A Constrained Nonlinear Inversion Approach to Quantitative Interpretation of Self-potential Anomalies Caused by Cylinders, Spheres and Sheet-like Structures

An interpretative method based on a nonlinearly mathematical optimization concept has been developed in this paper, in order to interpret self-potential anomalies (SP) due to horizontal cylinder, vertical cylinder, sphere and sheet-like structures. This interpretative method comprises three main steps. The first step is to formulate mathematically a nonlinearly constrained minimization problem (NCMP) to describe the geophysical problem related to the studied structure. The second one is to suggest an interior penalty function in order to convert the nonlinearly constrained minimization problem (NCMP) into a nonlinearly unconstrained minimization one (NUMP). The third step is to solve the converted nonlinearly unconstrained minimization problem (NUMP) by the well-known Hooke and Jeeves direct search algorithm in order to estimate the geophysical parameters of the studied structure, i.e., depth, polarization angle, electric dipole moment (magnitude of polarization) and geometric shape factor. The Hooke and Jeeves direct search algorithm is purposely chosen for being robust and its application to SP data allows a rapid convergence towards the optimal estimate of parameters. This interpretative method was first tested on theoretical synthetic models with different random noise, where a very close agreement was obtained between assumed and evaluated parameters. The validity of the proposed interpretative method is also tested on practical field examples taken from Turkey, India and Germany, where available SP data existed and was previously analyzed by different interpretative methods. The agreement between the results obtained by the developed method and those obtained by other published methods is good.