The Jacobian Conjecture is a long-standing open problem in algebraic geometry. Though the problem is inherently algebraic, it crops up in fields throughout mathematics including perturbation theory, quantum field theory and combinatorics. This thesis is a unified treatment of the combinatorial approaches toward resolving the conjecture, particularly investigating the work done by Wright and Singer. Along with surveying their contributions, we present new proofs of their theorems and motivate their constructions. We also resolve the Symmetric Cubic Linear case, and present new conjectures whose resolution would prove the Jacobian Conjecture to be true.

The Jacobian Conjecture is a long-standing open problem in algebraic geometry. Though the problem is inherently algebraic, it crops up in fields throughout mathematics including perturbation theory, quantum field theory and combinatorics. This thesis is a unified treatment of the combinatorial approaches toward resolving the conjecture, particularly investigating the work done by Wright and Singer. Along with surveying their contributions, we present new proofs of their theorems and motivate their constructions. We also resolve the Symmetric Cubic Linear case, and present new conjectures whose resolution would prove the Jacobian Conjecture to be true.

In this paper, we present a nonlinear curvelet-based sparsity-promoting formulation for three problems related to seismic noise, namely the ’good’, corresponding to noise generated by random sampling; the ’bad’, corresponding to coherent noise for which (inaccurate) predictions exist and the ’ugly’ for which no predictions exist. We will show that the compressive capabilities of curvelets on seismic data and images can be used to tackle these three categories of noise-related problems.

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This thesis examines the sensitivity of the marine controlled source electromagnetic (CSEM) method to sub-seafloor resistivity structure, with a focus on gas hydrate and free gas occurrences. Different analysis techniques are applied with progressive sophistication to a series of studies based on simulated and measured data sets. CSEM data are modelled in time domain for one-dimensional models with gas hydrate, free gas and/or permafrost occurrences. Linearized and non-linear inversion methods are considered to infer subsurface models from CSEM data. One study applies forward modelling and singular value decomposition to estimate uncertainties for permafrost models of the Beaufort Sea. This simulation study analyzes the resolution of the CSEM data for shallow water depth which is a challenging case because the electromagnetic signature of the air-water boundary may mask the sub-seafloor response. The results reveal a blind window as a function of water depth in which the CSEM data are insensitive to the sub-seafloor structure. However, the CSEM data are sensitive to the top and the bottom of the permafrost with increasing uncertainties with depth. The next study applies non-linear Bayesian inversion to CSEM data acquired in 2005/2006 on the Northern Cascadia margin to investigate sub-seafloor resistivity structure related to gas hydrate deposits and cold vents. Bayesian inversion provides a rigorous approach to estimate model parameters and uncertainties by probabilistically sampling of the parameter space. The resulting probability density function is interpreted here in terms of posterior median models, marginal and joint marginal probability densities for model parameters and credibility intervals. The Bayesian information criterion is applied to determine the amount of structure (number of layers) that can be resolved by the data. The parameter space is sampled with the Metropolis-Hastings algorithm in principal-component space. Non-linear, probabilistic inversion allows the analysis of unknown acquisition parameters such as time delays between receiver and transmitter clocks or unknown source amplitude. The estimated posterior median models and credibility intervals from Bayesian CSEM inversion are compared to reflection seismic data to provide a more complete geological interpretation. The CSEM data on the Northern Cascadia margin generally reveal a 1 to 3 layer sediment structure. Inversion results at the landward edge of the gas hydrate stability zone indicate a sediment unconformity as well as several potential cold vents which were previously unknown. The resistivities generally increase upslope due to sediment erosion