GRSS | IEEE - Ac · 2015. 7. 10. · Re ections F ran cois Soulat Director: Giulio Ru ni 1 Octob er 2003 Advisor: An toni Bro quetas 2 1 Starlab ... comp osed b J er^ ome Gourrion,

Do toral ThesisSea Surfa e Remote Sensingwith GNSS and Sunlight Re e tions

Fran� ois SoulatDire tor: Giulio RuÆni1 O tober 2003Advisor: Antoni Broquetas2

1StarlabEdi� i de l'Observatori FabraC. de l'Observatori s.n., Muntanya del Tibidabo08035 Bar elona, Spainhttp://starlab.es 2Universitat Polit�e ni a de CatalunyaDepart. of Signal Theory and Comm.Jordi Girona, 3108034 Bar elona, Spainhttp://www-ts .up .es

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A knowledgementsI would like to express my deep and sin ere gratitude to many people who made this thesispossible. Spe ial thanks are due to the two persons who have dire ted me during this thesis:Giulio RuÆni, head of Starlab, Spain, and Bertrand Chapron, head of the O eanographi Spa e Resear h Department at Ifremer, Fran e. I owe my most sin ere gratitude to the formerespe ially for his wide knowledge, en ouragement and personal guidan e. Many thanks to thelatter|king grasshopper|for his exhaustive advi e.I wish to express my warm and sin ere thanks to the Starlab team for the support theyprovided. I warmly thank Olivier Germain, Mar o Caparrini, Leo RuÆni and Ernest Mendozawho provided me, among others, dis ipline in my work and useful omments. Ana Maiquesand Ara eli Pi have been ex eptional with their unforgettable maternal attention.My sin ere thanks are also due to the GNSS-R ommunity, with a parti ular a knowl-edgement to Manuel Mart��n-Neira and Pierluigi Silvestrin from the European Spa e Resear hand Te hnology Centre, The Netherlands, and ESA/ESTEC, who supported together withStarlab most of this resear h. I am also grateful to Antonio Rius (IEEC), Estel Cardel-la h (JPL/NASA), Tony Elfouhaily (CNRS/IRPHE/IOA), Donald Thompson (Johns Hop-kins University/APL), and Jordi Vil�a from the Bar elona Port Authority.I also would like to thank the young Fren h team of Ifremer/DRO/OS omposed by J�eromeGourrion, Ni olas Reul and Olivier Ar her, as well as Fabri e Collard from Boost Te hnologies.My sin ere thanks are due to my tutor at the Universitat Polit�e ni a de Catalunya, AntoniBroquetas, who advised me during my thesis proje t, and to Adriano Camps (UPC/TSC) forhis kindness and support.Personal thanks to those who heered me on during these three years:Lauren e, Fran� oise, Jean-Louis, B�en�edi te, Christine, Paul, Eloise, Ali e, Andr�e, Marie-Ali e,Yvonne, Simon, Emile, Mi h�ele, Mar , Guy, Fan hon, Fran� ois, Edith, Isabelle, Guillaume,Ambroise, Aur�elien, Fru tu, Silvia, Marie, Antoine, J�ezabel, Line, the St�ephanes, St�ephanie,Bruno, Ignasi, Olivier, Chlo�e, Martin, Anne, Anne-Laure, Marianne, Astrild, Pauline, Ariane,Julien, F�eli ien, Ni ola, Catherine, Christian, Jason, Jordi (there are a lot!), Miguel-Angel,Ana, Carles, Ensaladilla Sound System, Tiboum, the Bass.3

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Abstra tThis thesis investigates the remote-sensing apabilities of GNSS re e tions (GNSS-R) foro eanographi appli ations, through theoreti al studies and experimental data analysis. We�rst provide an overview of the models related to involved physi al pro esses, su h as spe traland statisti al hara teristi s of random sea surfa e, ele tromagneti s attering and signalpro essing. From a theoreti al and numeri al point of views, we parti ularly fo us on the val-idation at L-band of the Geometri Opti s approximation of the Kir hho� s attering theoryover the o ean. A new model for the s atterometri bi-dimensional waveform, namely theDelay-Doppler Map (DDM), is then proposed. A GNSS-R simulation toolkit is developed,whi h enables the synthesis of the ele tromagneti re e ted �eld over o ean surfa e. Thissimulator is used for assessing the retrieval of sea-state from a oastal platform. Furthermore,we fo us on GNSS-R sensitivity to sea-surfa e dynami s, roughness and salinity through ex-perimental data analysis. Parti ular interest is put on the investigation of s atterometri appli ations analyzing data from an experimental ight under quite strong wind onditions.A omplete DDM inversion is arried out to infer dire tional sea-surfa e mean square slope(DMSS), whi h de�nes the size, the dire tion and the isotropy of a Gaussian slope PDF.This te hnique is validated and appears to be a very eÆ ient tool for GNSS-R s atterometry.Simultaneously gathered, solar bistati re e tion measurements (SORES) enhan e the under-standing of quasi-spe ular s attering through the bi-dimensional Tilt-Azimuth Mapping ofthe sea surfa e. Considered as a repetition of the famous Cox and Munk experiment, SORESinversion provides DMSS whi h are onsistent with L-band derived DMSS, and reveals a sig-ni� ant non-Gaussianity of the slope PDF. Based on a spe tral analysis, we investigate theimportant impa t of swell in addition to wind stress over DMSS. Both opti al and L-bandtotal MSS are 20% higher than what predi ts a wind-driven model for the observed windspeeds. We on lude that DMSS should be onsidered as a self-standing produ t and not besystemati ally linked to the near-surfa e wind ve tor. Additional ground experiments are re-ported in order to draw preliminary on lusions on the possible retrieval of sea-surfa e salinityand emissivity with GNSS-R. The impa t of salinity is observed through analysis of the inter-feren e between dire t signal along with re e ted signal, whi h shows a strong noise omparedto the expe ted estimated magnitude. Con erning the radiometri apability of GNSS-R, itis shown that the ratio of the dire t and re e ted signals allows the estimation of emissivity,although an a priori knowledge of sea-state is required.

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ContentsA knowledgements 3Abstra t 5Introdu tion 11I Bistati Sea Surfa e Remote Sensing with GNSS Signals 171 GNSS Signal Chara teristi s 191.1 Global Positioning System: GPS . . . . . . . . . . . . . . . . . . . . . . . . . . 191.1.1 Signal stru ture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.1.2 Signal dete tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.1.3 The modernization of GPS . . . . . . . . . . . . . . . . . . . . . . . . . 221.1.4 GPS re e tions overage . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.2 GLONASS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261.3 Galileo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 Sea Surfa e Model 292.1 Sea-surfa e spe tral model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.1.1 The involved geophysi al parameters . . . . . . . . . . . . . . . . . . . . 292.1.2 Elfouhaily et al. spe trum . . . . . . . . . . . . . . . . . . . . . . . . . . 302.1.3 Sea surfa e generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.1.4 Dis ussion on the use spe tral models for s atterometri purposes . . . . 312.1.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.2 Gaussian sea-surfa e statisti s . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.2.1 Statisti al hara terization of random surfa es . . . . . . . . . . . . . . 332.2.2 Slope distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.2.3 Impa t of urrent on sea-surfa e slope varian e . . . . . . . . . . . . . . 353 S attering Model 373.1 Bistati ele tromagneti models . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.1.1 Kir hho� approximation or Physi al Opti s . . . . . . . . . . . . . . . . 393.1.2 The Geometri Opti s or Stationary Phase Approximation . . . . . . . 403.1.3 Small Perturbation Method . . . . . . . . . . . . . . . . . . . . . . . . . 433.1.4 The Two-S ale (Composite or Hybrid) Model . . . . . . . . . . . . . . . 443.2 GO validation at L-band . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.2.1 Radius of urvature and s atterer size . . . . . . . . . . . . . . . . . . . 453.2.2 Paraboli surfa e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.2.3 Gaussian surfa e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487

8 CONTENTS3.2.4 Con lusions on GO assumption for GNSS-R . . . . . . . . . . . . . . . . 513.3 Waveform model: DDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.3.1 Integral expression of the power waveform . . . . . . . . . . . . . . . . . 523.3.2 Expression as a 2-D integral on sea-surfa e . . . . . . . . . . . . . . . . 583.3.3 Comparison with Zavorotny's and Pi ardi's models . . . . . . . . . . . . 584 GNSS-R Signal Pro essing 614.1 From raw data to Level 0: Delay-Doppler PRN ode despreading . . . . . . . . 614.1.1 Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.1.2 Tra king the 1-D waveforms . . . . . . . . . . . . . . . . . . . . . . . . . 674.1.3 2-D waveforms generation (DDM) . . . . . . . . . . . . . . . . . . . . . 674.2 From Level 0 to Level 0b data: in oherent averaging . . . . . . . . . . . . . . . 684.3 From Level 0b to Level 1 data: retra king . . . . . . . . . . . . . . . . . . . . . 68II GNSS-R Signal Simulations 715 GNSS-R Simulation Toolkit 735.1 Model 1: GRADAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.1.1 How to simulate the �eld for large o ean surfa es? . . . . . . . . . . . . 745.1.2 GNSS dire t and re e ted signals . . . . . . . . . . . . . . . . . . . . . . 765.1.3 GRADAS algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.1.4 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795.2 Model 2: Intermittent S reamers . . . . . . . . . . . . . . . . . . . . . . . . . . 825.2.1 Spe ular point spe i� ations . . . . . . . . . . . . . . . . . . . . . . . . 835.2.2 Code implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.2.3 Simulated data and analysis . . . . . . . . . . . . . . . . . . . . . . . . . 865.3 Con lusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 876 An Appli ation for Sea State Monitoring 896.1 Simulation spe i� ations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 896.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 906.2.1 Phase analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 906.2.2 Amplitude analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 906.2.3 Field derivative analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 926.3 Con lusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93III GNSS-R Airborne S atterometri Performan e Analysis 957 The Eddy Experiment Flight 977.1 Experimental ampaign . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 977.1.1 Instrumental set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 997.1.2 Satellite on�guration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1007.2 Ground truth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1008 S atterometry with Sunlight 1058.1 The data set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1058.2 Inversion pro edure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1078.2.1 The dire tional mean square slope . . . . . . . . . . . . . . . . . . . . . 1078.2.2 Forward model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

CONTENTS 98.2.3 Parameter sear h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1138.2.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1138.3 Wind and waves hara terization . . . . . . . . . . . . . . . . . . . . . . . . . . 1178.3.1 Spe tral analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1178.3.2 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1208.4 Comparison with Cox and Munk's measurements . . . . . . . . . . . . . . . . . 1258.5 Con lusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1289 S atterometry with GNSS-R 1319.1 The data set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1319.2 Retra king and inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1329.2.1 The forward model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1329.2.2 Inversion s heme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1339.2.3 Empiri al adjustments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1389.3 Analysis of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1389.3.1 Estimated DMSS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1389.3.2 Convergen e and residual . . . . . . . . . . . . . . . . . . . . . . . . . . 1409.4 Comparison of opti al and L-band derived DMSS . . . . . . . . . . . . . . . . . 1409.4.1 Link to wind speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1439.5 Con lusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143IV Other GNSS-R Appli ations 14710 Sensitivity to Salinity 14910.1 Experimental ampaign . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14910.1.1 Des ription . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14910.1.2 Instrumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15010.1.3 Satellites in view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15010.2 Theoreti al analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15010.2.1 Re eived power versus elevation angle . . . . . . . . . . . . . . . . . . . 15210.2.2 Diele tri onstant model . . . . . . . . . . . . . . . . . . . . . . . . . . 15310.2.3 How do the polarization oeÆ ients depend on salinity? . . . . . . . . . 15710.2.4 Power model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15910.3 Data analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15910.4 Dis ussion on salinity estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 16210.5 Con lusions on salinity retrieval with GNSS-R . . . . . . . . . . . . . . . . . . 16211 Emissivity Retrieval 16511.1 Experimental ampaign . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16511.2 Power model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16611.2.1 Re e ted to dire t signal ratio analysis . . . . . . . . . . . . . . . . . . . 16811.2.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16911.3 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16911.3.1 Re e ted to dire t signal ratio analysis . . . . . . . . . . . . . . . . . . . 17111.4 Data analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17111.5 Con lusions on radiometry with GNSS-R . . . . . . . . . . . . . . . . . . . . . 174Con lusions and Perspe tives 175

10 CONTENTSV Appendix 179A Relevant Symbols 181B A ronyms 183List of Figures 185List of Tables 191Bibliography 193

Introdu tionThe sea surfa e represents the interfa e between the o ean and the atmosphere, the two most ru ial systems governing the dynami s of limate and global hange of the planet. Improvedunderstanding of the oupled intera tions of the physi al pro esses taking pla e at the o ean-atmosphere interfa e is thus essential.It has long been re ognized that the retrieval of o eanographi data is a key omponentin modern meteorologi al analysis and fore asting, as well as an essential resour e for limateresear h. O eanographi studies su�ered during a long period from la k of measurements.Indeed �rst observations began with in-situ measurements su h as buoys, radars on-boardplanes or boats. During the last twenty years, o ean observation has in reased in eÆ ien ywith the use of remote-sensing instruments su h as s atterometers, syntheti aperture radarsand radar-altimeters on-board satellites. The intera tion between emitted waves and the seasurfa e an provide a geophysi al signature.The sea responds to atmospheri for ing on many s ales in both time and spa e. Remote-sensing of near-surfa e wind speed over the o ean by mi rowave te hniques relies on themodi� ation of short surfa e waves by the surfa e layer winds. This inter-relationship istypi ally sampled, formulated, and utilized on spatial s ales of 10-50 km: the mesos ale.In the last past few years, an original remote-sensing on ept has fo used on the passivebistati retrieval of geophysi al data. The term passive indi ates that the developed systemsdo not emit any signal, they just \listen". The term bistati means that emitter and re eiverare lo ated at two di�erent pla es in spa e. The underlying prin iple is the use of re e tedsignals originating from sour es of opportunity to infer properties of the re e ting surfa e, su has signi� ant wave height (SWH), mean sea level (MSL) and dire tional mean square slope(DMSS). It an potentially provide more information ompared to monostati approa hes,where emitter and re eiver are put on the same platform, due to the potential availability ofmultiple sour es.Here, we onsider Global Navigation Satellite Systems (GNSS) as sour es of opportunityemitting at L-band (�20 m). GNSS re e tions (GNSS-R) is an o�spring of bistati radarin whi h only the re eivers need to be deployed on a spe i� platform ( oastal, airborneor spa eborne); the emitters are already in pla e for other purposes. We emphasize thatGNSS systems are premium andidates for remote-sensing appli ations, given their overage,potential resolution, long-term availability and signal hara teristi s.Obje tivesThe main obje tive of the thesis is to analyze the GNSS-R apabilities in o ean surfa eremote-sensing. For this purpose, we aim at developing models with spe i� data produ ts, arrying out dedi ated experimental ampaigns. The use of GNSS signals re e ted o� thesea surfa e as a remote-sensing tool has generated onsiderable attention sin e the pioneering

12 INTRODUCTIONstudy of [Mart��n-Neira1993℄. Two lasses of appli ations have rapidly emerged in the om-munity: sea-surfa e altimetry, whi h aims at retrieving the mean sea level like lassi al radaraltimeters and sea-surfa e s atterometry for the determination of sea-roughness. The former onsists in an estimation of the relative delay between the dire t and re e ted signals, whereasthe latter|addressed in this dissertation|fo uses on the modi� ation of the s attered signalto infer sea-surfa e hara teristi s. In addition, parti ular interest has been put on anotherparameter usually provided by radiometers, the emissivity, whi h has not been yet investi-gated with GNSS-R. We will also onsider a omponent of the emissivity, salinity, althoughthe retrieval is mu h more hallenging.We will present for this purpose the models related to involved physi al pro esses, su h asspe tral and statisti al hara teristi s of the random sea surfa e, ele tromagneti s atteringand signal pro essing. We aim at determining the re e ted signal hara teristi s through themodeled return 2-D waveform, namely the Delay-Doppler Map (DDM). Several experimentswill be presented in order to investigate retrievals of spe i� geophysi al sea-surfa e parame-ters. The re eived signal an be treated with several pro essing strategies depending on thedesired produ t.We emphasize in this study the investigation on the dire tional sea-surfa e roughness,whi h is one key missing integrated element, essential to understand and quantify the o ean-atmosphere ux of energy and momentum. DMSS an help to better ontrol the width ofthe dire tional distribution of wave energy. It is a simple parameter|often related to thesea-surfa e wind ve tor|required to hara terize the sea-state degree of development. Thisparameter shall onstrain the sour e terms in wave energy evolution equation. It an bereadily assimilated and help routine fore ast of waves.In order to enhan e the understanding of GNSS-R response to sea-surfa e roughness,sunlight an also be used as a sour e of opportunity. The approa h is simply based on theobservation and analysis of the visible glitter pattern over the sea surfa e.The three key on epts to bear in mind in this dissertation are:� Bistati s attering: bistati remote-sensing di�ers from the monostati one by thefa t that transmitter and re eiver have di�erent lo ations.� Spe ular re e tions: they dominate radar forward s attering in the mi rowave do-main and at shorter wavelengths. S atterometry may be thus re-named spe ulometry.� Sea-surfa e roughness: it is mainly des ribed by the mean square slope (MSS)|orthe dire tional MSS (DMSS)|whi h is by de�nition the se ond order moment of thesea-surfa e slope probability density fun tion (PDF). The MSS does not provide anyinformation on the dire tivity of the sea-surfa e slope PDF, whereas the DMSS is aparameter of the 2-D PDF.ContextThe investigation in ludes theoreti al analysis and experimental ampaigns, both of whi hhave been arried out at Starlab within the s ope of proje ts for the European Spa e Agen y(ESA), the Institut Fran� ais de Re her he et d'Exploitation de la MER (IFREMER), andStarlab. Table 1 summarizes the geophysi al produ ts observed and modeled during thethesis for di�erent platforms ( oastal, airborne and spa eborne).A parti ular motivation of this study was to gather simultaneously opti al and L-band datasets from the same airborne platform during the PARIS- Experiment, as fully des ribed

INTRODUCTION 13in Chapter 7. This experimental ampaign was ondu ted by Starlab in September 2002along the Catalan Coast for the demonstration of GNSS-R altimetry. An air raft ying at1000 m olle ted roughly 1.5 hours of GNSS-R data along a tra k of 150 km total length.We investigate the full exploitation of the bi-dimensional DDM produ t to infer the threeparameters fully des ribing the sea-surfa e PDF.In addition to GNSS-R, high resolution opti al photographs of Sun glitter were taken,providing the so- alled SORES data (SOlar RE e tan e Spe ulometer). Sin e the famousstudy of [Cox and Munk1954℄, it is well known that sea-roughness an be inferred from su hdata and its availability thus provides an extra sour e of o-lo ated information. Sin e thereis a strong similarity between produ ts (DDM for GNSS-R and Tilt-Azimuth Map (TAM) forSORES) and models, the same inversion methodology was applied to both datasets.Platform Wavelength Experiment Contra t Produ tsCoastal L-band O eanPal (simulations)L-band RadiometrySalpex StarlabIfremerIfremer Sea-stateRadiometrySalinityAirborne L-band OPPSCAT IIPARIS- ESAESA Spe ulometryAltimetryAirborne Opti al PARIS- ESA Spe ulometrySpa eborne L-band PARIS-�(arti� ial data) ESA Spe ulometryAltimetryTable 1: Related experiments arried out during the thesis.State of the artMany GNSS-R appli ations have emerged re ently. A omplete overview of GNSS s atteringover the sea surfa e an be found in [RuÆni et al.1999℄ and [RuÆni et al.2001a℄. In ad-dition to the well-known altimetri and s atterometri measurements, attempts have takenpla e to retrieve new produ ts, su h as ground re e tivity properties ([Kavak et al.1996℄),i e dete tion ([Komjathy et al.2000a℄) and soil moisture determination ([Masters et al.2002,Masters et al.2003℄).As far as spe ulometry is on erned, the inversion of GNSS-R data requires (i) a parametri des ription of the sea surfa e, (ii) an ele tromagneti model for sea-surfa e s attering at L-band and (iii) the hoi e of a GNSS-R data produ t to be inverted. A review of these models an be found in the OPPSCAT proje ts supported by ESA. In the literature, it is oftenassumed that bistati s attering at L-band an be reasonably des ribed by the Geometri Opti s approximation of the Kir hho� theory. This assumption is studied and validated inthis dissertation. This important on lusion allows us to state that the most important featureof sea surfa e is the statisti s of small fa et slopes: the bi-dimensional slope PDF. Under aGaussian assumption, three parameters, en apsulated by DMSS, suÆ e to fully de�ne theellipsoidal shape of the slope PDF: the size (total MSS), the dire tion (Slope PDF azimuth)and the isotropy (Slope PDF isotropy).The bistati radar equation proposed by [Zavorotny and Voronovi h2000℄ is to date thereferen e model for the ommunity. Even though the sea surfa e is dire tly parametrized byDMSS, it is rarely presented as the geophysi al parameter of interest. Rather, most authorsprefer to link it to the near surfa e wind ve tor whi h is thought to be more expli it for

14 INTRODUCTIONo eanographi and meteorologi al users. Nevertheless, this link requires the assumption thatsea-roughness be wind-driven and no swell be present. Under su h assumptions, one anuse a given sea-elevation spe trum (like [Elfouhaily et al.1997℄) to infer wind speed from themeasurement of total MSS. We will argue that this assumption is restri tive.Generally, the produ t inverted in GNSS-R spe ulometry is a waveform, that is a 1-Ddelay mapping of the re e ted signal amplitude. Using one waveform, the wind speed an beinferred assuming an isotropi slope PDF (i.e., the PDF's shape is a ir le) [Garrison2002,Cardella h et al.2003, Komjathy et al.2000b℄. Attempts have also been made to measure thewind dire tion by �xing the PDF isotropy to some theoreti al value (around 0.7) and us-ing at least two satellites with di�erent azimuths [Zu�ada and Elfouhaily2000, Garrison2003℄.[RuÆni et al.2000℄ and [Elfouhaily et al.2002℄ proposed to work on a produ t of higher in-formation ontent: the Delay-Doppler Map (DDM), that is a 2-D delay-Doppler mapping ofthe re e ted signal amplitude. The provision of this extra dimension opens the possibilityto a full estimation of the DMSS. In parti ular, [Elfouhaily et al.2002℄ developed a rapid butsub-optimal method based on the moments of the DDM to estimate the full DMSS. However,this approa h negle ts the impa t of the bistati Woodward Ambiguity Fun tion modulationof the Delay-Doppler return.Plan of the dissertationThis dissertation is divided into four parts. The �rst part is, from a theoreti al point of view,dedi ated to bistati sea surfa e remote-sensing using GNSS frequen ies. We present thenin a se ond part a simulator for GNSS-R based on the knowledge of the previous hapters.As a third part, we investigate the GNSS-R airborne s atterometri performan e during theEddy Experiment Flight arried out in the frame of the ESA proje t PARIS- . An opti aldevi e was also used to enhan e the understanding of the L-band s attering in spe ular regime.Finally, a fourth part reports additional experimental and theoreti al studies on the apabilityto retrieve sea-surfa e diele tri onstant and emissivity with GNSS-R measurements. TheChapter stru ture is as follows:� PART I: Bistati Sea Surfa e Remote-Sensing with GNSS Signals.This part des ribes the me hanisms involved in bistati sea-surfa e s attering at GNSSfrequen ies. It starts in Chapter 1 with a brief des ription of the available and futureGNSS signals. A statisti al des ription of the sea surfa e is then presented in Chapter 2,as well as the sea spe trum model used for this study. The bistati ele tromagneti models are presented in Chapter 3. Then, the signal pro essing of GNSS re e tions andspe ulometri data produ ts are presented in Chapter 4. The stru ture of this part isillustrated in �gure 1.� PART II: GNSS-R Signal Simulations.We present a spa eborne simulator for GNSS-R in Chapter 5. An example of appli ationof su h simulator is then reported in Chapter 6 with the derivation of empiri al inversionlaws for sea-state monitoring at low altitude.� PART III: GNSS-R Airborne S atterometri Performan e Analysis.We provide in Chapter 7 the ight plan, instrumentation and set-up of the Eddy Ex-periment Flight. We report then in Chapter 8 a re-do of the Cox and Munk experimentwhi h aimed at DMSS retrieval through inversion of Tilt-Azimuth Map of Sun glit-ter opti al photographs. Furthermore, Chapter 9 is dedi ated to the �rst inversion ofGNSS-R full Delay-Doppler Map for the retrieval of the sea-surfa e dire tional meansquare slope. In addition, a omparison of the SORES and GNSS-R data inversions isundertaken.

INTRODUCTION 15

Figure 1: Stru ture of Part I.� PART IV: Other GNSS-R Appli ations.Two oastal radiometri experiments are presented as an attempt to retrieve sea-surfa esalinity and emissivity through GNSS-R power hara terization. Chapter 10 des ribesthe Salpex experimental ampaign and the related model to infer diele tri properties ofthe sea surfa e. The L-band Radiometry Experiment is then addressed in Chapter 11.The on lusions of the dissertation and perspe tives for future work are presented in page 175.The reader an �nd the list of relevant symbols and units used in this dissertation in Ap-pendix A, and a ronyms in Appendix B.

16 INTRODUCTION

Part IBistati Sea Surfa e RemoteSensing with GNSS Signals

17

Chapter 1GNSS Signal Chara teristi sThis Chapter provides an overview of Global Navigation Satellite Systems (GNSS) with afo us on the issues relevant to this study. It in ludes the present Global Positioning System(GPS) along with its future improvements, the Russian system GLONASS and the futureEuropean navigation satellite system, Galileo. We fo us espe ially on the former, sin e it isthe basis of all the experiments arried out during this study. The goal of this Chapter is todes ribe the stru ture of the signal to better understand the s attering models presented inChapter 3.1.1 Global Positioning System: GPSThe GPS operational onstellation, GPS-241, was designed to provide, from any �xed pointon the Earth surfa e and at any instant of time, three-dimensional navigation apabilities. It onsists of a onstellation of 24 satellites orbiting in 6 di�erent planes (in lined at 55o) at analtitude of about 20200 km over the Earth's surfa e. There are four satellites in ea h plane.The satellites have a period of 12 h sideral time.We depi t in the following se tion the GPS stru ture of the two presently available L1and L2 signals. Se tion (1.1.2) introdu es brie y the main omponents of a GPS re eiver inorder to dete t the in oming signal. An overview of the updates and future improvements ofthe whole system is then presented in se tion (1.1.3). Finally, GPS re e tions overage fromspa e is presented to highlight the great number of observations ompared to a monostati system.1.1.1 Signal stru tureThe stru ture of the GPS signal is based on spread spe trum signal te hniques. It onsistsin spreading the bandwidth of the data signal (modulation at 50 Hz for the navigation data)by multiplying it by a pseudo-random �1 signal sequen e of re tangular unit pulses (Pseudo-Random Noise PRN) at a mu h higher rate than the data stream, but syn hronous to it. Thegoal of su h spreading of the signal is to drop down the power spe tral density for a given totalradiated power, to in rease the toleran e to multi-path and jamming, in addition to providethe system with multiple a ess apability (by means of di�erent pseudo-random sequen esfor di�erent transmitters).Two main spreading signals are used: the Coarse/A quisition (C/A) and the Pre ise (P) ode. The C/A ode is the ivil GPS signal. It is a � 1 pseudo-random sequen e with a lo krate of 1.023 Mbps, and a short period of 1 ms allowing a rapid a quisition. The rate de�nes127 satellites are operational as of 11-2000 (see http://www.nav en.us g.gov/gps/).

20 CHAPTER 1. GNSS SIGNAL CHARACTERISTICSa hip length of approximately 9:775�10�7 se ond, i.e., a length of around �C=A=300 meters.Note that no information is arried by the unit pulse of the ode. For this reason, we allthem hip instead of bit, whi h is reserved for the navigation information unit pulse.The P ode is a � 1 pseudo-random sequen e with a lo k rate of 10.23 Mbps, and alonger period of exa tly 1 week. The transmitter-re eiver range measurement through this ode o�ers a better pre ision than C/A observables, sin e its hip length �P is ten timesshorter than �C=A.The whole signal stru ture lays in L-band, as shown in �gure 1.1. This avoids in parti ularrain e�e ts (observed at higher frequen ies). There are two emitted frequen ies: L1 and L2synthesizing respe tively 19 m and 24 m long waves.

Figure 1.1: Present GPS signals.The L1 signalThe GPS L1 signal (1575.42 MHz) for satellite i is given by [Parkinson and Spilker1996℄:SiL1(t) = p2P �XGi(t) �Di(t) � os(1t+ �) +q2Pp �XP i(t) �Di(t) � sin(1t+ �);where 1 is the L1 angular frequen y, � represents phase noise and os illator drift and P andPp are the C/A and P signals powers, respe tively. The term XGi(t) stands for the C/A odeand XP i(t) for the P ode. The binary data Di(t) is the navigation bit stream and also hasan amplitude of � 1 at 50 bps. It provides the user with ephemeris, almana s, orre tionsand some other information required to obtain a satisfa tory navigation and time transfersolution. The navigation data is formated into 30-bit word and has a 6 s sub-frame and a 30s frame period. Figure 1.2 presents the stru ture of the L1 signal.Ea h satellite i transmits unique C/A and P odes. The C/A ode is nominally 3 dBstronger than the P ode on L1. To avoid spoo�ng, the P ode (whi h is of publi knowledge)is repla ed by an en rypted ode (the Y ode) when the \anti-spoof" mode of operation(AS) is on. Sele tive availability (SA), on the other hand, is a purposeful degradation ofthe GPS lo ks and orbits. It has been turned o� on May 1st 2000. The resulting a ura yimprovement has lead to 22 meters horizontal, 33 meters verti al and 200 nanose onds (60light-meters) relative to UTC, all 95 per ent of time.

1.1. GLOBAL POSITIONING SYSTEM: GPS 21

Figure 1.2: L1 signal stru ture.The L2 signalThe L2 signal (1227.6 MHz) is bi-phase modulated by the C/A or the P ode, as sele ted byground ommand. In the normal P operation, it is given by:SiL2(t) = p2P2 �XP i(t) �Di(t) � os(2t+ �);where p2P2 represents the L2 signal amplitude, and XP i(t) is the P ode for the ith satellite,whi h is lo ked in syn hronism with the L1 odes. It is possible to re over the L2 arrierwithout knowing the ode, by squaring the signal or ross- orrelating L1 with L2 with a delayo�set that mat hes the L1 to L2 ionospheri delay di�eren e.Signal strength and patternThe minimum spe i�ed re eived signal strength at output from a user re eiver Right Hand Cir- ular Polarized 0dBi antenna for satellites above 5o of elevation is -160 dBw for the C/A om-ponent and -163 for the P ode at L1. For L2, both �gures be ome -166 dBw. Maximum �gures(with a 0.6 dB atmospheri loss) are never over -153 dBw ([Parkinson and Spilker1996℄,vol I,p. 82). The total ux density in a 4-kHz band is below the permitted -154 dBw/m2, thanksto the spread spe trum of GPS signals.The transmitted signal is Right-Hand Cir ularly Polarized (RHCP), with an ellipti itybetter than 1.2 dB for L1, and better than 3.2 dB for L2 within the angular range �14:3ofrom the antenna bore-sight (ellipti ity here de�ned as the semi-axis ratio).In terms of power Signal to Noise Ratio (SNR), the dire t re eived signal has arrierpower-to-noise density ratio with typi al values ranging between 39 and 52 dB-Hz, dependingon the geometry, real transmitted power and instrumental/propagation losses.1.1.2 Signal dete tionA positioning solution an be derived from the GPS signals without any a priori knowledgeof neither the re eiver lo ation nor the onstellation status. First, the re eiver operates witha old start sequen e orresponding to a sear h with a lean repli a of a PRN C/A ode intoseveral Doppler frequen ies and delays to �nd a orrelation peak. On e the peak is found, there eiver is able to tra k the C/A signal and thus to de ode the navigation data message. Thelo ation of other visible satellites is then feasible along with their delay and frequen y o�setsto appropriately adjust the PRN C/A repli as. The whole set of C/A signals gathered by there eiver antenna an be demodulated and the arriers lo ked: the primary GPS observables

22 CHAPTER 1. GNSS SIGNAL CHARACTERISTICSare tra ked. They in lude the pseudo-range and the phase of the GPS signal, as shown in�gure 1.3.The pseudo-range is a rough estimation of the geometri distan e (i.e., range) betweenthe emitter and the re eiver. This distan e is alled pseudo-range sin e it is not orre ted forthe multiple possible orrupting e�e ts, whi h are mainly the lo k errors and atmospheri delays (see below). Given an a priori delay, the Delay Lo k Loop (DLL) follows the variationof the time delay between transmitter and re eiver, dire tly related with the pseudo-rangethrough the speed of the light. Provided that the a priori delay is a urate enough and therate of hange in the delay is low, the DLL a ts as a maximum likelihood estimator. It usesthe derivative of the ode as a dis rimination fun tion, so that the produ t with the in omingsignal yields the error in the delay of the dis rimination fun tion, permitting its iterative orre tion. The un ertainty in the pseudo-range observable depends on the omputation ofthe dis riminating fun tion. High un ertainty is obtained for wide interval of derivation (ÆT inthe early-late di�eren e). The error in delay estimation is (see [Parkinson and Spilker1996℄):� ode = � sBDLLÆT2SNR0 ; (1.1)where ÆT has hip units, is the speed of the light, � the hip time length, BDLL thenoise bandwidth of the DLL and SNR0 the arrier power-to-noise density ratio (re eiverdependent). Fine early-late spa ing yields good pre ision, but on the ontrary, the redu tionof this spa ing length also orrupts the tra king of the dynami s of the signal. This errorranges between 0.3 m and 1 m for the C/A ode and is below 0.2 m for the P ode. The odemeasurements are not pre ise but a urate.The phase observable represents the history of the frequen y adjustments performed bythe Numeri ally Controlled Os illator (NCO) to maintain the lo k of the signal. This isequivalent to the integrated Doppler frequen y, due to the relative motion between emitterand re eiver. Therefore, this data does not provide the range itself, but a variation in the arrier y le, whi h an be expressed as a range variation by the relation: �Æ�=2�. It is thus alled the A umulated Delta Range (ADR). The tra king of this observable is done througha Phase Lo k Loop (PLL) and the pre ision is given by:�phase = �2�sBPLLÆT2SNR0 ; (1.2)where BPLL is the bandwidth of the PLL. Typi ally, the error magnitudes are very small(below 1 mm). It is a pre ise measurement but very ina urate, sin e it provides the integratedrate of hange of the range up to an unknown bias o�set.1.1.3 The modernization of GPSIn 1998, US Vi e President Gore announ ed that a se ond ivil signal will be broad aston the L2 arrier, and that beginning in 2005 a third ivil signal spe i� ally designed forsafety-of-life servi es will also be broad ast. The role of the ivil L2 signal is to provideneeded ionospheri orre tions. For non-di�erential real-time GPS users, the addition of thesenew signals provides redundan y, improved a ura y, availability and integrity, ontinuity ofservi e and resistan e to RF interferen e. For di�erential appli ations, they also assist in highpre ision appli ations. The next paragraphs are devoted to two papers re ently published inGPS World ([Shaw2000℄,[Dierendon k2000℄). Figure 1.4 shows the urrent and future ivilGPS frequen ies. For military appli ations, new odes are being developed, the M- odes,whi h will modulate the L1 and L2 arriers.

1.1. GLOBAL POSITIONING SYSTEM: GPS 23

Figure 1.3: Stru ture of a GPS re eiver.In [Ma Donald2002℄ a review of the development, status and urrent apabilities of GPS an be found. The author investigates also the impli ations of the GPS modernization andenhan ement a tivities and their relationship to the analogous European Galileo programa tivities presented below in se tion (1.3).Civil signal on L2The L2 signal is shared between ivil and military odes. The new L2 ivil signal (L2C) willbe transmitted by modernized IIR (IIR-M) s heduled to be laun hed in 2003. L2C is limitedto a single bi-phase omponent in phase quadrature with the P/Y ode. It is also limitedto a 1.023 MHz lo k rate in order to maintain spe tral separation from the new militaryM ode. The L2C signal ontains two odes of di�erent lengths. The moderate length ode(CM) has 10230 hips, repeats every 20 ms, and is modulated with message data. The long ode (CL) has 767250 hips, repeats every 1.5 s, and has no data modulation. The ompositesignal is lo ked at 1.023 MHz and alternates between hips of ea h ode ( hip-by- hip timemultiplexed). A detailed des ription of the signal an be found in [Fontana et al.2001℄.

Figure 1.4: Present and future ivil GPS frequen ies (from [Ma Donald2002℄'s paper).

