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ocean system reflectance
REMOTE SENS. ENVIRON. 43:179-192 (1993)
An Analytical Model for the Cloud-Free Atmosphere/Ocean System Reflectance E M. Brdon Laboratoire d'Optique Atmosph~rique, Villeneuve d'Ascq, France
A n analytical model for the estimation of spectral and directional reflectance over the ocean is developed. The model accounts for the first and second orders of reflection at the ocean surface, and/or scattering by the atmosphere and the water column. The model parameters are the optical thickness and angstr6m exponent of aerosols, the spectral albedo of the ocean, and the directional specular reflectance of the ocean surface (glitter), the latter being a function of the wind speed and direction. We show that our analytical model is su2fficient to predict all reflectance angular and spectral variations by comparing its results with a Monte Carlo simulation that accounts for the higher orders of scattering. Using this analytical model, actual bidirectional and spectral measurements are easily inverted in order to retrieve geophysical and optical parameters of the oceanatmosphere system (the application to the data acquired by the Airborne POLDER is developed in a companion article), INTRODUCTION Over the ocean, the sparsity of in situ measurement makes it necessary to use satellite observations for the estimate of key climatological and
Address correspondence to F. M. Br~on, Laboratoire d'Optique Atmosph~rique, USTL, Bat. P5, 59655 Villeneuve d'Ascq Cedex, France. Received 24 January1992; revised 23 May1992. 0034-4257 / 93 / $6.00 ©Elsevier Science Publishing Co. Inc., 1993 655 Avenue of the Americas, New York, IVY 10010
biological parameters such as cloud coverage, atmospheric aerosol content, sea surface state, or the ocean upper layer phytoplankton content. The requirement for global ocean monitoring has been stressed in the recent years, and a large consensus has been reached about the crucial role of the ocean in the expected climate change. POLDER(POLarizationandDirectionalityof the Earth Reflectance) is a new instrument devoted to the observation of the polarization and directionality of solar radiation reflected by the Earth-atmosphere system. It allows a multiangular, multiwavelength, multipolarization measurement of the Earth system reflectance. Its design and mission as part of the Japanese ADEOS platform, scheduled to be launched in 1996, has been described in Deschamps et al. (1990; 1993). An airborne version of the POLDER instrument has been implemented, and several dedicated flight experiments have been made or are planned to prepare for the spaceborne experiment. In order to take the full advantage of the POLDER angular and spectral measurement capabilities, it is necessary to develop a model that describes the angular and spectral variations of the observed reflectances. This model should be reasonably simple but must account for all important scatterings and/or reflections occurring in the ocean-atmosphere system. It also needs to be invertible so that optical parameters of the ocean-atmosphere system can be retrieved from POLDER observations. Extended radiative t r a n s fer simulations have been performed for the
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180 Br~on
ocean-atmosphere system (e.g., Plass et al., 1975), including polarization effects (e.g., Kattawar et al., 1973), but these do not meet our simplicity requirements, In the next section, the various scattering/ reflectance processes occurring in the atmosphere/ocean system are discussed. In the third section, a reflectance model is developed that estimates these processes. The model is validated against a Monte Carlo procedure in the fourth section. An inversion method for spectral and directional reflectance measurements is described in the fifth section. The sixth section summarizes and concludes. In a companion article (Br6on and Deschamps, 1993), the model and the inversion procedure are applied to POLDER measurements acquired over the Mediterranean Sea.
SCATTERING AND REFLECTANCE PROCESSES OCCURRING IN THE ATMOSPHERE/OCEAN SYSTEM In the atmosphere, solar radiation is scattered by molecules and aerosols. Molecular scattering closely follows Rayleigh's theory. The molecular optical thickness has a 2-4 dependence, where 2 is the wavelength. It decreases from 0.22 to 0.017 between 450 nm and 850 nm. Moreover, the molecular scattering phase function is very close to the theoretical Rayleigh's law: P(og) =~3 [1 + cos2(o9)].
(1)
Slight deviation from this law is observed, resulting from the nonsphericity of atmospheric molecules. Aerosol scattering is more complex and variable. Its scattering phase function and the spectral variation of its optical thickness greatly depend on the aerosol size distribution and refractive index. However, some constant characteristics are found: • Because the aerosol size distribution is of the order of, or larger than, the wavelength of visible light, the aerosol scattering phase function shows a large maximum in the forward scattering direction, The ratio of scattered energy in the forward hemisphere is typically of the order of 0.85.
• The aerosol phase function shows little variation in the visible and near-infrared wavelength range. This variation will therefore be neglected in the analytical model. • The aerosol optical thickness spectral variation is closely modeled by the following exponential formula: Taer2 ~" "/'aer500(/]- / 5 0 0 ) - y (2) (2 in nm), where raerS00is the aerosol optical thickness at 500 nm and ~, is the ~,ngstr6m exponent, which generally ranges from 0.5 to 2. The largest ,4,ngstr6m exponents are found for the smallest size distributions. Over the ocean, the typical values for raersoorange from 0.05 to 0.4. • The aerosol single scattering albedo, the ratio of scattering to extinction, is very close to 1, meaning no absorption. This is not the case for the urban type aerosols, but they are seldom found over the ocean. Aerosol absorption is then negligible when compared with scattering. The aerosol scattering phase function depends on the physical properties, size distribution, and refractive index of the aerosol. These properties may vary, leading to extremely diferent phase functions (Deirmendjian, 1969; Hansen and Travis, 1974). Different aerosol types have been proposed by the scientific community (WMO, 1986). The simple Henyey-Greenstein analytical function can also be used to approximate the more complex expression of the phase functions computed from the Mie theory: PFH_c(~) =
1 - G2 [1 + G e - 2G cos(o))] 3/2'
(3)
where o) is the scattering angle. G is the anisotropy factor, that is, the scattered radiation mean value of the scattering angle cosine. The phase function is isotropic for G = 0. It approximates the Dirak function for G = 1. In the visible and near-infrared spectral ranges, the atmospheric molecule absorption is mainly due to water vapor and ozone. Ozone absorption is relatively constant and occurs in the upper layers of the atmosphere. Scattering and ozone absorption processes can, therefore, be easily decoupled and modeled separately. However,
Analytical Model of Atmosphere/ Ocean Reflectance 181
water vapor absorption and aerosol scattering both occur mainly in the lower troposphere. At the surface, specular reflection follows the Fresnel law. The fraction of incident energy that is specularly reflected is about 2% at incidence angles up to 50 ° and then increases up to 100% at grazing i n c i d e n c e . F o r a perfectly flat o c e a n surface, the reflected direct solar radiation, also called glitter, would be contained in a very small solid angle around the specular direction. However, the wind generates large and small scale waves on the surface, and these broaden the glitter pattern. Specularly reflected radiance can then be observed further away from the specular direction. Ocean specular reflectances are closely related to the wave statistics, a model of which has b e e n proposed by Cox and Munk (1954). T h i s model is recalled and used in the Appendix i n order to express the glitter reflectance as a function of the wave slope distribution. The ocean surface roughness may modify the albedo of the ocean surface by changing the incidence angle and thus the Fresnel reflectance. Let us evaluate how the specular albedo varies with ocean surface roughness. We here assume an isotropic Gaussian wave slope distribution (see the Appendix) and neglect all atmospheric effects, The reflectance is averaged over all the possible wave slope orientations according to their distribution statistics. The results are presented in Figure 1. The no wind speed, "fiat surface," albedo is simply the water specular reflection coefficient, which increases with the incidence angle but not linearly. As a result, the ocean surface albedo increases with the wind speed at most solar zenith angles (< 65°). Note that the effect is very small at the lowest zenith angles because the specular reflectance is nearly constant at incidence angles less than 40 °. The relative change in the ocean surface albedo is less than 10% for a wind speed of 8 ms -1. These results agree with that of Austin (1974). It is clear that the surface roughness greatly affects the angular field of the reflected radiances. However, the above computations show that the reflected fraction of the downward irradiance is only weakly sensitive to the surface roughness, The radiance scattered by the water column has a large spectral variation and this variation depends on the upper layer concentration in marine algae and associated detritus (Morel, 1988).
12 . . . . I . . . . I . . . . ! . . . . . . . . . . . . . . . I Flat Surface I f_ 10 . . . . . . . . . . . i......................... !.....................T--ff-" ~ ws = 8 ms-1 || i ~g II .. .. .. .. .. ws=14ms-1 i i "7 'J! .~ 8 I | .-.~;?~--.._~ 6 .....................~........................~........................i......................i.........................i............~;~'-'~..................... ~. :: i i i i,¢" i = 4 .....................ii ........................Fi ........................i~:....................... -~"" i ...................... i iii ~ , L_._ 2 . ....... I .... i .... i .... i .... 0 0 . . . . 10 20 30 40 50 60 70 .............
...................
.....................
Solar Zenith Angle
Figure 1. Ocean specular albedo sensitivity to the surface roughness. The specular albedo is given as a function of solay zenith angle for a flat surface (lower curve), a wind speed of 8 m s- 1 (middle curve), and a wind speed of 14 m s-1 (upper curve). The surface slopes are distributed according to Eq. (Ag).
