| The magnetotelluric (MT) method was created in the 1950s. The last more than 50 years have seen its great progresses in theoretical research, data processing, interpretation, and applications. Like other geophysical methods, forward problems are the theoretical basis of MT, and inversion problems lie in the heart of MT data interpretation. At present, there are still some issues of MT forward problems which remain to be solved. This thesis focuses on the calculation of the auxiliary field in 2D MT forward modelling. The purpose is to raise the precision of forward calculation and provide MT inversion with more sufficient supports.First, this thesis reviews previous studies on MT forward problems. Since the MT method appeared, its forward problems have attracted much attention, yielding rapid development. Based on these results, a great number of inversion algorithms have been created for data interpretation. Currently although MT inversion is turning toward 3D problems, 2D inversion is still the primary tool in data interpretation. And both the speed and precision of 2D inversion depend on 2D forward solutions. It is found that when processing complex problems, the speed and precision of MT 2D forward calculation are sometimes not high enough for the requirement of the inversion problems. To solve this issue, a lot of efforts have been made, which use more times of interpolation of the element shape function. These studies focus, however, their attention on enhancing the calculation precision of the principal field, while little concerns on another important factor related the forward precision, i.e. the auxiliary filed.In 2D MT forward modelling, the auxiliary field is determined by calculating the vertical partial derivative of the principal field near the surface. For the calculation of the principal filed, the linear interpolation finite element method (FEM) is usually used, with little application of quadratic interpolation FEM due to its own problem. If making direct derivative calculation to the basic function of linear interpolation, the value of the auxiliary filed is a constant in the element, yielding a not high precision of the result. Some researchers have studied the issue how to improve the calculation precision of the auxiliary field.This thesis has analyzed the effect of the auxiliary field on the calculation precision. It suggests to construct a quadratic interpolation shape function in the surface grid for calculation of the auxiliary field, when employing the current available method of linear interpolation for calculating the principal field. Then the vertical partial derivative of this shape function is calculated to raise the precision of the auxiliary field, so that the precision of the final forward result can be improved. But constructing the quadratic interpolation shape function requires known filed values of more nodes of the model as well as higher precision. In this work, the direct iteration FEM (DIFEM) is utilized to solve the calculations of interpolation nodes.Essentially, DIFEM is the FEM based on the variational method. As the classical FEM has some disadvantages, such as complicated programming for formation, storage, and solution of stiff matrixes, large amount of storage, and slow calculation, the variational method uses the explicit iteration scheme similar to the finite difference method (FDM), so avoiding stiff matrix. In this way, some mature iteration algorithms of FDM can be used in calculation, making amount of storage of FEM decrease greatly, and more practicable for real problems. In the DIFEM, by the direct iterative scheme, the known values of the principal field at a few nodes are used to calculate the values of the principal field of more nodes, and then the quadratic interpolation shape functions are determined. As the FEM iteration is a special interpolation way based on the variational principle, it ensures the field values in the neighbourhoods of solved nodes to meet the conditions of controlling differential equations, making the field values of other nodes from interpolation have the same precision of the principal field. This is also attested by the error analysis.This study proposes an algorithm of quadratic interpolation for calculation of the auxiliary field. Its basic idea is to use the surface grid and principal field values of a few nodes, by subdivision of the grid and successive solution of DIFEM, and DIFEM calculation based on the variational principle, to obtain the principal field values of more nodes for constructing the quadratic shape function. Then quadratic interpolation basic functions are determined by these known principal field values and those interpolated, and derivative calculations are made to these basic functions. Finally, the auxiliary field is computed at a higher precision. On the basis of this theoretical analysis, programs for linear interpolation and quadratic interpolation are compiled. In conjunction with available DIFEM 2D MT forward modelling, this new algorithm is tested through model calculation. Comparing with the 2D forward modelling of the auxiliary field by the linear interpolation, the quadratic interpolation does not increase much amount of computation, and raises the precision of MD forward calculation.Besides, this work applies this new algorithm to 2D forward calculation of a uniform half space model. The quantitative result shows that using the quadratic interpolation can improve the ability of the forward result to fit the grid. When using the linear interpolation to calculate the auxiliary field, the cell size of the surface grid is confined to 1/3 of the skin effect depth, otherwise the precision of the forward modelling will be seriously influenced. With the new quadratic interpolation, for the same request of precision, the element size can be enlarged to one skin effect depth. On the basis of this result, the current design criterion for 2D forward grid has been improved. When the topography is complicated and linear interpolation is used for MT forward modelling, the element size of the grid must be very small to ensure the precision. But such fine grid division makes the number of grid elements grow dramatically, and expands the inversion space, not favourable for the implementation of 2D inversion. The quadratic interpolation can solve this issue to some extent. Using the improved forward model grid, even if the model is involved in complicated topography, it does not lower the precision of forward calculation while the number of grid elements is not increased much, so that the computation time is saved greatly. In this thesis, a simple model with topography is designed for forward calculation, in which both the linear and quadratic interpolations are utilized. The NLCG 2D inversion is performed to this model to achieve the final model. It demonstrates the advantage of the quadratic interpolation in treatment of complicated topography.
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