This paper develops an efficient hierarchical trans-dimensional (trans-D) Bayesian algorithm to invert magnetotelluric (MT) data for subsurface geoelectrical structure, with unknown geophysical model parameterization (the number of conductivity-layer interfaces) and data-error models parameterized by an auto-regressive (AR) process to account for potential error correlations. The reversible-jump Markov-chain Monte Carlo algorithm, which adds/removes interfaces and AR parameters in birth/death steps, is applied to sample the trans-D posterior probability density for model parameterization, model parameters, error variance and AR parameters, accounting for the uncertainties of model dimension and data-error statistics in the uncertainty estimates of the conductivity profile. To provide efficient sampling over the multiple subspaces of different dimensions, advanced proposal schemes are applied. Parameter perturbations are carried out in principal-component space, defined by eigen-decomposition of the unit-lag model covariance matrix, to minimize the effect of inter-parameter correlations and provide effective perturbation directions and length scales. Parameters of new layers in birth steps are proposed from the prior, instead of focused distributions centred at existing values, to improve birth acceptance rates. Parallel tempering, based on a series of parallel interacting Markov chains with successively relaxed likelihoods, is applied to improve chain mixing over model dimensions. The trans-D inversion is applied in a simulation study to examine the resolution of model structure according to the data information content. The inversion is also applied to a measured MT data set from south-central Australia.

For large-scale magnetotelluric (MT) forward modeling using vector finite-element methods, iterative solvers are commonly applied. However, as frequency decreases close to zero, the iterative solution process struggles to converge. This is mainly caused by the fact that the weak conductivity term in the curl-curl equation governing the electromagnetic (EM) diffusion in the earth fades during the iterative solution for EM fields. This can lead to violation of the divergence-free condition for current and false jumps in the calculated fields. We develop a regularization technique for vector finite-element MT forward modeling, in which a scaled grad-div operator for electrical fields is included in the curl-curl equation to enforce the divergence-free condition explicitly. Because of its use of basis functions, the direct edge element discretization of the scaled term can lead to zero coefficients. To address this, an equivalent substitute of the scaled grad-div operator based on potential representation of electrical fields is used, discretized with node elements. For one specific element, this is finally reduced to a scaled gradient of the surface integral of current across all the surfaces of the element, resulting in better connectivity of the linear system of equations. The correctness of our algorithm is verified with two synthetic models and an inversion model from real data. The numerical performance is compared to the traditional iterative divergence correction technique (and without the application of the divergence correction for one case) for each of the three models. The results indicate that our algorithm is generally more efficient and stable compared to the traditional technique for all models at all periods considered, with significant improvement at long periods.

Practical application of 3D magnetotelluric inversion requires efficient forward modeling of electromagnetic (EM) fields in the earth. To resolve realistic 3D structures, large computational domains and extremely large linear systems of equations are required. The iterative solvers, which are almost exclusively used to solve these systems, can be inefficient due to the abundant null space of the curl-curl operator. Multigrid (MG) solvers are considered a potentially efficient technique for solving such problems. However, due to the abundant null solution space and existence of the air layer, MG solvers can still converge slowly or even diverge. We have developed an efficient MG solver for finite-difference frequency-domain EM solution. In this algorithm, the excellent smoothing property of an efficient four-color cell-block Gauss-Seidel (GS) is exploited to remove the short-range errors effectively, and the interpolation and prolongation operators are used to handle the long-range errors. They work as a whole to speed the convergence of our algorithm remarkably. Because all of the nodes for the four-color cell-block GS are grouped into four colors and the edge components attached to different nodes in each color are completely decoupled, this can be used to develop a highly vectorized or parallelized algorithm. Another important property is that our algorithm is locally current divergence free, effectively eliminating spurious solutions in the null space of the curl-curl operator. The accuracy and efficiency of the algorithm are verified by comparing the numerical solutions obtained with our MG solver to those from the biconjugate gradient stabilized solver with different preconditioners based on synthetic models and a model from 3D inversion. Comparisons, in terms of iteration number and computational time, indicate that our algorithm is extremely stable and efficient relative to the other solvers. Our MG algorithm will be suitable for massively parallel computing as well.

