Galvanic distortion of magnetotelluric (MT) data due to small-scale surficial bodies or due to topography is one of the major factors that prevents accurate imaging of the subsurface. We present a 3-D algorithm for joint inversion of MT impedance tensor data and a frequency-independent full distortion matrix that circumvents this problem. We perform several tests of our algorithm on synthetic data affected by different amounts of distortion. These tests show that joint inversion leads to a better conductivity model compared to the inversion of the MT impedance tensor without any correction for distortion effects. For highly distorted data, inversion without any distortion correction results in strong artefacts and we cannot fit the data to the specified noise level. When the distortion is reduced, we can fit the data to an RMS of one, but still observe artefacts in the shallow part of the model. In contrast, in both cases our joint inversion can fit the data within the assumed noise level and the resulting models are comparable to the inversion of undistorted data. In addition, we show that the elements of the full distortion matrix can be well resolved by our algorithm. Finally, when inverting undistorted data, including the distortion matrix in the inversion only results in a minor loss of resolution. We therefore consider our new approach a promising tool for the general analysis of field MT data.

The objective of this paper is to construct a stable inversion scheme to determine the conductance of a buried inhomogeneous layer masked by sedimentary cover. We employ a model which incorporates a surface sheet of conductance S1 (x, y) and a buried sheet of conductance S2 (x, y). The buried sheet is embedded at a depth d in a layered earth of conductivity σ(z). The σ(z), d and S1 (x, y) are assumed to be known. We determine S2(x, y) by inverting the monochromatic horizontal electromagnetic (EM) field measured at the earth’s surface. To do this, we use Price’s matching conditions at the sheets and a stable continuation of EM field from the earth’s surface onto the buried sheet. Employing synthetic data we demonstrate that the designed inversion scheme generates a reasonable image of the buried structure. The scheme is fast, since it doesn’t require any forward modelling.

We find out if the ratio of cross-strait voltage to water transport for the Bering Strait (BS) is constant. For this purpose, we have developed a technique to construct BS water velocity maps. We have built up a three-dimensional conductivity model for the BS region. Using this model, we have simulated coast-to-coast voltage for the various velocity maps constructed. We have found that the voltage/transport ratio remains constant for electromagnetic field periods exceeding 2 days. We have estimated the ratio as 239 ± 11 mV/(km2 · m/s) assuming bottom sediment conductance to be 600 S. We conclude that measuring cross-strait voltage allows the monitoring of BS water transport.

Detecting a salt dome overhang is known to be problematic by seismic methods alone. We used magnetotellurics (MT) as a complementary method to seismics to investigate the detectability of a salt dome overhang. A comparison of MT responses for 3D synthetic salt models with and without overhang shows that MT is very sensitive to shallow salt structures and suggests that it should be possible to detect an overhang. To further investigate the resolution capability of MT for a salt dome overhang, we performed a 3D MT inversion study and investigated the impact of model parametrization and regularization. We showed that using the logarithms of the conductivities as model parameters is crucial for inverting data from resistive salt structures because, in this case, commonly used Tikhonov-type stabilizers work more equally for smoothing the resistive and conductive structures. The use of a logarithmic parametrization also accelerated the convergence and produced better inversion results. When the Laplace operator was used as a regularization functional, we still observed that the inversion algorithm allows spatial resistivity gradients. These spatial gradients are reduced if a regularization based on first derivatives in contrast to the Laplace operator is introduced. To demonstrate the favorable performance when logarithmic parametrization and gradient-based regularization are employed, we first inverted a data set simulated for a simple model of two adjacent blocks. Subsequently, we applied the code to a more realistic salt dome overhang detectability study. The results from the detectability study are encouraging and suggest that 3D MT inversion can be applied to decide whether the overhang is present in the shallow salt structure even in the case when only profile data are available. However, to resolve the overhang, a dense MT site coverage above the flanks of the salt dome is required.

We have improved the integral equation method for modeling 3D electromagnetic fields by using the separability of its inherent [Formula: see text] dyadic Green’s tensors. Conventional integral equation approaches exhibit a quadratic dependence on model size, at least for the vertical dimension. In contrast, our approach has a linear dependence on all three dimensions. We tested our method on an example of induction logging in deviated boreholes.