24 CHAPTER 1. GNSS SIGNAL CHARACTERISTICSThe new ivil L5 signalThe third ivilian signal, L5, will be added to the �rst Boeing Blo k IIF satellite along withC/A- ode on L2, and M- ode on L1 and L2. L5 is allo ated at 1176.45 MHz with a 24 MHzbandwidth. A higher hipping rate and a longer ode than the C/A to redu e self-interferen eare improved features of this signal. L5 power will be in reased by 6 dB ompared to the urrent L1, equally split in a phase data hannel (I) and a quadrature data-free hannel (Q)to improve resistan e to interferen e, espe ially from other pulse emitting systems in the sameband as L5. The latter in ludes Distan e Measuring Equipment (DME) systems already usedfor en route and terminal area air navigation and the military Joint Ta ti al InformationSystem (JTIDS) used for riti al military ommand and ontrol ommands. The data-free omponent of the new signal also provides more robust arrier phase tra king. This signalis lo ated in the Aeronauti al Radio Navigation Servi e (ARNS) band spe ially designed forthe mentioned aviation appli ations. A omplete des ription of this signal an be found in[ICD-GPS-7052002℄.The urrent GPS modernization e�orts will end in 2010, with a total deployment of up to12 GPS Blo k IIF satellites. For the period 2010{2030, a new generation of satellites, Blo kIII is planned.Satellite S hedule Laun hThe modernization of the GPS signals will be arried out through di�erent laun hes:� Blo k IIR modernization: presently, twelve IIR satellites are modernized (the IIR-M) to speed up the avaibility of the military M ode on L1 and L2 and the ivil odeon L2. The �rst laun h of a Blo k IIR-M satellite is foreseen in 2003 and it will arrythe new signal, as will all subsequent GPS satellites. Initial Operational Capability(IOC=18 properly pla ed satellites) is anti ipated in 2008 and Full Operational Capa-bility (FOC=24 satellites) for 2010.� Blo k IIF modernization: these satellites will be the fourth generation of satellitesand will ontinue with the L2C signal on L2 and the M ode on the L1 and L2 frequen ies,but will add a new ivil ode at the L5 frequen y. The �rst laun h of the Blo k IIF isforeseen for 2005. IOC is anti ipated for 2012 and FOC for 2015.� GPS-III: the goal of the GPS-III program is to deliver best-value a quisition andar hite tural solutions that will satisfy the urrently de�ned, yet evolving, military and ivilian needs for a spa e-based positioning, navigation and timing system through 2030.This system will have two other hannels that provide navigation signals for ivilian usein lo al, regional and national appli ations. One of the new ivil signals is expe tedto transmit higher power than the other two signals for improved re eption worldwide.The �rst of the new satellites is to be laun hed in 2010. IOC is expe ted for 2016 andFOC for 2018, with the entire onstellation expe ted to remain operational through atleast 2030.Impa t on GNSS-R measurementsAs far as GNSS-R is on erned, the main onsequen e of the future GPS modernization shall ome from the signal redundan y and real-time double frequen y appli ations. In parti ular,it shall impa t the ionospheri and lo k errors, and enhan e the interferometri apa ities ofthe system:

1.2. GLONASS 25� Ionospheri error: after SA, the next largest ontributor to the GPS positioning errorbudget is the atmosphere|mainly the ionosphere. Ionospheri e�e ts an be a uratelyestimated through the use of dual frequen y measurements. The use of C/A ode on theL2 frequen y in onjun tion with L1 will redu e the typi al ionospheri error of 7 metersto 1 m. This will result in a stand-alone a ura y as low as 8.5 meters (95 per entof the time). To implement the C/A ode on L2, the GPS Blo k II Replenishmentsatellite ontra t will be revised to dire t Lo kheed Martin to modify the last 12 Blo kIIR satellites.� Clo k error: after atmospheri errors, the next sour e in the GPS error budget areephemeris and lo k errors. In the urrent onstellation, the lo k and ephemeris er-rors ontribute approximately to 1.8 and 1.4 meters, for a ombined 2.3 meters UserEquivalent Range Error (UERE). A new te hnique, alled the A ura y ImprovementInitiative (AII), is expe ted to redu e the GPS lo k and ephemeris ontribution toUERE to approximately 1.25 meters. As a result, we will obtain a 6 meter or betterhorizontal a ura y 95 per ent of the time.� Interferometry: the ombination of two di�erent signals ( hosen between L1, L2 andL5) permits to synthesize larger wavelengths. Studies have been undertaken to investi-gate the improvement in altimetri measurements ([RuÆni and Soulat2000b℄).To summarize, initial operational apability for dual frequen y navigation will o ur in2008, with laun hes beginning in 2003, a status based on a onstellation of 18 properly pla edsatellites broad asting L2 C/A in phase-quadrature to the military P signal. The L5 signal willbe available on 18 GPS satellites by 2012, requiring the laun h of 6 to 12 GPS-III satellites.Stand-alone horizontal a ura y will improve from 100 to 6 meters or better. For non-dynami or non-real time appli ations, entimeter level a ura ies will be a hieved more qui kly and ost e�e tively than today, through the existen e of 3 wide lanes to resolve integer ambiguities.1.1.4 GPS re e tions overageBistati radar o�ers several advantages over monostati systems. One of them is the superior overage and lower ost than monostati systems. The geometry of bistati GNSS s attering,shown in �gure 1.5, highlights the high a ross tra k sampling possible due to the large numberof GNSS satellites produ ing spe ular re e tions.Studies on GPS re e tions overage from spa e have been arried out by [Cardella h2002℄and [Apari io et al.2000℄, and more re ently by Astrium and Starlab in the PARIS- proje t(see [PARIS Gamma2003℄).1.2 GLONASSGLONASS is the Russian GNSS system2, and is very similar to GPS. A 24 satellite onstel-lation was available in 1996, distributed in three orbital planes at an in lination angle of 64.8degrees|only 16 satellites are operating as of today. The orbits are nearly ir ular, with analtitude of 19100 km and a period of 11.25 h. The major di�eren e with GPS is that broad astfrequen ies are satellite-spe i� . The individual arrier frequen ies, f iL1 and f iL2 are:f iL1 = 1602:0000 + i � 0:5625 MHzf iL2 = 1246:0000 + i � 0:4375 MHz;2see http://www.glonass- enter.ru/ .

26 CHAPTER 1. GNSS SIGNAL CHARACTERISTICS

Figure 1.5: Geometry of multistati GNSS radar system (from V. U. Zavorotny, Bistati GPS Signal S atteringfrom an O ean Surfa e: Theoreti al Modeling and Wind Speed Retrieval from Air raft Measurements, 1999).where i=0,...,24 is the hannel number spe i� to ea h satellite [Hofmann-Wellenhof et al.1997℄.Some satellites have the same frequen ies but are pla ed in antipodal slots of orbit planes andthey never appear at the same time in user's view. The two odes modulated onto the arriersare the same for all satellites. The publi C/A ode has a period of 1 ms, a hip rate of 0.511MHz and is modulated on L1 only. The P- ode has a period of 1 s, a hip rate of 5.113 MHzand is modulated on both L1 and L2, although its use is not open to the publi . There is noanalog of Sele tive Availability in GLONASS, however.1.3 GalileoThe future European navigation satellite system, Galileo, will omplement the existing satel-lite navigation system, whi h presently relies mainly on GPS, the Ameri an Global PositioningSystem. Developed by the European Spa e Agen ies and the European Union on the basis ofequal o-funding, Galileo is designed to provide a omplete ivil system and s heduled to beoperational by 2008.The Galileo system will be built of around 30 satellites (27 operational and 3 reserve raft) o upying three ir ular earth orbits, in lined at 56o to the Equator, at an altitudeof 23616 km. This on�guration will provide ex ellent overage of the planet. Two Galileo ontrol enters will be established in Europe to ontrol satellite operations and manage thenavigation system. The ontra ts for the �rst Galileo satellites were signed on Friday 11 July2003 at ESTEC, the European Spa e Agen y's resear h and te hnology enter.The European Satellite Navigation System is urrently under design development. Thelast update on the system, dated September 2002, on the Galileo frequen ies and signal design an be found in [Hein et al.2002℄. Galileo will provide 10 navigation signals in RHCP in thefrequen y ranges 1164-1215 MHz (E5a and E5b), 1215-1300 MHz (E6) and 1559-1592 MHz(E2-L1-E1), whi h are part of the Radio Navigation Satellite Servi e (RNSS) allo ation. Anoverview is shown in �gure 1.6. The arrier frequen ies, as well as the frequen y bands thatare ommon to GPS and GLONASS are also highlighted.All the Galileo satellites will share the same nominal frequen y, making use of CodeDivision Multiple A ess ompatible with the GPS approa h.Six signals, in luding three data-less hannels, so- alled pilot tones (ranging odes notmodulated by data), will be a essible to all Galileo users on the E5a, E5b and L1 arrier

1.3. GALILEO 27frequen ies for Open Servi e (OS) and Safety-of-life Servi e (SoL). Two signals on E6 withen rypted ranging odes, in luding one data-less hannel will be a essible only to somededi ated users that gain a ess through a given Commer ial Servi e (CS) provider. Finally,two signals (one in E6 band and one in E2-L1-E1 band) with en rypted ranging odes anddata will be a essible to authorized users of the Publi Regulated Servi e (PRS).

Figure 1.6: Galileo frequen y plan (as of 2003) together with GPS and GLONASS systems.ARNS is the a ronym for Aeronauti al Ratio Navigation Servi e. This band is dedi ated tosafety-of-life servi es (i.e., ivil aviation). RNSS is the a ronym for Radio Navigation SatelliteServi e.

28 CHAPTER 1. GNSS SIGNAL CHARACTERISTICS

Chapter 2Sea Surfa e ModelThe purpose of this Chapter is to investigate sea-surfa e hara teristi s to get a good un-derstanding of the s attering pro ess. Indeed, the s attered �eld an be determined fromensemble averages of sea-surfa e heights. Following a Gaussian model, these ensemble av-erages an be analyti ally derived and ex lusively related to the two �rst moments of thedistribution. These moments an be omputed from the sea-surfa e spe trum.We �rst present in se tion (2.1) an overview of the sea spe tra, with spe ial attention tothe spe trum developed by [Elfouhaily et al.1997℄. A short dis ussion on the use of spe tralmodels to infer the sea-surfa e slope varian e is proposed. The sea-surfa e statisti s are thenreviewed in se tion (2.2) under the Gaussian assumption.2.1 Sea-surfa e spe tral modelThis se tion addresses the status on sea-surfa e spe trum models. An overview of the main omponents of su h spe tra is done, with a parti ular interest on the [Elfouhaily et al.1997℄'sspe trum. The simulation of sea surfa es is then easily done from the spe trum (se tion (2.1.3)).A short dis ussion on the use of spe tral representation of o ean to infer roughness (s atterom-etry) is �nally proposed (se tion (2.1.4)).2.1.1 The involved geophysi al parametersThere are three important aspe ts in wave generation:� the wind speed at ten meters over the surfa e U10 in m/s,� the duration of wind blowing a tion,� the fet h: the distan e to nearest shore. This is a measure of the e�e tive area in whi hwind is blowing on the water. The fet h on a small lake, for instan e, is very limited.The general features one expe ts to see in a wave spe tral model are:� A spe tral peak at radial frequen y !m that depends on wind speed U10 and fet h.� A high frequen y tail falling o� as !�5.A formula summarizing re ent work and englobing these two points is (see [Apel1995℄, page199): S(!) = �(U10; xf )g2(2�)4 e� 54� !!m ��4 F (!;U10)!5 ; (2.1)where g is the gravitational a eleration onstant and xf the fet h. Its features are:

30 CHAPTER 2. SEA SURFACE MODEL1. Waves having frequen ies below !m are traveling faster than the wind and have no netfor ing to grow.2. Waves at higher frequen ies are dominated by breaking and usp-like rests, whosefrequen y spe tra may be shown to behave as !�5.3. Wind speeds and limited fet h, a ounted for in �(U10; xf ), modify the overall energylevel of S(!) and hen e of wave height without a�e ting its shape.4. The fun tion F (!;U10) in reases the peakedness of the spe tral maximum and parametrizesthe weak nonlinear wave/wave s attering intera tions that ause the spe trum to bemore narrow and ohesive than it would be the ase of linear, independent Fourier om-ponents. The older [Pierson and Moskowitz1964℄ spe tral shape does not in lude thesefeatures.For instan e, in order to generate a fully developed sea|a saturated spe trum|a wind of 10m/s needs to blow with a duration of 18 hours over about 320 km of o ean. Under these onditions, SWH is about 2 meters and wave period will be about 7.5 se onds.Unlike ele tromagneti waves in air, o ean waves are dispersive. The dispersion relationfor surfa e waves is given by:!(~k) = �q(gk + k3�s=�) tanh kH; (2.2)where �s is the surfa e tension, � is the density, and H is the depth. At long wavelengths the apillary terms (i.e., waves of the order of the m) are negligible and the dispersion relationfor deep water be omes: !(~k) = �pgk tanh kH: (2.3)It is often emphasized that the use of radar data in the derivation of sea models should beavoided. Indeed, the use of radar data implies the hoi e of an ele tromagneti model, andto date, no satisfa tory model exists (see [Anderson1999, Elfouhaily et al.1997℄). Moreover,using a model would bias future model evaluation using su h spe tra.2.1.2 Elfouhaily et al. spe trumLet us dis uss the spe trum of [Elfouhaily et al.1997℄, sin e it is the most popular one in the ommunity. They emphasize that the models available do not meet fundamental riteria: theyshould properly a ount for di�erent fet h onditions, and agree with the in situ observationsof [Cox and Munk1954℄ (of wind-dependent mean square slopes), [J�ahne and Riemer1990℄and [Hara et al.1994℄ (gravity- apillary wave urvature laboratory measurements) in the highwavenumber regime.They further riti ize that these models su�er undesirable features su h as dis ontinu-ities a ross wavenumber limits, non-physi al tuning parameters, and non- entrosymmetri dire tional spreading fun tions. They propose to remedy this situation with a new model: atwo-dimensional wavenumber spe trum valid over all wave-numbers and analyti ally amenablefor use in ele tromagneti models. Radar data is ex luded from the model development. Thespe trum is derived for the ase of wind-generated seas, with wind and waves aligned. Theauthors begin by reviewing the several available full wavenumber models:1. The [Bjerkaas and Riedel1979℄ spe trum. This widely used spe trum is de�ned in fourseparate wavenumber ranges, and the onne tion a ross ranges is provided througharbitrary onstants to mat h the urves. This spe trum applies to fully developed sea

2.1. SEA-SURFACE SPECTRAL MODEL 31 onditions only. The �rst range, near the peak for gravity (long) waves orrespondsto the [Pierson and Moskowitz1964℄ spe trum. Note that [Fung and Lee1982℄ use asimpli�ed, two-regime, version of this spe trum.2. The [Donelan and Pierson1987℄ spe trum. This is a two-regime spe trum. In the lowwavenumber regime the long-wave spe trum resulting from the Joint North Sea WaveProje t (JONSWAP [Hasselmann1973℄) is used. It is well-a epted that this spe trumgives a reasonable des ription of fet h-limited wind wave development. Shorter wavesare theoreti ally derived by requiring an energeti balan e between wind input anddissipation due to vis ous damping and wave breaking.3. The [Apel1994℄ spe trum. He alled his spe trum the \Donelan-Banner-J�ahne" spe -trum. [Elfouhaily et al.1997℄ follow Apel's obje tive of building an analyti ally amenablemodel to satisfy the primary need of ele tromagneti models: the auto orrelation fun -tion or elevation energy spe trum. [Elfouhaily et al.1997℄ riti ize that this model is not onsistent with [Cox and Munk1954℄'s observations of the mean square slope statisti s,however.The [Elfouhaily et al.1997℄ energy spe trum an be summarized by the expression:(k; �) = 12�k�1[Sl(k) + Sh(k)℄[1 + �(k) os(2�)℄: (2.4)The term [1+�(k) os(2�)℄ is the entrosymmetri spreading fun tion. The term k�1[Sl(k)+Sh(k)℄ is the omni-dire tional part of the spe trum, divided into low and high frequen ies. Adetailed expression of ea h term is presented in the paper. The model ontains two parameters:U10 and km (asso iated to the phase velo ity m), the wave number of the dominant waveor the spe tral peak. These quantities are related to the inverse wave age through =U10= m � U10pkm=g. The fri tion velo ity u�, whi h is linked to U10 through a drag oeÆ ient[Dun an et al.1974℄), is also onsidered.2.1.3 Sea surfa e generationThe generation of sea surfa es is done in the following way. We de�ne �r a matrix of randomphases uniformly distributed between 0 and 2�; the sea height at ~r = (x; y) is then:�(~r; t) = FFT�1[q(kx; ky)ei(�r�!t)℄: (2.5)We use here the mentioned uni�ed spe trum from [Elfouhaily et al.1997℄. We re all thatthis spe trum is made without radar data and is derived for the ase of wind-generated seas,with wind and waves aligned. The time dependen e implements the wave propagation.For spe i� s enarios (namely airborne and spa eborne) the re eiver is moving above theobservation s ene. Instead of moving the re eiver, a tri k onsists in \moving" the o eanbelow it. For this purpose, we add another phase term in the integral. This phase ontainsthe re eiver speed ve tor. For a re eiver speed ~vr, the o ean heights are now de�ned as:�(~r + ~vrt; t) = Z [q(kx; ky)ei(�r+(~k� ~vr�!)t)℄ei~k�~rd~k: (2.6)

32 CHAPTER 2. SEA SURFACE MODEL2.1.4 Dis ussion on the use spe tral models for s atterometri purposesThe sea-surfa e MSS is histori ally used to derive wind speed estimates. However, it onsistsin the high-pass �ltered integration of the all sea elevation spe trum, whi h is not solelydependent on lo al wind onditions.In the open o ean, the uto� wavenumber for the equilibrium range at the lower frequen ypart of the spe trum is very lose to the spe tral peak km, whi h is generally assumed wind-dependent, through the relation: km = gU210 : (2.7)At the high wavenumber end, it is ontrolled by apillarity waves. It is assumed that the major ontribution to surfa e slopes omes from higher wavenumbers. Using a simpli�ed saturationspe trum (1/k4), we an give a good sea-surfa e MSS estimation [Phillips1966℄:�2s = Zk k2(~k)d~k = B Log k km = B Log k U210g ! ; (2.8)where k is the uto� frequen y and B a non-dimensional onstant (equal to 0.00405).Therefore, this model states that sea-surfa e roughness is purely a fun tion of wind speed.However, this result is based on wave motion alone. If the wave motion en ounters a urrentfor instan e, the hara teristi s of the waves will be di�erent as a onsequen e of the kinemati and dynami intera tion. The impa t of surfa e urrent on sea-surfa e roughness has beendis ussed by [Huang et al.1973℄ (see se tion (2.2.3)).During the Mid-Term review of the ESA Contra t No. 12934/98/NL/GD on the \Studyof the impa t of sea-state on nadir looking and side looking mi rowave ba ks atter", animportant point was made. Co-lo ated Topex/Buoy data sets (using 41 buoys) show thatthe orrelation between radar ross se tion �o|whi h is inversely proportional to sea-surfa eMSS|and U10 is higher for underdeveloped seas, and also higher for shorter radar wavelengths(Ku-band). The sea-state was de�ned a \fully developed sea" using the Pierson-Moskowitz ondition, �� > 0:02352 � U210, with the buoy data.Other interesting on lusion were that algorithms using both �o and �� redu e signi� antly theerror in wind speed retrieval (from 1.5 m/s down to 1.23 m/s) and that the dual-frequen y al-gorithms [Elfouhaily et al.1999a℄ do not seem to improve wind speed retrievals. The sea-stateimpa t on the s attering ross se tion has been deeply investigated in [Gourrion et al.2002a℄.As a on lusion of this short dis ussion, we emphasize that the sea-surfa e slope varian eis not a simple fun tion of wind. Other parameters su h as sea-state (SWH) and surfa e urrents may strongly impa t the roughness as well. The roughness estimations undertakenduring experimental ampaigns (see Chapter 7) illustrate this point.2.1.5 SummaryFor a good, up-to-date review of the state-of-the-art in spe trummodeling, see [Anderson1999℄.Here we summarize the important on lusions:� There is no a epted des ription of the wave spe trum.� A re ent attempt an be found in [Elfouhaily et al.1997℄, valid from 6 mm to 600 mwavelengths. This spe trum depends on U10, , and fri tion velo ity u�. No attempt ismade to in lude swell. This is a problem ommon to other models. No radar data havebeen used in this model.

2.2. GAUSSIAN SEA-SURFACE STATISTICS 33� For short wave (0.01 to 10 m) spe tra, where tension is important, the model developedby [Elfouhaily et al.1997℄ seems the best available, although short omings are pointedout. This is a strongly non-linear regime, and unexplored theoreti al ground abounds.� Wave breaking is in luded on models based on experimental data (opti al data is usedin [Elfouhaily et al.1997℄). Very poorly understood.� Wave breaking, urrents, sli ks and rain all a�e t the short wave spe trum, and experi-ments ontinue to provide more unexplained phenomena.2.2 Gaussian sea-surfa e statisti s2.2.1 Statisti al hara terization of random surfa esWe give here an overview of the important statisti al parameters used to des ribe the seasurfa e under the Gaussian assumption. We address the standard deviation of the surfa eheight variation (or RMS height) �� , the surfa e orrelation length l, the mean square slopeMSS=�2s and the mean radius of urvature r . However, higher order statisti al parameters,su h as the skewness, would also be needed to properly depi t sea-surfa e height statisti s.Note that spe i� ation of higher order statisti s are ne essary for non-Gaussian surfa es.Signi� ant Wave HeightLet �� � h�(~�)i be the average height of a surfa e �=�(~�), where ~� is the horizontal displa ementve tor. Then, the RMS height is de�ned by:�� � qh�� � ���2i: (2.9)We re all that from a spe tral point of view, it is given by the integral of the sea-surfa espe trum. We usually address the Signi� ant Wave Height (SWH) related to �� by:SWH = 4 �� : (2.10)Correlation lengthThe surfa e orrelation length l in some dire tion is given in terms of the auto orrelationfun tion of the surfa e. This is a measure of the spatial s ale asso iated to the statisti alrelationship between points in the surfa e.To have an idea of the behavior of the o ean model relative to wind speed, the auto orre-lation fun tion of sea heights that de�nes the auto orrelation length l is shown in �gure 2.1.It appears learly that l in reases with wind speed, in agreement with reality: for high windspeed the waves be ome more developed and roughness in reases also.Surfa e slope varian eOne an show that if the height distribution is Gaussian, then the distribution of slopes is alsoGaussian. Let A(x) be the auto orrelation fun tion of the surfa e (let us restri t ourselvesto the 1-D ase for now, the generalization is simple). One an show that the slope varian eof the surfa e (\average slope"|a misnomer) is given by ([Ulaby et al.1986℄, Appendix 12 F,page 1012, and [Fung1994℄, Appendix 2B, page 117):�2s = ��2�A00(0): (2.11)

34 CHAPTER 2. SEA SURFACE MODEL

Figure 2.1: Correlation fun tion for di�erent wind speeds (U10=1, 3, 5, ..., 17 m/s), withGaussian spe trum from [Elfouhaily et al.1997℄.For the more spe i� ase of a Gaussian auto orrelation fun tion with auto orrelation lengthl, A(x) = e�x2=l2 , this yields �s = p2��=l: (2.12)Sometimes the slope tan�0 is also used as a measure of the average slope varian e of thesurfa e, where tan�0 = 2��=l (see [Be kmann and Spizzi hino1963℄, page 89 and AppendixD, page 193).From a spe tral point of view, the sea-surfa e slope varian e is given by the integral of thespe trum multiplied by the wave-ve tor squared.Radius of urvatureAnother quantity whi h will be onsidered in s attering model requirements|see se tion (3.2.1)of Chapter 3|is the mean radius of urvature, whi h will also have a Gaussian distributionif the height distribution is Gaussian. One an show that if the auto orrelation fun tion isGaussian, the mean radius of urvature writes (see [Ulaby et al.1986℄ volume II, Appendix12F): r = l2��q�=24 = l�sq�=12: (2.13)In [Apel1994℄, some typi al values for the orrelation length, average radius of urvatureand o ean wavelength (this is in fa t the wavelength of o ean waves travelling at the windspeed) are given as fun tion of wind speed (these values belong to the o ean model dis ussedthere). For a wind speed of 6 m/s, they are respe tively l=2.4 m, r =0.8 m, and D=23 m,for instan e.2.2.2 Slope distributionWe here use an anisotropi Gaussian assumption to des ribe the 2-D slope probability densityfun tion: Ps(sx; sy) = 12�pdet(M) exp24�12 sxsy !yM�1 sxsy !35 ; (2.14)

2.2. GAUSSIAN SEA-SURFACE STATISTICS 35where sx = ��=�x and sy = ��=�y stand for the dire tional slopes in some frame xy andM isthe matrix of slope se ond order moments: the dire tional mean-square slopes (DMSS). Thexy frame mapped on sea-surfa e is de�ned as follows: it is entred on the spe ular point andhas its x axis aligned with the Transmitter-Re eiver line. Mean-square slopes along majorand minor prin ipal axes are often referred to as (respe tively) MSS up-wind (�2u) and MSS ross-wind (�2 ). The M matrix is then obtained via a simple rotation:M = " os � sin sin os # : " �2u 00 �2 # : " os sin � sin os # ; (2.15)where is the angle between x axis and the slope prin ipal axis.The three geophysi al parameters of interest are �2u, �2 and . They an be understoodas the three parameters of an ellipse (see �gure 2.2) representing the slope PDF mapped onsea-surfa e. In this dissertation, we will a tually onsider the equivalent set of parameters:� Total MSS, de�ned as: MSS = �2s = 2p�2u:�2 . This magnitude is a tually proportionalto ellipse area and an be interpreted in terms of wind speed, onsidering a parti ularwind-driven sea-surfa e spe trum, like the Elfouhaily's spe trum [Elfouhaily et al.1997℄.� Slope PDF isotropy (SPI), de�ned as SPI = �2 =�2u. When SPI=1, the slope PDF is fullyisotropi and the glistening zone is ir ular. Low values of SPI indi ate a highly dire tivePDF. Typi ally, SPI is expe ted to be around 0.65 for well developed sea-surfa e.� Slope PDF azimuth (SPA), de�ned as the dire tion of semi-major axis with respe t toNorth. As shown by �gure 2.2, this angle (modulo 2�) is SPA = � +�� , if � is thesatellite azimuth w.r.t. North.NORTH

EASTRX

TX

SPA

φ

ψ

Figure 2.2: Sket h of sea-surfa e slope PDF and related frames.2.2.3 Impa t of urrent on sea-surfa e slope varian eThis se tion aims at introdu ing the dependen e of MSS with urrent onditions. By takinginto a ount the surfa e urrent velo ity U and onsidering equation 2.8, the MSS writes

36 CHAPTER 2. SEA SURFACE MODEL[Huang et al.1973℄: �2s = Z k km B�1 + U �7 dkk : (2.16)Introdu ing the phase velo ities m and orresponding respe tively to the spe tral peakand uto� wavenumbers, we get (up to the se ond order):�2s = 2B(Log ����U + mU + ����+ 6� U U + m � U U + �� 152 "� U U + m�2 � � U U + �2#+ :::) :(2.17)Note that when U =0, equation 2.17 redu es to equation 2.8.

−0.5 0 0.5 10

0.05

0.1

0.15

0.2

Current speed (m/s)

Slo

pe v

ariance

U10

= 1 m/s

U10

= 5 m/s

U10

= 10 m/s

U10

= 15 m/s

U10

= 20 m/s

Figure 2.3: Variation of sea-surfa e slope varian e at L-band with urrent speed, with windspeed as a parameter (see equation 2.17).Due to the presen e of urrent velo ity in the above equation, the dependen e of �2s onwind speed be omes less obvious. Figure 2.3 shows the variations of sea-surfa e slope varian eat L-band with urrent speed, with wind speed as a parameter. The sharp slope of the urvesimmediately reveals the sensitive dependen e between �2s and urrent speed espe ially formoderate urrents, probably the prevailing state over most of the o ean.

Chapter 3S attering ModelThe purpose of this Chapter is to provide the basi s of the ommon ele tromagneti modelsinvolved in s attering over sea surfa es. Starting from the Kir hho� model and its onditions,we fo us then on the Geometri Opti s theory, whi h is validated at L-band through Monte-Carlo simulations in se tion (3.2). Furthermore, a waveform model for GNSS-R is proposedin se tion (3.3), following [Zavorotny and Voronovi h2000℄'s work.The signatures of o eani features whi h an be dete ted by a tive or passive mi rowavesensors like s atterometers, imaging radars and GNSS-R are asso iated with surfa e e�e ts:they result from an intera tion between the ele tromagneti waves and water waves. Theability of the o ean surfa e to s atter ele tromagneti power ba k towards the re eiver is hara terized by its normalized radar ba ks attering ross se tion (NRCS), whi h depends onthe frequen y used, polarization and in iden e and s attering angles.It is ne essary to have a lear pi ture of the physi al pro ess involved in the ele tromagneti o ean-surfa e intera tion in order to extra t the o ean variables su h as near-surfa e windspeed, wave height, wave slope, o ean wave spe trum, et . When al ulating the ross se tion,two sets of approximations are urrently used. The �rst set of approximations on erns thes attering theory, presented hereafter, and the se ond set enters in the statisti al des riptionused to ensemble average the resulting ross se tion|presented above in Chapter 2.Note that to predi t ele tromagneti s atter one should also onsider the hydro-dynami sof o ean waves. The statisti al distribution of sea-surfa e displa ement is Gaussian in thelinear approximation of a superposition of non-intera tive sinusoidal water waves. But thenon-linear oupling of the o ean waves produ es a ontinuous redistribution of energy andmomentum of the waves ausing the statisti s to be non-Gaussian. Besides the energy balan eof the o ean waves is ontrolled by several ompeting pro esses: the input from the wind, theenergy transfer by non-linear wave-wave intera tions and the energy lost by vis osity and wavebreaking. It is well known that waves riding on an non-homogeneous spatially varying �eld willhave their growth, dissipation, speed, wavelength and height modi�ed. These modi� ationswill thus dire tly enter in the amplitude strength and Doppler signature of the radar rossse tion. This is not however the purpose of the present study.3.1 Bistati ele tromagneti modelsAll available losed-form models for ele tromagneti re e ted �eld from random surfa es areasymptoti solutions of the Maxwell equations. Two limits are generally onsidered: 1) theKir hho� approximation ([Be kmann and Spizzi hino1963℄) and 2) the small perturbationmethod (SPM) ([Ri e1951℄). The former is obtained under the ondition of small slopes and

38 CHAPTER 3. SCATTERING MODELlong waves while the latter is derived for small slopes and short waves. The Kir hho� approx-imation des ribed below in se tion (3.1.1)|together with its high frequen y limit approx-imation (Geometri Opti s)|a urately models the quasi-spe ular s attering but does not onsider the sensitivity to polarization. On the ontrary, SPM, presented in se tion (3.1.3), arries the polarization fa tors but fails to reprodu e the near-spe ular regime, sin e it doesnot properly a ount for the longer s ale features.A "two-s ale" or omposite-surfa e model has been tested. This approa h, presented inse tion (3.1.4), ombines two " lassi al" methods of wave-propagation theory: the methodof small perturbations or tangent plane and the semi- lassi al approximation. However, inaddition to la king the ne essary a ura y for quantitative predi tions, this method usesa somewhat arbitrary s ale-dividing parameter that makes the solution ambiguous. Manyimportant hara teristi s of s attering, su h as polarization, angle, and dependen e of rossse tions on wind speed, annot be a ounted for over realisti ranges of parameter values. Amore a urate theory is required that is free of �tting parameters.For a long time, no solutions to this problem were known. [Holliday1987℄ proposed anapproa h that in ludes the polarization under both situations: spe ular and moderate-in idents attering. He showed in the ombination of the two limits that for the ase of ba ks atter inthe SPM limit, the de� ien y in the Kir hho� �eld an be orre ted through the in lusion ofthe next iterative orre tion to the surfa e urrent.Other authors have also investigated this issue, su h as [Rodriguez and Kim1992℄, [Tatarskii1993℄,[Voronovi h1994℄ and [Fung1994℄. More re ently, [Elfouhaily et al.1999b℄ generalized Holli-day's results to develop a bistati Kir hho� model that is polarization sensitive.The main assumption for these models is to onsider the surfa e as a perfe t ondu tor.Starting from the Stratton-Chu equations and assuming that the in oming in ident �eld is aplane wave ~Bi = ~B0exp(�iki � r), the total magneti �eld is:~B(r0) = ~Bi(r0)� IS ~J(r1)�rG(r0; r1) dA1; (3.1)where S is the surfa e des ribed by z=�(x) and x is the horizontal omponent of the three-dimensional ve tor r. The integrand in equation (3.1) is the ross produ t between the totalsurfa e urrent ~J(r) = n(r)� ~B(r) and the gradient of the Green's fun tion:G(r0; r1) = � 14� exp(ikjr0 � r1j)jr0 � r1j ; (3.2)where k is the ele tromagneti wave number. The omplexity of the above equation is thatthe total surfa e urrent is a fun tion of the total magneti �eld ~B(r). At the s atteringsurfa e it follows the integral equation:~J(r1) = ~Ji(r1)� 2n(r1)� IS ~J(r2)�rG(r1; r2) dA2; (3.3)where n is the unit ve tor normal to the s attering surfa e. It is dire tly determined a ordingto the lo al slopes of the sea surfa e:n = ez �r�p1 + (r�)2 = nz(ez �r�): (3.4)

3.1. BISTATIC ELECTROMAGNETIC MODELS 393.1.1 Kir hho� approximation or Physi al Opti sAn integral equation su h as equation (3.3) is de�ned by a kernel K, and symboli ally, it maybe des ribed in generality by: f = f0 +�Kf; (3.5)where f0 is the inhomogeneous (given) term, Kf stands for the onvolution of the kernel withthe unknown fun tion f , and � is a onstant. An iterative solution to this problem is givenby a Newmann series (see [Courant and Hilbert1989℄):f = f0 +�Kf = f0 +�K(f0 +�Kf) = ::::; (3.6)and so on. For instan e, to �rst order in �, the solution is f1 = f0 +�Kf0.We onsider in the Kir hho� approximation the �rst term in the iterative series solutionof the surfa e- urrent integral equation. In this �rst iteration, the total urrent ~J1 is for edto mat h the input urrent ~Ji:~J1(x1) � ~Ji(x1) = 2n(x1)� ~Bi(x1): (3.7)The needed assumption for this approximation is that the in ident and s attered �elds at thesurfa e be related (linearly) through the Fresnel re e tion oeÆ ient R, whi h is a fun tionof the lo al in iden e angle and polarization. In this way, the integral equations are redu edto integration of the in ident �eld over the surfa e.The �nal form of the Kir hho� �eld is given by [Elfouhaily et al.1999b℄ as follows:~Bs(r0) = ~B(r0)� ~Bi(r0) = � i2� eikr0r0 B0k ~P s(1) Z exp(�iqz�(x1))exp(�iq? � x1)dx1; (3.8)where ~P s(1) is an outgoing ve tor that depends on the in ident polarization, the in ident ands attering dire tions, the normal to the surfa e and the Fresnel oeÆ ient. The subs ript (1) isreferred to the �rst iteration of the surfa e- urrent integral equations. The ve tor ~q = (~q?; qz)is the s attering ve tor, de�ned as the di�eren e between the s attered wave number ~ks andthe in ident wave number ~ki|as shown in �gure 3.1. It is the ve tor normal to the plane thatwould spe ularly re e t the rays in the dire tion of observation. This ve tor oin ides withthe normal n for a spe ular fa et de�ned in the Geometri Opti s theory in se tion (3.1.2).This formulation is ompa t and shows learly that the re e ted �eld depends on thes attering ve tor, the polarization ve tor and the surfa e shape, of ourse.Limitation of the Kir hho� approximationPhysi ally, in this approximation we assume that the surfa e is smooth enough to be lo allyrepresented by planes. The validity of Kir hho� theory relies on a main aspe t: the radius of urvature r . Generally the requirement is:2kr os � >> 1; (3.9)where � is the in ident angle.It follows, from this requirement, that at suÆ iently large distan es from the point oftangen y| ompared to the in ident wavelength �|the tangent plane shall not be appre- iably distant from the surfa e. The radius of urvature should be greater than the in ident

40 CHAPTER 3. SCATTERING MODELn

q

i

s

Transmitter

Receiver

ScattererFigure 3.1: De�nition of the s attering ve tor ~q = k(~s�~i).wavelength. For the parti ular sea-surfa e ase, this radius of urvature is dominated by smalls ales, i.e., the apilarity-gravity waves.Toward grazing in iden e more omplex me hanisms that annot be a ounted for by theKir hho� approximation begin to play a role: shadowing, di�ra tion, multiple s attering, andtrapping by atmospheri du ts and waves. In fa t, all of the models fail in this regime to somedegree.Sin e a large r ( ompared to e�e tive wavelength) an be an operating onstraint in theKir hho� approximation, it is useful to rewrite in the ase of Gaussian statisti s of the seasurfa e. As seen in equation 2.13 (se tion (2.2) of Chapter 2):r � l=�s; (3.10)with l the surfa e orrelation length and �s the sea-surfa e slope RMS. We an also write:r � l2=�� ; (3.11)with �� the sea height RMS.This means that the large r ondition an be met if:i) l is large ompared to e�e tive wavelength (�= os �), or ifii) �s is small. This ondition an be met again if l is large (ba k to i) or if �� is small ompared to e�e tive wavelength.3.1.2 The Geometri Opti s or Stationary Phase ApproximationThe goal of the present se tion is to review the me hanisms involved in the Geometri Opti s(GO) theory, with an emphasis on the Stationary Phase Approximation (SPA).The Geometri Opti s approximation to the Kir hho� theory for ele tromagneti s atter-ing represents a high frequen y limit. The physi al pi ture an be understood in terms of aspe ular point (SP) model. The �eld at the re eiver is the superposition of the �elds gener-ated by a number of "mirrors" (not at mirrors, though) on the s attering surfa e whi h areoriented in the orre t manner. The radiation from ea h spe ular point is as oherent as thein oming radiation, but there is no oheren e in the phase relationship between the radiationout- oming from the di�erent mirrors. The result is that there is power at the re eiver, butthat the phase of the re eived signal is randomly distributed. The behavior of the resulting

3.1. BISTATIC ELECTROMAGNETIC MODELS 41phase will depend on the geometry, on the number and distribution of mirrors, and also onthe temporal variation of these quantities.Generally, the re e ted �eld is determined by a rapidly os illating integral over the roughsurfa e (see equation (3.8)). We will see that spe ular points are identi�ed with the riti alpoints of the integrand's phase. We will treat here the radiation oming from ea h mirrorseparately, as was mentioned above. We will use the Fraunhofer approximation (plane wavesall the way through) to estimate the out- oming �eld. To ful�ll the Fraunhofer requirements,the size of ea h mirror must be of the order of the �rst Fresnel zone, de�ned as the set ofpoints near the spe ular with ranges to the re eiver equal or less than the spe ular range plusa wavelength. In other way, the radius of this zone is p2�H , where H is the height of there eiver and � the in ident wavelength.Let the in oming �eld be des ribed by1:Uo(p) = eikrr : (3.12)The quasi-spe ular s attered �eld under Kir hho� approximation, the Fresnel integral, writes(see [Born and Wolf1993℄): U(p) = �i4� Z R � eik(r+s)rs (~q � n) dS; (3.13)where r is the range between re eiver and the surfa e and s the range between emitter andthe surfa e. The s attering geometry from �gure 3.2 leads to:kir = kir0 + ~ki � ~r: (3.14)The sign before the s alar produ t hanges as we onsider the out- oming ve tor ~ks. We an onsequently rewrite the phase of the integrand:k(r + s) � k(r0 + s0)� ~q � ~r: (3.15)In the far �eld both emitter and re eiver are very far ompared to the size of the s attererand sin e our integral is near a spe ular point with n � q, we �nd:U(p) = �i4� eik(r0+s0)r0s0 Z R � e�i~q�~r (~q � n) dS: (3.16)Now, let us fo us on a single spe ular point: imagine that there is only one spe ular pointon the surfa e. This means ~q � n = q. To perform the integral let us use a oordinate system inwhi h the z axis is parallel to ~q. In this oordinate system the tangent plane at the spe ularplane is therefore parallel to the x-y plane. Hen e, dS = d2~�n�ez = d2~�. We need to ompute:I = q Z R � e�iq �(x;y) d2~x: (3.17)Now we use the Stationary Phase Approximation that states that the main ontribution omesfrom the spe ular point. The basi idea of this approximation is to expand the argument ofthe exponential into a series about the spe ular point ~�spe (x; y):�(~�spe + ~�) = �(~�spe ) + r�jspe ~�+ 12! r2ij����spe �i�j + ::: (3.18)1Here we do a s alar treatment, think of U as a omponent of the �eld.

