T. M. El-Araby

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.

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...

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.

Shape and depth determinations from second moving average residual self-potential anomalies

We have developed a semi-automatic method to determine the depth and shape (shape factor) of a buried structure from second moving average residual self-potential anomalies obtained from observed data using filters of successive window lengths. The method involves using a relationship between the depth and the shape to 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 is read at the common intersection of the window curves. The validity of the method is tested on a synthetic example with and without random errors and on two field examples from Turkey and Germany. In all cases examined, the depth and the shape solutions obtained are in very good agreement with the true ones.

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.

A new approach to depth determination from magnetic anomalies

We have developed a semiautomatic method to determine the depth to shallow and deep‐seated structures from a magnetic anomaly profile. It involves using a relationship between the depths to two coaxial sources obtained by combining observations at symmetric points with respect to the coordinate of the sources center. For five established, fixed data points, the depth to the shallow structure is determined for each preassigned depth of the deep‐seated structure. The computed depths to the shallow structure are plotted against the computed depths to the deep‐seated structure, yielding a continuous, monotonically increasing depth curve. The spacing between the observations is then modified, producing several curves. The accepted estimates for the depths to both structures are read at the common intersection of these curves. The effective intensity and the angle of magnetization of both structures are also estimated.The proposed method was tested both on noisy synthetic and real magnetic data. In the case of ...

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.

Depth and shape solutions from second moving average residual magnetic anomalies

We have developed a simple and fast numerical method to simultaneously determine the depth and shape of a buried structure from second moving average residual anomalies obtained from magnetic data with filters of successive window lengths. The method is similar to Euler deconvolution, but it solves for depth and shape independently. The method involves using a nonlinear relationship between the depth to the source and the shape factor, and a combination of observations at five points with respect to the coordinate of the source centre with a free parameter (window length). The method is based on computing the standard deviation of the depths determined from all second moving average residual anomalies for each value of the shape factor. The standard deviation may generally be considered a criterion for determining the correct depth and shape of the buried structure. When the correct shape factor is used, the standard deviation of the depths is less than the standard deviation using incorrect values of the shape factor. This method can be applied to residuals, as well as the observed magnetic data consisting of the combined effect of a residual component due to a purely local structure and a regional component represented by a polynomial of up to fourth-order. The method is applied to synthetic data, with and without random errors, and tested on a field example from Brazil. In all cases, the shape and depth of the buried structures are found in good agreement with the actual ones.