DESIGN OF SPATIAL FILTERS AND THEIR APPLICATION TO HIGH‐RESOLUTION AEROMAGNETIC DATA

Geophysics ◽  
1972 ◽  
Vol 37 (1) ◽  
pp. 68-91 ◽  
Author(s):  
B. K. Bhattacharyya
Keyword(s):  
High Resolution ◽  
Field Data ◽  
Vertical Gradient ◽  
Double Fourier Series ◽  
Two Dimensional ◽  
Total Field ◽  
Spatial Filters ◽  
Aeromagnetic Survey ◽  
Potential Field Data ◽  
High Frequency Response

Methods for the design of spatial filters are discussed in this paper. For a given response of a one‐dimensional filter, the weighting coefficients are calculated by solving a set of simultaneous equations with a simple matrix inversion procedure. In the case of a two‐dimensional filter, the method for obtaining the coefficients of a double Fourier series representing a set of given values is used to design the spatial operator. The problems connected with the length of the operator and the choice of a suitable decay in the high‐frequency response are discussed in detail. In order to show the usefulness of these methods, the paper presents several examples of operators designed for computing the vertical gradient, the second vertical derivative, and downward continuation of potential field data. A two‐dimensional vertical gradient filter is applied to the total field data obtained during a high‐resolution aeromagnetic survey over an area in the Precambrian Shield of Northeastern Ontario. The calculated gradient maps are compared with maps showing measured gradient values. The quality of the calculated maps in defining trends, patterns, and detailed features of anomalies shows the feasibility of obtaining very accurate vertical gradient maps from observed total field data.

RECURSION FILTERS FOR DIGITAL PROCESSING OF POTENTIAL FIELD DATA

Geophysics ◽  
1976 ◽  
Vol 41 (4) ◽  
pp. 712-726 ◽  
Author(s):  
B. K. Bhattacharyya
Keyword(s):  
Frequency Domain ◽  
Potential Field ◽  
Field Data ◽  
Least Squares Method ◽  
Vertical Gradient ◽  
Two Dimensional ◽  
Single Variable ◽  
Potential Field Data ◽  
Recursive Filter ◽  
Rational Expression

Zero‐phase two‐dimensional recursive filters, with a specified frequency domain response, have been designed for processing potential field data. In the case of second vertical derivative filters, it is possible to use the rational approximation of symmetrical functions of a single variable for the design of a short recursive filter. The filter so designed has an excellent response in the frequency domain. For vertical gradient and continuation filters, a method is developed for obtaining, by the least‐squares method, a rational expression for a two‐dimensional symmetrical function. In order to ensure the stability of the recursive filter, the denominator of the rational expression is approximated by a product of two factors, each being a function of a single variable. Finally, to keep the error of the filter response as small as possible, an iterative procedure is used for adjusting the zeros of the denominator and then determining the coefficients of the numerator of the rational expression.

ADDITIONAL COMMENTS ON THE ANALYTIC SIGNAL OF TWO‐DIMENSIONAL MAGNETIC BODIES WITH POLYGONAL CROSS‐SECTION

Geophysics ◽  
1974 ◽  
Vol 39 (1) ◽  
pp. 85-92 ◽  
Author(s):  
Misac N. Nabighian
Keyword(s):  
Cross Section ◽  
Potential Field ◽  
Input Data ◽  
Field Data ◽  
Field Component ◽  
Analytic Signal ◽  
Two Dimensional ◽  
Field Profile ◽  
Total Field ◽  
Potential Field Data

In a previous paper (Nabighian, 1972), the concept of analytic signal of bodies of polygonal cross‐section was introduced and its applications to the interpretation of potential field data were discussed. The input data for the proposed treatment are the horizontal derivative T(x) of the field profile, whether horizontal, vertical, or total field component. As it is known, this derivative curve can be thought of as being due to thin magnetized sheets surrounding the causative bodies.

A comparison of methods for approximating the vertical gradient of one‐dimensional magnetic field data

Geophysics ◽  
1986 ◽  
Vol 51 (9) ◽  
pp. 1725-1735 ◽  
Author(s):  
J. W. Paine
Keyword(s):  
Magnetic Field ◽  
Fourier Transform ◽  
Field Data ◽  
Total Error ◽  
Vertical Gradient ◽  
Magnetic Data ◽  
Total Field ◽  
One Dimensional ◽  
Convolution Filter ◽  
Principal Factors

The vertical gradient of a one‐dimensional magnetic field is known to be a useful aid in interpretation of magnetic data. When the vertical gradient is required but has not been measured, it is necessary to approximate the gradient using the available total‐field data. An approximation is possible because a relationship between the total field and the vertical gradient can be established using Fourier analysis. After reviewing the theoretical basis of this relationship, a number of methods for approximating the vertical gradient are derived. These methods fall into two broad categories: methods based on the discrete Fourier transform, and methods based on discrete convolution filters. There are a number of choices necessary in designing such methods, each of which will affect the accuracy of the computed values in differing, and sometimes conflicting, ways. A comparison of the spatial and spectral accuracy of the methods derived here shows that it is possible to construct a filter which maintains a reasonable balance between the various components of the total error. Further, the structure of this filter is such that it is also computationally more efficient than methods based on fast Fourier transform techniques. The spacing and width of the convolution filter are identified as the principal factors which influence the accuracy and efficiency of the method presented here, and recommendations are made on suitable choices for these parameters.