42 CHAPTER 3. SCATTERING MODEL

r

0

z

x

i

r´

r

k

Figure 3.2: Ve tor geometry for s attering.At this point the linear terms of the series vanish, sin e spe ular fa ets are hara terized by azero-slope. In other words, as q gets larger the integral gets ontributions only very near thespe ular point|the ontributions farther out an el out. Indeed in the one dimension ase,the integral an be expressed as follows:I = q Z ���R � e�iq �(x) d2~x+ q Zx6=[��;�℄R � e�iq �(x) d2~x: (3.19)The phase of the se ond integral os illates rapidly for large q so that the integral tends to an el. On the ontrary, the phase around the spe ular point doesn't vary so fast.Ba k to the two-dimensional ase, we obtain with a development at the order two:I � qRspe e�iq �spe Z e�i q2 r2ij�jspe xixj d2~x (3.20)= qRspe e�i~q�~rspe 2�(�i)q det1=2 �r2ij����spe � (3.21)= iRspe e�i~q�~rspe 2�det1=2 �r2ij����spe � : (3.22)Now, it is not hard to show that det1=2 �r2ij����spe � = 1=prxry|the determinant just yieldsthe produ ts of the radii of urvature at the spe ular point (rx and ry are the radii of urvaturein x and y respe tively, here at the spe ular point).We obtain: I = iRspe e�i~q�~rspe 2�prxry: (3.23)Finally, we get the Spe ular �eld, de�ned as the re e ted ele tromagneti �eld under GOassumption (here for one spe ular point):U(p) = �i4� eik(r0+s0)r0s0 � iRspe ei~q�~r2�prxry (3.24)= Rspe 2 eik(rspe +sspe )rspe sspe prxry:

3.1. BISTATIC ELECTROMAGNETIC MODELS 43The oeÆ ient Rspe depends on the Fresnel oeÆ ients and on the lo al geometry. Thes attering ross se tion is de�ned as the ratio of the s attered power per unit area and thein ident power density. At the spe ular point it be omes:�ospe = jU j2=AojUoj2=4�r2o (3.25)= jU j2 4�rspe sspe jUoj2 lxly= jRspe j2�rxrylxly� jRspe j2�2s :The s attering ross se tion of the surfa e is the sum of ea h spe ular point ontribution. Thereader must keep in mind that we keep in the development of the integrand's phase the termsto the se ond order only. As we will see later through simulations in se tion (3.2.2), the GOmethod an be perfe tly applied for surfa es that have for highest orders the quadrati terms(paraboli ase).In the general ase of a Gaussian sea surfa e, the s attering ross se tion writes:�o = �jRj2 q4q4z Ps ��qxqz ; �qyqz � ; (3.26)where Ps is the Gaussian sea-surfa e slope PDF, presented in se tion (2.2.2) of Chapter 2.3.1.3 Small Perturbation MethodThe Small Perturbation Method (SPM) belongs to a large family of perturbation expansionsolutions to the wave equation. It is a partial di�erential equation boundary value problemapproa h. The basi idea is to �nd a solution in terms of plane waves that mat h the sur-fa e boundary onditions, whi h state that the tangential omponent of the �eld must be ontinuous a ross the boundary. The surfa e �elds are expanded in a perturbation series,~E = ~E(0) + ~E(1) + ~E(2) + � � � (3.27)where ~E(n) depends on the n-th power of the surfa e elevation � and its gradient, whi h isrelated to the mean squared slope �2s . In the expansion, for instan e, ~E(0) would be the surfa e�eld if the surfa e was at. The philosophy behind this approa h is that small e�e tive surfa e urrents on a mean surfa e repla e the role of the small-s ale roughness. So this method appliesto surfa es with small surfa e height variation and small surfa e slopes , i.e., when both ��and �s are small, independently of the radius of urvature. Therefore, the surfa e no longerneeds to be approximated by planes, as the Kir hho� approximation.The small s ale roughness is expanded in a Fourier series and the ontributions to the�eld are therefore analyzed in terms of the di�erent wavelength omponents. A onsequen eof this analysis is that frequen ies in the surfa e spe trum that mat h the Bragg ondition,thus ontributing oherently to the �eld, play a predominant role. Sin e Bragg s attering isfundamental to the operation of modern s atterometers, let us brie y detour to dis uss thisaspe t in more detail.Mi rowave frequen ies used in modern s atterometers are resonant to omponents of theo ean surfa e spe trum that are either very short gravity waves or surfa e tension waves whi h

44 CHAPTER 3. SCATTERING MODELride on top of larger o ean waves. The resonan e e�e t is so strong for in ident radiation fromabout 20o to 65o that apillary waves of the order of one millimeter of height play a dominantrole in the return signal, even if the underlying waves are several meters high. Even in this ase, however, non-Bragg s attering from longer o ean waves annot be negle ted.In order to assess the impa t of the Bragg s attering omponent, the spe trum of thes attering surfa e must be onsidered. Resonan e o urs with the omponents of this spe trumthat are multiples of �=2 sin �. In the ase of standard s atterometers, the wavelengths involvedare resonant with apillary waves, whi h are the �rst link in the wind-sea oupling hain ofevents. Bragg s attering at GNSS frequen ies ompetes with the dominant di�use s atteringat low elevation in iden e angles [Zavorotny et al., 1998b℄. In [Zu�ada el al., 1998℄, moreover,it is noted that Bragg s attering at GNSS frequen ies does not originate from apillary waves(whi h are sub- entimetri ).When Bragg s attering is the dominant me hanism, it provides a ri h tool to study thesurfa e. Indeed, �xing a Bragg frequen y �xes a surfa e spe tral frequen y and this an berelated|through the dispersion relation for o ean waves|to Doppler e�e ts in the s atteredsignal.To summarize, SPM is a good model for small slope statisti s, with both standard deviationof the sea-surfa e height and orrelation length below the ele tromagneti wavelength. It isthe most appropriate for Bragg s attering issues and polarimetri performan es.3.1.4 The Two-S ale (Composite or Hybrid) ModelWhen there are two relevant s ales in the surfa e roughness (one large and one small), theresults from the rough s ale model an be used by in oherently averaging the e�e tive in iden eand s attering angles with the large s ale slopes. This is done in four steps:1. The surfa e height u tuation � is divided into a large s ale and a small s ale u tuation,� = �l + �s, ea h with its own spe trum, Wl and Ws, and it is assumed that the totalspe trum (whi h is the sum of the two spe tra) is e�e tively one or the other dependingon whether the o ean wavelength is larger or smaller than a given wavelength.2. The small s ale spe trum is used to ompute the SPM s attering oeÆ ient �ort(�; �s; �s),the subs ript s standing for the s attered angles.3. Using the large s ale statisti s tilting angle () statisti s, through the probability dis-tribution fun tion P (), the SPM s attering oeÆ ient is averaged,�00rt(�; �s; �s) � Z dP ()�ort(R[�; �s℄; �s): (3.28)In this expression, R[�; �s℄ stands for the lo al in ident and s attering angles afterrotation to the lo ally tilted referen e frame. The original expressions for the tiltingangles given in [Valenzuela1978℄ are orre ted in re ent work by [Elfouhaily et al.1999b℄.4. Finally, using the large s ale portion of the spe trum the Kir hho� approximation rossse tion is added to �00rt, �oT = �00rt + �oK to a ount for the spe ular part missing in theSPM. Adding this last step (not ru ial for ba ks attering but important for forwards attering) makes the model a omposite model. See [Mar hand and Brown1998℄ for aslight modi� ation of this re ipe|it is argued that the Kir hho� portion needs to bedown-weighted due to the loss of s attering area aused by the small-s ale stru ture.

3.2. GO VALIDATION AT L-BAND 45One of the problems with this approa h is that the real surfa e is not orre tly des ribed bytwo s ales, and some re ipe must be used to divide the spe trum. In [Durden and Vese ky1990℄,an ensemble of 1-D small Gaussian surfa es (24�) is generated, and the ross se tion is al u-lated using the Moment Method|a omputationally intensive method impra ti al for large2-D surfa es like the o ean. The optimal wavelength for the spe trum split is found to beroughly at k=2, but is found to depend on in iden e angle. The authors state that the resultis in rough agreement with theoreti al results found by [Chen and Fung1988℄.In se tion (3.1), we have reviewed di�erent ele tromagneti models. The Kir hho� ap-proximation requires a large mean radius of urvature ompared to the in ident wavelength.The Geometri Opti s model is a high frequen y limit approximation of the Kir hho� �eld.The Small Perturbation approa h requires small mean slopes and heights ompared to wave-length, with no restri tions on the radius of urvature. The Two-S ale model uses these twoapproa hes to take into a ount basi ally the small s ales and the big s ales of the o eanspe trum. It seems to be the most su essful in terms of experimental data.However we keep the GO method during the present study sin e it is parameter-free. Thework done in the following se tion shows that its validation holds at GNSS frequen ies.3.2 GO validation at L-bandWe �rst de�ne in se tion (3.2.1) the requirements needed to apply the GO method, based onthe radius of urvature and size of the s atterer. We present then simulations arried out inorder to understand the validation of GO s attering over a paraboli surfa e in se tion (3.2.2)and over a Gaussian surfa e in se tion (3.2.3). This work refers to [RuÆni and Soulat2000b℄and [RuÆni and Soulat2000a℄.3.2.1 Radius of urvature and s atterer sizeFor the use of the GO method at L-band we have to onsider in addition of the joint prob-ability of slopes and heights the probability of radii of urvature. At the spe ular point theradius of urvature must be large enough and the height di�eren e between the asso iatedparabola|i.e., parabola with same radius of urvature|and the surfa e must be less than onewavelength. This de�nes a toleran e related to the radius of urvature in order to apply theGO method as shown in �gure 3.3. We an ompare this approa h to the atness toleran ede�ned by S hooley for the opti al ase, see [S hooley1962℄.If we assume that the z-axis is normal to the spe ular tangent plane, so that the slope iszero at the spe ular point, the ondition in the one dimensional ase is:�����(x)� ��(xspe ) + 12!r2�(xspe )(x� xspe )2����� < �; (3.29)where x < 10 �p(�rx)=(4�) and � is the in ident wavelength.In the 1-D ase, if we onsider one of the radii of urvature, rx or ry, the integral to be omputed from equation (3.20) is essentially of the form:I = Z lx�lx dx eiqx2=rx : (3.30)

46 CHAPTER 3. SCATTERING MODELλ

Surface

Associated parabola

SP

λ

Figure 3.3: Asso iated parabola for spe ular point determination.With the simple hange of variable u = xpq=rx, we obtain:I = qrx=q Z lxpq=rx�lxpq=rx eiu2du: (3.31)In this high frequen y approximation the s attering ve tor ~q has to be large. The importantquestion is how large q has to be ompared to the surfa e hara teristi s (s atterer size, radiusof urvature). For the integral to be reasonably lose to p�, we need lxpq=rx to be greaterthan 10, say. The inferior limit for lx is thus 10p�rx=4�.Finally, in the bidimensional ase, the requirements be ome for the hara teristi radiusL of the s atterer: 10 �s�prxry4� < L: (3.32)3.2.2 Paraboli surfa eFirst we onsider a paraboli surfa e in order to understand how its resolution and urvaturewere involved in the omputations of the re e ted �eld. The surfa e is bell shaped and theradii of urvature in x and y are equal (rx=ry). The re eiver is at nadir in iden e|onespe ular point is thus onsidered|with a height de�ned by:H = Ho + �(x; y); (3.33)where �(x; y) = �(x2 + y2)=(2prxry), and Ho is a onstant altitude. We ompute the s at-tered Fresnel and Spe ular �elds negle ting the phase and under the far-emitter approximation:UFresnel(p) = �i4� Z ei2kH0r0 exp"i2k (x2 + y2)2prxry #2kdS; (3.34)USpe ular(p) = ei2kH0prxry2H0 : (3.35)The Fresnel �eld should not depend on the wavelength and the resolution. Nevertheless,the os illations of the exponential in the integrand of equation (3.34) in rease with frequen y.We need onsequently a small resolution in the numeri al simulation as de�ned by followingequality: d�dx � resolution << �: (3.36)The impa t of simulation resolution in the Fresnel �eld omputation is shown in �gure 3.4 forthe bi-dimensional ase. The straight line stands for the Spe ular �eldmagnitude: prxry=(2H0),

3.2. GO VALIDATION AT L-BAND 47the exa t solution in this ase. We took a 10�10 m2 paraboli surfa e with a radius of ur-vature at the nadir point (in x and y) of (a) 5 meters and (b) 10 meters; the re eiver is atH=500 meters over the surfa e and the transmission is in L-band (�=19 m). A resolution of10 m seems to be the limit for a tabulation size of 10 meters. This limit de reases for smallerwavelengths and in reases for greater values of radius of urvature as shown in �gure 3.4(b).Indeed the above inequality is mu h respe ted be ause the derivative of � rea hes smallervalues.

Figure 3.4: Fresnel �eld magnitude relative to surfa e resolution for two di�erent radii of urvature: (a) rx=ry=5 m, (b) rx=ry=10 m. The re eiver is at 500 meters over a 10�10 m2paraboli surfa e.

Figure 3.5: Fresnel �eld magnitude relative to the tabulation size of the surfa e for twodi�erent radii of urvature: (a) rx=ry=5 m and (b) rx=ry=10 m. The resolution is 3 m andthe re eiver is at 500 meters transmitting in L-band.The size of the surfa e (the size of the s atterer) is also a very important physi al param-eter, as we dis ussed. Figure 3.5 shows the Fresnel �eld magnitude relative to the size of thes attering surfa e for two di�erent radii of urvature: (a) rx=ry=5 m and (b) rx=ry=10 m.

48 CHAPTER 3. SCATTERING MODELSimilarly, the straight line stands for the Spe ular �eld magnitude: prxry=(2H0). From �g-ure 3.5(a), we an he k the requirement of (3.32) for the size of the s atterer. The \spe ular"s atterer needs to be big enough. For the same paraboli surfa e and if the inequality (3.32)is veri�ed, the re eiver \sees" one spe ular point only and the Spe ular �eld magnitude shouldalso be prxry=(2H0) as expe ted in equation (3.24). Geometri Opti s is therefore expe tedto be a good approximation for high enough frequen ies.In addition, a simple simulation has been arried out to he k if the Geometri Opti s isstill available if we add very small ripples on the paraboli surfa e. The ripples are generatedby a sinusoidal signal hara terized by its amplitude and wavelength. The e�e t of rippleson the re e ted �eld is shown in �gure 3.6 for a 10�10 m2 paraboli surfa e (rx=ry=5 m).Under the urve, there is a di�eren e of �10% between Fresnel magnitude and the magnitude omputed on a perfe t paraboli surfa e in L-band.

Figure 3.6: E�e t of small ripples added on a very smooth paraboli wave. Under the urve,there is a di�eren e of �10% between Fresnel magnitude and the magnitude omputed on aperfe t paraboli surfa e in L-band.Physi ally we assume in the Kir hho� theory that the surfa e is at enough to be lo allyrepresented by planes. The ase of two s ale model is interesting be ause the GO approxi-mation is not valid any more. Indeed the di�ra tion aused by short s ale waves has a greatpart in the return signal. Their ontribution appears in the higher terms of the phase de-velopment. We emphasize that the GO method works for a perfe t paraboli surfa e but isdestroyed when small s ale features are added.3.2.3 Gaussian surfa eThe GO validity is undertaken by omparing the Fresnel and Spe ular �elds. The latter is omputed 1) applying equation 3.24 to all spe ular points dete ted on the sea surfa e and 2)summing in oherently their omplex �elds. We �rst fo us on the spe ular point dete tion andthen ompare the two �elds.Spe ular point determinationWe onsider the in ident radiation as a plane wave normal to a Gaussian surfa e and a re eiver entered above the s attering surfa e. Using Geometri Opti s, the point (x; y; �(x; y)) of the

3.2. GO VALIDATION AT L-BAND 49surfa e is onsidered to be spe ular if the slopes (in x and y) follow geometri al onditions,see [D.E.Freund 1997℄: d�dx = x�xrur + x�xsuszr��ur + zs��us = x� xrzr � � (3.37)d�dy = y�yrur + y�ysuszr��ur + zs��us = y � yrzr � � ; (3.38)where ur (resp. us) is the distan e between the point of the surfa e of oordinates (x; y; �)and the re eiver of oordinates (xr; yr; zr) (resp. the sour e, onsidered here very high). Ourapproa h onsists in the dete tion of slope variations around the expe ted spe ular slopesde�ned above. We ompute at ea h point P of the surfa e, and for both dire tions x andy, the expe ted spe ular slope S|i.e., the slope needed to re e t the in ident wave to there eiver|and the slopes of the fa ets lo ated just before (slope Sbefore) and after (slope Safter)the point P . The ondition for P to be a spe kle is that (Sbefore � S) and (S � Safter) mustbe of the same sign. This determination is attra tive be ause we do not need to de�ne anytoleran e on the spe ular slopes. Therefore it doesn't take in a ount the saddle points de�nedas in e tion points.Barri k derived the average number of spe ular points per unit of area n in terms of thesurfa e statisti s, see [Barri k1968℄:n � 1l2 exp �tan2 �s2 ! ; (3.39)where s2 is the mean square value of the total slope at a point of the two-dimensionally roughsurfa e, de�ned as s2 = D�2x + �2yE, and � is de�ned as the angle between the mean normalto the surfa e and the lo al surfa e normal at the spe ular point. For ba ks attering|i.e.,when the in ident angle � is zero|it is easy to see that the term in the exponential vanishes.Thus the spe ular points density is inversely proportional to the auto orrelation length ofthe surfa e. Figure 3.7 shows the spe ular point number relative to wind speed for theba ks attering ase. The underlying idea is that as the wind speed in reases the waves aremore developed and onsequently the auto orrelation length in reases.As an example, �gure 3.8 shows the spe ular points position|represented by a ross|ona Gaussian surfa e for a wind at 45o of the x-axis with two di�erent speeds. The surfa e isa 30 meter side square with 10 m resolution so that the surfa e is a matrix of 300 � 300pixels. Bla k areas stand for troughs and white areas for rest zones. For (a) U10=5 m/s, there eiver (500 meters high, at nadir in iden e) \sees" 1038 spe ular points while 881 in ase(b) U10=15 m/s.Fields omparisonBased on simulated Gaussian o ean surfa es, we have observed that there is no strong orre-lation between the Fresnel and Spe ular �elds, neither in magnitude nor in phase. However, avery important result is that the power from the Fresnel �eld orrelates well with the numberof spe ular points. Figure 3.9 illustrates how involved is the number of spe ular points inthe re e ted power omputed by the Fresnel integrals. The omputation has been done fora moving Gaussian surfa e based on [Elfouhaily et al.1997℄'s spe trum (5�5 m2 with 5 mresolution) during 2 se onds, under a 15 m/s wind and a fet h of 10 km; the re eiver is at 50meters over the surfa e.

50 CHAPTER 3. SCATTERING MODEL

Figure 3.7: Spe ular points number relative to wind speed for ba ks attering (25x25 metersurfa e with 5 m resolution).

(a) U10=5 m/s (b) U10=10 m/sFigure 3.8: Spe ular points position for (a) U10=5 m/s and (b) U10=15 m/s.

3.2. GO VALIDATION AT L-BAND 51

Figure 3.9: Correlation between the re e ted power and spe ular point number on a Gaussiano ean model in L-band (L1=19 m) within 2 se onds: 5�5 m2, 5 m resolution, U10=15 m/s,re eiver at nadir in iden e at 50 meters.It is rather lear that there is a orrelation between spe ular point number and the returnsignal power. The surfa e is quite well developed|U10=15 m/s|and onsequently smoothenough to apply the GO method.But is this orrelation big enough to state that ounting spe ular points an itself be usefulto des ribe the re e ted �eld? We have ompared the above orrelation in L-band to thatwith an in ident wavelength of 1 m, whi h is mu h better in terms of the high frequen y limitrequirements. In �gure 3.10, we see that the orrelation is mu h better for a 1 m in identwavelength than for the L-band ase, as expe ted.We also see that as the mean radius of urvature in reases with high wind speeds, i.e., therequirements of the Kir hho� theory are better respe ted, the orrelation in reases. Similarly,the di�ra tion terms e�e ts de rease with in reasing o ean s ale, so that their ontributionin the integrand's phase de reases with in reasing wind speed. These di�ra tion terms arethose negle ted by the usual stationary phase method whi h evaluates the re e ted �eld withterms of order two only in the integrand's phase.3.2.4 Con lusions on GO assumption for GNSS-RThe omputation of the re e ted �eld orresponds to a rapidly os illating integral over therough surfa e. The spe ular points are identi�ed with the riti al points of the integrand'sphase, as stated by the Stationary Phase Approximation. Thus, in the high frequen y limit,almost all the s attered �eld intensity omes from the spe ular points. It is exa t in theparaboli ase, provided that Kir hho� assumption applies and the size of the s atterer islarge enough. We saw that small perturbations added on the paraboli surfa e destroy theStationary Phase Approximation integrity.In the realisti ase of a large but �nite in ident wavenumber, the s attering from thespe ular point often arries the major ontribution to the re e ted signal. Therefore, inmany ir umstan es, the area that ontains the major ontribution to the re e ted �eld|thebright area| an be identi�ed as the part of the sea surfa e where the probability for spe ularre e tion is signi� ant.

52 CHAPTER 3. SCATTERING MODEL

Figure 3.10: Correlation between the spe ular points number and the re e ted power om-puted with the Fresnel integral, for an in ident wavelength L1=1 m or L2=20 m.Re all one more time that the GO method is: a) the Stationary Phase approa h, b) anapproximation to the �eld up to se ond order and ) in oherent averaging.We have not seen a good orrelation between the Fresnel �eld and the Spe ular �eld omputed dire tly from the spe ular points. There exists, however, a orrelation betweenthe spe ular point number and the re e ted power (points a) and )). In general the higherthe ele tromagneti frequen y and wind speed|i.e., higher roughness and larger s ales in theo ean|the better this orrelation is, as expe ted from our theoreti al understanding of theapproximation.Several on lusions an be extra ted from this study:� Most of the radiation omes from spe ular points, hen e the good orrelation betweenspe ular point number and �eld power. This means that the surfa e slope PDF willplay an important role in bistati s attering modeling at L-band.� The stationary phase approximation an be used but higher order terms must play a roleat L-band. These orre tions are usually alled di�ra tion orre tions. This explainsthe poor orrelation between the Spe ular and Fresnel �elds.� GO's su ess in GNSS-R work stems more from the use of spe ular point number statis-ti s than from the use of pre ise �eld al ulations based on the stationary phase approx-imation. That is, the su essful use of the PDF in GNSS s attering is not a validationof GO. In retrospe t, this is all evident: the lesson from GO for L-band is limited tounderstanding the importan e of the slope PDF to approximate the �eld. The slopePDF is the tool to ount spe ular points. Ne essary but not suÆ ient, of ourse.With this aveat, the GO model is thus further used in all L-band s attering pro essespresented in this dissertation, su h as the waveform model developed in the following se tion.3.3 Waveform model: DDMWe present in the following se tion the integral expression of the power GNSS waveform. A omparison is then done with the models developed by [Zavorotny and Voronovi h2000℄ and

3.3. WAVEFORM MODEL: DDM 53[Pi ardi et al.1998℄.3.3.1 Integral expression of the power waveformBistati s attering of a mono hromati waveIf the o ean surfa e is illuminated with a mono hromati spheri al wave, the omplete quasi-spe ular s attered �eld writes under Kir hho� approximation:u(; t) = i4� Z R ~q:n GtGrRtRr e�i(t�l(~r;t)= ) d~r; (3.40)with:� R : Fresnel oeÆ ient,� ~q : s attering ve tor,� n : normal to the surfa e,� Gt; Gr : transmitter and re eiver amplitude antenna patterns,� Rt; Rr : transmitter-o ean and o ean-re eiver distan es,� : arrier pulsation,� l(~r; t) = Rt+Rr : length of the path Transmitter to generi point (Rt) to Re eiver (Rr).At high altitudes, many of the above magnitudes do not vary signi� antly over the integrationarea. First Gt, Rt and R an reasonably be onsidered onstant. Then, for smooth o eansurfa e and in quasi-spe ular regime, we have n ' ez and ~q ' qez. Therefore, ~q:n ' 2 sin (where is the elevation of the satellite). Finally, for high enough altitudes, Gr and Rr arealso uniform fun tions. As a result, the Kir hho� integral simpli�es to:u(; t) = i sin 2� R GtGrRt0Rr0 Z e�i(t�l(~r;t)= ) d~r: (3.41)Re eiver and transmitter motions make the travel path time-dependent in fast-varying fun -tions. To �rst order, this e�e t an be written:l(~r; t) ' d(~r) + �2�!D(~r)t; for 0 < t < Ti; (3.42)where !D=2� is the Doppler ontribution of point ~r and Ti is the oherent integration timede�ned below. Finally, we an state that the s attered �eld for a mono hromati in identwave at is: u(; t) = i sin 2� R GtGrRt0Rr0 Z e�i(�!D(~r))t eid(~r)= d~r: (3.43)PRN modulation as a wave pa ketIn GNSS-R, the in ident wave is not a pure mono hromati wave. The PRN ode|let's nameit here a(t) for both C/A and P odes|a tually modulates the arrier, broadening the in identspe trum. Note that the arrier is =1.5 GHz and the bandpass is either 1 MHz (C/A ode)or 10 MHz (P ode). Sin e the Kir hho� s attering me hanism is linear, the outgoing �eld

54 CHAPTER 3. SCATTERING MODEL an be seen as the superposition of a series of mono hromati omponents s attered by theo ean surfa e: u(t) = Z +1�1 ~a(�) � u( + �; t) d�; (3.44)where ~a(�) is the Fourier baseband spe trum of the PRN ode and u( + �; t) is given byequation (3.43). Note that in the equation above, � appears as the frequen y broadening fromthe arrier .Code despreadingAs introdu ed in se tion (1.1.2), the GNSS re eiver performs signal dete tion through odedespreading, that is by orrelating over a period Ti the in oming signal with a lean modulatedrepli a of the PRN ode: y(�; !) = 1Ti ZTi u(t) � a(t� �)ei(�!)t dt: (3.45)The result of this pro essing is nothing but the omplex delay-Doppler map at time-frequen yshift (�; !).Combining equations (3.43), (3.44) and (3.45) leads to:y(�; !) = i sin 2� Ti R GtGrRt0Rr0 ZZZ ( + �) a(t� �) ~a(�)ei[(!D�!��)t+(+�)d= ℄ d� dt d~r: (3.46)Let us �rst perform the integration on �:Z ( + �) ~a(�)e�i�(t�d= ) d� ' a(t� d= ); (3.47)where we have assumed that the fa tor ( + �) is basi ally identi al to over the GNSSbandpass.The integration over t writes:ZTi a(t� d= ) a(t� �) ei(!D�!)t dt = Ti �(Æ�; Æ!); (3.48)where we have introdu ed: ( Æ� = � � d(~r)= Æ! = ! � !D(~r): (3.49)The fun tion � is known in radar pulse- ompression te hniques as the Woodward Ambigu-ity Fun tion (WAF) of pseudo-random sequen es, de�ning here the spatial resolution of thesystem. It is here assumed separable:�(Æ�; Æ!) = �(Æ�) � S(Æ!) (3.50)= �(Æ�) � sin (Æ! Ti): (3.51)The fun tion � is the triangle fun tion of width twi e a hip length and sin (x) is sin(�x)=�x.Finally, the despread signal writes:y(�; !) = i sin � R GtGrRt0Rr0 Z �(Æ�) � S(Æ!) eid= d~r: (3.52)

3.3. WAVEFORM MODEL: DDM 55Delay-Doppler mapLet us make expli it the expression of the delay and Doppler mapping on sea-surfa e (namelyd(~r)= and !D(~r)). Note that we will assume wD deterministi (Doppler mapping is omputedon a at surfa e) whereas d has a random ontribution through sea-elevation z=�(x; y).The distan es to any point (x; y; z) on the o ean surfa e write:8<: R2t = R2t0 �1 + 2x os Rt0 � 2 z sin Rt0 + x2+y2+z2R2t0 �R2r = R2r0 �1� 2x os Rr0 � 2 z sin Rr0 + x2+y2+z2R2r0 � ; (3.53)where Rt0 and Rr0 are the distan es of transmitter and re eiver to the spe ular point. Re-taining only signi� ant terms, we have:d(~r) ' d0 � 2 sin z + sin2 2 Rr0x2 + 12 Rr0 y2; (3.54)where d0 stands for the bistati path to spe ular point (Rt0 + Rr0), that is the parameter ofinterest for an altimetri appli ation.If we negle t the Doppler spread due to transmitter motion and only retain the ontributionof the re eiver with velo ity ~v, we have:!D = 2�� ~v: ~RrRr= 2�v os �� x+ y tan��Rr0 os pRr0 � 2x os + x2 + y2 ; (3.55)where � is the angle between the Transmitter-Re eiver line (x axis) and ~v. To �rst order:!D = !D0 + 2�v os �� �sin2 xRr0 + tan� yRr0� ; (3.56)where !D0 = �2�v os� os � is the Doppler of the spe ular point.Averaging in powerAt this point, it is time to point out whi h are the ontributions of the o ean surfa e and bythis mean, identify the random parameters in the above integral. There are a tually threesour es of randomness:� The most ru ial one is obviously d in the exponential term. Note that only this random ontribution was onsidered in [Zavorotny and Voronovi h2000℄. This approa h yieldsa statisti al sea surfa e model only featuring MSS.� The argument of � is also random through d. If it is taken into a ount, the statisti aldes ription of sea surfa e will also feature SWH, as in [Pi ardi et al.1998℄.� In prin iple, the argument of S is also random be ause wD slightly depends on thesea-surfa e topography through re eiver radial speed. However, this e�e t is really tinyand will not be onsidered here.In the following, we will therefore assume that Æ! is deterministi (wD is omputed on a atsurfa e) and that d is the only random variable:y(�; !) = i sin � R GtGrRt0Rr0 Z S(Æ!) �(� � d= ) eid= d~r: (3.57)

56 CHAPTER 3. SCATTERING MODELIt is then onvenient to introdu e the Fourier transform of �(�), that is the delay-spe trumof the despread ode: �(�) = Z ~�(�) ei�� d�; (3.58)so that we an write:y(�; !) = i sin � R GtGrRt0Rr0 Z ~�(�) ei�� Z S(Æ!) ei(��)d= d~r d�; (3.59)and hen e isolate the random ontribution in the exponential term. The average power signalis then de�ned as: Y (�; !) = h jy(�; !)j2 i: (3.60)That is: Y (�; !) = sin2 �2 jRj2 G2tG2rR2t0R2r0: Z d�1 Z d�2 ~�(�1) ~��(�2) ei�(�1��2): Z d~r1 Z d~r2 S(Æ!1) S�(Æ!2) h ei���1 d1���2 d2� i: (3.61)Plugging the delay mapping in Y (�; !) yields:Y (�; !) = sin2 �2 jRj2 G2tG2rR2t0R2r0: Z d�1 Z d�2 ~�(�1) ~��(�2) ei(��d0= )(�1��2): Z d~r1 Z d~r2 S(Æ!1) S�(Æ!2) e i2 Rr0 [(��1)(sin2 x21+y21)�(��2)(sin2 x22+y22)℄: h e�i 2 sin [(��1)z1�(��2)z2℄ i: (3.62)ApproximationsSo far, the integration is performed over the six variables x1, x2, y1, y2, �1 and �2. For ea h ouple of variables, we introdu e the mean and the di�eren e. For instan e:(x1; x2) �! ( Æx = x1 � x2�x = x1+x22 : (3.63)Note that the Ja obian of this transformation is unitary. In the following, we will fo us on theexpression of Y (�; !D0), i.e., the waveform obtained after exa t ompensation of the spe ularpoint Doppler 2. We have:Y (�; !D0) = sin2 �2 jRj2 G2tG2rR2t0R2r0 � Z d�� Z d(Æ�) Z d�x Z d(Æx) Z d�y Z d(Æy): ~�(�� + Æ�2 ) ~�(�� � Æ�2 ): ei(�� d0 )Æ�: S(�x+ Æx2 ; �y + Æy2 ) S(�x� Æx2 ; �y � Æy2 ): e �i2 Rr0 Æ�hsin2 (�x2+ Æx22 )+�y2+ Æy22 i: e i2 Rr0 2(���)(sin2 Æx��x+Æy��y): h e�i 2 sin [(���)Æz��zÆ�℄ i: (3.64)2Note that S(!D0 � !D) is then a fun tion of (x; y) only.

3.3. WAVEFORM MODEL: DDM 57We therefore end-up with a tough integral featuring six distin t terms where the variablesare oupled. Some approximations an reasonably be made. We remind that �� is bound bythe ompressed pulse bandwidth (1 MHz for C/A ode) , and that Æx and Æy are bound bythe o ean orrelation length (some tenth of meters).Approximation 1.1In the sixth and �fth term of (3.64), we assume:� �� ' : (3.65)As a matter of fa t, the maximum phase ontributions indu ed by �� is negligible in both ases.Approximation 1.2In the fourth term of (3.64), we assume:sin2 (�x2 + Æx22 ) + �y2 + Æy22 ' sin2 �x2 + �y2: (3.66)As a matter of fa t, the maximum phase ontributions indu ed by Æx and Æy are negligible.Approximation 1.3In the third term of (3.64), we assume:S(�x+ Æx2 ; �y + Æy2 ) S(�x� Æx2 ; �y � Æy2 ) ' S2(�x; �y); (3.67)be ause the orrelation length of sea surfa e is short and S is a slow-varying fun tion.Simpli�ed expressionA ording to the above approximations, the expression (3.64) rewrites:Y (�; !D0) ' sin2 �2 jRj2 G2tG2rR2t0R2r0 � Z d�� Z d(Æ�) Z d�x Z d(Æx) Z d�y Z d(Æy): ~�(�� + Æ�2 ) ~�(�� � Æ�2 ): ei(�� d0 )Æ�: S2(�x; �y): e �i2 Rr0 Æ�[sin2 �x2+�y2℄: e i Rr0 (sin2 Æx��x+Æy��y): h e�i 2 sin (Æz��zÆ�) i: (3.68)Geometri Opti sIn order to al ulate the ensemble average in the latter expression, we follow the standardpro edure of the Geometri Opti s limit. As des ribed in se tion (3.1.2), the essen e of thisapproa h is to onsider the sea-surfa e as a set of plane fa ets so that Æz an be expressed to�rst order as a fun tion of the lo al dire tional slope:Æz ' �~rz � Æ~r: (3.69)

58 CHAPTER 3. SCATTERING MODELWe are here dealing with two random pro esses of interest: the sea-surfa e elevations z andthe sea-surfa e dire tional slopes �~rz. If these are onsidered independent:h e�i 2 sin (Æz��zÆ�) i = h e�i 2 sin �~rz�Æ~r i � h ei 2 sin �z:Æ� i: (3.70)The above ensemble averages are then identi�ed as hara teristi fun tions of the randompro esses. Introdu ing the sea-elevation PDF P� and the sea-slope PDF Ps, we end-up with:h e�i 2 sin (Æz��zÆ�) i = ~Ps �2 sin Æ~r� � ~P� �2 sin Æ�� ; (3.71)where ~: denotes the Fourier Transform. The expression of Y (�; !) be omes:Y (�; !D0) ' sin2 �2 jRj2 G2tG2rR2t0R2r0 � Z d�� Z d(Æ�) Z d�x Z d(Æx) Z d�y Z d(Æy): ~�(�� + Æ�2 ) ~�(�� � Æ�2 ) ~P� �2 sin Æ�� ei��Æ�: S2(�x; �y) ~Ps �2 sin Æ~r� e i Rr0 (sin2 Æx��x+Æy��y); (3.72)with: �� = � � d0 � sin2 �x2 + �y22 Rr0 : (3.73)3.3.2 Expression as a 2-D integral on sea-surfa eIntegrationThe six-variable integral redu es to:Z d�x Z d�y S2(�x; �y) � J(�x; �y) �K(�x; �y); (3.74)with: J(�x; �y) = Z d�� Z d(Æ�) ~�(�� + Æ�2 ) ~�(�� � Æ�2 ) ~P� �2 sin Æ�� ei��Æ� ; (3.75)K(�x; �y) = Z d(Æx) Z d(Æy) ~Ps �2 sin Æ~r� e i Rr0 (sin2 Æx��x+Æy��y): (3.76)Integration of JIntegrating �rst on d��, we re ognize the auto orrelation of ~� and the se ond integral appearsas the Fourier transform of the produ t of two fun tions:J(�x; �y) = Z d(Æ�) h~� ? ~�i (Æ�) ~P� �2 sin Æ�� ei��Æ�/ ��2(�) ? P� � �2 sin �� (��)� �2e(��): (3.77)This result shows that the main impa t of sea-state is to introdu e an e�e tive ompressedpulse �2e de�ned as the onvolution between �2 and the sea-elevation PDF. As a result, thenominal ompressed pulse width � is broaden by an additive fa tor proportional to signi� antwave height.

3.3. WAVEFORM MODEL: DDM 59Integration of KWe re ognize here a Fourier transform:K(�x; �y) / Ps �sin �x2Rr0 ; 1sin �y2Rr0� : (3.78)2-D integral expressionThe waveform now writes as a 2-D integral over sea-surfa e:Y (�; !D0) / Z d�x Z d�y: S2(�x; �y) � �2e � � d0 � sin2 �x2 + �y22 Rr0 !: Ps �sin �x2Rr0 ; 1sin �y2Rr0� : (3.79)3.3.3 Comparison with Zavorotny's and Pi ardi's modelsComparison with Zavorotny's modelExpression (3.79) an be ompared to the model of [Zavorotny and Voronovi h2000℄, equation31. We have a 2D surfa e integral with three terms respe tively a ounting for the WAF inDoppler, the WAF in delay and �o. The two fundamental di�eren es are:� Sea-state has been in luded in this formulation through the e�e tive ompressed pulse�e. Note however that the sea-elevation PDF is very narrow ompared to the pulse andsea-state impa t is therefore quite tiny.� We have not retained the expli it expressions of delay-Doppler mapping but performedse ond and �rst order expansions instead. We feel one ould obtain the same expressionwith the full delay-Doppler expressions (not expanded).Comparison with Pi ardi's modelExpression (3.79) an be ompared to the model of [Pi ardi et al.1998℄. The improvementsof the proposed approa h are:� orre tion of a mistake. Compare equation (3.54) to their equation (4). They forgota sin term, thus addressing the monostati ase (iso-range are ir les) instead of thebistati (iso-ranges are ellipses).� in lusion of the Doppler spreading e�e t.� in lusion of the dire tional MSS.On the other hand, their model in ludes height distribution skewness.As a on lusion, we have presented a new model (whi h we will refer to as the GNSS-RStarlab model) for quasi-spe ular sea-surfa e s attering of L-band PRN-en oded signals, in afully bistati geometry. As in Zavorotny's model, sea-surfa e roughness is des ribed by theDMSS. Following Pi ardi, wave-height impa t was also in luded as a slight modi� ation ofZavorotny's model: an additional input parameter for sea-surfa e des ription is SWH. How-ever, due to its 300 m hip length, the C/A ode signal is not sensitive to SWH. Consequently,

60 CHAPTER 3. SCATTERING MODELZavorotny's model is suÆ ient in the present study. Nevertheless, the new model ould be- ome useful in the future when dealing with P- ode: the hip length of 30 m is, in this ase,mu h loser to typi al SWH and the latter will be ome signi� ant.