Typical values of the water column reflectance range from 0.003 to 0.025 at 450 nm, from 0.005 to 0.01 at 550 nm, and from 0.0001 to 0.002 at 700 nm. An increase of the algae concentration lowers water column reflectances at wavelengths below 550 nm because absorption by phytoplankton is the dominant process; it has the opposite effect above 550 nm, where the scattering by phytoplankton prevails over the absorption (except in the range 650-680 nm, which corresponds to another chlorophyl absorption band). The angular field of the radiance scattered by the water column is not well known because it is not easy to measure in situ. Underwater measurements by Smith (1974) indicate limited variation over the hemisphere. Computations by Plass et al. (1975) lead to similar conclusions. The angular distribution of upwelling radiance may affect the remote sensing of ocean color from space. More research is needed on that subject in order to normalize the directional observations by the current scanners or to take full advantage of the POLDER multiangular viewing capabilities. Using a Monte Carlo method to simulate the underwater radiative transfer in a limited number of cases, we have found little angular dependence in the water column reflectance, especially at viewing zenith angles less than 60 °, which are those angles accessible to most spaceborne and airborne sensors. Therefore, in our model, we
182
Br~on
have chosen to neglect the angular dependence of the water column reflectance, At large wind speeds, above ~- 7 m s- 1, whitecaps will appear and occupy part of the ocean surface. The whitecap reflectance is assumed spectrally flat in the visible range. Note that whitecaps are seldom observed from spaceborne visible sensors because strong winds are frequently associated with bad weather and, therefore, cloudiness. However, the frequency of their occurrence under clear sky conditions cannot be ruled out. Whitecaps have been reported during the airborne POLDER flight that is analyzed in the companion article. Note that, assuming isotropy of the whitecap reflectance, it has now the same nature as the water column reflectance and can be included in it. Then, in our model, the water column reflectance is not necessarily negligible in the near infrared spectral range,
cludes the scattering by atmospheric molecules and aerosols. The model also accounts for the ocean surface specular reflection and the water column reflectance. Atmospheric optical thickness is assumed to be small enough so that all photon paths involving more than two of the above processes can be neglected. An important assumption of our modeling is to neglect the interactions between molecular and aerosol atmospheric scatterings. The practical reason for this choice is that its estimate would require a convolution of both phase functions. This approximation can be justified by the fact that aerosol scattering is mostly in the forward direction. Therefore, aerosol-then-molecular scattering would have an angular distribution similar to the molecular-only scattering. Moreover, it involves two scattering processes that both correspond to relatively small optical thicknesses, so that taking into account the molecule-aerosol double scattering would involve a second-order term. The validation of this hypothesis will be obtained by comparison of our analytical model with a Monte Carlo method computation that accounts for all orders of scattering. The measured reflectance at flight level is modeled as the addition of nine different contributions (see Fig. 2): molecular single scattering (pl); molecular second order of scattering (p2 and P3); interactions between specular reflection and
THE ANALYTICAL REFI.FCTANCE MODEL Our objective is to develop a model that correctly predicts the angular and spectral reflectance variations of the ocean-atmosphere system when flying at a given altitude. We want this model to be simple and analytical so that it can be inverted with a limited computational burden. We assume the atmosphere to be cloud-free. The model in-
Figure 2. Graphic representation of scattering and reflectance processes occurring in the atmosphere / ocean system
and evaluated in the analytical model: molecule single scattering (Pl); molecule double scattering (p2 and p3); ocean specular reflectance (P4); specular reflectance and molecule scattering (p5 and p6); water column reflectance (pT); aerosol reflectance (Ps); specular reflectance and aerosol scattering (p9). The dashed line is for single scattering occurring above the instrument.
- ~
Modelized Scattering/Reflectance processes
Solar Directi Instrument Level
~
,
p2
p
p3
,
/ p8
/'"
.-
p9
Viewing Directi~ p5
/
/
// p6
p7
Analytical Model of Atmosphere / Ocean Reflectance 183 molecule scattering (P4 and ps); ocean surface specular reflectance or glitter (P6); water column reflectance (p7); aerosol reflectance (p8); and interactions between specular reflection and aerosol scattering (p9). Our model simulates the reflectance at the top of the atmosphere (satellite observation) or somewhere within the atmosphere (airborne measurements). Let us define aun =
Psurface- P~rcr~ esurface
(4)
and P~rcr~ aab = P~,,aace'
(5)
where PsurraCeis the surface atmospheric pressure and P,,r~r~ is the atmospheric pressure at the aircraft level. If ra is the atmospheric molecule optical thickness, run = a,nra and r~b = a~brx are then, respectively, the molecular optical thickness under and above the instrument. When applied to satellite observation, a~b shall be set to 0 and a,, to 1. The first-order Rayleigh scattering contribution to the measured reflectance is
[ -r~b\.l-exp[-run(l/gs+l/g~)] p, = exp~---~ ) x
4(/is +/iv)
P,~y(O~,0~, cp),
(6)
where enay is the Rayleigh scattering phase function and/t, and go are the cosines of the solar and viewing zenith angles, respectively, p~ is the single scattering reflectance below the instrument by the molecular layer of optical thickness run after the extinction above the instrument by the molecular layer of optical thickness r~b. In order to estimate the second order of molecular scattering, we must study two cases: i) the first scattering occuring above the instrument and ii) both seatterings oceuring under the instrument. First, we estimate the latter process. At flight level, the downward direct solar irradiance is /~=Fo exp(-r~b//C). (7) The fraction of/~ that is scattered twice below the instrument is 1 f [ " dr If~ (~) /t~ -~1 - exp dg +
f~
1 - exp
(
r~ ] /t / dg ru._-
(8)
or 1 l~,,[I I (~) ~ Jo 1 - exp -
]
dg dr.
(9)
The integration variable dr is the layer at which the first scattering occurs. The other integration variable, d/t, is the cosine of the radiance zenith angle after the first interaction. The first (resp. second) integral within the bracket is for radiance scattered upward (resp. downward) by the first molecular interaction. In this formulation, we do not consider any third-order scattering. Similarly, we neglect the angular dependence related to the scattering phase function. This assumption would not be appopriate with an aerosol-type phase function that greatly departs from isotropy, but it is reasonable with the Rayleigh phase function. On the other hand, we account for the fact that radiance first scattered to a nearhorizontal direction has a large probability of going through a second scattering. To our knowledge, the double integral in Eq. (9) is not analytically integrable. However, within the visible and near infrared spectral ranges, we are dealing with small values of run, typically of the order of 0.1 or less. For such small values, the double integral can be very well approximated as t"runll 1 J0 LJO -
dg]
= (r,n)2Cl(r,n) (10)
with C1(r)=0.4447-1.165 logxo('C). (11) Assuming isotropic scattering and neglecting any third-order scattering, the modeled reflectance has a 1/go angular dependence. The reflectance corresponding to the contribution by the second order of molecular scattering, when both interactions occur below the instrument, is then (r"~)~ e x p ( - ~-~,b)Cl(run). (12) P2 = 4/t,g~ The fraction of the downward irradiance that is first scattered above the aircraft and then again below the aircraft can be estimated by using a similar method, that is, the integral over the scattering level and the radiance zenith angle for the two molecular scatterings. As for the estimate of Pz, one encounters an integral that does not have an analytical solution. As above, we find that, for small values of the optical thickness, it can be approximated by
184 Br~on f~I ( ~2)1[ ( ~)1 /t 1 - exp 1 - exp dg = runr,b C 2 ( ~ )
(13)
with C~(r) = - 0.3025 - 2.2265 logl0(r).
(14)
This approximation is accurate provided that both optical thicknesses are small (of the order of 0.1 or less) and of the same order of magnitude. The former condition holds for atmospheric molecular optical thicknesses in the visible and near-infrared spectral regions. The latter condition is fulfilled provided that the instrument altitude is of the order of 5000 m. At very low altitudes (< 1000 m), or high flight levels like space observations, the reflectance contribution estimated in this section is neglected, The reflectance resulting from one scattering above the aircraft and one below is then evaluated as
"['abl'unt~(X~r, ,.., , /------~ nrab). (15) 8/~dz~ It is clear that our formulation of the second order of molecular scattering, as expressed in Eqs. (9) and (12), is not valid at large viewing angles (small /z~) because we did not account for the radiance reduction by a possible third scattering. However, the approximation holds for reasonable optical thicknesses and the viewing angle capabilities of most instruments (~u~>0.5). Let us now evaluate the reflectance due to single molecular scattering and specular reflection by the ocean surface. For this contribution, we shall assume that the ocean surface is flat. Although it is not strictly so, the departure from flatness does not greatly affect the amount and angular distribution of reflected radiance that goes through molecular scattering and specular reflection. The reasons are: i) The wave slopes are small (of the order of 10°); ii) the fraction of the irradiance that is reflected depends little on the surface roughness (see Fig. 1); and iii) the molecular scattering phase function is rather close to isotropy, which smooths any angular distribution deviation, Because the radiometer may not be at the top of the atmosphere, one should distinguish between the radiance that is first scattered and P3 =
then specularly reflected and the radiance that is specularly reflected and then scattered to the viewing direction. The first process leads to a reflectance equal to p4 =
exp( - ra l/J,) - exp( - ra IlUv) 4(gt,- go) r..\ p Ray(-- ~s,/~v, ¢0), (16) x exp - ~v)pSp(l~v)
whereas the second gives p5 = e x p f - ~ exp(- r..//~,)- e x p ( - r../l~v) \ /~J 4(/a,-#v) ×Psp(ltl~)PRay(_/As, lAv,(O), (17) where Psp is the water reflection coefficient given by Fresnel laws. These expressions were obtained by integrating the radiance reaching the instrument over the various levels where the scattering may occur. In Eqs. (16) and (17), the first term is for the downwelling path and the second term is for the upwelling path. The phase function to be used is simply that of single molecular scattering with a mirror sun direction (/g, is changed to - ~t,). Glitter reflectance or specular reflection of direct solar radiation by the surface greatly depends on the sea state, that is, the surface wave slope distribution. On the other hand, the specular reflection coefficient is independent of wavelength. We therefore model the glitter reflectance as ( rx +/~,raera~run+'/'aera\] exp ( ~v -'S'~,/t,, ¢), p0 = exp / (18) where raera is the spectral aerosol optical thickness and S depends on the sea surface state and viewing conditions, but not on the wavelength. The incident direct solar radiation that is reflected by the surface is attenuated by molecular and aerosol scattering, with a spectral dependence through their optical thicknesses. Note that, in this formulation, we assume that the aerosol layer is located mostly below the instrument. When this is not so, one reduces accordingly the aerosol optical thickness on the upwelling path (second exponential term). In the Appendix, we show how specular reflectance at the surface, S, is related to the wind speed and direction according to a wave slope distribution model.
