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The geomagnetic deep sounding (GDS) method is one of electromagnetic (EM) methods in geophysics that allows the estimation of the subsurface electrical conductivity distribution. This paper presents the inversion modeling of GDS data employing Markov Chain Monte Carlo (MCMC) algorithm to evaluate the marginal posterior probability of the model parameters. We used thin-sheet model to represent quasi-3D conductivity variations in the heterogeneous subsurface. The algorithm was applied to invert field GDS data from the zone covering an area that spans from eastern margin of the Bohemian Massif to the West Carpathians in Europe. Conductivity anomalies obtained from this study confirm the well-known large-scale tectonic setting of the area.

In geomagnetic deep sounding (GDS), we measure the natural Earth’s magnetic transient variations to infer large-scale subsurface conductivity distribution. Recent advances in magnetotelluric (MT) technique tend to put the attention to move towards local scale investigation of conductivity anomalies with more economic interests as in exploration for mineral, geothermal, or hydrocarbon. However, GDS is still considered as the most appropriate natural source electromagnetic (EM) method capable of imaging the Earth’s interior especially for tectonic study at the regional and continental scales (e.g., [

This paper describes the inversion modeling technique for GDS data in terms of conductivity distribution by using the Markov Chain Monte Carlo (MCMC) algorithm. The MCMC inversion algorithm has been applied for 1D modeling of MT [

We will first briefly review the concept and the formulation of the thin-sheet modeling and then describe the MCMC inversion algorithm. The result of inversion of real GDS data from Bohemian Massif-West Carpathians and also its interpretation will be discussed.

In electromagnetic (EM) geophysics, the thin-sheet modeling refers to an approximation of 3D conductivity variation by a thin layer with variable conductance, that is, integrated conductivity over the thickness of the thin layer. Such approximation is generally valid in the large-scale studies where the heterogeneities are confined in a layer with thickness much smaller than the penetration depth of EM fields. The thin-sheet modeling significantly simplifies the resolution of the Maxwell’s equations describing the EM fields in quasi-3D media. We used an algorithm employing integral equation method for thin-sheet EM modeling proposed by Vasseur and Weidelt [

A heterogeneous superficial thin-sheet model embedded in 1D or layered host medium. The resistivities and thicknesses are arbitrary, for the illustration purpose only.

The thin-sheet containing heterogeneities is discretized into rectangular uniform blocks in which the conductance is assumed constant. The total electric field in the

For a large number of blocks in the anomalous domain, a direct matrix inversion to resolve (

The normal electric field

Resolving (

where

Assuming that the parameters for 1D host medium are known, the inverse problem consists in determining the conductance of the thin-sheet discretized in blocks (see Figure

Equation (

Updating model parameters sequentially using (

The iterative process described earlier is commonly called Gibbs sampler that may be employed to approximate a probability density function (PDF). It leads to the approximation of the marginal posterior probability for model parameter

The MCMC inversion algorithm was tested to invert synthetic data associated with thin-sheet models with satisfactory results [

The geology of the study area covering Czech, Slovakia, and parts of Poland is relatively complex since it is composed of two different geological units, that is, the Bohemian Massif and the western part of the East Carpathians. The Bohemian Massif and the Carpathians are part of the Hercynian and the Alpine chains, respectively, that are fundamental elements of the geology of Europe. Figure

Simplified tectonic map of Carpathians and Pannonian Basin (adapted from [

The study of the Carpathian conductivity anomaly has a very long history since its discovery deduced from geomagnetic soundings in the late 70s. Newly acquired data in recent studies still exhibit roughly similar regularities, in particular those related to the West Carpathian anomaly (WCA). Kováčiková et al. [

The GDS data set used in our study is fundamentally the same as previous studies (e.g., [

The study area shown in WGS84 geographical coordinates system along with GDS stations (dots), country border outline (thin dashed line), the West Carpathian anomaly (red dashed line), and discretization grids (thin dotted line) of 9 by 18 blocks with 35 by 35 km width for the thin-sheet modeling.