Chapter 4GNSS-R Signal Pro essingWe des ribe in this Chapter the pro essing of the GNSS-R data. A step-by-step des ription ispresented from the raw data to the �nal produ t to be inverted: the Delay-Doppler Map. The orrelation of the raw GNSS signal with a lean PRN repli a is the basis of the pro essing.Due to the transmitter's motion and the possible re eiver's motion, the orrelation peak isfound when the orre t Doppler frequen y (Doppler enter) is inserted into the lean PRNrepli a.We provide hereafter the di�erent data produ t levels, as presented in �gure 4.1.LEVEL 0b

Incoherent averaging

LEVEL 1

Retracking

SCATTEROMETRYALTIMETRY

LEVEL 0

(amplitude, time, frequency)(amplitude, time)

1−D Waveforms 2−D Waveforms (DDM)

Figure 4.1: Data produ t levels in GNSS-R pro essing.4.1 From raw data to Level 0: Delay-Doppler PRN ode de-spreadingIn order to spe ify our de�nition of Level 0 data, we take into a ount that any futureGNSS-R payload will be more than a simple antenna front-end: the delay-Doppler PRN odedespreading that is presently performed on ground will be a tually arried out on board.

62 CHAPTER 4. GNSS-R SIGNAL PROCESSINGBased on this, our Level 0 data will be time series of Delay-Doppler Maps (DDMs) for bothdire t and re e ted GPS signals. This de�nition is a tually CEOS- ompliant:Unpro essed payload data in hronologi al order at full spa e/time resolution withall supplementary information to be used in subsequent pro essing (e.g., orbitaldata, health, time onversion, et .) appended, after removal of all ommuni ationartifa ts (e.g., syn hronization frames, ommuni ations headers, dupli ate data).We use the Starlab in-house software (Starlight), implemented in Matlab, to produ eDDMs time-series (one per PRN). The general strategy of the pro essing is to tra k thedelay-Doppler of dire t signal and then ompute DDMs for both dire t and re e ted signals.Those DDMs a tually represent omplex amplitude of in oming signals when pro essed witha delay-Doppler value slightly di�erent from the estimated delay-Doppler enter.The very high level ow harts of the pro essor is presented in �gure 4.2. The three basi sblo ks are:� the initialization: it orresponds to the estimation of the Doppler enter and all pa-rameters orrupting the signal extra tion (navigation bit, dire t-re e ted signals syn- hronization, ...),� the tra king: the orrelation pro ess itself,� the DDM generator: the generation of the delay waveforms at di�erent Dopplerfrequen ies around the Doppler enter.

TRACKER

DDM

Clean replica of PRN i

Raw data

Doppler frequency update rate

Coherent integration time

INITIALIZATION

Figure 4.2: The top high level ow hart of the GNSS-R pro essor for PRN i.The pro essor assumes as input data the available GPS signal (I or Q omponent), at a ertainIntermediate Frequen y (IF), sampled at a ertain rate fs, and with a two level quantization.Ea h signal is then represented by a bit stream, in natural-time sequen e. As an example,in the ase of data oming from the TurboRogue re eiver (like those in the ESA GPS-Rhardware), the IF is 308 kHz and the sampling frequen y is 20:456 MHz.

4.1. FROMRAWDATA TO LEVEL 0: DELAY-DOPPLER PRN CODEDESPREADING634.1.1 InitializationThe �rst module, the initializer, pre-pro esses the data in order to generate an ASCII �le ontaining all the information needed to tra k both dire t and re e ted signals (i.e., these ond blo k). The input of this blo k are the data �les, the PRN number and an additional�le ontaining all the parameters that will be used during the pro essing. It is assumed thatthe GPS satellites provided to the �rst blo k is a visible satellite.This blo k is subdivided into four modules applied on the dire t signal:� First Doppler estimation: this module is in harge of �nding a �rst estimation of theDoppler frequen y of the PRN under investigation.� Navigation bit syn hronization: taking bene�t of the previous outputs, this module �ndsthe position of the navigation bit inside the data stream of the dire t signal.� Fine Doppler estimation: on e the position of the navigation bit is known, a series of orrelations, un orrupted by the navigation bit transition, are performed. The output isa new and more pre ise value of the Doppler frequen y of the PRN under examination.FIRST DOPPLER ESTIMATION

NAVIGATION BIT SYNCHRONIZATION

FINE DOPPLER ESTIMATIONFigure 4.3: Initialization module of the GNSS-R pro essor.Let's des ribe the three parts of this module.First Doppler estimationThe goal of this module is to �nd a �rst estimation of the Doppler frequen y for a givenPRN and dire t signal data �le. The algorithm is based on the onsideration that orrelatingthe dire t signal with a lear repli a of the PRN ode under investigation, the orrelationpeak will be higher if the Doppler frequen y used to down- onvert the signal is loser to thereal one. Thus, performing many orrelations using di�erent Doppler frequen ies| hosen asto span the whole range of possible Doppler for the given situation|the one generating thehighest peak or, better said, the best signal-to-noise ratio during the orrelations, is assumedto be the a tual Doppler.The number of C/A ode periods used for the oherent integration is set to 3. If nosatisfying Doppler frequen y estimation is found, a se ond trial is done, using an in oherentaveraging of three su essive orrelations. This pro edure allows fast estimation of the Doppler

64 CHAPTER 4. GNSS-R SIGNAL PROCESSINGfrequen y when dealing with high SNR signals. Of ourse, the exe ution time in reases, asthe number of in oherently averaged waveforms in reases. This pro edure is robust also withrespe t to the possibility of �nding a navigation bit hange in the pie e of data analyzed bythis module.This pro edure provides a Doppler estimation whi h is in general within �40 Hz from thereal one. An example of the orrelating pro ess output is shown in �gure 4.4.

−20456 −10228 0 10228 20456 −20456 −10228 0 10228 20456 −20456

sample [~15 m]Figure 4.4: An example of orrelation peak in arbitrary units, for 5 ms of oherent integrationtime, using the PRN ode number 9.Navigation Bit Syn hronizationThe goal of this module is to �nd the position of the navigation bit inside the dire t signaldata stream.We re all that the GPS navigation ode represents the a tual data provided by ea h GPSsatellite to the users. It onsists in a 50 hertz data stream ontaining all the informationne essary to the GPS re eiver, su h as the ranging signal time of transmission, the satellite'sorbital elements, the ranging orre tions, status ags, et ... To su essfully orrelate a re eivedGPS signal with a lo al repli a, the navigation ode should be removed.To understand this method, onsider �gure 4.5. The top drawing represents the multipli- ation of two repli as of a PRN ode, that is the value of its auto orrelation at lag zero. Aswe know this value is one. The entral drawing represents the multipli ation between a PRN ode and its repli a multiplied by �1. In this ase the value of su h operation is �1. Thelast drawing represents the multipli ation between a PRN ode and its repli a after that these ond half of this ode has been multiplied by �1. The result is now zero.This third s enario illustrates what happens when the navigation bit transition appears rightin the middle of the portion of signal that we are oherently integrating, i.e., orrelating withits lear repli a. If the transition of the navigation bit o urs somewhere along the ode andnot in the middle, the maximum absolute value of the orrelation will be less than one.We take advantage of this behavior in order to dete t the navigation bit position withinthe re eived signal. Let's imagine that we use a 20 ms long repli a of a ertain C/A ode(equal to 20 C/A ode periods, that is equal to the duration of one navigation bit). Supposethat we are syn hronous with the navigation bit and that we start multiplying the signal andthe ode repli a. Sin e we are syn hronous the value of the multipli ation will be maximum.

4.1. FROMRAWDATA TO LEVEL 0: DELAY-DOPPLER PRN CODEDESPREADING65

Figure 4.5: E�e t of the navigation bit transition on the orrelation pro ess.Now we shift the repli a by one period, i.e. 1 ms, and we perform again the multipli ation.This time, the repli a will have its last C/A ode period inside the new navigation bit andthis will degrade the multipli ation value (if the navigation bit hanges). Iterating the pro essshifting the lear repli a of 1 ms ea h time, when the navigation bit transition rea hes themiddle of the ode, the result of the multipli ation between the two signals will rea h itsminimum value. After that, it will in rease again. This pro ess an be repeated so that manyestimations of the minimum values of the multipli ation of the signal with the repli a an bein oherently summed to provide a more reliable estimate. This pro edure is shown in �gure4.6.On e this syn hronization with the navigation bit is ompleted, the oherent integrationtime an be set to 20 ms without the fear for SNR losses. Of ourse this pro ess has to beiterated for every PRN ode present in the signal. Moreover, sin e the dire t data stream andthe re e ted one are not syn hronized, this pro edure has to be done separately for the dire tand the re e ted signal.The output of this module is the number of samples to remove from the dire t signal bitstream to be syn hronized with the navigation bit (and thus with the C/A ode) for the PRNunder investigation.Fine DopplerThis module is in harge of providing a better estimation of the Doppler frequen y of thesignal relative to the PRN ode under investigation.The algorithm is based on the fa t that the hange in the phase of the orrelation peakbetween two su essive orrelations (normally very small, say in the order of one tenth of a y le) should be, on the average, zero. If the Doppler frequen y used to down- onvert thesignal is wrong, the average of the delta phase will be di�erent from zero. The value of this

66 CHAPTER 4. GNSS-R SIGNAL PROCESSING

Figure 4.6: Example of the sear h for the navigation bit syn hronism. As the 20 ms long ode repli a slides on the signal (a, b, , d, e, f), the value of its produ t (green urve atthe bottom) with the signal (�rst row) depends on the position of the navigation bit (red)transition within the ode repli a.

4.1. FROMRAWDATA TO LEVEL 0: DELAY-DOPPLER PRN CODEDESPREADING67average is dire tly related to the error in the Doppler by the following equation:�fDoppler = < Æphase >Ti ; (4.1)with intuitive notation. The pre ision of this estimation is of the order of one Hertz.4.1.2 Tra king the 1-D waveformsThe tra king module simply orrelates the dire t and re e ted signals with the lean PRNrepli a. The waveform is omputed through the orrelation oeÆ ients. As shown in �gure4.7, two parameters are required in addition to the outputs of the initialization module:� the oherent integration time: it orresponds to the length in number of periods ofthe repli a. Typi al values are 1, 2, 5, 10 or 20 ms.� the Doppler update rate: this parameter indi ates how often the Doppler frequen yis updated while arrying out the orrelations along the data stream. Typi al values are0.25 or 0.5 se ond.

TIME TAGS

REFLECTED SIGNAL CORRELATION FUNCTIONS (WAVEFORMS)

DIRECT SIGNAL CORRELATION FUNCTIONS

CORRELATION PROCESS

DOPPLER FREQUENCY HISTORY

Starting Doppler

Code and navigation bit synchronization

INPUT PARAMETERS

Coherent integration time

Doppler frequency update rate

INITIALIZATION OUTPUT

Figure 4.7: Tra king module of the GNSS-R pro essor.The orrelation pro ess uses Fast Fourier Transforms (FFT). The output of this moduleare basi ally the orrelation fun tions of the dire t and re e ted signal, with a time tag andthe Doppler frequen y used.4.1.3 2-D waveforms generation (DDM)In the ase of s atterometri appli ations, we simply onstru t the Delay-Doppler Mapsthrough the generation of delay waveforms at di�erent Doppler frequen ies around the Doppler

68 CHAPTER 4. GNSS-R SIGNAL PROCESSING enter. Examples of DDMs for both dire t and re e ted signals are shown in �gure 4.8. They orrespond to an airborne s enario (see the Eddy Experiment Flight in Chapter 7). The oherent integration time was set to 20 ms to ensure a Doppler resolution of 50 Hz. Thedelta-delay spans -40 to 40 orrelation lags (i.e., +/- 1.95 �s) with a lag step (48.9 ns) andthe delta-Doppler spans -200 Hz to 200 Hz with a step of 20 Hz. The DDMs are plotted inamplitude times 103.

0

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2 (b) Re e ted signal DDMFigure 4.8: DDM of dire t (a) and re e ted (b) signals from the Eddy Experiment Flight(PARIS� ESA Contra t). The DDMs are plotted in amplitude times 103.4.2 From Level 0 to Level 0b data: in oherent averagingThe Level 0b data onsists in in oherently averaging Level 0 data over a given a umulationtime Ta. This pro ess aims at redu ing both thermal and spe kle noises by a fa tor of pN ,N being the number of DDMs. It is however limited by the blurring e�e t as Ta in reasesand the required resolution.4.3 From Level 0b to Level 1 data: retra kingFinally, in order to rea h the needed sub-sample a ura y for both altimetri and s atteromet-ri purposes, we rely on the use of the waveform (or orrelation fun tion) model developed inse tion (3.3) of Chapter 3. The natural model for retra king of the dire t signal waveforms isthe mean auto orrelation of the GPS C/A ode in presen e of additive Gaussian noise, whi ha ounts mainly for the re eiver noise. As far as the re e ted signal is on erned, the model isnot so straightforward. In fa t, the re e tion pro ess indu es modi� ations on the GNSS sig-nals whi h depend on sea surfa e onditions (dire tional roughness), re eiver-emitter-surfa ekinemati s and geometry, and antenna pattern. Among all these quantities, the least knownones are those related to the sea surfa e onditions. In prin iple, these quantities should be onsidered as free parameters in the model for the re e ted signal waveform and estimatedduring the retra king pro ess of the waveform.The ru ial step of retra king is also known in altimetry by the Radar Altimeter (RA) ommunity. On e a orrelation waveform is obtained for both the dire t and re e ted signals,

4.3. FROM LEVEL 0B TO LEVEL 1 DATA: RETRACKING 69the lapses an be estimated. We emphasize that this is not as trivial as onsidering themaximum sample of ea h waveform or the waveform entroid, for instan e, as the bistati re e tion pro ess deforms severely the signals and, in general, su h distortions will displa e thewaveform maximum or entroid lo ation. Moreover, the sampling rate of 20.456 MHz, whilemu h higher than the Nyquist rate for the C/A ode, is equivalent to an inter-sample spa ing of49 ns|about 15 light-meters. This oarseness in the lapse estimation is not suÆ ient to rea hthe �nal altimetri pre ision target. The main hallenge for GNSS-R using the GPS CoarseA quisition (C/A) ode is to provide sub-de imetri altimetry using a 300 m equivalent pulselength, something that an only be a hieved by intense averaging with due are of systemati e�e ts. For referen e, pulse lengths in monostati altimeters su h as Jason are almost threeorders of magnitude shorter.For these reasons, the temporal position of ea h waveform (the delay parameter) is es-timated via a Least Mean Squares model �tting pro edure (see [RuÆni et al.2003℄). Theimplementation of a urate waveform models (for dire t and re e ted signals) is fundamentalto retra king. Con eptually, a retra king waveform model allows for the transformation of there e ted waveform to an equivalent dire t one (or vi e-versa), and a oherent and meaningful omparison of dire t and re e ted waveforms for the lapse estimation.The retra king is a tually ne essary for s atterometri purposes, in order to invert orre tlythe DMSS, as shown in se tion (9.2) of Chapter 9.

70 CHAPTER 4. GNSS-R SIGNAL PROCESSING

Part IIGNSS-R Signal Simulations

71

Chapter 5GNSS-R Simulation ToolkitIn this Chapter, we present a software simulation tool| alled GRADAS (GNSS Re e tionsArti� ial Data Synthesiser)|able to produ e GNSS-R Arti� ial Data (AD) [Soulat et al.2002a℄.We show here simulations of spa eborne data re eived from a Low Earth Orbit platform(LEO). We onsider one GNSS satellite emitting at nadir the C/A ode in L1 and L2, inprevision of the implementation of C/A ode in L2 in 2003|see se tion (1.1.3) of Chapter 1.The inputs of this software are the position and velo ity of the re eiver (amplitude anddire tion), and sea-state model parameters (su h as signi� ant wave height, wind speed anddire tion). By Arti� ial Data, we mean syntheti data in the same format as the data gen-erated by the ESA Sony re orders used in all ESA ampaigns, with the extrapolation to thespa e s enario. A omplete des ription of the ESA material an be found in se tion (7.1.1)of Chapter 7. Thus, the output is a set of re orded data simulating the 20.456 Mbit/s datastreams produ ed by the Sony re order as if it was in orbit. The goal of this work is indeedto simulate data whi h an then be read and pro essed by the Starlab in-house software, withthe goal of assessing retrieval feasibility from spa e|by extra tion of the pseudo ranges andphases.In order to onstru t the simulator, many aspe ts of the problem|most presented inprevious Chapters|have to be taken into a ount:� Geometry,� In oming signal stru ture (L1, L2, PRN odes, navigation message),� Spe ular point lo ation,� Atmospheri e�e ts,� O ean model,� Ele tromagneti intera tion model,� Instrumental aspe ts (down onversion, sampling, re ording).Given the omplexity of the task, we will fo us not on absolute exibility or details (allgeometries, navigation message, et ). Rather, the goal of this work is to get samples of datafor a given s enario with exibility on o ean modeling: we used a Gaussian o ean surfa e.We present hereafter two di�erent approa hes. The �rst model developed in the next se tionis based on the Kir hho� approximation as the ele tromagneti intera tion model, where there e ted �eld is given by the Fresnel integral, as des ribed in se tion (3.1.1) of Chapter 3.The se ond one of se tion (5.2) onsiders a dire t appli ation of the GO approximation tothe Kir hho� theory (see se tion (3.1.2) of Chapter 3). The two models have been validatedthrough the arrier phase retrieval from our in-house GPS pre-pro essor.

74 CHAPTER 5. GNSS-R SIMULATION TOOLKIT5.1 Model 1: GRADASWe �rst on entrate on o ean simulations for huge areas. A spe ial attention will be paidon the diÆ ulty to simulate o ean surfa es within the First Chip Zone (FCZ) with requiredresolution (20 m). Indeed, its radius an rea h 17 km for a re eiver's altitude of 500 km, whi himplies tremendous memory and CPU requirements. We propose a demo rati sampling ofthe FCZ.Se tion (5.1.2) gives an overview of the stru ture of the GNSS signal. An expressionof the re e ted signal is proposed, whi h determines the basi s of the developed algorithmpresented in se tion (5.1.3). This latter se tion �rst des ribes the inputs of the algorithm andthe assumptions made and then presents a blo k diagram whi h details the di�erent tasksperformed by the simulator.Se tion (5.1.4) presents two kinds of validations of the proposed algorithm. First, thevalidity of the o ean surfa e sampling presented in se tion (5.1.1) is veri�ed. Then, thequality of the GNSS-R AD is on�rmed.5.1.1 How to simulate the �eld for large o ean surfa es?This se tion on entrates on the generation of sea surfa e. Before fo using on the synthesisof Arti� ial Data|meaning basi ally that we take into a ount the C/A ode|we presentin this se tion our approa h to generate o ean surfa es, when the re e ted �eld has to be omputed for a large area. This requires for the LEO s enario, a sampling of the area, thatwe validate with the notion of minimum representative pat h.Sea-surfa e elevations are generated as shown in se tion (2.1.3) of Chapter 2, and deter-mined by a Gaussian sea spe trum as proposed in [Elfouhaily et al.1997℄|see se tion (2.1.2).We re all that the inputs of the spe trum are the wind speed at 10 m above the surfa e U10(amplitude and dire tion) and the fet h. The spa e raft is moving over the surfa e at 7 km/s,say.The inputs of the implemented ode are the size of the sea pat h with the requiredresolution|a resolution of 20 m has been onsidered for this study, a ordingly to Paris-� analysis (see [RuÆni et al.2001b℄) on the resolution|and the time in rement.As mentioned above, it is not possible to simulate o ean surfa es over the total �rst hipzone in a LEO s enario. It represents the points on the surfa e whose delay with respe t tothe spe ular point is less or equal to one hip length � hip. For a nadir-looking re eiver at500 km and onsidering the C/A ode, this represents a disk of radius 17 km and hen e a85000�85000 matrix. To over ome this problem, it has been suggested in Paris-�, that thes attered �eld an be omputed only from a sample of \small" pat hes randomly distributedover the FCZ.Let us now fo us on the size of these \small" pat hes and introdu e the notion of minimumrepresentative pat h. Considering statisti al homogeneity of the sea-surfa e roughness over arelatively large area, we seek for a surfa e size over whi h the statisti al features of the re e ted�eld are ongruent to the average one oming from the whole zone de�ned by the WoodwardAmbiguity Fun tion (WAF). This would enable us to synthesize a big pat h with only severalsub-pat hes taken randomly, and thus ompute the re e ted �eld with a drasti ally redu edtime.For this purpose, we fo us now on the temporal spe trum of the re e ted �eld that ontainsmost of the information we need. Let FS be the �eld re e ted by the whole surfa e S and FAthe �eld re e ted by a sub-surfa e A of S. Their respe tive temporal Fourier transforms FS

5.1. MODEL 1: GRADAS 75and FA an be des ribed in the dis rete domain as:FS = 1T T�1Xj=0 e(�ikj 2�T )0�NS�1Xp=0 I(p; j)1A ; (5.1)FA = 1T T�1Xj=0 e(�ikj 2�T )0�NA�1Xp=0 I(p; j)1A ; (5.2)where I(p; j) is the usual integrand presented below in equation 5.6. The index p indi atesthe summation over the surfa e and j the summation over time. NS (respe tively NA) is thenumber of dis rete elementary s atterers that ompose the surfa e S (respe tively A). Understationary and homogeneous statisti al assumptions, the spe trum estimation gives:jFS j = sSA jFAj: (5.3)A

A

A

A

Decorrelated

1

2

3

f

SFigure 5.1: S hemati drawing of the minimum representative pat h.If we onsider now di�erent sub-surfa esAf of S (see �gure 5.1) and making the assumptionthat the heights distribution is de orrelated from a sub-surfa e to another, the temporalspe trum of the �eld re e ted by the surfa e S an be estimated by the in oherent sum of theFourier transforms of the �elds re e ted by the sub-pat hes Af .Simulations have been arried out to validate the spe trum estimation presented in equa-tion 5.3. We omputed the spe trum of the re e ted �eld for a 2.5 se ond period (with 2.5 msof time in rement) over a 200�200 m2 pat h (with 20 m of spatial resolution). The re eiveris very high (100 km) at nadir. Figure 5.2 shows the spe trum estimation omputed fromtwo sub-pat hes only, with their enters at 50 meters of the big pat h, symmetri ally lo atedrelative to it. We studied the normalized spe trum|see equation 5.3|for three di�erent sizesof the sub-pat hes: 12.8 m, 25.6 m and 51.2 m side long. The wind speed is 4 m/s, so thatthe length of the sub-pat hes is mu h longer than the orrelation length of the sea (the windsea peak is orresponding to a 7 m wavelength). We see that the estimation is quite good forthe 25.6 m and 51.2 m ases, and a bit worse for the 12.8 m ase. From this analysis, we an on lude that the sea surfa e re e ted �eld is suÆ iently homogeneous over the whole areato onsider the existen e of a minimum representative pat h that des ribes the total averagere e ted �eld quite well. As anti ipated, we also tested the spe trum estimation onsidering4 sub-pat hes (ea h 25.6 m side) with very good su ess.

76 CHAPTER 5. GNSS-R SIMULATION TOOLKIT

Figure 5.2: Spe trum estimation of the true temporal spe trum of the re e ted �eld over a200�200 m2 surfa e (thi k line) with 2 sub-pat hes only: 12.8 m side (dotted line), 25.6 mside (dashed line), 51.2 m side (dashed dot dot line).A demo rati sample of the �rst hip zone seems to be a good way to ompute the re e ted�eld.5.1.2 GNSS dire t and re e ted signalsDire t signalAs des ribed in Chapter 1, the dire t GNSS signal is a spheri al wave, modulated in phaseby a PRN ode, that propagates with a arrier frequen y f . It an be onsidered as quasi-mono hromati , be ause the ode bandwidth is mu h less than the arrier frequen y.The dire t signal after down- onversion at the re eiver, lo ated at ~Rr, an be written as:Ud( ~Rr; t) = Code(t)eikRdRd e�i2�fIF t; (5.4)where k is the L1/L2 wavenumber, Rd is the distan e between the transmitter and the re eiver,and fIF=308 kHz is the arrier frequen y in base-band (after down- onversion). We onsiderhere Code(t) as the C/A ode.Re e ted signalThe GNSS re e ted signal is modi�ed by the intera tion with the o ean surfa e. Note thatsu h pro ess has been analyzed and simulated in the s ope of [PIPAER2000℄ and PARIS-�=�ESA proje ts. We now omplete the modeling by introdu ing the ode itself and a large spa eintegration to ompute the re e ted �eld.Let us �rst introdu e some omments about signal Doppler frequen y ontent. We an onsider the surfa e frozen during 1 ms a ording to the dynami s of the sea elevations fornormal onditions. However, a sea-surfa e sampling of 10 kHz has been hosen to apture theDoppler e�e t introdu ed by the re eiver's velo ity (7 km/s). Let us remind that in the aseof the re e ted signal, another Doppler frequen y is added. An elementary s atterer in ~r attime t, indu es a Doppler frequen y:fD(~r; t) = [~vt �~i(~r; t)� ~vr � ~s(~r; t)℄=� + [~q(~r; t) � ~v(~r; t)℄=2�; (5.5)

5.1. MODEL 1: GRADAS 77where ~i is the unit ve tor of the in ident wave, ~s is the unit ve tor of the s attered wave, ~vtand ~vr are the velo ities of the transmitter and the re eiver respe tively, ~r is the position ofthe s atterer, ~q = k(~s�~i) is the s attering ve tor (see �gure 3.1 in se tion (3.1.1) of Chapter 3for ve tor de�nitions) and ~v the sea-surfa e velo ity.If we onsider the �rst hip zone at nadir, the Doppler bandwidth introdu ed by a satellitemoving at 7 km/s at 500 km altitude is 1.2 kHz (without onsidering transmitter's Doppler).Considering that the maximal orbital velo ity for a wave is 10 m/s, this implies a sea-surfa eindu ed Doppler of 100 Hz, whi h is mu h less than the one introdu ed by the satellite motion.Let us now onsider I(~r; t) the Fresnel integrand:I(~r; t) = eik(r+s)rs (~q � ~n) ; (5.6)where r and s are the distan es transmitter-s atterer and s atterer-re eiver respe tively. Whenintegrating over the spa e and onsidering the arrier already as the baseband pulsation !IF ,we obtain: Ur( ~Rr; t) = Z Code[t� �r(~r; t)℄I(~r; t)ei!IF tdS; (5.7)with �r(~r; t) the time delay between the re eiver and a s atterer at ~r at time t.It is ne essary to sample the ode at high rate. If sampled at the hip rate for instan e, the ode will have a �xed value for any s atterer of the �rst hip zone, be ause the range �r(~r)will be equal or less than the length of one hip � hip (293 m for C/A ode), by de�nitionof the FCZ! It is important to noti e at this point that at a ode sampling rate of 20.456MHz, the sea elevations do not impa t mu h the lo ation of a ode transition within the �rst hip zone. Indeed, the waves will introdu e variations of �r of the order of meters, whereasthe range resolution is about 15 m at 20.456 MHz. Therefore the delay �r will be omputedignoring the sea elevations, that is for a at surfa e. As a result, the delay �r be omes timeindependent.Finally, we have: Ur( ~Rr; t) = ei!IF t � Z Code[t� �r(~r)℄I(~r; t)dS: (5.8)5.1.3 GRADAS algorithmThe GRADAS algorithm implements equation 5.8 with IDL routines, taking into a ount thepat h sampling of the �rst hip zone and the high sampled C/A ode.Inputs/outputsThe main assumptions are summarized in �gure 5.3. There are two kinds of inputs:� the o ean surfa e features,� and the re eiver's position and velo ity.The geometry of this s enario is presented in �gure 5.4. The re eiver is looking at nadirand moving at the speed ~vr at a �xed altitude. The transmitter is �xed and emitting aspheri al wave. Sin e its altitude is very large, we onsider the in oming wave as plane.The output of this simulator is simply a bit data stream. We generate 0.5 s of data at20.456 MHz. This data set an be read by the Starlab in-house Starlight pro essor. Resultsof the pro essing of the data are presented in the following se tion.

78 CHAPTER 5. GNSS-R SIMULATION TOOLKIT

GRADAS ALGORITHM

Nadir case

First Chip Zone

Fixed transmitter

Incoming plane wave

AD bitstream

(0.5s)

OCEAN

RECEIVER

10 kHz sampling

INPUTS

5000 patches (25x25m,20cm resol.)

ASSUMPTIONS

OUTPUTS

PRE−PROCESSOR

U10

Height=500km

IF=308kHz

Vr=7km/sFigure 5.3: Inputs and outputs of the GRADAS simulator.

OCEAN SURFACE

1st chip zone

TRANSMITTER

RECEIVER

y

x

vr

Figure 5.4: LEO geometry for the GRADAS simulator.

5.1. MODEL 1: GRADAS 79From the Fresnel �eld to Arti� ial DataThe GRADAS algorithm is des ribed in the blo k diagram of �gure 5.5. After generating ademo rati sampling of N pat hes of the FCZ, the routine is based on two main loops. The�rst one onsiders one pat h after the other. The se ond one is over time. The sea elevationsare generated with di�erent random phases from one pat h to the other.Let us onsider one pat h randomly lo ated in the FCZ. As already mentioned, the sam-pling rate of the re e ted �eld is 10 kHz to apture the Doppler frequen y introdu ed by there eiver speed. Note that the �elds are omputed at both L1 and L2 frequen ies. The C/A ode sampled at 20.456 MHz is then inserted onto the time series of the Fresnel �eld for thatpat h. It is important to stress out at this point, that this insertion depends on the positionof the pat h. As we onsider for now the �rst hip zone, it is easy to see that, at a giventime, the ode an have a maximum of two values within the orresponding area. The 20.456MHz sampling rate of the C/A ode orresponds approximatively to 20 samples per hip (293m)|i.e., 15 m spatial resolution. That means that the �rst hip zone an be divided into 20zones, as shown in �gure 5.6. Let us imagine now that a ode transition in the GNSS signalis hitting the surfa e: the zone 1 in �gure 5.6 is rea hed �rst, be ause its distan e to thetransmitter is the smallest one. Then the transition is displa ed along the FCZ. For instan e,a pat h in the zone 2 will re eive a transition in the ode before a pat h lo ated in the zone3 and farther. This is a key point to insert the ode into the re e ted �eld.GRADAS is the sum over all the pat hes of the modulated ele tromagneti �elds. Thebase-band frequen y is then added to it. The amount of noise to add is determined by themission s enario, i.e., by the re eiver antenna gain whi h drives the expe ted SNR of thesignal. A ording to [Lowe1999℄, a spa e-borne s enario with a 25dB gain antenna wouldprodu e a single-sample voltage SNR of 0:0651. Finally, the data stream is the in-phase omponent of this �eld, sampled at one bit.It is important to stress out at this stage the introdu tion of the indu ed weighting fun tionon the ba ks attered �eld due to the GPS re eiver pro essing|the WAF fun tion des ribedin [Zavorotny and Voronovi h2000℄. This fun tion sele ts and weights the �eld in a given areaof the FCZ and weights the ontribution of ea h s atterer. The part of the o ean involved inthe re e tion pro ess is determined by the integration time used for orrelations. Therefore,the demo rati sampling an be redu ed to this area, whi h learly saves some omputationtime.The omputational time is of the order of 4 minutes to ompute 0.5 se ond of the re e ted�eld from one pat h. We used 6 omputers working in parallel to ompute the time series ofthe �elds. Then a routine is dedi ated to gather all these data and sum them, as shown inthe bottom of the blo k diagram of �gure 5.5.5.1.4 ValidationThe pro ess of validation for the output of this study, as learly omes out from the previousse tions, an be thought as a two step pro ess. The problem of generating an ele tromagneti �eld, ba ks attered by huge surfa es, was solved with the on ept of demo rati pat hing. A�rst step of validation is therefore needed at this level, in order to verify the validity of su happroa h. Further, the re e ted �eld is modi�ed so that it simulates the GNSS ba ks atteredsignal. Essentially, the presen e of a ode that modulates the arrier �eld must be a ountedfor. The way this modi� ation is arried out must also undergo a validation pro ess. Thesetwo kinds of validations are des ribed in the next se tions.

80 CHAPTER 5. GNSS-R SIMULATION TOOLKIT

Fresnel field computation: F(k,t)

~"Insert" code in F(k,t) acccording to position k: F(k,t)

Read C/A code

Sea surface generation at patch k

t=t+dt

k=k+1

GRADAS = kΣ F(k,t)

~

Democratic sampling of WAF zone

N patches (25x25m)

k=1patch

t=totime

eiw t

IF

GRADAS_I = Re[GRADAS]

NOISE

Sampling=20.456MHz

Sampling=10kHz

1 bit samplingFigure 5.5: GRADAS algorithm.

5.1. MODEL 1: GRADAS 81.

.

..

..

4

1

3

2

20

Figure 5.6: First hip zone: the 20.456 MHz sampling rate of the C/A ode orrespondsapproximatively to 20 samples ea h hip. The �rst hip zone is divided into 20 rings. We ansee an example of a pat h (25�25 m2) lo ated in zone 4.Re e ted �eld validationFirst, this se tion aims at giving a lear pi ture of the re e ted �eld without any C/A odemodulation. This �eld is the one ompared to the �eld retrieved by the pre-pro essor asdes ribed below. We would like to understand if the �eld generated by sampling the FCZ hasa physi al meaning. What should be the spe trum of the re e ted �eld?We present in �gure 5.7 the phase, the amplitude and the power spe trum of the re e ted�eld from 240 pat hes (6�400 m2) demo rati ally lo ated in the 200 Hz Doppler zone ofthe FCZ. We see learly, in this 0.5 se ond evolution, the large Doppler bandwidth, whi h istheoreti ally equal to 400 Hz for this Doppler zone. Therefore, the phase behavior for this 10m/s simulation behaves a ordingly to our expe tations.As observed, the spe trum is not ontinuous and peaks spa ed by a �xed frequen y of 273.4Hz are sub-sampling the Fourier transform. These peaks ome from the fa t that the signalis highly periodi . Let's remind that the re eiver motion is simulated by adding a phase termwhen generating the sea elevations. This tri k has the pe uliarity to \shift" the surfa e in su hway that after a ertain time|fun tion of the re eiver speed|the surfa e omes ba k to itsinitial position. This repetition time is 3.6 ms for a re eiver speed of 7 km/s, orrespondingto the 273.4 Hz spa ing frequen y. That means that it takes approximately 3.6 ms for as atterer to enter a 25�25 m2 pat h and to leave it. During that time, the sea waves haven'tpropagated very mu h, whi h gives the periodi ity in the �eld.To fa e this issue, we generated re tangular pat hes, so that the waves propagate in alonger range in order to in rease the repetition time. Typi ally, we used 6�400 m2 sidepat hes, giving a 17 Hz frequen y spa ing in the Fourier domain.

82 CHAPTER 5. GNSS-R SIMULATION TOOLKIT

Figure 5.7: From left to right are plotted the phase, the amplitude and the power spe trum ofthe re e ted �eld from 240 pat hes (6�400 m2) demo rati ally lo ated in the 200Hz Dopplerzone of the �rst hip zone. The re eiver moves at 7 km/s along the x-axis, its altitude is 500km. The sampling rate is 10 kHz, 0.5 se ond evolution, U10=10 m/s.Pre-pro essing of GRADASThe best way to validate the simulated AD is to pro ess them with an open loop GPS re eiver.This open loop re eiver behaves like a normal GPS re eiver, using the orrelation betweenthe re eived (simulated, in our ase) signal and a repli a of the C/A ode under investigation.The integration time used to ompute the orrelations is 20 millise onds. The C/A ode usedin these simulations is the PRN number 9.Figure 5.8 represents a situation with a wind speed U10=10 m/s. On the left-hand side,the red urve represents the phase of the �eld before adding the C/A ode and after 20 ms of oherent integration. The bla k urve represents the phase of the �eld as tra ked by the openloop GPS re eiver, with the same integration pro ess to be onsistent when omparing thetwo phases. The tra ked phase is obviously very similar to the original one. Clearly in thisexample, the phase is lost at some points of the urve, due to some fading as shown in theright-hand side. The rosses represent the phase di�eren e between the red and bla k urves.As observed, it is mainly onstant ex ept at the points where the amplitude is low (fadingso urring in the green urve).5.2 Model 2: Intermittent S reamersThe foundation of this model is a physi al understanding of the dominant me hanism in thenear spe ular forward s attering, namely, quasi-spe ular s attering. Is has been alled Inter-mittent S reamers (IS), sin e ea h spe ular point is onsidered as sour e of a time-limitedindependent tone. This is the result of the observation that GO-like s attering approxi-mation holds for near spe ular L-band bistati measurements, as on�rmed in the study ofse tion (3.2) of Chapter 3. As expe ted, the power return in this regime, as demonstrated byprevious work during the OPPSCAT proje t, is dominated by the sea-surfa e slope statisti s.While we assume here that a GO model is appropriate, we take into a ount the �nite sizeof the spe ular lobe (in GO this lobe has zero width). The goal here is to produ e a signalresembling losely the signals to be re eived from spa e within the �rst hip zone, but withoutthe omputational e�ort required for GRADAS.

5.2. MODEL 2: INTERMITTENT SCREAMERS 83

0 50 100 150 200 250 300 350 400 450 500−1.5

−1

−0.5

0

0.5

1g2_uphasemy_uphase

0 50 100 150 200 250 300 350 400 450 500−1

0

1

2phase difference

0 50 100 150 200 250 300 350 400 450 500−40

−30

−20

−10SNR

V

Figure 5.8: The lower line in the left-hand �gure, represents the phase of the ele tromagneti �eld generated with the demo rati pat hing of the o ean surfa e, before C/A ode modu-lation. In this example, 240 pat hes lo ated in the 200Hz Doppler zone are onsidered, there eiver is moving at 7 km/s, the wind speed is U10=10 m/s. The red urve is the phaseretrieved with the GPS open loop pro essor. On the right-hand side, the retrieved amplitudeis shown with the di�eren e in phase ( rosses) between the unwrapped phase. The relationbetween fadings and the lose of the phase is learly observed.The following are our assumptions:1. The power is dominated by spe ular re e tions.2. The power from ea h s attered is hara terized by the s atterer lifetime, radius of ur-vature, delay, Doppler and a random phase.3. Under a Gaussian assumption, all ne essary parameters an be dedu ed from an o eansurfa e elevation spe trum.We re all that under GO assumption the omplex re e ted �eld from one s atterer writes:U0(p) = Rspe 2 e�ik(rspe +sspe )rspe sspe prxry: (5.9)At this stage we see all the relevant features of the spe ular �eld: a random phase relatedto the Signi� ant Wave Height, and a magnitude related to the e�e tive radius of urvature.In order to produ e a sensible simulation we have to be able to generate random draws ofthese quantities based on an o ean model.5.2.1 Spe ular point spe i� ationsWe emphasize here the derivation of spe ular point statisti s under the Gaussian assumption,as introdu ed in se tion (2.2) of Chapter 2. Our attention is fo used on the �rst hip fromspa e, therefore the mean square slope will only enter in our simulation through the spe ularpoint density.We remind that in [Barri k1968℄ a previous result by Kodis ([Kodis1966℄) is extended tothe bistati ase with �nite ondu tivity. It is shown that taking ensemble averages of thesurfa e statisti s before or after omputation of the Kir hho� integral in the stationary phase

84 CHAPTER 5. GNSS-R SIMULATION TOOLKITapproximation integral is equivalent. Starting from equation 3.26 of Chapter 3, the radar ross se tion is expressed in the following|very intuitive|manner:�o = �n � hjrxryji � jR(�l)j2 (5.10)= � se 4 � � Ps(sxspe ; syspe ) � jR(�l)j2:We re all that n is the average number of spe ular points per unit area, hjrxryji is the averageabsolute value of the produ t of the prin ipal radii of urvature at spe ular point (i.e., thee�e tive area of ea h s atterer), and �l is the lo al angle of in iden e at spe ular points1.The following onditions for validity are stated: i) the radius of urvature everywhere on thesurfa e is mu h greater than the wavelength so that the tangent plane approximation an beapplied, ii) multiple s attering is negle ted and iii) k2�2� os �2 >> 1, so that the surfa e anbe onsidered rough and power an be summed in oherently.If Gaussian statisti s are assumed the result is2:�o = se 4 �tan2 �0 e� tan2 �tan2 �0 jR(� � �)j2: (5.11)We an translate this here to mean that at the spe ular (�=0):n � hjrxryji = 1tan2 �0 = 12�2s : (5.12)Re alling Barri k, at �=0, assuming a Gaussian orrelation fun tion with orrelationlength l, we have for an isotropi surfa e:n = 7:255�2l2 ; (5.13)and hjrxryji = 0:1378� l2�2�2s : (5.14)For a dire tional sea, we follow re ent developments by [Gardashov2000℄ helping to sim-plify a seminal analysis from [Longuet-Higgins1958℄. Under Gaussian statisti s, the ne essarystatisti al hara teristi s asso iated with the joint statisti al distribution of the mean and dif-ferential urvature of the surfa e an be simply derived from the 2nd and 4th order sea-surfa eelevation spe tral moments, de�ned as:mpq = Z Z kpxkqyS(kx; ky)dkxdky: (5.15)For our appli ation, the average number of fa ets entering a tangent plan approximation willbe evaluated by onsidering a �ltered o ean surfa e. To a good approximation for our purpose,this implies the above expe tation to be taken up to a given uto� wavenumber. Numeri ally,taking the uto� to orrespond to 60 m wavelength (3 times the L-band wavelength), theaverage number of fa ets and mean radius of urvature an be numeri ally evaluated.Under low wind onditions 5 m/s, we �nd n ' 1:6 m�2 and hjrxryji ' 16 m2. Under higherwind onditions 10 m/s, the average urvature remains almost the same and the averagenumber of fa ets slightly de reases to give n ' 1 m�2. For omparison, expe tations taken upto a uto� wavenumber orresponding to 5 m wavelength give n ' 106 m�2 and hjrxryji '0:08 m2 for the 10 m/s ase.1We have: os 2�l = ni � ns; also �l = �i��, where � is the lo al fa et in lination angle|the spe ular angle.2We re all that tan �0 = 2��=l and that the mean square slope is also related to this quantity, �s = p2��=l.

5.2. MODEL 2: INTERMITTENT SCREAMERS 855.2.2 Code implementationAn IDL ode has been developed in order to use the above presented GNSS signal o eanre e tion model. This relatively simple ode is stru tured as des ribed in �gure 5.9. The

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Figure 5.9: Flow hart of the ode implementing the IS-model.input parameters an be divided basi ally into two di�erent ategories:Signal related inputs Sea-surfa e related inputsGPS signal frequen ies grid dimensionsGPS C/A ode grid resolutionLEO re eiver satellite height wind speedre eiver antenna gain SWHsimulated signal time duration number of s reamerss reamers lifetimesurfa e mean square slopeEssentially, the ode generates the grid on the sea surfa e and throw randomly on this gridthe predetermined number of s reamers, a ording to the hara teristi s spe i�ed. At ea htime instant (the sampling frequen y is 20:456 MHz), the ontribution of all the s reamersare summed together. The time series of su h values is the time series of the simulatedele tromagneti �eld ba ks attered from the sea surfa e. This �eld undergoes two di�erentpro esses.

86 CHAPTER 5. GNSS-R SIMULATION TOOLKITOne pro ess onsists in adding to this �eld the e�e t of the GPS C/A ode and the noise.The amount of noise to add is determined by the mission s enario, i.e., by the re eiver antennagain whi h drives the expe ted SNR of the signal. Similarly to the GRADAS simulator, we onsider a single-sample voltage SNR of 0:0651. This is the value taken into onsideration inour simulations. S aling properly this value it is easily possible to simulate di�erent s enarios.Moreover, the simulated ele tromagneti �eld is also spatially weighted by a triangularfun tion (in the domain of time-delay), whi h emulate the e�e t of the GPS re eiver orrelationpro ess, i.e., the delay-depending fa tor of the WAF. This is done in order to have a �eld whosephase is omparable to the one retrieved with the GPS pro essor.The IS simulator has been implemented also in C++ language to improve its speed. The urrent version is able to generate one millise ond of GNSS-R data in 15:2 �N millise onds,where N is the number of independent s reamers onsidered on the sea surfa e.5.2.3 Simulated data and analysisThe GNSS-R syntheti data are fed into an open-loop GPS re eiver to tra k the arrierphase. An example of orrelation peak is shown in �gure 5.10. The red peak is the (s aled)auto orrelation of the C/A ode onsidered. The blue one is the orrelation of the learC/A ode with the simulated re e ted signal, i.e., the waveform. The result of this openloop pro essing is the series of the arrier phase values, whi h is then ompared to the phaseextra ted from the partial WAF-modi�ed �eld (i.e., the simulated physi al �eld after spatial�ltering with the triangular fun tion).

−4 −2 0 2 4 6chipFigure 5.10: The waveform (blue urve) resulting from the orrelation between the simulatedo ean-re e ted GPS data and a lear repli a of the orresponding C/A ode. In red, forpurpose of omparison, the auto orrelation of the same ode. The asymmetry of the waveformre e t the extension, in hips units, of the sea surfa e taken into onsideration in the signalgeneration pro ess. The number of hips is the time-delay unit on the x-axis. On the y-axis,arbitrary units.Let us now onsider a omplete simulation pro ess, applied to a well de�ned s enario.We will onsider the PETREL (PARIS Explorer for Tra king and RE e tometry in L-band)

5.3. CONCLUSIONS 87s enario. This mission has been proposed to the European Spa e Agen y (ESA) in 2002 by a onsortium of European ompanies, led by Starlab.The key parameters of this mission to be onsidered in this ase are:re eiving LEO satellite altitude 500 kmre eiving antenna gain 25The other inputs to the simulation are:grid dimensions 200�200grid resolution 20 mwind speed 4 m/sSWH 31 mnumber of s reamers 3000s reamers mean lifetime 10 mssurfa e mean square slope 0.01simulated signal time duration 200 msThe simulated GNSS-R data have been pro essed using 1 ms of oheren e integration timewith a subsequent of in oherent integration of 10 ms. The result of the pro essing of thissimulated data are shown in �gures number 5.11 and 5.12. It is lear that the phase tra king

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Figure 5.11: The phases as retrieved by the open-loop GPS pro essor (in bla k) and theone from the simulated physi al �eld (in red), after the spatial �ltering with the triangularfun tion. A y le slip in the tra ker an be observed.in this ondition has been su essful.5.3 Con lusionsGRADAS and IS are two related approa hes for simulation of re e ted data, and we arenow able to generate realisti arti� ial data of GNSS re e ted signal at 20.456 MHz as willbe gathered from a spa e-borne re eiver. We have used the data to begin optimization of aGNSS-R pro essor, with the aim of demonstrating, in-sili o, ode and phase tra king. After

88 CHAPTER 5. GNSS-R SIMULATION TOOLKIT

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Figure 5.12: The blue urve represents the di�eren e between the retrieved phase and the oneof the ele tromagneti �eld. The green urve represents the �eld amplitude, in dBv. Clearly,when a fading o urs, the tra king of the phase an fail. In normal onditions the phase iswell re onstru ted.pro essing of GRADAS data, the phase of the re e ted signal has been orre tly tra ked.That is, the virtual experimental loop is losed: generation of a physi al �eld, simulation ofre eption, sampling, and post-pro essing to return to the original input. This, at the veryleast, guarantees onsisten y of the approa h.An important aspe t of this work has been a thorough understanding of the GNSS signalstru ture and of the me hanism of re e tion from the sea surfa e. This knowledge allowsmodeling of the involved phenomena to a degree of reality that is enough for produ ingrealisti arti� ial data but, at the same time, not too detailed to result in an unbearable omputational burden.Of ourse, the �nal proof of the on ept will ome from a real spa e-borne experiment.

Chapter 6An Appli ation for Sea StateMonitoringThis Chapter provides an appli ation of the GNSS-R simulator developed in Chapter 5.Monte-Carlo simulations have been undertaken in order to infer an empiri al inversion lawrelating sea-surfa e parameters with statisti s of the omplex re e ted ele tromagneti �eld.These simulations onsider GNSS re e tions gathered from a oastal platform.As observed in these simulations, it appears that the re e ted �eld is quite sensitive tosea-surfa e dynami s, whi h are modeled by a wind-driven o ean. We see that the dynami sof the variations in phase and amplitude orrelate linearly with the sea-state. It is then shownthat looking at the statisti s of the derivative of the entire �eld improves the inversion law.6.1 Simulation spe i� ationsWe present simulations of GNSS-R from a oastal platform using the GRADAS simulator|seese tion (5.1). The in oming signal is assumed to be a plane wave boun ing o� a wind-drivensea surfa e, whi h is modeled by a 50�50 m2 pat h of Gaussian sea elevations. The surfa eresolution is set to 20 m. The re eiver is lo ated at 30 meters over one edge of the pat h,with the wind blowing in the dire tion of sight, away from it.One minute simulations have been undertaken with a time in rement of �t=2.5 ms fordi�erent wind speeds, ranging between 1 and 15 m/s. We have onsidered here the orre-sponding sea height RMS �� , or SWH (ranging from 0 to 90 m). We have fo used on thefuture L5 GNSS frequen y, whi h synthesizes a 25.5 m wavelength. The re e ted �eld, om-puted with the Fresnel integral, an be des ribed as a omplex phasor whose amplitude andphase are fun tion of time: F (t) = r(t)ei�(t): (6.1)As seen from a stati platform, the �eld s attering from a frozen o ean ould be representedas a stati omplex phasor. The motion of the o ean surfa e thus translates into motion ofthe phasor in the omplex plane. We an see su h a phasor representation of the re e tedele tromagneti �eld in �gure 6.1. The underlying idea is that the analysis of the dynami sof the re e ted phasor provides the key to estimating sea-surfa e parameters from su h stati platforms.Let us onsider for instan e the �eld at time t1: r(t1)ei�(t1) and at t2 = t1+�t: r(t2)ei�(t2).The goal of this study is to understand how high an be the variations in phase from �(t1)to �(t2) and the variations in amplitude from r(t1) to r(t2) as a fun tion of the sea-surfa e89

90 CHAPTER 6. AN APPLICATION FOR SEA STATE MONITORING

Figure 6.1: The dynami L-band ele tromagneti �eld in a omplex phasor representationprodu ed by a wind-driven o ean model.elevation RMS. Intuitively, these variations are quite small for a alm sea and in rease aswaves be ome big.Therefore, a fo us on the derivatives of both phase and amplitude has been undertaken,respe tively Æ� and Ær with varian es �2Æ� and �2Ær. Then, the derivative of the �eld is onsid-ered.6.2 Results6.2.1 Phase analysisFigure 6.2 presents the standard deviation �Æ� of the derivative of the phase in Hz ( y les/�t)as a fun tion of sea elevation RMS �� . The jitter in phase is obviously in reasing quite linearlywith SWH. It spans between 2 Hz and 5.5 Hz when SWH ranges from 0 to 90 m. However,it appears that the phase variations are not the best way to infer sea-surfa e features. Indeedhigh variations in phase o urring when the �eld magnitude is small has not the same meaningas big ex ursions in phase o uring on a �eld of high magnitude. In other words, the phasehas less physi al ontent when the magnitude of the �eld be omes low. Besides, as observedin our simulated time series, the tail of the distribution of Æ� is quite high: big y le slips mayappear from time to time. The un ertainty in the estimation of �Æ� is thus quite high.6.2.2 Amplitude analysisA �rst analysis has been made on the standard deviation �r of the amplitude r as a fun tionof SWH. As observed on �gure 6.3, the varian e in reases quite rapidly with �� and thensaturates. This saturation is rea hed when �� is around 6.4 m (dot dashed line in the �gure)whi h orresponds to a Signi� ant Wave Height of 25.6 m. This value is similar to thein ident wavelength L5. This would mean that the amplitude of the �eld stabilizes when theheight of the waves be omes greater than the in ident wavelength.A se ond analysis fo used on the standard deviation �Ær of the time derivative of theamplitude of the re e ted �eld, as shown on �gure 6.4. The derivative of the amplitude has

6.2. RESULTS 91

Figure 6.2: Standard deviation of the derivative of the re e ted phase (��) as a fun tion ofsea elevation RMS �� . L5, 1 minute simulation.

Figure 6.3: Standard deviation of the re e ted amplitude (�r) as a fun tion of sea elevationRMS �� . L5, 1 minute simulation.

92 CHAPTER 6. AN APPLICATION FOR SEA STATE MONITORING

Figure 6.4: Standard deviation of the derivative of the amplitude of the re e ted �eld (�Ær)as a fun tion of sea elevation RMS �� . L5, 1 minute simulation, with 4� error bars.been normalized as follow: Ær = H ÆÆt(jF � F j); (6.2)where F is the mean of the re e ted �eld, and H the re eiver height.As observed, two linear regimes an also be distinguished around the riti al value SWH=25.6 m. First, the parameter �Ær in reases rapidly with �� , whereas the in rease be omes a bitslowly in the se ond regime. On the ontrary to the �Æ� parameter, �Ær does not saturatewith SWH.Assuming that the derivative of the amplitude follows a Gaussian distribution, the stan-dard deviation of the estimation �Ær is �Ær=p2N , with N the number of independent measure-ments. The 1 minute errors bars are plotted on the graph (4� is represented). As observed,the regression line of the se ond regime has a slope of 32.51. Then, knowing that one minutesimulation gives a maximum in ertainty of 0.83 in the estimation of �Ær, we on lude a max-imum in ertainty in the SWH retrieval of 4� 0:83=32:51=10.2 m. Naturally, this �gure willde rease onsiderably for longer simulations. For instan e, this maximum un ertainty wouldde rease to 3.2 m after 10 minutes of simulation.6.2.3 Field derivative analysisLet us onsider now the derivative of the re e ted �eld, or speed ve tor:ÆF = (Ær + irÆ�)ei�: (6.3)Taking the absolute value of this omplex �eld, we obtain:jÆF j = qÆr2 + r2Æ�2: (6.4)Intuitively, this parameter depends on the orbital velo ities of the sea elevations. It ontainsboth phase and amplitude variations and does not present su h rare intense magnitudes, asobserved in the phase or amplitude alone. Therefore the un ertainty in SWH retrieval is

6.3. CONCLUSIONS 93improved. Figure 6.5 presents �jÆF j as a fun tion of �� . On the left hand side, 1 minuteestimations are plotted, together with the linear regression line. On the right hand side, 10se ond estimations are added (5 points for ea h value of ��). As observed, the parameter �jÆF jdoes not saturate with sea-state and follows a linear trend.

Figure 6.5: Left: Standard deviation �jÆF j of the amplitude of derivative of the re e ted �eldas a fun tion of sea elevation RMS �� . L5, 1 minute simulation. Right: Same plot with 10 s�jÆF j estimation (squares).6.3 Con lusionsWe reported oastal simulations as an attempt to derive empiri al laws for sea-state estima-tion. The GRADAS simulator has been used under a oastal s enario. We have analyzedphase, amplitude and derivative of the re e ted �eld as a fun tion of �� in order to deriveinversion urves.Under the assumption of a wind-driven o ean, several on lusions on SWH/wind speedinversion using the re e ted ele tromagneti �eld an be made:� The analysis of the derivative of the phase provides an inversion law through the sea-elevation RMS parameter.� The varian e of the amplitude saturates rapidly with SWH. However, an analysis of itsderivative showed that the �Ær parameter does not saturate with sea-state and presentstwo regimes whose separation is determined by the in oming wavelength magnitude. Inboth regimes, a linear inversion urve is empiri ally obtained. Under the assumption thatthis parameter has a Gaussian distribution, we found a maximum in ertainty in SWHof 10.2 m after one minute simulation. However, some rare events of high magnitudemay a�e t onsiderabely the quality of the estimations and longer times series wouldallow a better pre ision.� The derivative of the �eld, or the so- alled speed ve tor, is a more appropriate parameterto derive SWH. The u tuations �jÆF j versus SWH follow a linear trend for the wholerange of sea-states expe ted in a oastal area.

94 CHAPTER 6. AN APPLICATION FOR SEA STATE MONITORING

Part IIIGNSS-R Airborne S atterometri Performan e Analysis

95

Chapter 7The Eddy Experiment FlightAs part of ESA proje t PARIS- (TRP ETP 137.A|Phase 1), an experimental ight was ondu ted by Starlab on September 27th 2002 along the Catalan oast in order to gathersea-re e ted GPS signals. This experiment was arried out in order to demonstrate thepossibility of GNSS-R mesos ale mapping of the o ean surfa e from an airborne platform. Thes ienti� /te hnologi al goal was �rst to demonstrate the apability of mapping a mesos alephenomenon su h as eddies, with de imetri altimetri signatures. We intentionally do notemphasize here the altimetri pro essing. Rather, the goal of this Chapter is to provide allinformation required to validate the s atterometri inversions.We present hereafter the experimental ampaign in luding ight plan, instrumentationand set-up. We report then in se tion (7.2) the available ground truth as a basis of thevalidation of DMSS estimations presented in Chapters number 8 (opti al) and 9 (L-band).7.1 Experimental ampaignThe experiment onsisted in gathering GNSS-R data along a 140 km long tra k over the seasurfa e. The air raft was a Partenavia P-68 Observer, provided by the Catalan Cartographi Institute (ICC). The ight altitude was 1000 m and the nominal airspeed and heading were50 m/s (180 km/h or 100 knots) and 30o from North, respe tively. The tilt of the plane wasrequired to be less than 15o in order to maintain the maximum number of olle ted PRNswithout dis ontinuities (whi h would endanger Kinemati solution) and avoid phase patterndisturban es.Figure 7.1 shows the map of the ight traje tory. The tra k of the plane has been hosento follow the as ending y le tra k 187 of Radar Altimeter (RA) Jason. It is almost parallelto the Catalan oast, but only lose enough in the latitude range from 41.5o to 42.5o North.During its way, the tra k rosses the Palamos Canyon, a strong bathymetri feature thata�e ts the surfa e topography.Three main ground stations were deployed at Llafran , Creus and at the Bar elona airport(see �gure 7.1). These three temporal GPS ground stations provided additional navigationinformation to re over the traje tory of the ight in the most a urate way. P1 and P2 aretwo points along the Jason's tra k. The P1-P2 distan e is roughly 140 km. The total ightduration was 3h 43 mn and the time of a quisition was 2h 54 mn.A GPS two-buoy system, gathering double frequen y 1 Hz observables was also deployedon the Jason's ground-tra k, under the airplane traje tory. Unfortunately, time series of thesea level ould not be obtained from the GPS buoy observables during the whole experiment,sin e the sea onditions were too strong for buoy deployment (see �gure 7.2). However, the

98 CHAPTER 7. THE EDDY EXPERIMENT FLIGHT

Figure 7.1: Map of the ight traje tory. The blue line shows the Jason's tra k. The pointsP1 and P2 mark the experiment boundaries, while the green points indi ate the positions ofs heduled GPS buoy measurements.

Figure 7.2: Views of the sea surfa e during the experiment from the air and sea.

7.1. EXPERIMENTAL CAMPAIGN 99�rst 30 minutes of measurement allowed to �nd a signi� ant wave height in the middle of thetra k of around 1.7 m.7.1.1 Instrumental set-upThe instrument on�guration is presented in �gure 7.3. Both dire t and re e ted L1 signalsare olle ted at high sampling rate. The re eivers shared the internal re eiver lo k.

Figure 7.3: Air raft set-up on�guration.The equipment used during this ESA experiment was lent by the European Spa e Resear hand Te hnology Centre (ESTEC). It is presented in �gure 7.4 and in ludes:� A LHCP antenna, Allan Osborne Asso iates (ESTEC/ESA). The antenna is lo ated atthe bottom of the plane. It is nadir looking for GNSS-R signal re eption.� A RHCP antenna, Allan Osborne Asso iates (ESTEC/ESA). The antenna is lo ated atthe top of the plane. It is zenith looking for GNSS dire t signal re eption.� Two Turbo Rogue re eivers (ESTEC/ESA). These ommer ial re eivers have been mod-i�ed to extra t the data from their IF at 20.4 Mbps. They still a t as ommer ial re- eivers obtaining the GNSS observables. Ea h of these two re eivers were onne ted toa GNSS antenna (RHCP, LHCP).� Two SONY re orders (ESTEC/ESA): These re orders an work at 20.456 Mbps usinga single hannel. Ea h of them olle ted data from one Front End of the Turbo Roguere eiver. Before oming into the re order, the signal passes through a Digital LevelConverter (SONY SBS-LC1) in TTL input mode.

100 CHAPTER 7. THE EDDY EXPERIMENT FLIGHTWe also used the Inertial Navigation System (INS) data provided by ICC. This provides therate of hange of the angular speed of the air raft in its three omponents (heading, pit hand roll). This data is needed to pre isely position the nadir looking antenna from the zenithlooking antenna traje tory.The PCS an software was used to download the data from tapes to hard disk, where theywere automati ally saved and named with the extension \.BST" (bit stream). Then, word-by-word byte swapping (used in BST �le format) was removed, and the data �les were namedwith the extension \.bit". At this stage, the �les ontain the re eived bits, ordered a ordingto their a tual time of re eption. Those BST �les are referred to as raw data. The volume ofthese �les, onsidering that the signal sampling frequen y is 20.456 MHz, is of around 3 GBfor 1 minute of data (dire t and re e ted).

Figure 7.4: ESA equipment used during the Eddy Experiment Flight.7.1.2 Satellite on�gurationThe analysis of the data has been arried out on three PRNs, namely PRN number 08, 24and 27. They are the three highest satellites in view during the ight. Their elevations andazimuths are given in �gure 7.5. The plots of �gure 7.6 show the delay-Doppler mapping atthe beginning and end of the tra k.7.2 Ground truthThe ground truth is provided by a numeri al model developed by the Spanish National Me-teorology Institute (INM), and by Jason's GDR data.

7.2. GROUND TRUTH 101

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Figure 7.6: Map of the one- hip iso-delays and iso-Dopplers (-100 Hz, 0 and +100 Hz) forPRNs 08 (blue), 10 (red) and 24 (green) at the beginning (left) and end (right) of the tra k.The -3dB antenna pattern is also represented in bla k.Wind/wave modelWind ve tor (speed and dire tion) provided by INM is shown in �gure 7.7(a) and SWH in�gure 7.7(b) at 12 UTC of the day of PARIS- Experiment. The experiment started ataround 10 UTC. Therefore, these graphs give an overall pi ture of the sea-state during theexperiment. The bla k straight line orresponds to the observation tra k: it starts at latitude41.32o North and ends at latitude 42.58o North, whi h �ts the top of the graph.The sea-surfa e roughness was quite strong during the experiment, whi h was prin ipallydue to a strong wind oming from the Golfe du Lion. As observed along the tra k, the winddire tion represented by the green arrows varies quite signi� antly from 30o North to -30oNorth approximately. The wind speed varies between 9 and 13 m/s approximately along the onsidered tra k. It in reases regularly from latitude 41.32o North to latitude 42.2o North,and then starts de reasing slowly.The SWH in reases from around 1.5 to 2 meters along the observation tra k. The greenarrows indi ate the presen e and dire tion of wind generated waves, meaning that a youngsea was observed that day along the tra k. Developed swell is observed prin ipally South to

102 CHAPTER 7. THE EDDY EXPERIMENT FLIGHT

(a) Wind Field (b) SWHFigure 7.7: Wind �eld and SWH at 12 UTC in the observation area (data from INM).Baleares islands.Jason's GDR dataThe Jason radar altimeter essentially provides the sea-surfa e height from the elapsed timeof the radar pulse, from the satellite to the surfa e and ba k. In addition to giving estimateof sea-surfa e height, it also gives signi� ant wave height from the slope of the leading edgeof the return waveform and wind speed over the o ean from the strength of the returnedba ks attered power. Jason's data are sampled at 1 Hz with an a ura y of 20 km. Figure7.8 shows the MSS in Ku- and C-bands and wind speed as obtained by Jason the day of theexperiment. The SWH is shown in �gure 7.9.The sea-surfa e slope varian e is omputed with the ba ks atter oeÆ ient �o, throughthe relationship: MSSJason = ��o ; (7.1)where � is a proportionality onstant; here it in ludes the Fresnel re e tion oeÆ ient, slopestatisti al des ription and possible radar alibration o�set. In absen e of the last two pie esof information � is set here to 0.64 for both wavelengths, the nominal Fresnel oeÆ ient value.The wind speed obtained from Jason's GDR data is shown on �gure 7.8(b). It is al ulatedthrough a relationship with �o in Ku-band and the signi� ant wave height using the algorithmdeveloped by [Gourrion et al.2002b℄. The wind speed model fun tion is evaluated at 10 metersabove the sea surfa e, and is onsidered to be a urate to 2 m/s. The wind dire tion is givenby an interpolation of the ECMWF model with an a ura y of about 20o.

7.2. GROUND TRUTH 103

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Figure 7.9: The SWH from RA Jason in Ku-band (blue urve) and C-band (red urve).

104 CHAPTER 7. THE EDDY EXPERIMENT FLIGHT

Chapter 8S atterometry with SunlightAs a omplementary remote-sensing instrument, high resolution opti al photographs of Sunglitter were taken during the ight, providing the SORES data (SOlar RE e tan e Spe u-lometer, a Starlab a ronym). The goal of this Chapter is to perform Least-Square inversionof these data to obtain opti al estimates of dire tional sea-surfa e mean square slope. Theforward model and inversion te hnique have been developed within the OPPSCAT II proje t(see [Soulat et al.2002b℄ and [Soulat et al.2003℄).The Chapter begins with the presentation of the data set in se tion (8.1). The inversionpro edure is then reported in se tion (8.2) in luding forward model, parameter sear h andresults. A spe tral analysis of the images is then undertaken in se tion (8.3) to enhan e thesea surfa e hara terization. We depi t in parti ular the wind indu ed long waves wavelength,dire tion and mean square slope. Furthermore, simulations highlight the impa t of wave heightand wind/wave misalignment on the three parameters of the DMSS. Finally in se tion (8.4),the results are ompared to Cox and Munk's measurements.8.1 The data setThe photographs were taken from time to time along the tra k when the roll, pit h and yawof the plane were low enough. The �rst photograph was taken at 10:00 UTC. The Jason passon the area of interest was observed at 9:33 UTC. Therefore, the SORES data were very well o-lo ated in spa e and time with Jason. The Sun elevation was approximately 45o duringthe whole experiment. The plane over- ew the altimeter tra k both upward and downward.The amera was a Lei a dedi ated to aerial photography lent by ICC. Figure 8.1 showsthe amera together with the GPS LHCP antenna, both lo ated at the bottom of the plane.An inertial system provides the position and time of ea h snapshot. Figure 8.2 presents thelo alization of the snapshots numbered from 35 to 60. The �lm was a pan hromati AviphotPan 80. The fo al length is 15.2 m, and the photographi plate has an area of 23�23 m2.The aperture angle is onsequently 74.2o. Given the altitude of the plane, the observed s eneis a square with area 1.124�1.124 km2. The exposure time is �xed at 1/380 and aperture atf/4.The argenti photographs have been s anned by ICC. The original images are omposedof 16128�16128 squared pixels whi h size is around 260 Mb. Then, the images have beenbi-linearly resampled to 400�400 squared pixels, in order to be easily pro essed. An exampleof SORES raw data is shown in top of �gure 8.3.Note that the roughness of the sea surfa e being quite strong during the experiment,breaking events and foam o ur on many pi tures (see bottom of �gure 8.3). They have not

106 CHAPTER 8. SCATTEROMETRY WITH SUNLIGHT

Figure 8.1: High resolution amera and GPS LHCP antenna lo ated at the bottom of theplane.

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8.2. INVERSION PROCEDURE 107been orre ted for the sea-surfa e slope varian e estimation presented in this study.8.2 Inversion pro edure8.2.1 The dire tional mean square slopeTo date, and as systemati ally referen ed when modeling the sea-surfa e slope statisti s,results derived from the glitter-pattern of re e ted sunlight as photographed by Cox andMunk in 1951 [Cox and Munk1954℄ remain the most reliable dire t measurements of wind-dependent slope statisti s. We do not propose here the slope distribution as in se tion (2.2.2)of Chapter 2, but onsider higher order terms. Based on Cox and Munk analysis, we �t thelogarithm of sea-surfa e slope probability density fun tion Ps:log(Ps) = ao � a0os2 + a00os4 + s(a1 � a01s2) os�0 (8.1)+ s2(a2 + a02s2) os 2�0 + a3s3 os 3�0 + a4s4 os 4�0:The slope s = tan� is de�ned as the tangent of the tilt of steepest as ent of ea h sea-surfa efa et, �0 = � � �o is the azimuthal angle of the fa et a ording to the prin ipal axis �o,whi h is the dire tion of highest MSS en ountered in the 2-D probability density fun tion.See �gure 8.4 for the de�nition of these two angles. The �t oeÆ ients amn have then beenrelated to the expansion oeÆ ients pertaining the umulants of the slope distribution undera Gram-Charlier approximation. Note that ao in equation 8.1 is an arbitrary o�set due to theinability to fully resolve the steepest angles.The model is simpli�ed in this analysis by removing the �rst and third order moments andthe dire tionality in the fourth moment. Only part of the glitter pattern ould be a tuallyobserved given the quite low Sun elevation. The skewness parameter would thus not beobservable. Equation 8.1 simpli�es to:log(Ps) = ao + [�a0o + a2 os(2(� � �o))℄s2 + a00os4: (8.2)The slope ve tor is de�ned in the frame (X,Y,Z) shown in �gure 8.4 with the omponents:( sx = sin� tan �sy = os� tan � (8.3)In the (sx,sy) frame, the sea-surfa e slope PDF writes:Ps(sx; sy) = exp[ao � (a0o + a2 os 2�o)s2x (8.4)�(a0o � a2 os 2�o)s2y+(2a2 sin 2�o)sxsy+a00o(2s2xs2y + s4x + s4y)℄:There are four parameters of interest and one s aling fa tor:� ao is the normalization parameter,� a0o is the inverse overall MSS,� a2 addresses the anisotropy of the slope PDF,� a00o is for the fourth order moment,

108 CHAPTER 8. SCATTEROMETRY WITH SUNLIGHT

Figure 8.3: Top: Example of SORES raw data (photograph 47). Bottom: Example ofbreaking event observed during the experiment.

8.2. INVERSION PROCEDURE 109� �o provides the dire tion of steepest en ountered MSS relative to the Sun azimuth. Thisdire tion is then orre ted and given with respe t to the North.Given the three mentioned parameters a0o, a2 and �o, we investigate in this se tion:� the up-wind MSS: �2u = 12(a0o�a2) , whi h is the higher MSS en ountered in the 2-DPDF,� the Slope PDF Isotropy: SPI = �2 �2u = a0o�a2a0o+a2 ,� the Slope PDF Azimuth (SPA): de�ned as the dire tion of the up-wind MSS, simplygiven by �o minus the Sun azimuth (the azimuth angles of the TAM are referen ed withrespe t to the Sun azimuth, as presented below). There is however an ambiguity of 180oin this dire tion. The dire tion of the up-wind MSS is expe ted to be pretty lose tothe wind dire tion.Note that the Total MSS is related to the up-wind and ross-wind MSS by:Total MSS = 2q�2 � �2u: (8.5)The term a00o gives information on the departure from the Gaussianity of the PDF. We willanalyze this fourth order moment through the parameter b00o (used to remain onsistent withCox and Munk's nomen lature): b00o = a00oa02o : (8.6)A ording to their results, this parameter is not negligible and almost wind independent. For lean sea surfa e measurements, the suggested onstant value is 0.04.8.2.2 Forward modelA short overview of the me hanisms involved in the relation between the sea-surfa e slopePDF and the photograph intensity are summarized in this se tion. We �rst present how thesea-surfa e slope distribution and sea-surfa e radian e are related. Then, from the instrumentpoint of view, we relate the sea-surfa e radian e to the pixel intensity. The geometry of theexperiment is given in �gure 8.4. This development is based on the gridding of the sea surfa ewith ells of required tilts � and azimuths �: the Tilt-Azimuth Map (TAM).Geometry of the experimentHereafter is presented the geometry of the sea surfa e re e tion in �gure 8.4, as onsidered byCox and Munk. The in ident ray is in the plane (Y;Z), makes an angle � with (X;Y ), andforms an image on the verti al photographi plate. The angle between the in ident ray andthe re e ted one is 2!. The points ABCD de�ne a horizontal plane through A and AB0C 0D0the plane tangent to the sea surfa e. The tilt � is measured in the dire tion AC of steepestdes ent, and this dire tion makes an angle � lo kwise from the Sun. OO0 is parallel to theZ-axis and the plane of the photographi plate is parallel to (X;Z). The unit ve tors n, iand r, respe tively normal to the surfa e, along the in ident ray and along the re e ted ray, an be expressed as: n = (� sin� sin�;� os� sin�; os �)i = (0;� os �;� sin�) (8.7)r = (� sin � sin�;� os� sin�; os�):

110 CHAPTER 8. SCATTEROMETRY WITH SUNLIGHTA ording to the law of re e tion, these ve tors follow:r � i = (2 os!)n: (8.8)We an dedu e: os� = 2 os � os! � sin�; (8.9)and tan ��1 = 1tan� � os�2 sin� sin� os! ; (8.10)with: os! = n:r = �n:i = os � sin�� os� sin� os�: (8.11)Z

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Figure 8.4: Geometry of SORES sensing during the Eddy Experiment Flight.The Tilt-Azimuth Map (TAM)The way to read these equations is the following. We an �rst onsider the in ident ray boun -ing everywhere o� the surfa e with the same angle �, sin e the Sun is at very high distan e.This angle an be known at the moment an observer takes a pi ture. From equation 8.11, !be omes a fun tion of � and �, and equations number 8.9 and 8.10 de�ne a simple system oftwo equations with two unknown variables. Thus, a relation between the dete ted points onthe photographi plate and the wave slopes an be established.Then, iso-tilts and iso-azimuths on the photograph are plotted on ea h pro essed pho-tograph. We all it the Tilt-Azimuth Map (TAM). An example of TAM is shown below in�gure 8.5(a). Note that this mapping makes a strong analogy with the Delay Doppler Mappresented in the s atterometri GNSS-R data. The original intensity of the photograph isthus re-organized from the metri spa e (X,Y ) to the slope angle spa e (�,�).

8.2. INVERSION PROCEDURE 111From sea-surfa e slope distribution Ps to sea-surfa e radian e NAs explained by Cox and Munk in their well-do umented report [Cox and Munk1956℄, wehave to take into a ount the toleran e of slopes for the o urren e of a highlight at a �xedpoint on the surfa e, as related to the �nite size of the Sun. The area of the slope toleran eellipse at the point (x0,y0) with slope hara terized by the tilt �0 and azimuth �0 is:�t = 14 �'2 os3 �0 os!0 ; (8.12)where ' is the angular radius of the Sun and !0 the half angle between the in ident andre e ted rays. The re e ted radiant intensity from one spe ular point is:J = �(!0) H�'2�h os!0 os�0 ; (8.13)where �h designates the area of a single spe ular point proje ted onto a horizontal plane, Hthe Sun irradian e and � the re e tion oeÆ ient.Let P(�,�) be the likelihood for a spe ular point to o ur in a ertain area (�,�) of slope.By de�nition, it orresponds to the integration of the slope probability Ps over the toleran eellipse: P = Ps(�; �)�t(�; �). The re e ted radiant intensity from the entire sea surfa e withradian e N is then: N os� = PJ = Ps�tJ: (8.14)The sea-surfa e slope probability expression derived by Cox and Munk is:Ps = 4� os4 �NH os�: (8.15)A ording to [Stegelmann and Garvey1973℄, this expression must be divided by the orre tionterm sin� os � os! : (8.16)This orre tion omes from the assumption that spe ular points are urved rather than at,as assumed by Cox and Munk. However, it has not a signi� ant impa t in the present on�guration. Strong dis repan ies appear at lower Sun elevations.From sea-surfa e radian e N to pixel intensity IThe solid angle subtended by the amera aperture of diameter d is:�d2 os�4r2 ; (8.17)where r is the distan e between the fo used area and the amera. The ux re eived into thephotosensitive amera system N�d2�(�) os2 �4r2 ; (8.18)introdu es the transmission �(�), whi h allows for the instrument's behavior with the angleof the in oming opti al ray. Considering the re e tions of the in oming rays with the outerand inner fa e of the amera length, �(�) is proportional to (1��)3. Note that the vignettinge�e t has not been taken into a ount in �(�).

112 CHAPTER 8. SCATTEROMETRY WITH SUNLIGHTIt an be shown that the sea-surfa e radian e and the intensity on the photograph arerelated by: N � I�(�) os4 �; (8.19)where I is the pixel intensity ranging between 0 (bla k) and 1 (white).Finally we have: Ps � I os3 � os!�(1� �)3 sin� os3 �: (8.20)This probability is not absolute, sin e we have no absolute relation between N and I inequation 8.19. The intensity on the photograph is thus proportional to the sea-surfa e slopeprobability times a fun tion of the geometry (Sun elevation �) and the Tilt-Azimuth Mapping:I(�; �) � Ps(�; �) � f(�; �; �); (8.21)with f(�; �; �) = �(1� �)3 sin� os3 � os3 � os! : (8.22)However, this relationship is not valid for observed intensities far away from the glint blob.Indeed, the very infrequent slopes are masked by the ba kground re e tivity of the sea surfa e.Ba kground modelThe pixel intensity on the image omes prin ipally from the additive ontribution of sunglintand re e ted skylight. The sunlight s attered by parti les beneath the sea surfa e is assumednegligible and is not onsidered here. A model has been developed to remove re e tions ofsky radian e from the glint.The approa h onsists in onsidering ea h sea surfa e fa et spe ular be ause, for a givenlo ation of the re eiver, it always exists a \sour e" in the sky that satis�es the spe ularre e tion ondition. Let's onsider the ell (�i; �i) of the TAM. It orresponds to the slope omponents required to re e t the solar rays onto the amera. The radian e of the sea surfa edue to re e ted skylight in the ell (�i; �i) an thus be simply modeled by the integration ofintensity I of equation 8.21 over all the azimuths � and tilts � ex ept the azimuth �i and thetilt �i of the orresponding ell:Ib(�i; �i) = Z�6=�i Z� 6=�iK � Ps(�; �) � f(�; �; �)d�d�; (8.23)with K a onstant related to the sky radian e. The ba kground is thus parametrized withthe same parameters of interest of the slope PDF and a multipli ative onstant.Forward intensity modelFrom equations 8.21 and 8.23, the model intensity Im writes:Im(�; �) = Ps(�; �) � f(�; �; �) + Ib(�; �): (8.24)It is omposed of two nuisan e parameters (ao, K) and four parameters of interest (a0o, a2, a00oand �o). Figure 8.5 shows an example of a photograph with its asso iated TAM data Id.

8.2. INVERSION PROCEDURE 113

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40 (b) IdFigure 8.5: (a) Photograph 41 with the TAM. (b) Data intensity Id.8.2.3 Parameter sear hInversion was based on the least-square approa h, i.e., estimation is performed through aminimization of the root mean square error between model and data DDMs. The optimizationpro edure used to �t a TAM Id with our model Im has been arried out through the non-linear�tting Matlab fun tion nlinfit. This fun tion has been modi�ed so that a domain of sear hfor ea h parameter an be spe i�ed. The inputs of this fun tion are a �rst guess of the sixparameters and a domain of sear h for ea h parameter. The �rst guesses have been �xedto ao=3, a0o=20, a2=6, a00o=15, �o=180o and K=0. The domains of sear h are hosen withreferen e to Cox and Munk's observations:� ao 2 [0 : 5℄,� a0o 2 [0 : 50℄,� a2 2 [0 : 30℄,� a00o 2 [0 : 50℄,� �o 2 [0o : 180o℄,� K 2 [0 :1℄.Note that be ause of the date and time of the experiment, part of the sunglint is visible only.Therefore, the domain of sear h for �o is restri ted to half the spa e only.Figure 8.6 shows an example of data and best-�t model TAMs, together with the residualbetween model and data. As observed, the residual do not ex eed 10% and is below 4% mostof the time.8.2.4 ResultsThe obtained dire tional slope varian es along the tra k of the plane are presented in thefollowing �gures and summarized in Table 8.1. In this study, DMSS are represented by rosses for the as ending tra k and by diamonds for the des ending tra k.

114 CHAPTER 8. SCATTEROMETRY WITH SUNLIGHT

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8.2. INVERSION PROCEDURE 115In �gure 8.7, both estimated up-windMSS and slope PDF isotropy are plotted as a fun tionof photograph latitude. As observed during the as ending tra k, the up-wind mean squareslope in reases regularly from latitude 41.2o to approximately latitude 42.2o. A de reaseis then observed in the range of latitude [42.2o-42.6o℄. These variations are in very goodagreement with the wind speed provided by INM and with MSS provided by Jason as shownin �gures number 7.7(a) and 7.8(a), respe tively. At latitudes ranging between 42o and 42.2o,the magnitudes of the up-wind MSS obtained during the des ending tra k of the plane arelower than the ones obtained during the as ending tra k. Two hours have passed between the�rst photograph (number 35) and the last one (number 60). However, as shown after, thisvariation in MSS orresponds to wind variations less than 1.5 m/s.Photograph Latitude North (deg) a0o a2 a00o SPA from �o (deg North)35 41.3245o 13.8606 1.5815 12.6136 -3.9436o37 41.5585o 14.0545 3.4893 10.4035 -12.6559o38 41.6610o 13.3639 3.3965 9.9729 -11.2093o39 41.7760o 11.1930 2.4997 9.0526 -10.7038o40 41.9395o 10.6231 3.2986 7.4682 -13.4550o41 41.9498o 10.8427 3.0718 7.7430 -14.6193o42 42.0362o 10.9335 3.2780 7.5862 -17.3171o43 42.1263o 11.1635 3.2415 7.7810 -18.3593o44 42.1659o 11.3595 3.6843 7.1311 -18.4752o45 42.2897o 12.6817 4.0638 8.5308 -20.1206o47 42.5808o 13.8954 3.4875 11.5217 -10.7170o48 42.5037o 12.9548 2.8321 11.1340 -26.0437o49 42.3845o 13.5814 3.7823 10.0749 -19.1450o50 42.2890o 13.2319 3.5123 10.5604 -18.4675o51 42.1534o 12.1448 3.2049 9.2280 -9.0746o52 42.0819o 13.7373 4.3761 9.9098 -6.1770o53 41.9756o 12.6535 3.8245 9.1848 -4.1865o54 41.8704o 13.0652 4.0820 9.3435 1.3328o55 41.7962o 15.4996 5.8453 1.4376 -4.2403o58 41.7889o 13.9303 4.7390 9.7291 2.8801o59 41.7509o 12.8656 3.7523 9.6445 7.6377o60 41.7245o 12.9297 3.9557 9.4167 7.7232oTable 8.1: Parameter estimation in hronologi al order.The isotropy of the PDF de reases quite signi� antly from 0.8 to rea h a minimum valueof around 0.5 at latitude 42.2o, as shown in �gure 8.7(b). It is learly observed, when om-paring the behaviors of up-wind MSS and SPI, that the isotropy of the sea-surfa e slope PDFde reases with sea-surfa e roughness. However su h variations in SPI for the range of windspeeds onsidered are not expe ted by Elfouhaily's model, as shown below. We will show astrong orrelation between slope PDF isotropy and wind/waves misalignment.We show in �gure 8.8(a) the Slope PDF Azimuth whi h provides a good a priori of winddire tion, although they are not always aligned. SPA is aligned with North at the beginningof the tra k and rotates progressively to rea h -30o at the end of the tra k. Dis repan ies in

116 CHAPTER 8. SCATTEROMETRY WITH SUNLIGHTSPA between as ending and des ending tra ks an be observed, but they do not ex eed 15o.The estimated dire tion is in pretty good agreement with the ECMWF wind dire tion for thatparti ular day and with the INM model (see �gure 7.7(a)). We an already anti ipate thatan estimation bias appears when wind dire tion and wind-generated waves are not exa tlyaligned.

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(b) Slope PDF isotropyFigure 8.7: Up-wind mean square slope and slope PDF isotropy along the tra k.

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Ascending trackDescending track(b) b00oFigure 8.8: Slope PDF azimuth and b00o along the tra k.As observed in �gure 8.8(b), the parameter b00o does not vary signi� antly along the as- ending and des ending tra ks of the plane. The magnitudes are however higher than theaverage 0.04 observed by Cox and Munk|see below the dis ussion in se tion (8.4).When ompared to ground truth in a more pre ise way|as shown latter in se tion (8.4)|the estimation provides high sea-surfa e slope varian es for the onsidered wind speeds ob-served that day. The parameters a0o, a2 and a00o present indeed lower values ompared to

8.3. WIND AND WAVES CHARACTERIZATION 117the ones obtained by Cox and Munk for high wind speeds. This an be explained with twoarguments:� We extended the inversion pro edure to the whole image. Given the Sun elevation and amera aperture, we were able to dete t slopes up to 40o, whereas Cox and Munk did not onsider slopes greater than 25o, say. However, they in reased their slope varian e by afa tor of 23% in order to take into a ount the un ommon steep slopes they intentionallydid not measured due to SNR onsiderations.� As learly observed on the photographs, the glitter pattern is modulated by long waves.This modulation orrupted the slope varian e estimation and gave the motivation to hara terize these long waves.8.3 Wind and waves hara terizationAs observed on the photographs, the return intensity is modulated by a wind-generated swell.It is expe ted that long sea waves do have a non-negligible impa t on the DMSS. We depi tin this se tion the ontribution of the long waves in the DMSS through a spe tral analysis ofthe images and simulations.The spe tral analysis of the images has been undertaken in se tion (8.3.1) to estimate thedire tion and wavelength of the swell along the tra k of the plane. In addition, we estimatedthe slope varian e of the observed long waves.The simulations of se tion (8.3.2) depi t the impa t of long waves on the estimated sea-surfa e slope PDF. We fo us on the behavior of the up-wind MSS, SPI and SPA with respe tto the height and dire tion of the wind generated waves.8.3.1 Spe tral analysisWe intend in this se tion to estimate the sea long wave spe trum with a Fourier analysis ofthe return intensity. This requires usually the entering of the intensity around its mean. The entered intensity I at pixel (i,j) is de�ned in our ase by the return intensity at pixel (i,j)divided by the mean value obtained in the orresponding ell (�,�) the pixel is belonging to:I (i; j) = I(i; j)�I(�; �) : (8.25)Figures 8.9(a) and 8.9(b) show the entered intensity I in optimal region of photograph 47and its Fourier transform, respe tively.As shown before, the mean return intensity in ell (�,�) is proportional to the sea-surfa eslope PDF times a orre tion fa tor f whi h is simpli�ed here to 1= os4 � for further al u-lation: �I(�; �) � 1 os4 � � eaoe[�a0o+a2 os(2(���o))℄ tan2 �: (8.26)In spe ular regime, the amplitude modulation of the ba ks atter is due to the lo al tiltingof the short waves by underlying longer energy ontaining swell. So far, in wave spe trumretrieval from SAR images, the Modulation Transfer Fun tion (MTF) under a Gaussian as-sumption is de�ned by: MTF (k) = �1I dId tan � � ik�2 : (8.27)

118 CHAPTER 8. SCATTEROMETRY WITH SUNLIGHT

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1 (b)Figure 8.9: (a) Centered intensity I in optimal region of photograph 47. (b) 2-D Fouriertransform of I .It omes to: MTF (k) = " 4sin2 �tan � � 2tan �mss ! � ik#2 ; (8.28)with here: 1mss = a0o � a2 os[2(� � �o)℄: (8.29)The MTF provides the dynami of the de ay of the return intensity I, i.e., the de reasingo urren e of sea-surfa e slopes. It allows to des ribe the fa t that the ontrast in the imagedepends on the observation angle.Then, the relationship between the image power spe trum and the sea wave spe trumwrites: S(~k) = F 2MTF ; (8.30)where F is the Fourier transform of I .The sea wave spe trum estimated from photograph 47 is presented in �gure 8.10(a). Apassband �lter allows a better hara terization of the long wave peak (see �gure 8.10(b)).The wavelength and dire tion of the long waves are simply estimated by the position of thepeak in the 2-D spe trum.We also investigate from the estimated long wave spe tra the mean square slope msslrelated to the observed long waves. It done through the �tting of the integrated spe trumwith the signi� ant wave height (SWH) provided by Jason. This parameter sear h is motivatedby the simple assumption that the Total MSS is the sum of the long wave slope varian e msslplus a wind-related slope varian e mssw:Total MSS =mssw +mssl: (8.31)A lose look to the data has shown that equation 8.28 does not hold. Spe tral analyseshave been arried out on 128x128 squared pixel images at di�erent observation angles. The

8.3. WIND AND WAVES CHARACTERIZATION 119

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0.4 (b)Figure 8.10: (a) Sea elevation estimated spe trum S(k). (b) Passband �ltered S(k).integrated image spe trum presents three regimes along the tilt axis of the TAM, i.e. the angleof observation, as shown for photograph 37 in �gure 8.11. It is �rst an in reasing fun tion of�, then rea hes a 4o large plateau and �nally de reases.When the integrated spe trum does not depend on the observation angle, it implies fromequation 8.30 that the MTF does not vary as well. This means that in this range of slopesthe derivative of the re eived intensity (i.e., slope PDF) with respe t to the slopes is onstant.Consequently, the o urren e of slopes de reases in exp(� tan�=mss) in this range and notin exp(� tan2 �=mss) as in the Gaussian ase.Note that this plateau orresponds to the in e tion point of the PDF. As observed on allthe set of photographs, it shifts towards large slopes values as the roughness in reases.In this parti ular range of slopes, the signi� ant wave height an thus be de�ned by:SWH = 4�� = 4sZ S(~k)d~k = 4sZ F 2(~k)C2~k2 d~k: (8.32)The onstant C is an ad ho parameter repla ing the MTF and omputed so that SWH �tsthe values provided by Jason (see �gure 7.9). We dedu e �nally a slope varian e of the longwaves: mssl = 1C2 Z F 2(~k)d~k: (8.33)ResultsFinally, we obtained from the spe tral analysis the followings results:� The swell wavelength spans between 40 and 45 meters along the tra k.� As shown in �gure 8.12, the swell dire tion �s rotates quite linearly from about 40oNorth at latitude 41.6o to -60o North at latitude 42.6o. At ea h point of the tra kwhere waves have been hara terized, the wave orientation does not hange in timesin e there are no signi� ant di�eren es between the dire tions estimated during the

120 CHAPTER 8. SCATTEROMETRY WITH SUNLIGHT

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Figure 8.11: Integrated spe trum versus tilt �.as ending and des ending tra ks. However, these dire tions di�er from the slope PDFazimuths estimated through the TAM data inversion at low latitudes. As shown below,there is indeed a de oupling between wind and waves as we go South.� The long waves slope varian e mssl ranges between 0.008 and 0.016 as observed in�gure 8.13(a). The Total MSS and mssw are also plotted. These magnitudes are very onsistent with Vandemark's re ent measurements [Vandemark et al.2003℄ for oastalobservations. Their mssl in ludes waves with wavelengths greater than 2 m.We quanti�ed in per ent of the overall slope varian e the mssl ontribution of the longwaves in �gure 8.13(b). The long wave slope varian e represents more than 10% of theoverall slope varian e along the tra k and an rea h values up to 22%.

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Figure 8.12: Swell dire tion �s along the tra k from a spe tral analysis of the images.8.3.2 SimulationsWe investigate in this se tion the impa t of a swell on the Sun glitter through some simulatedphotographs. The goal is to quantify the long s ale slope omponent of the total sea-surfa eslope varian e.

8.3. WIND AND WAVES CHARACTERIZATION 121

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(b)Figure 8.13: (a) Total MSS, mssw and mssl versus latitude. (b) Long wave varian e ontri-bution in per ent to the overall slope varian e.The swell an be simply modeled by a sine fun tion with three parameters: height H,wavelength L and dire tion �s:�s(X;Y ) = H os �2�L (sin�sX + os�sY )� : (8.34)As observed from the spe tral analysis, the swell wavelength is �xed to L=45 meters. Were all that the indu ed wind sea-surfa e slope PDF proposed in se tion (8.2.1) is:Ps(�; �) = eao+[�a0o+a2 os(2(���o))℄ tan2 �: (8.35)A slope originally spe ular in the ell (�,�) of the TAM is now tilted along the swell byan angle �s. The slope (�,�) is to be expressed in the lo al frame of the swell ( ~Xs, ~Ys, ~Zs) atthat point of the surfa e, de�ned as:8><>: ~Xs = os�s ~X � sin�s~Y~Ys = os �s sin�s ~X + os�s os�s~Y � sin�s ~Z~Zs = sin�s sin�s ~X + sin�s os�s~Y + os �s ~ZThen, the unit ve tor ~u tangent to the spe ular fa et writes in frames ( ~X; ~Y ; ~Z) and ( ~Xs; ~Ys; ~Zs):~u = 8><>: sin� os � os� os �� sin� ( ~X;~Y ;~Z) = 8><>: os � sin(�� �s) os � os �s os(�� �s) + sin� sin�s os � sin�s os(�� �s)� sin� os �s ( ~Xs; ~Ys; ~Zs)The modi�ed slope PDF omes to:Ps(�; �0) = eao+[�a0o+a2 os(2(���o))℄ tan2 �0 ; (8.36)with: sin�0 = sin� os �s � os � sin�s os(�� �s): (8.37)

122 CHAPTER 8. SCATTEROMETRY WITH SUNLIGHT

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500Figure 8.14: Simulated photographs with the Eddy Experiment Flight amera spe i� ations.The up-wind MSS is 0.031. Top: without swell. Bottom: with swell (H=1 m).

8.3. WIND AND WAVES CHARACTERIZATION 123Simulations have been done based on equations 8.36 and 8.37 in order to highlight theimpa t of H and the relative orientation between wind and waves (�� = �s � �o) on theup-wind MSS, SPI and SPA. We �xed an initial up-wind MSS to 0.031 (a0o=20 and a2=4),whi h orresponds to an isotropy of 0.67. The bistati geometry is similar to the one of thephotographs, with a Sun elevation of 45o. However, both Sun azimuth and wind dire tion areassumed to be aligned with the top of the photograph (�o=0o). Figure 8.14 shows an exampleof simulated glint with and without long waves for that parti ular DMSS. The swell dire tionis 50o away from the wind dire tion with H=1 m.The use of the Matlab �tting fun tion is not appropriate in this ase, sin e the domainof sear h hanges signi� antly from one in rement of H or �� to the other. The parameterestimation has been done with a simple least square inversion. The results of the simulationsare presented below.Impa t of SWHWe fo us on the variations of the mean square slope with the height of wind-generated wavesfor a �xed orientation, whi h is hosen to be at �� = �s��o = 45o from the wind dire tion.As observed in �gure 8.15(a), the up-wind MSS in reases with H. For instan e, a SWH of 2m (H=1) will in rease the slope varian e by a fa tor 1.24.

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(b) Slope PDF isotropyFigure 8.15: Impa t of wave height H on �2u and SPI. �� = 45o.On the ontrary, the isotropy of the PDF de reases with the SWH and is lowered by afa tor 1.08 for SWH=2 m (see �gure 8.15(b)). We on lude that the SWH in reases theanisotropy of the sea-surfa e slope PDF, but not signi� antly.The di�eren e between the slope PDF azimuth and wind dire tion is plotted as a fun tionof the height of the waves in �gure 8.16. The deviation from the true wind dire tion in reasesquite signi� antly with H and an rea h 15o for SWH=2 m.Impa t of a wind/wave misalignmentThe parameter H is now �xed to 0.7 (blue urve) and 1 meter (red urve), and �� be omesthe varying parameter. Figure 8.17(a) shows that the up-wind MSS in reases of about 40%

124 CHAPTER 8. SCATTEROMETRY WITH SUNLIGHT

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Figure 8.16: Impa t of H on wind dire tion estimation. �� = 45o.when waves and wind are aligned (��=0) for a SWH of the order of 2 m (H=1 m). Itde reases then to the nominal value of 0.031 when the wave dire tion is perpendi ular to thewind dire tion. Note that shadowing e�e ts are not modeled here, and given the geometry ofthe s attering (Sun elevation is 45o) they should have a great impa t on the return intensity.A lower in rease than the 40% found in the mean square slope is expe ted.

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(b) Slope PDF isotropyFigure 8.17: Impa t of �� on �2u (a) and SPI (b). H is 0.7 (blue urve) and 1 meter (red urve). The bla k line orresponds to the nominal value of the analyzed parameter.The slope PDF isotropy equals 0.5 when wind and waves are aligned as shown in �gure8.17(b). As expe ted in that ase, the long wave tilting e�e t in reases the anisotropy of theslope PDF. When the wave orientation di�ers from the wind dire tion, the PDF be omesmore isotropi and SPI equals 0.8 for �� = 90o. We on lude that the relative orientation ofthe wind and waves be omes a ru ial parameter governing the sea-surfa e slope shape.The di�eren e angle between slope PDF azimuth and wind dire tion is maximum around�� = 40o to 60o. The estimation bias is then of the order of 15o as observed in �gure 8.18.

8.4. COMPARISON WITH COX AND MUNK'S MEASUREMENTS 125

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Figure 8.18: Impa t of �� on wind dire tion estimation. H is 0.7 (blue urve) and 1 meter(red urve).8.4 Comparison with Cox and Munk's measurementsWe ompare in this se tion the obtained �tting oeÆ ients with sea-surfa e roughness andwith the ones found by Cox and Munk. The relationship between DMSS and sea-surfa e hara teristi s su h as SWH and misalignment between wind and waves is dis ussed. Spe ialattention is then paid to the fourth order moment of the slope PDF.Sea-surfa e slope varian eThe estimated MSS de�ned by 1=a0o has been plotted as a fun tion of Jason's wind speed in�gure 8.19(a), together with the MSS omputed with the Cox and Munk's parameters a0o (red ir le). The latter in ludes a orre tion fa tor (the so- alled \blanket" fa tor) of 23%, whi hmultiplies their inferen e of an \in omplete" slope varian e 1=a0o to a ount for the infrequentand unmeasured steep slopes (see Cox and Munk, 1956, se tion 7.3 and se tion 9). A ordingto the mathemati al extrapolation pro edure, the mean square slope would then be simplyestimated via the dedu tion: MSS ' Ba0o ; (8.38)where B is the approximated blanket adjustment fa tor to take into a ount the very infre-quent steep slope omponents in the total varian e.As understood, this fa tor thus en ompasses the approximation of the ratio a00o=a02o , and iswind-insensitive. As proposed by Cox and Munk, the value is 1.23 leading to the often- iteddedu tion for lean surfa e:MSS lean = 0:003 + 0:005U12 ' 1:23a0o lean; (8.39)where U12 is the wind at a 12 meter altitude, g the a eleration gravity. This regression lineis also plotted in red. In addition, both Elfouhaily's and Kudryavtsev's models are plotted(see [Kudryavtsev et al.1999℄).A linear variation of the opti ally derived slope varian e with the wind speed is observed.The regression line has not been omputed sin e the range of wind speeds during the exper-iment was too small. The sea-surfa e slope varian es obtained during this experiment arehigher than the ones omputed by Cox and Munk. We �t perfe tly their data by redu ing

126 CHAPTER 8. SCATTEROMETRY WITH SUNLIGHT

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(b) Swell/PDF axis misalignmentFigure 8.19: (a) Total MSS versus wind speed. Cox and Munk slope varian es are alsoplotted (red ir les). (b) Angle di�eren e between slope PDF azimuth and swell dire tion(�� = �s � �o) versus wind speed.our estimated varian es by 20%. This high slope varian es may ome parti ularly from thesea-state. Indeed simulations have highlighted that the non-wind related SWH ould in reasethe MSS by around 20%. We have indeed observed during this experiment a stronger wind-generated swell (SWH�2 m) than the one observed by Cox and Munk (SWH�1.2 m) for highwind speeds.Furthermore, we have seen in se tion (8.3.1) that the estimated long wave slope varian emssl ontributes to more than 10% of the overall sea-surfa e slope varian e. It on�rms thefa t that at high wind speed onditions (9 to 13 m/s), high SWHs|i.e., high mssl|willin rease signi� antly the overall MSS.Slope PDF azimuthCox and Munk have found the slope PDF azimuth to be oriented with the wind in the whole setof data they pro essed. In our set of data, we dete ted dis repan ies between waves dire tionand SPA at low latitudes of the tra k. Figure 8.19(b) presents the di�eren es between thedire tions of the wind-generated waves and the slope PDF azimuths along the tra k. Swell andwind are approximately aligned at high latitudes, whereas di�eren es rea hing values of theorder of 40o are observed at the beginning of the tra k. Probably due to strong bathymetri variations in this area, the wind generated waves rotate signi� antly towards the Catalan oast as we go South, whereas the wind de reases and rotates slowly. Given the results of thesimulations, the dete tion of this di�eren e allows us to think that there is a bias betweenthe estimated SPA and the true wind dire tion ranging between 0o and 15o. This bias ouldexplain the dis repan y of the data with the ECMWF model at the beginning of the tra k.Slope PDF isotropyIn their well do umented report, Cox and Munk measured isotropies ranging from 0.54 to1 and noti ed that high values of this parameter (i.e., a quasi-omnidire tional slope PDF)were obtained for gusty winds of high variability. Therefore they related isotropy low valueswith the low variability of the wind. We observe in �gure 8.20(a) similar low values of about

8.4. COMPARISON WITH COX AND MUNK'S MEASUREMENTS 1270.5 in an area where the wind is strong (12-13 m/s) and parti ularly aligned with the waves.It is learly observed that the slope PDF isotropy de reases with wind speed, with highervariations than the ones predi ted by the Elfouhaily model (bla k urve). Indeed, this modeldoes not predi t signi� ant variations of the isotropy at this range of wind speeds.

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(b)Figure 8.20: (a) Slope PDF isotropy versus wind speed. (b) Slope PDF isotropy versus ��.We anti ipate that high values of SPI are obtained also when there is a misalignmentbetween wind and waves. Cox and Munk's observations do onsolidate this dependen ysin e the high isotropy value of 0.97 en ountered for a 8.6 m/s wind speed orresponds to amisalignment of 39o.The relative orientation of waves and wind and the SWH impa t signi� antly the isotropyof the sea-surfa e slope PDF, as understood from the simulations. It has been shown howeverthat SPI was mu h more sensitive to this misalignment than the SWH. Figure 8.20(b) showsindeed the pretty good orrelation between SPI and this misalignment. The slope PDFisotropy in reases from 0.5 to 0.6 when 30o of di�eren e between wind and wave orientationsis observed. Note that the small isotropies en ountered orrespond also with the highestSWHs.Furthermore, we observe in this �gure that the minimum of the isotropy does not o ur at�� = 0 as expe ted but at around 15o. This bias ould be related to the bias between slopePDF azimuth and true wind dire tion, sin e �� is not the exa t angle di�eren e betweenwind and waves but the di�eren e between SPA and waves.Departure from GaussianityThe analysis of parameter b00o learly shows that, in this high wind speed regime, it is higherthan the value 0.04 mentioned by Cox and Munk. The importan e of this value for theiranalysis omes from the ne essity to a ount for the missing data (a0os2 � 4) to evaluate thetotal slope varian e. Cox and Munk extrapolate their �t results out to a0os2 � 8 to estimatefor what they alled the in omplete slope varian es. This limit an be judged quite arbitrary,and is in fa t strongly depending upon the ratio a00o=a02o [Chapron et al.2000℄. When this valueis less or equal to the hosen onstant 0.04, the extrapolation pro edure an be tested and a

128 CHAPTER 8. SCATTEROMETRY WITH SUNLIGHTposteriori justi�ed. But, it is not the ase for higher values (in fa t obtained for wind speedshigher than 6 m/s).As observed in �gure 8.21(a), the fourth order moment a00o of the slope distribution isa de reasing fun tion of the wind in the range 9 - 13 m/s. The regression line omputedregardless of the outlier gives: a00o = 18:1942 � 0:7316U10 : (8.40)The parameter b00o (see �gure 8.21(b)) spans between 0.05 and 0.07 and presents slow in reasingvariations with wind speed, a ording to the regression line:b00o = a00oa02o = 0:0569 + 2� 10�4U10: (8.41)Note that some of the Cox and Munk's data fall into the obtained ranges of values.

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SORESRegression lineC&M (b) b00oFigure 8.21: Parameters a00o (a) and b00o (b) versus wind speed. Cox and Munk's data are alsoplotted (red ir les).The blanket adjustment fa tor used by Cox and Munk, along with its impli it relationto the PDF 4rth order orre tion, is potentially highly variable (sea-state, stability, urrent,sli k, ...). The strong sea-state observed during the present experiment may explain the quitehigh departure of the slope PDF from Gaussianity, depi ted by b00o=0.057, ompared to themean value of 0.04 omputed by Cox and Munk for a wider range of wind speeds (from 0.5 to14 m/s). In addition, Cox and Munk learly observed values lower than 0.04 for wind speedslower than 5 m/s and a SWH not ex eeding 1 m. Therefore, the parameter b00o ertainlydepends more on the sea-state than on the sea-surfa e roughness.As understood, this value impa ts the varian e estimation and the predi ted o urren eof the infrequent but very large o ean slope omponents. This fa tor does strongly impa tthe shape of the PDF, and thus the orresponding opti al and radar ross se tions in quasi-spe ular re e tions.

8.5. CONCLUSIONS 1298.5 Con lusionsWe have presented in this Chapter a re-do of the Cox and Munk experiment whi h aimed atDMSS retrieval through inversion of Tilt-Azimuth Map of Sun glitter opti al photographs. Aset of 25 photographs has been analyzed by the SORES pro essor along a path aligned withJason's tra k, lose to the Catalan oast. We have retrieved the DMSS pro�le along the tra kwith a very good agreement ompared to the altimeter data.Estimated sea-surfa e mean square slopes range between 0.04 and 0.08 and are linearlyproportional to the wind speed. Slope PDF Isotropies present quite strong variations from 0.8to 0.5. The wind dire tion agrees very well with the ground truth given by ECMWF modeland numeri al models from INM.A spe tral analysis has been arried out to extra t further parameters of the sea surfa e,su h as wave dire tion, wavelength and long wave slope varian e. These parameters havebeen estimated along the tra k, showing very onsistent results. These additional parametersallowed us to improve the understanding of the long wave impa t on the DMSS, whi h hasbeen enhan ed through simulations.In parti ular, we have seen that a SWH of 2 meters ould in rease the slope varian e ofabout 20%. The explanation omes from the tilting e�e t of the long waves on the small s aleroughness. It has been shown that the dire tion of the wind-generated waves with respe t tothe wind dire tion signi� antly impa ts the isotropy of the PDF. This e�e t has been learlyobserved experimentally. The dis repan y between wind and waves orientations learly lowersthe anisotropy of the slope PDF, i.e., in reases the SPI. In addition, we have shown that swelland wind misalignment an lead to estimation biases in the wind dire tion up to 15o.A omparison has been undertaken with Cox and Munk's results. The main di�eren e inthe pro essing of the data omes from the fa t that they limitated the �t of the sea-surfa eslope PDF to maximum slopes of about 20-25o, while we have here onsidered the whole imagewith dete table slopes up to 40o. The motivation of this approa h was to develop a ba k-ground model, whose parameters are minimized during the inversion pro edure. Therefore, wedon't need any orre tion or so- alled \blanket" fa tor|as introdu ed by Cox and Munk|toestimate the sea-surfa e slope varian e. However, the slope varian es are 20% higher than theones estimated by Cox and Munk for high wind speeds. This is onsistent with the fa t thatwe observed wave heights two times bigger than the ones observed by Cox and Munk for thesame wind onditions.The non negligible ontribution of the long wave slope varian e in the overall mean squareslope on�rms that the sea-state a�e ts onsiderably the estimation of the wind-related DMSS.Furthermore, a strong departure of the PDF to Gaussianity has been observed duringthis experiment. It has been suggested that the parameter b00o is strongly dependent to SWH.Further experimental data taken for di�erent wind speed and SWH onditions are needed to on�rm this statement.

130 CHAPTER 8. SCATTEROMETRY WITH SUNLIGHT

Chapter 9S atterometry with GNSS-RThis Chapter presents estimation of DMSS during the Eddy Experiment Flight by inversionof the bi-dimensional Delay-Doppler Maps. The method uses a Least-Square approa h andpermits to retrieve simultaneously at L-band the Total MSS, Slope PDF Isotropy and SlopePDF Azimuth. This work has been arried out in the frame of the OPPSCAT II ESA proje t(see [Germain et al.2003℄).The data set is �rst des ribed with the spe i� delay and Doppler resolutions. Se tion (9.2)reports then the retra king and inversion pro edures that have been arried out on the s at-terometri data produ ts. Furthermore, the estimated DMSS are analyzed in se tion (9.3)through their on�den e interval length. Finally, the results are ompared to SORES inversionand ground truth.9.1 The data setAs it would have been too omputationally extensive to analyze the full tra k, the latterwas divided into 46 10-se ond long ar s, sampled every 50 se onds. The �rst ar startsat SOW=468008.63. Given the speed of the air raft, one ar spans roughly 500 meters.Figure 9.1 shows a plot of those ar s while air raft kinemati parameters are presented on�gure 9.2.The �rst pro essing step onsists in performing a delay-Doppler PRN ode despreading to oherently dete t the dire t signal (from GPS emitter) and the re e ted signal (s attered bythe sea-surfa e). We used the Starlab in-house software (Starlight) to produ e three DDMstime-series (one per PRN), sampled into 46 ar s of 10 se onds ea h. The general strategy ofthe pro essing is to tra k the delay-Doppler of dire t signal and then ompute DDMs for bothdire t and re e ted signals. As des ribed in se tion (4.1) of Chapter 4, those DDMs a tuallyrepresent the omplex amplitude of in oming signals when pro essed with a delay-Dopplervalue slightly di�erent from the estimated delay-Doppler enter. The oherent integrationtime was set to 20 ms to ensure a Doppler resolution of 50 Hz. The delta-delay spans -40 to40 orrelation lags (i.e., +/- 1.95 �s) with a lag step (48.9 ns) and the delta-Doppler spans-200 Hz to 200 Hz with a step of 20 Hz. As a result, the Level 0 data for one PRN and onear is a omplex delay-Doppler-Time ube of size 81�21�500.We have hosen to set the a umulation time Ta to 10 se onds so that ea h ar is hara -terized by a unique DDM, resulting from the sum of the amplitudes of 500 omplex DDMs(or 500 \looks"). Thus, for ea h ar and ea h PRN, the Level 0b data is a real delay-Dopplermap of size 81�21.

132 CHAPTER 9. SCATTEROMETRY WITH GNSS-R

Figure 9.1: Map of the 46 10-se ond ar s onsidered along the tra k.9.2 Retra king and inversion9.2.1 The forward modelThe model onsiders the average amplitude of the DDM. Sin e amplitude data are approx-imately Rayleigh distributed, it an be shown that the average amplitude is lose to thesquare-root of the average power. Hen e, the average amplitude of the DDM writes:A(�; f) = q�:P (�; f) + g: (9.1)P (�; f) is the waveform model presented in se tion (3.3) of Chapter 3. As we do not laim tohave a alibrated model, we also have introdu ed an overall s aling parameter � in the latterequation in order to �t model to data. For delays lower than one- hip, the DDM amplitudeis not null but shows a onstant value g (often alled "grass level"). It an be shown thatthis value depends on the orrelation length of a thermal noise orrupting the signal and the oherent integration time (see se tion (11.2) of Chapter 11 for further details).To sum up, the proposed forward model features three parameters of interest and four\nuisan e parameters". Those are:� three DMSS hara terizing the Gaussian slope PDF: Total MSS, isotropy (SPI) andazimuth (SPA),� DDM delay-Doppler enters: � and f ,� overall s aling parameter: �,� grass level: g.Other parameters ne essary to run the forward model are re alled in Table 9.1.

9.2. RETRACKING AND INVERSION 133

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Figure 9.2: Altitude, heading (azimuth from North) and speed of the air raft during the 4610-se ond ar s.9.2.2 Inversion s hemeAs in the SORES pro edure, we used a least-square approa h. Let Ad(�; f) be the dataDDM and A�(�; f) the model DDM orresponding to a � ve tor of seven parameters (DMSSand nuisan e parameters). The root mean square error (RMSE) between data and model isde�ned as the root mean square of residual:RMSE(�) = sX�;f [Ad(�; f)�A�(�; f)℄2; (9.2)Geometry Air raft: Altitude, speed and heading provided at 1 HzSatellite: Elevation and azimuth provided at 1 HzInstrument Antenna Pattern: provided (see �gure 9.3)Band: L1 (19 m)GPS Code: C/APro essing Integration Time: 20 msA umulation Time: 10 sDoppler span: [ -200 Hz , 200 Hz ℄, 20 Hz stepDelay span: [-40 samples, 40 samples℄, 1 sample stepTable 9.1: Overview of the parameters ne essary for running the DDM forward model.

134 CHAPTER 9. SCATTEROMETRY WITH GNSS-R

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Figure 9.3: Power pattern of the LHCP antenna used to gather the GNSS-R signals (mea-surements provided by IEEC). It is a 3 dB antenna, having its ut-o� at 55o. It was mountedon the air raft so that air raft nose orresponds to azimuth 210o.and the estimator is the value minimizing this error:� = argmin� RMSE(�): (9.3)Theoreti ally, the absolute minimum of RMSE should be sear hed jointly over the sevenparameters. In pra ti e, this is not tra table and we have adopted the following sub-optimalstrategy. First, the grass level is not optimized but a priori set to a value measured on data.This value is the average DDM amplitude in the domain su h that delay is smaller than minusone- hip. Given the sampling, this value is estimated over 20�21�500 = 210; 000 samples andis therefore expe ted to be quite a urate. Se ond, the following two-step iterative algorithmwas applied:� In a �rst step, DMSS parameters are frozen while the three other parameters (s alingand enters) are optimized. This step is alled retra king: given a shape for model DDM,the optimal translation is sear hed for in order to �t data. We re all that in GNSS-Raltimetry, this operation is arried out over a waveform (a Doppler- ut in the DDM) to�ne-estimating lapse between dire t and re e ted signals.� In a se ond step, the delay-Doppler enters are frozen and the four other parameters(s aling and DMSS) are optimized. This step is alled inversion: given a position formodel DDM, the optimal shape is sear hed for in order to �t data.Numeri al optimization is arried out with a steepest-slope-des ent algorithm based on Levenberg-Marquardt type adjustment ([Levenberg1944, Marquardt1963℄). In pra ti e, the DMSS isoptimized by repeating this two-step sequen e until onvergen e.Two kinds of data are a tually delivered by this pro essing. Magnitudes like delay-Doppler enters are Level 1 data that an be pro essed further to infer geophysi al param-eters. In parti ular, the pro essing of delay enters is the subje t of GNSS-R altimetry

9.2. RETRACKING AND INVERSION 135to re over sea-surfa e height. This an be done through the sear h of delay entroids (see[Apari io et al.2002℄) or through the retra king of the delay waveform (see [RuÆni et al.2003℄and [Lowe et al.2002℄). Although we will fo us only on Level 2 data, namely the DMSS, wepresent hereafter the s atterometri bias whi h highlights how altimetri and s atterometri measurements are linked together, espe ially at low altitudes.The s atterometri biasThis iterative algorithm somehow makes the impli it assumption that DDM \position" isonly governed by enter values while its \shape" is only DMSS dependent. If the latterstatement is true, the former is only an approximation, for we know that at low altitude,DMSS also impa ts the delay-Doppler DDM enters: this e�e t is alled s atterometri biasin GNSS-R altimetry (see [Rius et al.2002℄). Figure 9.4 illustrates this bias by showing twodelay waveform models obtained for di�erent MSS, su h as MSS2 > MSS1. We re all thataltimetry measurement onsists in estimating the delay � between the dire t and re e teddelay waveforms. It appears that sea-surfa e roughness tends to in rease the delay: we have�2 > �1.MSS

2τ2estimated delay:

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REFLECTED SIGNALDIRECT SIGNAL

TimeFigure 9.4: S atterometri bias in the delay waveform.Sensitivity analyses have been arried out to depi t the s atterometri bias �� = �2 � �1with respe t to SWH and the three parameters of the DMSS: MSS, SPI and SPA. We �rstgenerated an initial waveform with a given set of nominal parameters, as shown in Table 9.2.Then, we generated several model waveforms with the same set of parameters ex ept onethat spans a range of values around the nominal one. For ea h value of this parameter, aretra king of the initial waveform with the model waveform has been undertaken to evaluatethe s atterometri bias. MSS SPA SPI SWH0.025 0o 0.55 2 mTable 9.2: Set of nominal parameters for sensitivity analysis of parameter �� .First, sensitivity of �� to signi� ant wave height is shown in �gure 9.5 for the onsideredPRNs. Unlike RA, the signi� ant wave height does not impa t signi� antly the delay estima-tion. At this strong sea-state regime (SWH�2 m) the error is less than 1 m for a dis repan yof 2 m in the SWH. This omes from the fa t that the hip length in GNSS is mu h longerthan the one used in RA. Therefore, the SWH parameter is not taken into a ount in theinversion.

