The aim of the presented research activitiesThe aim of the presented research activities

Is to develop new interpretation techniques for potential fields exploration methods

(gravity, magnetic, and

self potential).

The main goal of these new interpretation The main goal of these new interpretation methodsmethods

Is to find the model parameters that best describe the subsurface geologic structure which is responsible for the encountered geophysical anomaly.

Model choiceModel choice

- Either a fixed simple geometry model ( sphere, horizontal-cylinder, vertical cylinder, dike, semi-infinite slab) or an irregularly shaped body can be used.

Simple shapes vs. irregular shapesSimple shapes vs. irregular shapes

- The advantage of the fixed geometry methods over 2D and 3D continuous modeling and inversion methods is that they do not require essential information on the parameters like density contrast, magnetic susceptibility, current density, resistivity and an average range of depths as a part of the input.

- These proposed parameters can greatly affect the results. Slight changes in these parameters can yield different interpretation results.

- In the proposed new

interpretation techniques

this disadvantage is

omitted through the proposed automated unique parameter determination just by using the given

measured anomaly .

- The optimization problem for the model parameters is highly nonlinear.

- Increasing the number of parameters to be solved simultaneously also increases the dimensionality and complexity of the error surface, which increases the probability of the optimization stalling in a local minimum.

- Common sense dictates that the nonlinear optimization should be restricted to as few parameters as is consistent with obtaining useful results.

- This is why, the solution for only one unknown at a time using the well known least- squares optimization technique, is proposed.

- This is done by reducing the number of unknowns in any potential field expression of a simple geometrical body into one unknown only through manipulating and simplifying the expressions to eliminate some parameters by introducing other pieces of information like anomaly values at certain points on the measured profile. This is to end up with a nonlinear equation in the form of f ( z ) = 0, and could be solved using iteration methods.

- Other formulas are also derived for the other parameters and using the early determined parameter the rest could be reached through other individual optimization processes for each parameter.

- All developed methods are tested using synthetic data with and without random errors and approved to be stable and gives very good results.

- Also the applicability of the methods are tested on real field examples and are approved to match with the actual parameters obtained from drilling or previously published data.

Paper No. 1Paper No. 1

Three least-squares minimization approaches to depth, shape, and amplitude coefficient

determination from gravity data

by

E. M. Abdelrahman, H. M. El-Araby, T. M. El-Araby, and E. R. Abo-Ezz

Published in Geophysics Vol. 66, No. 4 (July-August 2001)

Paper Overview- This paper presents an automated method that

determines successively the depth (z), shape (q) and amplitude coefficient (k) (which is related to the radius and density contrast) of a buried structure approximated by sphere, horizontal cylinder, and vertical cylinder from the residual gravity anomaly.

- Using the value of the anomaly at the origin of the profile g (max) and taking the logarithm of the normalized anomaly values and with the help of its value at any other point Xi =A, it was possible to eliminate the unknowns q and k from the gravity anomaly expression, leaving one unknown only which is z.

- Applying the least-squares minimization approach, the best value of z that gives the closest fit of the calculated anomaly with the observed one can be estimated.

- Knowing the depth and applying the least-squares minimization approach twice again, the shape factor and the amplitude coefficient are determined individually using two separate simple linear equations.

- Measuring the goodness of fit between the observed and the computed gravity data for each A value by determining the least sum-squared differences will pinpoint the best calculated model parameters

- The method is applied to synthetic data with and without random errors. It gives exact values for noise-free data and a very small error for contaminated data.

- The method is also tested on a field example from the United States and has been approved to give a very good results compared to the drilling and seismic information initially published in literature.

Paper No. 2Paper No. 2

A new approach to depth determination from magnetic anomalies

by

E. M. Abdelrahman, H. M. El-Araby, T. M. EI-Araby, and K. S. Essa

Published in Geophysics VOL. 67, NO. 5 (SEPTEMBER-OCTOBER 2002)

Paper Overview- In this paper the depth is determined to both shallow

and deep-seated structures at the same time without applying separation techniques.

- The method can use either the total, vertical, or horizontal magnetic anomaly profiles for a variety of combinations of geometric sources (sphere, cylinder, dike, and geologic contact).

- The method is based on finding a relationship between the depths of the two coaxial sources from a combination of observations at symmetric points around the source center.

- The depth is determined using depth curves that are calculated by iteration technique applied on the derived depths relationship given in equation 7, which yields depth solutions z1 for all possible z2 values for fixed N and M spacing values. The depth curves associated with different spacing values should intersect at a single point.

- Not only the depth, but also the effective magnetic intensity and the effective angle of magnetization can be determined from the estimated index parameter (θ) and the amplitude coefficient (C) respectively. (equations 3, 4, 5, 6, and table 2)

- The method can also be used to estimate the source type by using the goodness of fit between the observed and calculated anomalies of the sum of the two sources.

- The Validity of the method

is tested on noisy synthetic

data and on a field example

from Turkey and has been

approved to be stable and

gives very good results.

