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2 Statstcal Propertes of Surface Slopes va Remote Sensng Josué Álvarez-Borrego 1 and Beatrz Martín-Atenza 3 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).

3 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

4 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)

5 54 Remote Sensng Advanced Technques and Platforms On the other sde, the detecton system pupl can be represented by the functon P d 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 I R 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 (11) We fnd then the gltter functon, gven by B rect o. 1 o (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).

6 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 (14) Defnng a 1 4 and o o b 1 4, we can wrte o o 1 b a I I x erf erf. (15) The varance of the ntenstes n the mage, I, s defned by (Papouls, 1981) I But, B B, then I x Ix and substtutng the expresson of I x I x B I 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, (18) whch s the requred relaton between the varance of the ntenstes n the mage, the varance of the surface slopes,. 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

7 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) C dc d, (19) 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.

8 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 (0) 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. (1) s I Table 1. Values of the mage varance n order to normalze the correlatons n fgure 3 for dfferent values for s.

9 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.

10 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)

11 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 1 1 o I Ix rect exp d. 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 1 1 b a I I x erf erf. 1 (31) The varance of the ntenstes n the mage Atenza, 010) I s defned by (Álvarez-Borrego & Martín-

12 Statstcal Propertes of Surface Slopes va Remote Sensng 61 1 I I x Ix B I p d 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) 1 1 b a 1 b a I erf erf erf erf 1 4, (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

13 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.

14 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 where, 1 CI s gven by 1 1 I C I B 1 B p 1 d 1 d 1 1 C and the,, (35) p s defned by p 1 1 C 1 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 1 1 b C a C ICI exp erf erf d, 1 a C 1 C where a o 1o 4 and o o b 1 4. (37) 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 (on-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):

15 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 I I x B p d 1 (38) where B s the gltter functon defned by equaton (4). p s the probablty densty functon, where a non-gaussan functon s consdered n one dmenson (Cureton, 010) exp , 6 4 p (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

16 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 o I rect exp d 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 a a I exp a 3 a 3. (41) b b exp b 3 3 b 6 4

17 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 a a I exp a 3 a b b exp b 3 3 b b a erf erf a a exp a a b 4 b exp b 3 3 b (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 I

18 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

19 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.

20 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, I C I B 1 B p 1 d 1 d 1 1 I C and the,, (43) p s defned by (Cureton, 010) p 1 1 C 1 1, exp 1/ 1 C 1 C C 1 1, C where and are the skewness, moments of 1 and. 1 and (44) 1 are the relatonshp between the 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 b 1 1 ICI ua v A Ba C d 1 a 1 exp exp, erf ub v erf ua v A B C (45) where u 1 1 C,

21 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 1 C 3 6 C 3 3, 1 1 C C a 1 C 3 6 C 3 3, A3 C 4 3C 3 C, 4 1C 30 1C B3 C 1 C 1 C, 8 C 3 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).

22 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.

23 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 d 0 o, 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.

24 Statstcal Propertes of Surface Slopes va Remote Sensng Acknowledgments Ths work was partally supported by COACyT wth grant o 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, ISS X Álvarez-Borrego, J. (1993). Wave heght spectrum from sun glnt patterns: an nverse problem. Journal of Geophyscal Research, Vol.98, o.c6, pp , ISS Álvarez-Borrego, J. (1995). Some statstcal propertes of surface heghts va remote sensng. Journal of Modern Optcs, Vol.4, o., pp , ISS Álvarez-Borrego, J. & Machado M. A. (1985). Optcal analyss of a smulated mage of the sea surface. Appled Optcs, Vol.4, o.7, pp , ISS 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, o.10, pp , ISS Bréon, F. M. & Henrst. (006). Spaceborn observatons of ocean glnt reflectance and modelng of wave slope dstrbutons. Journal Geophyscal Research, Vol.111, CO6005, ISS Chapman, R. D. & Iran G. B. (1981). Errors n estmatng slope spectra from wave mages. Appled Optcs, Vol.0, o.0, pp , ISS 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 , ISS Cox, C. & Munk W. (1954a). Statstcs of the sea surface derved from sun gltter. Journal Marne Research, Vol.13, o., pp , ISS 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, o.11, pp , ISS Cox, C. & Munk W. (1955). Some problems n optcal oceanography. Journal of Marne Research, Vo.14, pp , ISS 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, o.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, o.9, pp , ISS 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, o.1, pp , ISS

25 74 Remote Sensng Advanced Technques and Platforms Gaskll, J. D. (1978). Lnear systems, Fourer transform, and optcs. John Wley & Sons. ISB , ew York, USA Longuet-Hggns, M. S. (196). The statstcal geometry of random surfaces. Proceedngs Symposum Appled Mathematcs th Hydrodynamc Instablty, pp Longuet-Hggns, M. S.; Cartwrght, D. E. & Smth,. D. (1963). Observatons of the drectonal spectrum of sea waves usng the motons of a floatng buoy, In: Ocean Wave Spectra, Prentce-Hall, Englewood Clffs,. J. (Ed.), Munk, W. (009). An nconvenent sea truth: spread, steepness, and skewness of surface slopes. Annual Revew of Marne Scences, Vol.1, pp , ISS Papouls, A. (1981). Probablty, Random Varables, and Stochastc Processes, chapter 9, McGraw- Hll, ISB , ew York, USA Peppers,. & Ostrem, J. S. (1978). Determnaton of wave slopes from photographs of the ocean surface: A new approach. Appled Optcs, Vol.17, o.1, pp , ISS X Plant, W. J. (003). A new nterpretaton of sea-surface slope probablty densty functons. Journal of Geophyscal Research, Vol.108, o.c9, 395, ISS Stlwell, D. Jr. (1969). Drectonal energy spectra of the sea from photographs. Journal of Geophyscal Research, Vol.74, o.8, pp , ISS Stlwell, D. Jr. & Plon, R. O. (1974). Drectonal spectra of surface waves from photographs. Journal of Geophyscal Research, Vol.79, o.9, pp , ISS

26 Remote Sensng - Advanced Technques and Platforms Edted by Dr. Bors Escalante ISB Hard cover, 46 pages Publsher InTech Publshed onlne 13, June, 01 Publshed n prnt edton June, 01 Ths dual concepton of remote sensng brought us to the dea of preparng two dfferent books; n addton to the frst book whch dsplays recent advances n remote sensng applcatons, ths book s devoted to new technques for data processng, sensors and platforms. 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. How to reference In order to correctly reference ths scholarly work, feel free to copy and paste the followng: Josué Álvarez-Borrego and Beatrz Martín-Atenza (01). Statstcal Propertes of Surface Slopes va Remote Sensng, Remote Sensng - Advanced Technques and Platforms, Dr. Bors Escalante (Ed.), ISB: , InTech, Avalable from: InTech Europe Unversty Campus STeP R Slavka Krautzeka 83/A Rjeka, Croata Phone: +385 (51) Fax: +385 (51) InTech Chna Unt 405, Offce Block, Hotel Equatoral Shangha o.65, Yan An Road (West), Shangha, 00040, Chna Phone: Fax:

27 01 The Author(s). Lcensee IntechOpen. Ths s an open access artcle dstrbuted under the terms of the Creatve Commons Attrbuton 3.0 Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted.