Parallel three-dimensional magnetotelluric inversion using adaptive finite-element method. Part I: theory and synthetic study

This paper presents a distributed magnetotelluric inversion scheme based on adaptive finite-element method (FEM). The key novel aspect of the introduced algorithm is the use of automatic mesh refinement techniques for both forward and inverse modelling. These techniques alleviate tedious and subjective procedure of choosing a suitable model parametrization. To avoid overparametrization, meshes for forward and inverse problems were decoupled. For calculation of accurate electromagnetic (EM) responses, automatic mesh refinement algorithm based on a goal-oriented error estimator has been adopted. For further efficiency gain, EM fields for each frequency were calculated using independent meshes in order to account for substantially different spatial behaviour of the fields over a wide range of frequencies. An automatic approach for efficient initial mesh design in inverse problems based on linearized model resolution matrix was developed. To make this algorithm suitable for large-scale problems, it was proposed to use a low-rank approximation of the linearized model resolution matrix. In order to fill a gap between initial and true model complexities and resolve emerging 3-D structures better, an algorithm for adaptive inverse mesh refinement was derived. Within this algorithm, spatial variations of the imaged parameter are calculated and mesh is refined in the neighborhoods of points with the largest variations. A series of numerical tests were performed to demonstrate the utility of the presented algorithms. Adaptive mesh refinement based on the model resolution estimates provides an efficient tool to derive initial meshes which account for arbitrary survey layouts, data types, frequency content and measurement uncertainties. Furthermore, the algorithm is capable to deliver meshes suitable to resolve features on multiple scales while keeping number of unknowns low. However, such meshes exhibit dependency on an initial model guess. Additionally, it is demonstrated that the adaptive mesh refinement can be particularly efficient in resolving complex shapes. The implemented inversion scheme was able to resolve a hemisphere object with sufficient resolution starting from a coarse discretization and refining mesh adaptively in a fully automatic process. The code is able to harness the computational power of modern distributed platforms and is shown to work with models consisting of millions of degrees of freedom. Significant computational savings were achieved by using locally refined decoupled meshe

Robust and scalable 3-D geo-electromagnetic modelling approach using the finite element method

We present a robust and scalable solver for time-harmonic Maxwell's equations for problems with large conductivity contrasts, wide range of frequencies, stretched grids and locally refined meshes. The solver is part of the fully distributed adaptive 3-D electromagnetic modelling scheme which employs the finite element method and unstructured non-conforming hexahedral meshes for spatial discretization using the open-source software deal.II. We use the complex-valued electric field formulation and split it into two real-valued equations for which we utilize an optimal block-diagonal pre-conditioner. Application of this pre-conditioner requires the solution of two smaller real-valued symmetric problems. We solve them by using either a direct solver or the conjugate gradient method pre-conditioned with the recently introduced auxiliary space technique. The auxiliary space pre-conditioner reformulates the original problem in form of several simpler ones, which are then solved using highly efficient algebraic multigrid methods. In this paper, we consider the magnetotelluric case and verify our numerical scheme by using COMMEMI 3-D models. Afterwards, we run a series of numerical experiments and demonstrate that the solver converges in a small number of iterations for a wide frequency range and variable problem sizes. The number of iterations is independent of the problem size, but exhibits a mild dependency on frequency. To test the stability of the method on locally refined meshes, we have implemented a residual-based a posteriori error estimator and compared it with uniform mesh refinement for problems up to 200 million unknowns. We test the scalability of the most time consuming parts of our code and show that they fulfill the strong scaling assumption as long as each MPI process possesses enough degrees of freedom to alleviate communication overburden. Finally, we refer back to a direct solver-based pre-conditioner and analyse its complexity in time. The results show that for multiple right-hand sides the direct solver-based pre-conditioner can still be faster for problems of medium size. On the other hand, it also shows non-linear growth in memory, whereas the auxiliary space method increases only linearly.ISSN:0956-540XISSN:1365-246

