REMOTE SENSING ADVANCED TECHNIQUES AND PLATFORMS. Edited by Boris Escalante-Ramírez

Transcription

1 REMOTE SENSING ADVANCED TECHNIQUES AND PLATFORMS Edted by Bors Escalante-Ramírez

2 Remote Sensng Advanced Technques and Platforms Edted by Bors Escalante-Ramírez Publshed by InTech Janeza Trdne 9, Rjeka, Croata Copyrght 01 InTech All chapters are Open Access dstrbuted under the Creatve Commons Attrbuton 3.0 lcense, whch allows users to download, copy and buld upon publshed artcles even for commercal purposes, as long as the author and publsher are properly credted, whch ensures maxmum dssemnaton and a wder mpact of our publcatons. After ths work has been publshed by InTech, authors have the rght to republsh t, n whole or part, n any publcaton of whch they are the author, and to make other personal use of the work. Any republcaton, referencng or personal use of the work must explctly dentfy the orgnal source. As for readers, ths lcense allows users to download, copy and buld upon publshed chapters even for commercal purposes, as long as the author and publsher are properly credted, whch ensures maxmum dssemnaton and a wder mpact of our publcatons. Notce Statements and opnons expressed n the chapters are these of the ndvdual contrbutors and not necessarly those of the edtors or publsher. No responsblty s accepted for the accuracy of nformaton contaned n the publshed chapters. The publsher assumes no responsblty for any damage or njury to persons or property arsng out of the use of any materals, nstructons, methods or deas contaned n the book. Publshng Process Manager Dragana Manestar Techncal Edtor Mroslav Tadc Cover Desgner InTech Desgn Team Frst publshed June, 01 Prnted n Croata A free onlne edton of ths book s avalable at Addtonal hard copes can be obtaned from orders@ntechopen.com Remote Sensng Advanced Technques and Platforms, Edted by Bors Escalante-Ramírez p. cm. ISBN

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4 Contents Preface IX Secton 1 Analyss Technques 1 Chapter 1 Chapter Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Characterzng Forest Structure by Means of Remote Sensng: A Revew 3 Hooman Latf Fuson of Optcal and Thermal Imagery and LDAR Data for Applcaton to 3-D Urban Envronment and Structure Montorng 9 Anna Brook, Marjke Vandewal and Eyal Ben-Dor Statstcal Propertes of Surface Slopes va Remote Sensng 51 Josué Álvarez-Borrego and Beatrz Martín-Atenza Classfcaton of Pre-Fltered Multchannel Remote Sensng Images 75 Vladmr Lukn, Nkolay Ponomarenko, Dmtry Fevralev, Benot Vozel, Kacem Chehd and Andry Kurekn Estmaton of the Separable MGMRF Parameters for Thematc Classfcaton 99 Rolando D. Navarro, Jr., Joselto C. Magada and Enrco C. Parngt Low Rate Hgh Frequency Data Transmsson from Very Remote Sensors 13 Pau Bergada, RosaMa Alsna-Pages, Carles Vlella and Joan Ramon Regué A Contrbuton to the Reducton of Radometrc Mscalbraton of Pushbroom Sensors 151 Chrstan Rogaß, Danel Spengler, Mathas Bochow, Karl Segl, Angela Lausch, Danel Doktor, Sgrd Roessner, Robert Behlng, Hans-Ulrch Wetzel, Kata Urata, Andreas Huen and Hermann Kaufmann

5 VI Contents Chapter 8 Chapter 9 Chapter 10 Dfferental Absorpton Mcrowave Radar Measurements for Remote Sensng of Barometrc Pressure 171 Roland Lawrence, Bn Ln, Steve Harrah and Qlong Mn Energy Effcent Data Acquston n Wreless Sensor Network 197 Ken C.K. Lee, Mao Ye and Wang-Chen Lee Three-Dmensonal Lneament Vsualzaton Usng Fuzzy B-Splne Algorthm from Multspectral Satellte Data 13 Maged Marghany Secton Sensors and Platforms 33 Chapter 11 COMS, the New Eyes n the Sky for Geostatonary Remote Sensng 35 Han-Dol Km, Gm-Sl Kang, Do-Kyung Lee, Kyoung-Wook Jn, Seok-Bae Seo, Hyun-Jong Oh, Joo-Hyung Ryu, Herve Lambert, Ivan Lane, Phlppe Meyer, Perre Coste and Jean-Lous Duquesne Chapter 1 Hyperspectral Remote Sensng Usng Low Flyng Arcraft and Small Vessels n Coastal Lttoral Areas 69 Charles R. Bostater, Jr., Gaelle Coppn and Floran Levaux Chapter 13 CSIR NLC Moble LIDAR for Atmospherc Remote Sensng 89 Svakumar Venkataraman Chapter 14 Actve Remote Sensng: Ldar SNR Improvements 313 Yasser Hassebo Chapter 15 Smart Staton for Data Recepton of the Earth Remote Sensng 341 Mykhaylo Palamar Chapter 16 Atmospherc Propagaton of Terahertz Radaton 371 Janquan Yao, Ran Wang, Haxa Cu and Jngl Wang Chapter 17 Chapter 18 Road Feature Extracton from Hgh Resoluton Aeral Images Upon Rural Regons Based on Mult-Resoluton Image Analyss and Gabor Flters 387 Hang Jn, Marc Mska, Edward Chung, Maoxun L and Yanmng Feng Hardware Implementaton of a Real-Tme Image Data Compresson for Satellte Remote Sensng 415 Albert Ln

