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

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.

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On: “Gravity Interpretation Using the Fourier Integral”, by M. E. Odegard and J. W. Berg, Jr. (GEOPHYSICS, June 1965, p. 127–138)

Geophysics ◽

10.1190/1.1440532 ◽

1975 ◽

Vol 40 (2) ◽

pp. 356-357

Author(s):

Jay Gopal Saha

Keyword(s):

Fourier Transform ◽

Gravity Anomaly ◽

Fourier Integral ◽

Two Dimensional ◽

Transform Method ◽

Reverse Problem ◽

Vertical Fault ◽

Vertical Step ◽

The Fourier Transform ◽

Gravity Interpretation

In their paper, Odegard and Berg claim that from the gravity anomaly due to a two‐dimensional vertical fault the density, the throw, and the depth can be determined uniquely by a Fourier transform method. It is true that the solution of the reverse problem for a two‐dimensional vertical step is theoretically unique. The derivation of the Fourier transform by the authors, however, is erroneous.

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

Canadian Journal of Earth Sciences ◽

10.1139/e70-042 ◽

1970 ◽

Vol 7 (2) ◽

pp. 512-516 ◽

Cited By ~ 7

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.

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An analysis of the spectrum of a gravity anomaly over an inclined dike

Geophysics ◽

10.1190/1.1442017 ◽

1985 ◽

Vol 50 (9) ◽

pp. 1500-1501

Author(s):

B. N. P. Agarwal ◽

D. Sita Ramaiah

Keyword(s):

Fourier Transform ◽

Gravity Anomaly ◽

Fourier Spectrum ◽

Fourier Transforms ◽

Weighted Average ◽

Window Function ◽

Average Spectrum ◽

Amplitude Spectra ◽

Distance Data ◽

The Fourier Transform

Bhimasankaram et al. (1977) used Fourier spectrum analysis for a direct approach to the interpretation of gravity anomaly over a finite inclined dike. They derived several equations from the real and imaginary components and from the amplitude and phase spectra to relate various parameters of the dike. Because the width 2b of the dike (Figure 1) appears only in sin (ωb) term—ω being the angular frequency—they determined its value from the minima/zeroes of the amplitude spectra. The theoretical Fourier spectrum uses gravity field data over an infinite distance (length), whereas field observations are available only for a limited distance. Thus, a set of observational data is viewed as a product of infinite‐distance data with an appropriate window function. Usually, a rectangular window of appropriate distance (width) and of unit magnitude is chosen for this purpose. The Fourier transform of the finite‐distance and discrete data is thus represented by convolution operations between Fourier transforms of the infinite‐distance data, the window function, and the comb function. The combined effect gives a smooth, weighted average spectrum. Thus, the Fourier transform of actual observed data may differ substantially from theoretic data. The differences are apparent for low‐ and high‐frequency ranges. As a result, the minima of the amplitude spectra may change considerably, thereby rendering the estimate of the width of the dike unreliable from the roots of the equation sin (ωb) = 0.

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FOURIER TRANSFORMS OF FINITE LENGTH THEORETICAL GRAVITY ANOMALIES

Geophysics ◽

10.1190/1.1440804 ◽

1977 ◽

Vol 42 (7) ◽

pp. 1450-1457 ◽

Cited By ~ 5

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.

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THEORETICAL TRANSFORMS OF THE GRAVITY ANOMALIES OF TWO IDEALIZED BODIES

Geophysics ◽

10.1190/1.1440842 ◽

1978 ◽

Vol 43 (3) ◽

pp. 631-633 ◽

Cited By ~ 4

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.