Swarm Satellites in EM Induction Studies

There exists no single EM induction source that is sensitive to the whole mantle.We need to integrate several natural sources in order to bridge across the scales.On the global scale, the most promising methodology is to combine magnetospheric and ocean tidal signals to image mantle under both continents and oceans

Satellite tidal magnetic signals constrain oceanic lithosphere-asthenosphere boundary

The tidal flow of electrically conductive oceans through the geomagnetic field results in the generation of secondary magnetic signals, which provide information on the subsurface structure. Data from the new generation of satellites were shown to contain magnetic signals due to tidal flow; however, there are no reports that these signals have been used to infer subsurface structure. We use satellite-detected tidal magnetic fields to image the global electrical structure of the oceanic lithosphere and upper mantle down to a depth of about 250 km. The model derived from more than 12 years of satellite data reveals a ≈72-km-thick upper resistive layer followed by a sharp increase in electrical conductivity likely associated with the lithosphere-asthenosphere boundary, which separates colder rigid oceanic plates from the ductile and hotter asthenosphere.ISSN:2375-254

Satellite Tidal Magnetic Signals Constrain Oceanic Lithosphere-Asthenosphere Boundary Earth Tomography with Tidal Magnetic Signals

The tidal flow of electrically conductive oceans through the geomagnetic field results in the generation of secondary magnetic signals, which provide information on the subsurface structure. Data from the new generation of satellites were shown to contain magnetic signals due to tidal flow; however, there are no reports that these signals have been used to infer subsurface structure. Here we use satellite-detected tidal magnetic fields to image the global electrical structure of the oceanic lithosphere and upper mantle down to a depth of about 250 km. The model derived from more than 12 years of satellite data reveals an Approximately 72 km thick upper resistive layer followed by a sharp increase in electrical conductivity likely associated with the lithosphere-asthenosphere boundary, which separates colder rigid oceanic plates from the ductile and hotter asthenosphere

Dreidimensionale Controlled-Source-Elektromagnetik-Inversion unter Verwendung von modernen Rechenkonzepten