along the slope. Inversion results on the middle slope infer several vent systems close to well-known Bullseye vent in agreement with ongoing interdisciplinary observations. Finally, a trans-dimensional (trans-D) Bayesian inversion is applied to CSEM data acquired in 2012 in the German North Sea to investigate possible free gas occurrences. Trans-D inversion treats the number of layers as an additional unknown sampled probabilistically in the inversion. %over the parameter space by evaluating probabilistically the transition to a higher or lower number of interfaces. Parallel tempering is applied to increase sampling efficiency and completeness. Inversion results for the German North Sea yield resistivities at the seafloor which are typical for marine deposits, while resistivities at greater depth increase slightly and can be correlated with a transition from fine-grained marine deposits (Holocene age) to coarse-grained, glacial sediments (Pleistocene age), which is observed in a sediment core. The depths of layer interfaces estimated from CSEM inversion match the seismic reflector related to the contrast between the two depositional environments. The CSEM survey targeted a strong, phase-reversed, inclined seismic reflector within the glacial sediments, potentially indicating free gas. While interface-depth estimates from CSEM inversion do not correlate closely with this reflector, resistivities are generally elevated above the strong seismic amplitudes and the thickness of the resistive layer follows the trend of the inclined reflector. However, the uncertainties of deeper interface depth estimates increase significantly and overlap with the targeted reflector at some of the measurement sites. Relatively low resistivities of a third layer correlate with sediments of late-Miocene origin with a high gamma-ray count indicating an increased amount of fine-grained sediments with organic material. The interface at the bottom of the third layer has wide uncertainties which relates to the penetration limit of the CSEM array. / Graduate

Amplitude Variation with Oset (AVO) inversion is one of the techniques that is being used to estimate subsurface physical parameters such as P-wave velocity, S-wave velocity, and density or their attributes. AVO inversion is an ill-conditioned problem which has to be regularized in order to obtain a stable and unique solution. In this thesis, a Bayesian procedure that uses a Multivariate Cauchy distribution as a prior probability distribution is introduced. The prior includes a scale matrix that imposes correlation among the AVO attributes and induces a regularization that provokes solutions that are sparse and stable in the presence of noise. The performance of this regularization is demonstrated by both synthetic and real data examples using linearized approximations to the Zoeppritz equations. / Geophysics

The Skellefte District (SD) is one of the richest metallogenic mining areas in Sweden. The main deposits consist of volcanic-hosted massive sulphides (VHMS) rich in zinc, copper, lead, gold and silver, that have been explored and mined for more than a century. Considering that technological advancements allow deeper mining, and that today new discoveries rarely occur, renewed efforts are now directed at locating targets at greater depths.   Thus, current exploration strategies need to be adapted, and a better understanding of regional scale structures is necessary. To address these questions the project VINNOVA 4D modeling of the Skellefte District was launched. Its main purpose is to unravel the regional structures and tectonic setting of the SD. To accomplish this, new geological and geophysical data have been acquired in two key localities. This thesis presents the contribution from 2D and 3D inversion of magnetotelluric (MT) data and seismic reflection data. The main findings include: conductive hydrothermally altered zones within the otherwise resistive rocks of the Skellefte Group, the depth extension of early and postorogenic intrusions, prominent shear zones in the central part of the district, and enhanced reflectivity and conductivity at the base of the Skellefte Group throughout the SD. Even though the application of these methods is challenged by the complex geological setting of the SD, it is shown that after a careful processing and analysis of the data, they are able to provide a robust image of the deep subsurface. Additionally, the combination of reflection seismics and MT has proved to be a powerful tool for hypothesis