Three least‐squares minimization approaches to depth, shape, and amplitude coefficient determination from gravity data

Geophysics ◽  
2001 ◽  
Vol 66 (4) ◽  
pp. 1105-1109 ◽  
Author(s):  
E. M. Abdelrahman ◽  
H. M. El‐Araby ◽  
T. M. El‐Araby ◽  
E. R. Abo‐Ezz
Keyword(s):  
Least Squares ◽  
Shape Factor ◽  
Least Squares Method ◽  
Gravity Data ◽  
Synthetic Data ◽  
Gravity Anomalies ◽  
Maximum Error ◽  
The United States ◽  
Random Errors ◽  
Amplitude Coefficient

Three different least‐squares approaches are developed to determine, successively, the depth, shape (shape factor), and amplitude coefficient related to the radius and density contrast of a buried structure from the residual gravity anomaly. By defining the anomaly value g(max) at the origin on the profile, the problem of depth determination is transformed into the problem of solving a nonlinear equation, [Formula: see text]. Formulas are derived for spheres and cylinders. Knowing the depth and applying the least‐squares method, the shape factor and the amplitude coefficient are determined using two simple linear equations. In this way, the depth, shape, and amplitude coefficient are determined individually from all observed gravity data. A procedure is developed for automated interpretation of gravity anomalies attributable to simple geometrical causative sources. The method is applied to synthetic data with and without random errors. In all the cases examined, the maximum error in depth, shape, and amplitude coefficient is 3%, 1.5%, and 7%, respectively. Finally, the method is tested on a field example from the United States, and the depth and shape obtained by the present method are compared with those obtained from drilling and seismic information and with those published in the literature.

A least‐squares derivatives analysis of gravity anomalies due to faulted thin slabs

Geophysics ◽  
2003 ◽  
Vol 68 (2) ◽  
pp. 535-543 ◽  
Author(s):  
El‐Sayed M. Abdelrahman ◽  
Hesham M. El‐Araby ◽  
Tarek M. El‐Araby ◽  
Eid Ragab Abo‐Ezz
Keyword(s):  
Least Squares ◽  
Least Squares Method ◽  
Gravity Data ◽  
Bouguer Anomaly ◽  
Theoretical Data ◽  
Gravity Anomalies ◽  
Random Errors ◽  
Fault Structure ◽  
Amplitude Coefficient ◽  
Buried Fault

This paper presents two different least‐squares approaches for determining the depth and amplitude coefficient (related to the density contrast and the thickness of a buried faulted thin slab from numerical first‐, second‐, third‐, and fourth‐horizontal derivative anomalies obtained from 2D gravity data using filters of successive graticule spacings. The problem of depth determination has been transformed into the problem of finding a solution to a nonlinear equation of the form f(z) = 0. Knowing the depth and applying the least‐squares method, the amplitude coefficient is determined using a simple linear equation. In this way, the depth and amplitude coefficient are determined individually from all observed gravity data. The depths and the amplitude coefficients obtained from the first‐, second‐, third‐, and fourth‐ derivative anomaly values can be used to determine simultaneously the actual depth and amplitude coefficient of the buried fault structure and the optimum order of the regional gravity field along the profile. The method can be applied not only to residuals but also to the Bouguer anomaly profile consisting of the combined effect of a residual component due to a purely local fault structure (shallow or deep) and a regional component represented by a polynomial of any order. The method is applied to theoretical data with and without random errors and is tested on a field example from Egypt.

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.

A least‐squares minimization approach to depth determination from moving average residual gravity anomalies

Geophysics ◽  
1993 ◽  
Vol 58 (12) ◽  
pp. 1779-1784 ◽  
Author(s):  
El‐Sayed M. Abdelrahman ◽  
Tarek M. El‐Araby
Keyword(s):  
Least Squares ◽  
Gravity Data ◽  
Moving Average ◽  
Gravity Anomalies ◽  
The United States ◽  
Random Errors ◽  
Minimization Method ◽  
Residual Gravity ◽  
Average Residual ◽  
Least Squares Minimization

