, Propagation dans des grilles WRF

, Génération des grilles de N

L. Sorties-du-modèle-méso-Échelle and W. De, indice de réfraction (et donc le coindice) sur un ensemble de points repérés par leurs coordonnées en latitude, longitude et altitude. Puisque cette thèse se restreint à la propagation en deux dimensions, les grilles qui vont être utilisées dans cette sous-section sont obtenues en fixant la latitude dans des grilles de N en latitude, longitude et altitude. C'est un choix arbitraire que de fixer la latitude

, sous-section correspondent à la simulation du modèle WRF dans la région des Caraïbes, montrée en Fig. 3.20. Les latitudes sont comprises entre 11, Les données utilisées pour les simulations de cette

, 24 montrent les coupes de N à différentes altitudes Il en ressort que la variation de N à altitude constante est significative, variant de plusieurs dizaines d'unités. En revanches, ces variations sont plus faibles que les variations verticales montrées par les Fig. 3.25 à 3, Les Fig. 3.21 à 3

, Figure 3.20 ? Illustration de la zone géographique des Caraïbes, modélisée par le modèle WRF pour obtenir les grilles de coindice N

, Modélisation d'une liaison satellitaire en radio-occultation et inversion en amplitude

.. Géométrie-de-la-radio-occultation, , p.104

.. Géométrie-de-la-radio-occultation, , p.109

.. , Méthodes d'inversion des données de radio-occultation, p.110

.. Méthode-d-'inversion-de-sokolovskiy, , p.112

G. Modélisation-du-problème-pour-la-méthode, 112 4.3.1 Modélisation de la configuration radio-occultation en deux dimensions, p.116

.. De-réfraction, 116 4.4.1 Validation de la méthode sur un gradient canonique 116 4.4.2 Inversion des résultats en amplitude pour l'inversion d'un profil quelconque 120 4, p.126

S. , , p.126

P. Position-du, , p.143

. .. Démonstration-de-l-'équation-d-'onde-en-coordonnées-curvilignes and .. Frenet, 143 B.2.2 Coordonnées liées au rayon central, p.146

.. , Expression du facteur d'échelle h en fonction de l'indice de réfraction du milieu, p.146

.. Équation-eikonale-et-Équation-de-transport, , p.146

E. , Résolution de l'équation eikonale par le fonctionnel, p.148

B. Ahluwalia, R. Lewis, and J. Boersma, Uniform Asymptotic Theory of Diffraction by a Plane Screen, SIAM Journal on Applied Mathematics, vol.16, issue.4, pp.783-807, 1968.
DOI : 10.1137/0116065

]. B. Ahma-99, G. Ahmad, and . Tyler, Systematic errors in atmospheric profiles obtained from Abelian inversion of radio occultation data: Effects of large-scale horizontal gradients, Journal of Geophysical Research: Atmospheres, vol.77, issue.D4, pp.3971-3992, 1999.
DOI : 10.1175/1520-0477(1996)077<0019:GSOTAF>2.0.CO;2

R. Anthes, C. Rocken, and Y. Kuo, Applications of COSMIC to Meteorology and Climate, Terrestrial, Atmospheric and Oceanic Sciences, vol.11, issue.1, pp.115-156, 2000.
DOI : 10.3319/TAO.2000.11.1.115(COSMIC)

URL : https://doi.org/10.3319/tao.2000.11.1.115(cosmic)

]. J. Arno-86 and . Arnold, Geometrical theories of wave proapgation : a contemporary review, IEE Proceedings, vol.133, issue.2, pp.165-188, 1986.

, Ray method of computation of the intensity of wave fronts. doklady akademii nauk sssr Ed, 1956.