6 Contents VII Chapter 19 Progress Research on Wreless Communcaton Systems for Underground Mne Sensors 49 Larb Talb, Ismal Ben Mabrouk and Mourad Nedl Chapter 0 Cold Gas Propulson System An Ideal Choce for Remote Sensng Small Satelltes 447 Assad Ans

7 Preface Nowadays t s hard to fnd areas of human actvty and development that have not profted from or contrbuted to remote sensng. Natural, physcal and socal actvtes fnd n remote sensng a common ground for nteracton and development. From the end-user pont of vew, Earth scence, geography, plannng, resource management, publc polcy desgn, envronmental studes, and health, are some of the areas whose recent development has been trggered and motvated by remote sensng. From the technologcal pont of vew, remote sensng would not be possble wthout the advancement of basc as well as appled research n areas lke physcs, space technology, telecommuncatons, computer scence and engneerng. Ths dual concepton of remote sensng brought us to the dea of preparng two dfferent books. The present one s devoted to new technques for data processng, sensors and platforms, whle the accompanyng book s meant to dsplay recent advances n remote sensng applcatons. From a strct perspectve, remote sensng conssts of collectng data from an object or phenomenon wthout makng physcal contact. In practce, most of the tme we refer to satellte or arcraft-mounted sensors that use some sort of electromagnetc radaton to gather geospatal nformaton from land, oceans and atmosphere. The growng dversty of human actvty has motvated the desgn of new sensors and platforms as well as the development of new methodologes that can process the enormous amount of nformaton that s beng generated daly. Collected nformaton, however, represents only a footprnt of the object or the phenomenon we are nterested n. In order for the end-user to be able to nterpret and use ths nformaton, the data has to be processed so that t does not longer represent a dgtal number, but a physcalrelated value. Among the tasks that usually must be carred out on ths data, we fnd several numercal correctons and calbratons: geometrcal, dgtal elevaton, atmospherc, radometrc, etc. Moreover, dependng on the end-user applcaton, data may need to be fltered, compressed, transmtted, fused, classfed, nterpolated, etc. The problem s even more complex when we thnk of the varety of sensors and satelltes that have been desgned and launched. We are talkng about a large dversty that ncludes passve or actve sensors; panchromatc, multspectral or hyperspectral sensors; all of them wth spatal resolutons that range from a couple of centmeters to several klometers, to menton a few examples. In summary, dfferent methodologes and technques for data processng must be desgned and customzed accordng, not only to the specfc applcaton, but also to the sensor and satellte characterstcs.

8 X Preface We do not ntend ths book to cover all aspects of remote sensng technques and platforms, snce t would be an mpossble task for a sngle volume. Instead, we have collected a number of hgh-qualty, orgnal and representatve contrbutons n those areas. The frst part of the book s devoted to new methodologes and technques for data processng n remote sensng. The reader wll fnd nterestng contrbutons n forest characterzaton, data fuson, surface slopes statstcal propertes, multchannel and Markovan classfcaton, road feature extracton, mscalbraton correcton, barometrc pressure measurements, wreless sensors networks and lneament vsualzaton. The second part of the book gathers chapters related to new sensors and platforms for remote sensng, ncludng the new COMS satellte, hyperspectral remote sensng, moble LIDAR for atmospherc remote sensng, SNR mprovements n LIDAR, a smart staton for data recepton, terahertz radaton propagaton, HF data transmsson for very remote sensng, hardware mage compresson, wreless communcatons for underground sensors, and cold gas propulson for remote sensng satelltes. I wsh to express my deepest grattude to all authors who have contrbuted to ths book. Wthout ther strong commtment ths book would not have become such a valuable pece of nformaton. I am also thankful to InTech edtoral team who has provded the opportunty to publsh ths book. Bors Escalante-Ramírez Natonal Autonomous Unversty of Méxco, Faculty of Engneerng, Mexco Cty, Mexco

