The discrete Fourier transform in Cartesian coordinates has proved to be invaluable in many disciplines. However, less theory has been developed for functions that are best described in polar coordinates. In this thesis, a new discrete 2D-Fourier transform in polar coordinates is proposed and tested by numerical simulations. Guidelines for choosing sample size are developed. Furthermore, to be as useful as its Cartesian counterpart, improvements are made to reduce the computing time.

The Fourier transform is one of the most useful tools in science and engineering and can be expanded to multi-dimensions and curvilinear coordinates. Multidimensional Fourier transforms are widely used in image processing, tomographic reconstructions and in fact any application that requires a multidimensional convolution. By examining a function in the frequency domain, additional information and insights may be obtained. In this thesis, the development of a symbolic computer algebra toolbox to compute two dimensional Fourier transforms in polar coordinates is discussed. Among the many operations implemented in this toolbox are different types of convolutions and procedures that allow for managing the toolbox effectively. The implementation of the two dimensional Fourier transform in polar coordinates within the toolbox is shown to be a combination of two significantly simpler transforms. The toolbox is also tested throughout the thesis to verify its capabilities.

The Fourier transform is one of the most useful tools in science and engineering and can be expanded to multi-dimensions and curvilinear coordinates. Multidimensional Fourier transforms are widely used in image processing, tomographic reconstructions and in fact any application that requires a multidimensional convolution. By examining a function in the frequency domain, additional information and insights may be obtained. In this thesis, the development of a symbolic computer algebra toolbox to compute two dimensional Fourier transforms in polar coordinates is discussed. Among the many operations implemented in this toolbox are different types of convolutions and procedures that allow for managing the toolbox effectively. The implementation of the two dimensional Fourier transform in polar coordinates within the toolbox is shown to be a combination of two significantly simpler transforms. The toolbox is also tested throughout the thesis to verify its capabilities.

The Fourier transform is one of the most useful tools in science and engineering and can be expanded to multi-dimensions and curvilinear coordinates. Multidimensional Fourier transforms are widely used in image processing, tomographic reconstructions and in fact any application that requires a multidimensional convolution. By examining a function in the frequency domain, additional information and insights may be obtained. In this thesis, the development of a symbolic computer algebra toolbox to compute two dimensional Fourier transforms in polar coordinates is discussed. Among the many operations implemented in this toolbox are different types of convolutions and procedures that allow for managing the toolbox effectively. The implementation of the two dimensional Fourier transform in polar coordinates within the toolbox is shown to be a combination of two significantly simpler transforms. The toolbox is also tested throughout the thesis to verify its capabilities.

The Fourier transform is one of the most useful tools in science and engineering and can be expanded to multi-dimensions and curvilinear coordinates. Multidimensional Fourier transforms are widely used in image processing, tomographic reconstructions and in fact any application that requires a multidimensional convolution. By examining a function in the frequency domain, additional information and insights may be obtained. In this thesis, the development of a symbolic computer algebra toolbox to compute two dimensional Fourier transforms in polar coordinates is discussed. Among the many operations implemented in this toolbox are different types of convolutions and procedures that allow for managing the toolbox effectively. The implementation of the two dimensional Fourier transform in polar coordinates within the toolbox is shown to be a combination of two significantly simpler transforms. The toolbox is also tested throughout the thesis to verify its capabilities.

