Bayesian inverse modelling provides a top-down technique of verifying emissions and uptake of carbon dioxide (CO₂) from both natural and anthropogenic sources. It relies on accurate measurements of CO₂ concentrations at appropriately placed sites and "best-guess" initial estimates of the biogenic and anthropogenic emissions, together with uncertainty estimates. The Bayesian framework improves current estimates of CO₂ fluxes based on independent measurements of CO₂ concentrations while being constrained by the initial estimates of these fluxes. Monitoring, reporting and verification (MRV) is critical for establishing whether emission reducing activities to mitigate the effects of climate change are being effective, and the Bayesian inverse modelling approach of correcting CO₂ flux estimates provides one of the tools regulators and researchers can use to refine these emission estimates. South Africa is known to be the largest emitter of CO₂ on the African continent. The first major objective of this research project was to carry out such an optimal network design for South Africa. This study used fossil fuel emission estimates from a satellite product based on observations of night-time lights and locations of power stations (Fossil Fuel Data Assimilations System (FFDAS)), and biogenic productivity estimates from a carbon assessment carried out for South Africa to provide the initial CO₂ flux estimates and their uncertainties. Sensitivity analyses considered changes to the covariance matrix and spatial scale of the inversion, as well as different optimisation algorithms, to assess the impact of these specifications on the optimal network solution. This question was addressed in Chapters 2 and 3. The second major objective of this project was to use the Bayesian inverse modelling approach to obtain estimates of CO₂ fluxes over Cape Town and surrounding area. I collected measurements of atmospheric CO₂ concentrations from March 2012 until July 2013 at Robben Island and Hangklip lighthouses. CABLE (Community Atmosphere Biosphere Land Exchange), a land-atmosphere exchange model, provided the biogenic estimates of CO₂ fluxes and their uncertainties. Fossil fuel estimates and uncertainties were obtained by means of an inventory analysis for Cape Town. As an inventory analysis was not available for Cape Town, this exercise formed an additional objective of the project, presented in Chapter 4. A spatially and temporally explicit, high resolution surface of fossil fuel emission estimates was derived from road vehicle, aviation and shipping vessel count data, population census data, and industrial fuel use statistics, making use of well-established emission factors. The city-scale inversion for Cape Town solved for weekly fluxes of CO₂ emissions on a 1 km × 1 km grid, keeping fossil fuel and biogenic emissions as separate sources. I present these results for the Cape Town inversion under the proposed best available configuration of the Bayesian inversion framework in Chapter 5. Due to the large number of CO₂ sources at this spatial and temporal resolution, the reference inversion solved for weekly fluxes in blocks of four weeks at a time. As the uncertainties around the biogenic flux estimates were large, the inversion corrected the prior fluxes predominantly through changes to the biogenic fluxes. I demonstrated the benefit of using a control vector with separate terms for fossil fuel and biogenic flux components. Sensitivity analyses, solving for average weekly fluxes within a monthly inversion, as well as solving for separate weekly fluxes (i.e. solving in one week blocks) were considered. Sensitivity analyses were performed which focused on how changes to the prior information and prior uncertainty estimates and the error correlations of the fluxes would impact on the Bayesian inversion solution. The sensitivity tests are presented in Chapter 6. These sensitivity analyses indicated that refining the estimates of biogenic fluxes and reducing their uncertainties, as well as taking advantage of spatial correlation between areas of homogeneous biota would lead to the greatest improvement in the accuracy and precision of the posterior fluxes from the Cape Town metropolitan area.