9 3 Statstcal Propertes of Surface Slopes va Remote Sensng Josué Álvarez-Borrego 1 and Beatrz Martín-Atenza 1 CICESE, Dvsón de Físca Aplcada, Departamento de Óptca Facultad de Cencas Marnas, UABC Méxco 1. Introducton The complexty of wave moton n deep waters, whch can damage marne platforms and vessels, and n shallow waters, same that can afflct human settlements and recreatonal areas, has gven orgn to a long-term development n laboratory and feld studes, the conclusons of whch are used to desgn methodology and set bases to understand wave moton behavor. Va remote sensng, the use of radar mages and optcal processng of aeral photographs has been used. The nterest n wave data s manfold; one element s the nherent nterest n the drectonal spectra of waves and how they nfluence the marne envronment and the coastlne. These wave data can be readly and accurately collected by aeral photographs of the wave sun glnt patterns whch show reflectons of the Sun and sky lght from the water and thus offer hgh-contrast wave mages. In a seres of artcles, Cox and Munk (1954a, 1954b, 1955) studed the dstrbuton of ntensty or gltter pattern n aeral photographs of the sea. One of ther conclusons was that for constant and moderate wnd speed, the probablty densty functon of the slopes s approxmately Gaussan. Ths could be taken as an ndcaton that n certan crcumstances, the ocean surface could be modeled as a Gaussan random process. Smlar observatons by Longuet-Hggns et al. (1963) (cted by Longuet-Hggns (196)) wth a floatng buoy, whch flters out the hgh-frequency components, come consderably closer to the Gaussan dstrbuton. Other authors (Stlwell, 1969; Stlwell & Plon, 1974) have studed the same problem consderng a sea surface llumnated by a contnuous sky lght wth no azmuthal varatons n sky radance. Dfferent models of sky lght have been used emphaszng the exstence of a nonlnear relatonshp between the slope spectrum and the correspondng wave mage spectrum (Peppers & Ostrem, 1978; Chapman & Iran, 1981). Smulated sea surfaces have been analyzed by optcal systems to understand the optcal technque n order to obtan best qualtatve nformaton of the spectrum (Álvarez-Borrego, 1987; Álvarez-Borrego & Machado, 1985).

10 5 Remote Sensng Advanced Technques and Platforms Fuks and Charnotsk (006) derved the jont probablty densty functon of surface heght and partal second dervatves for an ensemble of specular ponts at a random rough Gaussan sotropc surface at normal ncdence. However, n a real physcal stuaton, consderaton of Gaussan statstcs can be a very good approxmaton. Cox and Munk (1956) observed that the center of the gltter pattern mages had shfted downwnd from the grd center. Ths shft can be assocated wth an up/downwnd asymmetry of the wave profle (Munk, 009). Surfaces of small postve slope are more probable than those of negatve slope; large postve slopes are less probable than larger negatve slopes, thus permttng the restrant of a zero mean slope (Bréon & Henrst, 006). Accordng wth Longuet-Hggns (1963) the sea surface slopes have a Gaussan probablty functon to a frst approxmaton. In the next approxmaton skewness s taken nto account. The kurtoss s zero, as are all the hgher cumulants. In the next approxmaton, the dstrbuton s gven taken nto account the kurtoss. Walter Munk (009) wrtes that the skewness appears to be correlated wth a rather sudden onset of breakng for wnds above 4 m s -1 and he does not thnk that skewness comes from parastc capllares. Chapron et al. (00) suggest that the actual waves form under nearbreakng condtons, along wth the varyng populaton and length scales for these breakng events, should also contrbute to the skewness. In ths chapter we wll consder two dfferent cases to analyze statstcal propertes of surface slopes va remote sensng: frst we assume the fluctuaton of the surface slopes to be statstcally Gaussan and the second case we assume the fluctuaton of the surface slopes to be statstcally non-gaussan. We, also, assume that the surfaces are llumnated by a source, the Sun, of a fxed angular extent,, and maged through a lens that subtends a very small sold angle. Wth these consderatons, we calculated ther mages, as they would be formed by a sgnal clppng detector. In order to do ths, we defne a gltter functon, whch operates on the slope of the surfaces. In the frst case we consder two stuatons: the detector lne of sght angle, d, s constant for each pont on the surface and d s varable for each pont n the surface. In the second case, wth non-gaussan statstcs, we consder d varable for each pont n the surface only, because we consder that ths case s more realstc.. Geometry of the model (Gaussan case consderng a constant detector angle) The physcal stuaton s shown n fgure 1. The surface x s llumnated by a unform ncoherent source S of lmted angular extent, wth wavelength. Its mage s formed n D by an aberraton free optcal system. The ncdence angle, s, s defned as the angle between the ncdence angle drecton and the normal to the mean surface. Then, n fgure 1, s, represents the mean angle subtended by the source S and d represents the mean angle subtended by the optcal system of the detector wth the normal to the mean surface. The apparent dameter of the source s and of the detector s d. Lght from the source s reflected on the surface just one tme and, dependng on the slope, the lght reflected wll or wll not be part of the mage. In broad terms, the mage conssts of brght and dark regons that we call a gltter pattern. represents the angle between the x axs and the surface, and

11 Statstcal Propertes of Surface Slopes va Remote Sensng 53 Fg. 1. The detector s located n the zenth of each reflecton pont n the profle. represents the angle between the normal to the plane and the source S. Ths angle s gven by s, and the specular angle s gven by r. From ths two equatons we can wrte. (1) r s Because the source has a fnte sze, there are several ncdence drectons whch are specular reflected to the camera. The drectons, os (where ths angle s the angular dmenson of the Sun), where there are ncdence rays whch are determned by the condton s os s, () n other words, the source s angularly descrbed by the functon, os, can be wrtten lke os rect os s, (3) where rect(.) represents the rectangle functon (Gaskll, 1978). So, the projecton of ths source on the detector, after reflecton, s gven by s s, (4) where equaton (1) s taken nto account. r R rect, (5)