136 CHAPTER 9. SCATTEROMETRY WITH GNSS-RHowever, �gures number 9.6, 9.7 and 9.8 show the drasti impa t of sea-surfa e roughness(respe tively the wind strength related to MSS, SPI and SPA) on �� . For instan e, a bias of1 meter in delay is rea hed when:� the wind speed di�ers by 2 m/s,� SPI is 0.25 lower than nominal,� or SPA di�ers by 25o.

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9.2. RETRACKING AND INVERSION 137

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138 CHAPTER 9. SCATTEROMETRY WITH GNSS-RDDM waveform, whi h in ludes in addition to the glistening zone, the Woodward AmbiguityFun tion and antenna pattern. In the ase of a low altitude re eiver, the glistening zone orresponds to the smallest support and thus modulates strongly the return waveform. Notethat these e�e ts will signi� antly redu e at high altitude, sin e the glistening area is largerthan the �rst hip zone. In a spa eborne s enario, the bias be omes onstant.We an anti ipate that GNSS-R s atterometry from spa e will be quite hallenging. Themain issue addresses the spatial resolution, for the sensitivity of the DDM to DMSS appearsat higher delays and Doppler frequen ies than those limited by the �rst hip zone. Therefore,the orresponding spatial s ales of the sea surfa e are too large ompared to sea-surfa eroughness variations. As part of the OPPSCAT II ESA proje t, Starlab developed a multi-look te hnique whi h aims at produ ing high-resolution pro�les of absolute mean square slope[Germain and RuÆni2002℄.9.2.3 Empiri al adjustmentsThe �rst guesses provided to the algorithm were:� DMSS are initialized to values onsistent with expe ted sea-state: MSS=0.025, SPA=0o,SPI=0:65.� Delay-Doppler enters are set to zero.� S aling is set to 1 and both data and model DDMs are normalized to have their maximato 1.The �t was not performed over the total delay-Doppler domain. As a matter of fa t, weknow that all useful signal is omprised in the delay range [-1 hip,+1 hip℄. Therefore, onlythis domain was used.9.3 Analysis of the resultsIn this se tion, we present the results obtained after inversion. Note that omparison withground truth as well as geophysi al interpretations are beyond the s ope of this se tion andwill be addressed below in se tion (9.4).9.3.1 Estimated DMSSFigure 9.9 shows the estimated DMSS together with their formal on�den e interval length(CIL). The Total MSS in reases along the tra k of the plane. PRNs 08 and 24 are prettysimilar, whereas PRN 10 seems to be biased by a positive onstant. As shown by the CIL,Total MSS is better estimated with PRNs of high elevation, whereas azimuth is better apturedwith grazing (and therefore more dire tion-sensitive) PRNs. Furthermore, there is a very goodinter-PRN agreement in estimation of isotropy: the three PRNs do have the same trend forSPI.Note that the estimation of SPA is degenerate in two parti ular ases: when the transmitteris at zenith or when the re eiver moves towards the transmitter [Cardella h and RuÆni2000℄.In these two ases, the delay-Doppler lines that map the glistening zone are symmetri aroundthe re eiver dire tion. Hen e, one annot distinguish between a slope PDF and its mirrorimage about the re eiver dire tion. Here, PRN 08 has its elevation omprised between 74 and83 degrees. It is then very likely that the SPA estimated for this PRN is degenerate. Forthis reason, we have added on the plot of �gure 9.12 the mirror image of the SPA about there eiver dire tion (30o from North). We also note that the azimuth of PRN 10 de reases downto 230o at the end of the tra k, quite lose from 210o (aligned with the re eiver's dire tion).

9.3. ANALYSIS OF THE RESULTS 139

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140 CHAPTER 9. SCATTEROMETRY WITH GNSS-RPRN Total MSS (�10�3) SPA (deg) SPITrend CIL Trend CIL Trend CIL08 28 to 32 0.5 < 0 to 65 70 < 0.75 to 0.65 to 0.8 0.2 <10 32 to 39 0.8 < 20 to 35 30 < 0.9 to 0.6 to 0.8 0.04 <24 28 to 28 1 < -20 to -20 5 < 0.75 to 0.5 to 1 0.07 <Table 9.3: Analysis of estimated DMSS of �gure 9.9.9.3.2 Convergen e and residualFigure 9.10 illustrates the fast onvergen e of the optimization. After ten steps of retra king-inversion, a RMSE minimum is obtained and estimated values are stable. The graphs learlyshow the monotoni variations of the three parameters of DMSS as well as delay and Doppler enters.Figure 9.11 gives an example of DDMs aspe ts as well as the residual stru ture. Higherror zones generally orrespond to higher delays or intermediate Doppler frequen ies. Letus emphasize that the �nal RMSE never ex eeds 2.5%. Note that these graphs resemble thegraphs of �gure 8.6, illustrating the omplete analogy between the TAM and DDM sea-surfa emappings.9.4 Comparison of opti al and L-band derived DMSSFigure 9.12 presents Total MSS, Slope PDF Azimuth and Slope PDF Isotropy along theair raft tra k between latitudes 41.2o and 42.2o. Bla k diamonds stand for SORES estimationsand the blue, red and green urves for GNSS-R ones (PRNs 08, 10 and 24, respe tively).Additional information when available in Jason's GDR data has been added, su h as MSS inKu- and C-bands and wind dire tion from ECMWF. Swell dire tion from the SORES spe tralanalysis has also been onsidered.Total MSSTotal MSS has been plotted in log-s ale in order to ompare di�erent frequen y measurementsmore easily. The ommon trend for all bands is that slope varian e in reases with latitudeuntil rea hing a relative plateau. Measurements of PRNs 08 and 24 show good agreementwhile PRN 10 seems to be somewhat up-shifted. As expe ted, we observe that the level anddynami of MSS de rease with longer wavelength: opti al, Ku-, C- and L-band, in this order.Nevertheless, the level and dynami of GNSS-R plots (espe ially PRN 10) seem a bit largefor L-band measurements, when ompared to C-band. We re all however that Jason's MSShave been obtained through the relationship MSS=�=�o, � being an empiri al parametera ounting for alibration o�sets. Here, we have set �=0.45 for C-band and 0.95 for Ku-bandbut an un ertainty ertainly remains on this parameter, making the overall levels of Jason'splots unsure.Slope PDF AzimuthThe Slope PDF Azimuth estimation shows very onsistent results in both L-band and opti alregimes when omparing to the wind dire tion provided by the ECMWF model. Followingwind propagation, the ground truth states that wind dire tion is around -20o from North athigh latitudes of the tra k and in reases progressively when going South to rea h a value ofabout 30o at latitude 41.2o.

9.4. COMPARISON OF OPTICAL AND L-BAND DERIVED DMSS 141

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142 CHAPTER 9. SCATTEROMETRY WITH GNSS-R

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9.5. CONCLUSIONS 143A ording to ECMWF data and SORES spe tral analysis, wind and swell were slightlymisaligned. PRN 08 (or its mirror image) mat hes very well the swell dire tion. So does PRN10 along most of the tra k. This result underlines the fa t that GNSS-R is not sensitive towind only and that swell has a strong impa t too. PRN 24 has a di�erent behavior, in linewith SORES. Those two measurements agree relatively better with wind dire tion, althougha dis repan y of 30o is observed at the beginning of the tra k.Slope PDF IsotropyIt is worthy reminding that Elfouhaily's spe trum predi ts a SPI value of 0.65, hardly sensitiveto wind speed. Here, we note that SPI varies quite signi� antly along the tra k for both GNSS-R and SORES. The important departure observed from the 0.65 nominal value is probablya signature of an immature sea and presen e of strong swell. Note that further resear hshould be undertaken in order to better understand the potential information ontained inthis produ t.9.4.1 Link to wind speedOn �gure 9.13, we have plotted the estimated total MSS versus Jason's wind speed togetherwith two theoreti al links:� Elfouhaily's sea-height spe trum, integrated for di�erent ut-o� wavelengths,� An empiri al model proposed by Katzberg for L-band, based on a modi� ation of Coxand Munk's relationship: MSS=0:9:10�3q9:48U10 + 6:07U210.We see that both SORES and GNSS-R estimations follow the Elfouhaily's model trend (in-tegrated at 3 wavelengths) but give higher values of MSS (from to 20 to 40% up-shifted).A tually, estimations of PRNs 08 and 24 are very well �tted by Elfouhaily's spe trum in-tegrated down to one wavelength only. The 20% dis repan y an be explained by a strongsea-state with a SWH twi e as high as the one observed during the Cox and Munk's experi-ment (2 m ompared to 1 m) for the same wind onditions. At any rate, those results tendto show that the wind to MSS link is not straightforward and that the MSS should probablybe onsidered as a self-standing produ t for o eanographi users.9.5 Con lusionsWe have reported the �rst inversion of GNSS-R full Delay-Doppler Map for the retrieval ofthe sea-surfa e dire tional mean square slope. In addition, we ompared the estimated DMSSwith those obtained by the SORES data inversion.Our results show that both opti al and L-band total MSS are 20% higher than what pre-di ted by Elfouhaily's model for the observed wind speed (9 to 13 m/s). The SPA estimatedby GNSS-R mat hes the swell dire tion with good a ura y for at least 2 out of 3 PRNs. Ageophysi al produ t has been put forward: the slope PDF isotropy whi h an be related towind/wave misalignement as well as sea degree of development. The analysis highlighted theimportant impa t of sea-state (SWH) in addition to wind stress over DMSS. Spe ulometry be-ing sensitive to slope pro esses over a wide range of s ales, the link between DMSS and wind isnot straightforward: total MSS and SPA are de�nitely impa ted by swell. Quantitatively, the20% bias observed in SORES results an be explained by SWH. Other geophysi al parameters ould a�e t the sea-surfa e roughness. For instan e, it was found by [Huang et al.1973℄ thatMSS is extremely sensitive to the hange of urrent onditions. They even suggested to use the

144 CHAPTER 9. SCATTEROMETRY WITH GNSS-Rsea-surfa e roughness parameter as a measure of lo al urrent hanges. Consequently, DMSS an and should be studied as an independent parameter, of intrinsi independent geophysi alvalue.Let us �nally emphasize that the ight was not optimized for spe ulometry (1000 maltitude, 50 m/s speed) and that higher/faster ights are needed in the future in order to onsolidate the on ept of DDM inversion for DMSS estimation.

9.5. CONCLUSIONS 145

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146 CHAPTER 9. SCATTEROMETRY WITH GNSS-R

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Part IVOther GNSS-R Appli ations

147

Chapter 10Sensitivity to SalinityThe goal of this Chapter is to analyze the sensitivity of GNSS re e tions to salinity througha simple ground experiment. The Salpex Experiment (a Starlab/Ifremer Contra t) fo usedon diele tri mapping sensitivity and methods at L-band (see [Soulat and RuÆni2003℄). Thegoal is to extra t absolute re e tan e measurements from a salty surfa e by measuring thepower interferen e pattern at di�erent in iden e angles. The main strategy is to exploit thelong time s ales available to extra t measurements, whi h allows, in prin iple, to average outother e�e ts.This report is divided into three main se tions. The �rst se tion presents the Salpexexperimental ampaign, where we studied re e tions from a at surfa e, over a swimming pool,with di�erent salinities. Se tion (10.2) addresses the theoreti al analysis of water diele tri onstant dete tion. For this purpose the re eived power is modeled. Se tion (10.3) is dedi atedto the data analysis.10.1 Experimental ampaignThe ampaign gathered re e ted GNSS signals over a small swimming pool along with dire tsignals with one Right Hand Cir ularly Polarized (RHCP) antenna. Note that a similarexperiment was arried out by [Kavak et al.1996℄.10.1.1 Des riptionAs shown in �gure 10.1, the antenna is deployed to gather interferen es between dire t andre e ted GNSS signals. The antenna is at h=56 m above the surfa e (d=15 m). Theoutput of the antenna is then gathered by a GPS re eiver. It is expe ted, as presented in thefollowing se tion, to get an interferen e pattern in power with the variations of the elevationof the satellite.Two kinds of data sets have been re orded with the same experimental set-up:� Salted water: the �rst re ordings took pla e the 25th and 26th of November 2002, witha water ondu tivity ranging between 42 and 43 mS (29 to 30 psu). The water surfa etemperature ranged between 9 and 11.3 degrees Celsius.� Pure water: the se ond set of re ordings took pla e the 28th and 29th of November2002, with a water ondu tivity ranging between 1.1 and 1.2 mS (1 psu). The watersurfa e temperature ranged between 10 and 14 degrees Celsius.These experimental ampaigns have been arried out for the same on�guration of the satel-lites in view (the GPS onstellation has a period of one sideral day, whi h orresponds to 24

150 CHAPTER 10. SENSITIVITY TO SALINITYhours minus 4 minutes). The two re ordings made for ea h s enario is an attempt to he kthe repeatability of the interferen e pattern and get an estimation of the noise.Direct Signal

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Figure 10.1: Geometry of the Salpex Experiment.10.1.2 InstrumentationThe material used for this ampaign (see �gure 10.2) is:� a small swimming pool, ontaining approximately 1 m3 of salted/pure water (we thankPablo Sedo for lending us his swimming pool),� a tripod where the antenna is �xed,� one RHCP antenna, Allan Osborne Asso iates (ESTEC/ESA),� one TurboRogue re eiver (ESTEC/ESA), delivering the GNSS observables,� one portable omputer (Starlab).10.1.3 Satellites in viewTwo satellites have been used to analyze the data sets. PRNs 15 and 17 seemed to be theappropriate andidates a ording to the available spa e window plotted in green in �gure 10.3.This green mask represents the area where the GPS signal re e tions are supposed to be freeof shadowing phenomena due to the environment of the swimming pool, and therefore only thesatellite within this mask an be taken into onsideration for further analysis. Ea h oloredar represents the position of a GPS satellite during the experiment. The PRN number is losed to the beginning of the data. The lo kwise angles stand for satellite azimuths and the enter point of the graph orresponds to 90o elevation angle.10.2 Theoreti al analysisIn this experiment the re e ted signal writes, under the Fresnel assumption, simply as thepolarization matrix times the in ident signal:" EsREsL # = " URR ULRURL ULL # " EiREiL # : (10.1)

10.2. THEORETICAL ANALYSIS 151

Figure 10.2: Salpex experimental ampaign.

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152 CHAPTER 10. SENSITIVITY TO SALINITYThe polarization matrix, expressed in the ir ular basis, redu es to:" URR ULRURL ULL # = 12 " R? +Rjj R? �RjjR? �Rjj R? +Rjj # ; (10.2)where the lo al Fresnel oeÆ ients are given by:R? = sin �p�� os2 sin +p�� os2 (10.3)Rjj = � sin �p�� os2 � sin +p�� os2 :The parameter � is the omplex diele tri onstant of sea water and is the elevation angle.The latter is de�ned as the angle between the in ident �eld dire tion and the water surfa eplane: =90o for a satellite at zenith.In this ase, the de-polarization only omes from the diele tri nature of seawater. Assuming that the surfa e of the water in the swimming pool is at, we on lude thatthe re eived power P is proportional to the squared Fresnel oeÆ ients.10.2.1 Re eived power versus elevation angleThe total �eld re eived by the antenna is the sum of dire t signal and spe ularly re e tedsignal. The re e ted signal arrives with a phase shift that an be written � = 4�h=� sin ,with � the wavelength (we re all here �=0.19 m). The interferen e between the in ident ands attered �eld writes: " EiREiL #+ " EsREsL # ei�; (10.4)where EiR (EiL) is the right (left) omponent of the in ident omplex �eld, EsR (EsL) is theright (left) omponent of the s attered omplex �eld.Introdu ing the polarization matrix, we get:" EiREiL #+ " URR ULRURL ULL # " EiREiL # ei� = " EiR + (URREiR + ULREiL)ei�EiL + (URLEiR + ULLEiL)ei� # : (10.5)The emitted GPS signal is assumed purely Right Hand Cir ular Polarized and the re e tiono� the sea-surfa e makes it mostly Left Hand Cir ular Polarized at re eption. With EiL=0,the interferen e �eld be omes: EiR " 1 + URRei�URLei� # : (10.6)The re eived power is modulated by the gain pattern of the RHCP antenna in both rightpolarization (GR) and left polarization (GL). We �nally obtain:P / ���GR(1 + URRei�) +GLURLei����2 : (10.7)We re all that:� � is related to and h by the relation � = 4�h=� sin ,� the polarization oeÆ ients URR and URL are fun tions of and the water diele tri onstant � (whi h is fun tion of salinity S and surfa e temperature T ),

10.2. THEORETICAL ANALYSIS 153� the antenna gains GR and GL are fun tion of the elevation and the azimuth of thesatellite.During our experimental ampaigns, h is known and �xed, S and T measured. The re eivedpower be omes a fun tion of only. De�ning the polarization parameters URR = jURRj ei�RRand URL = jURLj ei�RL , equation 10.7 rewrites:P / GR2 +GR2 jURRj2 +GL2 jURLj2 (10.8)+2GR2 jURRj os(�+ �RR) + 2GRGL jURLj os(�+ �RL)+2GRGL jURRj jURLj os(�RL � �RR):10.2.2 Diele tri onstant modelA more detailed dis ussion about the properties of the diele tri onstant � an be found inUlaby et al. [Ulaby et al.1986℄, (vol III, p2019). In general, � is omplex, onsisting of a realpart, �0, and an imaginary part, �00: � = �0 � j�00: (10.9)Usually, �0 is referred as the permittivity of the material, whereas �00 is o asionally referredto as the diele tri loss fa tor of the material.In the mi rowave range, the permittivity of o ean water is a omplex fun tion of thetemperature T , salinity S, and also of mi rowave frequen y f . Usually, the Debye or moreadequate Cole-Cole relaxation models are used for mi rowave remote sensing appli ations.The basi idea is to onsider that the permittivity of sea water will be a slight deviation fromthat of pure water due to addition of ioni onstituents. At low mi rowave frequen ies (f <20 GHz), the diele tri onstant of sea water an thus be modeled by a Debye relaxation plusa ondu tivity term of the form:�sw(T; S; f) = �1sw(T; S) + �osw(T; S)� �1sw(T; S)1� 2�f�sw(T; S) + j �i(T; S)2��of ; (10.10)where� �1sw(T; S) is the high-frequen y (or opti al) limiting value of the diele tri onstant ofsaline water,� �osw(T; S) is the stati value of the diele tri onstant of saline water,� �sw(T; S) is the relaxation time in se onds,� �o is the permittivity of free spa e (�o = 8:854 � 10�12 F�m�1),� �i(T; S) is the ioni ondu tivity of the aqueous saline solution in S�m�1.These Debye parameters depend upon the temperature T and the on entration of theioni salt. We hoose to represent the on entration in terms of S, but it an be representedin term of hlorinity or normality. Previous works whi h des ribe models of permittivityof sea water and aqueous saline solutions are those of Stogryn, [Klein and Swift1977℄ andSwift, Aggarwal and Johnston, Singh et al., and [Ellison et al.1998℄. A omplete and riti aldis ussion of the literature survey on erning the diele tri properties of sea water and aqueousioni solutions an be found in [Ellison et al.1996℄.Brie y, the basi assumption behind all the proposed interpolation models is that for a�xed temperature, and in the frequen y range 0-100 GHz, the permittivity of pure water and

154 CHAPTER 10. SENSITIVITY TO SALINITYthat of an aqueous saline solution (sea water or a more simple substan e) is des ribed bya simple Debye fun tion. The se ond assumption is that for a given ele trolyte, the Debyeparameters are related to the Debye parameters of pure water by a linear fun tion of the on entration.In the present work, we will only onsider the model developed by Klein and Swift (basedon Stogryn's model). Aqueous solution of NaCl have been used to estimate the permittivitydata. Stogryn reasoned that sin e NaCl is the prin ipal ioni onstituent of sea water, thediele tri behavior of sea water will not be signi� antly di�erent of that of an aqueous NaClsolution of the same salinity. Moreover, Stogryn points out that there is no eviden e toindi ate that �1sw depends on salinity and onsider the variation of this Debye parameter withtemperature to be very low so that it is onsidered onstant. Klein and Swift used exa tlythe same analyti al expressions than Stogryn for the Debye parameters but modi�ed themnumeri ally by taking into a ount the measured data of Ho and Hall whi h were performedat 1.43 GHz for an NaCl solution.The following graphs show �0 and �00 as a fun tion of temperature for a few representativesalinities (�gure 10.4) and as fun tion of salinity for a few representative temperatures (�gure10.5), for Klein and Swift's model at 1.575 GHz.

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10.2. THEORETICAL ANALYSIS 155

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156 CHAPTER 10. SENSITIVITY TO SALINITYPenetration depthFor a plane wave propagating in a lossy medium in the z dire tion, the ele tri -�eld intensityat a point z is given by: E(z) = Eo exp(�Cz); (10.11)where Eo is the �eld intensity at z=0, and:C = A+ jB; (10.12)where C, A and B are the propagation, absorption and phase onstants of the medium. Theyare related to � by: A = kojIm(p�)j; (10.13)B = koRe(p�):where ko = 2�=�o is the wave number in free spa e, and �o is the free-spa e wavelength inmeters. Ignoring s attering losses in the medium, the power density S(z) at a point is givenby: S(z) = So exp(��az); (10.14)where �a is the power absorption oeÆ ient and is related to A by:�a = 2A: (10.15)Often, �a and A are ea h expressed in dF�m�1 through the relation:�a(dB �m�1) = 4:34�a(Np �m�1): (10.16)A related quantity of interest in remote sensing is the penetration depth Æp. For a frees atterer medium: Æp = 1=�a = �o4�Im(p�) : (10.17)The depth of the skin-layer Æp (depth of the ele tromagneti wave's penetration) at frequen y

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Figure 10.7: Penetration depth in m versus salinity for di�erent temperatures. Klein's modelat 1.575 GHz.

10.2. THEORETICAL ANALYSIS 157f hanges in relation to the water parameter S and T , sin e � is dependent on this parameters.It an be al ulated by the formula 10.17.The al ulations show that the depth of penetration of millimeter and entimeter mi- rowaves in the o ean water is equal to Æp � (0:01 � 0:1)�o. In this wavelength range, thevalue of Æp weakly depends on salinity and temperature of the water. But in the de imeterrange, the depth of the skin-layer depends essentially on salinity and temperature, and anbe equal to several entimeters (see �gure 10.7).One an see from these urves that understanding the physi s of the skin-layer, and parti -ularly the verti al distributions of salinity and temperature in this layer is ru ial for remotesensing of salinity using L-band GNSS signals.10.2.3 How do the polarization oeÆ ients depend on salinity?Using the diele tri onstant model presented above, we analyze the polarization oeÆ ientsURR and URL in amplitude and phase. Figure 10.8 and �gure 10.9 present the amplitude andphase of URR and URL, respe tively, versus the sine of the satellite's elevation for di�erentsalinities and temperatures.The temperature does not a�e t signi� antly the magnitude of the polarization oeÆ ients.However, it shifts the phase by a few degrees.The oeÆ ient jURRj de reases with the elevation angle, so that at nadir the right om-ponent of the in ident �eld is totally onverted into the left omponent. The phase of URRtends to stabilize as the elevation in reases, and there is a di�eren e of about 10o at nadirbetween pure and salted water. On the ontrary, jURLj in reases with elevation angle andslightly in reases with salinity. The phase term behaves di�erently from the ase S=1 psuto S=30 psu at low elevation angles (we observe higher sensitivity for the salted water), andstabilizes at higher elevation. The di�eren e in �RL between the pure and salted ase at nadiris about 2o. This represents a very small phase shift of about one millimeter.

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S=30 psu (b) PhaseFigure 10.8: Amplitude and phase of URR versus sin . Both pure water ase (blue urve)and salted water ase (red urve) are plotted for T=10o C (full line) and T=20o C (dashedline).

158 CHAPTER 10. SENSITIVITY TO SALINITYIt is expe ted for a perfe t ondu tor (and still assuming the total in-plane s attering)that we get URR=0 and URL=-1. This means that the phase of URL should tend to �180oas the imaginary part of � tends to in�nity. On the ontrary, we observe in �gure 10.9(b) a lear departure of �RL with the value �180o for a higher salinity. This behavior is due tothe fa t that in the range S=0 to 40 psu, the permittivity and diele tri loss fa tor are ofthe same order of magnitude. The graph on �gure 10.10(a) illustrates the behavior of �RLat nadir as fun tion of the real part of � for di�erent values of the imaginary part. Similarly,the behavior of �RL at nadir as fun tion of the imaginary part of � for di�erent values of thereal part is presented in �gure 10.10(b). These two graphs highlight the impa t of the relativemagnitudes between the permittivity and the diele tri loss fa tor.

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(b) PhaseFigure 10.9: Amplitude and phase of URL versus sin . Both pure water ase (blue urve) andsalted water ase (red urve) are plotted for T=10o C (full line) and T=20o C (dashed line).

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(b)Figure 10.10: (a) Phase of URL at nadir versus permittivity. Diele tri loss fa tor is 0, 50 and100. (b) Phase of URL at nadir versus diele tri loss fa tor. Permittivity is 0, 65 and 85.

10.3. DATA ANALYSIS 159To on lude, all the information needed to derive sea-surfa e diele tri onstant is on-tained in the polarization oeÆ ients. They present a slight dependen y to salinity both inamplitude and phase. The analysis of these oeÆ ients showed two regimes in the phase. Atransition regime o urs for low elevation angles, where the phase variation is quite sensitiveto the salinity. For higher elevation angles, the phase stabilizes and a phase shift of about 0.2 m should be dete ted in the ross polarization oeÆ ient URL, in order to dete t a variationin salinity of �S=30 psu.10.2.4 Power modelThe e�e ts of the water surfa e salinity on the re eived power are shown in �gure 10.11. Themodeled power presented in equation 10.7 is plotted as a fun tion of the satellite elevationangle. The plots are done for S=1 psu (blue urve) and 30 psu (red urve), T=11o Celsius, andh=0.56 m, whi h orresponds approximately to the experimental onditions. It orresponds,a ording to the used model, to � = 82:3579+i10:8664 for S=1 psu and � = 75:5106+i47:1052for S=30 psu.The antenna gain is taken into a ount onsidering elevation and azimuth variations ofPRN 17. We have he ked that water surfa e temperature, with ranges observed during ourexperiment, does not impa t signi� antly the power interferen e pattern. As observed, weexpe t the e�e ts of salinity on the re eived power to be very small.

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Figure 10.11: Power model versus elevation angle for two di�erent values of salinity (S=1 psuand S=30 psu). The experimental onditions are similar to the Salpex ones. T=11oC. PRN17.10.3 Data analysisWe present in this se tion the results of the measured power as a fun tion of the satellite'selevation. The power is derived from the SNR (in amplitude) provided by the TurboRoguere eiver for ea h sele ted satellite. The interferen e pattern is shown for pure water (1psu)and salted water (30 psu) with PRN 15 ( f �gure 10.12(a)) and PRN 17 ( f �gure 10.12(b)).

160 CHAPTER 10. SENSITIVITY TO SALINITYSeveral observations an be made:� The data repeat quite well. The repeatability is better for salted water, as shown in �gure10.13, where the di�eren e between the two measurements (with same experimental onditions) are plotted both for pure and salted water. This is onsistent with the fa tthat a higher ondu tivity of the surfa e produ es less noisy data.� The drops observed with a salted water appear to be deeper than the ones observedfor a pure water. This is also onsistent with the fa t that a high ondu tive mediumprodu es deeper fades in the interferen e pattern.� For a given salinity, the patterns di�er from one PRN to the other. The azimuth of thesatellite seems to orrupt strongly the interferen e pattern.� The position of some drops seems to orrespond with the ones predi ted by the model.However, there are drops that are not predi ted by the model, espe ially for high eleva-tion angles.� The main trend of the interferen e pattern with elevation angle seems di�erent from themodel. Indeed, the modeled power de reases of about 12 dB in the range 0 to 60o, asshown in �gure 10.11. Moreover, this trend hanges with the PRN; it is thus sensitiveto its azimuth.

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(b) PRN 17Figure 10.12: Interferen e pattern as a fun tion of the sine of the elevation for pure water(blue urve) and salted water (red urve).It is obvious that the model presented in �gure 10.11 doesn't �t the observed data. Thedrops position and shape do not orrespond to the ones in the modeled interferen e pattern.The presen e of additional drops|i.e., higher frequen y in the signal|provides the proof ofthe presen e of re e tions with the ground. The model has been then modi�ed by taking intoa ount these re e tions:P 0 / ���GR(1 + URRwei�w + URRgei�g) +GL(URLwei�w + URLgei�g )���2 ; (10.18)where subs ript w stands for water ontribution and g stands for ground ontribution. A�rst attempt to �t the experimental data has been done using a ground diele tri onstant at

10.3. DATA ANALYSIS 161

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(b) PRN 17Figure 10.13: Repeatability of the data. Plot of the power di�eren e between two measure-ments with same experimental onditions, for both pure water (blue urve) and salted water(red urve).

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Figure 10.14: Fit of the Salpex experimental data using model presented in equation 10.18,when the ground ontribution is taken into a ount. The data orrespond to a salted water(30 psu). PRN 17.

162 CHAPTER 10. SENSITIVITY TO SALINITYL-band of about �g = 3+ i0:02 (from http://home.earthlink.net/�jimlux/radio/soildiel.htm).Figure 10.14 presents the �t of the data taking into a ount the ground ontribution in ludedin the formula 10.18. The �t spans for elevation angles ranging from 24o to 50o.It is lear that we have dete ted a non negligible ontribution of the ground. The watersurfa e area (delimited by the swimming pool) is not large enough to ontain the wholefootprint of the re e tion, although the antenna has been positioned lose to the surfa e onthis purpose. This ontribution shall explain mostly the observed sensitivity of the trend ofthe interferen e pattern with the azimuth of the satellite in view. As a matter of fa t, it isquite diÆ ult to model the ground re e tions, a ording to the omplexity of the lutter andobsta les around the swimming pool (trees, grass, ben h, ...).10.4 Dis ussion on salinity estimationIn radiometry, salinity is measured through the brightness temperature of the sea surfa e. Asimple model for brightness temperature TB introdu es the sea-surfa e emissivity e:TB = e � T: (10.19)At a �xed temperature in a smoothed o ean surfa e (i.e., assuming the e�e ts of wind-indu edsurfa e roughness, wind generated foam and white aps negligible) in reasing salinity is asso- iated with de reasing emissivity and the lower the in iden e angle, the higher the sensitivityof brightness temperature to salinity. It an be shown that for a 20o surfa e temperature a1% hange in brightness temperature orresponds to a variation in salinity of 2 psu.In bistati remote sensing, salinity is measured through the magnitude squared of theFresnel re e tion oeÆ ient, whi h is related to emissivity:jRj2 = 1� e: (10.20)We de�ne the per ent hange in power � between S=40 psu and S=20 psu at T=20oC as:� = 100 � jURL(S = 40psu; T = 20oC; )j2 � jURL(S = 20psu; T = 20oC; )j2jURL(S = 30psu; T = 20oC; 90o)j2 : (10.21)Figure (10.4) shows this per ent hange in power as a fun tion of the sine of elevation. Asexpe ted, the sensitivity is higher around the Brewster angle and rea hes 5% of hange inmeasured power to get a variation in salinity of 20 psu. In other words, a 1% hange in power orresponds to a variation in salinity of 4 psu. This is twi e what we obtain with urrentradiometers, i.e, half the sensitivity.This sensitivity may still hold to get valuable information on sea-surfa e salinity. Anexperiment onsisting in a �xed ground GNSS antenna re ording long time power series wouldbe interesting to on�rm this �gure and also understand how sea-surfa e roughness orruptsthe salinity estimation.10.5 Con lusions on salinity retrieval with GNSS-RThe Salpex Experiment has been a �rst attempt to determine whether it is possible to deter-mine water surfa e diele tri onstant through the use of GNSS re e tions. We have analyzedthe power oming from a RHCP antenna that gathers interferen es between dire t GNSSsignal along with re e ted signals over a alm water surfa e.

10.5. CONCLUSIONS ON SALINITY RETRIEVAL WITH GNSS-R 163

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Figure 10.15: Per ent hange in power between S=40 psu and S=20 psu as a fun tion of sineof elevation angle (see equation 10.21).This was a simple approa h, and more omplex set-ups an ertainly be devised. However,our �rst goal was to perform a simple analysis of the impa t of salinity on the signals, anddevelop the models.The derived model uses the Klein and Swift's model for the diele tri onstant, and statesthat the salinity impa t on the return power is very poor. All the information needed toderive sea-surfa e diele tri onstant is ontained in the polarization oeÆ ients. The anal-ysis of these oeÆ ients showed two regimes in the phase. A transition regime o urs forlow elevation angles, where the phase variation is quite sensitive to the salinity. For higherelevation angles, the phase stabilizes and a phase shift of about 0.2 m should be dete ted inthe ross polarization oeÆ ient URL in order to dete t a variation in salinity of �S=30 psu.Thus, di�erential phase analysis of the return power ould in prin iple provide anothermean for salinity determination, given the fa t that the instrument is a urate enough todete t small hanges in the phase. However, a target of <1 mm in altimetry is a hard one,even from a �xed platform and after 1 month averaging (re all that salinity is in prin ipleslowly varying). This means that using an altimetri approa h for salinity monitoring will be hallenging.In turn, we an also on lude from this analyti al study that salinity will not impa t singlefrequen y L-band altimetri measurements beyond the level of a few mm.Another approa h is to fo us on the re e ted power. We have seen that in order to besensitive to salinity we should aim at a 1% relative power pre ision after 1 month averaging.The term \relative" is important. In prin iple, the best approa h, exploiting the te hnique'sstrengths, is to fo us on the re e ted over dire t power, a ratio more immune to equipmentvariability.The analysis of the data shows a noisy interferen e pattern giving lear fade events with avery good repeatability. The noise, whi h is mu h larger than the predi ted salinity impa t,has been partially understood by the presen e of re e tions oming from the ground and

164 CHAPTER 10. SENSITIVITY TO SALINITYother multipaths. A better �t of the data has been done by taking into a ount the ground ontribution, whi h orrupts quite signi� antly the interferen e pattern.Neither the apparatus nor the experimental onditions were appropriate to judge whetherwe an rea h a 1% relative power pre ision after 1 month averaging. A se ond Salpex ex-periment using more pre ise equipment near the open sea or a larger swimming pool wouldimprove strongly the quality of the data.

Chapter 11Emissivity RetrievalThis Chapter addresses the analysis of re e ted GNSS signals gathered during the experi-mental ampaign arried out in the frame of the L-band Radiometry Experiment, an IfremerContra t (see [Soulat and RuÆni2002℄). The purpose of this analysis is to provide some pre-liminary on lusions on the apability of a GNSS pro essor as a radiometer. In the followingse tion, we provide the des ription of the experiment. A power model is then proposed inse tion (11.2), and estimations of the parameters of interest are derived. For radiometri purposes, we introdu e the parameter K whi h represents the ratio of dire t and re e tednoise varian es. A simulation has been arried out to he k the validity of the estimators, inse tion (11.3). The experimental data are then analyzed in se tion (11.4).11.1 Experimental ampaignThe experiment took pla e at the Bar elona harbor the 4rth of September 2002. We thankJordi Vil�a from the Bar elona Port Authority. Two antennas were deployed, as shown in�gure 11.1, over a 30 meter high building lo ated on a pier. The material was similar to theone used during the Eddy Experiment Flight (see Chapter 7) and the Salpex Experiment(see Chapter 10). For the purpose of sea-surfa e emissivity hara terization, the experiment onsisted in gathering GNSS-R over a alm sea, for two di�erent antenna on�gurations:� Part A: both RHCP and LHCP antennas are zenith pointing.� Part B: RHCP antenna is zenith pointing and LHCP antenna is nadir pointing.The underlying idea is that the signature of sea-surfa e emissivity should be observed by omparing the statisti s of the LHCP antenna output signal between Part A and Part B. Thewind speed at ten meters was around 4 m/s, so that sea-surfa e roughness variability e�e tswere redu ed.Thirty se onds of data have been analyzed for ea h part of the experiment. An integrationtime of 10 ms has been used in Starlab's GNSS pro essor. PRN 01 was the only good andidate to make the orrelations, sin e re e tions were free of shadowing given its elevationand azimuth. Figure 11.2 shows the time serie of the peak power (in dB) using PRN 01, duringPart A and Part B, respe tively. The orresponding satellite elevation is 21o (its azimuth is61o) during Part A, and 20o (azimuth 62o) during Part B. Several main observations an bemade:� The peak power from RHCP antenna is quite steady in both Part A and Part B, asexpe ted.

166 CHAPTER 11. EMISSIVITY RETRIEVAL� In Part A, we observe many fadings in the power gathered by the LHCP antenna.However, some good orrelation peaks appear from time to time.� In Part B, the orrelation peaks of the LHCP antenna are mu h higher. The down-looking antenna has gathered re e ted signals.

COASTAL PIERREFLECTED SIGNAL

DIRECT SIGNAL

SEA SURFACE

Figure 11.1: Geometry of the experiment during Part B.

