DEVELOPMENT OF MATLAB BASED 3D INVERSION ALGORITHM FOR MT AND DCR DATA

Abstract

In this study, I present the development of a MATLAB based computer code, AP3DMT, for modeling and inversion of 3D Magnetotelluric (MT) and Direct Current Resistivity (DCR) data. This code can be used to invert just the MT data or the DCR, or invert both data sets simultaneously. A 3D MT forward modeling code based on finite difference (FD) method for solving the vector Helmholtz equation, expressed in electrical field, is developed. The staggered grid is used for accurate simulation of nodal electric fields. The electric fields can be computed either using total field or primary/secondary approach. For efficiency, in terms of memory and computation time, the matrix equation is solved using iterative solver with incomplete LU decomposition as pre-conditioner. The convergence of iterative solver is further improved by using the static divergence correction. The developed code, AP3DMT, comprises two independent components: grid generator code and modeling/inversion code. The grid generator code performs model discretization and acts as an interface by generating various I/O files. The inversion code performs core computations in modular form 􀀀 forward modeling, data functionals, sensitivity computations and regularization. These modules can be readily extended to other similar inverse problems like Controlled-Source EM (CSEM), DCR (implemented by the author). The modular structure of the code provides a framework useful for implementation of new applications and inversion algorithms. The use of MATLAB and its libraries makes it more compact and user friendly. The inversion code includes Gauss-Newton optimization (model space as well as data space) and non-linear inversion using conjugate gradient. In both these schemes, Jacobian is not computed explicitly, rather product of Jacobian (or its transpose) with a vector is computed. Special emphasis is given on the block representation of Jacobian for a multi-frequency, multi-component data set and its product with a vector using its three components. It is shown how, for Jacobian, the matrix formed by differentiation of system matrix is made independent of frequency to optimize the operations. A coarse grained parallelization is implemented over number of frequencies for both forward modeling and sensitivity computations. iii The code developed in this study, has been tested on several published models. To demonstrate the versatility of the code, the accuracy of the simulated responses is verified by comparing the responses obtained using a different code and inversion was performed for two complex synthetic models. Further, the code was tested on field data. The data set was acquired over the past decade using Broadband MT survey by our group along the Roorkee-Gangotri profile in the Indo-Gangetic plain, Sub Himalayan and Lesser Himalayan region. The dataset was inverted using ModEM and using AP3DMT by the author. For comparison a 3D diagram of the ratio of inverted cell conductivities in two cases is presented. During the development of divergence correction routine, we observed that this module can be extended to DCR modeling. The developed 3D DCR forward modeling code is based on FD method. The nodal potential are simulated on a normal grids. To strike a balance between computational time and accurate solutions two nodes are used between adjacent electrodes. For removal of singularity due to a point source primary/secondary approach is used. The primary potentials are computed analytically for half space. For secondary potentials, the matrix equation is solved using iterative solver or direct solver depending on the size of the matrix and available resources. The accuracy of the simulated response is verified by comparing the responses with published results. The relevant portions of original AP3DMT code was modified to incorporate DCR inversion. Since, AP3DMT had a modular structure, new modification were made only in forward problem and Jacobian with other portions unchanged. The product of Jacobian with a vector is efficiently managed. The versatility and capabilities of the inversion code was tested against two different models. To make further progress in data interpretation, 3D joint inversion of MT and DCR is developed. The modular structure of AP3DMT code proves to be very useful in performing this task. Since, both these methods has different depth of penetrations model discretization is very crucial. For the joint inversion with each set having different number of data points, data weights needs to be re-calculated, if not then the more numerous data set of one type can cause the influence of another data on the imaging outcome to become insignificant. Two different schemes based on number of data points and gradient are tested. The joint inversion resolves the model better as compared to inversion of individual data sets. The importance of joint inversion is further demonstrated through a synthetic model. Also, both schemes of data weights re-computations is compared on the same model.