12 54 Remote Sensng Advanced Technques and Platforms On the other sde, the detecton system pupl can be represented by the functon d P rect. (6) d The ntensty lght I, arrvng to the detecton plane D depends on the overlap between the functons P, and can be approxmated by R and In practcal stuatons, d R I P d. (7) d s so smaller than, that we can to approxmate P where s the Drac delta, of ths way The lght reflecton wll arrve to the detector D when and because, we have r s I Rd, d r. (8) rect r d r, (9) s d s d. (10) 4 4 Defnng tan, /and tan, and usng the relatonshp s d tan 4 tan 1 tan 4, vald for small 4, we obtan the next condton for the slopes o o 1o o 1 o. 4 4 We fnd then the gltter functon, gven by o Brect. 1 o (11) (1) Ths expresson (eq. 1) tell us that the geometry of the problem selects a surface slope regon and encodes lke brght ponts n the mage (gltter pattern).

13 Statstcal Propertes of Surface Slopes va Remote Sensng 55.1 Relatonshp among the varances of the ntenstes n the mage, surface slopes and surface heghts The mean of the mage, I, may be wrtten (Papouls, 1981) where B s defned by equaton (1) and I Ix ( ) B p d, (13) p s the probablty densty functon n one dmenson, where n a frst approxmaton a Gaussan functon s consdered. Substtutng n p, we have equaton (13) the expressons for B and 1 o I I( x) rect exp d. 1/ 1 o Defnng a 1 4 and o The varance of the ntenstes n the mage, o b 1 4, we can wrte o 1 b a I I x erf erf. o I, s defned by (Papouls, 1981) (14) (15) I But, B B, then I x Ix and substtutng the expresson of I I x I x B p d. (16), therefore Ix I I x I I 1, (17) Ix, equaton (15), n equaton (17), we have 1 b a 1 b a I erf erf erf erf, whch s the requred relaton between the varance of the ntenstes n the mage, the varance of the surface slopes,. (18) I, and The relaton (18) s shown n fgure for some typcal cases, usng the geometry descrbed above, wth 0 o d and 0.68 o. In the horzontal axs we have the varance of the surface slopes,, and n the vertcal axs we have the varance of the ntenstes of the mage, I. In the fgure we can observe the dependence of ths relatonshp wth the angular poston of the source, s. In fgure we also can observe that for small ncdence angles (0-10 degrees) and small values of varance of the surface slopes, t s possble to obtan bgger values n the varance of the ntenstes n the mage. From equaton (18), we can see that ths behavor s

14 56 Remote Sensng Advanced Technques and Platforms ndependent of any surface heght power spectrum that we are analyzng, because ths relaton depends on the probablty densty functon of the surface slopes and the geometry of the experment only. Fg.. Relatonshp between the varance of the surface slopes wth the varance of the ntenstes n the mage. In certan cases, fgure, f we have data correspondng to a s value only, t s not possble to obtan the varance of the surface slopes,, because for a value of I we wll have two possble values of. To solve ths problem, t s necessary to analyze mages whch correspond at two or more ncdence angles and to select a slope varance value whch s consstent wth all these data. The relatonshp between and can be derved from (Papouls, 1981) dc C, (19) d f we know the correlaton functon of the surface heghts (ths wll be shown n next secton of ths chapter). Here, C s the correlaton functon of the surface heghts and C s the correlaton functon of the surface slopes.. Relatonshp between the correlaton functon of the ntenstes n the mage and of the surface heghts Our analyss nvolves three random processes: the surface profle, x, ts surface slopes, x, and the mage, Ix. Each process has a correlaton functon and t was shown (Álvarez-Borrego, 1993) that these three functons hold a relatonshp.

15 Statstcal Propertes of Surface Slopes va Remote Sensng 57 The relatonshp between correlaton functons of the surface heghts, C, and the surface slopes, C, s gven by equaton (19), and the relatonshp between C and the correlaton functon of the ntenstes n the mage, C, s gven by (Álvarez-Borrego, 1993) I B1B 1 C 1 ICI exp d 1/ 1d. 1 C 1 C In order to acheve the nverse process, usng equaton (19) and equaton (0), these two equatons must meet certan condtons. For example, t s requred that there exsts one to one correspondence among the amount nvolved. Usng equaton (19) the processed data can be numercally ntegrated twce, such that we obtan nformaton of the correlaton functon of the surface heghts, C, from the correlaton functon of the surface slopes, C. Although equaton (0) s a more complcated expresson, we cannot obtan an analytcal result from t. A frst ntegral can be analytcally solved and for the second t s possble to obtan the soluton by numercal ntegraton. Resolvng the frst ntegral analytcally, equaton (0) can be wrtten lke b bc ac ICI exp erf erf d, 4 a 1C 1C where a o 1o 4 and o o b 1 4. So, a relatonshp between values of the correlaton functon of the ntenstes n the mage, CI, and the values of the correlaton functon of the surface slopes takes, C, can be obtaned (Fgure 3). In ths case, to small angles we can fnd hgher values for the correlaton functon of the ntenstes n the mage. In all the cases, the angular poston of the camera or detector, d, s zero and =0.03. The correlaton functons of fgure 3 are normalzed. Also, from equaton (19), t s possble to obtan the correlaton functon of the surface heghts, C, from C and the requre nverse process to determne the correlaton functon of the surface heghts s completed. A theoretcal varance I can be calculated from equaton (1). We wrote n Table 1 the values of the mage varance n order to normalze the correlatons n fgure 3 for dfferent values for s. (0) (1) s I Table 1. Values of the mage varance n order to normalze the correlatons n fgure 3 for dfferent values for s.

16 58 Remote Sensng Advanced Technques and Platforms Fg. 3. Relatonshp between the correlaton functon of the surface slopes and the correlaton functon of the ntenstes n the mage. 3. Geometry of the model (Gaussan case consderng a varable detector angle) A more real physcal stuaton s shown n fgure 4. The surface, x, s llumnated by a unform ncoherent source S of lmted angular extent, wth wavelength. Its mage s formed n D by an aberraton-free optcal system. The ncdence angle s s defned as the angle between the ncdence angle drecton and the normal to the mean surface and represents the mean angle subtended by the source S. d corresponds to the angle subtended by the optcal system of the detector wth the normal to pont of the surface,. e. 1 x d tan, () H where H s the heght of the detector and x s the nterval between surface ponts. We can see that n ths more realstc physcal stuaton, angle d s changng wth respect to each pont n the surface. It s worth notcng that a varable d does not restrct the sensor feld of vew. s the angle subtended between the normal to the mean surface and the normal to the slope for each pont n the surface 1 s d s 1 x tan. (3) H The apparent dameter of the source s. Lght from the source s reflected on the surface for just one tme, and, dependng on the slope, the lght reflected wll or wll not be part of the mage. Thus, the mage conssts of brght and dark regons that we call a gltter pattern.

17 Statstcal Propertes of Surface Slopes va Remote Sensng 59 Fg. 4. Geometry of the real physcal stuaton. Counterclockwse angles are consdered as postve and clockwse angles as negatve. The gltter functon can be expressed as (Álvarez-Borrego & Martín-Atenza, 010) where o B rect, (4) 1 o o 1o o 1 o, 4 4 (5) tan, (6) s d tan o. (7) The nterval characterzed by equaton (5) defnes a specular band where certan slopes generate brght spots n the mage. Ths band has now a nonlnear slope due to the varaton of d wth respect to each pont of the surface (Fgure 5). Combnng equatons (5) (7), the slope nterval, where a brght spot s receved by the detector, s s 1 1x s 1 1x tan tan. (8) H 4 H Relatonshps among the varances of the ntenstes n the mage and surface slopes The mean of the mage I may be wrtten as (Álvarez-Borrego & Martín-Atenza, 010)

18 60 Remote Sensng Advanced Technques and Platforms Fg. 5. All the random processes nvolved n our analyss. The specular band corresponds to brght regons n the mage. I x B p d (9), I where B s the gltter functon defned be equaton (4). p s the probablty densty functon, where a Gaussan functon s consdered n one dmenson. Substtutng n p, we have equaton (9) the expressons for B and N 1 1 o I Ix rect exp d. N 1 1 o (30) The detector angle d s a functon of the poston x ; thus, the specular angle s a functon of the dstance x from the nadr pont of the detector n 0 to the pont n (equaton ). Defnng a o 1o 4 and o o b 1 4, we can wrte N 1 1 b a I I x erf erf. N 1 The varance of the ntenstes n the mage I s defned by (Álvarez-Borrego & Martín- Atenza, 010) (31)

