Three-dimensional individual and joint inversion of direct current resistivity and electromagnetic data

The objective of our studies is the combination of electromagnetic and direct current (DC) resistivity methods in a joint inversion approach to improve the reconstruction of a given conductivity distribution. We utilize the distinct sensitivity patterns of different methods to enhance the overall resolution power and ensure a more reliable imaging result. In order to simplify the work with more than one electromagnetic method and establish a flexible and state-of-the-art software basis, we developed new DC resistivity and electromagnetic forward modeling and inversion codes based on finite elements of second order on unstructured grids. The forward operators are verified using analytical solutions and convergence studies before we apply a regularized Gauss-Newton scheme and successfully invert synthetic data sets. Finally, we link both codes with each other in a joint inversion. In contrast to most widely used joint inversion strategies, where different data sets are combined in a single least-squares problem resulting in a large system of equations, we introduce a sequential approach that cycles through the different methods iteratively. This way, we avoid several difficulties such as the determination of the full set of regularization parameters or a weighting of the distinct data sets. The sequential approach makes use of a smoothness regularization operator which penalizes the deviation of the model parameters from a given reference model. In our sequential strategy, we use the result of the preceding individual inversion scheme as reference model for the following one. We successfully apply this approach to synthetic data sets and show that the combination of at least two methods yields a significantly improved parameter model compared to the individual inversion results.Ziel der vorliegenden Arbeit ist die gemeinsame Inversion (\"joint inversion\") elektromagnetischer und geoelektrischer Daten zur Verbesserung des rekonstruierten Leitfähigkeitsmodells. Dabei nutzen wir die verschiedenartigen Sensitivitäten der Methoden aus, um die Auflösung zu erhöhen und ein zuverlässigeres Ergebnis zu erhalten. Um die Arbeit mit mehr als einer Methode zu vereinfachen und eine flexible Softwarebasis auf dem neuesten Stand der Forschung zu etablieren, wurden zwei Codes zur Modellierung und Inversion geoelektrischer als auch elektromagnetischer Daten neu entwickelt, die mit finiten Elementen zweiter Ordnung auf unstrukturierten Gittern arbeiten. Die Vorwärtsoperatoren werden mithilfe analytischer Lösungen und Konvergenzstudien verifiziert, bevor wir ein regularisiertes Gauß-Newton-Verfahren zur Inversion synthetischer Datensätze anwenden. Im Gegensatz zur meistgenutzten \"joint inversion\"-Strategie, bei der verschiedene Daten in einem einzigen Minimierungsproblem kombiniert werden, was in einem großen Gleichungssystem resultiert, stellen wir schließlich einen sequentiellen Ansatz vor, der zyklisch durch die einzelnen Methoden iteriert. So vermeiden wir u.a. eine komplizierte Wichtung der verschiedenen Daten und die Bestimmung aller Regularisierungsparameter in einem Schritt. Der sequentielle Ansatz wird über die Anwendung einer Glättungsregularisierung umgesetzt, bei der die Abweichung der Modellparameter zu einem gegebenen Referenzmodell bestraft wird. Wir nutzen das Ergebnis der vorangegangenen Einzelinversion als Referenzmodell für die folgende Inversion. Der Ansatz wird erfolgreich auf synthetische Datensätze angewendet und wir zeigen, dass die Kombination von mehreren Methoden eine erhebliche Verbesserung des Inversionsergebnisses im Vergleich zu den Einzelinversionen liefert

On Algorithms Based on Joint Estimation of Currents and Contrast in Microwave Tomography

This paper deals with improvements to the contrast source inversion method which is widely used in microwave tomography. First, the method is reviewed and weaknesses of both the criterion form and the optimization strategy are underlined. Then, two new algorithms are proposed. Both of them are based on the same criterion, similar but more robust than the one used in contrast source inversion. The first technique keeps the main characteristics of the contrast source inversion optimization scheme but is based on a better exploitation of the conjugate gradient algorithm. The second technique is based on a preconditioned conjugate gradient algorithm and performs simultaneous updates of sets of unknowns that are normally processed sequentially. Both techniques are shown to be more efficient than original contrast source inversion.Comment: 12 pages, 12 figures, 5 table

Stochastic estimation of hydraulic transmissivity fields using flow connectivity indicator data