L. Boithias, Propagation des ondes radio electriques dans l'environnement terrestre, 1983.

]. C. Bourlier and V. Fabbro, Radar propagation modeling using the boundary integral equations in a maritime environment with a duct, 2014 International Radar Conference, 2014.
DOI : 10.1109/RADAR.2014.7060314

URL : https://hal.archives-ouvertes.fr/hal-00982752

L. Brekhovskikh, Waves in Stratified Media. moscow, ussr : academy of science Ed, 1957.

K. Budden, Waves in the Ionosphere. cambridge, england : cambridge univ. press Ed, 1961.

]. R. Burkholder and P. Pathak, Analysis of EM penetration into and scattering by electrically large open waveguide cavities using Gaussian beam shooting, Proceedings of the IEEE, pp.1401-1411, 1991.
DOI : 10.1109/5.104215

M. Catedra, J. Perez, F. De-adana, and O. Gutierrez, Efficient ray-tracing techniques for three-dimensional analyses of propagation in mobile communications: application to picocell and microcell scenarios, IEEE Antennas and Propagation Magazine, vol.40, issue.2, pp.15-28, 1998.
DOI : 10.1109/74.683539

V. Cerveny, M. M. Popov, and I. Psencik, Computation of wave fields in inhomogeneous media-Gaussian beam approach, Geophys. J. Astron. Soc, vol.3, pp.508-522, 1982.

, Modélisation électromagnétique des radômes par des techniques basées sur les faisceaux gaussiens, 2004.

M. Chahine, D. Mccleese, P. Rosenkranz, and D. Staelin, Interaction mechanisms within the atmosphere, in Manual of Remote Sensing, Am. Soc. of Photogramm, pp.165-230, 1983.

]. S. Choi and W. Chun, A ray tracing simulation to visualize atmospheric effect on wave propagation using radiosonde information, 2015 IEEE 5th Asia-Pacific Conference on Synthetic Aperture Radar (APSAR), 2015.
DOI : 10.1109/APSAR.2015.7306250

H. Chou, P. Pathak, and R. Burkholder, Novel Gaussian beam method for the rapid analysis of large reflector antennas, IEEE Transactions on Antennas and Propagation, vol.49, issue.6, pp.880-893, 2001.
DOI : 10.1109/8.931145

J. Crank and P. Nicolson, A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type, Advances in Computational Mathematics, vol.226, issue.3, pp.207-226, 1997.
DOI : 10.1007/BF02127704

, Daub 92] I. Daubechies Mathematical Thoery of Optics, 1992.

M. Dottling, A. Jahn, S. Didascalou, and W. Wiesbeck, On the Approximate Boundary Conditions for the Electromagnetic Field on the Surface of Well Conducting Bodies " . Investigations on Propagation of Radio Waves, pp.27-37, 2001.

M. Dottling, A. Jahn, S. Didascalou, and W. Wiesbeck, Two- and three-dimensional ray tracing applied to the land mobile satellite (LMS) propagation channel, IEEE Antennas and Propagation Magazine, vol.43, issue.6, pp.27-37, 2001.
DOI : 10.1109/74.979492

D. Pawlak, Metop : The Space Segment for Eumetsat's Polar System, ESA Bulletin, vol.102, pp.6-18, 2000.

P. Einziger, S. Ray, and M. Saphira, Gabor representation and aperture theory, Journal of the Optical Society of America A, vol.3, issue.4, pp.508-522, 1986.
DOI : 10.1364/JOSAA.3.000508

, Beam series representation and the parabolic approximation : the frequency domain, Journal of optical society of America, vol.5, pp.1883-1892, 1988.

]. V. Fabb-14a, M. Fabbro, C. Noblet, and . Bourlier, Propagation modeling using the Split Step Fourier Method Ground boundary conditions analysis and acceleration by GPU, 2014.

]. V. Fabb-14b, M. Fabbro, C. Noblet, and . Bourlier, Propagation modeling using the Split Step Fourier Method Ground boundary conditions analysis and acceleration by GPU, 2014.

J. , F. Jr, W. Gustafson, R. Easter, J. Barnard et al., Evolution of Ozone, Particulates, and Aerosol Direct Forcing in an Urban Area Using a New Fully-Coupled Meteorology, Chemistry, and Aerosol Model Journal of Geophysical, vol.11, 2006.

M. Feit and J. Fleck, Light propagation in graded-index optical fibers, Applied Optics, vol.17, issue.24, pp.3990-3998, 1978.
DOI : 10.1364/AO.17.003990

W. Felsen, M. Kuperman, H. Porter, and . Schmidt, Computational Ocean Acoustics, AIP series in Modern Acoustics and Signal, 1994.

V. Fock, Electromagnetic Diffraction and Propagation Problems, 1965.

, Ètude de la propagation dans une atmosphère inhomoggène dans les directions horizontale et verticale par la mméthode de l'équation parabolique, AGARD CP 453, pp.1-13, 1989.

, Theory of communication, Journal of the institute of electric engineering, vol.93, pp.429-457, 1946.

R. Galas and W. Koehler, A binary exchange format for GPS data, Physics and Chemistry of the Earth, Part A: Solid Earth and Geodesy, vol.26, issue.6-8, pp.645-648, 2001.
DOI : 10.1016/S1464-1895(01)00114-4

V. Galdi, L. Felsen, and D. Castanon, Quasi ray Gaussian algorithm for timeharmonic two-dimensional scattering by moderately rough interfaces, IEEE Transactions on Antennas and Propagation, vol.49, issue.8, 2001.

V. Galdi, L. Felsen, and D. Castanon, Time-domain radiation from large two-dimensional apertures via narrow-waisted gaussian beams, IEEE Transactions on Antennas and Propagation, vol.51, issue.1, pp.78-88, 2003.
DOI : 10.1109/TAP.2003.808520

P. Gegout, P. Oberlé, C. Desjardins, J. Moyard, and P. Brunet, Ray-Tracing of GNSS Signal Through the Atmosphere Powered by CUDA, HMPP and GPUs Technologies, IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing, vol.7, issue.5, pp.1592-1602, 2014.
DOI : 10.1109/JSTARS.2013.2272600

I. Ghannoum, Etudes d'outils de calcul de propagation radar en milieu complexe (milieu urbain, présence de multi-trajets) par des techniques de faisceaux Gaussiens

, Thèse de l'Institut National des Télécommunications, 2010.

M. Gorbunov, A. Gurvich, and A. Schmakov, Back-propagation and radio-holographic methods for investigation of sporadic ionospheric E-layers from Microlab-1 data, International Journal of Remote Sensing, vol.4, issue.4, pp.675-685, 2002.
DOI : 10.1175/1520-0477(1996)077<0019:GSOTAF>2.0.CO;2

M. Gorbunov, A. Gurvich, and L. Bengtsson, Advanced algorithms of inversion of GPS/MET Satellite data and their application to reconstruction of temperature and humidity, Max-Planck-Inst. fur Meteorol, p.211, 1996.

G. Grell, S. Peckham, R. Schmitz, S. Mckeen, G. Frost et al., Fully coupled ???online??? chemistry within the WRF model, Atmospheric Environment, vol.39, issue.37, pp.6957-6975, 2005.
DOI : 10.1016/j.atmosenv.2005.04.027

G. Hajj, E. Kurkinski, L. Romans, W. Bertiger, and S. Leroy, A technical description of atmospheric sounding by GPS occultation, Journal of Atmospheric and Solar-Terrestrial Physics, vol.64, issue.4, pp.451-469, 2002.
DOI : 10.1016/S1364-6826(01)00114-6

G. Haltiner and R. Williams, Numerical Prediction and Dynamic Meteorology, 1980.

]. R. Hard-73a, F. Hardin, and . Tappert, Application of the split-step Fourier method to the numerical solution of non linear and variable coefficient wave equations, SIAM Rev, vol.15, p.423, 1973.

]. R. Hard-73b, F. Hardin, and . Tappert, Application of the split-step Fourier method to the numerical solution of non linear and variable coefficient wave equations, SIAM Rev, vol.15, p.423, 1973.

]. S. Healy, Radio occultation bending angle and impact parameter errors caused by horizontal refractive index gradients in the troposphere: A simulation study, Journal of Geophysical Research: Atmospheres, vol.104, issue.D11, pp.875-889, 2001.
DOI : 10.1029/1999JD900450

J. Hocke, Inversion of GPS meteorology data, Annales Geophysicae, vol.15, issue.4, pp.443-450, 1997.
DOI : 10.1007/s00585-997-0443-1

URL : https://hal.archives-ouvertes.fr/hal-00316218

, Four-Dimensional Variational Data Assimilation for WRF : Formulation and Preliminary Results, Monthly Weather Review, vol.137, issue.1, pp.299-314, 2009.