The main goals of this thesis are: (i) developing a 3D inversion scheme for controlled-source electromagnetic (CSEM) data in frequency-domain using modern computational concepts; (ii) creating a methodology to invert real datasets including those collected using a three-phase transmitter configuration developed within the research group I have been working with. Three- dimensional inversion of non-stationary electromagnetic fields is a challenging task that, given problems of practical interest, can only be handled through using massively parallel platforms. Therefore, I developed a parallel distributed inversion scheme that runs on modern supercomputers and clusters. Since memory that is required to store vectors, matrices and arrays, as well as workload are distributed evenly among processes, good scalability of the numerical scheme is achieved. A number of technical and numerical challenges were addressed in the implementation. First, I use a direct solver for 3D forward modeling. Since direct solvers have not been widely used for 3D inversions, analysis of advantages and drawbacks of this decision from both technical and numerical points of view is presented. Second, in inversion I calculate and store the Jacobian matrix explicitly. Analysis of memory and time complexities shows that, if a direct solver is used for forward modeling, the calculation of the full Jacobian is feasible for a large amount of practical cases. Advantages of this approach are demonstrated in the following chapters. For the first time in 3D CSEM inversion I study the implicit regularization effect resulting from incomplete solutions of linear least- squares problems using Krylov subspace techniques. This study gives an insight into sources of inverse problem instability. Moreover, several explicit stabilizing functionals have been implemented. In addition to well-known smoothing regularization, this work includes minimum-norm and focusing stabilizers. The application of different regularization techniques helps explore the model space and estimate the bias of individual regularization techniques. Finally, the algorithm developed is applied to the land-based CSEM data collected across the Ketzin CO$_2$ storage formation. I start with data preparation and aim to extract a representable subset of data that permits many inversion runs within reasonable computational time and effectively contains most of subsurface information. Then, the design of inversion and forward modeling grids is presented. A proper discretization of the domain has to (i) minimize negative boundary effects, (ii) provide accurate model responses and (iii) be able to adequately capture subsurface structures resolved during inversion. Different regularization techniques, starting models and several approaches to avoid numerical singularities arising in land-based CSEM studies are studied and compared. The preferred model is selected based on overall misfit. I provide the analysis of data fit, resolution and depth penetration for the preferred model. Finally, it is compared with a regional geological model and shows a good agreement with respect to both structure and lithology. The methodology devised in this work can be generally followed when dealing with real CSEM datasets.Hauptziele der vorliegenden Doktorarbeit sind: (i) Entwicklung eines 3D- Inversions-schemas für CSEM-Daten im Frequenzbereich (CSEM: Controlled Source Electromagnetic); (ii) Schaffung einer Methodik für die Arbeit mit realen Datensätzen, darunter auch solchen, die unter Verwendung einer in unserer Forschungsgruppe neu entwickelten Dreiphasen-Transmitterkonfiguration aufgenommen wurden. Die dreidimensionale Inversion nicht stationärer elektromagnetischer Felder ist eine Herausforderung, die angesichts der in Praxis gegebenen Probleme nur mit Einsatz umfangreicher paralleler Plattformen zu bewältigen ist. Daher habe ich ein vollständig parallel verteiltes Inversionsschema entwickelt, das auf modernen Supercomputern und Clustern verwendet werden kann. Da der für Vektoren, Matrizen und Felder benötigter Speicherplatz sowie die Rechnerauslastung gleichmäßig zwischen verschiedenen Prozessen verteilt werden, wird eine gute Skalierbarkeit des numerischen Schemas erreicht. Bei der Implementierung der Inversion war eine Reihe technischer und numerischer Herausforderungen zu meistern. Erstens verwende ich direkte Löser für die 3D-Vorwärts-modellierung. Da diese nicht häufig für 3D-Inversionen eingesetzt werden, habe ich die Vor- und Nachteile dieser Wahl sowohl in technischer als auch in numerischer Hinsicht analysiert. Zweitens berechne und speichere ich bei der Inversion explizit die Jacobi-Matrix. Die Analyse von Speicher- und Zeitbedarf zeigt, dass die Berechnung der voll- ständigen Jacobi-Matrix für eine große Anzahl praktischer Anwendungen durchgeführt werden kann, wenn direkte Löser für 3D-Vorwärtsmodellierung verwendet wurden. Die Vorzüge dieses Ansatzes werden in den folgenden Kapiteln diskutiert. Erstmals bei der 3D-CSEM-Inversion untersuche ich mithilfe des Krylovraum Verfahrens den impliziten Regularisierungseffekt, der sich aus der Unvollständigkeit der Lösungen bei der Methode der linearen kleinsten Quadrate ergibt. Diese Untersuchung beleuchtet die Ursachen der Instabilität von inversen Problemen. Zusätzlich zur bekannten Regularisierung durch Glättung wurden mehrere Stabilisatoren, wie z.B. minimale Norm oder Fokussierung, implementiert und getestet. Die Anwendung unterschiedlicher Regularisierungstechniken ermöglicht eine umfassende Erforschung des Modellraums sowie die Abschätzung der Verzerrung bei den einzelnen Regularisierungstechniken. Schließlich wird der entwickelte Algorithmus auf die CSEM-Daten angewandt, welche über der CO2-Speicherungsformation in Ketzin aufgenommen wurden. Beginnend mit der Aufbereitung der Daten ist das Ziel eine repräsentative Teilmenge der Daten auszuwählen, welche eine Vielzahl von Inversionsdurchläufen in einer angemessenen Zeit erlaubt und dabei die Mehrzahl der Untergrundinformationen enthält. Danach wird das Design der Inversion und des Modelgrids vorgestellt. Eine saubere Diskretisierung des Inversionsbereichs muss (i) negative Randeffekte minimieren, (ii) genaue Modelantworten liefern und (iii) in der Lage sein, die durch Inversion aufgelöste Untergrundstrukturen ausreichend zu erfassen. Weiterhin werden unterschiedliche Regularisierungstechniken und Ausgangsmodelle sowie verschiedene Ansätze zur Vermeidung von numerischen Singularitäten untersucht, die für landbasierte CSEM-Studien typisch sind. Das bevorzugte Model wird anhand des Gesamt-Misfits ausgewählt und durch Analyse der Datenanpassung, Auflösung und Eindringtiefe überprüft. Schließlich wird das Model mit dem regionalen geologischen Modell verglichen, mit dem es in Bezug auf die Struktur wie auch Lithologie eine gute Übereinstimmung zeigt. Die in dieser Arbeit entwickelte Inversionsmethodik ist nicht nur für den untersuchten Datensatz gültig, sondern kann allgemein für die Auswertung der landbasierten CSEM-Daten eingesetzt werden

