for a qualitative study of the anomaly shapes. However, for quantitative modeling purposes, the graphically produced residual is most suitable, since it can be successfully fitted by models that are consistent with the known surface geology and measured density values. The location, spatial extent, and the amplitudes of the analytically produced residual anomalies, in many areas, show poor correlation with the surface measurements, rendering these maps less satisfactory.

GRATICULE SPACING VERSUS DEPTH DISCRIMINATION IN GRAVITY INTERPRETATION

Very serious distortions in both magnitude and extension of local gravity anomalies result from the still widely used 9-point “residual” and 17-point “second derivative” graticules. Although these types of residual maps are very useful for recognizing and pinpointing the existence of interesting local anomalies, the distorted results cannot be used to derive geologic interpretations of significant reliability. A practical procedure based on changes in anomaly magnitudes from two (or more) different grid spacings, effectively overcomes shortcomings of previous interpretation methods.

LEAST SQUARES POLYNOMIAL FITTING TO GRAVITATIONAL DATA AND DENSITY PLOTTING BY DIGITAL COMPUTERS

The fitting of a nth order polynomial in x and y to gravity data by least squares is discussed. A consideration of the normal equations for the general case shows certain simplifications resulting from rectangularity in data distribution. Some sample residual maps are constructed. Density plotting, made possible by the digital computer, is described and illustrated. It is shown that this process can serve as a substitute for contouring when only a qualitative picture is desired.

SOME GEOMETRICAL PROPERTIES OF RESIDUAL MAPS

From well known mathematical theory it can be demonstrated that most contour maps may be considered to be built up by the superposition of a double infinity of elementary undulating surfaces, each of which has the form of a horizontal sinusoidally corrugated sheet, infinite in extent. These elementary surfaces may have all possible wave lengths, orientations, amplitudes, and phases. Several examples are given of simple mosaic‐type composite maps built up by combining only two such elementary surfaces in different ways. These resemble geophysical contour maps in many significant respects. Residual maps are often prepared by using a template procedure for computing the residual value at any point as a linear combination of several neighborhood values interpolated from the original map. An expression is derived for the Fourier transform of any residual map prepared in this way. This transform gives the amplitude spectrum of the residual map in terms of the amplitude spectrum of the original map and the geometry of the template pattern. It is applied to the special case of an original map of the two‐component mosaic variety mentioned above. The results are presented quantitatively in the form of attenuation, or filter, curves which show the amplitudes of the residual anomalies for various sizes and shapes of original anomalies, and for several different residual templates. The geometrical significance of “second derivative” maps is discussed, and it is shown that they may be prepared by a process which is a limiting case of applying a residual template pattern of very simple type. Attenuation curves are presented for several kinds of residual templates when applied to an idealized original contour map consisting of a single anomaly of various shapes. These filter curves are very similar to those for original maps of the simple mosaic type. It is concluded that, since most geophysical maps may be considered to be of a kind intermediate between these two extreme types, the attenuation curves given here may be useful for designing residual templates which will have desired selective characteristics.