While the bulk of topography on Earth is generated and maintained by variations in the thickness and density of crust and lithosphere, a significant time-variable contribution is expected as a result of convective flow in the underlying mantle. For over three decades, this dynamic topography has been calculated numerically from inferred density structure and radial viscosity profiles. Resulting models predict ±2 km of long wavelength (i.e., ~ 20,000 km) dynamic topography with minor contributions at wavelengths shorter than ~ 5,000 km. Recently, observational studies have revealed that, at the longest wavelengths, dynamic topography variation is ~ 30% that predicted, with ±1 km amplitudes recovered at shorter wavelengths. Here, the existing database of water-loaded basement depths is streamlined, revised and augmented. By fitting increasingly sophisticated thermal models to a combined database of these oceanic basement depths and corrected heat flow measurements, the average thermal structure of oceanic lithosphere is constrained. Significantly, optimal models are consistent with invariable geochemical and seismological constraints whilst yielding similar values of mantle potential temperature and plate thickness, irrespective of whether heat flow, subsidence or both are fit. After recalculating residual depth anomalies relative to optimal age-depth subsidence and combining them with continental constraints from gravity anomalies, a global spherical harmonic representation is generated. Although, long wavelength dynamic topography increases by ~ 40% in the revised observation-based model, spectral analysis confirms that a fundamental discrepancy between observations and predictions remains. Significantly, residual depth anomalies reveal a ~4,000 km-scale eastward tilt across the Indian Peninsula. This asymmetry extends onshore from the high-elevation Western Ghats in the west to the Krishna-Godavari floodplains in the east. Calibrated inverse modelling of drainage networks suggest that the tilt of the peninsula grew principally in Neogene times with vertical motions linked to asthenospheric temperature anomalies. Uplift rates of up to 0.1 mm a⁻¹ place important constraints on the spatio-temporal evolution of dynamic topography and suggest that rates of transient vertical motion exceed those predicted by many modelling studies. Most numerical models excise the upper ~ 300 km of Earth's mantle and are unable to reconstruct the wavelength and rate of uplift observed across Peninsular India. By contrast, through conversion of upper mantle shear wave velocities to density using a calibrated anelastic parameterisation, it is shown that shorter wavelength (i.e., ≤ 5,000 km) dynamic topography, can mostly be explained by ±150°C asthenospheric temperature anomalies. Inclusion of anelastically corrected density structure in whole-mantle instantaneous flow models also serves to reduce discrepancy between predictions and observations of dynamic topography at long wavelengths. Residual mismatch between observations and predictions is further improved if the basal 300-600 km of large low shear wave velocity regions in the deep mantle are geochemically distinct and negatively buoyant. Finally, inverse modelling of geoid, dynamic topography, gravity and core-mantle boundary topography observations using adapted density structure suggests that geodynamic constraints can be acceptably fit using plausible radial viscosity profiles, contradicting a long-standing assertion that modest long wavelength dynamic topography is incompatible with geoid observations.

Des simulateurs numériques sont couramment utilisés pour la prédiction et l'optimisation de l'exploitation d'aquifères et pour la détermination de paramètres physiques (e.g perméabilité) par calcul inverse. L'équation de Richards, décrit l'écoulement d'un fluide dans un milieu poreux non saturé. C'est une équation aux dérivées partielles non linéaires, dont la résolution numérique en grande dimension 3D est très coûteuse et en particulier pour du calcul inverse.Dans ce travail, une méthode de réduction de modèle (ROM) est proposée par une décomposition orthogonale propre (POD) afin de réduire significativement le temps de calcul, tout maîtrisant la précision. Une stratégie de cette méthode est de remplacer localement dans l'algorithme d'optimisation, le modèle fin par un modèle réduit type POD. La méthode de Petroc-Galerkin POD est d'abord appliquée à l'équation de Richards et testée sur différents cas, puis adaptée en linéarisant les termes non linéaires. Cette adaptation ne fait pas appel à une technique d'interpolation et réduit le temps de calcul d'un facteur [10;100]. Bien qu'elle ajoute de la complexité du ROM, cette méthode évite d'avoir à ajuster les paramètres du noyau, comme c'est le cas dans les méthodes du POD par interpolation. Une exploration des propriétés d'interpolation et d'extrapolation inhérentes aux méthodes intrusives est ensuite faite. Des qualités d'extrapolation intéressantes permettent de développer une méthode d'optimisation nécessitant de petits plans d'expériences (DOE). La méthode d'optimisation recrée localement des modèles précis sur l'espace des paramètres, en utilisant une classification à vecteurs de support non