DESIGN OF SMALL OPERATORS FOR THE CONTINUATION OF POTENTIAL FIELD DATA

Geophysics ◽  
1972 ◽  
Vol 37 (3) ◽  
pp. 488-506 ◽  
Author(s):  
Irshad R. Mufti
Keyword(s):  
Potential Field ◽  
Field Data ◽  
Grid Point ◽  
Frequency Characteristics ◽  
Potential Fields ◽  
Two Dimensional ◽  
Potential Field Data ◽  
Computational Work ◽  
Considerable Loss ◽  
Equivalent Operators

Two‐dimensional continuation of potential fields is commonly achieved by employing a continuation operator which consists of a number of coefficients operating upon uniformly gridded field data. To obtain accurate results, the size of the operator has to be quite large. This not only requires a lot of computational work, but also causes a considerable loss of information due to the reduced size of the field obtained after continuation. Small‐size “equivalent” operators were designed which are free from these drawbacks but yield accurate results. In order to demonstrate the efficiency of these operators, a potential field was continued upward by using 31×31 Tsuboi coefficients. This required 961 multiplications for computing the continued field at each grid point. When procedure was repeated using the equivalent operator, the number of multiplications required for each grid point was reduced to 15, the size of the resulting map was much larger, but the results in both cases were practically identical in accuracy. Frequency characteristics of the equivalent operators and the continuation of data very close to the boundary of the field map are discussed.

BICUBIC SPLINE INTERPOLATION AS A METHOD FOR TREATMENT OF POTENTIAL FIELD DATA

Geophysics ◽  
1969 ◽  
Vol 34 (3) ◽  
pp. 402-423 ◽  
Author(s):  
B. K. Bhattacharyya
Keyword(s):  
Field Data ◽  
Spline Interpolation ◽  
Amplitude Spectrum ◽  
Accurate Method ◽  
Fourier Series Expansion ◽  
Small Scale ◽  
Spline Functions ◽  
Two Dimensional ◽  
Potential Field Data ◽  
Bicubic Spline

A method for the generation of bicubic spline functions is presented in this paper. From this method it becomes apparent that these functions derive their potential strength in accurate and reliable representation of two‐dimensional data by maintaining continuity of the variable and its slope and curvature throughout the area of observation. The results obtained by computing horizontal and vertical derivatives with model and field data illustrate the exceptional accuracy achieved with spline functions. The piecewise cubic polynomial functions expressing observed data analytically in space are used to estimate amplitude and phase spectra of magnetic anomalies. At relatively long wavelengths the amplitude spectrum thus calculated displays remarkable similarity with the true spectrum and is found to be superior to that obtained with two‐dimensional Fourier series expansion. A cubic spline method is also presented for computing values of an observed variable at equispaced points along two orthogonal directions with the help of irregularly distributed data. The interpolation technique applied to field data shows high resolution by maintaining the separation of neighboring anomalies and the small‐scale features. The shapes, peaks, and troughs of both large and small amplitude anomalies are faithfully reproduced. The gradients of the magnetic field do not undergo any appreciable distortion. It can thus be concluded that cubic splines are a reliable and accurate method of interpolation.

Reduction of potential field data to a horizontal plane

Geophysics ◽  
1990 ◽  
Vol 55 (5) ◽  
pp. 549-555 ◽  
Author(s):  
Mark Pilkington ◽  
W. E. S. Urquhart
Keyword(s):  
Potential Field ◽  
Horizontal Plane ◽  
Field Data ◽  
Synthetic Data ◽  
Mirror Image ◽  
Source Distribution ◽  
Aeromagnetic Survey ◽  
Potential Field Data ◽  
Equivalent Sources ◽  
Topographic Gradients

Most existing techniques for potential field data enhancement and interpretation require data on a horizontal plane. Hence, when observations are made on an irregular surface, reduction to a horizontal plane is necessary. To effect this reduction, an equivalent source distribution that models the observed field is computed on a mirror image of the observation surface. This irregular mirror image surface is then replaced by a horizontal plane and the effect of the equivalent sources is computed on the required horizontal level. This calculated field approximates the field reduced to a horizontal plane. The good quality of this approximation is demonstrated by two‐dimensional synthetic data examples in which the maximum errors occur in areas of steep topographic gradients and increased magnetic field intensity. The approach is also applied to a portion of a helicopter‐borne aeromagnetic survey from the Gaspé region in Quebec, Canada, where the results are a horizontal shifting of anomaly maxima of up to 150 m and changes in anomaly amplitudes of up to 100 nT.

Automated interpretation of horizontal magnetic gradient profile data

Geophysics ◽  
1992 ◽  
Vol 57 (2) ◽  
pp. 288-295 ◽  
Author(s):  
D. L. Marcotte ◽  
C. D. Hardwick ◽  
J. B. Nelson
Keyword(s):  
Vertical Gradient ◽  
Horizontal Gradient ◽  
Two Dimensional ◽  
Total Field ◽  
Profile Data ◽  
Magnetic Gradient ◽  
Inversion Methods ◽  
Automated Interpretation ◽  
A New Technique ◽  
Magnetic Gradients

A new technique has been developed to estimate the vertical magnetic gradient [Formula: see text] along a profile from measurements of the two orthogonal horizontal magnetic gradients [Formula: see text]. In addition, a means of identifying two‐dimensional anomaly sources and the angle of the structure relative to the profile direction is shown to be a function of [Formula: see text]. This angle can be used to correct interpretations from vertical gradient or total field inversion methods which assume source structures oriented perpendicular to the profile direction. A modified Werner deconvolution algorithm for vertical gradient data incorporating these features has been applied to both real and simulated horizontal gradient data.

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