Analytical Model of Atmosphere/ Ocean Reflectance 185
We now evaluate the water column contribution to the reflectance. As mentioned in the previous section, there is assumed an isotropic angular distribution for the water-leaving radiance just above the surface. As aerosol scattering is mostly concentrated in the forward direction, a large fraction of the downward irradiance does reach the ocean surface even after it has been scattered by the aerosols. Therefore, we model only the downward irradiance attenuation by molecular scattering, and we neglect the effect of aerosol scattering on it. We then model the water column reflectance contribution to the measured reflectance as [1
1 exp(_ za/][exp(_ run/+ run] lz,/J[ \ l~v/ 2/z~/
p~=Ro~e~ +-~
(19) where Ro~ is the spectral reflectance of the water column, including white caps. The first bracket is the irradiance ratio that reaches the ocean surface. We assume here that half the irradiance once scattered by molecules reaches the surface (single-scattering approximation). The second bracket includes the direct radiance from the ocean to the instrument and that which is scattered once. For the latter, we assume isotropic single scattering and neglect higher order scattering. Note that Roc~a is here the diffuse ocean reflectance measured above the surface. A correction to the more generally used reflectance under the surface would therefore be necessary for comparison purposes. As shown in the previous section, aerosol optical thickness is an exponential function of the wavelength. The aerosol contribution to the measured reflectance is then modeled as
p8 ~aeraeaer(llls'[llv' (p) =
-r
= 4/.tgt~
~5--~]
~ (20)
This formulation is used, rather than that of single scattering [Eq. (6)] because it directly accounts for most of the multiple scattering that occurs within the large forward peak of the aerosol scattering phase function. The second order of aerosol scattering out of this forward peak is neglected. The phase function is selected in accordance with the investigated situation. As deduced from Eq.
(20), the parameter of importance is not the phase function nor the aerosol optical thickness but rather their product in the backscattering hemisphere. The last term of our analytical reflectance model estimates the radiance specularly reflected by the ocean suface and scattered by the aerosol layer. As for the corresponding molecular scattering, we assume that the ocean surface is fiat. We then obtain p0 =
Taer2
r
[psp(/Zs)+ psp(/~o)] Paer( --/~s, U~, ~). (21)
The approximation of ocean surface flatness here does not hold as well as it does for the molecular scattering (P4 and Ps). This is because the aerosol scattering phase function, in contrast to the molecular one, is far from isotropic. Therefore, close to the center of glitter, that is, the specular direction, one needs an angular convolution of the glitter reflectance and the phase function to estimate p9 precisely. However, away from the specular direction, the reflectance estimate as expressed by P9 is a good approximation and is an alternative to the time-consuming convolution. Hemispheric convolution of the phase function and the glitter reflectance is clearly contradictory to our model objectives. Alternatives depend on the surface conditions. • If the ocean surface is very smooth (low wind speed), the specular reflectance is confined within a narrow solid angle. The contribution to the reflectance, as expressed by Eq. (21), is then accurate. • If the ocean surface is rough (large wind speed), one may neglect, close to the specular direction, the aerosol effect on the glitter reflectance. This is possible because most aerosol scattered radiance deviates only a little. One therefore sets p9 to zero and neglects the aerosol reduction of direct glitter radiance in P6. The two approximations partly compensate each other. The directional reflectance of the ocean-atmosphere system is modeled as the sum of the nine p~ terms. The model parameters are the
186 Br~on
aerosol optical thickness rae~, the ]kngstr6m exponent y, the spectral water column reflectance Ro~ea, and the glitter reflectance at the surface S for each viewing condition. The Appendix shows that the function S(/zs,/zo, ~) is related to the wave slope distribution Pws(Z~,Z~)according to
S(lzs, lZo,¢p)= npsp(X)Pws(Z~,Zy)
(22) 4/~s/zo cos4(/~) ' where Zx and Zy are the two components of the wave slope, 1~ is the wave tilt, and X is the incidence angle. Zx, Zy, and/~ can be obtained from /zs,/to, and ¢pwith the help of simple trigonometry, Cox and Munk (1954) give an expression for Pws as a function of the surface wind speed and direction only. When used in the direct mode or for validation purposes (next section), we shall use the Cox and Munk parameterization for the glitter reflectance. When used for the inversion of reflectance measurements, however, the glitter reflectance is a parameter for each direction of observation. As this approach implies at least one unknown per observation direction, it is clearly feasible only if several spectral measurements are available,
MODEL VALIDATION Several approximations have been made in order to develop the model described in the previous section. Some of them, such as isotropicity of oceanic reflectance, the nonabsorbing property of aerosols, or the phase function invariance with wavelength, can only be validated by comparison with measurements. On the other hand, the neglect of third- and further-order reflectance / scattering processes, as well as the neglect of aerosol / molecule radiative interaction, can be evaluated with the help of a more detailed model that accounts for all orders of scattering. A Monte Carlo model was developed for the study of angular distribution of reflectance from broken clouds (Br~on, 1992). We adapted this model for the ocean/surface/atmosphere system simulation. The geometry is plane parallel. The atmosphere is constituted of two layers, the upper one with molecules only and the lower one with mixed aerosols and molecules. The ocean surface slope distribution follows the anisotropic parame-
terization by Cox and Munk (1954). The radiance reflected by the inner ocean is isotropic. The Monte Carlo method, hereafter referred to as MC, follows the path of many photons. It makes use of random numbers for the determination of the optical path between two scatterings, the angle of diffusion, the orientation of the ocean surface, or the choice between transmission and refraction at the surface. An improvement of the computational efficiency is achieved by applying to each photon a "weight" according to its history. The sum of the photon weights that exit the lower atmospheric layer in the upward direction is directly related to the reflectance measured at the boundary between the two atmospheric layers. Figures 3, 4, and 5 show the results of both the MC and our analytical model simulations. These are reflectances over a constant azimuth direction and for varying zenith angles. The analytical model results are shown as a line whereas MC estimates are depicted by dots. Note that the latter should be considered as an average over an angular bin. Any lack of smoothness in the MC estimates is due to the statistical uncertainty inherent in this computation method. Let us first discuss the case of a very clear atmosphere (no aerosols). The computations are made for a wavelength of 450 nm, and the total atmospheric optical thickness is, therefore, 0.22. We set the solar zenith angle to 40 ° , the ocean reflectance to 1.5%, the wind speed to 8 m s -1, and the wind direction relative to the sun to 150 ° . The atmospheric pressure at the instrument level is assumed to be 560 Hpa. Figures 3a, 3b, and 3c are the results of our comparisons for the principal plane, the 30-210 ° azimuth plane, and the perpendicular plane. The reflectance variations with zenith angle are all correctly predicted. The most noticeable features are: * A large reflectance increase toward limb viewing. This increase is related to the increased atmospheric optical path. * A large reflectance maximum in the principal plane toward the sun. Note that the viewing zenith angle for the local maximum is not equal to the solar zenith angle: The reflectance maximum direction is shifted from the specular direction to a larger zenith angle. Note also the local
Analytical Model of Atmosphere/ Ocean Reflectance 1 8 7
3o . . . . . . . . . . . . . . . . . .
j ...... i 25 ......... Wavelength:450 nm ...........' ~ ' ~ ~iiiiiii "~ 20 ~ ~\ ~ 15
m i n i m u m of reflectance correctly modeled at about 70 ° zenith viewing angle. • A local reflectance m a x i m u m close to zenith viewing in the perpendicular plane. This m a x i m u m is related to the glitter pattern thatspeedseXtendsto that direction. For wind lower than that chosen here (8 m s-l), the glitter pattern would be narrower, and the local m a x i m u m found here may not be present.
Principal Plane
~
l0 ............o .............;................~.................i.............. .....-.......-....~...............-.~.--.---.-. ..... oo ~ ~ 5 -80
-60
-,40
(a)
-20
0
20
40
60
80
ViewingZenith Angle 25 . . . . . .
, ................. i
Plane 30 °- 210 ° Wavelength: 450 nm 20 ........................................................................................................................................
" 15
! i
lo ' - - ~ o
i i
i i
i i o !°~ i 5 7. . . . . . i , , , ~,, -8o -60 -40 -20 (b)
i o i i ~ ~ ) 7 ......