The use of geophysical testing methods has considerable potential to be a cost effective and accurate technique to assess near-surface soil conditions. Multi channel analysis of surface waves (MASW) test is a geophysical non-intrusive test that uses the dispersive characteristic of Rayleigh waves to estimate low strain shear modulus and damping coefficient of near-surface soil. Also, this technique is used to detect underground voids. Recently, MASW technique has gained more attention, partly because of its ease of use and partly because of the significant improvements in data acquisition systems. The theories of MASW test consider the effect of horizontal soil layering, though the effect of lateral inhomogeneities (i. e. cavities and voids), inclined layering and inverse layering (i. e. a layered system in which the top layers are stiffer than the bottom ones) are not addressed properly in these theories. <br /><br /> The objective of this dissertation is to investigate the effect of lateral inhomogeneities on the propagation of Rayleigh waves in an elastic half-space excited by a transient loading. The results can be applied to locate underground cavities using MASW test and to improve the MASW analysis techniques. In lieu of theoretical solutions, two and three dimensional numerical models are constructed to simulate the MASW test. To assure the quality of the obtained data, numerical models are calibrated with Lamb solution. Voids with different sizes and embedment depths are inserted in the medium. Responses along the surface as well as inside the medium are recorded and analyzed in time, frequency, spatial and frequency-wave number domains. Different material types and sources are used to generalize the results. Afterwards, the combined effect of void and layered systems on the surface responses are studied. To verify the results, experimental field and laboratory data are presented and the trends are compared to the numerical results. <br /><br /> It is found that the void starts to vibrate in response to the Rayleigh wave excitation. Due to the vibration of the void energy partitioning occurs. Part of the incident energy is reflected in the form of Rayleigh wave. Another part is converted to body waves, and spread into the medium. The transferred part of the energy is attenuated and has smaller amplitudes. Finally, a part of energy is trapped in the void region and bounces back and forth between the void boundaries, until it damps. The trapped energy is associated to higher modes of Rayleigh waves and excited Lamb waves. The effect of trapped energy is seen as a region in the vicinity of the void with concentrated energy, in frequency domain. The extents of this region depends on the void size, and the frequency content of the incident energy. Thus, in some cases it is possible to correspond the size of the model to the extents of the region with energy concentration. <br /><br /> A new technique is proposed to determine the location of a void, and estimate its embedment depth. The technique is called Attenuation Analysis of Rayleigh Waves (AARW), and is based on the observed damping effect of the void on the surface responses. For verification, the results are compared to experimental field and laboratory data. The observations are in good agreement with the observed numerical results. Further, the AARW technique showed to be a promising tool for void detection.

The use of geophysical testing methods has considerable potential to be a cost effective and accurate technique to assess near-surface soil conditions. Multi channel analysis of surface waves (MASW) test is a geophysical non-intrusive test that uses the dispersive characteristic of Rayleigh waves to estimate low strain shear modulus and damping coefficient of near-surface soil. Also, this technique is used to detect underground voids. Recently, MASW technique has gained more attention, partly because of its ease of use and partly because of the significant improvements in data acquisition systems. The theories of MASW test consider the effect of horizontal soil layering, though the effect of lateral inhomogeneities (i. e. cavities and voids), inclined layering and inverse layering (i. e. a layered system in which the top layers are stiffer than the bottom ones) are not addressed properly in these theories. <br /><br /> The objective of this dissertation is to investigate the effect of lateral inhomogeneities on the propagation of Rayleigh waves in an elastic half-space excited by a transient loading. The results can be applied to locate underground cavities using MASW test and to improve the MASW analysis techniques. In lieu of theoretical solutions, two and three dimensional numerical models are constructed to simulate the MASW test. To assure the quality of the obtained data, numerical models are calibrated with Lamb solution. Voids with different sizes and embedment depths are inserted in the medium. Responses along the surface as well as inside the medium are recorded and analyzed in time, frequency, spatial and frequency-wave number domains. Different material types and sources are used to generalize the results. Afterwards, the combined effect of void and layered systems on the surface responses are studied. To verify the results, experimental field and laboratory data are presented and the trends are compared to the numerical results. <br /><br /> It is found that the void starts to vibrate in response to the Rayleigh wave excitation. Due to the vibration of the void energy partitioning occurs. Part of the incident energy is reflected in the form of Rayleigh wave. Another part is converted to body waves, and spread into the medium. The transferred part of the energy is attenuated and has smaller amplitudes. Finally, a part of energy is trapped in the void region and bounces back and forth between the void boundaries, until it damps. The trapped energy is associated to higher modes of Rayleigh waves and excited Lamb waves. The effect of trapped energy is seen as a region in the vicinity of the void with concentrated energy, in frequency domain. The extents of this region depends on the void size, and the frequency content of the incident energy. Thus, in some cases it is possible to correspond the size of the model to the extents of the region with energy concentration. <br /><br /> A new technique is proposed to determine the location of a void, and estimate its embedment depth. The technique is called Attenuation Analysis of Rayleigh Waves (AARW), and is based on the observed damping effect of the void on the surface responses. For verification, the results are compared to experimental field and laboratory data. The observations are in good agreement with the observed numerical results. Further, the AARW technique showed to be a promising tool for void detection.