19 Statstcal Propertes of Surface Slopes va Remote Sensng 61 N 1 I I x Ix B I p d N 1. (3) However, B = B, then I x = Ix ; therefore Ix I I x I I Substtutng the equaton (31) n equaton (33), we have 1. (33) N 1 1 b a 1 b a I erf erf erf erf N 1 4N, (34) whch s the requred relatonshp between the varance of the ntenstes n the mage and the varance of the surface slopes. The relatonshp between the varance of the surface slopes and the varances of the ntenstes of the mage for dfferent s angles (10 o -50 o ) s shown n fgure 6 (equaton 34). The detector s located as shown n fgure 4 and the subtended angle by the source s 0.68 o. When the camera detector s at H=100 m the behavor of the curves look smlar to the curves shown n Álvarez-Borrego & Martín-Atenza, 010 (fgure 6a). In ths case, we also can observe that, for bg ncdence angles (40 o 50 o ) and small values of varance of the surface slopes, t s possble to obtan bgger values n the varance of the ntenstes n the mage. If we analyze the fgure 6j we can observe that ncreases for lower s values (10 o -0 o ). These results match wth the results presented by Álvarez-Borrego n Fgure 6j was made consderng an H=1000 m. The reason for ths match s that the condton proposed by Álvarez-Borrego n 1993 consders a d value constant (see fgure ). Ths condton s smlar to have the sensor camera to an H value very hgh where the surface slopes values are consdered almost constant. I Fgure 6 shows how these relatonshps ( versus ) are changng whle H s beng bgger. Dark lnes show lmt extremes for s of 10 o and 50 o. It can be seen that when H s ncreasng to 00 m the lne of 50 o starts to decay and start to cross wth the others. In so far as H goes up, the lnes, wth larger s go down untl the order of the curves change. The explanaton for ths s very smple: f the camera stays at H=100 m, t wll receve more reflecton of lght at large s, because the geometry of reflecton. When H ncreases, the camera wll receve less lght reflecton of large ncdence angles but wll have more lght reflecton for small ncdence angles. Therefore, when the camera s at a larger heght, wll have more reflecton from lght ncdence angles smaller than lght of larger ncdence angles. Thus we can say that the results presented by Álvarez-Borrego n 1993, Cureton et al., 007 and Álvarez-Borrego & Martín-Atenza n 010 are correct for the Gaussan case. In certan cases, f we have data correspondng to one s value, t s not possble to obtan a sngle value for the varance of the surface slopes. To solve ths problem, t s necessary to analyze mages whch correspond at two or more ncdence angles and to select a slope varance value whch s consstent wth all these data (Álvarez-Borrego, 1995). I I

20 6 Remote Sensng Advanced Technques and Platforms Fg. 6. Relatonshp between the varance of the surface slopes and the varance of the ntenstes of the mage for dfferent H values. From equaton (34), we can see that ths relaton depends on the probablty densty functon of the surface slopes and the geometry of the experment only.

21 Statstcal Propertes of Surface Slopes va Remote Sensng Relatonshp between the correlaton functons of the ntenstes n the mage and of the surface slope The relatonshp between the correlaton functon of the surface slopes correlaton functons of the ntenstes n the mage s gven by where, 1 N N 1 1 I C I B 1 B p 1 d 1 d N 1 N 1 CI C and the,, (35) p s defned by 1 1 C 1 p 1, exp. (36) 1/ 1 C 1 C Although t s possble to obtan an analytcal relatonshp for the frst ntegral, for the second ntegral the process must be numerc. Thus, eq. (35) can be wrtten lke b N N 1 1 b C a C ICI N exp erf erf d, 1 N a C 1 C (37) where a o 1o 4 and o o b 1 4. In order to avod computer memory problems, the data pont profle was dvded nto nto a number of consecutve ntervals. The value of d vares pont to pont n the profle. For each nterval and for each s value, the relatonshp between the correlaton functons CI and C was calculated. Then, the several computed relatonshps for each s value were averaged. In ths case we used a value of =0.03. The correlaton functon of the ntenstes n the mage s not normalzed. Smlar to the behavor of the varances, when H ncreases the behavor of the curves have a smlar process. A theoretcal varance I can be calculated from equaton (37). We wrote n Table the values of the mage varance n order to normalze the correlatons n fgure 7 for dfferent values for s and H (100, 500, 1000 and 5000 m). 4. Geometry of the model (Non-Gaussan case consderng a varable detector angle) The model, consderng d as varable, s shown n fgure 4. We thnk ths s a more realstc stuaton. 4.1 Relatonshps among the varances of the ntenstes n the mage and surface slopes consderng a non-gaussan probablty densty functon The mean of the mage I may be wrtten as (Álvarez-Borrego & Martín-Atenza, 010):

22 64 Remote Sensng Advanced Technques and Platforms H s I Table. Values of the mage varance n order to normalze the correlatons n fgure 7 for dfferent values for s and H. 1 N I I x B p d N 1 (38) s the probablty densty functon, where a non-gaussan functon s consdered n one dmenson (Cureton, 010) where B s the gltter functon defned by equaton (4). p exp p , 6 4 (39) where 3 s the skewness, 4 s the kurtoss and s the standard devaton of the surface slopes. Substtutng n equaton (38) the expressons for B and p, we have

23 Statstcal Propertes of Surface Slopes va Remote Sensng 65 Fg. 7. Relatonshp between the correlaton functon of the surface slopes and the correlaton functon of the ntenstes n the mage N o I rect exp d. 4 N 1 1 o (40) The detector angle d s a functon of the poston x, thus, the specular angle s a functon of the dstance x from the nadr pont of the detector, n = 0, to the pont n = (see equaton ()). Wrtng agan a o 1o 4 and o o b 1 4, we can wrte b a erf 1 3 erf 8 N a a I exp a 3 a 3 N. (41) b b exp b 3 3 b 6 4

