Nonlinear Unconstrained Inversion of Geomagnetic Data Using Sub-Problem Trust-Region Algorithm

This paper presents the application of the trust-region algorithm to an unconstrained non-linear inversion of magnetic data. Inversion of geophysical data aims at getting physical attributes or model parameters. Geophysical inverse problems can be generally solved using two approaches: global search methods and gradient-based methods. The global search methods approach a global optimum while the gradient-based methods approach a local optimum point. However, the cost of global search method is higher with a huge number of the model parameters. In gradient-based methods, which are based on Jacobian and Hessian matrix, choosing improper initial values may lead to local minimums, and consequently, estimation of a subsurface model which may be far from the reality of the earth. To overcome this issue, the trust-region algorithm is proposed. The proposed method possesses a high convergence rate and it crosses local minimums and its numerical computations are much less compared to global algorithms. To catch the new iteration in nonlinear problems, the line search and trust-region strategies are utilized. Both of these strategies usually employ the quadratic structure of the objective function and the Taylor expansion. They control descent condition with the step length and the search direction. The trust-region algorithm has proved to be more efficient and has better convergent properties especially for ill-posed inverse problems in comparison to the line search strategy. It has a potential to cover the global minimum under certain conditions. In trust-region method, an approximate model is constructed near the current iteration and the solution of the approximation model is taken as the next iteration. Compared to the line search algorithm, the trust-region method only trusts the approximate model in a region near the current iterate. This is reasonable, because for general non-linear functions, local approximate model scan only fits the original function locally. The region that the approximate model is trusted is called the trust-region. In the line search method, the search direction is first determined followed by the step length, while in the trust-region algorithm the step length is limited to the radius of trust-region followed by the determination of the search direction. Due to its strong convergence properties and robustness, trust-region methods have been studied in many disciplines.Similar to every geophysical approach, interpretation of geomagnetic data is along with non-uniqueness and identical magnetic responses could be produced by different geometric shapes, thus the existence of initial information based on geological data is mandatory to achieve the model near to the reality of earth. The assumption of stable magnetic susceptibility makes it possible to achieve nonlinear equations of simple shapes. In this paper, we used the trust-region algorithm to solve nonlinear inverse potential field problems stemmed from simple-shape anomalies. First, an algorithm that adjoins the trust-region method to least squares problems is presented. Then, to verify the efficiency of the proposed methods, some synthetic and real numerical experiments are provided. Comparing the results derived from the trust-region method and those of the Levenberg-Marquardt, we can discover that the proposed strategy outperforms the Levenberg-Marquardt algorithm in terms of the rate of convergence and accuracy. The paper is structured as follow: In section 2, the theory of the trust-region algorithm as well as an algorithm that links the trust-region algorithm to least-squares problems is presented. Section 3 consists of the verification of the performance of the trust-region algorithm in minimization of two standard mathematical benchmarks and synthetic and real magnetic data derived from simple-shape anomalies such as thin sheet, horizontal cylinder, and faults. Section 4 summarizes the results of this study.