Quasi-Newton Method For Two-Dimensional Magnetotelluric Inversion

Inversion is a key step in the magnetotelluric (MT) data processing and interpretation. MT inversion techniques have been transformed from1D into2D, or even3D currently. Several3D MT inversion algorithms have been developed by some authors in recent years, most of which have succeeded in inversion of synthetic models. However, the performances of these methods are still questionable when they are used to deal with field datasets. Due to the complexity of the real electrical structure, the nonuniqueness of the inverse problem will be tremendously magnified when conducting3D inversion of real data, and more iterations will be required to obtain reasonable results, which would certainly lead to unacceptable long running time. With the rapid growth of computers as well as more applications of new technologies such as parallel computation, the long computing time of3D inversion will no doubt be avoided. Nevertheless, it is necessary for geophysicists to exploit new numerical algorithms, since there are still many excellent mathematical algorithms that have not been applied to geophysical problems, and perhaps some improvements will be gained from them. On the other hand, the existing methods may fail as the target areas become increasingly complicated, and compare the inversion results of different algorithms with each other is an efficient way to reduce nonuniqueness.Since they are descended from Newton’s methods, Quasi-Newton methods (QN) still attain a good convergence rate like Newton’s methods, but they do not require the computation of the second derivative of the objective function, which is referred to as Hessian. Particularly, the limited-memory Quasi-Newton method (LMQN) is considered as one of two most suitable approaches for large-scale optimization (the other is nonlinear conjugate gradient method), and it has good prospects for application in geophysical inversion.In this paper, we attempt to apply QN strategies to MT2D inversion, and this study is the foundation of using QN method to solve3D MT inversion problems. This paper is organized as follows:First, we review the present situation of MT numerical modeling and inversion. Advantages and disadvantages of the three most widely-used approaches, finite-di(?)erence, finite-element and integral equation methods are compared with each other, and some new technologies are mentioned. Future researches on MT inversion of next period of time are considered. Second, we present a2D staggered grid finite difference method for MT modeling, which is actually a simple2D version of the well-known3D staggered grid FD method. In contrast to traditional FD, the electric and magnetic fields are sampled in a different way. Difference equations for two modes are derived from the integral forms of Maxwell’s equations. This algorithm is tested on several theoretical models. In comparison with analytic solutions and numerical solutions obtained from the well-known2D MT modeling code PW2DI, its solutions are equivalently accurate.In the third part, we briefly introduce the basic idea of MT inversion, which is known as regularization theory. It then makes a mathematical classification of several main inversion methods according to the numerical optimization algorithms they employed so that the clear relationship among them can be acquired. In addition, their main merits and drawbacks are compared with each other.A brief introduction to the theory of Quasi-Newton method is given in part4. Then we exhaustively show how to calculate sensitivity via solving quasi-forward problems, including the computation of the full sensitivity and the product of it with any vector, which are significant issues in linearized, iterated methods that make use of the gradient of the objective function. To choose regularization parameters, a gradually-descend manner is regarded as simple and reasonable, and suitable for this QN algorithm. We give the whole process of our inversion algorithm.In section5, by doing some numerical experiments on synthetic models, we discuss the regularization parameter’s impact on the inversion results and demonstrate the efficiency of our inversion algorithm applied to synthetic data. In addition, we apply our algorithm to the inversion of a real AMT dataset of a surveying line in Xinyuan, Xinjiang Province. The inverse model is compared with that obtained from commercial software SCS2D of Zonge. The large-scale features of the two models coincide with each other, which implies that our algorithm is practical for inverting field data.Finally, we summarize the achievement and drawbacks of our research. The potential improvements are brought out.