ALBERT

Description: We developed a novel parallel domain-decomposition approach for 3D large-scale electromagnetic induction modeling in the earth. We used the edge-based finite-element method and unstructured meshes. Unstructured meshes were divided into sets of nonoverlapping subdomains. We used the curl-curl electric field equation to carry out the analysis. In each subdomain, the electric field was discretized by first-order vector shape functions along the edges of tetrahedral elements. The tangential components of the magnetic field on the interfaces of the subdomains were defined as a set of Lagrange multipliers. The unknown Lagrange multipliers were solved from an interface problem defined on the interfaces of the subdomains. With the availability of the Lagrange multipliers, the electric field values in each subdomain were solved independently. Three synthetic examples were evaluated to verify our code. Excellent agreement with previously published solutions was obtained. Synthetic examples revealed that our domain decomposition technique is scalable with respect to the number of subdomains and robust with regard to frequency and the heterogeneous distribution of material parameters, i.e., electric conductivity, electric permittivity, and magnetic permeability.

Description: The Okavango Delta of northern Botswana is one of the world's largest inland deltas or megafans. To obtain information on the character of sediments and basement depths, audiomagnetotelluric (AMT), controlled-source audiomagnetotelluric (CSAMT) and central-loop transient electromagnetic (TEM) data were collected on the largest island within the delta. The data were inverted individually and jointly for 1-D models of electric resistivity. Distortion effects in the AMT and CSAMT data were accounted for by including galvanic distortion tensors as free parameters in the inversions. By employing Marquardt–Levenberg inversion, we found that a 3-layer model comprising a resistive layer overlying sequentially a conductive layer and a deeper resistive layer was sufficient to explain all of the electromagnetic data. However, the top of the basal resistive layer from electromagnetic-only inversions was much shallower than the well-determined basement depth observed in high-quality seismic reflection images and seismic refraction velocity tomograms. To resolve this discrepancy, we jointly inverted the electromagnetic data for 4-layer models by including seismic depths to an interface between sedimentary units and to basement as explicit a priori constraints. We have also estimated the interconnected porosities, clay contents and pore-fluid resistivities of the sedimentary units from their electrical resistivities and seismic P -wave velocities using appropriate petrophysical models. In the interpretation of our preferred model, a shallow ~40 m thick freshwater sandy aquifer with 85–100 m resistivity, 10–32 per cent interconnected porosity and 〈13 per cent clay content overlies a 105–115 m thick conductive sequence of clay and intercalated salt-water-saturated sands with 15–20 m total resistivity, 1–27 per cent interconnected porosity and 15–60 per cent clay content. A third ~60 m thick sandy layer with 40–50 m resistivity, 10–33 per cent interconnected porosity and 〈15 per cent clay content is underlain by the basement with 3200–4000 m total resistivity. According to an interpretation of helicopter TEM data that cover the entire Okavango Delta and borehole logs, the second and third layers may represent lacustrine sediments from Paleo Lake Makgadikgadi and a moderately resistive freshwater aquifer comprising sediments of the recently proposed Paleo Okavango Megafan, respectively.

Description: The Okavango Delta of northern Botswana is one of the world's largest inland deltas or megafans. To obtain information on the character of sediments and basement depths, audiomagnetotelluric (AMT), controlled-source audiomagnetotelluric (CSAMT) and central-loop transient electromagnetic (TEM) data were collected on the largest island within the delta. The data were inverted individually and jointly for 1-D models of electric resistivity. Distortion effects in the AMT and CSAMT data were accounted for by including galvanic distortion tensors as free parameters in the inversions. By employing Marquardt–Levenberg inversion, we found that a 3-layer model comprising a resistive layer overlying sequentially a conductive layer and a deeper resistive layer was sufficient to explain all of the electromagnetic data. However, the top of the basal resistive layer from electromagnetic-only inversions was much shallower than the well-determined basement depth observed in high-quality seismic reflection images and seismic refraction velocity tomograms. To resolve this discrepancy, we jointly inverted the electromagnetic data for 4-layer models by including seismic depths to an interface between sedimentary units and to basement as explicit a priori constraints. We have also estimated the interconnected porosities, clay contents and pore-fluid resistivities of the sedimentary units from their electrical resistivities and seismic P -wave velocities using appropriate petrophysical models. In the interpretation of our preferred model, a shallow ~40 m thick freshwater sandy aquifer with 85–100 m resistivity, 10–32 per cent interconnected porosity and 〈13 per cent clay content overlies a 105–115 m thick conductive sequence of clay and intercalated salt-water-saturated sands with 15–20 m total resistivity, 1–27 per cent interconnected porosity and 15–60 per cent clay content. A third ~60 m thick sandy layer with 40–50 m resistivity, 10–33 per cent interconnected porosity and 〈15 per cent clay content is underlain by the basement with 3200–4000 m total resistivity. According to an interpretation of helicopter TEM data that cover the entire Okavango Delta and borehole logs, the second and third layers may represent lacustrine sediments from Paleo Lake Makgadikgadi and a moderately resistive freshwater aquifer comprising sediments of the recently proposed Paleo Okavango Megafan, respectively.

