E. R. Abo-Ezz

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...

New least-squares algorithm for model parameters estimation using self-potential anomalies

We have developed a new least-squares minimization approach to depth determination from self-potential (SP) data. By defining the anomaly value at the origin and at any two symmetrical points around the origin on the profile, the problem of depth determination from the residual SP anomaly has been transformed into finding a solution to a nonlinear equation of the form f(z)=0. Procedures are also formulated to estimate the polarization angle, amplitude coefficient and the shape of the buried structure (shape factor). The method is simple and can be used as a rapid method to estimate parameters that produced SP anomalies. The method is tested on synthetic data with and without random errors. It is also applied to a field example from Turkey. In all cases, the model parameters obtained are in good agreement with actual ones.

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

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 ...

A least-squares variance analysis method for shape and depth estimation from gravity data

We have developed a simple method to estimate the shape (shape factor) and the depth of a buried structure simultaneously from modified first moving average residual anomalies (second moving average residuals) obtained from gravity data using filters of successively greater window lengths. The method is based on computing the variance of the depths determined from all second moving average residual anomaly profiles using the least-squares method for each shape factor. The minimum variance is used as a criterion for determining the correct shape and depth of the buried structure. When the correct shape factor is used, the variance of the depths is always less than the variances computed using wrong shape factors. The method is applied to synthetic data with and without random errors, complex regional anomalies and interference from neighbouring structures, and tested on a field example from the USA.

Parametric inversion of residual magnetic anomalies due to simple geometric bodies

We have developed a simple method to determine the depth, inclination parameter and amplitude coefficient of a buried structure from a residual magnetic anomaly profile using a new formula representing the magnetic anomaly expressions produced by most geological structures. The method is based on defining the anomaly value at the origin and four characteristic points and their corresponding distances on the anomaly profile. Using all possible combinations of the four characteristic points and their corresponding distances, a procedure is developed for automated determination of the best fit model parameters including the shape (shape factor) of the buried structure from magnetic data. The method was applied to synthetic data with and without random errors and tested on two field examples from Canada and India. In both cases, the model parameters obtained by the present method, particularly the shape and depth of the buried structures, were found to be in good agreement with the actual parameters. The present method has the capability of avoiding highly noisy data points and enforcing the incorporation of points of the least random errors to enhance the interpretation results. We present a method to determine the model parameters of a buried structure from a residual magnetic anomaly profile using a new formula representing the magnetic anomaly expressions produced by most geological structures. The method has the capability of avoiding highly noisy data points to enhance the interpretation results.

A least-squares minimization approach for model parameters estimate by using a new magnetic anomaly formula

A new linear least-squares approach is proposed to interpret magnetic anomalies of the buried structures by using a new magnetic anomaly formula. This approach depends on solving different sets of algebraic linear equations in order to invert the depth (z), amplitude coefficient (K), and magnetization angle (θ) of buried structures using magnetic data. The utility and validity of the new proposed approach has been demonstrated through various reliable synthetic data sets with and without noise. In addition, the method has been applied to field data sets from USA and India. The best-fitted anomaly has been delineated by estimating the root-mean squared (rms). Judging satisfaction of this approach is done by comparing the obtained results with other available geological or geophysical information.

A least-squares depth-horizontal position curves method to interpret residual SP anomaly profiles

In this paper, we have developed a least-squares analysis method to estimate not only the depth and shape but also to determine the horizontal position of a buried structure from the residual SP anomaly profile. The method is based on normalizing the residual SP anomaly using three characteristic points and their corresponding distances on the anomaly profile and then determining the depth for each horizontal position of the buried structure using the least-squares method. The computed depths are plotted against the assumed horizontal positions on a graph. The solution for the depth and the horizontal position of the buried structure is read at the common intersection of the curves. Knowing the depth and the horizontal position and applying the least-squares method, the shape factor is determined using a simple linear equation. Procedures are also formulated to estimate the polarization angle and the electric dipole moment. The method is semi-automatic and it can be applied to short or long residual SP anomaly profiles. The method is applied to synthetic data with and without random noise. The validity of the method is tested on a field example from Turkey. In all cases, the model parameters obtained are in good agreement with the actual ones.

Higher derivatives analysis of 2-D magnetic data

This paper presents a new approach for determining the depth of a buried structure from numerical second‐, third‐, and fourth‐horizontal‐derivative anomalies obtained from 2-D magnetic 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 z = f(z). Formulas have been derived for a horizontal cylinder and a dike. The depths obtained from the second‐, third‐, and fourth‐derivative anomaly values can be used to determine simultaneously the actual depth to the buried structure and the optimum order of the regional magnetic field along the profile. This powerful technique can solve two major potential field problems: regional residual separation and depth determination. The method is applied to theoretical data with and without random errors and is tested on a field example from Arizona.

A least-squares standard deviation method to interpret magnetic anomalies due to thin dikes

We have developed a least-squares method to determine simultaneously the depth and the horizontal position (origin) of a buried thin dike that extends in both strike direction and down dip (2D) and in which the depth is much greater than the thickness from horizontal gradients obtained numerically from magnetic data using filters of successive window lengths. The method involves using a relationship between the depth and the horizontal position of the source and a combination of windowed observations. The method is based on computing the standard deviation of the depths determined from all horizontal gradient anomalies for each horizontal position. The standard deviation may generally be considered as a criterion for determining the correct depth and the horizontal position of the buried dike. When the correct horizontal position value is used, the standard deviation of the depths is less than the standard deviation using incorrect horizontal position values. This method can be applied to residuals as well as to the observed magnetic data. The method is applied to synthetic examples with and without random errors. The present method was able to provide both the depth and horizontal position of the source accurately. The practical utility of the method is tested on an outcropping dike in Canada.