وارون‌سازی نامقید غیرخطی داده‌های مغناطیسی با استفاده از الگوریتم زیرمسئله ناحیه اعتماد

مقاله حاضر به وارون‌سازی نامقید غیرخطی داده های مغناطیسی به کمک الگوریتم منطقه مورد اعتماد می پردازد. روش های معمول وارون‌سازی مبتنی بر ماتریس تغییرات و ماتریس هسین، در صورت انتخاب نامناسب مقدار اولیه، به کمینه های محلی و درنتیجه، تخمین مدلی دور از واقعیت زمین منجر می شوند. برای غلبه بر این مشکل، روش منطقه مورد اعتماد پیشنهاد می شود که خواص همگرایی بسیار مناسبی دارد. در این الگوریتم، اطلاعات گردآوری شده درباره تابع هدف، برای ساختن یک مدل ساده تر از آن استفاده می شود. در این روش، محدود کردن مدل به ناحیه اطراف نقطه کنونی، سبب می شود مدل در آن ناحیه، رفتاری شبیه به تابع هدف داشته باشد؛ ازاین‌رو به جای کمینه سازی تابع هدف می توان مدل را در یک ناحیه کمینه کرد. در این مقاله، الگوریتم منطقه مورد اعتماد با روش لونبرگ مارکوارت مقایسه می شود که از یک الگوریتم جستجوی خطی بهره می برد. برای بهینه سازی توابع استاندارد ریاضی  که علاوه بر کمینه سراسری، کمینه های محلی نیز دارند از الگوریتم مذکور استفاده شده است تا توانایی آن، بررسی و نتایج آن با روش جستجوی خطی نیوتن مقایسه شود. همچنین الگوریتم پیشنهادی برای وارون سازی داده های مصنوعی و واقعی بی هنجاری مغناطیسی با اشکال هندسی گسل، استوانه و صفحه نازک به‌کار گرفته شده و نتایج آن با نتایج وارون سازی روش لونبرگ مارکوارت مقایسه شده است. در هر دو حالت، مدل‌سازی عددی، کارایی و برتری این روش را نسبت به روش لونبرگ مارکوارت به ترتیب در بهینه سازی و وارون سازی نشان می دهد.

Nonlinear Unconstrained Inversion of Geomagnetic Data Using SubProblem TrustRegion Algorithm

This paper presents the application of the trustregion algorithm to an unconstrained nonlinear 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 gradientbased methods. The global search methods approach a global optimum while the gradientbased methods approach a local optimum point. However, the cost of global search method is higher with a huge number of the model parameters. In gradientbased 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 trustregion 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 trustregion 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 trustregion algorithm has proved to be more efficient and has better convergent properties especially for illposed inverse problems in comparison to the line search strategy. It has a potential to cover the global minimum under certain conditions. In trustregion 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 trustregion method only trusts the approximate model in a region near the current iterate. This is reasonable, because for general nonlinear functions, local approximate model scan only fits the original function locally. The region that the approximate model is trusted is called the trustregion. In the line search method, the search direction is first determined followed by the step length, while in the trustregion algorithm the step length is limited to the radius of trustregion followed by the determination of the search direction. Due to its strong convergence properties and robustness, trustregion methods have been studied in many disciplines.     Similar to every geophysical approach, interpretation of geomagnetic data is along with nonuniqueness 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 trustregion algorithm to solve nonlinear inverse potential field problems stemmed from simpleshape anomalies. First, an algorithm that adjoins the trustregion 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 trustregion method and those of the LevenbergMarquardt, we can discover that the proposed strategy outperforms the LevenbergMarquardt algorithm in terms of the rate of convergence and accuracy. The paper is structured as follow: In section 2, the theory of the trustregion algorithm as well as an algorithm that links the trustregion algorithm to leastsquares problems is presented. Section 3 consists of the verification of the performance of the trustregion algorithm in minimization of two standard mathematical benchmarks and synthetic and real magnetic data derived from simpleshape anomalies such as thin sheet, horizontal cylinder, and faults. Section 4 summarizes the results of this study.