This is the peer reviewed version of the following article: [Freixas, G., D. Fernàndez-Garcia, and X. Sanchez-Vila (2017), Stochastic estimation of hydraulic transmissivity fields using flow connectivity indicator data, Water Resour. Res., 53, 602–618, doi:10.1002/2015WR018507], which has been published in final form at http://onlinelibrary.wiley.com/doi/10.1002/2015WR018507/abstract. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving.Most methods for hydraulic test interpretation rely on a number of simplified assumptions regarding the homogeneity and isotropy of the underlying porous media. This way, the actual heterogeneity of any natural parameter, such as transmissivity ( math formula), is transferred to the corresponding estimates in a way heavily dependent on the interpretation method used. An example is a long-term pumping test interpreted by means of the Cooper-Jacob method, which implicitly assumes a homogeneous isotropic confined aquifer. The estimates obtained from this method are not local values, but still have a clear physical meaning; the estimated math formula represents a regional-scale effective value, while the log-ratio of the normalized estimated storage coefficient, indicated by math formula, is an indicator of flow connectivity, representative of the scale given by the distance between the pumping and the observation wells. In this work we propose a methodology to use math formula, together with sampled local measurements of transmissivity at selected points, to map the expected value of local math formula values using a technique based on cokriging. Since the interpolation involves two variables measured at different support scales, a critical point is the estimation of the covariance and crosscovariance matrices. The method is applied to a synthetic field displaying statistical anisotropy, showing that the inclusion of connectivity indicators in the estimation method provide maps that effectively display preferential flow pathways, with direct consequences in solute transport.Peer ReviewedPostprint (published version

New Methods for Inferring Past Climatic Changes from Underground Temperatures

In this thesis new methods have been developed for the recovery of past surface temperature variations from underground temperature-depth profiles. This has been undertaken from a Bayesian standpoint with an emphasis on model comparison, which allows differently parameterised inverse models (inferred past temperature histories) to be automatically constructed and compared in the light of the data and the prescribed prior information. In the first contribution a new method for inverting temperature-depth profiles is presented which relies on trans-dimensional Bayesian sampling. The temperature histories are parameterised in terms of a variable number of linear segments over time. Relying on the natural parsimony of Bayesian inference, whereby simpler models which can adequately explain the data are preferred, the complexity or roughness of the temperature histories can be determined without the need for explicit a priori smoothing. This method therefore allows a more objective inference of the past temperature changes. These concepts are extended to the spatial domain in the following chapter using the method of Bayesian partition modelling. This seeks to find the posterior distribution of the number and spatial distribution of independent temperature histories given a spatially distributed ensemble of temperature-depth profiles. The results from application to 23 real boreholes in the UK are discussed in detail and show a clear preference for 8 or 9 independent (and mostly contrasting) temperature histories. It is thus concluded that the majority of these data cannot be considered as reliable sources of palaeoclimate reconstruction. A 3D finite element heat transfer forward model is developed in the latter part of the thesis, and is used to simulate underground temperatures. This forward model is linked to the first of the two Bayesian inverse methods described above. The effect of the reduction in average ground surface temperature with altitude is included in the forward model and inversion of the resultant profiles using a 1D forward model is shown to give significant discrepancies in the inferred temperature histories. Finally the inversion results from the Bayesian formulation are compared with those using a conventional gradient descent method. The thesis concludes with some possibilities for future research in this field which builds upon the work presented herein

Structural joint inversion of electrical and seismic tomography data

This research project has been focused on the achievement of the structural joint inversion of two geophysical methods. The final target is to obtain a high resolution characterization of the shallow subsurface. The aim of determining petrophysical properties, structural boundaries, etc, can be obtained through the integration of different information that derives from various geophysical methods. In fact, since each method is sensitive to a specific physical property, their integration can lead to an accurate final model. However, if such integration is conducted individually inverting the data sets, the final model will be affected by the resolution limitations of each method. For this reason, an important tool has been developed in geophysical applications: the joint inversion. Two different approaches can be used to carry out the joint inversion: the petrophysical one, in which a petrophysical relationship is used, and the structural one, in which a structural similarity between models is imposed (Gallardo and Meju, 2004). Specifically, I decided to implement the algorithm for the structural joint inversion and specifically the structural approach developed by Gallardo and Meju (2003, 2004), since from literature it results to be the most robust method in the joint inversion (Moorkamp, 2017). In this process, an objective function that includes the objective function of each geophysical method is build and simultaneously minimized. In conclusion, the joint inversion may improve the resolution of each geophysical model and bring to models that are more accurate and easier to interpret. Specifically, in this thesis, the electrical resistivity tomography (ERT) and the seismic refraction tomography (SRT) have been used to carry out the joint inversion. Both these high-resolution methods can be crucial in environmental and engineering applications, as for the geotechnical characterization of a site or for the detection of hydrological resources. Since the resistivity range overlaps for the different materials, resistivity measurements cannot be related to a specific soil or rock. Because of that, it would be better to obtain other information, for example from the seismic tomography. In fact, this method allows not only the reconstruction of the seismic wave velocities with depth, but also to obtain a good lateral resolution. Instead, the Ground Penetrating Radar (GPR) has not been considered since it presents some limits in the investigation depth, due to the high attenuation of electromagnetic energy in porous conductive media. In addition to the integrated inversion, another goal has been obtained in this thesis: the implementation of the forward modeling for the seismic method and specifically, the Multistencils Fast Marching Method (MSFMM). This method can be seen as an extension of the FMM, that is considered from literature the fastest and the most efficient method for the solution of the eikonal equation and accordingly for the computation of the first arrivals traveltimes. In particular, the MSFMM improves the accuracy and the efficiency of the FMM, since it considers also the information that derives from the diagonal directions. Both the algorithms, the one of the joint inversion and the one of the forward modeling for the seismic method, have been implemented in Python language and integrated in the open-source software pyGIMLi