The limited-memory quasi-Newton method with simple bounds is used to develop a novel, fully 3D magnetotelluric (MT) inversion technique. This nonlinear inversion is based on iterative minimization of a classical Tikhonov regularized penalty function. However, instead of the usual model space of log resistivities, the approach iterates in a model space with simple bounds imposed on the conductivities of the 3D target. The method requires storage proportional to [Formula: see text], where [Formula: see text] is the number of conductivities to be recovered and [Formula: see text] is the number of correction pairs (practically, only a few). These requirements are much less than those imposed by other Newton methods, which usually require storage proportional to [Formula: see text] or [Formula: see text], where [Formula: see text] is the number of data to be inverted. The derivatives of the penalty function are calculated using an adjoint method based on electromagnetic field reciprocity. The inversion involves all four entries of the MT impedance matrix; the [Formula: see text] integral equation forward-modeling code is used as an engine for this inversion. Convergence, performance, and accuracy of the inversion are demonstrated on synthetic numerical examples. After investigating erratic resistivities in the upper part of the model obtained for one of the examples, we conclude that the standard Tikhonov regularization is not enough to provide consistently smooth underground structures. An additional regularization helps to overcome the problem.

3D numerical studies have shown that the vertical voltage above the Baikal deep-water fault is detectable and that respective transfer functions, Ez-responses, are sensitive to the electrical resistivity changes of the fault, i.e. these functions appear actually informative with respect to the resistivity «breath» of the fault. It means that if the fault resistivity changed, conventional electromagnetic instruments would be able to detect this fact by measurement of the vertical electric field, Ez, or the vertical electric voltage just above the fault as well as horizontal magnetic field on the shore. Other electromagnetic field components (Ex, Ey, Hz) do not seem to be sensitive to the resistivity changes in such a thin fault (as wide as 500 m). On the other hand, such changes are thought to be able to indicate a change of a stress state in the earthquake preparation zone. Besides, the vertical profile at the bottom of Lake Baikal is suitable for electromagnetic monitoring of the fault electrical resistivity changes. Altogether, the vertical voltage above the deep-water fault might be one of earthquake precursors.

The limited-memory quasi-Newton optimization method with simple bounds has been applied to develop a novel fully three-dimensional (3-D) magnetotelluric (MT) inversion technique. This nonlinear inversion is based on iterative minimization of a classical Tikhonov-type regularized penalty functional. But instead of the usual model space of log resistivities, the approach iterates in a model space with simple bounds imposed on the conductivities of the 3-D target. The method requires storage that is proportional to n(cp) x N, where the N is the number of conductivities to be recovered and n(cp) is the number of the correction pairs (practically, only a few). This is much less than requirements imposed by other Newton type methods (that usually require storage proportional to N x M, or N x N, where M is the number of data to be inverted). Using an adjoint method to calculate the gradients of the misfit drastically accelerates the inversion. The inversion also involves all four entries of the MT impedance matrix. The integral equation forward modelling code x3d by Avdeev et al. ([1, 2]) is employed as an engine for this inversion. Convergence, performance and accuracy of the inversion are demonstrated on a 3D MT synthetic, but realistic, example.

We apply a limited-memory quasi-Newton (QN) method to the 1D magnetotelluric (MT) inverse problem. Using this method we invert a realistic synthetic MT impedance data set calculated for a layered earth model. The calculation of gradients based on the adjoint method speeds up the inverse problem solution many times. In addition, regularization stabilizes the QN inversion result and a few correction pairs are sufficient to produce reasonable results. Comparison with the L-BFGS-B algorithm shows similar convergence rates. This study is a first step towards the solution of large-scale electromagnetic problems, with a full treatment of the 3D conductivity structure of the earth.

A limited memory quasi-Newton (QN) method with simple bounds is applied to a 1-D magnetotelluric (MT) problem. The method is used to invert a realistic synthetic MT impedance dataset, calculated for a layered earth model. An adjoint method is employed to calculate the gradients and to speed up the inverse problem solution. In addition, it is shown that regularization stabilizes the QN inversion result. We demonstrate that only a few correction pairs are enough to produce reasonable results. Comparison with inversion based on known L-BFGS-B optimization algorithm shows similar convergence rates. The study presented is a first step towards the solution of large-scale electromagnetic problems with a full 3-D conductivity structure of the Earth.

