The ionosphere is a dynamic and ionized medium. Specification of the ionospheric electron density is important for radio systems operating up to a few GHz. Such systems include communication, navigation and surveillance operations. Computerized Ionospheric Tomography (CIT) is a technique that allows specification of the electron density in the ionosphere. CIT, unlike medical tomography, has geometric limitations such as uneven and sparse distribution of ground-based receivers and limited-angle observations. The inversion is therefore underdetermined and to overcome the geometric limitations of the problem, regularization techniques need to be used. In this thesis the horizontal variation of the ionosphere is represented using wavelet basis functions. Wavelets are chosen because the ground based ionospheric instrumentation is unevenly distributed and hence there is an expectation that the resolution of the tomographic image will change across a large region of interest. Wavelets are able to represent structures with different scale and position efficiently, which is known as Multi Resolution Analysis (MRA). The theory of sparse regularization allows the usage of a small number of basis functions with minimum loss of information. Furthermore, sparsity through wavelets can better differentiate between noise and actual information. This is advantageous because it increases the efficacy to resolve the structures of the ionosphere at different spatial horizontal scale sizes. The basis set is also extended to incorporate time dependence in the tomographic images by means of three-dimensional wavelets. The methods have been tested using both simulated and real observations from the Global Navigation Satellite System (GNSS). The simulation was necessary in order to have a controllable environment where the ability to resolve different scale structures would be tested. The further analysis of the methods required also the use of real observations. They tested the technique under conditions of temporal dynamics that would be more difficult to reproduce with simulations, which often tend to be valid in quiet ionospheric behaviors. Improvements in the detection and reconstruction of ionospheric structures were illustrated with sparse regularization. The comparison was performed against two standard methods. The first one was based on spherical harmonics in space, whilst the second relied on a time-dependent smoothing regularization. In simulation, wavelets showed the possibility to resolve small-scale structures better than spherical harmonics and illustrated the potential of creating ionospheric maps at high resolution. In reality, GNSS satellite orbits allow satellite to receiver datasets that traverse the ionosphere at a few hundred km per second and hence a long time window of typically half an hour may be required to provide observations. The assumption of an unchanging ionosphere is only valid at some locations under very quiet geomagnetic conditions and at certain times of day. For this reason the theory was extended to include time dependence in the wavelet method. This was obtained by considering two approaches: a time-smooth regularization and three-dimensional wavelets. The wavelet method was illustrated on a European dataset and demonstrated some improvements in the reconstructions of the main trough. In conclusion wavelets and sparse regularization were demonstrated to be a valid alternative to more standard methods.

In this study, we aim to develop new likelihood based method for estimating parameters of ordinary differential equations (ODEs) / delay differential equations (DDEs) models. Those models are important for modeling dynamical processes that are described in terms of their derivatives and are widely used in many fields of modern science, such as physics, chemistry, biology and social sciences. We use our new approach to study a distributed delay differential equation model, the statistical inference of which has been unexplored, to our knowledge. Estimating a distributed DDE model or ODE model with time varying coefficients results in a large number of parameters. We also apply regularization for efficient estimation of such models. We assess the performance of our new approaches using simulation and applied them to analyzing data from epidemiology and ecology.

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La SER (Surface Équivalente Radar) est une grandeur caractérisant le pouvoir rétrodiffuseurd’une cible soumise à un champ électromagnétique. Dans de nombreuses applications,il est capital d’analyser et de contrôler la SER. L’imagerie 3D est l’outil adapté pourlocaliser et caractériser en trois dimensions les principaux contributeurs à la SER. Cependant,ce traitement est un problème de synthèse de Fourier qui n’est pas inversible car il y aplus d’inconnues que de données. Les méthodes conventionnelles telles que le Polar FormatAlgorithm, consistant en un reformatage des données avec complétion de zéro suivi d’unetransformée de Fourier inverse rapide, fournissent des résultats de qualité limitée.Dans ce travail, nous proposons une nouvelle méthode haute résolution. Elle est dénomméeSPRITE (pour SParse Radar Imaging TEchnique) et permet d’accroître considérablementla qualité des cartes de rétro-diffusion estimées. Elle repose sur une régularisation duproblème consistant en la prise en compte d’informations a priori de parcimonie et d’uneinformation de support. La solution est alors définie comme le minimiseur d’un critère pénaliséet contraint. L’optimisation est assurée par l’algorithme primal-dual ADMM (AlternatingDirection Method of Multiplier) dont une adaptation aux spécificités du problème mène à descalculs efficaces à l’aide de transformées de Fourier rapides.Finalement, la méthode est évaluée sur des données synthétiques et réelles. Comparativementà la méthode conventionnelle, la résolution est drastiquement accrue. Les images 3Dproduites sont alors un outil particulièrement adapté à l’analyse et au contrôle de SER. / The RCS (Radar Cross Section) is a quantity which characterizes the scattering power ofa target exposed to an electromagnetic field. Its analysis and control are important in manyapplications. 3D imaging is a suitable tool to accurately locate and characterize in 3D themain contributors to the RCS. However, this is a non-invertible Fourier synthesis problembecause the number of unknowns is larger than the number of data. Conventional methodssuch as the Polar Format Algorithm, which consists of data reformatting including zeropaddingfollowed by a fast inverse Fourier transform, provide results of limited quality.In this work, we propose a new high resolution method, named SPRITE (for SParse RadarImaging TEchnique), which considerably increases the quality of the estimated RCS maps. Itis based on a regularization scheme that accounts for information of sparsity and support. Thesolution is then defined as the minimizer of a penalized and constrained criterion. Optimizationis ensured by an appropriate adaptation of the ADMM (Alternating Direction Methodof Multiplier) algorithm that is able to quickly perform calculations using fast Fourier transforms.Finally, the method is evaluated on both simulated and real data. Compared to the conventionalmethod, the resolution is significantly increased and the images can support a betterRCS analysis and control.