Global 3-D imaging of mantle conductivity based on inversion of observatory C-responses—II. Data analysis and results

The global 3-D electrical conductivity distribution in the mantle (in the depth range between 400 and 1600 km) is imaged by inverting C-responses estimated on a global net of geomagnetic observatories. Very long time-series (up to 51 years; 1957-2007) of hourly means of three components of the geomagnetic field from 281 geomagnetic observatories are collected and analysed. Special attention is given to data processing in order to obtain unbiased C-responses with trustworthy estimates of experimental errors in the period range from 2.9 to 104.2 d. After careful inspection of the obtained C-responses the data from 119 observatories are chosen for the further analysis. Squared coherency is used as a main quality indicator to detect (and then to exclude from consideration) observatories with a large noise-to-signal ratio. During this analysis we found that—along with the C-responses from high-latitude observatories (geomagnetic latitudes higher than 58°)—the C-responses from all low-latitude observatories (geomagnetic latitudes below 11°) also have very low squared coherencies, and thus cannot be used for global induction studies. We found that the C-responses from the selected 119 mid-latitude observatories show a huge variability both in real and imaginary parts, and we investigated to what extent the ocean effect can explain such a scatter. By performing the systematic model calculations we conclude that: (1) the variability due to the ocean effect is substantial, especially at shorter periods, and it is seen for periods up to 40 d or so; (2) the imaginary part of the C-responses is to a larger extent influenced by the oceans; (3) two types of anomalous C-response behaviour associated with the ocean effect can be distinguished; (4) to accurately reproduce the ocean effect a lateral resolution of 1°× 1° of the conductance distribution is needed, and (5) the ocean effect alone does not explain the whole variability of the observed C-responses. We also detected that part of the variability in the real part of the C-responses is due to the auroral effect. In addition we discovered that the auroral effect in the C-responses reveals strong longitudinal variability, at least in the Northern Hemisphere. Europe appears to be the region with smallest degree of distortion compared with North America and northern Asia. We found that the imaginary part of the C-responses is weakly affected by the auroral source, thus confirming the fact that in the considered period range the electromagnetic (EM) induction from the auroral electrojet is small. Assuming weak dependence of the auroral signals on the Earth's conductivity at considered periods, and longitudinal variability of the auroral effect, we developed a scheme to correct the experimental C-responses for this effect. With these developments and findings in mind we performed a number of regularized 3-D inversions of our experimental data in order to detect robust features in the recovered 3-D conductivity images. Although differing in details, all our 3-D inversions reveal a substantial level of lateral heterogeneity in the mantle at the depths between 410 and 1600 km. Conductivity values vary laterally by more than one order of magnitude between resistive and conductive regions. The maximum lateral variations of the conductivity have been detected in the layer at depths between 670 and 900 km. By comparing our global 3-D results with the results of independent global and semi-global 3-D conductivity studies, we conclude that 3-D conductivity mantle models produced so far are preliminary as different groups obtain disparate results, thus complicating quantitative comparison with seismic tomography or/and geodynamic models. In spite of this, our 3-D EM study and most other 3-D EM studies reveal at least two robust features: reduced conductivity beneath southern Europe and northern Africa, and enhanced conductivity in northeastern Chin

Global 3-D imaging of mantle electrical conductivity based on inversion of observatory C-responses—I. An approach and its verification

We present a novel frequency-domain inverse solution to recover the 3-D electrical conductivity distribution in the mantle. The solution is based on analysis of local C-responses. It exploits an iterative gradient-type method—limited-memory quasi-Newton method—for minimizing the penalty function consisting of data misfit and regularization terms. The integral equation code is used as a forward engine to calculate responses and data misfit gradients during inversion. An adjoint approach is implemented to compute misfit gradients efficiently. Further improvements in computational load come from parallelizing the scheme with respect to frequencies, and from setting the most time-consuming part of the forward calculations—calculation of Green′s tensors—apart from the inversion loop. Convergence, performance, and accuracy of our 3-D inverse solution are demonstrated with a synthetic numerical example. A companion paper applies the strategy set forth here to real dat

General formalism for the efficient calculation of the Hessian matrix of EM data misfit and Hessian-vector products based upon adjoint sources approach