Paper No. 3Paper No. 3

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

by

E.M. Abdelrahman, H.M. El-Araby, T.M. El-Araby, and E.R. Abo-Ezz

Published in Geophysics VOL. 68, NO. 2 (MARCH-APRIL 2003)

Paper Overview

- This paper presents a full-automatic method that uses two different least-squares approaches to determine the depth and amplitude coefficient ( related to the density contrast and the thickness) of a buried faulted thin slab, successively from numerical first-, second-, third-, and fourth-derivative anomalies obtained from observed 2D gravity data using filters of successive graticule spacings.

- The least-squares minimization approach is used to derive four non-linear equations relating the depth of the fault to its first, second, third, and fourth gravity anomaly horizontal derivative. Solving these equations using simple iteration method will result in four solutions of the depth value.

- Using the four computed depth values, we can find another four values for the amplitude coefficient by solving four linear equations related to the first, second, third, and fourth derivative anomalies.

- The model parameters obtained from the different numerical derivative anomaly values can be used to determine simultaneously the actual model parameters of the buried fault and the optimum order of the regional

gravity field along the profile as follows:

- If z1 = z2 and k1 = k2then z1 and k1 are the true depth and amplitude coefficientand the regional is represented by a zero-order polynomial.

- If z1 ≠ z2 = z3 and k1 ≠ k2 = k3then z2 and k2 are the true depth and amplitude coefficientand the regional is represented by a first-order polynomial.

- If z1 ≠ z2 ≠ z3 = z4 and k1 ≠ k2 ≠ k3 = k4then z3 and k3 are the true depth and amplitude coefficientand the regional is represented by a second-orderpolynomial.

- The validity of the method is tested on theoretical examples with and without random errors. The exact values of z and k were obtained for noise free data. While for 5% random errors contaminated data the depth and amplitude coefficient are within 4.5%

- The method is also applied on a field example from south Aswan, Egypt and matches good with the previously published results.

Paper No. 4Paper No. 4

A least-squares minimization approach to depth determination from magnetic data

by

E.M. Abdelrahman, H.M. El-Araby, T.M. El-Araby, and K.S. Essa

Published in Pure and Applied Geophysics VOL. 160,(2003)

Paper Overview

- This paper presents a least-squares minimization approach to depth determination from residual magnetic anomalies of simple geometrical shaped models like, sphere, horizontal cylinder, dike, and geologic contact.

- By defining the anomaly value T(0) at the origin and the anomaly value T(N) at any other distance (N) on the profile, the magnetic expression is reduced to have only one unknown which is the depth.

- Using the least-squares minimization method a non-linear equation in one unknown only (z) is obtained and then solved by iteration technique and the depth z is determined for each N value.

- Procedures are also formulated to estimate the effective magnetization intensity and the effective magnetization inclination.

- Selecting the true model parameters is done by measuring the goodness of fit between the observed and the computed magnetic anomaly for each N value by evaluating the root –mean-square error.

- The method is applied to synthetic data with and without random errors and approved to be stable and gives very good results.

- The method has been applied to two field examples from Canada and Arizona. The estimated depths were found to be in good agreement with actual values.

Paper No. 5Paper No. 5

New methods for shape and depth determinations from SP data

by

E.M. Abdelrahman, H.M. El-Araby, A.G. Hassaneen, and M.A. Hafez

Published in Geophysics VOL. 68, No. 4 (July-August 2003)

Paper Overview - This paper presents two different methods for

depth and shape determination of a simple geometrical shaped body (namely, sphere, horizontal and vertical cylinder) from self potential anomalies.

- The first method uses the numerical horizontal second-,third-, and fourth-derivative anomalies obtained from observed SP data using filters of successive window lengths to determine simultaneously the depth and shape of the buried structure and the optimum polynomial order for representing the regional SP anomaly.

- For a fixed window length, the computed depths for different values of q are plotted against the shape factor representing continuous window curves.

- The solution of the depth and shape is read at the common intersection of the window curves for different window lengths.

- This is repeated three times for the second-, third -and fourth-derivative to determine (z2, q2) , (z3,q3) and (z4,q4) respectively.

- These estimated depths and shapes computed from different derivative anomalies can be used to determine the optimum order of the regional anomaly and the actual parameters as follows:

Case I: If z2 = z3 and q2 = q3

Then the regional is zero- or first-order polynomial.

The true parameters are z2 and q2.

Case II: If z2 ≠ z3 = z4 and q2 ≠ q3 = q4

Then the regional is a second-order polynomial.

The true parameters are z3 and q3.

- The procedure depends on the fact that the p-order derivative removes the effect of p-1 order (and less) regional anomaly.

- The second method is 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.

- The angle of polarization Ө and the electric dipole moment K can be eliminated from the SP anomaly expression using the anomaly value at the origin V(0) and the zero anomaly distance x0.

- Also the shape factor q is eliminated from the SP anomaly expression by introducing R(a), that is , the self potential at any point (a) on the x-axis other than zero.