testing and to develop the general understanding of the configuration and history of the SD. Furthermore, two 3D inversion models of MT data are presented and compared with the results of standard 2D determinant inversions. The 3D procedure shows significant improvements in data fit and is able to constrain better the observed model features. Although 3D inversion of MT data is not yet a run of the mill scheme and issues like model assessment and galvanic distortion effects need to be further addressed, results from complex environments with areal coverage, are already superior to those from 2D inversions. / Skellefteåfälten är ett av de viktigaste malmdistrikten i Sverige. Malmkropparna består av vulkaniskvärda Massiva Sulfider (VHMS) rika på Zink, Koppar, Bly, Guld och Silver, och har utforskats och brutits i mer än ett sekel. Med tanke på att de senaste tekniska framstegen tillåter djupare brytning, och att nya upptäckter är ovanliga idag, riktas nya ansträngningar mot att lokalisera malm på större djup. Aktuella prospekteringsstrategier måste därför anpassas, och en bättre förståelse av regionala strukturer är nödvändig. För att lösa dessa frågor lanserades projektet VINNOVA 4D modeling of the Skellefte District. Dess främsta syfte är att utreda de regionala strukturerna och det tektoniska läget av Skelleftefältet. För att uppnå detta, har nya geologiska och geofysiska data insamlats vid två viktiga platser i distriktet. Denna avhandling presenterar bidrag från inversionsmodellering i 2D och 3D av magnetotelluriska (MT) data samt resultaten av en reflektionsseismisk profil. De viktigaste resultaten är: bra ledande hydrotermiskt förändrade zoner inom de annars resistiva bergarterna i Skellefte-gruppen, djupet till tidiga och postorogeniska intrusioner, framstående skjuvzoner i den centrala delen av området, och ökad reflektionsförmåga och konduktivitet vid basen av Skellefte-gruppen i hela fältet. Även om tillämpningen av dessa metoder utmanas av fältens komplexa geologiska läge, visas det efter en noggrann bearbetning och analys av data att de ger en robust bild av den lite djupare berggrunden. Dessutom har kombinationen av reflektionsseismik och MT visat sig vara ett kraftfullt verktyg för hypotesprövning och för att utveckla den allmänna förståelsen av Skelleftefältet och dess historia. Därutöver presenteras två 3D inversionsmodeller av MT data och jämförs sedan med resultaten från 2D determinantinversioner. 3D tekniker visar betydande förbättringar av datapassform och begränsar observerade anomalier bättre. Även om 3D inversion av MT data ännu inte är en vanlig teknik och frågor som modellbedömning och galvaniska distorsionseffekter måste behandlas ytterligare, är resultat från komplexa miljöer med lagom yttäckning redan överlägsna. / VINNOVA 4D modeling of the Skellefte District

Th. doct. ing.--Bordeaux 1, 1983. N°: 348.

Les volcans sont des édifices complexes et hétérogènes. Imager leur structure demande des méthodes adaptées au contexte ainsi qu'à la nature, à la qualité et à la quantité des données disponibles.Ce travail porte sur l'imagerie géophysique de l'île volcanique de Basse-Terre en Guadeloupe, par inversion de données gravimétriques et magnétiques, par tomographie d'ondes de surface extraites de corrélations de bruit sismique et par tomographie de temps d'arrivée des séismes de l'arc des Petites Antilles. Nous obtenons ainsi les premiers modèles tridimensionnels de l'île de Basse-Terre en termes de densité, aimantation et vitesses sismiques. La gravimétrie et le magnétisme conduisent à des modèles de densité et d'aimantation avec une très bonne résolution latérale dans les premiers kilomètres, mais avec une résolution verticale très faible. Les ondes de surface issues des corrélations de bruit présentent une bonne résolution verticale dans les premiers kilomètres, mais résolvent mal les variations latérales de vitesses. La tomographie de temps d'arrivée permet d'imager les vitesses sismiques jusqu'à des profondeurs élevées, avec une bonne résolution latérale, mais une mauvaise résolution verticale. Nous proposons une méthode pour inverser conjointement les données sismologiques et gravimétriques pour produire un modèle de vitesses et de densité qui bénéficie des résolutions complémentaires des différents jeux de données.Nos résultats soulignent une tendance NNO-SSE, avec des vitesses et des densités qui diminuent vers le sud en moyenne, conformément à la migration du volcanisme du nord vers le sud au cours de l'histoire géologique de