3-D electromagnetic (EM) studies of the Earth have advanced significantly over the past decade. Despite a certain success of the 3-D EM inversions of real data sets, the quantitative assessment of the recovered models is still a challenging problem. It is known that one can gain valuable information about model uncertainties from the analysis of Hessian matrix. However, even with modern computational capabilities the calculation of the Hessian matrix based on numerical differentiation is extremely time consuming. Much more efficient way to compute the Hessian matrix is provided by an ‘adjoint sources' methodology. The computation of Hessian matrix (and Hessian-vector products) using adjoint formulation is now well-established approach, especially in seismic inverse modelling. As for EM inverse modelling we did not find in the literature a description of the approach, which would allow EM researchers to apply this methodology in a straightforward manner to their scenario of interest. In the paper, we present formalism for the efficient calculation of the Hessian matrix using adjoint sources approach. We also show how this technique can be implemented to calculate multiple Hessian-vector products very efficiently. The formalism is general in the sense that it allows to work with responses that arise in EM problem set-ups either with natural- or controlled-source excitations. The formalism allows for various types of parametrization of the 3-D conductivity distribution. Using this methodology one can readily obtain appropriate formulae for the specific sounding methods. To illustrate the concept we provide such formulae for two EM techniques: magnetotellurics and controlled-source sounding with vertical magnetic dipole as a sourc

Accelerating EM integral equation forward solver for global geomagnetic induction using SVD based matrix compression method

We develop an singular value decomposition-based compression of the Green's function matrix of an electromagnetic integral equation forward solver for global geomagnetic induction, on top of an fast Fourier transform reduction of the system to a block-diagonal form. With this approach, the memory usage and CPU time of Krylov subspace iterative solutions are significantly reduced at a very small cost of accuracy, making the accelerated forward solver well suited for 3-D inversions as well as forward simulations with multiple source

3-D EM inversion of ground based geomagnetic Sq data. Results from the analysis of Australian array (AWAGS) data

We present the first inversion of geomagnetic Sq data in a framework of 3-D conductivity models. This problem has been considered as immensely difficult due to the complex spatial structure of the Sq source which, in addition, varies with season and solar activity. Recently, we developed a 3-D electromagnetic (EM) inversion solution that allows one to work in a consistent manner with data that originates from sources, irrespective of their spatial complexity. In this paper, we apply our 3-D EM inversion scheme to Sq data collected during the Australian Wide Array of Geomagnetic Stations project. Within this project, three components of the geomagnetic field were recorded between 1989 November and 1990 December with the use of 53 portable vector magnetometers. The instruments were distributed over the Australian mainland with an average spacing of 275 km between sites. Inverting this unique—in a sense of its spatial regularity, density and long operational time—data set, we recovered the 3-D conductivity distribution beneath Australia at upper mantle depths (100-520 km). This depth range was justified in the paper from resolution studies using checkerboard tests. In addition, we performed extensive modelling to estimate quantitatively the influence of various factors on Sq signals, namely from hypothetical anomalies, inaccuracy in the source, ocean, and model discretization. As expected, the ocean (coastal) effect appeared to be the largest so that it has to be accounted for during 3-D inversion as accurately as possible. Our 3-D inversions—of data from either single or multiple days—revealed a strong offshore conductor near the south-east coast of Australia, which persists at all considered depths. Varying in details, this anomaly is remarkably robust irrespective of the considered day(s). We compared our results to those obtained from a different inversion scheme and an independent induction data set, and observed encouraging similarity. Combination of the two results suggests, that this conductor continues to the base of the mantle transition zone at 660 km. The nature of this anomaly is not fully understood but one possible explanation is that it is attributed to a reservoir responsible for three hotspots in the regio

Global 3-D EM inversion of Sq variations based on simultaneous source and conductivity determination: concept validation and resolution studies

We present a novel global 3-D electromagnetic (EM) inverse solution that allows to work in a unified and consistent manner with frequency-domain data that originate from ionospheric and magnetospheric sources irrespective of their spatial complexity. The main idea behind the approach is simultaneous determination of the source and conductivity distribution in the Earth. Such a determination is implemented in our solution as a looped sequential procedure that involves two steps: (1) determination of the source using a fixed 3-D conductivity model and (2) recovery of a 3-D conductivity model using a fixed source. We focus in this paper on analysis of Sq data and numerically verify each step separately and combined using data synthesized from 3-D models of the Earth induced by a realistic Sq source. To determine the source we implement an approach that makes use of a known conductivity structure of the Earth with non-uniform oceans. Based on model studies we show that this approach outperforms the conventional potential method. As for recovery of 3-D conductivity in the mantle, our inverse scheme relies on a regularized least-square formulation, exploits a limited-memory quasi-Newton optimization method and makes use of the adjoint source approach to calculate efficiently the misfit gradient. We perform resolution studies with checkerboard conductivity structures at depths between 10 and 1600 km for different inverse setups and conclude from these studies that: (1) inverting Z component gives much better results than inverting all (X, Y and Z) components; (2) data from the Sq source allows for resolving 3-D structures in depth range between 100 and 520 km; (3) the best resolution is achieved in the depth range of 100-250 k