linéaire pour délimiter la zone où le modèle est suffisamment précis, la région de confiance. Les méthodes sont appliquées sur un cas d'école en milieu non saturé régit par l'équation de Richards, ainsi que sur un aquifère situé dans le "Table Mountain Group" près de la ville du Cap en Afrique du Sud. / The Richards equation describes the movement of an unsaturated fluid through a porous media, and is characterised as a non-linear partial differential equation. The equation is subject to a number of parameters and is typically computationnaly expensive to solve. To determine the parameters in the Richards equation, inverse modelling studies often need to be undertaken. As a solution to overcome the computational expense incurred in inverse modelling, the use of Proper Orthogonal Decomposition (POD) as a Reduced Order Modelling (ROM) method is proposed in this thesis to speed-up individual simulations. The Petrov-Galerkin POD approach is initially applied to the Richards equation and tested on different case studies. However, due to the non-linear nature of the Richards equation the method does not result in significant speed up times. Subsquently, the Petrov-Galerkin method is adapted by linearising the nonlinear terms in the equation, resulting in speed-up times in the range of [10,100]., The adaptation, notably, does not use any interpolation techniques, favouring an intrusive, but physics-based, approach. While the use of intrusive POD approaches add to the complexity of the ROM, it avoids the problem of finding kernel parameters typically present in interpolative POD approaches. Furthermore, the interpolative and possible extrapolation properties inherent to intrusive PODROM's are explored. The good extrapolation propertie, within predetermined bounds, of intrusive POD's allows for the development of an optimisation approach requiring a very small Design of Experiments (DOE). The optimisation method creates locally accurate models within the parameters space usign Support Vector Classification. The limits of the locally accurate model are called the confidence region. The methods are demonstrated on a hypothetical unsaturated case study requiring the Richards equation, and on true case study in the Table Mountain Group near Cape Town, South Africa.

This study has two parts. In the first part we develop a computational approach to the solution of an inverse modelling problem concerning the material properties of electrolytes used in Lithium-ion batteries. The dependence of the diffusion coefficient and the transference number on the concentration of Lithium ions is reconstructed based on the concentration data obtained from an in-situ NMR imaging experiment. This experiment is modelled by a system of 1D time-dependent Partial Differential Equations (PDE) describing the evolution of the concentration of Lithium ions with prescribed initial concentration and fluxes at the boundary. The material properties that appear in this model are reconstructed by solving a variational optimization problem in which the least-square error between the experimental and simulated concentration values is minimized. The uncertainty of the reconstruction is characterized by assuming that the material properties are random variables and their probability distribution estimated using a novel combination of Monte-Carlo approach and Bayesian statistics. In the second part of this study, we carefully analyze a number of secondary effects such as ion pairing and dendrite growth that may influence the estimation of the material properties and develop mathematical models to include these effects. We then use reconstructions of material properties based on inverse modelling along with their uncertainty estimates as a framework to validate or invalidate the models. The significance of certain secondary effects is assessed based on the influence they have on the reconstructed material properties. / Thesis / Doctor of Philosophy (PhD)

No / The use of neural networks (NNs) in all aspects of process engineering activities, such as modelling, design, optimization and control has considerably increased in recent years (Mujtaba and Hussain, 2001). In this work, three different types of nonlinear control strategies are developed and implemented in batch reactors using NN techniques. These are generic model control (GMC), direct inverse model control (DIC) and internal model control (IMC) strategies. Within the control strategies, NNs have been used as dynamic estimator, dynamic model (forward model) and control (inverse model). An exothermic complex reaction scheme in a batch reactor is considered to explain all these control strategies and their robustness. A dynamic optimization problem with a simple model is solved a priori to obtain optimal operation policy in terms of the reactor temperature with an objective to maximize the desired product in a given batch time. The resulting optimal temperature policy is used as set-point in the control study. All types of controllers performed well in tracking the optimal temperature profile and achieving target conversion to the desired product. However, the NNs used in DIC and IMC controllers need training beyond the nominal operating condition to cope with uncertainties better.