- The unknown depth (Z) can be obtained by least-squares minimization applied on the logarithm of the normalized observed SP anomaly.

- Using the computed depth, the shape factor q can also be determined through another least-squares minimization process.

- Once zc and qc are known, the polarization angle and the electric dipole moment can be determined from the derived equations.

- Then the most appropriate source parameter solutions are determined using the measure of goodness of fit between the observed and the computed SP data for each (a) value.

- Finally, the method is tested using theoretical data with and without random noise and on a known field example from Germany. In all cases, the depth and shape solutions obtained are in good agreement with the actual ones.

Paper No. 6Paper No. 6

Quantitative interpretation of numerical horizontal Quantitative interpretation of numerical horizontal magnetic gradients over dipping dikesmagnetic gradients over dipping dikes

by

Hesham M. El-Araby

Published in Bull. Fac. Sci. Cairo Univ. VOL. 71(2003)

Paper Overview

- This paper utilizes the numerical horizontal magnetic gradients obtained from total intensity magnetic anomaly profile of dipping dike geological structures using filters of successive window lengths to determine the depth, width, dipping angle, and amplitude coefficient of the dike.

- By defining the horizontal gradient magnetic anomaly values at three points on the profile, a simple formula relating the depth to half-width of the dike is derived.

- For a fixed window length, the depth is determined iteratively using this simple formula for different half-width values.

- The computed depths are plotted against the half-width values representing a continuous window curve.

- The solution for the depth and half-width of the buried structure is read at the common intersection of the window curves.

- This method can be applied to true residuals as well as to short magnetic profile length, that includes linear regional effects and the results will not be affected.

- The method is applied to synthetic data with and without random errors and is approved to be fair and exact for the noise free data, while giving an acceptable range of errors for the case of noisy data.

- Moreover, when the error in profile origin position determination is studied, the method is approved to be not sensitive to it.

- Two field examples from Canada have also been analyzed and interpreted by the proposed method, and the results are in good agreement with the previously published results.

Paper No. 7Paper No. 7

A new method for complete interpretation of A new method for complete interpretation of self-potential anomaliesself-potential anomalies

by

Hesham M. El-Araby

Published in Journal of Applied Geophysics, VOL. 55 (2004)

Paper overview - This paper presents a least-squares approach

to determine the shape of a buried polarized body in a form of a sphere, horizontal cylinder, or vertical cylinder from a self-potential (SP) anomaly profile.

- By defining the anomaly value at three points on the profile, one at the origin and the others at any two symmetrical points around the origin, the problem of the shape-factor determination is transformed into the problem of finding a solution of a nonlinear equation.

- Procedures are also formulated to complete the quantitative interpretation by finding the depth, polarization angle, and the electric dipole moment.

- The validity of the method has been tested on synthetic data with and without random noise. The obtained parameters are in congruence with the model parameters when using noise free synthetic data. After adding ± 5% random error in the synthetic data, the maximum error in model parameters is less than ± 5%.

- The present method has the capability of avoiding noisy data points and enforcing the incorporation of points free from random errors to enhance the interpretation results.

- Further analysis showed that, the method is approved to be not sensitive to the error in profile origin position determination.

- Two field examples from Turkey have also been analyzed and interpreted by the proposed method, and an acceptable agreement has been noticed between the obtained results and other published results.

Paper No. 8Paper No. 8

An iterative least-squares minimization An iterative least-squares minimization approach to depth determination from approach to depth determination from

gravity anomaliesgravity anomalies

by

Hesham M. El-Araby

Published in Bull. Fac. Sci. Cairo Univ. VOL. 68(2000)

Paper overview

- In this paper a new least-squares approach is developed to determine the depth to a buried structure approximated by sphere, 2-D horizontal cylinder, vertical cylinder, or 2-D thin faulted layer from its residual gravity anomaly.

- By defining the anomaly value g(max) at the origin, and evaluating the logarithmic normalized form of the gravity expression, we eliminate all unknowns except z.

- The unknown z can be obtained by minimization in a least-squares sense and the solution of the derived nonlinear equation can be obtained by iteration methods. The amplitude coefficient A can then be determined.

- The method is applied to synthetic data with and without random errors and approved to be stable and give good results.

- The validity and accuracy of the method is tested on two field examples from Sweden and Senegal, the depth obtained compares well with other methods given in literature.

Paper No. 9Paper No. 9

A new method for shape and depth A new method for shape and depth determinations from gravity datadeterminations from gravity data

by

E.M. Abdelrahman, T.M. El-Araby, H. M. El-Araby, and E.R. Abo-Ezz

Published in Geophysics VOL. 66, NO.6 (November-December 2001)

Paper overview

- This method determines simultaneously the shape and depth of a buried structure represented by sphere, horizontal cylinder, or vertical cylinder from gravity anomalies using filters of successive window lengths.

- For a fixed window length, the depth is determined for each shape factor, and then 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.

- 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.