l'île. Nous observons de faibles densités, vitesses sismiques et aimantations au niveau du dôme du volcan de la Soufrière ainsi que dans la région de Bouillante sur la côte ouest, reflétant la forte altération hydrothermale de ces zones. Deux anciens centres éruptifs majeurs, caractérisés par de fortes densités et vitesses sismiques, sont identifiés : l'un au niveau de la vallée de Beaugendre, l'autre au nord au niveau de la vallée de Petite Plaine. / Volcanoes are complex and heterogeneous edifices. Imaging their structures requires methods that are adapted to the context and to the nature, quality and quantity of available data. This work focuses on the geophysical imaging of the volcanic island of Basse-Terre in Guadeloupe, by performing inversions of gravimetric and magnetic data, ambient noise based surface wave tomography, and travel time tomography from earthquakes of the Lesser Antilles arc. We therefore obtain the first three-dimensional models of Basse-Terre island in terms of density, magnetisation and seismic velocities. Gravimetry and magnetism lead to density and magnetisation models with a very good lateral resoltuion within the first kilometres but with very low vertical resolution. Surface wave tomography from noise correlations shows a good vertical resolution within the first kilometres, but poorly resolves the lateral velocity variations. Travel time tomography allows imaging seismic velocities up to great depths with a good lateral resolution but a poor vertical resolution. We propose a method to invert jointly the seismological and gravimetric data in order to produce a velocity and density model that takes advantage of the complementary resolutions associated with the different datasets.Our results outline a NNW-SSE trend, with velocities and densities roughtly decreasing southwards, in accordance with the southward migration of volcanism across the geological history of the island. We observe low densities, seismic velocities and magnetisations in the area of the dome of the Soufrière volcano as well as in the Bouillante area along the western coast, reflecting the high hydrothermal alteration of these areas. Two major old eruptive centres are identified in the areas of Beaugendre Valley and Petite Plaine Valley.

L'inversion de forme d'onde complète (FWI) est un processus non-linéaire et mal posé d’ajustement de données, dans notre cas, issues d’acquisitions simiques. Cette technique cherche à reconstruire, à partir d’un modèle initial obtenu à faible nombre d’onde (faible résolution), des paramètres constitutifs contrôlant la propagation des ondes à grands nombres d’ondes (forte résolution). Durant ce processus itératif, certains artéfacts peuvent altérer la qualité du modèle reconstruit. Afin de diminuer ces artéfacts et d’assurer une reconstruction des paramètres qui soit cohérente d’un point de vue géologique, différentes techniques de pré-conditionnement ou de régularisation peuvent être proposées.Cette thèse se focalise sur le potentiel de nouveaux filtres multi-dimensionnels construits dans l’espace des nombres d’ondes et orientés suivant les structures géologiques. Une stratégie de pré-conditionnement a été mise au point à l’aide de ces filtres et a été appliquée avec succès à la problématique FWI. La formulation analytique 1D de l’opérateur inverse de covariance laplacienne (Tarantola, 2005) constitue la base de la formulation d’opérateurs de dimension supérieure qui sont validés ici en les comparants avec l’opérateur analytique de covariance laplacienne 1D. Nous avons utilisé cette fonction analytique inverse 1D comme la base de filtrage de dimension supérieure, via l’addition de multiples fonctions inverses orientées orthogonalement. Ces fonctions laplaciennes inverses additionnelles (AIL) sont obtenues pour des configurations 2D et 3D après discrétisation par des techniques de différences finies. Nous montrons que l’on peut calculer un filtre en nombre d’onde de manière rapide et robuste en résolvant le système linéaire associé à ces opérateurs inverses. Lorsque des pentes sont inclues à l’étape de discrétisation par différences finies, il est alors possible d’utiliser ces opérateurs comme des filtres en nombre d’ondes orientés vers les structures géologiques, ceci avec une grande efficacité.Ce filtre (AIL) montre des propriétés rapides de convergence et des performances indépendantes du vecteur à filtrer. Nous montrons notamment comment ce filtre peut être utilisé comme un opérateur utile pour le gradient associé à la FWI. Le