Une meilleure connaissance du cycle biogéochimique du méthane est un élémentfondamental dans la compréhension du changement climatique actuel. La modélisationinverse est une des méthodes permettant d’estimer les sources et puits de méthaneen combinant l’information venant d’observations atmosphériques, d’une connaissancea priori des flux de méthane, et d’un modèle de chimie-transport. Cependant, leserreurs liées à la modélisation du transport atmosphérique sont apparues comme unelimitation de plus en plus dominante de cette méthode suite à l’augmentation dunombre et de la diversité des observations.Après avoir montré que l’impact des erreurs de transport sur les inversions des flux deméthane pouvait être important, j’ai cherché à améliorer les capacités de la versionoffline de LMDz, modèle de transport utilisé pour simuler le transport atmosphériquedans le système inverse du LSCE. Pour cela, j’ai intégré des développements récents(paramétrisation de la convection profonde, de la diffusion verticale et du mélangenon-local dans la couche limite) et raffiné la résolution horizontale et verticale.En exploitant les différentes versions disponibles de LMDz, neuf inversions atmosphériquesont été réalisées, estimant les sources et puits de méthane entre 2006 et 2012.Deux périodes de fortes émissions ont été mises en évidence : en 2007 et en 2010,qui ont principalement été attribuées à des anomalies dans les régions tropicales et enChine, où des événements climatiques majeurs ont été observés (Amérique du Sud etAsie du Sud-Est) et où le développement économique se poursuit à un rythme soutenu(Chine), même si les émissions de certains inventaires sont surestimées. / A better knowledge of the methane biogeochemical cycle is fundamental for a betterunderstanding of climate change. Inverse modelling is one powerful tool to derivemethane sources and sinks by optimally combining information from atmospheric observations of methane mixing ratios, from process-based models and inventories ofmethane emissions and sinks, and from a chemistry-transport model used to link emissionsto atmospheric mixing ratios. However, uncertainties related to the modelling ofatmospheric transport are becoming a serious limitation for inverse modelling due tothe increasing number and type of observations.After showing that the impact of transport errors on current atmospheric inversionscould be significant, I tried to improve the representation of atmospheric transport inthe inverse system used at LSCE. Thus, I have tested new physical parameterizations(deep convection, vertical diffusion and non-local transport within the boundary layer)in the LMDz model and adapted it to finer horizontal and vertical resolutions. Thesedevelopments were integrated into the inverse system.Nine inversions have been performed using the different versions of LMDz in order toestimate methane emissions over the period 2006-2012. Two years of strong methaneemissions have been highlighted in 2007 and in 2010. These anomalies have beenmainly attributed to anomalies in the Tropics and in China, where major climate eventshave been observed (Tropical South America and South East Asia) and where economicdevelopment is carrying on with a fast pace (China), even if emissions magnitude andtrend reported in inventories are found to be overestimated.

Les circonstances pouvant conduire à un rejet incontrôlé de polluants dans l'atmosphère sont variées : il peut s'agir de situations accidentelles, par exemples des fuites ou explosions sur un site industriel, ou encore de menaces terroristes : bombe sale, bombe biologique, notamment en milieu urbain. Face à de telles situations, les objectifs des autorités sont multiples : prévoir les zones impactées à court terme, notamment pour évacuer les populations concernées ; localiser la source pour pouvoir intervenir directement sur celle-ci ; enfin déterminer les zones polluées à plus long terme, par exemple par le dépôt de polluants persistants, et soumises à restriction de résidence ou d'utilisation agricole. Pour atteindre ces objectifs, des modèles numériques peuvent être utilisés pour modéliser la dispersion atmosphérique des polluants. Après avoir rappelé les processus physiques qui régissent le transport de polluants dans l'atmosphère, nous présenterons les différents modèles à disposition. Le choix de l'un ou l'autre de ces modèles dépend de l'échelle d'étude et du niveau de détails (topographiques notamment) désiré. Nous présentons ensuite le cadre général (bayésien) de la modélisation inverse pour l'estimation de sources. Le principe est l'équilibre entre des informations a priori et des nouvelles informations apportées par des observations et le modèle numérique. Nous mettons en évidence la forte dépendance de l'estimation du terme source et de son incertitude aux hypothèses réalisées sur les statistiques des erreurs a priori. Pour cette raison nous proposons plusieurs méthodes pour estimer rigoureusement ces statistiques. Ces méthodes sont appliquées sur des exemples concrets : tout d'abord un algorithme semi-automatique est proposé pour la surveillance opérationnelle d'un parc de centrales nucléaires. Un second cas d'étude est la reconstruction des termes sources de césium-137 et d'iode-131 