J. Klemp, W. Skamarock, and J. Dudhia, Conservative Split-Explicit Time Integration Methods for the Compressible Nonhydrostatic Equations, Monthly Weather Review, vol.135, issue.8, pp.2897-2913, 2007.
DOI : 10.1175/MWR3440.1

, Electromagnetic Theory and Geometrical Optics. new york : wileyinterscience Ed, 1965.

A. Kliore, D. Cain, G. Fjeldbo, B. Seidel, M. Sykes et al., The atmosphere of Mars from mariner 9 radio occultation measurements, Icarus, vol.17, issue.2, pp.484-516, 1972.
DOI : 10.1016/0019-1035(72)90014-0

H. Ko, J. Sari, and J. Skura, Anomalous wave propagation through atmospheric ducts, John Hopkins APL Tech. Dig, vol.4, pp.12-26, 1983.

H. Kogelnick, Laser Beams and Resonators, Applied Optics, vol.5, issue.10, pp.1550-1567, 1966.
DOI : 10.1364/AO.5.001550

R. Kouyoumjian and P. Pathak, A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface, Proc. IEEE, pp.1448-1461, 1974.
DOI : 10.1109/PROC.1974.9651

Y. Kravstov and . Orlov, Geometrical Optics for Inhomogeneous Media, 1990.

E. Kurkinski, G. Hajj, S. Leroy, and B. Herman, The GPS Radio Occultation Technique, TAO, vol.11, issue.1, pp.53-114, 2000.

E. Kurkinski, G. Hajj, K. Hardy, J. Schofield, and R. Linfield, Observing Earth's atmosphere with radio occultation measurements using the Global Positioning System, Journal of Geophysical Research: Atmospheres, vol.123, issue.44, pp.23429-23465, 1997.
DOI : 10.1175/1520-0493(1995)123<2229:AOARRU>2.0.CO;2

]. R. Laprise, The Euler Equations of Motion with Hydrostatic Pressure as an Independent Variable, Monthly Weather Review, vol.120, issue.1, pp.197-207, 1992.
DOI : 10.1175/1520-0493(1992)120<0197:TEEOMW>2.0.CO;2

, Utilisation du lancer de rayons pour le calcul de l'interaction d'un rayonnement électromagnétique avec des objets complexes métalliques et diélectriques, 2004.

M. Leontovich and V. Fock, Solution of propagation of electromagnetic waves along the Earth's surface by the method of parabolic equations, J. Phys. USSR, vol.10, pp.13-23, 1946.

L. Leung, Y. Kuo, and J. Tribbia, Research Needs and Directions of Regional Climate Modeling Using WRF and CCSM, Bulletin of the American Meteorological Society, vol.87, issue.12, pp.1747-1751, 2006.
DOI : 10.1175/BAMS-87-12-1747

, Parabolic equation methods for electromagnetic wave propagation, 2000.

M. Li and J. Kiang, A ray tracing technique for radio occultation, IEEE iWEM2011, 2011.
DOI : 10.1109/iWEM.2011.6021487

M. Liang, K. Xu, G. Kundel, J. Grell, and . Kain, Regional Climate Model Simulation of U.S.???Mexico Summer Precipitation Using the Optimal Ensemble of Two Cumulus Parameterizations, Journal of Climate, vol.20, issue.20, pp.5201-5207, 2007.
DOI : 10.1175/JCLI4306.1

]. G. Liang and H. Bertoni, A new approach to 3-D ray tracing for propagation prediction in cities, IEEE Transactions on Antennas and Propagation, vol.46, issue.6, pp.853-863, 1998.
DOI : 10.1109/8.686774

G. Lindal, H. Hotz, D. Sweetnam, Z. Shippony, J. Brenckle et al., Viking radio occultation measurements of the atmosphere and topography of Mars: Data acquired during 1 Martian year of tracking, Journal of Geophysical Research, vol.33, issue.B14, pp.8443-8456, 1979.
DOI : 10.1016/0019-1035(78)90182-3

Y. Liou, A. Pavelyev, J. Wickert, T. Schmidt, and A. Pavelyev, Analysis of atmospheric and ionospheric structures using the GPS/MET and CHAMP radio occultation database: a methodological review, GPS Solutions, vol.70, issue.4, pp.122-143, 2005.
DOI : 10.1029/92JD01679

, Modélisation d'antennes et de systèmes focaux par décomposition sur une famille de faisceaux gaussiens, 2001.

