Studies Of Multigrid Algorithm For 3D Geo-Electromagnetic Modeling And Its Applications

Geo-electromagnetic methods are widely used and play an important role in engineering geological survey, mining exploration and detection of the deep structure of earth's crust and mantle. With the growing complexity of the exploration targets, the requirements for investigation of 3D fine structure become more and more urgent, which greatly challenges the efficiency of 3D geophysical numerical modeling for complex underground structures. However, the convergence rates of standard iteration solvers (such as incomplete cholesky conjugate gradient, abbr. ICCG) for forward modeling will be severely affected by large scale of non-uniform grids, big conductivity contrast, and complex electrical structure and so on, since the condition number of the coefficient matrix becomes worse. Multigrid method is of high numerical efficiency in solving linear equations arisen from boundary value problem of partial differential equation (PDE), which is newly developed by the computational mathematicians in the past decades, but seldom used in 3D geo-electromagnetic modeling so far.First, the geometric multigrid (GMG) method is used to solve linear equation systems arisen from 3D Poisson equation and 3D Helmholtz equation, and compared to ICCG method that widely used in 3D geo-electromagnetic modeling at present. The convergence rate of GMG shows independency of the number of unknowns, while ICCG is apparently slow down as the number of unknowns growing. Moreover, GMG algorithm is much faster than ICCG algorithm in the same conditions, showing its superior efficiency.However, the GMG method mentioned above has difficulties when applied to 3D DC resistivity modeling. The modeling of DC resistivity method with point source is an open boundary problem, with highly non-uniform 3D grids. Furthermore, big conductivity contrast (discontinuous electrical interface) and singularity of point source are also inevitable. Because of these problems, the GMG method tends to lose its intrinsic high efficiency. Zhebel(2006) improves the standard GMG method in her mathematical PH.D. thesis, using matrix-dependent multigrid (MDMG) method to solve 3D DC resistivity modeling problem with conductivity contrast, it seems that the convergence performance is not good enough.Algebraic multigrid (AMG) method is developing very fast in the past ten years, its application in computational fluid mechanics shows it remains its intrinsic high efficiency with non-uniform grids and discontinuous coefficients. In this paper, we developed a fast 3D DC resistivity modeling algorithm using AMG to solve large linear equation system derived from finite difference simulation, and systematically compared our results with those from ICCG. Numerical calculations for a large number of models show the convergence rate of AMG algorithm is independent of the number of unknowns or the grid size, the conductivity contrast, the area of computation and the size of the inhomogeneity. The iteration number for AMG to reach convergence is around 7 to 8 times, and its computational time increases slowly and linearly as the number of unknowns grows. While the convergence curves of ICCG algorithm are oscillating, it requires hundreds or thousands of iterations to reach convergence with the influences mentioned above, and its computational time increases fast and non-linearly as the number of unknowns grows. AMG method is about 2 to 3 times faster than ICCG method for 3D DC resistivity modeling with a single point source, furthermore, AMG is about 8 times faster than ICCG for multi-sources case (21 sources for example), shows much more efficiency. Based on the fast algebraic multigrid solver, 3D DC resistivity modeling is applied to tunnel forward prediction, which is very important to the security of workers in underground mining engineering. Because of limit investigation space, weak anomalous signal in front of the tunnel, and the influences of tunnel cavity itself and the anomalies around the tunnel, the tunnel forward prediction remains a challenging task. In this paper, we carry out 3D DC resistivity modeling for a large number of underground tunnel models and analyze the relationship between the actual position of the anomaly in front of the tunnel and the position of the characteristic point(e.g. minimum point) on apparent resistivity curve. A quantitative prediction model is established, for the first time, to predict the actual position of the anomaly in front the tunnel. The influence of the size of the anomaly on observed apparent resistivity is also investigated, which could be used to qualitatively estimate the size of the anomaly in front of the tunnel. Meanwhile, a detail study on the identification and location of the anomalies around the tunnel is also carried out. The result would be very useful for the practice of tunnel forward prediction.The effective conductivity of complex multi-phase medium is important to the interpretation of logging data, and also it is the foundation to simulate conductivity model of the deep earth's crust and mantle which is composed of multi-phase minerals. A new method is developed to calculate the effective conductivity of the multi-phase medium using 3D resistivity modeling and inversion. With a wide range of variation of conductivity contrast (up to 10~8) of two-phase medium, the effective conductivities of two-phase medium calculated by our method are in good agreement with those from finite element modeling program ElecFEM3D, which shows our results is reliable.Both applications in tunnel forward prediction and effective conductivity of multi-phase medium involve large scale of 3D non-uniform grids, big conductivity contrast and 3D resistivity modeling for a large number of models, requiring huge amount of numerical works. Thus, fast AMG solver for 3D resistivity modeling plays a very important role in both applications.Finally, algebraic multigrid is applied to solve large linear equations with complex coefficients and consequently 3D magnetotelluric(MT) modeling using AMG is also investigated. Our results show that AMG algorithm obtains its intrinsic high efficiency for solving large linear equations with complex coefficients. However, we need a deep probe into the mathematical problems of AMG to get a better performance for solving 3D MT modeling.