The application of Walsh transforms to interpret gravity anomalies due to some simple geometrically shaped causative sources: A feasibility study

Geophysics ◽  
1990 ◽  
Vol 55 (7) ◽  
pp. 843-850 ◽  
Author(s):  
R. K. Shaw ◽  
B. N. P. Agarwal
Keyword(s):  
Fourier Transform ◽  
Linear Time ◽  
Gravity Data ◽  
Gravity Anomalies ◽  
Horizontal Cylinder ◽  
Walsh Transform ◽  
Walsh Functions ◽  
The Fourier Transform ◽  
Depth Extent ◽  
Good Agreement

Walsh functions are a set of complete and orthonormal functions of nonsinusoidal waveform. In contrast to sinusoidal waveforms whose amplitudes may assume any value between −1 to +1, Walsh functions assume only discrete amplitudes of ±1 which form the kernel function of the Walsh transform. Because of this special nature of the kernel, computation of the Walsh transform of a given signal is simpler and faster than that of the Fourier transform. The properties of the Fourier transform in linear time are similar to those of the Walsh transform in dyadic time. The Fourier transform has been widely used in interpretation of geophysical problems. Considering various aspects of the Walsh transform, an attempt has been made to apply it to some gravity data. A procedure has been developed for automated interpretation of gravity anomalies due to simple geometrical causative sources, viz., a sphere, a horizontal cylinder, and a 2-D vertical prism of large depth extent. The technique has been applied to data from the published literature to evaluate its applicability, and the results are in good agreement with the more conventional ones.

Shape and depth solutions from gravity data using correlation factors between successive least‐squares residuals

Geophysics ◽  
1993 ◽  
Vol 58 (12) ◽  
pp. 1785-1791 ◽  
Author(s):  
El‐Sayed M. Abdelrahman ◽  
Hesham M. El‐Araby
Keyword(s):  
Least Squares ◽  
Shape Factor ◽  
Gravity Data ◽  
Bouguer Anomaly ◽  
Vertical Cylinder ◽  
Gravity Anomalies ◽  
Horizontal Cylinder ◽  
Amplitude Coefficient ◽  
Regional Component ◽  
Correlation Factors

The gravity anomaly expression produced by most geologic structures can be represented by a continuous function in both shape (shape factor) and depth variables with an amplitude coefficient related to the mass. Correlation factors between successive least‐squares residual gravity anomalies from a buried vertical cylinder, horizontal cylinder, and sphere are used to determine the shape and depth of the buried geologic structure. For each shape factor value, the depth is determined automatically from the correlation value. The computed depths are plotted against the shape factor representing a continuous correlation curve. The solution for the shape and depth of the buried structure is read at the common intersection of correlation curves. This method can be applied to a Bouguer anomaly profile consisting of a residual component caused by local structure and a regional component. This is a powerful technique for automatically separating the Bouguer data into residual and regional polynomial components. This method is tested on theoretical examples and a field example. In both cases, the results obtained are in good agreement with drilling results.

FOURIER TRANSFORMS OF FINITE LENGTH THEORETICAL GRAVITY ANOMALIES

Geophysics ◽  
1977 ◽  
Vol 42 (7) ◽  
pp. 1450-1457 ◽  
Author(s):  
Robert D. Regan ◽  
William J. Hinze
Keyword(s):  
Fourier Transform ◽  
Analytical Solution ◽  
Finite Length ◽  
Fourier Transforms ◽  
Gravity Anomalies ◽  
Mathematical Structure ◽  
Practical Application ◽  
Convolution Integrals ◽  
The Fourier Transform ◽  
Interpretation Method

The mathematical structure of the Fourier transformations of theoretical gravity anomalies of several geometrically simple bodies appears to have distinct advantages in the interpretation of these anomalies. However, the practical application of this technique is dependent upon the transformation of an observed gravity anomaly of finite length. Ideally, interpretation methods similar to those for the transformations of the theoretical gravity anomalies should be developed for anomalies of a finite length. However, the mathematical complexity of the convolution integrals in the transform calculations of theoretical anomaly segments indicate that no general closed analytical solution useful for interpretation is available. Thus, in order to utilize the Fourier transform interpretation method, the data must be of sufficient length for the finite transform to closely approximate the theoretical transforms.

