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The pathways of fluids and mantle-originated carbon dioxide in the seismically active Ohře (Eger) Rift system appearing as mofettes at the surface are currently subject to investigation, especially by the International Continental Scientific Drilling Program “Drilling the Eger Rift”. If the aquifers show significant contrast in electrical resistivity to the host rocks, they can be investigated with geo-electromagnetic methods. However, imaging complex fluid and CO₂ pathways in detail in near-surface structures is challenging, because, in contrast to the background stratigraphy, they are often oriented in near-vertical directions. Therefore, we aim to investigate how the shallow aquifer structures can be examined best with an inductive electromagnetic method. For this purpose, we collected radio-magnetotelluric data in the Hartoušov mofette field and evaluated them by two- and three-dimensional inversions. Data from a nearby magnetotelluric station, drill hole data, gas flux measurements and electrical resistivity tomography models were used to assess the reliability and robustness of our inversion results. We concluded that the near-surface fluid reservoirs are adequately depictable, while the migration paths of gaseous CO₂ cannot be traced properly due to a lack of resistivity contrast. Our model analyses suggest that imaging the given geological setting with fluids and gases ascending in anastomosing pathways benefits from a fine-scale three-dimensional inversion approach because the fluids mostly appear as local conductive reservoir-like anomalies, which can be falsely projected onto the profiles during inversion in two dimensions. The resistivity models contribute with detailed images of the near-surface aquifers to the geodynamic model of the Ohře Rift.

We develop a three-dimensional inversion code to image the resistivity distribution of the subsurface from frequency-domain controlled-source electromagnetic data. Controlled-source electromagnetic investigations play an important role in many different geophysical prospecting applications. To evaluate controlled-source electromagnetic data collected with complex measurement setups, advanced three-dimensional modelling and inversion tools are required.We adopt a preconditioned non-linear conjugate gradient algorithm to enable three-dimensional inversion of impedance tensor and vertical magnetic transfer function data produced by multiple sets of two independent active sources. Forward simulations are performed with a finite-element solver. Increased sensitivities at source locations can optionally be counteracted with a weighting function in the regularization term to reduce source-related anomalies in the resistivity model. We investigate the capabilities of the inversion code using one synthetic and one field example. The results demonstrate that we can produce reliable subsurface models, although data sets from single pairs of independent sources remain challenging.

Controlled-source electromagnetic methods are applied to survey the electrical resistivity distribution of the subsurface. This work compares normalized electromagnetic field components and transfer functions such as impedance tensors and vertical magnetic transfer functions generated by two independent source polarizations as input data for three-dimensional inversion. As most other available inversion codes allow for inverting only one of the mentioned input data types, it is unclear which data type is preferable for controlled-source electromagnetic inversion. Our three-dimensional non-linear conjugate gradient inversion code can handle both input data types, facilitating a comparison of normalized electromagnetic field components and transfer functions inversion. Examining inversion results for a three-dimensional synthetic model with two anomalies, we infer that the transfer functions inversion is favourable for recovering the overall resistivity distribution below the receiver sites in fewer iterations. The inversion of normalized electromagnetic field components produces a sharper image of the anomalies and may be capable of detecting the resistivity distribution below the extended sources, which comes at the price of introducing a more heterogeneous background resistivity in the model.

We advance a previously established method for 2D inversion of electromagnetic data, in which the smoothness constraints are locally reweighted according to the seismic envelope fractional gradients (SEFGs). As the first step of our modifications, seis-mic envelope values are edited and normalized. This results in the rejection of noise-contaminated parts, clipping the envelope outliers and increasing the reflectivity power of weak seismic signals. Second, we introduce a weighting matrix in the normali-zation process to incorporate prior information in the con-strained inversion regarding the relative contrasts of electrical resistivity in the given parts of the model. Third, due to normal-izing the SEFG, there is no need to search for an optimum sta-bilization factor to modify local smoothing weights, and hence, we set it to a constant value. Finally, an Occam inversion with additional Levenberg-Marquardt damping is used to mitigate possible artifacts in the resistivity model generated by reflection seismic constraints. In applying the proposed scheme to a syn-thetic example, interfaces of various geologic units are restored as sharp boundaries. In addition, artifacts generated in the origi-nal approach are effectively mitigated thanks to the applied nor-malization process. For the first time, we apply the method to radio-magnetotelluric (RMT) and controlled-source audio-mag-netotelluric (CSAMT) field data acquired along a profile across a known aquifer in Heby, Sweden. Our inversion models com-pare favorably to previously presented results from seismic and geoelectric data. By modifying the smoothness weights based on SEFG, the depth to the bedrock is recovered well, being con-strained by the corresponding interfaces in the seismic image. The resistivity models from seismically constrained inversions of RMT and CSAMT data reveal steeply dipping and possibly fractured bedrock underneath the valley-shaped aquifer in the area. This interpretation is verified by borehole logs.