24 66 Remote Sensng Advanced Technques and Platforms The varance of the ntenstes n the mage equaton (41) n equaton (33) we have I s defned by equaton (33). Substtutng b a erf erf N a a I exp a 3 a 3 N b b exp b 3 3 b 6 4 b a erf erf N a a exp a 3 a 3 N b 4 b exp b 3 3 b 6 4 whch s the requred relatonshp between the varance of the ntenstes n the mage and the varance of the surface slopes when a non-gaussan probablty densty functon s consdered. The relatonshp between the varance of the surface slopes and the varances of the ntenstes of the mage for dfferent s angles (10 o -50 o ) s shown n fgures 8 and 9 (equaton 4). Fgures 8 and 9 show ths relatonshp consderng the skewness and the skewness and kurtoss n the non-gaussan probablty densty functon respectvely. We can see that the behavor of the curves looks very smlar to the Gaussan case (fgure 6). The values for skewness and kurtoss were taken from a Table showed by Plant (003) from data gven by Cox & Munk (1956), for a wnd speed of 13.3 m/s wth the wnd sensor at 1.5 m on the sea surface level. The curves ncludng the skewness and skewness and kurtoss are lttle hgher for small values of compared wth the Gaussan case (fgure 6) except when s s below 40 o where the Gaussan and non-gaussan cases (consderng skewness only) are nverted to small surface slope varances, and these results show that I ncreases for hgher s values (fgures 8a and 9a). Cox & Munk (1956) reported values of 0.04 and 0.05 lke maxmum values of the surface slopes n the wnd drecton and values of 0.03 n the cross wnd drecton for wnd speed bgger than 10 m/s. Thus, we thnk that n the range for from the behavor of the curves look very clear and separate each one of the other (fgures 8a and 9a). If we analyze the fgures 8j and 9j we can observe that I ncreases for lower s values (10 o -0 o ). Fgures 8 and 9 show how these relatonshps ( versus ) are changng whle H s beng bgger, where the skewness and skewness and kurtoss are beng consdered. These curves have the same behavor lke n the Gaussan case and the explanaton for ths nverson s the same as explaned before. I (4) I

25 Statstcal Propertes of Surface Slopes va Remote Sensng 67 Fg. 8. Relatonshp between the varance of the surface slopes and the varance of the ntenstes of the mage, for dfferent H values consderng a non-gaussan probablty densty functon where the skewness has been taken account only. About the non-gaussan case we can conclude that the man dfference wth the Gaussan case s the less hgher values of the varance of the ntenstes of the mage for small values of surface slope varance when s s n the 40 o 50 o range when H=100 m. In addton, when

26 68 Remote Sensng Advanced Technques and Platforms Fg. 9. Relatonshp between the varance of the surface slopes and the varance of the ntenstes of the mage, for dfferent H values consderng a non-gaussan probablty densty functon where the skewness and kurtoss have been taken account. H=1000 m ths condton s nverted, we can fnd less smaller values of the varance of the ntenstes of the mage for small values of surface slope varance when s s n the 10 o 0 o range. In the other angles, n both cases, t s not possble to see sgnfcant dfferences between the values 10 o 30 o when H=100 m and 30 o 50 o when H=1000 m.

27 Statstcal Propertes of Surface Slopes va Remote Sensng Relatonshp between the correlaton functons of the ntenstes n the mage and of the surface slope consderng a non-gaussan probablty densty functon As mentoned before, our analyss nvolves three random processes: the surface profle x, ts surface slopes x and the mage Ix. Each process has a correlaton functon and t was shown n (Álvarez-Borrego, 1993) that these three functons are related. The relatonshp between the correlaton functon of the surface slopes correlaton functon of the ntenstes n the mage C s gven by where, 1 N N 1 1 I C I B 1 B p 1 d 1 d N 1 N 1 I C and the,, (43) p s defned by (Cureton, 010) 1 1 C 1 p 1, exp 1/ 1 C 1 C C (44) 1 1, C where and are the skewness, and are the relatonshp between the moments of 1 and. Although t s possble to obtan an analytcal relatonshp for the frst ntegral, for the second ntegral the process must be numerc. Thus, equaton (43) can be wrtten lke exp ub v A1 B1b C N b N 1 1 ICI ua v A Ba C d N 1 N a 1 exp exp, erf ub v erf ua v A B C (45) where u 1 1 C,

28 70 Remote Sensng Advanced Technques and Platforms v C 1 C, 1 C A1 C 3 3C 3 A, 1 1 C 30 1 B1 C 3 3 B, 1 1 C C1 b C C 3, 1 1 C C a C C 3, A3 C 4 3C 3 C, 4 1C 30 1C B3 C 1 C 1 C, 8 C3 4. Fgure 10 shows graphcally the relatonshp between the normalzed correlaton functon of the surface slopes C n and the normalzed correlaton functon of the ntenstes of the mage C I. In ths case a n 0.03 was used. When H ncreases the behavor of the curves have a smlar process lke the varance curves. When H=100 m (Fgure 10a) the behavor of the curves for s of 10 o 0 o have an unusual behavor for low surface slope varances when compared wth Gaussan case. Ths s because the nverson of the curves starts to lower values of H. In order to avod memory computer problems, the data ponts profle was dvded nto a number of consecutve ntervals. The value of d vares pont to pont n the profle. For each nterval and for each s value, the relatonshp between the correlaton functons CI and C was calculated. Then, all the computed relatonshps for each s value were averaged. A theoretcal varance I can be calculated from equaton (45). We wrote n Table 3 the values of the mage varance n order to normalze the correlatons n fgure 10 for dfferent values for s and H (100, 500, 1000 and 5000 m).

