Nonlinearly Constrained Optimization Theory to Interpret Magnetic Anomalies Due to Vertical Faults and Thin Dikes

Abstract

A new and simple method based on a nonlinearly mathematical optimization concept has been proposed in this research to interpret magnetic anomalies due to vertical faults and thin dikes. This proposed interpretative method consists of three main steps. The first step is to formulate nonlinearly constrained optimization problems to describe the geophysical problems related to the studied structures. The second step is to suggest an interior penalty function in order to convert these nonlinearly constrained optimization problems into nonlinearly unconstrained optimization ones. The third step is to solve the converted nonlinearly unconstrained optimization problems by using the famous Hooke and Jeeves's algorithm in order to estimate the geophysical parameters of the studied structures such as: depth, amplitude coefficient, and index parameter. The Hooke and Jeeves's algorithm is purposely chosen for being robust and also its application to magnetic data converges rapidly towards the optimal estimation of parameters. This method was first tested on theoretical models with different random noise, where a very close agreement was obtained between the assumed and evaluated parameters.The validity of this new method was also tested on practical field examples taken from Australia, India, United States, and Brazil, where available magnetic data existed and was previously analyzed by different interpretative methods. The agreement between the results obtained by our developed method and those obtained by the other geophysical methods is good. The advantages of this newly proposed method, compared with the other published interpretative methods, also have been discussed and demonstrated.

A Robust Nonlinear Inversion for the Interpretation of Magnetic Anomalies Caused by Faults, Thin Dikes and Spheres Like Structure Using Stochastic Algorithms

“…The comparison between observed and theoretical computed anomalies clearly indicates the close agreement between them. Table 4 shows a comparison between the results of our interpretation method and those obtained by GAY (1963), ABDELRAHMAN (1997), SHARAFELDIN (1996), andTLAS (2004). The p-value found for 95% confidence level is 0.91.…”

“…The same anomaly is also analyzed and interpreted according to the method of ABDELRAHMAN and SHARAFELDIN (1996), ABDELRAHMAN and HASSAN-EIN (2000), SILVA (1989), and ASFAHANI and TLAS (2004). The results of these different methods are comparable as shown in Table 5.…”

“…A close agreement is noticed between the observed and the computed anomaly, which proves the efficiency of the proposed method. The anomaly is also analyzed and interpreted according to the method of QURESHI and NALAYE (1978), the method of ATCHUTA RAO and RAM BABU (1983) and the method of ASFAHANI and TLAS (2004). The results of these methods are comparable as shown in Table 3.…”

“…Recently, ASFAHANI and TLAS (2004) developed an interpretative method based on the nonlinearly constrained least-squares minimization for interpreting magnetic anomalies due to faults and thin dike-like structures. This method is to minimize an objective function, taken as the squared Euclidean distance between magnetic field data and synthetic values, on an unbounded subset contained in the space of parameters.…”

Global nonlinear optimization for the interpretation of source parameters from total gradient of gravity and magnetic anomalies caused by thin dyke

“…This anomaly was interpreted by several authors (Tlas and Asfahani, 2015, Abdelrahman and Essa, 2015, Asfahani and Tlas, 2007, Asfahani and Tlas, 2004, Abdelrahman and Sharafeldin, 1996, Gay, 1963 assuming a thin dyke model. The anomaly is interpreted using VFSA to obtain the different parameters.…”