~J~i ~ ~ ~. . . . . . . . . . . 20 40 60
80
Viewing ZenithAngle
15 13
i
ll
,~ 9 7
5-.80 (c)
--60
-..40
-20
0
20
40
60
80
Viewing Zenith Angle
Figure3. Comparison between the analytical model reflectance estimate (line) and that of a Monte Carlo method (dots). The wind speed is 8 m s-l; the solar zenith angle is 40 ° . The atmosphere only contains molecules. The wavelength is 450 nm.The reflectance is given as a function of viewing zenith angle in the principal plane (a), the 30 ° azimuth plane (b), and the perpendicular plane (c). See text for details,
Some slight discrepancies in the reflectance m a g n i t u d e are found. W h e n c o m p a r e d with MC simulations, the m o d e l e d reflectances are overestimated in the center glitter direction by about 1% (reflectance units). This discrepancy may be explained by the large azimuth gradient in reflectance in this direction and the fact that MC results are averaged over an angular bin (10 ° in the azimuth direction). For other viewing directions, the reflectance seems to be slightly u n d e r e s t i m a t e d by the analytical model. T h e exact m a g n i t u d e of this underestimate is difficult to estimate because of the statistical uncertainties i n h e r e n t in the MC method. It is of the order of 0.2-0.3% for zenith viewing angles smaller than 50 °. In order to establish which reflectance process is responsible for the bias, their various orders of m a g n i t u d e are discussed. For a zenith viewing angle of 20 ° in the perpendicular plane, single scattering accounts for a reflectance of 3.65%, the reflectance from the ocean is 1.22%, specular reflectance is 0.05%, second-order Rayleigh scattering amounts to 0.46 and 0.38 (p2 and p3), and specular reflectance/ molecular scattering (ps+p6) to 0.15%. The expressions for single scattering and specular reflectance are exact. T h e approximation for the ocean reflectance estimate cannot explain the differences. The order of magnitude for specular r e f l e c t a n c e / m o l e c u l a r scattering shows that any further scattering order of surface / atmosphere radiative interaction is negligible. The slight bias results, therefore, from the approximations for the estimate of multiple molecular scattering. A likely cause is the neglect of the phase-function anisotropy. An exact c o m p u t a t i o n of double Rayleigh scattering, which would involve the convolution of two phase functions, is clearly not within the scope of our m o d e l which is designed to be analyt-
188 Br~on
3O •.
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Figure 5. Same as Figure 4 but for a w a v e l e n g t h of 670
optical thickness 0.15, a solar zenith angle of 35 °, a w i n d speed of 5 m s- l, and a w a v e l e n g t h of 500 nm.
nm. At this w a v e l e n g t h , the aerosol optical thickness is 0.097.
ical. The accuracy achieved with the analytical form is sufficient to correctly reproduce the reflectance angular variations. If greater accuracy is needed, one must perform a more precise estimate of double Rayleigh scattering. Note that, as Rayleigh optical thickness is constant, the precise
computation could be performed once and for all. A greater accuracy could therefore be achieved with a limited computational burden. We now add the aerosol layer to our atmospheric model. The aerosol layer optical thickness raer~ is 0.15, and its Angstr6m coefficient is 1.5.
Analytical Model of Atmosphere~Ocean Reflectance 189
The aerosol phase function is given by the Henyey-Greenstein model with an asymmetry parameter G equal to 0.7. The surface wind is 5 m s -1 and the wind direction is unchanged. The ocean reflectance is 1.5% at 500 nm and 0.2% at 670 nm. The aerosol optical thickness is small compared with typical values over the oceans. However, the Henyey-Greenstein phase function is similar to a truncated "real" phase function. Thus, the product optical thickness-phase function has the correct order of magnitude, Figures 4 and 5 show the comparison between MC and the analytical model estimates for wavelengths of 500 nm and 670 nm, respectively. We present the reflectance angular variations in the principal plane, the 30-210" azimuth plane, and the perpendicular plane. These comparisons show main characteristics similar to the "moleculesonly~ case: The main angular variations are correctly reproduced. The reflectance is overestimated by the analytical model in the glitter direction and slightly underestimated in the other directions, The aerosol layer smooths the angular variations of the reflectance. It is especially obvious in the glitter direction (Figs. 4a and 5a). The glitter pattern is not correctly reproduced by the analytical model. This is because the estimate of reflectance resulting from specular reflection and aerosol scattering assumes a fiat surface. As mentioned before, this hypothesis leads to accurate results when considering molecular scattering because the Rayleigh phase function is close to isotropy. However, an accurate estimate of the aerosol contribution would require a convolution of the glitter pattern at the surface and the Rayleigh phase function is close to isotropy. However, an accurate estimate of the aerosol contribution would require a convolution of the glitter pattern at the surface and the Rayleigh phase function, Note that the model approximation of a fiat surface would be more accurate for lower wind speeds. The small bias between the MC and the analytical model estimates is troublesome. If the measurement objective is the retrieval of ocean color, an accuracy greater than the analytical model uncertainty is needed. A more precise estimate of double scattering or an empirical correction for the analytical model bias would then be necessary, For other purposes, such as the aerosol optical
thickness and Angst6m coefficient estimate, the model accuracy is sufficient.
MODEL INVERSION We designed our bidirectional reflectance model to be simple and analytical so that it is easily inverted against airborne measurements such as POLDER's. We will now describe the inversion procedure: Let us define Nob~as the number of available measurement directions and Na the number of spectral bands. There are Nobs+ N~ + 2 unknowns, whereas the number of observations is NaNobs. Moreover, the parameters are bounded by physically possible values: Far from the glitter viewing direction, the specular reflectance tends to zero. Similarly, the ocean diffuse reflectance for the longer wavelengths can reasonably be set to very small values, therefore reducing the number of unknowns. However, these include reflectance from white caps and, therefore, they cannot be neglected in the case of a rough sea. A first guess for the model parameters can be obtained as follows: • Far from the glitter direction, we neglect the direct specular reflection. We also neglect the longer wavelength ocean reflectances, or their differences if white cap reflectance is suspected. These wavelength reflectances measurements are then used to obtain raer and y. Note that most theoretical aerosol phase functions show a local maximum in the backscattering direction. The aerosol signal is, therefore, larger when the measurement direction is antispecular. This viewing direction is preferred for this step of the method. • Using the same measurement direction, we can then obtain a first guess for the shorter wavelength ocean reflectances. • Using the aerosol parameters' first guesses and the 850 nm reflectance measurements, a first guess for the surface specular reflectance can be derived for each viewing direction. From this first guess set, the reflectance model
190
Brdon
is inverted with an iterative method; a function is defined as a normalized squared difference between the model and the measurements: F(r .... y, Sk, Pt~e~ 3~) f7~/~ i 2| 1 / Nob -- s -N 2- / O b %i - Moda~ - " 2_J 2 . J -,L--S-_i ~ ] . (23) N~NobsL,=lj~l~ Obs~j
has been developed in order to take full advantage of POLDER multidirectional reflectance measurements. Its input parameters are the aerosol optical thickness, the aerosol /kngstr6m coefficient, the spectral ocean albedo, and the ocean surface directional reflectance. The model can be inverted against directional spectral measurements in order to retrieve the above parameters
This function is numerically computed for the parameter set, and then repeated with each parameter increased and reduced by a given increment. The F-function partial derivatives are thus obtained, and the model parameters are modified accordingly in order to reduce F. The parameter increments are progressively reduced, and this process is repeated until all second-order derivatives are positive. Because an accurate first guess was obtained, this iterative method rapidly converges to the sought parameters. A product of the inversion is the ocean specular reflectance in many directions. These are corrected for the atmospheric effects. The Appendix shows that the specular reflectance is related to the wave slope distribution as in Eq. (22). The final step of the inversion procedure is therefore a fit of the slope distribution as retrieved by the inversion to the Cox and Munk (1954) model. The two model parameters are the wind speed and the wind direction relative to the sun. A crude estimate of the wind speed is easily obtained assuming that the wave slope distribution is isotropic and follows a Gaussian distribution as in Eq. (A9). An estimate of the wind direction and a better retrieval of the wind speed are then sought: For discrete values of the wind direction and the wind speed around our first estimate, we compute the correlation between the retrieved wave slope distribution and the model values. The wind speed and direction that correspond to the best correlation then become our estimates,
as well as the wind speed and direction. The model expresses the measured reflectance as a sum of nine terms that account for the most significant scattering/reflectance processes. It is then very well suited to estimate the relative importance of these processes. The model accounts neither for successive scattering from molecules and aerosols, nor for all third and higher orders of scattering/reflectance. It does account, however, for all firstorder processes and all significant second-order processes. We have shown, through comparison with a Monte Carlo model that accounts for all scattering/reflectance orders, that the analytical model accurately reproduces the spectral and angular reflectance variations. The largest errors are found in the glitter direction where an accurate reflectance estimate would necessitate a convolution of the aerosol phase function and the glitter pattern at the surface. It is clear that, as the atmopsheric optical thickness increases, the neglect of the higher scattering orders becomes more critical. Similarly, the analytical model would fail to accurately estimate the atmosphere/ocean reflectance for large solar and / or viewing angles. As said before, our model has been specifically designed to take full advantage of midatmosphere airborne POLDER observations. This instrument provides a full description of the reflectance angular distribuiton at wavelengths suitable for ocean color monitoring and aerosol observation. Scheduled to be launched aboard the ADEOS platform in 1996, several airborne campaigns over the ocean have been carried out or are planned. These include Medimar, whose measurements are used in the companion article (Brdon and Deschamps, 1993), RACER in the Antarctic ocean in December 1991, and SOFIA-ASTEX over the Atlantic during summer 1992. These will provide opportunities to validate and use the analytical algorithm and inversion procedure.
