| DC resistivity method is one of the oldest geophysical methods, and has been widely used in the field of geophysical exploration. There are so many achievements in2.5-dimensional and three-dimensional forward modeling and inversion problem for DC resistivity method. In this paper, a new set of wave number has been calculated in the condition of large pole distance for2.5-D forward problem, and then discussed some problem in2.5-D forward process in detail. For the simulation of three-dimensional electric field, hexahedron has been split into five tetrahedrons, in order to achieve the simulation of the undulating terrain.Firstly, the2.5-D variational equation of the electric field of a point source has been deduced. And then use the rectangle grid within the subdivision triangle mesh to adapt the flat or topographic conditions. Next mainly given the process of2.5-D optimization method to select the wave number, and calculate two sets of wave numbers under the condition of the polar distance is extend to1000m, respectively, contain7and9numbers, and the relative error of these two set of wave number is under0.1%in fourier inverse transformation. Then use the finite element method analog uniform earth, and layered earth geoelectric section sounding curves with the polar distance less than1000m. Compared the analytical solution with the one by using the wave number given in the article, the relative error is less than3%, and with the one by using the wave number given by scholars before, the result is deviate from the analytical solution when the polar distance is over200m, and the relative error is30%at most. In the meanwhile, the accuracy of the result of7wave numbers and the result of9wave numbers are almost the same, the relative error between the two is less than1%while modeling layered earth, so while modeling, the7wave numbers is recommended. Then use the three-dimensional finite element simulation method mentioned in this paper to simulate two-dimensional model, compared to the results under different wave numbers, the simulation results verify the correctness of the wave numbers calculated in this paper again.And then discuss in detail the2.5-D simulation acceleration method:(1) using the homogeneous boundary conditions, while modeling different source point, we need to do matrix factorization once, and then solve the equation, get the answer, this would save a lot of time. Example analyzed to verify the effectiveness of this method.(2) use reciprocity and principle of superposition for the electric field, in order to reduce the times of forward problem we need to solve, and then contrast the result of the array used in fact and the result of using reciprocity and superposition principle and analytical solutions, then found that the three simulation method’s error is less than3%, so the reciprocity and the superposition principle can be used to reduce the times of forward modeling. In this article, superposition principle is used, because the use of this method is relatively simple to understand. And for a variety of devices, the forward simulation times and the number of electrodes is consistent.For two-dimensional polarization forward problem, respectively, to achieve the equivalent resistivity method, as well as the polarization is calculated according to Seigel theory. Mainly analyzes the process of calculating the partial derivatives of the apparent resistivity to the resistivity of underground grid, and gives the solving process of two-dimensional partial derivative matrix. Respectively, using the equivalent resistivity method, as well as the Seigel theory to simulate two-dimensional model, comparing the two methods under different device simulation results verify the correctness of these two methods.In the article, an improvement has being made in the process of using Cholesky decomposition to solve the sparse equation. According to the symmetry and sparse of the matrix after decompose, using a method of eliminate column first for the process of back substitution which take up more time throughout the total process of solution. In order to verify the availability of the proposed method, set the finite element mesh as124×20, calculated100times, the solution speed use this method is five times faster than the traditional one. And the more units, the more obvious.For the issue of three-dimensional point source, the hexahedron has being used in the overall subdivision, so to facilitate the implementation of the program. And hexahedron has been split into five tetrahedrons, in order to achieve the simulation of the undulating terrain and tilt body, this split compare to the split of one hexahedron into six tetrahedrons, there is one less unit, of course, improve the speed. Then given the unit matrix calculation method when use the subdivision mentioned before. By analog the layered earth and2-dimensional mountain terrain, compared to the analytical solution and experimental results in soil tank to verify the correctness of this three-dimensional finite element.In the last part of the article, compare the result of using three-dimensional finite element simulate three-dimensional exception with the one of using two-dimensional finite element method to approximate the same model. The abnormal appearance of different profile and the graph appearance is different, and the error between this two kind of simulate is up to20%. Then use the three-dimensional program to simulate three-dimensional valley terrain, compared to the response of the two-dimensional program to simulate the approximate two-dimensional terrain in different devices. When use pole pole array, three-dimensional and two-dimensional results is the same, and be able to demonstrate the appearance of the valley, when use three-pole device, dipole and symmetrical quadrupole device, two-dimensional response of the device and the three-dimensional response, although with the same appearance, but the value is deviate from each other, especially in the slope and the valley, the result of two-dimensional analog appear violently transition. Finally, the paper analyzes the simulation of two-dimensional terrain, using asymmetric mesh, the simulation results will produce a certain error, it is recommended that symmetrical mesh used in the simulation. |
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