We have developed a least‐squares minimization method to estimate the depth of a buried structure from moving average residual gravity anomalies. The method involves fitting simple models convolved with the same moving average filter as applied to the observed gravity data. As a result, our method can be applied not only to residuals but also to the Bouguer gravity data of a short profile length. The method is applied to synthetic data with and without random errors. The validity of the method is tested in detail on two field examples from the United States and Senegal.

scholarly journals A Fast Interpretation Method for Inverse Modeling of Residual Gravity Anomalies Caused by Simple Geometry

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Khalid S. Essa
Keyword(s):  
Gravity Anomaly ◽  
Shape Factor ◽  
Vertical Cylinder ◽  
Gravity Anomalies ◽  
Depth Estimation ◽  
Model Parameters ◽  
Fast Method ◽  
Random Errors ◽  
Residual Gravity ◽  
Amplitude Coefficient

An inversion technique using a fast method is developed to estimate, successively, the depth, the shape factor, and the amplitude coefficient of a buried structure using residual gravity anomalies. By defining the anomaly value at the origin and the anomaly value at different points on the profile, the problem of depth estimation is transformed into a problem of solving a nonlinear equation of the form . Knowing the depth, the shape factor can be estimated and finally the amplitude coefficient can be estimated. This technique is applicable for a class of geometrically simple anomalous bodies, including the semiinfinite vertical cylinder, the infinitely long horizontal cylinder, and the sphere. The efficiency of this technique is demonstrated with gravity anomaly due to a theoretical model, in each case with and without random errors. Finally, the applicability is illustrated using the residual gravity anomaly of Mobrun ore body, situated near Noranda, QC, Canada. The interpreted depth and the other model parameters are in good agreement with the known actual values.

A least‐squares approach to depth determination from gravity data

Geophysics ◽  
1983 ◽  
Vol 48 (3) ◽  
pp. 357-360 ◽  
Author(s):  
O. P. Gupta
Keyword(s):  
Nonlinear Equation ◽  
Least Squares ◽  
Gravity Data ◽  
Synthetic Data ◽  
Numerical Approach ◽  
Random Errors ◽  
Depth Determination ◽  
Vertical Fault ◽  
Horizontal Cylinders ◽  
Least Squares Approach

The present paper deals with a numerical approach to determine the depth of a buried structure from the residual anomaly. The problem of depth determination has been transformed into the problem of finding a solution of a nonlinear equation of the form [Formula: see text]. Formulas have been derived for a sphere, vertical and horizontal cylinders, and for a vertical fault (thin plate approximation). The procedure is applied to synthetic data with and without random errors. Finally, a field example is presented in which the depth to a fault is estimated at 3.8 km and verified from drilling results.

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.

A least‐squares minimization approach to depth determination from numerical horizontal gravity gradients

Geophysics ◽  
1995 ◽  
Vol 60 (4) ◽  
pp. 1259-1260 ◽  
Author(s):  
El‐Sayed Mohamed Abdelrahman ◽  
Sharafeldin Mahmoud Sharafeldin
Keyword(s):  
Least Squares ◽  
Least Squares Method ◽  
Gravity Data ◽  
Gravity Anomalies ◽  
Depth Estimation ◽  
Horizontal Cylinder ◽  
Theory And Practice ◽  
Gravity Gradient ◽  
Quantitative Interpretation ◽  
Gravity Gradients

The sphere and the horizontal cylinder models can be very useful in quantitative interpretation of gravity data measured in a small area over buried structures. Several graphical and numerical methods have been developed by many workers for interpreting the residual gravity anomalies caused by these models to find the depth of most geologic structures. Excellent reviews are given in Saxov and Nygaard (1953) and Bowin et al. (1986). The numerical approaches (Odegard and Berg, 1965; Gupta, 1983; Sharma and Geldart, 1968; Lines and Treitel, 1984; and Shaw and Agarwal, 1990) may have advantages in theory and practice over graphical depth estimation techniques (Pick et al., 1973: Nettleton, 1976; Telford et al., 1976). However, effective quantitative interpretation procedures using the least‐squares method based on the analytical expression of simple numerical horizontal gravity gradient anomalies are yet to be developed.

scholarly journals A new method for complete quantitative interpretation of gravity data due to dipping faults