SUMMARY AND CONCLUSION In this article, we have first developed an analytical radiative transfer model that predicts the directional reflectance as measured by an airborne or spaceborne directional radiometer. The model
Analytical Model of Atmosphere / Ocean Reflectance 191
APPENDIX: OCEAN SPECULAR REFI,ECTANCE AND WAVE SLOPE DISTRIBUTION
within d[~. The Jacobean of Eq. (A2) and simple trigonometry show that
Pws(Zx,Zy) $Zx 6Zy
Let Pws(Z~,Zy) describe the wave slope distribution. Z~ and Zv are the two components of the surface slope: aZ
Pw~(Z~,Z~) 6fi' = Pw~(Z~,Z¢) 6 ~ eos3(fl) 4 cos3(fl) cos(x)"
The total reflected irradiance measured perpendicular to the reflected rays is then
aZ
(A1)
Z,~= O---x-' ZY=O---y"
6F=psp(X)]O S Pws(Z~,Zy) ~n 4 cos'(#)
where Z is the surface height. Pws(Z~,Zy)6Z~ 6Zy is the probability for the wave surface slope to be such that its two components are in the interval (Z~ -1-~Zx / 2,Zy ± 6Zy / 2). Simple trigonometry shows that Z~ and Z~ are related to the incident and reflected rays, defined by the solar and viewing direction, by
(A7)
cos(0s) '
where psp(x) is the water specular reflectance as given by the Fresnel laws. The directional reflectance, dependent on the wave slope distribution, is therefore
R(O~,Or, ~a,WS) nPw~(Z~,Zy)psp(x)
=4 cos'(#) eos(0s) eos(Ov)" Z~ = sin(0v)cos(fp) - sin(Os) cos(0v) + cos(0~) ' sin(0v)sin(fp) Zy = cos(0o) + cos(0s)
(A2)
(the x axis is chosen in the direction of the sun). Let us consider a large horizontal surface S over the ocean. We consider the portion of the corresponding ocean surface whose slope lies in the interval Z~ ± gZ~ / 2,Zy ± 6Zy / 2). Its actual surface that is not projected in the horizontal is
OS = Pws(Z~,Zy)S6Z~gZy cos(fl)
Let J0 be the solar irradiance at the surface. The total flux intercepted by the surface portion is then cos(0~)
'
(A8)
Cox and Munk (1954) measurements showed that P~ is related to the wind speed and the wind direction. Regardless of the wind direction, it is shown that
Pws(Zx,Zy)-~ !7[0-2 exp
(
O-2
]
(A9)
where a 2 depends linearly on the wind speed (WS) following tr2 = 0.003 + 5.12 x 10 -3 WS
(m
s -~)
(A10)
One can also account for the wind direction posing:
'
where fl is the wave tilt, related to Z~ and Zy through tan(t) = ~ + ~ . (A3)
OF_Jo cos(x) 6S
(A6)
(A4)
where X is the incidence angle. Z is related to the incident and refracted rays directions through cos(2x) = eos(0s) cos(0v) - sin(O~) sin(0o) cos((a). (A5) As the two components of the wave slope vary within (Z~ ± ~SZ~/ 2,Z u± ~SZy/ 2), the normal to the surface varies within 6~', and the reflected ray
~=Z~/a~,
tl=Zu/a,,
(All) where Zc and Z, are the wave slope components crosswind and upwind, respectively. The wave slope probability can then be modeled as a GramCharlier series: exp[
ews(Z~,Z,)=
('~ +2 r/2)]
2nactr, C21 2 C03 3 1 --~-(~ - 1)t/---~-(t/ - 3r/)
+ C4o(~,_ 6~2 + 3) + Co4(q4_ 6q2 + 3) 24 24 +-~--. - 1)(r/z - 1) + " • (A12)
192
Br~on
where a~, a~, and the cij coefficients depend linearly on the wind speed [see Cox and Munk (1954) for t h e analytical expressions].
This research was conducted while the author was at the Meteorological Research Institute, Japan, under an STA (Science and Technology Agency of Japan)fellowship. We thank the Centre National dT~tudes Spatiales for funding assistance, We are indepted to two anonymous reviewers whose editorial comments greatly improved this article's readability.
REFERENCES Austin, R. W. (1974), The remote sensing of spectral radiance from below the ocean surface, in Optical Aspects of Oceanography (N. G. Jerlov and E. S. Nielsen, Eds.), Academic, London, Chap. XIV, pp. 317-344. Br6on, F. M. (1992), Reflectance of broken cloud fields: simulation and parameterization, J. Atmos. Sci. 49:12211232. Br6on, F. M., and Deschamps, P. Y. (1993), Optical and physical parameter retrieval from POLDER measurements over the ocean using an analytical model, Remote Sens. Environ. 43:193-207. Cox, C., and Munk, W. (1954), Measurements of the roughness of the sea surface from photographs of the sun's glitter, J. Opt. Soc. Am. 44:838-850. Deirmendjian, D. (1969), Electromagnetic Scattering on Spherical Polydispersions, Elsevier, New York. Deschamps, P. Y., Herman, M., Podaire, A., Leroy, M., La-
porte, M., and Vermande, P. (1990), A spatial instrument for the observation of polarization and directionality of Earth reflectances: POLDER, in Remote Sensing for the Nineties, Proc. lOth IGARSS, 20-24 May 1990, Washington, DC, IEEE Catalog No. 90CH2825-8, IEEE, New York, Vol. III, pp. 1769-1774. Deschamps, E Y., and the POLDER science team (1993), The POLDER mission: Instrument characteristics and scientific objectives, IEEE Trans. Geoscience and Rem. Sens., forthcoming. Hansen, J. E., and Travis, L. D. (1974), Light scattering in planetary atmosphere, Space Sci. Rev. 16:527-610. Kattawar, G. W., Plass, G. N., and Guinn, J. A. (1973), Monte Carlo calculations of the polarization of radiation in the Earth's atmosphere-ocean system, J. Phys. Oceanogr. 3: 353-372. Morel, A. (1980), In-water and remote measurements of ocean color, Boundary Layer Meteorol. 18:177-201. Morel, A. (1988), Optical modeling of the upper ocean in relation to its biogenous matter content (Case 1 waters), J. Geophys. Res. 93(C9):10,749-10,768. Plass, G. N., Kattawar, G. W., and Guinn, J. A. (1975), Radiative transfer in the Earth's atmosphere and ocean: influence of ocean waves, Appl. Opt. 14:1924-1936. Smith, R. C. (1974), Structure of solar radiation in the upper layers of the sea, in Optical Aspects of Oceanography (N. G. Jerlov and E. G. Nielsen, Eds.), Academic, London, pp. 95-119. WMO (1986), A Preliminary Cloudless Standard Atmosphere for Radiation Computation, Radiation Commission WCP 112, WMO/TD No. 24.
An Analytical Model for the Cloud-Free Atmosphere/Ocean System Reflectance E M. Brdon Laboratoire d'Optique Atmosph~rique, Villeneuve d'Ascq, France
A n analytical model for the estimation of spectral and directional reflectance over the ocean is developed. The model accounts for the first and second orders of reflection at the ocean surface, and/or scattering by the atmosphere and the water column. The model parameters are the optical thickness and angstr6m exponent of aerosols, the spectral albedo of the ocean, and the directional specular reflectance of the ocean surface (glitter), the latter being a function of the wind speed and direction. We show that our analytical model is su2fficient to predict all reflectance angular and spectral variations by comparing its results with a Monte Carlo simulation that accounts for the higher orders of scattering. Using this analytical model, actual bidirectional and spectral measurements are easily inverted in order to retrieve geophysical and optical parameters of the oceanatmosphere system (the application to the data acquired by the Airborne POLDER is developed in a companion article), INTRODUCTION Over the ocean, the sparsity of in situ measurement makes it necessary to use satellite observations for the estimate of key climatological and
Address correspondence to F. M. Br~on, Laboratoire d'Optique Atmosph~rique, USTL, Bat. P5, 59655 Villeneuve d'Ascq Cedex, France. Received 24 January1992; revised 23 May1992. 0034-4257 / 93 / $6.00 ©Elsevier Science Publishing Co. Inc., 1993 655 Avenue of the Americas, New York, IVY 10010
biological parameters such as cloud coverage, atmospheric aerosol content, sea surface state, or the ocean upper layer phytoplankton content. The requirement for global ocean monitoring has been stressed in the recent years, and a large consensus has been reached about the crucial role of the ocean in the expected climate change. POLDER(POLarizationandDirectionalityof the Earth Reflectance) is a new instrument devoted to the observation of the polarization and directionality of solar radiation reflected by the Earth-atmosphere system. It allows a multiangular, multiwavelength, multipolarization measurement of the Earth system reflectance. Its design and mission as part of the Japanese ADEOS platform, scheduled to be launched in 1996, has been described in Deschamps et al. (1990; 1993). An airborne version of the POLDER instrument has been implemented, and several dedicated flight experiments have been made or are planned to prepare for the spaceborne experiment. In order to take the full advantage of the POLDER angular and spectral measurement capabilities, it is necessary to develop a model that describes the angular and spectral variations of the observed reflectances. This model should be reasonably simple but must account for all important scatterings and/or reflections occurring in the ocean-atmosphere system. It also needs to be invertible so that optical parameters of the ocean-atmosphere system can be retrieved from POLDER observations. Extended radiative t r a n s fer simulations have been performed for the
1 79
180 Br~on
ocean-atmosphere system (e.g., Plass et al., 1975), including polarization effects (e.g., Kattawar et al., 1973), but these do not meet our simplicity requirements, In the next section, the various scattering/ reflectance processes occurring in the atmosphere/ocean system are discussed. In the third section, a reflectance model is developed that estimates these processes. The model is validated against a Monte Carlo procedure in the fourth section. An inversion method for spectral and directional reflectance measurements is described in the fifth section. The sixth section summarizes and concludes. In a companion article (Br6on and Deschamps, 1993), the model and the inversion procedure are applied to POLDER measurements acquired over the Mediterranean Sea.