J. Maciel and L. Fielsen, Discretized gabor-based beam algorithm for time-harmonic radiation from two-dimensional truncated planar aperture distributions-I: formulation and solution, IEEE Transactions on Antennas and Propagation, vol.50, issue.12, pp.1751-1759, 2002.
DOI : 10.1109/TAP.2002.807419

J. Maciel, Y. Hamaraty, and L. Fielsen, Complex rays for radiation from discretized aperture distribution, IEEE Transactions on Antennas and Propagation, vol.35, pp.1031-1044, 1987.

J. Maciel and L. Felsen, Systematic study of fields due to extended apertures by Gaussian beam discretization, IEEE Transactions on Antennas and Propagation, vol.37, issue.7, pp.884-892, 1989.
DOI : 10.1109/8.29383

]. D. Marcuse, Light transmission optics " . Can Nostrand Reinhold, 1982.

D. Martine and J. Bowen, Long-wave optics, IEEE Transactions on Microwave Theory and Techniques, vol.41, issue.10, pp.1676-1690, 1993.
DOI : 10.1109/22.247911

, Efficient calculation of Gaussian-beam seismograms for two-dimensional inhomogeneous media, Geophys. J. Astron. Soc, vol.79, pp.153-166, 1984.

S. Namba, General Theory on the Propagation of Radio Waves in the Ionized Layer of the Upper Atmosphere, Proceedings of the Institute of Radio Engineers, pp.238-262, 1933.
DOI : 10.1109/JRPROC.1933.227599

A. Pavelyev, Y. Liou, J. Wickert, A. Gavrik, and C. Lee, Eikonal acceleration technique for studying of the Earth and planetary atmospheres by radio occultation method, Geophysical Research Letters, vol.30, issue.3, 2009.
DOI : 10.1007/978-3-662-09041-1_19

A. Pavelyev, K. Zhang, Y. Liou, A. Pavelyev, C. Wang et al., Principle of Locality and Analysis of Radio Occultation Data, IEEE Transactions on Geoscience and Remote Sensing, vol.51, issue.6, 2013.
DOI : 10.1109/TGRS.2012.2225629

C. L. Pekeris, Accuracy of the Earth-Flattening Approximation in the Theory of Microwave Propagation, Physical Review, vol.203, issue.7-8, p.518, 1946.
DOI : 10.1098/rsta.1904.0013

M. B. Porter and H. P. Bucker, Gaussian beam tracing for computing ocean acoustic fields, The Journal of the Acoustical Society of America, vol.82, issue.4, pp.1349-1359, 1987.
DOI : 10.1121/1.395269

S. Priebe, M. Kannicht, M. Jacob, and T. Kurner, Ultra broadband indoor channel measurements and calibrated ray tracing propagation modeling at THz frequencies, Journal of Communications and Networks, vol.15, issue.6, pp.547-558, 2013.
DOI : 10.1109/JCN.2013.000103

L. Rao, . P. Carin, and . Ray, A Hybrid (Parabolic Equation)-(Gaussian Beam) Algorithm for Wave Propagation Through Large Inhomogeneous Regions, Broadband Complex Refractive Indices of Ice and Water, pp.700-709, 1972.

, Analysis and validation of GPS/MET data in the neutral atmosphere, J. Geophys. Res, vol.102, issue.29, pp.849-866, 1997.

, Interaction d'une onde électromagnétique avec des structures de grande taille par la méthodes des rayons gaussiens, Thèse de l'Université Paul Sabatier, 2001.

P. Schoott, F. Lemaître, and O. Pascal, Use of Gaussian beam to compute antenna pattern, Annales des télécommunications, vol.57, issue.78, pp.775-797, 2002.