Le phénomène de givrage qui se produit durant les vols d’aéronefs, lors de leurs traversées de la troposphère (zone dans laquelle la température peut descendre jusqu’à -60°C) se manifeste par une accrétion de glace sur différentes parties de l’appareil (voilures, réacteurs, sondes de mesure…) mettant en péril la sécurité de celui-ci. Ce phénomène constitue alors une problématique majeure pour la sécurité de l’aviation civile, c’est pourquoi il est nécessaire de développer de nouvelles techniques de mesure afin de détecter et d’éviter les zones à risques. Notre intérêt s’est porté sur l’imagerie interférométrique en défaut de mise au point, une technique optique offrant de nombreux avantages (large champ de mesure, gamme de tailles étudiée étendue [50 μm : quelques millimètres], distance particule/appareil de mesure de plusieurs dizaines de centimètres…). Au cours de ces travaux de thèse, nous avons développé un dispositif d’Imagerie Interférométrique de Particules (IIP) multi-vues permettant de caractériser les cristaux de glace en suspension dans latroposphère. En effet, par comparaison avec des mesures obtenues sur des images nettesenregistrées simultanément, nous avons validé l’utilisation de l’IIP multi-vues pour l’estimation dedimensions de particules de glace avec un taux d’erreur inférieur à 20%. Nous avons égalementproposé différentes approches permettant d’estimer les volumes des particules de glace et mis enévidence la « signature » typique d’une gouttelette en cours de givrage à partir d’images d’IIP.Dans une seconde partie, nous avons validé l’utilisation de l’IIP multi-vues pour le cas complexe oùles interférogrammes d’une paire de particules de glace proches l’une de l’autre se recouvrent etdiscuté des phénomènes de Moiré qui peuvent apparaître et perturber nos mesures. En outre, nousavons étendu le domaine d’utilisation de l’IIP à des milieux moins dilués. Enfin, dans une dernière partie, nous avons développé un dispositif expérimental innovant nous permettant d’effectuer des mesures d’IIP expérimentales à partir de particules « programmées » sur une matrice de micro-miroirs (DMD). / The icing phenomenon that occurs during aircraft flights during their troposphere crossings (an area where the temperature can drop to -60 ° C) is manifested by an accretion of ice on different parts of the apparatus (wings, reactors, measurement probes ...) endangering the safety of this latter. This phenomenon is therefore a major problem for the safety of civil aviation, that's why it is necessary to develop new measurement techniques to detect and avoid risk's areas. Our interest has been on interferometric out-of-focus imaging, an optical technique offering many advantages (wide measurement field, range of studied sizes extended [50 μm: a few millimeters], distance particle / measuring device several tens of centimeters ...). During this thesis, we have developed a multi-view Interferometric Particle Imaging (IPI) device to characterize suspended ice crystals in the troposphere. Indeed, by comparison with measurements obtained on in-focus images recorded simultaneously, we have validated the use of multi-view IIP for the estimation of ice particle dimensions with an error rate lower than 20%. We also proposed different approaches to estimate ice particle volumes and highlighted the typical "signature" of a droplet during icing from IIP images.In a second part, we validated the use of multi-view IIP for the complex case where the interferograms of a pair of ice particles close to each other overlap and discuss about Moiré phenomena that may appear and disrupt our measurements. In addition, we have extended the field of use of IIP to less diluted media. Finally, in the last part, we have developed an innovative experimental device allowing us to perform experimental IIP measurements from particles "programmed" on a matrix of micro-mirrors (DMD : Digital Micromirror Device).