SCATTERING AND REFLECTANCE PROCESSES OCCURRING IN THE ATMOSPHERE/OCEAN SYSTEM In the atmosphere, solar radiation is scattered by molecules and aerosols. Molecular scattering closely follows Rayleigh's theory. The molecular optical thickness has a 2-4 dependence, where 2 is the wavelength. It decreases from 0.22 to 0.017 between 450 nm and 850 nm. Moreover, the molecular scattering phase function is very close to the theoretical Rayleigh's law: P(og) =~3 [1 + cos2(o9)].
(1)
Slight deviation from this law is observed, resulting from the nonsphericity of atmospheric molecules. Aerosol scattering is more complex and variable. Its scattering phase function and the spectral variation of its optical thickness greatly depend on the aerosol size distribution and refractive index. However, some constant characteristics are found: • Because the aerosol size distribution is of the order of, or larger than, the wavelength of visible light, the aerosol scattering phase function shows a large maximum in the forward scattering direction, The ratio of scattered energy in the forward hemisphere is typically of the order of 0.85.
• The aerosol phase function shows little variation in the visible and near-infrared wavelength range. This variation will therefore be neglected in the analytical model. • The aerosol optical thickness spectral variation is closely modeled by the following exponential formula: Taer2 ~" "/'aer500(/]- / 5 0 0 ) - y (2) (2 in nm), where raerS00is the aerosol optical thickness at 500 nm and ~, is the ~,ngstr6m exponent, which generally ranges from 0.5 to 2. The largest ,4,ngstr6m exponents are found for the smallest size distributions. Over the ocean, the typical values for raersoorange from 0.05 to 0.4. • The aerosol single scattering albedo, the ratio of scattering to extinction, is very close to 1, meaning no absorption. This is not the case for the urban type aerosols, but they are seldom found over the ocean. Aerosol absorption is then negligible when compared with scattering. The aerosol scattering phase function depends on the physical properties, size distribution, and refractive index of the aerosol. These properties may vary, leading to extremely diferent phase functions (Deirmendjian, 1969; Hansen and Travis, 1974). Different aerosol types have been proposed by the scientific community (WMO, 1986). The simple Henyey-Greenstein analytical function can also be used to approximate the more complex expression of the phase functions computed from the Mie theory: PFH_c(~) =
1 - G2 [1 + G e - 2G cos(o))] 3/2'
(3)
where o) is the scattering angle. G is the anisotropy factor, that is, the scattered radiation mean value of the scattering angle cosine. The phase function is isotropic for G = 0. It approximates the Dirak function for G = 1. In the visible and near-infrared spectral ranges, the atmospheric molecule absorption is mainly due to water vapor and ozone. Ozone absorption is relatively constant and occurs in the upper layers of the atmosphere. Scattering and ozone absorption processes can, therefore, be easily decoupled and modeled separately. However,
Analytical Model of Atmosphere/ Ocean Reflectance 181
water vapor absorption and aerosol scattering both occur mainly in the lower troposphere. At the surface, specular reflection follows the Fresnel law. The fraction of incident energy that is specularly reflected is about 2% at incidence angles up to 50 ° and then increases up to 100% at grazing i n c i d e n c e . F o r a perfectly flat o c e a n surface, the reflected direct solar radiation, also called glitter, would be contained in a very small solid angle around the specular direction. However, the wind generates large and small scale waves on the surface, and these broaden the glitter pattern. Specularly reflected radiance can then be observed further away from the specular direction. Ocean specular reflectances are closely related to the wave statistics, a model of which has b e e n proposed by Cox and Munk (1954). T h i s model is recalled and used in the Appendix i n order to express the glitter reflectance as a function of the wave slope distribution. The ocean surface roughness may modify the albedo of the ocean surface by changing the incidence angle and thus the Fresnel reflectance. Let us evaluate how the specular albedo varies with ocean surface roughness. We here assume an isotropic Gaussian wave slope distribution (see the Appendix) and neglect all atmospheric effects, The reflectance is averaged over all the possible wave slope orientations according to their distribution statistics. The results are presented in Figure 1. The no wind speed, "fiat surface," albedo is simply the water specular reflection coefficient, which increases with the incidence angle but not linearly. As a result, the ocean surface albedo increases with the wind speed at most solar zenith angles (< 65°). Note that the effect is very small at the lowest zenith angles because the specular reflectance is nearly constant at incidence angles less than 40 °. The relative change in the ocean surface albedo is less than 10% for a wind speed of 8 ms -1. These results agree with that of Austin (1974). It is clear that the surface roughness greatly affects the angular field of the reflected radiances. However, the above computations show that the reflected fraction of the downward irradiance is only weakly sensitive to the surface roughness, The radiance scattered by the water column has a large spectral variation and this variation depends on the upper layer concentration in marine algae and associated detritus (Morel, 1988).
12 . . . . I . . . . I . . . . ! . . . . . . . . . . . . . . . I Flat Surface I f_ 10 . . . . . . . . . . . i......................... !.....................T--ff-" ~ ws = 8 ms-1 || i ~g II .. .. .. .. .. ws=14ms-1 i i "7 'J! .~ 8 I | .-.~;?~--.._~ 6 .....................~........................~........................i......................i.........................i............~;~'-'~..................... ~. :: i i i i,¢" i = 4 .....................ii ........................Fi ........................i~:....................... -~"" i ...................... i iii ~ , L_._ 2 . ....... I .... i .... i .... i .... 0 0 . . . . 10 20 30 40 50 60 70 .............
...................
.....................
Solar Zenith Angle
Figure 1. Ocean specular albedo sensitivity to the surface roughness. The specular albedo is given as a function of solay zenith angle for a flat surface (lower curve), a wind speed of 8 m s- 1 (middle curve), and a wind speed of 14 m s-1 (upper curve). The surface slopes are distributed according to Eq. (Ag).
Typical values of the water column reflectance range from 0.003 to 0.025 at 450 nm, from 0.005 to 0.01 at 550 nm, and from 0.0001 to 0.002 at 700 nm. An increase of the algae concentration lowers water column reflectances at wavelengths below 550 nm because absorption by phytoplankton is the dominant process; it has the opposite effect above 550 nm, where the scattering by phytoplankton prevails over the absorption (except in the range 650-680 nm, which corresponds to another chlorophyl absorption band). The angular field of the radiance scattered by the water column is not well known because it is not easy to measure in situ. Underwater measurements by Smith (1974) indicate limited variation over the hemisphere. Computations by Plass et al. (1975) lead to similar conclusions. The angular distribution of upwelling radiance may affect the remote sensing of ocean color from space. More research is needed on that subject in order to normalize the directional observations by the current scanners or to take full advantage of the POLDER multiangular viewing capabilities. Using a Monte Carlo method to simulate the underwater radiative transfer in a limited number of cases, we have found little angular dependence in the water column reflectance, especially at viewing zenith angles less than 60 °, which are those angles accessible to most spaceborne and airborne sensors. Therefore, in our model, we
182
Br~on
have chosen to neglect the angular dependence of the water column reflectance, At large wind speeds, above ~- 7 m s- 1, whitecaps will appear and occupy part of the ocean surface. The whitecap reflectance is assumed spectrally flat in the visible range. Note that whitecaps are seldom observed from spaceborne visible sensors because strong winds are frequently associated with bad weather and, therefore, cloudiness. However, the frequency of their occurrence under clear sky conditions cannot be ruled out. Whitecaps have been reported during the airborne POLDER flight that is analyzed in the companion article. Note that, assuming isotropy of the whitecap reflectance, it has now the same nature as the water column reflectance and can be included in it. Then, in our model, the water column reflectance is not necessarily negligible in the near infrared spectral range,
cludes the scattering by atmospheric molecules and aerosols. The model also accounts for the ocean surface specular reflection and the water column reflectance. Atmospheric optical thickness is assumed to be small enough so that all photon paths involving more than two of the above processes can be neglected. An important assumption of our modeling is to neglect the interactions between molecular and aerosol atmospheric scatterings. The practical reason for this choice is that its estimate would require a convolution of both phase functions. This approximation can be justified by the fact that aerosol scattering is mostly in the forward direction. Therefore, aerosol-then-molecular scattering would have an angular distribution similar to the molecular-only scattering. Moreover, it involves two scattering processes that both correspond to relatively small optical thicknesses, so that taking into account the molecule-aerosol double scattering would involve a second-order term. The validation of this hypothesis will be obtained by comparison of our analytical model with a Monte Carlo method computation that accounts for all orders of scattering. The measured reflectance at flight level is modeled as the addition of nine different contributions (see Fig. 2): molecular single scattering (pl); molecular second order of scattering (p2 and P3); interactions between specular reflection and
THE ANALYTICAL REFI.FCTANCE MODEL Our objective is to develop a model that correctly predicts the angular and spectral reflectance variations of the ocean-atmosphere system when flying at a given altitude. We want this model to be simple and analytical so that it can be inverted with a limited computational burden. We assume the atmosphere to be cloud-free. The model in-
Figure 2. Graphic representation of scattering and reflectance processes occurring in the atmosphere / ocean system
and evaluated in the analytical model: molecule single scattering (Pl); molecule double scattering (p2 and p3); ocean specular reflectance (P4); specular reflectance and molecule scattering (p5 and p6); water column reflectance (pT); aerosol reflectance (Ps); specular reflectance and aerosol scattering (p9). The dashed line is for single scattering occurring above the instrument.
