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.