2019 ◽  
Vol 49 (2) ◽  
pp. 133-151
Author(s):  
El-Sayed Abdelrahman ◽  
Mohamed Gobashy ◽  
Eid Abo-Ezz ◽  
Tarek El-Araby
Keyword(s):  
Least Squares ◽  
Gravity Data ◽  
Theoretical Data ◽  
Lower Portion ◽  
Model Parameters ◽  
Random Errors ◽  
Thin Slab ◽  
Simple Method ◽  
Amplitude Coefficient ◽  
Dip Angle

Abstract We have developed a simple method to determine completely the model parameters of a buried dipping fault from gravity data (depths to the centers of the upper and lower portions of the faulted thin slab, dip angle, and amplitude coefficient). The method is based on defining the anomaly values at the origin and at four symmetrical points around the origin on the gravity anomaly profile. By defining these five pieces of information, the dip angle is determined for each value of the depth of the lower portion of the faulted thin slab by solving iteratively one nonlinear equation of the form f(α)= 0. The computed dip angles are plotted against the values of the depth representing a continuous depth-dip curve. The solution for the depth to the lower portion of the faulted thin slab (down-thrown block) and the dip angle of the buried fault is read at the common intersection of the depth-dip curves. Knowing the depth to the center of the lower portion of the faulted layer and the dip angle, the problem of determining the depth to the center of the upper portion of the faulted slab (up-thrown block) is transformed into the problem of solving iteratively a nonlinear least-squares equation, f(z) = 0. Because the depths and the dip angle are known, the amplitude coefficient, which depends on the thickness and density contrast of the thin slab, is determined using a linear least-squares equation. The method is applied to theoretical data with and without random errors. The validity of the method is tested on real gravity data from Egypt. In all cases examined, the model parameters obtained are in good agreement with the actual ones and with those given in the published literature.

A new method for shape and depth determinations from gravity data

Geophysics ◽  
2001 ◽  
Vol 66 (6) ◽  
pp. 1774-1780 ◽  
Author(s):  
El‐Sayed M. Abdelrahman ◽  
Tarek M. El‐Araby ◽  
Hesham M. El‐Araby ◽  
Eid R. Abo‐Ezz
Keyword(s):  
Shape Factor ◽  
Gravity Data ◽  
Theoretical Data ◽  
The United States ◽  
Random Errors ◽  
Simple Method ◽  
Shape Factors ◽  
Family Of Curves ◽  
The Relationship ◽  
Good Agreement

We have developed a simple method to determine simultaneously the shape and depth of a buried structure from residualized gravity data using filters of successive window lengths. The method is similar to Euler deconvolution, but it solves for shape and depth independently. The method involves using a relationship between the shape factor and the depth to the source and a combination of windowed observations. The relationship represents a parametric family of curves (window curves). For a fixed window length, the depth is determined for each shape factor. The computed depths are plotted against the shape factors, representing a continuous, monotonically increasing curve. The solution for the shape and depth of the buried structure is read at the common intersection of the window curves. This method can be applied to residuals as well as to the Bouguer gravity data of a short or long profile length. The method is applied to theoretical data with and without random errors and is tested on a known field example from the United States. In all cases, the shape and depth solutions obtained are in good agreement with the actual ones.

BOUGUER GRAVITY CORRECTIONS USING A VARIABLE DENSITY

Geophysics ◽  
1962 ◽  
Vol 27 (5) ◽  
pp. 616-626 ◽  
Author(s):  
F. S. Grant ◽  
A. F. Elsaharty
Keyword(s):  
Least Squares ◽  
Gravity Data ◽  
Gravity Anomalies ◽  
Variable Density ◽  
Bouguer Gravity ◽  
Method Of Least Squares ◽  
Surface Conditions ◽  
Worked Example ◽  
Local Gravity

The principle of density profiling as a means of determining Bouguer densities is studied with a view to extending it to include all of the data in a survey. It is regarded as an endeavor to minimize the correlation between local gravity anomalies and topography, and as such it can be handled mathematically by the method of least squares. In the general case this leads to a variable Bouguer density which can be mapped and contoured. In a worked example, the correspondence between this function and the known geology appears to be good, and indicates that Bouguer density variations due to changing surface conditions can be used routinely in the reduction of gravity data.

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