Parallel three-dimensional magnetotelluric inversion using adaptive finite-element method. Part I: theory and synthetic study

This paper presents a distributed magnetotelluric inversion scheme based on adaptive finite-element method (FEM). The key novel aspect of the introduced algorithm is the use of automatic mesh refinement techniques for both forward and inverse modelling. These techniques alleviate tedious and subjective procedure of choosing a suitable model parametrization. To avoid overparametrization, meshes for forward and inverse problems were decoupled. For calculation of accurate electromagnetic (EM) responses, automatic mesh refinement algorithm based on a goal-oriented error estimator has been adopted. For further efficiency gain, EM fields for each frequency were calculated using independent meshes in order to account for substantially different spatial behaviour of the fields over a wide range of frequencies. An automatic approach for efficient initial mesh design in inverse problems based on linearized model resolution matrix was developed. To make this algorithm suitable for large-scale problems, it was proposed to use a low-rank approximation of the linearized model resolution matrix. In order to fill a gap between initial and true model complexities and resolve emerging 3-D structures better, an algorithm for adaptive inverse mesh refinement was derived. Within this algorithm, spatial variations of the imaged parameter are calculated and mesh is refined in the neighborhoods of points with the largest variations. A series of numerical tests were performed to demonstrate the utility of the presented algorithms. Adaptive mesh refinement based on the model resolution estimates provides an efficient tool to derive initial meshes which account for arbitrary survey layouts, data types, frequency content and measurement uncertainties. Furthermore, the algorithm is capable to deliver meshes suitable to resolve features on multiple scales while keeping number of unknowns low. However, such meshes exhibit dependency on an initial model guess. Additionally, it is demonstrated that the adaptive mesh refinement can be particularly efficient in resolving complex shapes. The implemented inversion scheme was able to resolve a hemisphere object with sufficient resolution starting from a coarse discretization and refining mesh adaptively in a fully automatic process. The code is able to harness the computational power of modern distributed platforms and is shown to work with models consisting of millions of degrees of freedom. Significant computational savings were achieved by using locally refined decoupled meshes.ISSN:0956-540XISSN:1365-246

Global 3-D Electrical Conductivity Model of the World Ocean and Marine Sediments

This study presents global 3-D electrical conductivity models of the world ocean and marine sediments. Electrical conductivity of the ocean was calculated by invoking the Equation of State of Seawater (TEOS-10) with temperature and salinity data retrieved from the World Ocean Atlas and a series of high-resolution regional ocean climatology data sets. The resolution of the ocean conductivity atlas varies between 0.urn:x-wiley:15252027:media:ggge22631:ggge22631-math-0001 and 0.urn:x-wiley:15252027:media:ggge22631:ggge22631-math-0002 globally. The conductivity of marine sediments was estimated by using compaction and thermal gradient models constrained by real observations on a 5-arc-minute global marine sediment thickness grid. I present numerical simulations of electromagnetic (EM) induction responses that demonstrate a significant effect of 3-D electrical conductivity of the ocean and marine sediments on EM responses for a broad range of frequencies. I show that both marine and land-based surveys designed for subsurface conductivity imaging or Space Weather modeling will benefit from inclusion of more realistic conductivity models of the ocean and seabed sediments.ISSN:1525-202