- ~
Modelized Scattering/Reflectance processes
Solar Directi Instrument Level
~
,
p2
p
p3
,
/ p8
/'"
.-
p9
Viewing Directi~ p5
/
/
// p6
p7
Analytical Model of Atmosphere / Ocean Reflectance 183 molecule scattering (P4 and ps); ocean surface specular reflectance or glitter (P6); water column reflectance (p7); aerosol reflectance (p8); and interactions between specular reflection and aerosol scattering (p9). Our model simulates the reflectance at the top of the atmosphere (satellite observation) or somewhere within the atmosphere (airborne measurements). Let us define aun =
Psurface- P~rcr~ esurface
(4)
and P~rcr~ aab = P~,,aace'
(5)
where PsurraCeis the surface atmospheric pressure and P,,r~r~ is the atmospheric pressure at the aircraft level. If ra is the atmospheric molecule optical thickness, run = a,nra and r~b = a~brx are then, respectively, the molecular optical thickness under and above the instrument. When applied to satellite observation, a~b shall be set to 0 and a,, to 1. The first-order Rayleigh scattering contribution to the measured reflectance is
[ -r~b\.l-exp[-run(l/gs+l/g~)] p, = exp~---~ ) x
4(/is +/iv)
P,~y(O~,0~, cp),
(6)
where enay is the Rayleigh scattering phase function and/t, and go are the cosines of the solar and viewing zenith angles, respectively, p~ is the single scattering reflectance below the instrument by the molecular layer of optical thickness run after the extinction above the instrument by the molecular layer of optical thickness r~b. In order to estimate the second order of molecular scattering, we must study two cases: i) the first scattering occuring above the instrument and ii) both seatterings oceuring under the instrument. First, we estimate the latter process. At flight level, the downward direct solar irradiance is /~=Fo exp(-r~b//C). (7) The fraction of/~ that is scattered twice below the instrument is 1 f [ " dr If~ (~) /t~ -~1 - exp dg +
f~
1 - exp
(
r~ ] /t / dg ru._-
(8)
or 1 l~,,[I I (~) ~ Jo 1 - exp -
]
dg dr.
(9)
The integration variable dr is the layer at which the first scattering occurs. The other integration variable, d/t, is the cosine of the radiance zenith angle after the first interaction. The first (resp. second) integral within the bracket is for radiance scattered upward (resp. downward) by the first molecular interaction. In this formulation, we do not consider any third-order scattering. Similarly, we neglect the angular dependence related to the scattering phase function. This assumption would not be appopriate with an aerosol-type phase function that greatly departs from isotropy, but it is reasonable with the Rayleigh phase function. On the other hand, we account for the fact that radiance first scattered to a nearhorizontal direction has a large probability of going through a second scattering. To our knowledge, the double integral in Eq. (9) is not analytically integrable. However, within the visible and near infrared spectral ranges, we are dealing with small values of run, typically of the order of 0.1 or less. For such small values, the double integral can be very well approximated as t"runll 1 J0 LJO -
dg]
= (r,n)2Cl(r,n) (10)
with C1(r)=0.4447-1.165 logxo('C). (11) Assuming isotropic scattering and neglecting any third-order scattering, the modeled reflectance has a 1/go angular dependence. The reflectance corresponding to the contribution by the second order of molecular scattering, when both interactions occur below the instrument, is then (r"~)~ e x p ( - ~-~,b)Cl(run). (12) P2 = 4/t,g~ The fraction of the downward irradiance that is first scattered above the aircraft and then again below the aircraft can be estimated by using a similar method, that is, the integral over the scattering level and the radiance zenith angle for the two molecular scatterings. As for the estimate of Pz, one encounters an integral that does not have an analytical solution. As above, we find that, for small values of the optical thickness, it can be approximated by
184 Br~on f~I ( ~2)1[ ( ~)1 /t 1 - exp 1 - exp dg = runr,b C 2 ( ~ )
(13)
with C~(r) = - 0.3025 - 2.2265 logl0(r).
(14)
This approximation is accurate provided that both optical thicknesses are small (of the order of 0.1 or less) and of the same order of magnitude. The former condition holds for atmospheric molecular optical thicknesses in the visible and near-infrared spectral regions. The latter condition is fulfilled provided that the instrument altitude is of the order of 5000 m. At very low altitudes (< 1000 m), or high flight levels like space observations, the reflectance contribution estimated in this section is neglected, The reflectance resulting from one scattering above the aircraft and one below is then evaluated as
"['abl'unt~(X~r, ,.., , /------~ nrab). (15) 8/~dz~ It is clear that our formulation of the second order of molecular scattering, as expressed in Eqs. (9) and (12), is not valid at large viewing angles (small /z~) because we did not account for the radiance reduction by a possible third scattering. However, the approximation holds for reasonable optical thicknesses and the viewing angle capabilities of most instruments (~u~>0.5). Let us now evaluate the reflectance due to single molecular scattering and specular reflection by the ocean surface. For this contribution, we shall assume that the ocean surface is flat. Although it is not strictly so, the departure from flatness does not greatly affect the amount and angular distribution of reflected radiance that goes through molecular scattering and specular reflection. The reasons are: i) The wave slopes are small (of the order of 10°); ii) the fraction of the irradiance that is reflected depends little on the surface roughness (see Fig. 1); and iii) the molecular scattering phase function is rather close to isotropy, which smooths any angular distribution deviation, Because the radiometer may not be at the top of the atmosphere, one should distinguish between the radiance that is first scattered and P3 =
then specularly reflected and the radiance that is specularly reflected and then scattered to the viewing direction. The first process leads to a reflectance equal to p4 =
exp( - ra l/J,) - exp( - ra IlUv) 4(gt,- go) r..\ p Ray(-- ~s,/~v, ¢0), (16) x exp - ~v)pSp(l~v)
whereas the second gives p5 = e x p f - ~ exp(- r..//~,)- e x p ( - r../l~v) \ /~J 4(/a,-#v) ×Psp(ltl~)PRay(_/As, lAv,(O), (17) where Psp is the water reflection coefficient given by Fresnel laws. These expressions were obtained by integrating the radiance reaching the instrument over the various levels where the scattering may occur. In Eqs. (16) and (17), the first term is for the downwelling path and the second term is for the upwelling path. The phase function to be used is simply that of single molecular scattering with a mirror sun direction (/g, is changed to - ~t,). Glitter reflectance or specular reflection of direct solar radiation by the surface greatly depends on the sea state, that is, the surface wave slope distribution. On the other hand, the specular reflection coefficient is independent of wavelength. We therefore model the glitter reflectance as ( rx +/~,raera~run+'/'aera\] exp ( ~v -'S'~,/t,, ¢), p0 = exp / (18) where raera is the spectral aerosol optical thickness and S depends on the sea surface state and viewing conditions, but not on the wavelength. The incident direct solar radiation that is reflected by the surface is attenuated by molecular and aerosol scattering, with a spectral dependence through their optical thicknesses. Note that, in this formulation, we assume that the aerosol layer is located mostly below the instrument. When this is not so, one reduces accordingly the aerosol optical thickness on the upwelling path (second exponential term). In the Appendix, we show how specular reflectance at the surface, S, is related to the wind speed and direction according to a wave slope distribution model.
Analytical Model of Atmosphere/ Ocean Reflectance 185
We now evaluate the water column contribution to the reflectance. As mentioned in the previous section, there is assumed an isotropic angular distribution for the water-leaving radiance just above the surface. As aerosol scattering is mostly concentrated in the forward direction, a large fraction of the downward irradiance does reach the ocean surface even after it has been scattered by the aerosols. Therefore, we model only the downward irradiance attenuation by molecular scattering, and we neglect the effect of aerosol scattering on it. We then model the water column reflectance contribution to the measured reflectance as [1
1 exp(_ za/][exp(_ run/+ run] lz,/J[ \ l~v/ 2/z~/
p~=Ro~e~ +-~
(19) where Ro~ is the spectral reflectance of the water column, including white caps. The first bracket is the irradiance ratio that reaches the ocean surface. We assume here that half the irradiance once scattered by molecules reaches the surface (single-scattering approximation). The second bracket includes the direct radiance from the ocean to the instrument and that which is scattered once. For the latter, we assume isotropic single scattering and neglect higher order scattering. Note that Roc~a is here the diffuse ocean reflectance measured above the surface. A correction to the more generally used reflectance under the surface would therefore be necessary for comparison purposes. As shown in the previous section, aerosol optical thickness is an exponential function of the wavelength. The aerosol contribution to the measured reflectance is then modeled as
p8 ~aeraeaer(llls'[llv' (p) =
-r
= 4/.tgt~
~5--~]
~ (20)
This formulation is used, rather than that of single scattering [Eq. (6)] because it directly accounts for most of the multiple scattering that occurs within the large forward peak of the aerosol scattering phase function. The second order of aerosol scattering out of this forward peak is neglected. The phase function is selected in accordance with the investigated situation. As deduced from Eq.