consécutifs à l'accident de la centrale nucléaire de Fukushima Daiichi. En ce qui concerne la localisation d'une source inconnue, deux stratégies sont envisageables : les méthodes dites paramétriques et les méthodes non-paramétriques. Les méthodes paramétriques s'appuient sur le caractère particulier des situations accidentelles dans lesquelles les émissions de polluants sont généralement d'étendue limitée. La source à reconstruire est alors paramétrisée et le problème inverse consiste à estimer ces paramètres, en nombre réduit. Dans les méthodes non-paramétriques, aucune hypothèse sur la nature de la source (ponctuelle, localisée, ...) n'est réalisée et le système cherche à reconstruire un champs d'émission complet (en 4 dimensions). Plusieurs méthodes sont proposées et testées sur des situations réelles à l'échelle urbaine avec prise en compte des bâtiments, pour lesquelles les méthodes que nous proposons parviennent à localiser la source à quelques mètres près, suivant les situations modélisées et les méthodes inverses utilisées / Uncontrolled releases of pollutant in the atmosphere may be the consequence of various situations : accidents, for instance leaks or explosions in an industrial plant, or terrorist attacks such as biological bombs, especially in urban areas. In the event of such situations, authorities' objectives are various : predict the contaminated zones to apply first countermeasures such as evacuation of concerned population ; determine the source location ; assess the long-term polluted areas, for instance by deposition of persistent pollutants in the soil. To achieve these objectives, numerical models can be used to model the atmospheric dispersion of pollutants. We will first present the different processes that govern the transport of pollutants in the atmosphere, then the different numerical models that are commonly used in this context. The choice between these models mainly depends of the scale and the details one seeks to take into account.We will then present the general framework of inverse modeling for the estimation of source. Inverse modeling techniques make an objective balance between prior information and new information contained in the observation and the model. We will show the strong dependency of the source term estimation and its uncertainty towards the assumptions made on the statistics of the prior errors in the system. We propose several methods to estimate rigorously these statistics. We will apply these methods on different cases, using either synthetic or real data : first, a semi-automatic algorithm is proposed for the operational monitoring of nuclear facilities. The second and third studies concern the source term estimation of the accidental releases from the Fukushima Daiichi nuclear power plant. Concerning the localization of an unknown source of pollutant, two strategies can be considered. On one hand parametric methods use a limited number of parameters to characterize the source term to be reconstructed. To do so, strong assumptions are made on the nature of the source. The inverse problem is hence to estimate these parameters. On the other hand non-parametric methods attempt to reconstruct a full emission field. Several parametric and non-parametric methods are proposed and evaluated on real situations at a urban scale, with a CFD model taking into account buildings influence on the air flow. In these experiments, some proposed methods are able to localize the source with a mean error of some meters, depending on the simulated situations and the inverse modeling methods

L'efficience intrinsèque d'utilisation de l'eau (Wi) définit comme le rapport de l'assimilation nette de CO2 (A) sur la conductance stomatique à la vapeur d'eau (gs) est un estimateur au niveau foliaire du compromis fait par la plante entre l'accumulation de biomasse et sa consommation en eau. De nombreuses études ont révélé une forte diversité inter et intra-spécifique de ce trait intégré dans le temps dont l'origine est encore mal connue. Les travaux réalisés sur les variations journalières de A et gs ont jusqu'à maintenant révélé un rôle plus important de la diversité de gs dans la diversité de Wi. Une approche de modélisation inversée a permis de décomposer les variations de gs, observées lors de cinétiques journalières, sous la forme de paramètres décrivant les réponses stomatiques aux différentes variables microclimatiques. Comparé aux modèles décrivant les variations de gs en régime permanent, le développement d'un nouveau modèle dynamique a permis d'ajouter une dimension temporelle essentielle décrivant la réponse temporelle des stomates aux variations microclimatiques. La diversité des réponses temporelles des stomates détectée ne semble pas dépendre de leur densité ou de leur taille. Elle présente toutefois une asymétrie entre l'ouverture et la fermeture des stomates qui participe à la diversité des variations journalières de gs et impacte le bilan hydrique journalier au niveau du plant entier. Ainsi, on peut distinguer deux composantes aux variations journalières de Wi liées à gs : une composante temporelle due à la lente réponse des stomates et une autre composante due aux différences de perception des variations du microclimat / Intrinsic water use