Sheng Fang

Quasi-linear approximation in 3-D electromagnetic modeling

The Born approximation in electromagnetic (EM) numerical modeling has limited application for solving 3-D electromagnetic induction problems, because in structures with high conductivity contrasts and at high frequencies, this approximation is inaccurate. In this paper, we develop a new and relatively simple approximation for the EM field called a quasi‐linear approximation, which is based on the evaluation of the anomalous field Ea by a linear transformation of the normal (primary) field: Ea=λˆEn, where λˆ is called the electrical reflectivity tensor. The reflectivity tensor inside inhomogeneities can be approximated by a slowly varying function that can be determined numerically by a simple optimization technique. The new approximation gives an accurate estimate of the EM response for conductivity contrasts of more than one hundred to one, and for a wide range of frequencies. It also opens the possibility for fast 3-D electromagnetic inversion.

Quasi-analytical approximations and series in electromagnetic modeling

The quasi‐linear approximation for electromagnetic forward modeling is based on the assumption that the anomalous electrical field within an inhomogeneous domain is linearly proportional to the background (normal) field through an electrical reflectivity tensor λ⁁. In the original formulation of the quasi‐linear approximation, λ⁁ was determined by solving a minimization problem based on an integral equation for the scattering currents. This approach is much less time‐consuming than the full integral equation method; however, it still requires solution of the corresponding system of linear equations. In this paper, we present a new approach to the approximate solution of the integral equation using λ⁁ through construction of quasi‐analytical expressions for the anomalous electromagnetic field for 3-D and 2-D models. Quasi‐analytical solutions reduce dramatically the computational effort related to forward electromagnetic modeling of inhomogeneous geoelectrical structures. In the last sections of this paper...

Three‐dimensional quasi‐linear electromagnetic inversion

One of the most challenging problems of electromagnetic (EM) geophysical methods is developing three-dimensional (3-D) EM inversion techniques. This problem is of utmost importance in practical applications because of the 3-D nature of the geological structures. The main difficulties in 3-D inversion are related to (1) limitations of 3-D forward modeling codes available and (2) ill-posedness of the inversion procedures in general. The multidimensional EM inversion techniques existing today can handle only simple models and typically are very time consuming. We developed a new approach to a rapid 3-D EM inversion. The forward scattering problem is solved using a new quasi linear (QL) approximation of the existing integral equation algorithms, developed for various sources of excitation. The QL approximation for forward modeling is based on the assumption that the anomalous field is linearly related to the normal field in the inhomogeneous domain by an electrical reflectivity tensor. We introduce also a modified material property tensor which is linearly proportional to the reflectivity tensor and the complex anomalous conductivity. The QL approximation generates a linear equation with respect to the modified material property tensor. The solution of this equation is called “a quasi-Born inversion”. We apply the Tikhonov regularization for the stable solution of this problem. The next step of the inversion includes correction of the results of the quasi Born inversion: after determining a modified material property tensor, we use the electrical reflectivity tensor to evaluate the anomalous conductivity. Thus the developed inversion scheme reduces the original nonlinear inverse problem to a set of linear inverse problems, which is why we call this approach “a QL inversion”. Synthetic examples (with and without random noise) of inversion demonstrate that the algorithm for inverting 3-D EM data is fast and stable.

Electromagnetic inversion using quasi‐linear approximation

Three-dimensional electromagnetic inversion continues to be a challenging problem in electrical exploration. We have recently developed a new approach to the solution of this problem based on quasi-linear approximation of a forward modeling operator. It generates a linear equation with respect to the modified conductivity tensor, which is proportional to the reflectivity tensor and the complex anomalous conductivity. We solved this linear equation by using the regularized conjugate gradient method. After determining a modified conductivity tensor, we used the electrical reflectivity tensor to evaluate the anomalous conductivity. Thus, the developed inversion scheme reduces the original nonlinear inverse problem to a set of linear inverse problems. The developed algorithm has been realized in computer code and tested on synthetic 3-D EM data. The case histories include interpretation of a 3-D magnetotelluric survey conducted in Hokkaido, Japan, and the 3-D inversion of the tensor controlled-source audio magnetotelluric data over the Sulphur Springs thermal area, Valles Caldera, New Mexico, U.S.A.

Quasi‐linear series in three‐dimensional electromagnetic modeling

We have recently introduced a quasi-linear (QL) approximation for the solution of the three-dimensional (3-D) electromagnetic modeling problem. In this paper we discuss an approach to improving its accuracy by considering the QL approximations of the higher-order. This approach can be considered the natural generalization of the Born series. We use the modified Greens operator with the norm less than 1 to ensure the convergence of the higher orders QL approximations to the true solution. This new approach produces the converged QL series, which makes it possible to estimate the accuracy of the original QL approximation without direct comparison with the rigorous full integral equation solution. It also opens principally new possibilities for fast and accurate 3-D EM modeling and inversion.

Quasi-linear Series in 3-D Electromagnetic Modeling

Summary We have recently introduced a quasi-linear (QL) approximation for the solution of the 3-D electromagnetic modeling problem. In this paper we discuss an approach to improving its accuracy by considering the QL approximations of the higher orders. This approach can be considered the natural generalization of the Born series. We use the modified Green’s operator with the norm less than 1 to ensure the convergence of the higher orders QL approximations to the true solution. This new approach produces the always converged QL series, which makes it possible to estimate the accuracy of the conventional QL approximation without direct comparison with the rigorous full integral equation (IE) solution. It also opens principally new possibilities for fast and accurate 3-D EM modelling and inversion.