Gravity and magnetic fields at the Earth's surface contain anomalies from sources of various size and depth. To interpret these fields, it is desirable to separate anomalies caused by certain features from anomalies caused by others. How to separate the anomalies depends on what type of features is of interest to the interpretor. In considering the wavelength of gravity anomalies, it is clear that a small body will cause an anomaly of short wavelength while a large body will cause an anomaly of long wavelength. Deep bodies will generate long wavelength anomalies while similar bodies near the surface will result in anomalies of shorter wavelengths. If anomalies could be separated by their wavelengths, certain features may become apparent that would otherwise be hidden. Such a method, the wave number filtering method using the Fast Fourier Transform, is presented here. Although this method can be applied to any type of potential field data, only the gravity field is discussed.
A contour map of gravity data may show many anomalies of various sizes and shapes. Certain trends or features may be difficult to see because of the complexity of the map. If the geologist is interested in analyzing local features in the gravity field, large scale regional features may be distorting the picture. If the geologist is only interested in deep seated features, the near surface anomalies may obscure them. By either isolating or enhancing portions of the gravity field, analysis may be simplified.
Many methods have been used to isolate or enhance various aspects of the gravity field. If it is desirable to separate the long wavelength regional trend from the short wavelength residual field, a numerical method using low order polynomials fitted to the data can be used to approximate the regional trend. The remainder can be assumed to be the residual field. This method usually does not give good results. Another graphical method is to separate the regional from residual field by hand. Large scale trends are simply estimated, and it is assumed that these trends are caused by large, regional, perhaps even continental, structures. Removing the regional trend leaves the residual field. It is assumed that this field is due to small bodies nearer the surface (Griffin, 1949; Nettleton, 1954). Isolating certain aspects of the gravity field by this method can give good results, but is very subjective. Neither of these methods are completely satisfactory. A more objective method of isolating the gravity field is desired.
As a potential field, the gravity field can be represented by an equation that satisfies Laplace's equation. The gravity field can therefore be isolated or enhanced purely theoretically by manipulation of this gravity equation. If the potential field is known everywhere at one elevation, it can be calculated at any other elevation outside the source region. This is called continuation filtering. Derivatives of the gravity equation permit different aspects of the field to be observed. Any order derivative of the gravity field can be taken in any direction, but the first and second derivative in the Z direction have been shown to be the most useful. In some cases, the first and second horizontal derivatives is also of use. These are refered to as Nth order derivative filtering. In addition, the gravity field can be thought of as a Fourier series, an infinite summation of sine and cosine waves of harmonic wavelengths. If these wave-lengths are known, then a specific range of wavelengths may be isolated. This is called band-pass filtering. In a two-dimensional map, if the orientation of these wavelengths are also known, then a specific range of orientations can be isolated as well. This is called strike-pass filtering.
It will be shown that these filtering methods can be performed easily and quickly by a wavenumber filtering method. The gravity field may be isolated or enhanced in any of a number of ways by filtering in the frequency or wavenumber domain rather than in the space domain. The wavenumber filtering method is preferable to other filtering methods because of its speed as well as its accuracy. This is made possible with the ability to transform a data set of anomaly values over a region to a set of amplitudes at specific wavenumbers with the Fast Fourier Transform computer algorithm. While convolving data with a filter in the space domain requires large amounts of computer time, performing the same type of filtering requires only direct multiplication in the wavenumber domain. The construction of the filter depends on which manner the the gravity field is to be isolated or enhanced. The purpose of this thesis is to develop a detailed analysis of the wavenumber filtering method and show its application to a geologic problem. This thesis is designed to provide a tutorial description of the method. The filtering computer programs SETUP and FILTER listed in Appendix A are modified and improved versions of programs from Reed (1980).
Chapter II of this paper describes the derivation of various filters from the gravity equation in the wavenumber domain. These filters are the Upward and Downward Continuation, and the Vertical and Horizontal Derivatives. The shape and characteristics of each type of filter is shown. In addition to filters derived from the gravity equation, working in the wavenumber domain allows two other types of filtering methods: band-pass filtering and strike-pass filtering. These filtering methods allow the isolation of anomalies of specific size and orientation. The application of all wavenumber filtering methods to a computer program is discussed. Chapter II also describes the FFT method and the possible errors that may arise, such as leakage and Gibb's phenomenon. A new methods of limiting these errors is presented.
To show the application of the wavenumber filtering method, a tectonically complex region of southern California has been selected for study. This region centers on the Transverse Ranges and includes portions of the Coast Ranges, Mojave Basin, Peninsular Ranges, and the California Continental Borderland. The region includes the North American-Pacific plate boundary at the San Andreas fault. Chapter III shows the preparation and filtering of the gravity data of this region. The relationship between features in the gravity field and the various provinces are discussed. Analysis of the gravity field allows boundaries of terranes to be proposed.
In chapter IV a specific problem of southern California tectonics is discussed. The formation of the Western Transverse Ranges has yet to be described adequately. Two-dimensional forward gravity modeling is performed along selected profiles across the Transverse Ranges and Borderland. From analyzing the gravity field a new tectonic model is proposed to explain the formation of the Western Transverse Ranges and the California Borderland.