Khalid S. Essa

Gravity data interpretation using the s-curves method

An algorithm is developed for a fast quantitative interpretation of gravity data generated by geometrically simple causative bodies. This algorithm utilizes numerical fourth horizontal derivatives computed from the observed gravity anomaly, using filters of successive window lengths to estimate the depth and shape of a buried structure. For a fixed window length, the depth is estimated 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 (s-curve). The estimated solution for the depth and shape of the buried structure is read at the common intersection of the s-curves. The technique can be applied not only to the true residuals, but also to the measured Bouguer gravity data. The method is applied to synthetic data with and without random errors and two field examples from Egypt and the USA. In all cases examined, the estimated depths and other model parameters are found to be in good agreement with the actual values.

Magnetic interpretation using a least-squares, depth-shape curves method

We have developed a least-squares approach to depth determination from residual magnetic anomalies caused by simple geologic structures. By normalizing the residual magnetic anomaly using three characteristic points and their corresponding distances on the anomaly profile, the problem of determining depth from residual magnetic anomalies has been transformed into finding a solution to a nonlinear equation of the form z = f(z). Formulas have been derived for spheres, horizontal cylinders, thin dikes, and contacts. The method is applied to synthetic data with and without random noise. We have also developed a method using depth-shape curves to simultaneously define the shape and depth of a buried structure from a residual magnetic anomaly profile. The method is based on determining the depth from the normalized residual anomaly for each shape factor using the least-squares method mentioned above. The computed depths are plotted against the shape factors on a graph. The solution for the shape and depth of the buried structure is read at the common intersection of the depth-shape curves. The depth-shape curves method was successfully tested on theoretical data with and without random noise and applied to a known field example from Ontario.

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.

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

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-parameters are in good agreement with the known actual values.

A new algorithm for gravity or self-potential data interpretation

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.

A new inversion algorithm for estimating the best fitting parameters of some geometrically simple body to measured self-potential anomalies

Abstract We have developed a new least-squares inversion approach to determine successively the depth (z), polarization angle, and electric dipole moment of a buried structure from the self-potential (SP) anomaly data measured along a profile. This inverse algorithm makes it possible to use all the observed data when determining each of these three parameters. The problem of the depth determination has been parameterised from the forward modelling operator, and transformed into a nonlinear equation in the form ξ(z) = 0 by minimising an objective functional in the least-squares sense. Using the estimated depth and applying the least-squares method, the polarization angle is then determined from the entire observed data by a linear formula. Finally, knowing the depth and polarization angle, the dipole moment is expressed by a linear equation and is computed using the whole measured data. This technique is applicable for a class of geometrically simple anomalous bodies, including the semi-infinite vertical cylinder, the infinitely long horizontal cylinder, and the sphere. The method is tested and verified on numerical examples with and without random noise. It is also successfully applied to two real datasets from mineral exploration in Germany and Turkey, and we have found that the estimated depths and the other SP model parameters are in good agreement with the known actual values.

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.

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 rapid technique for estimating the depth and width of a two-dimensional plate from self-potential data

Rapid techniques for self-potential (SP) data interpretation are of prime importance in engineering and exploration geophysics. Parameters (e.g. depth, width) estimation of the ore bodies has also been of paramount concern in mineral prospecting. In many cases, it is useful to assume that the SP anomaly is due to an ore body of simple geometric shape and to use the data to determine its parameters. In light of this, we describe a rapid approach to determine the depth and horizontal width of a two-dimensional plate from the SP anomaly. The rationale behind the scheme proposed in this paper is that, unlike the two- (2D) and three-dimensional (3D) SP rigorous source current inversions, it does not demand ap rioriinformation about the subsurface resistivity distribution nor high computational resources. We apply the second-order moving average operator on the SP anomaly to remove the unwanted (regional) effect, represented by up to a third-order polynomial, using filters of successive window lengths. By defining a function F at a fixed window length (s) in terms of the filtered anomaly computed at two points symmetrically distributed about the origin point of the causative body, the depth (z) corresponding to each half-width (w) is estimated by solving a nonlinear equation in the form ξ(s, w, z) = 0. The estimated depths are then plotted against their corresponding half-widths on a graph representing a continuous curve for this window length. This procedure is then repeated for each available window length. The depth and half-width solution of the buried structure is read at the common intersection of these various curves. The improvement of this method over the published first-order moving average technique for SP data is demonstrated on a synthetic data set. It is then verified on noisy synthetic data, complicated structures and successfully applied to three field examples for mineral exploration and we have found that the estimated depth is in good agreement with the known value reported in the literature.