Research On 3-D Controlled Source Electromagnetic Method Inversion Using Nonlinear Conjugate Gradients

Three dimensional inversion is the most effective method in data processing of electromagnetic method, which can get the correct geoelectric structure avoiding many problems in one dimensional inversion and two dimensional inversion. Many geophysics pursuit of full three dimensional inversion, but it's complexity make it still under development. Especially in controlled source electromagnetic method, the study is stay in theory abroad. Domestic three dimensional controlled source electromagnetic research is in forward modeling, and for such situation, this paper make use of staggered grid finite difference to achieve forward modeling, and nonlinear conjugate gradient method for three dimensional inversion.Firstly, this paper introduces a method to generalize the description of Green's functions from various basic sources buried in stratified earths. The key of the theory is based on calculating the vector potential's amplitude coefficient on the virtual interface passing through the source and paralleling to the other interfaces. In order to get the generalized expression, we firstly separate Green's function into TE mode and TM mode, then a virtual interface parallel to layer interfaces is inserted through source. The continuity of the tangential electric field and the discontinuity of the tangential magnetic field on the virtual interface are then utilized to obtain the initial amplitude coefficient on the interface. Lastly, the amplitude coefficients in each layer will be recurred from the initial amplitudes using the propagation theory of TE mode and TM mode. The obtained initial amplitudes of vector potential on the virtual interface can be decomposed into the amplitude coefficient of whole space Green's function (source intensity) and the impact factor from model parameters; and for different source, to get the corresponding Green's function, only the amplitude coefficient of whole space is necessary to be changed. The methodology unifies Green's functions in electromagnetic exploration, simplifies the computation and gets better numerical stability. On this basis, this paper discusses how to applying the two-level approximate discrete complex image method to fast and accurately compute the low frequency Green's functions. Through numerical simulation, the approximate parameters in the compution of low frequency electromagnetc field is determined as follows:(1) Sampling number N of spectral green's function in the two-approximate integral interval are ten times of the number of approximate polynomial M; (2) The integral interval is about 40/r, where r is the offset; (3) The conjunction value L1 of the two level integral interval is Q times of the ratio of the total size of the integral interval L2 and the sampling number N1 of the second integral interval; (4) the optimal value of Q is decided by the cross point of the spectral Green's function approximate error plot about Q and the horizontal axis. The application of this principle to the calculation of the electromagnetic fields from a finite bipolar source shows that the calculated apparent resistivity agree with direct integral result very well, this reveals that the the proposed strategy is feasible.Then, this paper use finite difference method based on the secondary field to achieve three dimensional modeling of controlled source electromagnetic. This paper analysis the secondary field and total field's morphological characteristics of low resistivity abnormal body and high resistivity abnormal with different frequences, and summed as:for low resistivity abnormal body, the total electric field Ex become small just above it, and on both sides of the abnormal body along the x-axis, the total electric field Ex become large; for the high resistivity abnormal body, its law to the contray. From the result of many abnormal bodys, it can be seen that the low resistivity abnormal body is easer to be distinguished then the high resistivity abnormal body, which can be a reference when select background resistivity in inversion.Based on the staggered grid finite difference modeling, we use nonlinear conjugate gradient method to solve the problem of three dimensional inversion. When in the inversion calculation, nonlinear conjugate gradient method doesn't need to compute the sensitivity matrix explicitly, but only to calculate the product of matrix and vector, which just one forward modeling and one adjoint forward modeling. Although, another 3 or 4 forward problems should be done to evaluate the optimal step lenth, thus, there are totally about 5 time forward modeling in each iteration, the whole computation timen is much less than directly sensitivity matrix computation method. The inversion results of simple abnormal body show that nonlinear conjugate gradient method can get the correct position,size and resistivity of abnormal body. This paper also discusse the layered abnormal body and the abnormal body outside the surveyed area. For the layered abnormal body, we can only get the geoelectric structure in the surveyed area; for the little abnormal body outside, inversion give the right position of the abnormal body, and doesn't affect the inversional resistivity in the surveyed area. This shows that three dimensional inversion eliminate the shadow effect; field data inversion shows that three dimensional inversion give the same results as two dimensional inversion, which prove the algorithm of three dimensional nonlinear conjugate gradient method in this paper is reliable.