We present a finite volume (FV) algorithm for the magnetotelluric (MT) forward modeling in 3-D conductivity structures with general anisotropy. The electric and magnetic fields are discretized on a conventional staggered grid, which cannot directly address the full-tensor conductivity. To overcome this difficulty, an interpolation scheme is employed to average different components of the electric field to the same position. We formulate the algorithm in pure matrix form and implement it in a new language, Julia, making the programming process highly efficient and leading to a code with excellent readability, maintainability and extendability. The validity of the FV Julia code is demonstrated using a layered 1-D anisotropic model. For this model, the FV code provides accurate results, and the computational cost is reasonable. The iterative solvers QMR and BiCGStab preconditioned with the electromagnetic potential (A-ø) system exhibit good convergence rate for a wide range of periods. The direct solvers MUMPS and PARDISO are highly efficient for small model sizes. For a relatively large model size with 2.18 millions unknowns, the linear system of one period can be solved by MUMPS within 360 seconds with multiple threads involved in the computation, and the memory usage is only 11.6 GB in the “Out-of-Core” mode. We further calculate MT responses of a 3-D model with dipping and horizontal anisotropy, respectively. The results suggest that electrical anisotropy can have significant influence on the MT response.

Presently, the 3-D inversion technique has started playing a more important role in controlled-source electromagnetic (CSEM) data interpretation. With the development of hardware and computation algorithm, 3-D inversion technique has developed rapidly during the past decades. In this article, we present a newly developed 3-D parallelized inversion algorithm in the frequency domain with hexahedral discretization. Within the framework of this approach, we use the finite-element method (FEM) in the forward modeling and Gauss-Newton optimization technique in the inversion. We solve the forward modeling and adjoint problem efficiently with Math Kernel Library (MKL) Pardiso parallel direct solver. Considering the fact that the forward modeling and sensitivity calculation are frequency independent, we further parallelize the algorithm over frequency using Message Passing Interface (MPI) to speed up the modeling and inversion process. The sensitivity matrix is calculated explicitly, which enables us to estimate the optimized regularization parameter easily based on the spectral radius estimation. We proposed a new roughness operator for hexhedral discretization which works well for CSEM inversion problems. We applied the developed algorithm to several realistic CSEM models. The inversion results demonstrate the effectiveness and stability of our inversion scheme.

Material nonlinearity analyses are widely used to determine the safety of structural components or engineering structures. Although advanced computer hardware technology has considerably improved the computational performance of such analyses, large and complex emerging structures and expensive computational process still attract the attention of researchers toward finding more efficient and accurate numerical solution methods. This paper combines an inelasticity-separated (IS) concept with the finite-element method (FEM) to establish a novel and efficient framework (IS-FEM) for structures with material nonlinear behavior that only occurs within certain small local domains. The IS concept presented in this paper begins by decomposing the strain on a nonlinear material into its linear-elastic and inelastic components and runs through the whole framework. Based on the principle of virtual work, a novel governing equation for structures with an IS form is derived by treating the decomposed inelastic strain as additional degrees of freedom. Moreover, the changing stiffness matrix in the classical FEM is expressed as the sum of the invariant global linear elastic stiffness matrix and another changing inelastic stiffness matrix with a small rank, and it represents the material nonlinearity of local domains so that the efficient solution method can be applied to perform nonlinear analyses via the mathematical Sherman–-Morrison–Woodbury (SMW) formula. Because the unchanging global stiffness matrix is used throughout the whole nonlinear analysis and the computational effort only focuses on a small dimension matrix that represents the local inelastic behavior, the efficiency of the proposed IS-FEM is improved greatly. The proposed method is validated against the results of classical FEM via three separate numerical examples and its value and potential for use in any material nonlinearity analyses are also demonstrated.

In this study, the complex frequency-shifted perfectly matched layer (CFS-PML) in stretching Cartesian coordinates is successfully applied to 3-D frequency-domain marine controlled-source electromagnetic (CSEM) field modelling. The Dirichlet boundary, which is usually used within the traditional framework of EM modelling algorithms, assumes that the electric or magnetic field values are zero at the boundaries. This requires the boundaries to be sufficiently far away from the area of interest. To mitigate the boundary artefacts, a large modelling area may be necessary even though cell sizes are allowed to grow toward the boundaries due to the diffusion of the electromagnetic wave propagation. Compared with the conventional Dirichlet boundary, the PML boundary is preferred as the modelling area of interest could be restricted to the target region and only a few absorbing layers surrounding can effectively depress the artificial boundary effect without losing the numerical accuracy. Furthermore, for joint inversion of seismic and marine CSEM data, if we use the PML for CSEM field simulation instead of the conventional Dirichlet, the modelling area for these two different geophysical data collected from the same survey area could be the same, which is convenient for joint inversion grid matching. We apply the CFS-PML boundary to 3-D marine CSEM modelling by using the staggered finite-difference discretization. Numerical test indicates that the modelling algorithm using the CFS-PML also shows good accuracy compared to the Dirichlet. Furthermore, the modelling algorithm using the CFS-PML shows advantages in computational time and memory saving than that using the Dirichlet boundary. For the 3-D example in this study, the memory saving using the PML is nearly 42 per cent and the time saving is around 48 per cent compared to using the Dirichlet.

An alternating joint inversion method for controlled-source electromagnetic (CSEM) and seismic data is developed. The structural constraint is used for correlating and constraining the electromagnetic (EM) resistivity and seismic velocity parameters during the inversion. The structural coupling used is the joint total variation (JTV) constraint, which is incorporated into the objective function of the individual EM or seismic inversion to enforce the structural similarity between the resistivity and velocity. In this paper, the conventional cross-gradient constraint is not preferred as it can only be used for enforcing structural similarity for 2-D or 3-D case since it is always zero for 1-D joint inversion. The JTV constraint can be applied for 1-D joint inversion, as well as for the 2-D/3-D case, which is of a broader interest. The improved Gauss-Newton (GN) is used for minimizing the objective function and for reconstructing the subsurface resistivity and velocity. The alternating joint inversion algorithm is applied for integrating land CSEM data with cross-well seismic data for subsurface reservoir evaluation and water-oil identification. Numerical examples show that the developed joint inversion can improve the inversion results significantly over those from the separate EM or seismic inversion.