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Figure 11.2: Time series of the peak power (in dB), using PRN 01, of RHCP antenna (blue urve) and LHCP antenna (red urve). Left: Part A, the orresponding satellite elevation is21o (its azimuth is 61o). Right: Part B, the elevation is 20o (azimuth 62o).11.2 Power modelThis se tion presents a statisti al analysis of the dire t and re e ted squared waveforms, basedon a simple power model introdu ing a spe kle noise model. We derive analyti expressions

11.2. POWER MODEL 167of the 1-D mean power waveform and its normalized varian e. Similarly, the statisti s of theratio of the two signals is presented.First, we re all that the waveform is generated through a orrelation pro ess de�ned by:Ca;b(�) = 1Ti Z Ti0 a(t)b�(t� �)dt; (11.1)where Ti is the oherent integration time.Se ond, the dire t return power in delay is omputed from the squared omplex orrelation oeÆ ients (in two hannels I and Q) of the repli a R with the in oming signal S+NdI+jNdQ.We assume NdI and NdQ to be two white Gaussian noises with mean zero and varian e �2d.The dire t squared orrelation oeÆ ient thus writes:~Pd(�) = 0� CS+NdI ;R(�)qCS+Nd;S+Nd(�)CR;R(�)1A2 +0� CNdQ;R(�)qCS+Nd;S+Nd(�)CR;R(�)1A2 : (11.2)Applying the large noise assumption that hara terizes GNSS signals (�d is greater than thesignal strength) to equation 11.2, we obtain:~Pd(�) = [CS;R(�) +CNdI ;R(�)℄2�2d + C2NdQ;R(�)�2d : (11.3)CS;R(�) is here a deterministi triangle fun tion, depending on the integration time Ti, the hip length, the antenna gain and the geometry (emitter elevation, re eiver position, ...). Asimple expression is: CS;R(�) = Ad�(�), with Ad the maximum amplitude of the waveform.We re all that GNSS re e tions ome essentially from the ontribution of s attering pointson the sea surfa e. The averaged ontribution of these di�erent s atterers introdu es fadingphenomena, whi h an be des ribed by a spe kle noise. We an then de�ne the re e ted poweras: ~Pr(�) = hCpnS;R(�) + CNrI ;R(�)i2�2r + C2NrQ;R(�)�2r ; (11.4)where, similarly, NrI and NrQ are two white Gaussian noises with mean zero and varian e�2r . We assume n to be a spe kle noise following a Gamma Law with mean �=1 and order L,su h as: P(n) = LL��(L) �n��L�1 e�Ln� : (11.5)The parameter L is de�ned as the shape parameter of the distribution. It is linked to thedegree of development of the sea surfa e: for a fully developed spe kle, L equals 1. We have:hni = � = 1hn2i = 1L + 1: (11.6)Considering the ensemble average over su essive waveforms, we get respe tively for the dire tand re e ted powers: h ~Pdi = A2d�2�2d +p2��nTi ; (11.7)h ~Pri = A2r�2�2r +p2��nTi ; (11.8)

168 CHAPTER 11. EMISSIVITY RETRIEVALwhere �n is the orrelation length of the noise.Sin e the GNSS-R system is at low altitude in this experiment, the Delay-Doppler Mappingover the sea surfa e is mu h wider than the portion of the surfa e ontributing to the re e tionpro ess (the glistening zone). The orrelation waveform of the re e ted signal an thus beapproximated by a triangle fun tion, like the dire t one. Note, that the noise power is inverselyproportional to the number of independent samples, de�ned by the ratio of the oherentintegration time and the noise orrelation time.We now onsider the normalized varian e de�ned as:NV (�) = h ~P 2i � h ~P i2h ~P i2 ; (11.9)and we obtain for the dire t and re e ted waveforms, respe tively:NVd(�) = p8�A2d�2�2d �nTi + 2� � �nTi �2A4d�4�4d +p8�A2d�2�2d �nTi + 2� � �nTi �2 ; (11.10)NVr(�) = 1L A4r�4�4r +p8�A2r�2�2r �nTi + 2� � �nTi �2A4r�4�4r +p8�A2r�2�2r �nTi + 2� � �nTi �2 : (11.11)We introdu ed, in the above expressions, the ratios A2d�2=�2d and A2r�2=�2r , whi h an be seenas a Signal to Noise Ratio. In the grass (i.e., low SNR), we have for both dire t and re e tedsignals: NV (�) � 1: (11.12)For time lags around lag zero (i.e., high SNR), we have:NVd(�) � p8� �2dA2d�(�)2 �nTi ; (11.13)NVr(�) � 1L: (11.14)As expe ted, equation 11.13 states that the normalized varian e gets higher values for highnoise power. In the s ope of this study, the parameter of interest is �2r . It is not possible toinfer this parameter from equation 11.14. Note that the normalized varian e is 1 when thespe kle is fully developed (L=1).We have seen, in this �rst analysis, that the normalized varian es are fun tion of the ratioA2�2=�2. Due to a la k of knowlegde on A, the study of the ratio of the two signals (dire tand re e ted) seems more appropriate. It introdu es, as shown, the ratio of the two noisevarian es.11.2.1 Re e ted to dire t signal ratio analysisWe fo us now on the statisti al des ription of the ratio of the two waveforms:~Pratio(�) = ~Pr(�)~Pd(�) : (11.15)

11.3. SIMULATIONS 169This analysis was motivated by the fa t that, as mentioned above, the re e ted power has no(or almost no) trailing edge for a low re eiver platform. However, the intensity has droppeddown with a fa tor �, due to the re e tion pro ess:Ar = p�Ad: (11.16)The parameter � is essentially a fun tion of the sea-state and the Fresnel oeÆ ients.At lag zero, we have: h ~Pratio(0)i � �K : (11.17)We have introdu ed here the parameter K = �2r=�2d. Equation 11.17 is the basis of emissivityretrieval, through the estimation of K. It is however orrupted by the � parameter whi h issea-state dependent.11.2.2 SummaryWe presented a power model for GNSS-R data olle ted at low altitude. The proposed pa-rameters are:� �2d and �2r : the noise varian es of the dire t and re e ted signals, respe tively. Theseparameters are the ones to be estimated in the s ope of this study. Their ratio K =�2r=�2d is the appropriate parameter to be investigated.� L: shape parameter of the Gamma Law. It provides the degree of development of thespe kle noise. It is sea-state dependent.� �: attenuation fa tor due to the re e tion pro ess. It is also sea-state dependent.11.3 SimulationsSimulations have been arried out to he k the validity of the previous equations, and verifythat there is no undesired e�e t due to the one-bit sampling. We fo us espe ially on equationsnumber 11.8, 11.13 and 11.17.We onsider, as an input of the orrelator, the following signal:S(t) = CA(t)ej!t +NI(t) + jNQ(t); (11.18)where CA is the C/A ode of PRN 01, NI and NQ two random variables with mean zero andvarian e �2. The arrier frequen y we onsider here is the IF frequen y: 308 kHz. This signalis one-bit sampled with a sampling frequen y of 20.456 MHz. These frequen ies are in linewith the ESA equipment spe i� ations. Before making the orrelation between the signal andthe repli a, we remove the arrier and onsider the signal S0 = S(t)e�i!t.The result of the mean squared orrelation over 1000 realizations is shown in �gure 11.3,for di�erent noise varian es �2. The hoi e of these varian es is related to the observed peakvalue in the data. The width of the waveform is approximately 40 samples sin e the C/A ode hip length is about 1 mi rose ond (we re all that there are 1023 hips in one C/A odeperiod, one hip lasts 1/1023 ms).As observed, the mean power drops down when the C/A ode is orrupted by a randomnoise. This in agreement with equation 11.7 (or 11.8). Furthermore, the signal is onstantin the grass whatever the value of �, as shown in the right hand side graph of �gure 11.3.This is also onsistent with equation 11.7. However, we note that the noise varian e an be

170 CHAPTER 11. EMISSIVITY RETRIEVAL

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Figure 11.3: Left: Mean power h ~P (�)i for di�erent noise levels (�=10, 20, 40 and 60). The hoi e of these varian es is related to the observed peak value in the data. The peak of orrelation is one for a lean signal. Right: Zoom of the squared orrelation in the grass,before lag zero (peak position). 1000 realizations, PRN 01, Ti=10 ms.estimated through the orrelation pro ess itself (without any normalization) if the data aresampled at 2 bits (or more).Figure 11.4 shows the variations of the mean peak power versus the inverse noise varian e1=�2. As expe ted, it follows the relation:h ~P (� = 0)i / 1�2 : (11.19)

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11.4. DATA ANALYSIS 171as 1=�(�)2 ( f equation 11.13). The normalized varian e at lag zero is plotted versus �2 onthe right hand side of �gure 11.5, whi h shows that equation 11.13 is valid.

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Figure 11.5: Left: Normalized varian e of the power for di�erent noise levels (�=10, 20, 40and 60). Right: Normalized varian e at lag zero versus �2. 1000 realizations, PRN 01, Ti=10ms.11.3.1 Re e ted to dire t signal ratio analysisWe onsider now in our simulations the power ratio of two GNSS signals:� the dire t signal with varian e �2d,� re e ted signals with di�erent varian es �2r ranging from �2d to 5�2d .As observed in �gure 11.6, we easily he k that the variations of the mean peak power areproportional to K�1, whi h validates equation 11.17. Hen e, looking at the ratio of the twosignals seems powerful if the attenuation fa tor is known. A model for � is therefore needed.11.4 Data analysisFigures number 11.7 and 11.8 present the mean power gathered by the RHCP antenna (thi k urve) and the LHCP antenna (dashed urve) during Part A and Part B, respe tively. Asexpe ted, the peak power re eived by the LHCP antenna is quite low during Part A. It ismu h higher during Part B, sin e the GNSS signal be omes mostly LHCP at the re e tion.As far as the noise is on erned, we �rst observe, in both on�gurations, that noise levels areslightly di�erent from one antenna to another. We also note that the noise level of the dire tsignal hanges a bit from Part A to Part B.Figure 11.9 shows the normalized varian e of the squared orrelation for the RHCP antenna(thi k urve) and the LHCP antenna (dashed urve). The urves are onsistent with the modeland simulations in term of shape and relative magnitudes. We parti ularly noti e that thenormalized varian e of the LHCP antenna in Part A is one all along the waveform. The SNRis indeed not high enough. During Part B, however, the normalized varian e behaves like1=�(�)2 and is above the RHCP urve.

172 CHAPTER 11. EMISSIVITY RETRIEVAL

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11.4. DATA ANALYSIS 173

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174 CHAPTER 11. EMISSIVITY RETRIEVAL11.5 Con lusions on radiometry with GNSS-RA simple GNSS-R ground experiment has been arried out to hara terize the signal powerand relate its statisti s to emissivity. For this purpose, a GNSS-R power model has beenpresented with theoreti al estimation of the parameters of interest su h as the ratio K of thedire t and re e ted signal noise varian es. The model has been validated through simulations,spe i� ally to verify that it is valid in the ase of one-bit sampled data, as those gatheredby the ESA equipment. Experimental data have been then analyzed demonstrating that theinvestigation of the statisti s between su essive waveforms ould be e�e tive for sea-surfa eemissivity measurement.The radiometri solution using GNSS-R ould be improved by onsidering the followingpoints:� As observed in the derived equations, sea-state dependent parameters are orruptingthe estimation of the emissivity noise. Models for parameters � and L are nedeed toimprove this estimation.� The use of another re ording devi e, whi h would sample the data at two bits, say,would de�nitely improve the thermal noise estimation, by a lose analysis of the grassof the orrelation.

Con lusions and Perspe tivesThis study investigated the GNSS-R remote-sensing apabilities for o eanographi appli a-tions. We depi ted GNSS-R sensitivity to sea-surfa e geophysi al parameters, su h as di-re tional mean square slope, salinity and emissivity, through theoreti al and experimentalanalyses. Parti ular emphasis was put on the spe ulometri appli ation, through the �rstinversion of GNSS-R full Delay-Doppler Maps. Solar bistati re e tions was also analyzedthrough a similar sea-surfa e mapping (Tilt-Azimuth Map) in order to enhan e the under-standing of spe ular s attering.The �rst part of this dissertation depi ted the me hanisms involved in the bistati s at-tering pro ess at L-band. It in luded the knowledge of GNSS signal hara teristi s, ele tro-magneti s attering model, random sea-surfa e features and basi s of pro essing of dire t andre e ted signals. From a theoreti al point of view, we parti ularly fo used on the validationat L-band of the GO approximation of the Kir hho� s attering theory, through simulations.A new model of the s atterometri 2-D waveform model (DDM) was then proposed. TheDDM data produ t is the basis of sea-surfa e roughness parameter inversion.In the se ond part, a simulator of GNSS-R signals was developed from the knowledgea quired in Part I. This simulator is omposed of two possible models (GRADAS and IS)based on the Kir hho� theory and GO model. An appli ation of this simulator was reportedfor the study of SWH retrieval from a oastal platform. It was shown that the behavior ofthe re e ted �eld is sensitive to sea-surfa e height dynami s. In parti ular, the simulationswith wind-driven o ean spe trum showed that the RMS of the amplitude of the derivative ofthe omplex �eld in reases linearly with SWH. This is a promising future line of resear h.The third part of the study provided experimental results on s atterometri GNSS-Rairborne apabilities, based on a ight experiment arried out in the s ope of the ESA PARIS- proje t. First, we reported a repetition of the Cox and Munk experiment, through theinversion of opti al high-resolution images whi h have been simultaneously gathered duringthe experiment (SORES data). The derived MSS enhan ed the understanding of GNSS-Rs atterometri measurements. In parti ular, the analysis highlighted the important impa tof sea-state (SWH) in addition to wind stress over DMSS, and a non-negligeable departurefrom Gaussianity of the sea-surfa e slope PDF. We observed that estimated opti al MSS an be larger by up to 20% than MSS predi ted by a wind-driven spe trum. The SORESte hnology also permitted to measure the swell dire tion and asso iated long wave varian ethrough spe tral analysis of the photographs.

176 CONCLUSIONS AND PERSPECTIVESAn inversion pro edure for DDM has then been presented and DMSS revealed to be very onsistent with models and ground truth, meaning that the developed inversion method wase�e tive. Our results showed that both opti al and L-band total MSS are 20% higher thanwhat predi ted by Elfouhaily's model for the observed wind speed (9 to 13 m/s). The SPAestimated by GNSS-R mat hed the swell dire tion with good a ura y for most ases. A newgeophysi al produ t has been put forward: the slope PDF isotropy whi h an be related towind/wave misalignment as well as sea degree of development.We emphasized that these two forward s attering instruments (GNSS-R and SORES)are sensitive to DMSS, the dire tional probability distribution of spe ular s atterers, andonly indire tly to wind speed and dire tion: the relationship is modulated by other physi alphenomena a�e ting DMSS, su h as swell magnitude and dire tion. This s atterometri analysis showed that the sea-surfa e roughness is a self-standing produ t, whi h an notbe dire tly asso iated to the wind. The roughness of the o ean surfa e an be seen as a ombination of several physi al pro esses. Therefore, roughness variations observed on thesurfa e do not ome systemati ally from the wind stress.Finally, spe i� studies on other possible GNSS-R appli ations have been reported ina fourth part. They provided some preliminary on lusions on the possible salinity andemissivity retrievals, through simple ground experiments.The goal of the Salpex Experiment was to perform a simple analysis of the impa t ofsalinity on the signals, and develop the models to ompare it to. On this purpose, we haveanalyzed the power oming from an RHCP antenna that gathers interferen es between dire tGNSS signals along with re e ted signals over a alm water surfa e. The analysis of the datashowed a noisy interferen e pattern giving lear fading events with a very good repetivity.The main observation was that the noise is mu h larger than the predi ted salinity impa t,whi h is theoreti ally very tiny. A phase approa h ould in prin iple provide the means forsalinity determination, given the fa t that the instrument is a urate enough to dete t small hanges in the phase. However, the needed target of <1 mm in altimetry is a very hallengingone, even from a �xed platform and after 1 month averaging. Another approa h was to fo uson the re e ted power. We have seen that in order to be sensitive to salinity we should aimat a 1% relative power pre ision after 1 month averaging. In prin iple, the best approa h isto fo us on the re e ted over dire t power, a ratio more immune to equipment variability.The purpose of the L-band Radiometry Experiment was to provide some preliminary on lusions on the apability of the GNSS pro essor as a radiometer. We provided a powermodel and algorithms to estimate the parameters of interest (emissivity noise varian e). Themodel has been he ked and validated through simulations. The ratio of dire t and re e tedsignals appears to be the more appropriate way to infer emissivity, by estimating the ratio ofthe noise varian es. However, its estimation relies on an a priori knowledge of the sea-state.From the experien e of this study, s atterometry with GNSS-R requires more ight ex-periments, in order to onsolidate the on ept of GNSS-R sea-roughness retrieval. Let usemphasize that the Eddy Experiment Flight was not optimized for spe ulometry (1000 maltitude, 50 m/s speed) and that higher/faster ights are needed. The strength of this te h-nique would re-enfor e signi� antly the remote-sensing ommunity. It appears learly, forinstan e, that future radiometers su h as the Soil Moisture and O ean Salinity ESA mission(SMOS)|working also at L-band|will require a good knowledge of DMSS for alibration.

CONCLUSIONS AND PERSPECTIVES 177As far as the emissivity and salinity retrievals are on erned, more experimental ampaignsare needed over a real sea surfa e.Regardless of the appli ation, altimetry, spe ulometry or radiometry, the important thingto keep in mind is that GNSS-R is a di�eren ing approa h (re e ted-dire t signals). Engineer-ing e�orts should fo us on stability of the system and maximal exploitation of the di�eren ings heme.

178 CONCLUSIONS AND PERSPECTIVES

Part VAppendix

179

Appendix ARelevant SymbolsSymbol Units Name� rad, degrees Sea-surfa e slope azimuth� rad, degrees Sea-surfa e slope tilt rad, degrees Satellite elevation� rad, degrees In iden e angle with respe t to the normal to the surfa e� meters Sea-surfa e height� meters Emitted wavelength (� 20 m at GNSS frequen ies)� Sea-water diele tri onstant� rad, degrees Sun elevation� se onds Time delay between dire t and re e ted GNSS signals�P se onds P ode hip length (� 30 m)�C=A se onds C/A ode hip length (� 300 m)�o S attering oeÆ ient�� meters Sea-surfa e RMS height�2s or MSS Sea-surfa e slope varian e�2 Sea-surfa e slope varian e in the rosswind dire tion�2u Sea-surfa e slope varian e in the upwind dire tionA(~�) Sea-surfa e height auto orrelation fun tionl meters Sea-surfa e orrelation lengthP� Sea-surfa e height PDFPs Sea-surfa e slope PDF~q = (~q?; qz) rad.m�1 S attering ve torR Fresnel oeÆ ientr meters Mean radius of urvature of surfa esx Sea-surfa e slope along x-axissy Sea-surfa e slope along y-axisS psu Sea-surfa e salinityT degrees Celsus Sea-surfa e temperatureTa se onds In oherent integration timeTi se onds Coherent integration timeUheight m/s Wind speed measured at height m above sea level

182 APPENDIX A. RELEVANT SYMBOLS

Appendix BA ronymsAD Arti� ial DataARNS Aeronauti al Radio Navigation Servi eCEOS Committee on Earth Observation SatellitesCIL Con�den e Interval LengthDDM Delay-Doppler MapDMSS Dire tional Mean Square SlopeECMWF European Centre for Medium-Range Weather Fore astsESA European Spa e Agen yESTEC European Spa e Resear h and Te hnology CentreFCZ First Chip ZoneFFT Fast Fourier TransformGDR Geophysi al Data Re ordsGNSS Global Navigation Satellite SystemGNSS-R GNSS Re e tionsGO Geometri Opti sGPS Global Positioning SystemGRADAS GNSS Re e tions Arti� ial Data SynthesiserICC Catalan Cartographi InstituteIEEC Catalan Spa e Resear h InstituteINM (Spanish) National Meteorologi al InstituteINS Inertial Navigation SystemIS Intermittent S reamersLEO Low Earth OrbitLHCP Left Hand Cir ularly PolarizedMSL Mean Sea LevelMSS Mean Square SlopeMTF Modulation Transfer Fun tionNWP Numeri al Weather Predi tion modelNRCS Normalized Radar Cross Se tionPARIS PAssive Re e tometry and Interferometry SystemPDF Probability Density Fun tionPO Physi al Opti sPRN Pseudo Random Noise

184 APPENDIX B. ACRONYMSRA Radar AltimeterRHCP Right Hand Cir ularly PolarizedRMS Root Mean SquareRMSE Root Mean Square ErrorRNSS Radio Navigation Satellite Servi eSAR Syntheti Aperture RadarSMOS the Soil Moisture and O ean Salinity missionSNR Signal to Noise RatioSORES SOlar RE e tan e Spe ulometerSOW Se ond Of the WeekSP Spe ular PointSPA Slope PDF AzimuthSPA Stationary Phase ApproximationSPI Slope PDF IsotropySPM Small Perturbation MethodSSH Sea Surfa e HeightSWH Signi� ant Wave HeightTAM Tilt-Azimuth MappingT/P Topex/PoseidonUTC Coordinated Universal TimeWAF Woodward Ambiguity Fun tion

List of Figures1 Stru ture of Part I. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.1 Present GPS signals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.2 L1 signal stru ture. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.3 Stru ture of a GPS re eiver. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.4 Present and future ivil GPS frequen ies (from [Ma Donald2002℄'s paper). . . . . . . . . . 231.5 Geometry of multistati GNSS radar system (from V. U. Zavorotny, Bistati GPS SignalS attering from an O ean Surfa e: Theoreti al Modeling and Wind Speed Retrieval from Air raft Measurements, 1999). 251.6 Galileo frequen y plan (as of 2003) together with GPS and GLONASS systems.ARNS is the a ronym for Aeronauti al Ratio Navigation Servi e. This band isdedi ated to safety-of-life servi es (i.e., ivil aviation). RNSS is the a ronymfor Radio Navigation Satellite Servi e. . . . . . . . . . . . . . . . . . . . . . . . 272.1 Correlation fun tion for di�erent wind speeds (U10=1, 3, 5, ..., 17 m/s), withGaussian spe trum from [Elfouhaily et al.1997℄. . . . . . . . . . . . . . . . . . . 332.2 Sket h of sea-surfa e slope PDF and related frames. . . . . . . . . . . . . . . . 352.3 Variation of sea-surfa e slope varian e at L-band with urrent speed, with windspeed as a parameter (see equation 2.17). . . . . . . . . . . . . . . . . . . . . . 363.1 De�nition of the s attering ve tor ~q = k(~s�~i). . . . . . . . . . . . . . . . . . . 403.2 Ve tor geometry for s attering. . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.3 Asso iated parabola for spe ular point determination. . . . . . . . . . . . . . . 463.4 Fresnel �eld magnitude relative to surfa e resolution for two di�erent radii of urvature: (a) rx=ry=5 m, (b) rx=ry=10 m. The re eiver is at 500 metersover a 10�10 m2 paraboli surfa e. . . . . . . . . . . . . . . . . . . . . . . . . . 473.5 Fresnel �eld magnitude relative to the tabulation size of the surfa e for two dif-ferent radii of urvature: (a) rx=ry=5 m and (b) rx=ry=10 m. The resolutionis 3 m and the re eiver is at 500 meters transmitting in L-band. . . . . . . . . 473.6 E�e t of small ripples added on a very smooth paraboli wave. Under the urve,there is a di�eren e of �10% between Fresnel magnitude and the magnitude omputed on a perfe t paraboli surfa e in L-band. . . . . . . . . . . . . . . . . 483.7 Spe ular points number relative to wind speed for ba ks attering (25x25 metersurfa e with 5 m resolution). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.8 Spe ular points position for (a) U10=5 m/s and (b) U10=15 m/s. . . . . . . . . 503.9 Correlation between the re e ted power and spe ular point number on a Gaus-sian o ean model in L-band (L1=19 m) within 2 se onds: 5�5 m2, 5 mresolution, U10=15 m/s, re eiver at nadir in iden e at 50 meters. . . . . . . . . 503.10 Correlation between the spe ular points number and the re e ted power om-puted with the Fresnel integral, for an in ident wavelength L1=1 m or L2=20 m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51185

186 LIST OF FIGURES4.1 Data produ t levels in GNSS-R pro essing. . . . . . . . . . . . . . . . . . . . . 614.2 The top high level ow hart of the GNSS-R pro essor for PRN i. . . . . . . . . 624.3 Initialization module of the GNSS-R pro essor. . . . . . . . . . . . . . . . . . . 634.4 An example of orrelation peak in arbitrary units, for 5 ms of oherent inte-gration time, using the PRN ode number 9. . . . . . . . . . . . . . . . . . . . 644.5 E�e t of the navigation bit transition on the orrelation pro ess. . . . . . . . . 654.6 Example of the sear h for the navigation bit syn hronism. As the 20 ms long ode repli a slides on the signal (a, b, , d, e, f), the value of its produ t (green urve at the bottom) with the signal (�rst row) depends on the position of thenavigation bit (red) transition within the ode repli a. . . . . . . . . . . . . . . 664.7 Tra king module of the GNSS-R pro essor. . . . . . . . . . . . . . . . . . . . . 674.8 DDM of dire t (a) and re e ted (b) signals from the Eddy Experiment Flight(PARIS� ESA Contra t). The DDMs are plotted in amplitude times 103. . . 685.1 S hemati drawing of the minimum representative pat h. . . . . . . . . . . . . 755.2 Spe trum estimation of the true temporal spe trum of the re e ted �eld overa 200�200 m2 surfa e (thi k line) with 2 sub-pat hes only: 12.8 m side (dottedline), 25.6 m side (dashed line), 51.2 m side (dashed dot dot line). . . . . . . . 765.3 Inputs and outputs of the GRADAS simulator. . . . . . . . . . . . . . . . . . . 785.4 LEO geometry for the GRADAS simulator. . . . . . . . . . . . . . . . . . . . . 785.5 GRADAS algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.6 First hip zone: the 20.456 MHz sampling rate of the C/A ode orrespondsapproximatively to 20 samples ea h hip. The �rst hip zone is divided into 20rings. We an see an example of a pat h (25�25 m2) lo ated in zone 4. . . . . 815.7 From left to right are plotted the phase, the amplitude and the power spe trumof the re e ted �eld from 240 pat hes (6�400 m2) demo rati ally lo ated inthe 200Hz Doppler zone of the �rst hip zone. The re eiver moves at 7 km/salong the x-axis, its altitude is 500 km. The sampling rate is 10 kHz, 0.5 se ondevolution, U10=10 m/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825.8 The lower line in the left-hand �gure, represents the phase of the ele tromag-neti �eld generated with the demo rati pat hing of the o ean surfa e, beforeC/A ode modulation. In this example, 240 pat hes lo ated in the 200HzDoppler zone are onsidered, the re eiver is moving at 7 km/s, the wind speedis U10=10 m/s. The red urve is the phase retrieved with the GPS open looppro essor. On the right-hand side, the retrieved amplitude is shown with thedi�eren e in phase ( rosses) between the unwrapped phase. The relation be-tween fadings and the lose of the phase is learly observed. . . . . . . . . . . . 835.9 Flow hart of the ode implementing the IS-model. . . . . . . . . . . . . . . . . 855.10 The waveform (blue urve) resulting from the orrelation between the simulatedo ean-re e ted GPS data and a lear repli a of the orresponding C/A ode.In red, for purpose of omparison, the auto orrelation of the same ode. Theasymmetry of the waveform re e t the extension, in hips units, of the seasurfa e taken into onsideration in the signal generation pro ess. The numberof hips is the time-delay unit on the x-axis. On the y-axis, arbitrary units. . . 865.11 The phases as retrieved by the open-loop GPS pro essor (in bla k) and the onefrom the simulated physi al �eld (in red), after the spatial �ltering with thetriangular fun tion. A y le slip in the tra ker an be observed. . . . . . . . . . 87

LIST OF FIGURES 1875.12 The blue urve represents the di�eren e between the retrieved phase and theone of the ele tromagneti �eld. The green urve represents the �eld amplitude,in dBv. Clearly, when a fading o urs, the tra king of the phase an fail. Innormal onditions the phase is well re onstru ted. . . . . . . . . . . . . . . . . 886.1 The dynami L-band ele tromagneti �eld in a omplex phasor representationprodu ed by a wind-driven o ean model. . . . . . . . . . . . . . . . . . . . . . . 906.2 Standard deviation of the derivative of the re e ted phase (�Æ�) as a fun tionof sea elevation RMS �� . L5, 1 minute simulation. . . . . . . . . . . . . . . . . 916.3 Standard deviation of the re e ted amplitude (�r) as a fun tion of sea elevationRMS �� . L5, 1 minute simulation. . . . . . . . . . . . . . . . . . . . . . . . . . 916.4 Standard deviation of the derivative of the amplitude of the re e ted �eld (�Ær)as a fun tion of sea elevation RMS �� . L5, 1 minute simulation, with 4� errorbars. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 926.5 Left: Standard deviation �jÆF j of the amplitude of derivative of the re e ted�eld as a fun tion of sea elevation RMS �� . L5, 1 minute simulation. Right:Same plot with 10 s �jÆF j estimation (squares). . . . . . . . . . . . . . . . . . . 937.1 Map of the ight traje tory. The blue line shows the Jason's tra k. The pointsP1 and P2 mark the experiment boundaries, while the green points indi atethe positions of s heduled GPS buoy measurements. . . . . . . . . . . . . . . . 987.2 Views of the sea surfa e during the experiment from the air and sea. . . . . . . 987.3 Air raft set-up on�guration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 997.4 ESA equipment used during the Eddy Experiment Flight. . . . . . . . . . . . . 1007.5 Elevation and azimuth of three GPS satellites in view during the experiment. . 1017.6 Map of the one- hip iso-delays and iso-Dopplers (-100 Hz, 0 and +100 Hz) forPRNs 08 (blue), 10 (red) and 24 (green) at the beginning (left) and end (right)of the tra k. The -3dB antenna pattern is also represented in bla k. . . . . . . 1017.7 Wind �eld and SWH at 12 UTC in the observation area (data from INM). . . . 1027.8 (a) Mean square slope from Jason along the tra k in both Ku-band (blue urve)and C-band (red urve). (b) Wind speed from Jason along the tra k. . . . . . . 1037.9 The SWH from RA Jason in Ku-band (blue urve) and C-band (red urve). . . 1038.1 High resolution amera and GPS LHCP antenna lo ated at the bottom of theplane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1068.2 Lo alization of the snapshots along the tra k of the plane. Ea h photograph islabeled by a number ranging from 35 to 63. . . . . . . . . . . . . . . . . . . . . 1068.3 Top: Example of SORES raw data (photograph 47). Bottom: Example ofbreaking event observed during the experiment. . . . . . . . . . . . . . . . . . . 1088.4 Geometry of SORES sensing during the Eddy Experiment Flight. . . . . . . . . 1108.5 (a) Photograph 41 with the TAM. (b) Data intensity Id. . . . . . . . . . . . . . 1138.6 Examples of data and best-�t model TAMs. Top: Data TAMs. Middle:Best-�t model TAMs. Bottom: Data-model residual. . . . . . . . . . . . . . . 1148.7 Up-wind mean square slope and slope PDF isotropy along the tra k. . . . . . . 1168.8 Slope PDF azimuth and b00o along the tra k. . . . . . . . . . . . . . . . . . . . . 1168.9 (a) Centered intensity I in optimal region of photograph 47. (b) 2-D Fouriertransform of I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1188.10 (a) Sea elevation estimated spe trum S(k). (b) Passband �ltered S(k). . . . . . 1198.11 Integrated spe trum versus tilt �. . . . . . . . . . . . . . . . . . . . . . . . . . . 1208.12 Swell dire tion �s along the tra k from a spe tral analysis of the images. . . . . 120

188 LIST OF FIGURES8.13 (a) Total MSS, mssw and mssl versus latitude. (b) Long wave varian e on-tribution in per ent to the overall slope varian e. . . . . . . . . . . . . . . . . . 1218.14 Simulated photographs with the Eddy Experiment Flight amera spe i� ations.The up-wind MSS is 0.031. Top: without swell. Bottom: with swell (H=1 m).1228.15 Impa t of wave height H on �2u and SPI. �� = 45o. . . . . . . . . . . . . . . . 1238.16 Impa t of H on wind dire tion estimation. �� = 45o. . . . . . . . . . . . . . . 1248.17 Impa t of �� on �2u (a) and SPI (b). H is 0.7 (blue urve) and 1 meter(red urve). The bla k line orresponds to the nominal value of the analyzedparameter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1248.18 Impa t of �� on wind dire tion estimation. H is 0.7 (blue urve) and 1 meter(red urve). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1258.19 (a) Total MSS versus wind speed. Cox and Munk slope varian es are alsoplotted (red ir les). (b) Angle di�eren e between slope PDF azimuth andswell dire tion (�� = �s � �o) versus wind speed. . . . . . . . . . . . . . . . . 1268.20 (a) Slope PDF isotropy versus wind speed. (b) Slope PDF isotropy versus ��. 1278.21 Parameters a00o (a) and b00o (b) versus wind speed. Cox and Munk's data are alsoplotted (red ir les). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1289.1 Map of the 46 10-se ond ar s onsidered along the tra k. . . . . . . . . . . . . 1329.2 Altitude, heading (azimuth from North) and speed of the air raft during the46 10-se ond ar s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1339.3 Power pattern of the LHCP antenna used to gather the GNSS-R signals (mea-surements provided by IEEC). It is a 3 dB antenna, having its ut-o� at 55o. Itwas mounted on the air raft so that air raft nose orresponds to azimuth 210o. 1349.4 S atterometri bias in the delay waveform. . . . . . . . . . . . . . . . . . . . . . 1359.5 Sensitivity analysis of delay enter to SWH. The nominal value is 2 m. . . . . . 1369.6 Sensitivity analysis of delay enter to wind speed. The nominal value is 10 m/s(i.e., MSS = 0.025). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1369.7 Sensitivity analysis of delay enter to sea-surfa e slope PDF Isotropy (SPI).The nominal value is 0.55. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1379.8 Sensitivity analysis of delay enter to sea-surfa e slope PDF Azimuth (SPA).The nominal value is 0o from North. . . . . . . . . . . . . . . . . . . . . . . . . 1379.9 Estimated DMSS along the 46 10-se ond ar s for PRNs 08 (blue), 10 (red)and 24 (green). First olumn: estimated values. Se ond olumn: formal on�den e interval length (CIL) of estimated values. First row: Total MSS.Se ond row: Slope PDF Azimuth. Third row: Slope PDF Isotropy. . . . . 1399.10 Evolution of DMSS, delay enter, Doppler enter and RMSE values throughthe onvergen e pro ess (here for PRN 10). Iterations number 0, 2, 4, 6, 8,10 are respe tively plotted in blue, red, green, bla k, lear blue and magenta.First olumn: DMSS values with, from top to bottom, the Total MSS, SPAand SPI. Se ond olumn: From top to bottom, delay enter, Doppler enterand RMSE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1419.11 Examples of data and best-�t model DDMs. Those orrespond to Ar 01. First olumn: PRN 08. Se ond olumn: PRN 24. First row: Data DDMs.Se ond row: Best-�t Model DDMs. Third row: Data-model residual. . . . 1429.12 DMSS estimated along the air raft tra k. First row: Total MSS (in dB).Se ond row: Slope PDF Azimuth. Third row: Slope PDF Isotropy. . . . . 1459.13 Total MSS versus Jason's wind speed. . . . . . . . . . . . . . . . . . . . . . . . 14610.1 Geometry of the Salpex Experiment. . . . . . . . . . . . . . . . . . . . . . . . . 150

LIST OF FIGURES 18910.2 Salpex experimental ampaign. . . . . . . . . . . . . . . . . . . . . . . . . . . . 15110.3 Satellites in view during Salpex experiment. Ea h olored ar represents theposition of a GPS satellite during the experiment. The PRN number is losedto the beginning of the data. The green mask represents the area where theGPS signal re e tions are supposed to be free of shadowing phenomena due tothe environment of the swimming pool, and therefore only the satellite withinthis mask an be taken into onsideration for further analysis. . . . . . . . . . . 15110.4 Dependen e of the real and imaginary parts of the diele tri onstant as fun -tion of temperature T for di�erent salinities (in psu). Klein and Swift's modelsat 1.575 GHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15410.5 Dependen e of the real and imaginary parts of the diele tri onstant as fun -tion of salinity S for di�erent temperatures. Klein and Swift's model at 1.575GHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15510.6 Dependen e of the re e tivity in parallel polarization as a fun tion of salinityS for di�erent elevation angles. . . . . . . . . . . . . . . . . . . . . . . . . . . . 15510.7 Penetration depth in m versus salinity for di�erent temperatures. Klein'smodel at 1.575 GHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15610.8 Amplitude and phase of URR versus sin . Both pure water ase (blue urve)and salted water ase (red urve) are plotted for T=10o C (full line) and T=20oC (dashed line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15710.9 Amplitude and phase of URL versus sin . Both pure water ase (blue urve)and salted water ase (red urve) are plotted for T=10o C (full line) and T=20oC (dashed line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15810.10(a) Phase of URL at nadir versus permittivity. Diele tri loss fa tor is 0, 50and 100. (b) Phase of URL at nadir versus diele tri loss fa tor. Permittivityis 0, 65 and 85. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15810.11Power model versus elevation angle for two di�erent values of salinity (S=1 psuand S=30 psu). The experimental onditions are similar to the Salpex ones.T=11oC. PRN 17. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15910.12Interferen e pattern as a fun tion of the sine of the elevation for pure water(blue urve) and salted water (red urve). . . . . . . . . . . . . . . . . . . . . . 16010.13Repeatability of the data. Plot of the power di�eren e between two measure-ments with same experimental onditions, for both pure water (blue urve) andsalted water (red urve). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16110.14Fit of the Salpex experimental data using model presented in equation 10.18,when the ground ontribution is taken into a ount. The data orrespond to asalted water (30 psu). PRN 17. . . . . . . . . . . . . . . . . . . . . . . . . . . . 16110.15Per ent hange in power between S=40 psu and S=20 psu as a fun tion of sineof elevation angle (see equation 10.21). . . . . . . . . . . . . . . . . . . . . . . . 16311.1 Geometry of the experiment during Part B. . . . . . . . . . . . . . . . . . . . . 16611.2 Time series of the peak power (in dB), using PRN 01, of RHCP antenna (blue urve) and LHCP antenna (red urve). Left: Part A, the orresponding satelliteelevation is 21o (its azimuth is 61o). Right: Part B, the elevation is 20o (azimuth62o). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16611.3 Left: Mean power h ~P (�)i for di�erent noise levels (�=10, 20, 40 and 60). The hoi e of these varian es is related to the observed peak value in the data.The peak of orrelation is one for a lean signal. Right: Zoom of the squared orrelation in the grass, before lag zero (peak position). 1000 realizations, PRN01, Ti=10 ms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

190 LIST OF FIGURES11.4 Simulated mean peak power at lag zero versus inverse noise varian e 1=�2. Aline with slope 1 is also plotted. 1000 realizations, PRN 01, Ti=10 ms. . . . . . 17011.5 Left: Normalized varian e of the power for di�erent noise levels (�=10, 20, 40and 60). Right: Normalized varian e at lag zero versus �2. 1000 realizations,PRN 01, Ti=10 ms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17111.6 Mean of the ratio of two power waveforms (with noise �2d and �2r ) at lag zeroversus K�1, with K = �2r=�2d . �d = 60, and �r ranges between 60 and 300. . . . 17211.7 Left: Mean power of RHCP antenna (thi k urve) and LHCP antenna (dashed urve), PRN 01, Part A. Right: Zoom of the squared orrelation in the grass,after lag zero. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17211.8 Left: Mean power of RHCP antenna (thi k urve) and LHCP antenna (dashed urve), PRN 01, Part B. Right: Zoom of the squared orrelation in the grass,after lag zero. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17311.9 Normalized varian e of the squared orrelation for the RHCP antenna (thi k urve) and the LHCP antenna (dashed urve), PRN 01. Left: Part A. Right:Part B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

List of Tables1 Related experiments arried out during the thesis. . . . . . . . . . . . . . . . . 138.1 Parameter estimation in hronologi al order. . . . . . . . . . . . . . . . . . . . 1159.1 Overview of the parameters ne essary for running the DDM forward model. . 1339.2 Set of nominal parameters for sensitivity analysis of parameter �� . . . . . . . . 1359.3 Analysis of estimated DMSS of �gure 9.9. . . . . . . . . . . . . . . . . . . . . 138

191

192 LIST OF TABLES

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