Khalid Essa | Cairo University (original) (raw)

Papers by Khalid Essa

Research paper thumbnail of 2.5D regularized inversion for the interpretation of residual gravity data by a dipping thin sheet: numerical examples and case studies with an insight on sensitivity and non-uniqueness

Earth, Planets and Space, 2015

Research paper thumbnail of A New Method for Depth and Shape Determinations from Magnetic Data

Pure and Applied Geophysics

We present in this paper a new formula representing the magnetic anomaly expressions produced by ... more We present in this paper a new formula representing the magnetic anomaly expressions produced by most geological structures. Using the new formula we developed a simple and fast numerical method to determine simultaneously the depth and shape of a buried structure from second-horizontal derivative anomalies obtained from magnetic data with filters of successive window lengths. The method involves using a nonlinear relationship between the depth to the source and the shape factor and a combination of observations at four points with respect to the coordinate of the source center with a free parameter (window length). The relationship represents a parametric family of curves (window curves). For a fixed free parameter, 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 wind...

Research paper thumbnail of A simple method for depth determination from self-potential anomalies due to two superimposed structures

Exploration Geophysics, 2015

Research paper thumbnail of A Simple Iterative Approach to Shape and Depth Estimation from SP Residual Anomalies

Journal of King Abdulaziz University-Earth Sciences, 2002

We have developed a simple numerical approach to define shape and depth from residual SP anomalie... more We have developed a simple numerical approach to define shape and depth from residual SP anomalies caused by simple geologic structures. By defining the anomaly value at the origin V(0) and the anomaly values at two equidistant points from the origin on the profile (V(N) and V(-N)), the problem of depth determination from residual SP anomalies has been transformed into finding a solution to a nonlinear equation for each shape factor. The computed depths are plotted against the shape factors on a graph. All points for each two equidistant points from the origin are connected by a continuous curve (depth curve). The solution for the shape and depth of the buried structure is read at the common intersection of the depth curves. The method is applied to theoretical data without and with (10%) random noise and tested on a field example from Colorado.

Research paper thumbnail of A New Algorithm for Depth Determination from Total Magnetic Anomalies due to Spheres

Pure and Applied Geophysics, 2008

We have developed an automatic method to determine the depth of a buried sphere from numerical se... more We have developed an automatic method to determine the depth of a buried sphere from numerical second horizontal derivative anomalies obtained from total field magnetic data. The method is based on using a relationship between the depth and a combination of observations at symmetric points with respect to the coordinate of the projection of the center of the source in the plane of the measurement points with a free parameter (graticule spacing). The problem of depth determination has been transformed into the problem of finding a solution of a nonlinear equation of f(z) = 0. Procedures are also formulated to determine the magnetic moment and the effective angle of magnetization. The method is applied to synthetic examples with and without random errors and tested on a field example from Senegal. In all cases, the depth solutions are in good agreement with the actual ones.

Research paper thumbnail of A Least-squares Window Curves Method for Interpretation of Magnetic Anomalies Caused by Dipping Dikes

Pure and Applied Geophysics, 2007

We have developed a least-squares method to determine simultaneously the depth and the width of a... more We have developed a least-squares method to determine simultaneously the depth and the width of a buried thick dipping dike from residualized magnetic data using filters of successive window lengths. The method involves using a relationship between the depth and the half-width of the source and a combination of windowed observations. The relationship represents a family of curves (window curves). For a fixed window length, the depth is determined for each half-width value by solving one nonlinear equation of the form f (z) = 0 using the least-squares method. The computed depths are plotted against the width values representing a continuous curve. The solution for the depth and the width of the buried dike is read at the common intersection of the window curves. The method involves using a dike model convolved with the same moving average filter as applied to the observed data. As a result, this method can be applied to residuals as well as to measured magnetic data. Procedures are also formulated to estimate the amplitude coefficient and the index parameter. The method is applied to theoretical data with and without random errors. The validity of the method is tested on airborne magnetic data from Canada and on a vertical component magnetic anomaly from Turkey. In all cases examined, the model parameters obtained are in good agreement with the actual ones and with those given in the published literature.

Research paper thumbnail of A least-squares depth-horizontal position curves method to interpret residual SP anomaly profiles

Journal of Geophysics and Engineering, 2006

In this paper, we have developed a least-squares analysis method to estimate not only the depth a... more 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.

Research paper thumbnail of A least-squares window curves method to interpret gravity data due to dipping faults

Journal of Geophysics and Engineering, 2013

We have developed a least-squares method to determine simultaneously the depth and the dip angle ... more We have developed a least-squares method to determine simultaneously the depth and the dip angle of a buried fault from first moving average residual gravity anomalies using filters of successive window lengths. The method involves using a relationship between the depth and the dip angle of the source and a combination of windowed observations. The relationship represents a family of curves (window curves). For a fixed window length, the depth is determined for each dip angle value by solving one nonlinear equation of the form f (z) = 0 using the least-squares method. The computed depths are plotted against the dip angle values representing a continuous curve. The solution for the depth and the dip angle of the buried fault is read at the common intersection of the window curves. The method involves using a dipping faulted thin slab model convolved with the same moving average filter as applied to the observed data. As a result, this method can be applied to residuals as well as to measured gravity data. The method is applied to theoretical data with and without random errors. The validity of the method is tested on 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.

Research paper thumbnail of A least-squares variance analysis method for shape and depth estimation from gravity data

Journal of Geophysics and Engineering, 2006

We have developed a simple method to estimate the shape (shape factor) and the depth of a buried ... more 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

Research paper thumbnail of Self-potential data interpretation using standard deviations of depths computed from moving-average residual anomalies

Geophysical Prospecting, 2006

We have developed a least-squares minimization approach to determine simultaneously the shape (sh... more We have developed a least-squares minimization approach to determine simultaneously the shape (shape factor) and the depth of a buried structure from self-potential (SP) data. The method is based on computing the standard deviation of the depths determined from all moving-average residual anomalies obtained from SP data, using filters of successive window lengths for each shape factor. The standard deviation may generally be considered a criterion for determining the correct depth and shape factor of the buried structure. When the correct shape factor is used, the standard deviation of the depths is less than the standard deviations computed using incorrect shape factors. This method is applied to synthetic data with and without random errors, complicated regionals and interference from neighbouring sources, and is tested on a known field example from Turkey. In all cases, the shape and depth solutions obtained are in a good agreement with the actual values.

Research paper thumbnail of A least-squares minimisation approach to depth, index parameter, and amplitude coefficient determination from magnetic anomalies due to thin dykes

Exploration Geophysics, 2003

ABSTRACT We have developed a least-squares approach to determine, successively, the depth, index ... more ABSTRACT We have developed a least-squares approach to determine, successively, the depth, index parameter, and amplitude coefficient of a buried thin dyke, using moving-average residual anomalies obtained from magnetic data using filters of successive graticule spacings. By defining the moving-average residual anomaly value at the origin on the profile, the problem of depth determination is transformed into the problem of solving a nonlinear equation, f(z) = 0. Knowing the depth and applying the least-squares method, the index parameter is determined by solving a nonlinear equation of the form l(q) = 0. Finally, knowing the depth and the index parameter, the amplitude coefficient is determined in a least-squares sense using a simple linear equation. In this way, the depth, index parameter, and amplitude coefficient are determined individually from all observed magnetic data. We have developed a procedure for automated interpretation of magnetic anomalies attributable to thin dykes. We apply the method to synthetic data with random errors, complicated regionals, and interference from neighbouring magnetic rocks, and we test it on two field examples from Brazil and Canada.

Research paper thumbnail of Quantitative Interpretation of Self-Potential Anomalies of Some Simple Geometric Bodies

Pure and Applied Geophysics, 2009

A quantitative method of interpreting self-potential anomaly caused by a spherical ore body using... more A quantitative method of interpreting self-potential anomaly caused by a spherical ore body using downward continuation method is presented. Master curves to determine the depth, radius and angle of polarization have been prepared.

Research paper thumbnail of A Fast Interpretation Method for Inverse Modeling of Residual Gravity Anomalies Caused by Simple Geometry

Journal of Geological Research, 2012

Research paper thumbnail of A rapid algorithm for self-potential data inversion with application to mineral exploration

ASEG Extended Abstracts, 2007

Research paper thumbnail of New fast least-squares algorithm for estimating the best-fitting parameters due to simple geometric-structures from gravity anomalies

Journal of advanced research, 2014

A new fast least-squares method is developed to estimate the shape factor (q-parameter) of a buri... more A new fast least-squares method is developed to estimate the shape factor (q-parameter) of a buried structure using normalized residual anomalies obtained from gravity data. The problem of shape factor estimation is transformed into a problem of finding a solution of a non-linear equation of the form f(q) = 0 by defining the anomaly value at the origin and at different points on the profile (N-value). Procedures are also formulated to estimate the depth (z-parameter) and the amplitude coefficient (A-parameter) of the buried structure. The method is simple and rapid for estimating parameters that produced gravity anomalies. This technique is used for a class of geometrically simple anomalous bodies, including the semi-infinite vertical cylinder, the infinitely long horizontal cylinder, and the sphere. The technique is tested and verified on theoretical models with and without random errors. It is also successfully applied to real data sets from Senegal and India, and the inverted-par...

Research paper thumbnail of A simple formula for shape and depth determination from residual gravity anomalies

This paper presents a simple method for shape and depth determination of a buried structure from ... more This paper presents a simple method for shape and depth determination of a buried structure from residual gravity anomalies along profile. The method utilizes the anomaly values of the origin and characteristic points of the profile to construct a relationship between the shape factor and depth of the causative source. For fixed points, the depth is determined for each shape factor. The computed depths are then plotted against the shape factor representing a continuous monotonically increasing curve. The solution for the shape and depth of the buried structure is then read at the common intersection point of the depth curves. This method is applied to synthetic data with and without random errors. Finally, the validity of the method is tested on two field examples from the USA.

Research paper thumbnail of Depth and shape solutions from second moving average residual magnetic anomalies

Exploration Geophysics, 2015

Research paper thumbnail of A Least-squares Approach to Depth Determination from Numerical Horizontal Self-potential Gradients

Pure and Applied Geophysics, 2004

We have developed a least-squares minimization approach to depth determination of a buried ore de... more We have developed a least-squares minimization approach to depth determination of a buried ore deposit from numerical horizontal gradients obtained from self-potential (SP) data using filters of successive window lengths (graticule spacings). The problem of depth determination from SP gradients has been transformed into the problem of finding a solution to a nonlinear equation of the form f ðzÞ ¼ 0. Formulas have been derived for vertical and horizontal cylinders and spheres. Procedures are also formulated to estimate the electrical dipole moment and the polarization angle. The method is applied to synthetic data with and without random noise. Finally, the validity of the method is tested on two field examples. In both cases, the depth obtained is found to be in a very good agreement with that obtained from drilling information.

Research paper thumbnail of Least-squares Minimization Approaches to Interpret Total Magnetic Anomalies Due to Spheres

Pure and Applied Geophysics, 2007

We have developed three different least-squares approaches to determine successively: the depth, ... more We have developed three different least-squares approaches to determine successively: the depth, magnetic angle, and amplitude coefficient of a buried sphere from a total magnetic anomaly. By defining the anomaly value at the origin and the nearest zero-anomaly distance from the origin on the profile, the problem of depth determination is transformed into the problem of finding a solution of a nonlinear equation of the form f(z)=0. Knowing the depth and applying the least-squares method, the magnetic angle and amplitude coefficient are determined using two simple linear equations. In this way, the depth, magnetic angle, and amplitude coefficient are determined individually from all observed total magnetic data. The method is applied to synthetic examples with and without random errors and tested on a field example from Senegal, West Africa. In all cases, the depth solutions are in good agreement with the actual ones.

Research paper thumbnail of A new algorithm for gravity or self-potential data interpretation

Journal of Geophysics and Engineering, 2011

An inversion algorithm is developed to estimate the depth and the associated model parameters of ... more An inversion algorithm is developed to estimate the depth and the associated model parameters of the anomalous body from the gravity or self-potential (SP) whole measured data. The problem of the depth (z) estimation from the observed data has been transformed into a nonlinear equation of the form F(z) = 0. This equation is then solved for z by minimizing an objective functional in the least-squares sense. Using the estimated depth, the polarization angle and the dipole moment or the depth and the amplitude coefficient are computed from the measured SP or gravity data, respectively. The method is based on determining the root mean square (RMS) of the depths estimated from using all s-values for each shape factor. The minimum RMS is used as a criterion for estimating the correct shape and depth of the buried structure. When the correct shape factor is used, the RMS of the depths is always less than the RMS computed using wrong shape factors. The proposed approach is applicable to a class of geometrically simple anomalous bodies, such as the semi-infinite vertical cylinder, the dike, the horizontal cylinder and the sphere, and it is tested and verified on synthetic examples with and without noise. This technique is also successfully applied to four real datasets for mineral exploration, and it is found that the estimated depths and the associated model parameters are in good agreement with the actual values.

Research paper thumbnail of 2.5D regularized inversion for the interpretation of residual gravity data by a dipping thin sheet: numerical examples and case studies with an insight on sensitivity and non-uniqueness

Earth, Planets and Space, 2015

Research paper thumbnail of A New Method for Depth and Shape Determinations from Magnetic Data

Pure and Applied Geophysics

We present in this paper a new formula representing the magnetic anomaly expressions produced by ... more We present in this paper a new formula representing the magnetic anomaly expressions produced by most geological structures. Using the new formula we developed a simple and fast numerical method to determine simultaneously the depth and shape of a buried structure from second-horizontal derivative anomalies obtained from magnetic data with filters of successive window lengths. The method involves using a nonlinear relationship between the depth to the source and the shape factor and a combination of observations at four points with respect to the coordinate of the source center with a free parameter (window length). The relationship represents a parametric family of curves (window curves). For a fixed free parameter, 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 wind...

Research paper thumbnail of A simple method for depth determination from self-potential anomalies due to two superimposed structures

Exploration Geophysics, 2015

Research paper thumbnail of A Simple Iterative Approach to Shape and Depth Estimation from SP Residual Anomalies

Journal of King Abdulaziz University-Earth Sciences, 2002

We have developed a simple numerical approach to define shape and depth from residual SP anomalie... more We have developed a simple numerical approach to define shape and depth from residual SP anomalies caused by simple geologic structures. By defining the anomaly value at the origin V(0) and the anomaly values at two equidistant points from the origin on the profile (V(N) and V(-N)), the problem of depth determination from residual SP anomalies has been transformed into finding a solution to a nonlinear equation for each shape factor. The computed depths are plotted against the shape factors on a graph. All points for each two equidistant points from the origin are connected by a continuous curve (depth curve). The solution for the shape and depth of the buried structure is read at the common intersection of the depth curves. The method is applied to theoretical data without and with (10%) random noise and tested on a field example from Colorado.

Research paper thumbnail of A New Algorithm for Depth Determination from Total Magnetic Anomalies due to Spheres

Pure and Applied Geophysics, 2008

We have developed an automatic method to determine the depth of a buried sphere from numerical se... more We have developed an automatic method to determine the depth of a buried sphere from numerical second horizontal derivative anomalies obtained from total field magnetic data. The method is based on using a relationship between the depth and a combination of observations at symmetric points with respect to the coordinate of the projection of the center of the source in the plane of the measurement points with a free parameter (graticule spacing). The problem of depth determination has been transformed into the problem of finding a solution of a nonlinear equation of f(z) = 0. Procedures are also formulated to determine the magnetic moment and the effective angle of magnetization. The method is applied to synthetic examples with and without random errors and tested on a field example from Senegal. In all cases, the depth solutions are in good agreement with the actual ones.

Research paper thumbnail of A Least-squares Window Curves Method for Interpretation of Magnetic Anomalies Caused by Dipping Dikes

Pure and Applied Geophysics, 2007

We have developed a least-squares method to determine simultaneously the depth and the width of a... more We have developed a least-squares method to determine simultaneously the depth and the width of a buried thick dipping dike from residualized magnetic data using filters of successive window lengths. The method involves using a relationship between the depth and the half-width of the source and a combination of windowed observations. The relationship represents a family of curves (window curves). For a fixed window length, the depth is determined for each half-width value by solving one nonlinear equation of the form f (z) = 0 using the least-squares method. The computed depths are plotted against the width values representing a continuous curve. The solution for the depth and the width of the buried dike is read at the common intersection of the window curves. The method involves using a dike model convolved with the same moving average filter as applied to the observed data. As a result, this method can be applied to residuals as well as to measured magnetic data. Procedures are also formulated to estimate the amplitude coefficient and the index parameter. The method is applied to theoretical data with and without random errors. The validity of the method is tested on airborne magnetic data from Canada and on a vertical component magnetic anomaly from Turkey. In all cases examined, the model parameters obtained are in good agreement with the actual ones and with those given in the published literature.

Research paper thumbnail of A least-squares depth-horizontal position curves method to interpret residual SP anomaly profiles

Journal of Geophysics and Engineering, 2006

In this paper, we have developed a least-squares analysis method to estimate not only the depth a... more 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.

Research paper thumbnail of A least-squares window curves method to interpret gravity data due to dipping faults

Journal of Geophysics and Engineering, 2013

We have developed a least-squares method to determine simultaneously the depth and the dip angle ... more We have developed a least-squares method to determine simultaneously the depth and the dip angle of a buried fault from first moving average residual gravity anomalies using filters of successive window lengths. The method involves using a relationship between the depth and the dip angle of the source and a combination of windowed observations. The relationship represents a family of curves (window curves). For a fixed window length, the depth is determined for each dip angle value by solving one nonlinear equation of the form f (z) = 0 using the least-squares method. The computed depths are plotted against the dip angle values representing a continuous curve. The solution for the depth and the dip angle of the buried fault is read at the common intersection of the window curves. The method involves using a dipping faulted thin slab model convolved with the same moving average filter as applied to the observed data. As a result, this method can be applied to residuals as well as to measured gravity data. The method is applied to theoretical data with and without random errors. The validity of the method is tested on 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.

Research paper thumbnail of A least-squares variance analysis method for shape and depth estimation from gravity data

Journal of Geophysics and Engineering, 2006

We have developed a simple method to estimate the shape (shape factor) and the depth of a buried ... more 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

Research paper thumbnail of Self-potential data interpretation using standard deviations of depths computed from moving-average residual anomalies

Geophysical Prospecting, 2006

We have developed a least-squares minimization approach to determine simultaneously the shape (sh... more We have developed a least-squares minimization approach to determine simultaneously the shape (shape factor) and the depth of a buried structure from self-potential (SP) data. The method is based on computing the standard deviation of the depths determined from all moving-average residual anomalies obtained from SP data, using filters of successive window lengths for each shape factor. The standard deviation may generally be considered a criterion for determining the correct depth and shape factor of the buried structure. When the correct shape factor is used, the standard deviation of the depths is less than the standard deviations computed using incorrect shape factors. This method is applied to synthetic data with and without random errors, complicated regionals and interference from neighbouring sources, and is tested on a known field example from Turkey. In all cases, the shape and depth solutions obtained are in a good agreement with the actual values.

Research paper thumbnail of A least-squares minimisation approach to depth, index parameter, and amplitude coefficient determination from magnetic anomalies due to thin dykes

Exploration Geophysics, 2003

ABSTRACT We have developed a least-squares approach to determine, successively, the depth, index ... more ABSTRACT We have developed a least-squares approach to determine, successively, the depth, index parameter, and amplitude coefficient of a buried thin dyke, using moving-average residual anomalies obtained from magnetic data using filters of successive graticule spacings. By defining the moving-average residual anomaly value at the origin on the profile, the problem of depth determination is transformed into the problem of solving a nonlinear equation, f(z) = 0. Knowing the depth and applying the least-squares method, the index parameter is determined by solving a nonlinear equation of the form l(q) = 0. Finally, knowing the depth and the index parameter, the amplitude coefficient is determined in a least-squares sense using a simple linear equation. In this way, the depth, index parameter, and amplitude coefficient are determined individually from all observed magnetic data. We have developed a procedure for automated interpretation of magnetic anomalies attributable to thin dykes. We apply the method to synthetic data with random errors, complicated regionals, and interference from neighbouring magnetic rocks, and we test it on two field examples from Brazil and Canada.

Research paper thumbnail of Quantitative Interpretation of Self-Potential Anomalies of Some Simple Geometric Bodies

Pure and Applied Geophysics, 2009

A quantitative method of interpreting self-potential anomaly caused by a spherical ore body using... more A quantitative method of interpreting self-potential anomaly caused by a spherical ore body using downward continuation method is presented. Master curves to determine the depth, radius and angle of polarization have been prepared.

Research paper thumbnail of A Fast Interpretation Method for Inverse Modeling of Residual Gravity Anomalies Caused by Simple Geometry

Journal of Geological Research, 2012

Research paper thumbnail of A rapid algorithm for self-potential data inversion with application to mineral exploration

ASEG Extended Abstracts, 2007

Research paper thumbnail of New fast least-squares algorithm for estimating the best-fitting parameters due to simple geometric-structures from gravity anomalies

Journal of advanced research, 2014

A new fast least-squares method is developed to estimate the shape factor (q-parameter) of a buri... more A new fast least-squares method is developed to estimate the shape factor (q-parameter) of a buried structure using normalized residual anomalies obtained from gravity data. The problem of shape factor estimation is transformed into a problem of finding a solution of a non-linear equation of the form f(q) = 0 by defining the anomaly value at the origin and at different points on the profile (N-value). Procedures are also formulated to estimate the depth (z-parameter) and the amplitude coefficient (A-parameter) of the buried structure. The method is simple and rapid for estimating parameters that produced gravity anomalies. This technique is used for a class of geometrically simple anomalous bodies, including the semi-infinite vertical cylinder, the infinitely long horizontal cylinder, and the sphere. The technique is tested and verified on theoretical models with and without random errors. It is also successfully applied to real data sets from Senegal and India, and the inverted-par...

Research paper thumbnail of A simple formula for shape and depth determination from residual gravity anomalies

This paper presents a simple method for shape and depth determination of a buried structure from ... more This paper presents a simple method for shape and depth determination of a buried structure from residual gravity anomalies along profile. The method utilizes the anomaly values of the origin and characteristic points of the profile to construct a relationship between the shape factor and depth of the causative source. For fixed points, the depth is determined for each shape factor. The computed depths are then plotted against the shape factor representing a continuous monotonically increasing curve. The solution for the shape and depth of the buried structure is then read at the common intersection point of the depth curves. This method is applied to synthetic data with and without random errors. Finally, the validity of the method is tested on two field examples from the USA.

Research paper thumbnail of Depth and shape solutions from second moving average residual magnetic anomalies

Exploration Geophysics, 2015

Research paper thumbnail of A Least-squares Approach to Depth Determination from Numerical Horizontal Self-potential Gradients

Pure and Applied Geophysics, 2004

We have developed a least-squares minimization approach to depth determination of a buried ore de... more We have developed a least-squares minimization approach to depth determination of a buried ore deposit from numerical horizontal gradients obtained from self-potential (SP) data using filters of successive window lengths (graticule spacings). The problem of depth determination from SP gradients has been transformed into the problem of finding a solution to a nonlinear equation of the form f ðzÞ ¼ 0. Formulas have been derived for vertical and horizontal cylinders and spheres. Procedures are also formulated to estimate the electrical dipole moment and the polarization angle. The method is applied to synthetic data with and without random noise. Finally, the validity of the method is tested on two field examples. In both cases, the depth obtained is found to be in a very good agreement with that obtained from drilling information.

Research paper thumbnail of Least-squares Minimization Approaches to Interpret Total Magnetic Anomalies Due to Spheres

Pure and Applied Geophysics, 2007

We have developed three different least-squares approaches to determine successively: the depth, ... more We have developed three different least-squares approaches to determine successively: the depth, magnetic angle, and amplitude coefficient of a buried sphere from a total magnetic anomaly. By defining the anomaly value at the origin and the nearest zero-anomaly distance from the origin on the profile, the problem of depth determination is transformed into the problem of finding a solution of a nonlinear equation of the form f(z)=0. Knowing the depth and applying the least-squares method, the magnetic angle and amplitude coefficient are determined using two simple linear equations. In this way, the depth, magnetic angle, and amplitude coefficient are determined individually from all observed total magnetic data. The method is applied to synthetic examples with and without random errors and tested on a field example from Senegal, West Africa. In all cases, the depth solutions are in good agreement with the actual ones.

Research paper thumbnail of A new algorithm for gravity or self-potential data interpretation

Journal of Geophysics and Engineering, 2011

An inversion algorithm is developed to estimate the depth and the associated model parameters of ... more An inversion algorithm is developed to estimate the depth and the associated model parameters of the anomalous body from the gravity or self-potential (SP) whole measured data. The problem of the depth (z) estimation from the observed data has been transformed into a nonlinear equation of the form F(z) = 0. This equation is then solved for z by minimizing an objective functional in the least-squares sense. Using the estimated depth, the polarization angle and the dipole moment or the depth and the amplitude coefficient are computed from the measured SP or gravity data, respectively. The method is based on determining the root mean square (RMS) of the depths estimated from using all s-values for each shape factor. The minimum RMS is used as a criterion for estimating the correct shape and depth of the buried structure. When the correct shape factor is used, the RMS of the depths is always less than the RMS computed using wrong shape factors. The proposed approach is applicable to a class of geometrically simple anomalous bodies, such as the semi-infinite vertical cylinder, the dike, the horizontal cylinder and the sphere, and it is tested and verified on synthetic examples with and without noise. This technique is also successfully applied to four real datasets for mineral exploration, and it is found that the estimated depths and the associated model parameters are in good agreement with the actual values.