SUMMARY: The fast and accurate 3-D magnetotelluric (MT) forward modelling is core engine of the interpretation and inversion of MT data. In this study, we develop an improved extrapolation cascadic multigrid method (EXCMG) to solve the large sparse complex linear system arising from the finite-element (FE) discretization on non-uniform orthogonal grids of the Maxwell’s equations using potentials. First, the vector Helmholtz equation and the scalar auxiliary equation are derived from the Maxwell’s equations using Coulomb-gauged potentials. The weighted residual method is adopted to discretize the weak formulation and assemble the FE equation. Secondly, carefully choosing the preconditioned complex stable bi-conjugate gradient method (BiCGStab) as multigrid smoother, we develop an improved EXCMG method on non-uniform grids to solve the resulting large sparse complex non-Hermitian linear systems. Finally, several examples including three standard testing models (COMMEMI3D-1, COMMEMI3D-2 and DTM1.0) and a topographic model are used to validate the accuracy and efficiency of the proposed multigrid solver. Numerical results show that the proposed EXCMG algorithm greatly improves the efficiency of 3-D MT forward modelling, is more efficient than some existing solvers, such as Pardiso, incomplete LU factorization preconditioned biconjugate gradients stabilized method (ILU-BiCGStab) and flexible generalized minimum residual method with auxiliary space Maxwell preconditioner (FGMRES-AMS), and capable to simulate large-scale problems with more than 100 million unknowns.

SUMMARY: Joint inversion of magnetotelluric (MT) and geomagnetic depth sounding (GDS) responses can constrain the crustal and mantle conductivity structures. Previous studies typically use either deterministic inversion algorithms that provide limited information on model uncertainties or using stochastic inversion algorithms with a predetermined number of layers that is generally not known a priori. Here, we present a new open-source Bayesian framework for the joint inversion of MT and GDS responses to probe 1D layered Earth’s conductivity structures. Within this framework, model uncertainties can be accurately estimated by generating numerous models that fit the observed data. A trans-dimensional Markov Chain Monte Carlo (MCMC) method is employed to self-parametrize the model parameters, where the number of layers is treated as an inversion parameter that is determined automatically by the data. This adaptability can overcome the under or over-parametrization problem and may be able to automatically detect the conductivity discontinuities in the Earth’s interior. To accelerate the computations, a large number of Markov chains with different initial states can be run simultaneously using the MPI parallel technique. Synthetic data sets are used to validate the feasibility of our method and illustrate how separate and joint inversions, as well as various priors affect the posterior model distributions. The trans-dimensional MCMC algorithm is then applied to jointly invert the MT and GDS responses estimated at the Tucson geomagnetic observatory, North America. Our results not only contain model uncertainty estimates but also indicate two distinct conductivity discontinuities at around 85 and 440 km, which are likely related to the lithosphere-asthenosphere boundary and the upper interface of the mantle transition zone, respectively.

SUMMARY: The traditional gravity forward modelling methods for solving partial differential equations (PDEs) only can yield second-order accuracy. When computing the gravity field vector and gradient tensor from the obtained potential, those numerical differentiation approaches will inevitably lose accuracy. To mitigate this issue, we propose an efficient and accurate 3-D forward modelling algorithm based on a fourth-order compact difference scheme. First, a 19-point fourth-order compact difference scheme with general meshsizes in x-, y- and z-directions is adopted to discretize the governing 3-D Poisson’s equation. The resulting symmetric positive-definite linear systems are solved by the pre-conditioned conjugate gradient algorithm. To obtain the first-order (i.e. the gravity field vector) and second-order derivatives (i.e. the gravity gradient tensor) with fourth-order accuracy, we seek to solve a sequence of tridiagonal linear systems resulting from the above mentioned finite difference discretization by using fast Thomas algorithm. Finally, two synthetic models and a real topography relief are used to verify the accuracy of our method. Numerical results show that our method can yield a nearly fourth-order accurate approximation not only to the gravitational potential, but also to the gravity field vector and its gradient tensor, which clearly demonstrates its superiority over the traditional PDE-based methods.