E. M. Abdelrahman

On the least‐squares residual anomaly determination

This paper discusses an approach to determine the least‐squares optimum order of the regional surface which, when subtracted from the Bouguer gravity anomaly data, minimizes distortion of the residual component of the field. The least‐squares method was applied to theoretical composite gravity fields each consisting of a constant residual component (sphere or vertical cylinder) and a regional component of different order using successively increasing orders of polynomial regionals for residual determination. The overall similarity between each two successive residual maps was determined by computing the correlation factor between the mapped variables. Similarity between residual maps of the lowest orders, verified by good correlation, may generally be considered a criterion for determining the optimum order of the regional surface and consequently the least distorted residual component. The residual map of the lower order in this well‐correlated doublet is considered the most plausible one and may be used...

A least‐squares approach to depth determination from self‐potential anomalies caused by horizontal cylinders and spheres

We have developed a least-squares approach to depth determination from self-potential anomalies caused by horizontal cylinders and spheres. By defining the zero-anomaly distance and the anomaly value at the origin on the profile, the problem of depth determination from self-potential data has been transformed into finding a solution to a nonlinear equation. Procedures are also formulated to estimate the electric dipole moment and the polarization angle. The error in the depth parameter estimation introduced by data errors was also studied through imposing 1 to 10% errors in the zero-anomaly distance and the anomaly value at the origin in one synthetic profile caused by a sphere. When the zero-anomaly distance and the anomaly value at the origin possess errors of equal magnitude and of the same signs, the results will not differ much from the true values. When errors have opposite signs, the maximum error in depth is 10%. Finally, the validity of the method is tested on a field example from Ergani Copper district, Turkey.

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

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.

New methods for shape and depth determinations from SP data

We have extended our earlier derivative analysis method to higher derivatives to estimate the depth and shape (shape factor) of a buried structure from self‐potential (SP) data. We show that numerical second, third, and fourth horizontal‐derivative anomalies obtained from SP data using filters of successive window lengths can be used to simultaneously determine the depth and the shape of a buried structure. The depths and shapes obtained from the higher derivatives anomaly values can be used to determine simultaneously the actual depth and shape of the buried structure and the optimum order of the regional SP anomaly along the profile. The method is semi‐automatic and it can be applied to residuals as well as to observed SP data.We have also developed a method (based on a least‐squares minimization approach) to determine, successively, the depth and the shape of a buried structure from the residual SP anomaly profile. By defining the zero anomaly distance and the anomaly value at the origin, the problem o...

A new method for shape and depth determinations from gravity data

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.

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

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, f(z)=0. 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 coeffic...

Derivative analysis of SP anomalies

Numerical second horizontal derivative self-potential (SP) anomalies obtained from SP data using filters of successive window lengths (graticule spacings) can be used to determine the shape and depth of a buried structure. For a fixed window length, the depth is determined using a simple formula for each shape factor. The computed depths are plotted against the shape factors on a graph. All points for a fixed window length are connected by a continuous curve (window curve). The solution for the shape and depth of the buried structure is read at the common intersection of the window curves. The method is applied to theoretical data with and without random noise and tested on a field example from Turkey.

A least‐squares minimization approach to shape determination from gravity data

The gravity anomaly expression produced by most geologic structures can be represented by a continuous function of both shape (shape‐factor) and depth‐related variables with an amplitude coefficient related to mass (Abdelrahman and El‐Araby, 1993). Few methods have been developed to determine the shape of the buried geologic structure from residual gravity anomaly profiles. These methods include a Walsh transform approach (Shaw and Agarwal, 1990) and the employment of a correlation factor between successive least‐squares residuals (Abdelrahman and El‐Araby 1993). In the present note, a least‐squares minimization approach to shape‐factor determination from a residual gravity anomaly profile is presented. The problem of the shape‐factor determination is transformed into the problem of finding a solution of a nonlinear equation of the form f(q) = 0.

Shape and depth solutions from numerical horizontal self-potential gradients

Abstract The self-potential anomaly expression produced by most geologic structures can be represented by a continuous function in shape (shape factor), depth and angle of polarization variables with an amplitude coefficient known as the electric dipole moment. Numerical horizontal self-potential gradients obtained from self-potential data using filters of successive window lengths can be used to determine the shape and depth of buried structures. For a fixed window length, the depth is determined using a simple formula for each shape factor. The computed depths are plotted against the shape factor representing a continuous window curve. The solution for the shape and depth of the buried structure is read at the common intersection of the window curves. The method is applied to synthetic data and tested on two field examples from Turkey.

A least-squares minimization approach to depth determination for numerical horizontal 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.