The limited-memory quasi-Newton optimization method with simple bounds has been applied to develop a novel fully three-dimensional (3-D) magnetotelluric (MT) inversion technique. This nonlinear inversion is based on iterative minimization of a classical Tikhonov-type regularized penalty functional. But instead of the usual model space of log resistivities, the approach iterates in a model space with simple bounds imposed on the conductivities of the 3-D target. The method requires storage that is proportional to n(cp) x N, where the N is the number of conductivities to be recovered and ncp is the number of the correction pairs ( practically, only a few). This is much less than requirements imposed by other Newton type methods ( that usually require storage proportional to N x M, or N x N, where M is the number of data to be inverted). Using an adjoint method to calculate the gradients of the misfit drastically accelerates the inversion. The inversion also involves all four entries of the MT impedance matrix. The integral equation forward modelling code x3d by Avdeev et al. [ 1, 2] is employed as an engine for this inversion. Convergence, performance and accuracy of the inversion are demonstrated on a 3D MT synthetic, but realistic, example.

The whole subject of three-dimensional (3-D) electromagnetic (EM) modelling and inversion has experienced a tremendous progress in the last decade. Accordingly there is an increased need for reviewing the recent, and not so recent, achievements in the field. In the first part of this review paper I consider the finite-difference, finite-element and integral equation approaches that are presently applied for the rigorous numerical solution of fully 3-D EM forward problems. I mention the merits and drawbacks of these approaches, and focus on the most essential aspects of numerical implementations, such as preconditioning and solving the resulting systems of linear equations. I refer to some of the most advanced, state-of-the-art, solvers that are today available for such important geophysical applications as induction logging, airborne and controlled-source EM, magnetotellurics, and global induction studies. Then, in the second part of the paper, I review some of the methods that are commonly used to solve 3-D EM inverse problems and analyse current implementations of the methods available. In particular, I also address the important aspects of nonlinear Newton-type optimisation techniques and computation of gradients and sensitivities associated with these problems.

During the last decade a number of one-dimensional (1-D) conductivity profiles have been constructed for the upper and mid-mantle of the North Pacific Ocean region. These profiles differ significantly, and from our point of view it is still unclear which profile is the best candidate for the upper and mid-mantle conductivity reference model for this region. Keeping the differences in mind, and inspired by recent findings that the ocean effect is a major contributor to the anomalous behaviour of C-responses up to the period of 20 days (especially at coastal observatories), the goal of this paper has been three-fold: (1) to understand, on the basis of systematic model studies using 3-D ocean models, which of the published 1-D upper and mid-mantle profiles is in best agreement with the available observations. (2) To try to reduce the misfit between the observed and modelled responses by using dense grids in modelling, by considering 3-D models which include not only an inhomogeneous surface layer but also inhomogeneous deeper structures. (3) To derive an alternative 1-D upper and mid-mantle section for the North Pacific Ocean by carefully selecting the data for interpretation and by using 3-D models that are as realistic and detailed as possible.In order to perform the simulations using realistic 3-D models on a routine basis a novel 3-D 'spherical' forward solution has been elaborated in this paper. The solution combines the modified iterative-dissipative method with a conjugate gradient iteration and allows one to compute efficiently the electromagnetic fields in full 3-D spherical models with very high lateral contrasts of conductivity and for very dense grids.During the 3-D simulations a systematic shift of observed C-responses at Honolulu compared with those at other observatories was detected. The reason for this shift is still unclear. Even if a very detailed grid of 0.3 x 0.3 is used, the 3-D simulations using a model of the inhomogeneous surface has no notable ocean effect to C-responses at this site.An attempt has been made to reduce the misfit between observed and modelled C-responses by incorporating a 3-D model with inhomogeneous lithosphere and upper mantle. However, this has resulted only in a slight change in responses.The main conclusion drawn from our 3-D model studies is that there exists a significant disagreement between observed and 3-D modelled C-responses if the published 1-D sections with conducting uppermost 400 km are considered as the upper and mid-mantle sections. Our 3-D simulations and reinterpretation of the data also confirm the recent finding that the upper and mid-mantle beneath North Pacific Ocean in the depth range down to 400 km is much more resistive than hitherto assumed.

We have developed a new series of 3D forward modeling codes for various geoelectromagnetics (geoEM) applications, including: (1) induction logging in deviated wells, (2) grounded and airborne controlled-source EM, (3) magnetotellurics (MT), and (4) global induction studies. This series of codes has proven to be effective way to simulate geoEM fields in complex 3D environment. Among the main features of these codes are the ability to work on PC platforms, and the combination of a fast, but exact state-of-the-art integral equation approach.