29 Statstcal Propertes of Surface Slopes va Remote Sensng 71 Fg. 10. Relatonshp between the correlaton functon of the surface slopes and the correlaton functon of the ntenstes n the mage. The curves correspond to dfferent values of s.

30 7 Remote Sensng Advanced Technques and Platforms H s I Table 3. Values of the mage varance n order to normalze the correlatons n fgure 10 for dfferent values for s and H. 5. Conclusons We derve the varance of the surface heghts from the varance of the ntenstes n the mage va remote sensng consderng a gltter functon gven by equaton (1) when the geometry consder a detector angle of 0 o d, and consderng a gltter functon gven by the equaton (4) consderng a geometrcally mproved model wth varable detector lne of sght angle, gven by fgure 4. In ths last case, we consder Gaussan statstcs and non- Gaussan statstcs. We derve the varance of the surface slopes from the varance of the ntenstes of remote sensed mages for dfferent H values. In addton, we dscussed the determnaton of the correlaton functon of the surface slopes from the correlaton functon of the mage ntenstes consderng Gaussan and non-gaussan statstcs. Analyzng the varances curves for Gaussan and non-gaussan case t s possble to see the behavor of the curves for dfferent ncdent angles when H ncreases. Ths behavor agrees wth the results presented by Álvarez-Borrego (1993) and Geoff Cureton et al. 007, and Álvarez-Borrego and Martn-Atenza (010) for the Gaussan case. These new results solve the nverse problem when t s necessary to analyze the statstcal of a real sea surface va remote sensng usng the mage of the gltter pattern of the marne surface.

31 Statstcal Propertes of Surface Slopes va Remote Sensng Acknowledgments Ths work was partally supported by CONACyT wth grant No and SEP- PROMET/103.5/10/501 (UABC-PTC-5). 7. References Álvarez-Borrego, J. (1987). Optcal analyss of two smulated mages of the sea surface. Proceedngs SPIE Internatonal Socety of the Optcal Engneerng, Vol.804, pp.19-00, ISSN X Álvarez-Borrego, J. (1993). Wave heght spectrum from sun glnt patterns: an nverse problem. Journal of Geophyscal Research, Vol.98, No.C6, pp , ISSN Álvarez-Borrego, J. (1995). Some statstcal propertes of surface heghts va remote sensng. Journal of Modern Optcs, Vol.4, No., pp , ISSN Álvarez-Borrego, J. & Machado M. A. (1985). Optcal analyss of a smulated mage of the sea surface. Appled Optcs, Vol.4, No.7, pp , ISSN X Álvarez-Borrego, J. & Martn-Atenza, B. (010). An mproved model to obtan some statstcal propertes of surface slopes va remote sensng usng varable reflecton angle. IEEE Transactons on Geoscence and Remote Sensng, Vol.48, No.10, pp , ISSN Bréon, F. M. & Henrst N. (006). Spaceborn observatons of ocean glnt reflectance and modelng of wave slope dstrbutons. Journal Geophyscal Research, Vol.111, CO6005, ISSN Chapman, R. D. & Iran G. B. (1981). Errors n estmatng slope spectra from wave mages. Appled Optcs, Vol.0, No.0, pp , ISSN X Chapron, B.; Vandemark D. & Elfouhaly T. (00). On the skewness of the sea slope probablty dstrbuton. Gas Transfer at Water Surfaces, Vol.17, pp , ISSN Cox, C. & Munk W. (1954a). Statstcs of the sea surface derved from sun gltter. Journal Marne Research, Vol.13, No., pp , ISSN Cox, C. & Munk W. (1954b). Measurements of the roughness of the sea surface from photographs of the Sun s gltter. Journal of the Optcal Socety of Amerca, Vol.4, No.11, pp , ISSN Cox, C. & Munk W. (1955). Some problems n optcal oceanography. Journal of Marne Research, Vo.14, pp , ISSN Cox, C. & Munk. W. (1956). Slopes of the sea surface deduced from photographs of sun gltter. Bulletn of the Scrpps Insttuton of Oceanography, Vol.6, No.9, pp Cureton, G. P. (010). Retreval of nonlnear spectral nformaton from ocean sunglnt. PhD thess, Curtn Unversty of Technology, Australa, March Cureton, G. P.; Anderson, S. J.; Lynch, M. J. & McGann, B. T. (007). Retreval of wnd wave elevaton spectra from sunglnt data. IEEE Transactons on Geoscence and Remote Sensng, Vol.45, No.9, pp , ISSN Fuks, I. M. & Charnotsk, M. I. (006). Statstcs of specular ponts at a randomly rough surface. Journal of the Optcal Socety of Amerca, Optcal Image Scence, Vol.3, No.1, pp , ISSN

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