2D Magnetotelluric Modeling And Inversion With Topography Using Adaptive Unstructured Finite Element

Magnetotelluric sounding (MT) is a frequency domain electromagnetic sounding method, which is usually used for the research of deep geological structure through observing the natural electromagnetic field with a wide frequency range. As one of the synthetical geological and geophysical method, MT has a developing history for about 50 years in China. It can be used to detect the material structure of crust and upper mantle, oil finding,gas and coal exploration. In addition to this, it also plays a significant role in solving engineering and environmental geological problems, the result turns out to be efficient.Magnetotelluric method is usually carried out on the topographic regions, so the effect of terrian is an unavoidable problem to geophysicists. The distribution of MT receivers are restricted by many factors especially in chinese mountanious areas. Most of our receivers just can be placed in hollows or somewherer like this, the terrian effect is more complicated when the mountain is irregular.Generally, there are three ways to do the geophysical modeling:analytical solution, laboratory simulation and numerical simulation. With the rapid development of computer technology, numerical simulation stands out from the three methods. It depends on partial differential equations and bounds condition to get the electromagnetic field. Althouth it is a kind of approximate solution, it suits complicated material distribution and topography. So it is widely used and becomes the main way for geophysical forward modeling. In our paper, we adopt the finite element to complete our research.Structured grid such as rectangle cells or triangle cells generated from rectangle cell using dianogal has its limit simulating complicated model, The regular cell edges are usually oriented horizontally and vertically, sloping and highly variable structures are usually approximated using a series of discrete jumps or stair steps in the model parameter values. Although these can be made sufficiently small to ensure accuracy, the resulting grid may have cells with very large aspect ratios, which are well known to seriously degrade the convergence of the Krylov solvers commonly used for the numerical linear system. Additionally, fine discretization in the model center extends both vertically and horizontally to the grid edges, greatly increasing the number of model cells and hence increasing the computer memory and run-time requirements.For the solusion of above problems, our paper introduce unstructured triangular grids which permit efficient discretization of complex modelling such as those containing topography, dipping layers and multiple scale structures etc. For the accuracy of model forwarding, we present posterior error method also called Double Weighted Residual (DWR)to estimate the difference between the finife element and true solution, if the error is huge we will refine the mesh.Meanwhile, parallel computing is introduced to short the run-time. Parallel technique is a kind of algorithm that can conduct mulriple orders at the same time. It aims to improving computing speed and solving huge and complicated problems through expand the question scale. Parallel technique can be divide into two type, one is parallel in time and the other one is parallel in space. Parallel in time refers to pipelining, while parallel in space refers to using multiprocessing. Our paper adopt parallel in space, We divide receivers into different groups, every group is arranged with one processor and start calculating at the same time. On the other hand, we also put the nearby frequences into a group, so a group of frenquence are the same refinement mesh not every frequence has one.Based on the above theory, we adopt adaptive finite element to study the MT response on sloping surface. Starting from the Maxwell equations, we derive the wave equation also called Helmholtz.Then the outer and inner boundary condition are discussed and TE and TM mode get their definition in differential form.We don’t solve the differential equation directly, instead we decompose the electromagnetic fields into primary and secondary components. Primary filed can be easily get from the homogeneous half-space analytic solution. The primary advantage to this approach lies in the fact that the FE grid only needs to accurately capture the secondary field variations, which tend to be more localized than total field variations; thus, a grid with fewer elements can often be used. The total fields are found by summing primary and secondary fileds. Finaly the apparent resistivity and phase is easily got.We built topographic model and flat model, caculating their forward respons respectively under the same situation. The results are compared and analyzed to find the terrian effect on MT exploration. We combine the adaptive finite element with OCCAM algorithm which is widely used since been proposed, the adptive finite element calculates when the model matrix updatas in the iteration process. The essence of OCCAM inversion is least squares with smooth constrain, the most differenc it has from others is that OCCAM will use different regularization parameter in every model updata process, while others will not they use the fixed regularization parameter from the star to the end. The fitting error is usually big in the early iterations, so OCCAM choose the model with smallest fitting error, in the later iterations, there existing several regularization parameter satisfy the fitting error, so we choose the smallest norm model. This inversion strategy, I got to say, makes OCCAM inversion unprecedented stable, it often get convergence in a few iterations.Before the inversion, we will discuss how to do the static shifs under terrian condition. Static shifts means offset of MT apparent resistivity caused by surface current disturbance. Heterogeneous electrical property around receivers and structure and topography such factors will arouse in surface current disturbance. so there is a guess that if we conduct inversion with topography, maybe we don’t need static shifts ever. We will discuss this in our paper.Four models, metal ore, thin layer metal ore, fold and fault, have been built to see how the adaptive finite element OCCAM works. First, we do the forward and get the respond data adding some random noise. Then we use the synthetic data to do the inversion. The result will be compared and analyzed. At last, the application of practical data from the Southwest of China, shows that adaptive unstructured finite element method is an available tool to analyze MT data of complex features such as steep surface topography.