T. Senior, Impedance boundary conditions for imperfectly conducting surfaces, Applied Scientific Research, Section B, vol.8, issue.1
DOI : 10.1007/BF02920074

, Appli. Sci. Res, vol.8, pp.418-436, 1960.

T. Senior and J. Volakis, Scattering by an imperfect right-angled wedge, IEEE Transactions on Antennas and Propagation, vol.34, issue.5, pp.681-688, 1986.
DOI : 10.1109/TAP.1986.1143864

, Sieg 86] A. Siegman, 1986.

W. Skamarock and J. Klemp, A Time-Split Non-Hydrostatic Atmospheric Model for Research and NWP Applications, Journal of Computational Physics : Special Issue on Environmental Modeling, vol.227, issue.7, pp.3465-3485, 2008.

P. Smulders, Geometrical optics model for millimetre-wave indoor radio propagation, Electronics Letters, vol.29, issue.13, pp.1174-1176, 1993.
DOI : 10.1049/el:19930785

S. Sokolovskiy, Inversions of radio occultation amplitude data, Radio Science, vol.27, issue.3, pp.97-105, 2000.
DOI : 10.1016/0019-1035(76)90014-2

S. Sokolovskiy, W. Schreiner, C. Rocken, and D. Hunt, Detection of high-altitude ionospheric irregularities with GPS/MET, Geophysical Research Letters, vol.15, issue.4, 2002.
DOI : 10.1029/1999RS002203

F. Tappert, The parabolic equation method, pp.224-287, 1977.

, Unio 90] I. T. Union Propagation in non-ionized media, Reports of the CCIR, vol.5, 1990.

, Preliminary results of dual-frequency radio occultation of the Martian ionosphere with the aid of Mars-5 spacecraft in 1974, Cosmich. Issled, vol.13, pp.48-51, 1975.

H. Wang, W. Skamarock, and G. Feingold, Evaluation of Scalar Advection Schemes in the Advanced Research WRF Model Using Large-Eddy Simulations of Aerosol???Cloud Interactions, Monthly Weather Review, vol.137, issue.8, pp.2547-2558, 2009.
DOI : 10.1175/2009MWR2820.1

D. Wang, C. Barker, T. Snyder, and . Hamill, A Hybrid ETKF???3DVAR Data Assimilation Scheme for the WRF Model. Part I: Observing System Simulation Experiment, Monthly Weather Review, vol.136, issue.12, pp.5116-5131
DOI : 10.1175/2008MWR2444.1

]. A. Webster, Raypath parameters in tropospheric multipath propagation, IEEE Transactions on Antennas and Propagation, vol.30, issue.4, pp.796-800, 1992.
DOI : 10.1109/TAP.1982.1142840

W. Skamarock, Time-Splitting Methods for Elastic Models Using Forward Time Schemes, Monthly Weather Review, vol.130, pp.2088-2097, 2002.

J. Wickert, G. Beyerle, G. Hajj, V. Schwieger, and C. Reigber, GPS radio occultation with CHAMP: Atmospheric profiling utilizing the space-based single difference technique, Geophysical Research Letters, vol.26, issue.48, pp.115-156, 2002.
DOI : 10.1016/S1464-1895(01)00113-2

H. C. Yeh, -. Chiu, M. Chen, J. Liu, and Y. Liou, Ray tracing simulation for GPS radio occultation in non-spherically symmetric atmosphere with ECMWF analysis, 2012 IEEE International Geoscience and Remote Sensing Symposium, p.801, 2014.
DOI : 10.1109/IGARSS.2012.6350352

E. Zauderer, Complex argument Hermite???Gaussian and Laguerre???Gaussian beams, Journal of the Optical Society of America A, vol.3, issue.4, pp.465-469, 1986.
DOI : 10.1364/JOSAA.3.000465

F. Zou, B. Vandenberghe, M. Wang, Y. Gorbunov, S. Kuo et al., A ray-tracing operator and ots adjoint for the use of GPS/MET refractuib angle measurements, J. Geophys. Res, vol.104, issue.22, pp.301-318, 1999.