Description: The near juxtaposition of the Makgadikgadi Basin (Botswana), the world’s largest saltpan complex, with the Okavango Delta, one of the planet’s largest inland deltas (technically an alluvial megafan), has intrigued explorers and scientists since the middle of the 19 th century. It was clear from early observations that the Makgadikgadi Basin once contained a huge lake, paleo–Lake Makgadikgadi. Several authors have since speculated that this lake also covered wide regions to the north and west of the Makgadikgadi Basin. Our interpretation of unusually high-quality helicopter time-domain electromagnetic (HTEM) data indicates that paleo–Lake Makgadikgadi extended northwestward at least into the region presently occupied by the Okavango Delta. The total area of paleo–Lake Makgadikgadi exceeded 90,000 km 2 , larger than Earth’s most extensive freshwater body today, Lake Superior (North America). Our HTEM data, constrained by ground-based geophysical and borehole information, also provide evidence for a paleo-megafan underlying paleo–Lake Makgadikgadi sediments.

Description: Probabilistic inversion methods based on Markov chain Monte Carlo (MCMC) simulation are well suited to quantify parameter and model uncertainty of nonlinear inverse problems. Yet, application of such methods to CPU-intensive forward models can be a daunting task, particularly if the parameter space is high dimensional. Here, we present a 2-D pixel-based MCMC inversion of plane-wave electromagnetic (EM) data. Using synthetic data, we investigate how model parameter uncertainty depends on model structure constraints using different norms of the likelihood function and the model constraints, and study the added benefits of joint inversion of EM and electrical resistivity tomography (ERT) data. Our results demonstrate that model structure constraints are necessary to stabilize the MCMC inversion results of a highly discretized model. These constraints decrease model parameter uncertainty and facilitate model interpretation. A drawback is that these constraints may lead to posterior distributions that do not fully include the true underlying model, because some of its features exhibit a low sensitivity to the EM data, and hence are difficult to resolve. This problem can be partly mitigated if the plane-wave EM data is augmented with ERT observations. The hierarchical Bayesian inverse formulation introduced and used herein is able to successfully recover the probabilistic properties of the measurement data errors and a model regularization weight. Application of the proposed inversion methodology to field data from an aquifer demonstrates that the posterior mean model realization is very similar to that derived from a deterministic inversion with similar model constraints.

Description: We have developed a novel goal-oriented adaptive mesh refinement approach for finite-element methods to model plane wave electromagnetic (EM) fields in 3-D earth models based on the electric field differential equation. To handle complicated models of arbitrary conductivity, magnetic permeability and dielectric permittivity involving curved boundaries and surface topography, we employ an unstructured grid approach. The electric field is approximated by linear curl-conforming shape functions which guarantee the divergence-free condition of the electric field within each tetrahedron and continuity of the tangential component of the electric field across the interior boundaries. Based on the non-zero residuals of the approximated electric field and the yet to be satisfied boundary conditions of continuity of both the normal component of the total current density and the tangential component of the magnetic field strength across the interior interfaces, three a-posterior error estimators are proposed as a means to drive the goal-oriented adaptive refinement procedure. The first a-posterior error estimator relies on a combination of the residual of the electric field, the discontinuity of the normal component of the total current density and the discontinuity of the tangential component of the magnetic field strength across the interior faces shared by tetrahedra. The second a-posterior error estimator is expressed in terms of the discontinuity of the normal component of the total current density (conduction plus displacement current). The discontinuity of the tangential component of the magnetic field forms the third a-posterior error estimator. Analytical solutions for magnetotelluric (MT) and radiomagnetotelluric (RMT) fields impinging on a homogeneous half-space model are used to test the performances of the newly developed goal-oriented algorithms using the above three a-posterior error estimators. A trapezoidal topographical model, using normally incident EM waves at both MT and RMT frequencies, is adopted to further test the convergence of the newly developed algorithms against a surface integral approach. Next, the 3D-1 benchmark model from the COMMEMI project is used to show the efficiency of the goal-oriented adaptive algorithm and to compare our solutions against volume integral solutions and other finite-element solutions. For all three test cases, we found that the error estimator using face jumps of normal components of current density embedded in the goal-oriented adaptive refinement procedure shows the most robust performance.