efficiency (Wi), defined as the ratio between net CO2 assimilation rate (A) and stomatal conductance to water vapour (gs), is a leaf level estimator of the trade-off between biomass accumulation and water loss at the plant level. A number of studies have shown a strong inter and intra-specific diversity, usually using a time integrated estimator of this trait. However, the origin of this diversity is not yet well known. Up to now, research on the daily variations of Wi have shown a stronger influence of gs on the diversity of Wi as compared to A. An inverse modelling approach has allowed partitioning the variations of gs observed during daily time-courses into parameters, which describe the stomatal responses to different microclimatic variables. Compared to steady-state gs models, the development of a new dynamic model of gs has allowed adding a necessary temporal dimension, which describes the temporal response of stomata to environmental variations. The observed diversity of these temporal stomatal responses was not related to stomatal density or size. The temporal responses of stomata were shown to be asymmetric between opening and closing, which impacts the observed diversity of gs during daily time courses as well as whole plant water relations. Overall these results suggest two components that determine the variations of Wi related to gs during daily time courses: one component due to the temporal response of stomata in itself, and one component which is due to differences in the sensing of microclimate variations

In this thesis we study a parameter identification problem for a stationary diffusion equation posed in heterogeneous media. This problem is closely related to the Calderón problem with anisotropic conductivities. The anisotropic case is particularly difficult and is ill-posed both in regards to uniqueness of solution and stability on the data. Since the present problem is posed in heterogeneous media, we can take advantage of multiscale modelling and the tools of homogenization theory in the study of the inverse problem, unlike the original Calderón problem. We investigate the possibilities of combining the theory of the Calderón problem with homogenization theory in order to obtain a well-posed parameter identification. We find that homogenization theory indeed can be used to make progress towards a well-posed identification of the diffusion coefficient. The success of the method is, however, dependent both on the precise structure of the heterogeneous media and on the modelling of the measurements in the invese problem framework. We have in mind a particular problem formulation which is motivated by an experiment to determine effective coefficients of materials used in food packaging. This experiment comes with a set of requirements on both the heterogeneous media and on the method for making measurements that, unfortunately, are in conflict with the currently available results for well-posedness. We study also an optimization approach to solving the inverse problem under these application specific requirements. Some progress towards well-posedness of the optimization problem is made by proving existence of minimizer, again with homogenization theory playing a key role in obtaining the result. In a proof-of-concept computational study this optimization approach is implemented and compared to two other optimization problems. For the two tested heterogeneous media, the only optimization method that manages to identify reasonably well the diffusion coefficient is the one which makes use of homogenization theory.

This study has two parts. In the first part we develop a computational approach to the solution of an inverse modelling problem concerning the material properties of electrolytes used in Lithium-ion batteries. The dependence of the diffusion coefficient and the transference number on the concentration of Lithium ions is reconstructed based on the concentration data obtained from an in-situ NMR imaging experiment. This experiment is modelled by a system of 1D time-dependent Partial Differential Equations (PDE) describing the evolution of the concentration of Lithium ions with prescribed initial concentration and fluxes at the boundary. The material properties that appear in this model are reconstructed by solving a variational optimization problem in which the least-square error between the experimental and simulated concentration values is minimized. The uncertainty of the reconstruction is characterized by assuming that the material properties are random variables and their probability distribution estimated using a novel combination of Monte-Carlo approach and Bayesian statistics. In the second part of this study, we carefully analyze a number of secondary effects such as ion pairing and dendrite growth that may influence the estimation of the material properties and develop mathematical models to include these effects. We then use reconstructions of material properties based on inverse modelling along with their uncertainty estimates as a framework to validate or invalidate the models. The significance of certain secondary effects is assessed based on the influence they have on the reconstructed material properties. / Thesis / Doctor of Philosophy (PhD)