In 2-D magnetotelluric modelling, the standard application of Dirichlet boundary conditions (BC) may severely diminish the solution accuracy, because the unknown scattered part of the electromagnetic field is erroneously reflected at the domain boundary. Therefore, we adapt the total and scattered field decomposition (TSFD) to geophysical modelling, enabling the application of fully absorbing boundary methods, here perfectly matched layers (PML), to the scattered field. Our novel TSFD divides the modelling domain into two regions. In the total-field region containing the area of interest, the solution is computed for the total field. In the scattered-field region containing the boundaries, the solution is obtained for the scattered field, which is fully absorbed by PML at the boundaries. The plane-wave source is excited at the TSFD interface between both regions. Thus, boundary reflections are eradicated leading to superior solution accuracy, and boundaries can be placed closer to the receivers, shrinking the computational problem. Especially for challenging models with strong lateral changes, the solution accuracy of the TSFD is superior to that of the standard Dirichlet approach. Owing to the linearity of Maxwell's equations, the inaccuracy introduced to the electric and magnetic fields by using Dirichlet BC can be expected to partly cancel out in the magnetotelluric transfer functions, for example the impedance tensor. In this work, we quantify this cancellation effect. The inaccuracy is less than typical measurement errors in the vast majority of apparent resistivity and phase data, even, when the primary fields are strongly inaccurate.

We develop an efficient and robust iterative framework suitable for solving the linear system of equations resulting from the spectral element discretisation of the curl-curl equation of the total electric field encountered in geophysical controlled-source electromagnetic applications. We use the real-valued equivalent form of the original complex-valued system and solve this arising real-valued two-by-two block system (outer system) using the generalised conjugate residual method preconditioned with a highly efficient block-based PREconditioner for Square Blocks (PRESB). Applying this preconditioner equates to solving two smaller inner symmetric systems which are either solved using a direct solver or iterative methods, namely the generalised conjugate residual or the flexible generalised minimal residual methods preconditioned with the multigrid-based auxiliary-space preconditioner AMS. Our numerical experiments demonstrate the robustness of the outer solver with respect to spatially variable material parameters, for a wide frequency range of five orders of magnitude (0.1-10’000 Hz), with respect to the number of degrees of freedom, and for stretched structured and unstructured as well as locally refined meshes. For all the models considered, the outer solver reaches convergence in a small (typically < 20) number of iterations. Further, our numerical tests clearly show that solving the two inner systems iteratively using the indicated preconditioned iterative methods is computationally beneficial in terms of memory requirement and time spent as compared to a direct solver. On top of that, our iterative framework works for large-scale problems where direct solvers applied to the original complex-valued systems succumb due to their excessive memory consumption, thus making the iterative framework better suited for large-scale 3D problems. Comparison to a similar iterative framework based on a block-diagonal and the auxiliary-space preconditioners reveals that the PRESB preconditioner requires slightly fewer iterations to converge yielding a certain gain in time spent to obtain the solution of the two-by-two block system.

For forward modelling of realistic 3-D land-based controlled-source electromagnetic (EM) problems, we develop a parallel spectral element approach, blending the flexibility and versatility of the finite element method in using unstructured grids with the accuracy of the spectral method. Complex-shaped structures and topography are accommodated by using unstructured hexahedral meshes, in which the elements can have curved edges and non-planar faces. Our code is the first spectral element algorithm in EM geophysics that uses the total field formulation (here that of the electric field). Combining unstructured grids and a total field formulation provides advantages in dealing with topography, in particular, when the transmitter is located on rough surface topography. As a further improvement over existing spectral element methods, our approach does not only allow for arbitrary distributions of conductivity, but also of magnetic permeability and dielectric permittivity. The total electric field on the elements is expanded in terms of high-order Lagrangian interpolants, and element-wise integration in the weak form of the boundary value problem is accomplished by Gauss–Legendre–Lobatto quadrature. The resulting complex-valued linear system of equations is solved using the direct solver MUMPS, and, subsequently, the magnetic field is computed at the points of interest by Faraday’s law. Five numerical examples comprehensively study the benefits of this algorithm. Comparisons to semi-analytical and finite element results confirm accurate representation of the EM responses and indicate low dependency on mesh discretization for the spectral element method. A convergence study illuminates the relation between high order polynomial approximation and mesh size and their effects on accuracy and computational cost revealing that high-order approximation yields accurate modelling results for very coarse meshes but is accompanied by high computational cost. The presented numerical experiments give evidence that 2nd and 3rd degree polynomials in combination with moderately discretized meshes provide better trade-offs in terms of computational resources and accuracy than lowest and higher order spectral element methods. To our knowledge, our final example that includes pronounced surface topography and two geometrically complicated conductive anomalies represents the first successful attempt at using 2nd order hexahedral elements supporting curved edges and non-planar faces in controlled-source EM geophysics.