Description: Helicopter time-domain electromagnetic (HTEM) surveying has historically been used for mineral exploration, but over the past decade it has started to be used in environmental assessments and geologic and hydrologic mapping. Such surveying is a cost-effective means of rapidly acquiring densely spaced data over large regions. At the same time, the quality of HTEM data can suffer from various inaccuracies. We developed an effective strategy for processing and inverting a commercial HTEM data set affected by uncertainties and systematic errors. The delivered data included early time gates contaminated by transmitter currents, noise in late time gates, and amplitude shifts between adjacent flights that appeared as artificial lineations in maps of the data and horizontal slices extracted from inversion models. Multiple processing steps were required to address these issues. Contaminated early time gates and noisy late time gates were semiautomatically identified and eliminated on a record-by-record basis. Timing errors between the transmitter and receiver electronics and inaccuracies in absolute amplitudes were corrected after calibrating selected HTEM data against data simulated from accurate ground-based TEM measurements. After editing and calibration, application of a quasi-3D spatially constrained inversion scheme significantly reduced the artificial lineations. Residual lineations were effectively eliminated after incorporating the transmitter and receiver altitudes and line-to-line amplitude factors in the inversion process. The final inverted model was very different from that generated from the original data provided by the contractor. For example, the average resistivity of the thick surface layer decreased from $$\sim 1800$$ to $$\sim 30\hbox{ \hspace{0.17em} }\hbox{ \hspace{0.17em} }\mathrm{\Omega m}$$ , the depths to the layer boundaries were reduced by 15%–23%, and the artificial lineations were practically eliminated. Our processing and inversion strategy is entirely general, such that with minor system-specific modifications it could be applied to any HTEM data set, including those recorded many years ago.

Description: We have developed a generalized and stable surface integral formula for 3-D uniform inducing field and plane wave electromagnetic induction problems, which works reliably over a wide frequency range. Vector surface electric currents and magnetic currents, scalar surface electric charges and magnetic charges are treated as the variables. This surface integral formula is successfully applied to compute the electromagnetic responses of 3-D topography to low frequency magnetotelluric and high frequency radio-magnetotelluric fields. The standard boundary element method which is used to solve this surface integral formula quickly exceeds the memory capacity of modern computers for problems involving hundreds of thousands of unknowns. To make the surface integral formulation applicable and capable of dealing with large-scale 3-D geo-electromagnetic problems, we have developed a matrix-free adaptive multilevel fast multipole boundary element solver. By means of the fast multipole approach, the time-complexity of solving the final system of linear equations is reduced to O ( m log m ) and the memory cost is reduced to O ( m ), where m is the number of unknowns. The analytical solutions for a half-space model were used to verify our numerical solutions over the frequency range 0.001–300 kHz. In addition, our numerical solution shows excellent agreement with a published numerical solution for an edge-based finite-element method on a trapezoidal hill model at a frequency of 2 Hz. Then, a high frequency simulation for a similar trapezoidal hill model was used to study the effects of displacement currents in the radio-magnetotelluric frequency range. Finally, the newly developed algorithm was applied to study the effect of moderate topography and to evaluate the applicability of a 2-D RMT inversion code that assumes a flat air–Earth interface, on RMT field data collected at Smørgrav, southern Norway. This paper constitutes the first part of a hybrid boundary element-finite element approach to compute the electromagnetic field inside structures involving complex 3-D conductivity and permittivity distributions.

Description: Surface-based monitoring of mass transfer caused by injections and extractions in deep boreholes is crucial to maximize oil, gas and geothermal production. Inductive electromagnetic methods, such as magnetotellurics, are appealing for these applications due to their large penetration depths and sensitivity to changes in fluid conductivity and fracture connectivity. In this work, we propose a 3-D Markov chain Monte Carlo inversion of time-lapse magnetotelluric data to image mass transfer following a saline fluid injection. The inversion estimates the posterior probability density function of the resulting plume, and thereby quantifies model uncertainty. To decrease computation times, we base the parametrization on a reduced Legendre moment decomposition of the plume. A synthetic test shows that our methodology is effective when the electrical resistivity structure prior to the injection is well known. The centre of mass and spread of the plume are well retrieved. We then apply our inversion strategy to an injection experiment in an enhanced geothermal system at Paralana, South Australia, and compare it to a 3-D deterministic time-lapse inversion. The latter retrieves resistivity changes that are more shallow than the actual injection interval, whereas the probabilistic inversion retrieves plumes that are located at the correct depths and oriented in a preferential north–south direction. To explain the time-lapse data, the inversion requires unrealistically large resistivity changes with respect to the base model. We suggest that this is partly explained by unaccounted subsurface heterogeneities in the base model from which time-lapse changes are inferred.