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ESTIMASI KETEBALAN SEDIMEN DENGAN ANALISIS POWER SPECTRAL PADA DATA ANOMALI GAYABERAT

GEOMATIKA ◽

10.24895/jig.2017.23-2.649 ◽

2018 ◽

Vol 23 (2) ◽

pp. 65 ◽

Cited By ~ 1

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

ABSTRAK Penelitian dengan analisis power spectral 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 power spectral dengan mentransformasikan data dari domain jarak ke dalam domain bilangan gelombang memanfaatkan transformasi Fourier. Hasil penelitian dengan menggunakan metode transformasi Fourier 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. Kata kunci: power spectral, anomali gayaberat, ketebalan sedimen ABSTRACT 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. Keywords: power spectral, gravity anomaly, sediment thickness

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Reply by authors to discussion by J. G. Saha

Geophysics ◽

10.1190/1.1486599 ◽

1975 ◽

Vol 40 (2) ◽

pp. 357-357

Author(s):

M. E. Odegard ◽

J. W. Berg

Keyword(s):

Fourier Transform ◽

Transformation Formula ◽

Vertical Fault ◽

The Fourier Transform

As Saha has pointed out, the Fourier transform for the vertical fault given in our paper (Odegard and Berg, 1965) is incorrect and the correct transform is given by Davis (1971). In our derivation of the Fourier transform the transformation [Formula: see text] was used. This transformation is correct only for [Formula: see text] with the transformation for [Formula: see text] given by [Formula: see text].

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

Geophysics ◽

10.1190/1.1442898 ◽

1990 ◽

Vol 55 (7) ◽

pp. 843-850 ◽

Cited By ~ 38

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.

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GRAVITY ANOMALIES OF A FAULT CUTTING A SERIES OF BEDS

Geophysics ◽

10.1190/1.1440124 ◽

1970 ◽

Vol 35 (4) ◽

pp. 708-712 ◽

Cited By ~ 3

Author(s):

Bijon Sharma ◽

Mahesh P. Vyas

Keyword(s):

Gravity Anomaly ◽

Gravity Anomalies ◽

Simple Expression ◽

Uniform Density ◽

Arbitrary Angle ◽

Single Block ◽

Inclined Fault ◽

Unusual Situation

The gravity anomaly due to a single horizontal semi‐infinite block terminated by a vertical or dipping fault has been discussed by several authors previously. Geldart, Gill, and Sharma (1966) gave a new and simple expression for calculating the gravity anomaly due to a block cut by an inclined fault at an arbitrary angle and used this expression to obtain the gravity anomaly due to a fault cutting a single bed. In their derivation the effects of both the upthrown and the downthrown blocks were taken into consideration. It is, however, only in the unusual situation that faults cut a single bed of uniform density. More often, faults cut a series of beds of different densities and thicknesses. If the densities and the thicknesses of the various beds differ greatly, an interpretation based upon replacement of the series of beds by a single bed of uniform density may be highly erroneous. Starting from the single block expression given by Geldart et al, an expression can be derived giving the gravity anomaly due to a fault cutting a series of beds having different densities and thicknesses.

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INTERPRETATION OF GRAVITY ANOMALIES DUE TO FINITE INCLINED DIKES USING FOURIER TRANSFORMATION

Geophysics ◽

10.1190/1.1440713 ◽

1977 ◽

Vol 42 (1) ◽

pp. 51-59 ◽

Cited By ~ 18

Author(s):

V. L. S. Bhimasankaram ◽

R. Nagendra ◽

S. V. Seshagiri Rao

Keyword(s):

Fourier Transform ◽

High Frequency ◽

Fourier Transformation ◽

Synthetic Data ◽

Gravity Anomalies ◽

Frequency Range ◽

Unknown Parameters ◽

Estimated Parameters ◽

The Fourier Transform ◽

Theoretical Considerations

The Fourier transform of the gravity field due to a finite dipping dike is derived and its real and imaginary components are separated. Analysis of these two functions in a certain high‐frequency range yields simple relations that can be used to estimate the unknown parameters of the dike. The theoretical considerations are tested on synthetic data after performing the discrete Fourier transform (DFT), and the validity of the method of interpretation is established from a comparison of the actual and estimated parameters.

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