pré-conditionnement du gradient peut atténuer les effets du problème mal-posé qui vont s’étendre dans l’espace des modèles. Deux exemples synthétiques (Valhall et Marmousi) calculés dans l’espace des fréquences sont proposés dans cette thèse. Le pré-conditionnement AIL s’avère efficace pour atténuer d’une part la signature mal-posée provenant de la présence de bruit ambient dans les données observées et d’autre part d’artéfacts liés aux effets de repliement spatial liés aux conditions d’imagerie par FWI. La possibilité d’inclure des pentes permet de filtrer de manière préférentielle en considérant des pendages géologiques. Cette stratégie de filtrage permet l’atténuation d’artéfacts, tout en préservant le contenu en nombre d’ondes de la stratigraphie orthogonale au pendage.Un cas réel d’inversion 2D FWI est finalement abordé permettant tout d’abord d’illustrer la sensibilité des résultats d’inversion au modèle initial. Celui-ci est d’importance majeure, particulièrement dans les régions profondes dépassant la pénétration maximale des ondes transmises. L’application de la technique FWI à cette acquisition sismique a permis d’améliorer de manière significative la cohérence sur une image migrée par renversement du temps (RTM). Nous montrons également que le pré-conditionneur AIL permet une décroissance significative du nombre de tirs requis à modéliser dans la boucle d’inversion, sans pour autant dégrader le contenu en nombre d’onde des structures géologiques principales dans les résultats finaux obtenus après inversion. / Full waveform inversion (FWI) is a non-linear, ill-posed, local data fitting technique. FWI looks to moves from an initial, low-wavenumber representation of the earth parameters to a broadband representation. During this iterative process a number of undesirable artifacts can map into our model parameter reconstruction. To mitigate these artifacts and to ensure a geologically consistent model parameter reconstruction, various preconditioning and/or regularization strategies have been proposed.This thesis details the construction of new, efficient, multi-dimensional, structurally-orientated wavenumber filters. A preconditioning strategy has been devised using these filters that we have successfully applied to FWI. The 1D analytical inverse Laplacian covariance operator (Tarantola, 2005) forms the basis of higher dimensional operators and is initially validated by comparing to the 1D analytical Laplacian covariance operator. We use this analytical 1D inverse function as the basis for higher dimensional filtering via the addition of multiple, orthogonally orientated inverse functions. These additive inverse laplacian functions (AIL) are shown in 2D and 3D configurations and are discretized using finite-difference techniques. We show that one can calculate, a rapid and robust wavenumber filter, by solving the linear system associated with these inverse operators. When dip is included at the finite difference discretization stage, it is possible to use these operators as highly efficient, structurally orientated wavenumber filters.The AIL filter is shown to be rapid to converge and its performance is independent of the vector to be filtered. We show, that the filter can be a useful preconditioning operator for the FWI gradient. Preconditioning the gradient can mitigate against ill-posed effects mapping into the model-space. Two synthetic (Valhall and Marmousi) frequency domain FWI example are shown in this thesis. The AIL preconditioner has success at mitigating the ill-posed imprint coming from ambient noise in the observed data and also artifacts from spatial aliasing effects in the FWI imaging condition. The ability to include dip, allows one to preferentially filter along geological dip. This filtering strategy allows the mitigation of artifacts, while simultaneously preserving the stratigraphic based wavenumber content that is orthogonal to dip.A 2D, real data FWI case-study is also shown and we highlight the sensitivity of the inversion result to the initial model. The initial model is of key importance, this especially true in the areas deeper than the maximum penetration of transmitted waves. The application of FWI on this line is able to significantly improve gather alignment on a RTM, migrated image. We also see that the AIL preconditioner can allows us to significantly decrease the number of shot records we are required to model in our inversion workflow without degrading the key geological wavenumber content in the final inversion result.