We developed a 3-D forward modelling code, which simulates controlled source electromagnetic problems in frequency domain using edge-based finite elements and a total electric field approach. To evaluate electromagnetic data acquired across complex subsurface structures, software performing accurate 3-D modelling is required, especially for incorporation in inversion approaches. Our modelling code aims at finding a good compromise between the necessary solution accuracy at the points of interest and the general problem size by using a goal-oriented mesh refinement strategy designed for models of variable electric conductivity and magnetic permeability. To formulate an improved error estimator suitable for controlled source electromagnetic problems, we developed literature approaches of mesh refinement further targeting three aspects. First, to generate a roughly homogeneously fine mesh discretization around all receiver sites, our new error estimator weights the adjoint source term by the approximate decay of the electric field with increasing distance from the primal source using the expression for a homogeneous half-space. This causes almost no additional computational cost. Second, the error estimator employed in the refinement approach can be optimized for models with pronounced conductivity and magnetic permeability contrasts as often encountered in, for example, mineral prospecting scenarios by optionally including terms that measure the continuity of the normal component of current flow and the tangential component of the magnetic field across interfaces of abutting elements. Third, to avoid amplitude-dependent over-refining of the mesh, we formulate our element-wise error estimators relative to the local amplitude of the electromagnetic field. In this work, we evaluate the implemented adaptive mesh refinement approach and its solution accuracy comparing our solutions for simple 1-D models and a model with 3-D anomalies to semi-analytic 1-D solutions and a second-order finite-element code, respectively. Furthermore, a feasibility study for controlled-source electromagnetic measurements across ferrous mineral deposits is conducted. The numerical experiments demonstrate that our new refinement procedure generates problem-specific finite-element meshes and yields accurate solutions for both simple synthetic models and realistic survey scenarios. Especially for the latter, characteristics of our code, such as the possibility of modelling extended sources as well as including arbitrary receiver distributions and detailed subsurface anomalies, are beneficial.

Long-period magnetotelluric (MT) data can be used to interpret upper mantle temperature, hydrogen content, and the presence of partial melt, all of which strongly influence mantle viscosity. We have collected the first long-period MT data in Svalbard and have combined them with preexisting broadband MT data to produce a model of the electrical resistivity of Svalbard's upper mantle. Asthenospheric resistivities are low compared to stable continental settings but more comparable to young oceanic asthenosphere, suggesting that the physical state of Svalbard's upper mantle is controlled by its proximity to the Mid-Atlantic Ridge. Interpretation of the MT model using a petrologically constrained genetic algorithm approach shows that partial melt is present in the uppermost asthenosphere beneath Svalbard. This is the first direct evidence of partial melt in Svalbard's asthenosphere from deep geophysical soundings. Viscosities calculated from the geophysical data show a low viscosity layer (similar to 10(18) Pa s) coincident with the partial melt layer, underlain by a higher viscosity layer (similar to 10(20) Pa s) extending to the transition zone. Viscosities calculated from glacial isostatic adjustment (GIA) data in Svalbard show a considerable range due mainly to uncertainties in past ice sheet models. Improved constraints on Svalbard's mantle viscosity from geophysical data may help to improve these GIA models.

Exact computation of the gravitational field and gravitational gradient tensor for a general mass body is a core routine to model the density structure of the Earth. In this study, we report on the existence of closed-form solutions of the gravitational potential, gravitational field and gravitational gradient tensor for a general polyhedral mass body with a polynomial density function of arbitrary non-negative integer orders that can simultaneously vary in both horizontal and vertical directions. Our closed-form solutions of the gravitational potential and the gravitational field are singularity-free, which implies that the observation sites can have arbitrary geometric relationships with polyhedral mass source bodies. However, weak logarithmic singularities exist on the edges of polyhedra for the gravitational gradient tensor. A simple prismatic mass body with polynomial density contrast varying in the vertical direction and a complicated dodecahedral mass body with quartic-order density contrasts were tested to verify the accuracy of the newly derived closed-form solutions. For the gravitational potential, gravitational fields and gradient tensors, our closed-form solutions are in excellent agreement with previously published analytical solutions and Gaussian numerical quadrature solutions.