Mapping 3-D mantle electrical conductivity from space: a new 3-D inversion scheme based on analysis of matrix Q-responses

We present a novel 3-D frequency-domain inversion scheme to recover 3-D mantle conductivity from satellite magnetic data, for example, provided by the Swarm mission. The scheme is based on the inversion of a new set of electromagnetic transfer functions, which form an array that we denote as matrix Q-response and which relate external (inducing) and internal (induced) coefficients of the spherical harmonic expansion of the time-varying magnetic field of magnetospheric origin. This concept overcomes the problems associated with source determination inherent to recent schemes based on direct inversion of internal coefficients. Matrix Q-responses are estimated from time-series of external and internal coefficients with a newly elaborated multivariate analysis scheme. An inversion algorithm that deals with matrix Q-responses has been developed. In order to make the inversion tractable, we elaborated an adjoint approach to compute the data misfit gradient and parallelized the numerical code with respect to frequencies and elementary sources, which describe the external part of the magnetic field of magnetospheric origin. Both parts of the scheme have been verified with realistic test data. Special attention is given to the issue of correlated noise due to undescribed source

Three-dimensional electrical conductivity structure beneath Australia from inversion of geomagnetic observatory data: evidence for lateral variations in transition-zone temperature, water content and melt

In this paper, we report the 3-D electrical conductivity distribution beneath the Australian continent in the depth range 410-1600 km, which we have imaged by inverting C-response estimates from a regional network of geomagnetic observatories. The inversion scheme is based on a quasi-Newton optimization method while the forward algorithm relies on an integral-equation approach. To properly account for the ocean effect in responses at coastal observatories we included a high-resolution (1°×1°) fixed thin laterally varying surface conductance layer. As starting model in the inversion we considered a laboratory-based 3-D conductivity model of the region obtained from seismic surface wave data and thermodynamic modelling. This model provides a good fit to observed C-response estimates supporting its choice as initial model. The most striking feature of the obtained 3-D model is a high-conductivity anomaly in the lower part of the mantle transition zone (MTZ; 520-660 km depth) beneath southeastern Australia implying considerable lateral as radial heterogeneity in the conductivity structure. The high-conductivity region appears to be 0.5-1 log units more conductive than previous global and other regionalized 1-D models. Further analysis using laboratory-based conductivity models combined with thermochemical phase equilibrium computations shows that the strong conductivity anomaly implies water contents of around 0.1 wt per cent in the upper part and >0.4 wt per cent in the lower part of the MTZ. This implies a large MTZ water reservoir that likely totals one to three times that which currently resides in the oceans. The amount of water in the lower MTZ appears to exceed the experimentally determined water storage capacity of the main lower MTZ mineral ringwoodite, which, as a result, undergoes dehydration-induced partial melting. Including contributions to conductivity from a thin melt layer (20 km thick) located in the mid-MTZ increases conductivity locally in the melt layer to ∼1 S m−1, that is, about 0.5 log units more conductive than the average surrounding mantle. This provides an adequate explanation for the strong conductivity anomalies observed beneath part of the continent and points to lateral variations in melt in the MT

Handling complex source structures in global EM induction studies: from C-responses to new arrays of transfer functions

The C-response is a conventional transfer function in global electromagnetic induction research and is traditionally determined from observations of magnetic variations in the vertical and horizontal components. Its interpretation relies on the assumption that the source of the variations is well approximated by a large-scale symmetric (magnetospheric) ring current, described by a single spherical harmonic. However, there is growing evidence for a more complex structure of this source. In this paper, we investigate the variability of C-responses due to sources different from the dominating large-scale symmetric ring current. We show that the effect is significant and persists at all periods. Describing the magnetospheric source by a single spherical harmonic coefficient thus injects substantial errors into the estimated responses. To overcome the problem, we introduce arrays of alternative transfer functions that relate the components of the magnetic variation to different spherical harmonic coefficients. These transfer functions can handle a complex spatial structure of the magnetospheric source. Compared to C-responses, we observe a significant increase in the coherencies relating input and output quantities of the new transfer functions, especially at high latitudes. This increases the usability of observatory magnetic data for the recovery of global 3-D mantle conductivity structur