Ground surface temperature histories inferred from 15 boreholes temperature profiles: comparison of two approaches

Understanding the climate change processes requires application of special methodologies for revealing a ground surface temperature history (GSTH). It was proved by different authors that the GSTH may bedetermined on the basis of analysis of the temperature field observed in short boreholes. In this paper, the authors analyze four mathematical models describing the GSTH: (1) sudden change, (2) linear increase,(3) square root of time increase and (4) exponential increase. Fifteen borehole temperature profiles from Europe, Asia and North America were selected in three groups based on their geographical proximity. Aftercareful analysis of temperature-depth profiles in these boreholes it was found out that two models (linear increase and square root of time increase) provide the best fit with field data. The calculated warming ratesin the 20th century were compared with those obtained by a few parameters estimation (FPE) technique

Das unstetige Galerkinverfahren für Strömungen mit freier Oberfläche und im Grundwasserbereich in geophysikalischen Anwendungen

Free surface flows and subsurface flows appear in a broad range of geophysical applications and in many environmental settings situations arise which even require the coupling of free surface and subsurface flows. Many of these application scenarios are characterized by large domain sizes and long simulation times. Hence, they need considerable amounts of computational work to achieve accurate solutions and the use of efficient algorithms and high performance computing resources to obtain results within a reasonable time frame is mandatory. Discontinuous Galerkin methods are a class of numerical methods for solving differential equations that share characteristics with methods from the finite volume and finite element frameworks. They feature high approximation orders, offer a large degree of flexibility, and are well-suited for parallel computing. This thesis consists of eight articles and an extended summary that describe the application of discontinuous Galerkin methods to mathematical models including free surface and subsurface flow scenarios with a strong focus on computational aspects. It covers discretization and implementation aspects, the parallelization of the method, and discrete stability analysis of the coupled model.Für viele geophysikalische Anwendungen spielen Strömungen mit freier Oberfläche und im Grundwasserbereich oder sogar die Kopplung dieser beiden eine zentrale Rolle. Oftmals charakteristisch für diese Anwendungsszenarien sind große Rechengebiete und lange Simulationszeiten. Folglich ist das Berechnen akkurater Lösungen mit beträchtlichem Rechenaufwand verbunden und der Einsatz effizienter Lösungsverfahren sowie von Techniken des Hochleistungsrechnens obligatorisch, um Ergebnisse innerhalb eines annehmbaren Zeitrahmens zu erhalten. Unstetige Galerkinverfahren stellen eine Gruppe numerischer Verfahren zum Lösen von Differentialgleichungen dar, und kombinieren Eigenschaften von Methoden der Finiten Volumen- und Finiten Elementeverfahren. Sie ermöglichen hohe Approximationsordnungen, bieten einen hohen Grad an Flexibilität und sind für paralleles Rechnen gut geeignet. Diese Dissertation besteht aus acht Artikeln und einer erweiterten Zusammenfassung, in diesen die Anwendung unstetiger Galerkinverfahren auf mathematische Modelle inklusive solcher für Strömungen mit freier Oberfläche und im Grundwasserbereich beschrieben wird. Die behandelten Themen umfassen Diskretisierungs- und Implementierungsaspekte, die Parallelisierung der Methode sowie eine diskrete Stabilitätsanalyse des gekoppelten Modells