The observed data are magnetic transfer function at 143 single stations for the periods of 20, 32, 64, and 98 minutes. The data processing is described in [

The GDS data represented as real part (a) and imaginary part (b) of the magnetic inductions vectors. The real part was inverted in accordance with Parkinson’s convention such that the induction arrows point towards more conductive zones. The reference magnitude of the induction arrows is given in the bottom-left corner of each map.

For the inversion, instead of interpolating the GDS data into more regular grid, for each block, we selected the datum located closest to the center of the block. In addition, we chose the most representative and coherent datum by evaluating the difference in a least-squared sense of the datum with its neighbors at the same block (if they exist). In this way we avoid introducing spurious effects that may be caused by interpolation process. However, there were blocks with no data at all that will be less constrained in the inversion process (see Figure

The real part of GDS data (arrows) and the calculated data (arrows without arrowhead) showing the fitness of the model response to the observed data. In blocks with only data plotted, both observed and calculated induction arrows coincide although their magnitude might be different.

From previous studies [

Starting from normal conductance (i.e., 20 Siemens/m) the inversion was performed up to 20 iterations. The posterior model was obtained by averaging the conductances of each block from the last 15 iterations. We present in Figure

The inverse model is presented in Figure

Conductance map from inversion of GDS data with initial block size (a) and after rediscretization of blocks (b). Interpreted anomalies include West Carpathian anomaly (WCA), Bohemian Massif anomaly (BMA), Moravian-Silesian lineament (MSL), and Labe lineament (LL).

In order to propose more meaningful interpretation, we rediscretized the thin-sheet conductance map obtained from the inversion into blocks with a half of their initial size, that is, approximately 17.5 by 17.5 km. The new interpolated conductance is obtained by employing a simple 5 by 5 block 2D moving average filter. The resulting model (see Figure

The result confirms the existence of the well-known conductivity anomalies, that is, the Bohemian Massif anomaly (BMA) and the West Carpathian anomaly (WCA). More detailed examination of the result shows that there is a lateral shift between the surface curved features of the West Carpathians and the conductive anomaly from the model. This anomaly is interpreted rather as the effect of the root of the dipping contact between geological entities forming the West Carpathians at depth (more than 10 km). This may also reflect the fact that we determine the integrated conductivity of the thin plate, which may include the contribution of anomalies located at different depths between 20 and 30 km.

Our model also highlights conductive zones other than previously known anomalies. By considering also the qualitative interpretation of the induction vector maps, other less obvious anomalies found coincide with the tectonic features present in this area such as the Moravian-Silesian lineament (MSL) and the Labe lineament (LL) [

The resistivity corresponding to the conductance of the anomaly of a thin plate with thickness of 10 km is less than 10 Ohm·m. Other estimates indicate the resistivity lower than 5 Ohm·m resistivity. Several hypotheses have been proposed to explain the presence of such a conducting medium in a very resistive host (between 500 and 1000 Ohm·m) more than 20 km deep. One that is most plausible is to assume that the conductive body is sedimentary rocks saturated with saline water. The presence of saline water at high temperature in porous rocks greatly increases the conductivity of the latter. This hypothesis favoured in Jankowski et al. [

The thin-sheet modeling employing MCMC inversion algorithm has been applied to GDS data from the area covering the Bohemian Massif and the West Carpathians with satisfactory result. The conductance map obtained from such inversion modeling correlates well with the regional geology of the study area. The result allows more quantitative analysis of the conductivity anomalies present in the area rather than only qualitative interpretation based on the pattern of the induction vectors or contoured map of the magnetic transfer function values.

The MCMC algorithm applied in this study is relatively generic and would be applicable to most geophysical inverse problems with appropriate parameterization. However, the MCMC algorithm remains a computer intensive method since it has to resolve the forward modeling a substantial number of times to explore the model space. Due to limited number of iterations that can be performed with our current computation resources, the posterior quantities leading to the model uncertainty is still underexploited. With the advances in computation processing power, in the near future, it would be possible to perform more thorough analysis of the inverse model uncertainty from the posterior probability function. It would be also possible to employ a full 3D EM modeling to avoid limitation of the thin-sheet modeling in representing complex subsurface heterogeneities.

The authors greatly acknowledge Václav Červ and his colleagues from the Institute of Geophysics, Prague, Czech Republic, for providing the GDS data for this study.