A GENERAL EXPRESSION FOR THE FOURIER TRANSFORM OF THE GRAVITY ANOMALY DUE TO A FAULT

Geophysics ◽  
1977 ◽  
Vol 42 (7) ◽  
pp. 1458-1461 ◽  
Author(s):  
Bijon Sharma ◽  
T. K. Bose
Keyword(s):  
Fourier Transform ◽  
Gravity Anomaly ◽  
General Expression ◽  
Fault Plane ◽  
Gravity Anomalies ◽  
Arbitrary Angle ◽  
Vertical Fault ◽  
The Fourier Transform

The application of the method of the Fourier transform in interpreting gravity anomalies of faults has so far been based upon the Fourier transform of the gravity anomaly due to a single semi‐infinite block cut by a vertical fault. A general expression for the Fourier transform of the fault anomaly is here derived which is valid for an arbitrary angle of inclination of the fault plane. For deriving the general expression, the gravity anomaly of the fault is first separated into a constant and a variable term. The transforms of the two terms are calculated separately and then added to give the general expression for the Fourier transform of the fault anomaly.

THEORETICAL TRANSFORMS OF THE GRAVITY ANOMALIES OF TWO IDEALIZED BODIES

Geophysics ◽  
1978 ◽  
Vol 43 (3) ◽  
pp. 631-633 ◽  
Author(s):  
Robert D. Regan ◽  
William J. Hinze
Keyword(s):  
Fourier Transform ◽  
Frequency Domain ◽  
Three Dimensional ◽  
Gravity Anomalies ◽  
Vertical Line ◽  
The Fourier Transform ◽  
Line Elements ◽  
Distribution Of Magnetization

Odegard and Berg (1965) have shown that the interpretational process can be simplified for several idealized bodies by utilizing the Fourier transform of the resultant theoretical gravity anomalies. Additional studies relating similar conclusions for other idealized bodies have been reported by Gladkii (1963), Roy (1967), Sharma et al (1970), Davis (1971), Eby (1972), and Saha (1975), and a summary of the spatial and frequency domain equations is given in Regan and Hinze (1976, Table 1); however, the transforms of the three‐dimensional prism and vertical line elements, often utilized in interpretation, have not been previously examined in this manner. Although Bhattacharyya and Chen (1977) have developed and utilized the transform of the 3-D prism in their method for determining the distribution of magnetization in a localized region, it is still of value to present the interpretive advantages of the transform equation itself.

scholarly journals ESTIMASI KETEBALAN SEDIMEN DENGAN ANALISIS POWER SPECTRAL PADA DATA ANOMALI GAYABERAT

GEOMATIKA ◽  
2018 ◽  
Vol 23 (2) ◽  
pp. 65 ◽  
Author(s):  
Mila Apriani ◽  
Admiral Musa Julius ◽  
Mahmud Yusuf ◽  
Damianus Tri Heryanto ◽  
Agus Marsono
Keyword(s):  
Spectral Analysis ◽  
Fourier Transform ◽  
Gravity Anomaly ◽  
Spectral Method ◽  
Gravity Data ◽  
Sediment Thickness ◽  
Transform Method ◽  
Fourier Transform Method ◽  
Power Spectral ◽  
The Fourier Transform

<p align="center"><strong>ABSTRAK</strong></p><p> </p><p>Penelitian dengan analisis <em>power spectral</em> data anomali gayaberat telah banyak dilakukan untuk estimasi ketebalan sedimen. Dalam studi ini penulis melakukan analisis spektral data anomali gayaberat wilayah DKI Jakarta untuk mengetahui kedalaman sumber anomali yang bersesuaian dengan ketebalan sedimen. Data yang digunakan berupa data gayaberat dari BMKG tahun 2014 dengan 197 lokasi titik pengukuran yang tersebar di koordinat 6,08º-6,36º LU dan 106,68º-106,97º BT. Studi ini menggunakan metode <em>power spectral</em>  dengan mentransformasikan data dari domain jarak ke dalam domain bilangan gelombang memanfaatkan transformasi <em>Fourier</em>. Hasil penelitian dengan menggunakan metode transformasi <em>Fourier  </em>menunjukkan bahwa ketebalan sedimen di Jakarta dari arah selatan ke utara semakin besar, di sekitar Babakan ketebalan diperkirakan 92 meter, sekitar Tongkol, Jakarta Utara diperkirakan 331 meter.</p><p><strong> </strong></p><p><strong>Kata kunci</strong>: <em>power spectral</em>, anomali gayaberat, ketebalan sedimen</p><p align="center"><strong><em> </em></strong></p><p align="center"><strong><em>ABSTRACT</em></strong></p><p><em> </em></p><p><em>Studies of spectral analysis of gravity anomaly data have been carried out to estimate the thickness of sediment. In this study the author did spectral analysis of gravity anomaly data of DKI Jakarta area to know the depth of anomaly source which corresponds to the thickness of sediment. The data used in the form of gravity data from BMKG 2014 with 197 locations of measurement points spread in coordinates 6.08º - 6.36º N and 106.68º - 106.97º E. This study used the power spectral method by transforming the data from the distance domain into the wavenumber domain utilizing the Fourier transform. The result of the research using Fourier transform method shows that the thickness of sediment in Jakarta from south to north is getting bigger, in Babakan the thickness of the sediment is around 92 meter, in Tongkol, North Jakarta is around 331 meter.</em></p><p><strong><em> </em></strong></p><p><strong><em>Keywords</em></strong><em>: </em><em>power spectral, gravity anomaly, sediment thickness</em><em></em></p>

A least‐squares minimization approach to invert gravity data

Geophysics ◽  
1991 ◽  
Vol 56 (1) ◽  
pp. 115-118 ◽  
Author(s):  
E. M. Abdelrahman ◽  
A. I. Bayoumi ◽  
H. M. El‐Araby
Keyword(s):  
Fourier Transform ◽  
Fast Fourier Transform ◽  
Gravity Data ◽  
Spline Approximation ◽  
Gravity Anomalies ◽  
Approximation Techniques ◽  
Local Gravity ◽  
Fourier Transform Methods ◽  
Minimization Approach ◽  
Least Squares Minimization

The interpretation of gravity data often involves initial steps to eliminate or attenuate unwanted field components in order to isolate the desired anomaly (e.g., residual‐regional separations). These initial filtering operations include, for example, the radial weights methods (Griffin, 1949; Elkins, 1951; Abdelrahman et al., 1990), the fast Fourier transform methods (Bhattacharyya, 1965; Clarke, 1969; Meskó, 1969, 1984, Botezatu, 1970), the rational approximation techniques (Agarwal and Lal, 1971) and recursion filters (Bhattacharyya, 1976), and the bicubic spline approximation techniques (Bhattacharyya, 1969; Inoue, 1986). The derived local gravity anomalies are then geologically interpreted to derive depth estimates, often without properly accounting for the uncertainties introduced by the filtering process. When filters are applied to observed data, the filters often cause serious distortions in the shape of the gravity anomalies (Hammer, 1977). Thus the filtered gravity anomalies generally yield unreliable geologic interpretations (Rao and Radhakrishnamurthy, 1965; Hammer, 1977; Abdelrahman et al., 1985, 1989.

GRAVITY PROFILE INTERPRETATION USING THE FOURIER TRANSFORM

Geophysics ◽  
1974 ◽  
Vol 39 (6) ◽  
pp. 862-866 ◽  
Author(s):  
S. J. Collins ◽  
A. R. Dodds ◽  
B. D. Johnson
Keyword(s):  
Fourier Transform ◽  
Horizontal Cylinder ◽  
Direct Interpretation ◽  
Data Points ◽  
Lateral Extent ◽  
The Fourier Transform ◽  
Interpretation Method ◽  
Data Length ◽  
Gravity Profile

A number of attempts have been made to perform direct interpretation of gravity profiles using the Fourier transform of the profile. Of these, the methods of Odegard and Berg (1965) and Sharma et al. (1970) appear to be most applicable. The purpose of this study was to take one of the proposed models (Odegard and Berg’s horizontal cylinder) and determine the applicability of the interpretation method in terms of the number and lateral extent of the data points. The relative accuracies of the estimates of the depth and mass of a cylinder were determined as criteria for estimating the effects of data length and number of data points. In addition, the interpretation was extended to include the separation of two cylinders.

