Abstract
In this research, a new numerical method, called the hybrid finite difference–finite element (hybrid FD–FE) method, is developed to solve 2-D magnetotelluric modeling by taking advantage of both the finite difference (FD) and finite element (FE) methods. With the hybrid FD–FE method, the model is first discretized as rectangular blocks and separated into two zones: the FD and FE zones. The FD zone is set for the subregions where topography or bathymetry does not appear. The FD approximation, which is fast, accurate and requires less memory resources, is then applied. For the FE zones where topography or bathymetry exists, the rectangular blocks are transformed into quadrilateral elements to handle the topography or bathymetry appropriately. Then, the FE approximation with quadrilateral elements, which is more accurate for topography or bathymetry zones, is applied. The system of equations for the hybrid FD–FE method is then formed according to the FD and FE schemes. The obtained system is a combination of the FD and FE equations. Three numerical methods are applied to test models with and without topography and bathymetry. The accuracy and efficiency in terms of errors, computational time and memory storage are presented, compared and discussed. The numerical experiments indicate that the FD scheme has a shorter computational time than the other schemes when modeling without topography and retains accuracy equivalent to that of the FE method, whereas FE is more practical when modeling with topography and bathymetry. However, our proposed hybrid FD–FE method is efficient in both situations. Without topography or bathymetry, its efficiency and accuracy approach those of the FD scheme. With topography and bathymetry, the hybrid FD–FE method is as accurate as FE, but its speed is slightly slower than that of FD. In terms of memory storage, the hybrid FD–FE method consumes slightly more storage than the FD method. This hybrid FD–FE method can be further extended and implemented for 3-D magnetotelluric modeling for more efficient computation.
Highlights
Topography or bathymetry zones may be encountered during magnetotelluric surveys
We present a hybrid finite difference– finite element method to incorporate topography and bathymetry for 2-D magnetotelluric modeling
Note that the finite element (FE) with fine mesh is excluded in this comparison. These results indicate that the hybrid method is more powerful than the FE and finite difference (FD) methods when topography exists in the model
Summary
Topography or bathymetry zones may be encountered during magnetotelluric surveys. These zones can affect the apparent resistivity and phase obtained from two-dimensional magnetotelluric (MT) surveys. The disadvantages of FE are greater consumption of memory storage and longer computation time Combining these two methods to exploit their advantages for solving magnetotelluric modeling is of interest and a challenge. Applying the Dirichlet boundary conditions obtained by solving the 1-D problem, rearranging and grouping all coefficients in (5) together, the obtained system of equations is given by. Quadrilateral element‐based finite element method In addition to the FD approach, the FE scheme is a powerful numerical method that is used to approximate the solution of boundary-valued problems of partial differential equations. The approximation of field φby using the four neighbor nodes is demonstrated (right bottom) applying Dirichlet boundary conditions and assembling coefficients and unknowns, the obtained system of equations is given by. The impedances, apparent resistivity and phases are computed at each site
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