Interpretation of gravity anomalies of dike-like bodies by Fourier transformation

1970 ◽  
Vol 7 (2) ◽  
pp. 512-516 ◽  
Author(s):  
Bijon Sharma ◽  
L. P. Geldart ◽  
D. E. Gill
Keyword(s):  
Fourier Transform ◽  
Detailed Analysis ◽  
Gravity Anomaly ◽  
Fourier Transformation ◽  
Amplitude Spectrum ◽  
Gravity Anomalies ◽  
Objective Method ◽  
The Fourier Transform

An objective method is presented for interpreting the gravity anomalies of a dike using the Fourier transform of the observed gravity anomaly function. The amplitude spectrum of the transformed function contains information about the depth, inclination, and the thickness of the dike. The usefulness of the Fourier transform technique is illustrated by a detailed analysis of the gravity anomaly of a dike.

scholarly journals Depth and shape solutions from residual gravity anomalies due to simple geometric structures using a statistical approach

2017 ◽  
Vol 47 (2) ◽  
pp. 113-132 ◽  
Author(s):  
El-Sayed Abdelrahman ◽  
Mohamed Gobashy
Keyword(s):  
Gravity Data ◽  
Random Noise ◽  
Synthetic Data ◽  
Vertical Cylinder ◽  
Gravity Anomalies ◽  
Horizontal Cylinder ◽  
Random Errors ◽  
Residual Gravity ◽  
Characteristic Points ◽  
The Usa

AbstractWe have developed a simple and fast quantitative method for depth and shape determination from residual gravity anomalies due to simple geometrical bodies (semi-infinite vertical cylinder, horizontal cylinder, and sphere). The method is based on defining the anomaly value at two characteristic points and their corresponding distances on the anomaly profile. Using all possible combinations of the two characteristic points and their corresponding distances, a statistical procedure is developed for automated determination of the best shape and depth parameters of the buried structure from gravity data. A least-squares procedure is also formulated to estimate the amplitude coefficient which is related to the radius and density contrast of the buried structure. The method is applied to synthetic data with and without random errors and tested on two field examples from the USA and Germany. In all cases examined, the estimated depths and shapes are found to be in good agreement with actual values. The present method has the capability of minimizing the effect of random noise in data points to enhance the interpretation of results.

scholarly journals Spin-density correlator and its Fourier transform in the dynamic spin-fluctuation theory

2016 ◽  
pp. 474-486
Author(s):  
Г.В. Парадеженко ◽  
Н.Б. Мельников ◽  
Б.И. Резер
Keyword(s):  
Fourier Transform ◽  
Spin Density ◽  
Spin Fluctuation ◽  
Fortran Program ◽  
Fluctuation Theory ◽  
Magnetic Contribution ◽  
Dynamic Spin ◽  
The Fourier Transform ◽  
Program Modules ◽  
Good Agreement

Выведена формула, связывающая магнитный вклад в сечение рассеяния поляризованных нейтронов с фурье-образом коррелятора спиновой плотности. Описан метод расчета коррелятора спиновой плотности и его фурье-образа в динамической теории спиновых флуктуаций. Приведены расчетные формулы и программные модули на языке Фортран. На примере железа показано, что численные результаты для фурье-образа коррелятора хорошо согласуются с экспериментом. A formula relating the magnetic contribution to a polarized neutron scattering cross-section with the Fourier transform of the spin-density correlator is derived. A method of calculating the spin-density correlator and its Fourier transform in the dynamic spin-fluctuation theory is described. Calculation formulas and Fortran program modules are presented. By the example of iron, it is shown that the numerical results obtained for the Fourier transform of the spin-density correlator are in good agreement with experimental data.

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