(20), the parameter of importance is not the phase function nor the aerosol optical thickness but rather their product in the backscattering hemisphere. The last term of our analytical reflectance model estimates the radiance specularly reflected by the ocean suface and scattered by the aerosol layer. As for the corresponding molecular scattering, we assume that the ocean surface is fiat. We then obtain p0 =
Taer2
r
[psp(/Zs)+ psp(/~o)] Paer( --/~s, U~, ~). (21)
The approximation of ocean surface flatness here does not hold as well as it does for the molecular scattering (P4 and Ps). This is because the aerosol scattering phase function, in contrast to the molecular one, is far from isotropic. Therefore, close to the center of glitter, that is, the specular direction, one needs an angular convolution of the glitter reflectance and the phase function to estimate p9 precisely. However, away from the specular direction, the reflectance estimate as expressed by P9 is a good approximation and is an alternative to the time-consuming convolution. Hemispheric convolution of the phase function and the glitter reflectance is clearly contradictory to our model objectives. Alternatives depend on the surface conditions. • If the ocean surface is very smooth (low wind speed), the specular reflectance is confined within a narrow solid angle. The contribution to the reflectance, as expressed by Eq. (21), is then accurate. • If the ocean surface is rough (large wind speed), one may neglect, close to the specular direction, the aerosol effect on the glitter reflectance. This is possible because most aerosol scattered radiance deviates only a little. One therefore sets p9 to zero and neglects the aerosol reduction of direct glitter radiance in P6. The two approximations partly compensate each other. The directional reflectance of the ocean-atmosphere system is modeled as the sum of the nine p~ terms. The model parameters are the
186 Br~on
aerosol optical thickness rae~, the ]kngstr6m exponent y, the spectral water column reflectance Ro~ea, and the glitter reflectance at the surface S for each viewing condition. The Appendix shows that the function S(/zs,/zo, ~) is related to the wave slope distribution Pws(Z~,Z~)according to
S(lzs, lZo,¢p)= npsp(X)Pws(Z~,Zy)
(22) 4/~s/zo cos4(/~) ' where Zx and Zy are the two components of the wave slope, 1~ is the wave tilt, and X is the incidence angle. Zx, Zy, and/~ can be obtained from /zs,/to, and ¢pwith the help of simple trigonometry, Cox and Munk (1954) give an expression for Pws as a function of the surface wind speed and direction only. When used in the direct mode or for validation purposes (next section), we shall use the Cox and Munk parameterization for the glitter reflectance. When used for the inversion of reflectance measurements, however, the glitter reflectance is a parameter for each direction of observation. As this approach implies at least one unknown per observation direction, it is clearly feasible only if several spectral measurements are available,
MODEL VALIDATION Several approximations have been made in order to develop the model described in the previous section. Some of them, such as isotropicity of oceanic reflectance, the nonabsorbing property of aerosols, or the phase function invariance with wavelength, can only be validated by comparison with measurements. On the other hand, the neglect of third- and further-order reflectance / scattering processes, as well as the neglect of aerosol / molecule radiative interaction, can be evaluated with the help of a more detailed model that accounts for all orders of scattering. A Monte Carlo model was developed for the study of angular distribution of reflectance from broken clouds (Br~on, 1992). We adapted this model for the ocean/surface/atmosphere system simulation. The geometry is plane parallel. The atmosphere is constituted of two layers, the upper one with molecules only and the lower one with mixed aerosols and molecules. The ocean surface slope distribution follows the anisotropic parame-
terization by Cox and Munk (1954). The radiance reflected by the inner ocean is isotropic. The Monte Carlo method, hereafter referred to as MC, follows the path of many photons. It makes use of random numbers for the determination of the optical path between two scatterings, the angle of diffusion, the orientation of the ocean surface, or the choice between transmission and refraction at the surface. An improvement of the computational efficiency is achieved by applying to each photon a "weight" according to its history. The sum of the photon weights that exit the lower atmospheric layer in the upward direction is directly related to the reflectance measured at the boundary between the two atmospheric layers. Figures 3, 4, and 5 show the results of both the MC and our analytical model simulations. These are reflectances over a constant azimuth direction and for varying zenith angles. The analytical model results are shown as a line whereas MC estimates are depicted by dots. Note that the latter should be considered as an average over an angular bin. Any lack of smoothness in the MC estimates is due to the statistical uncertainty inherent in this computation method. Let us first discuss the case of a very clear atmosphere (no aerosols). The computations are made for a wavelength of 450 nm, and the total atmospheric optical thickness is, therefore, 0.22. We set the solar zenith angle to 40 ° , the ocean reflectance to 1.5%, the wind speed to 8 m s -1, and the wind direction relative to the sun to 150 ° . The atmospheric pressure at the instrument level is assumed to be 560 Hpa. Figures 3a, 3b, and 3c are the results of our comparisons for the principal plane, the 30-210 ° azimuth plane, and the perpendicular plane. The reflectance variations with zenith angle are all correctly predicted. The most noticeable features are: * A large reflectance increase toward limb viewing. This increase is related to the increased atmospheric optical path. * A large reflectance maximum in the principal plane toward the sun. Note that the viewing zenith angle for the local maximum is not equal to the solar zenith angle: The reflectance maximum direction is shifted from the specular direction to a larger zenith angle. Note also the local
Analytical Model of Atmosphere/ Ocean Reflectance 1 8 7
3o . . . . . . . . . . . . . . . . . .
j ...... i 25 ......... Wavelength:450 nm ...........' ~ ' ~ ~iiiiiii "~ 20 ~ ~\ ~ 15
m i n i m u m of reflectance correctly modeled at about 70 ° zenith viewing angle. • A local reflectance m a x i m u m close to zenith viewing in the perpendicular plane. This m a x i m u m is related to the glitter pattern thatspeedseXtendsto that direction. For wind lower than that chosen here (8 m s-l), the glitter pattern would be narrower, and the local m a x i m u m found here may not be present.
Principal Plane
~
l0 ............o .............;................~.................i.............. .....-.......-....~...............-.~.--.---.-. ..... oo ~ ~ 5 -80
-60
-,40
(a)
-20
0
20
40
60
80
ViewingZenith Angle 25 . . . . . .
, ................. i
Plane 30 °- 210 ° Wavelength: 450 nm 20 ........................................................................................................................................
" 15
! i
lo ' - - ~ o
i i
i i
i i o !°~ i 5 7. . . . . . i , , , ~,, -8o -60 -40 -20 (b)
i o i i ~ ~ ) 7 ......
~J~i ~ ~ ~. . . . . . . . . . . 20 40 60
80
Viewing ZenithAngle
15 13
i
ll
,~ 9 7
5-.80 (c)
--60
-..40
-20
0
20
40
60
80
Viewing Zenith Angle
Figure3. Comparison between the analytical model reflectance estimate (line) and that of a Monte Carlo method (dots). The wind speed is 8 m s-l; the solar zenith angle is 40 ° . The atmosphere only contains molecules. The wavelength is 450 nm.The reflectance is given as a function of viewing zenith angle in the principal plane (a), the 30 ° azimuth plane (b), and the perpendicular plane (c). See text for details,
Some slight discrepancies in the reflectance m a g n i t u d e are found. W h e n c o m p a r e d with MC simulations, the m o d e l e d reflectances are overestimated in the center glitter direction by about 1% (reflectance units). This discrepancy may be explained by the large azimuth gradient in reflectance in this direction and the fact that MC results are averaged over an angular bin (10 ° in the azimuth direction). For other viewing directions, the reflectance seems to be slightly u n d e r e s t i m a t e d by the analytical model. T h e exact m a g n i t u d e of this underestimate is difficult to estimate because of the statistical uncertainties i n h e r e n t in the MC method. It is of the order of 0.2-0.3% for zenith viewing angles smaller than 50 °. In order to establish which reflectance process is responsible for the bias, their various orders of m a g n i t u d e are discussed. For a zenith viewing angle of 20 ° in the perpendicular plane, single scattering accounts for a reflectance of 3.65%, the reflectance from the ocean is 1.22%, specular reflectance is 0.05%, second-order Rayleigh scattering amounts to 0.46 and 0.38 (p2 and p3), and specular reflectance/ molecular scattering (ps+p6) to 0.15%. The expressions for single scattering and specular reflectance are exact. T h e approximation for the ocean reflectance estimate cannot explain the differences. The order of magnitude for specular r e f l e c t a n c e / m o l e c u l a r scattering shows that any further scattering order of surface / atmosphere radiative interaction is negligible. The slight bias results, therefore, from the approximations for the estimate of multiple molecular scattering. A likely cause is the neglect of the phase-function anisotropy. An exact c o m p u t a t i o n of double Rayleigh scattering, which would involve the convolution of two phase functions, is clearly not within the scope of our m o d e l which is designed to be analyt-
188 Br~on
3O •.
' ' ' 1 ' ' ' 1 ' ' ' 1 ' ' ' 1 .
.
. . . . . .
.
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! Principal Plane 25--.Wavelength=500nm .-.
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,~
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............. ~ ................. i ................. i ................. i ................. ~-. . . .
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= 500
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~................. ~. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . .
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Plane 30 <'- 210 ° .... W a v e l e n g t h
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i /i\ i ................. i...._.L9..9..,~....... L.............. i ~ ~
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.
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..................................................................
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=
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................ ~................. i ................ ~. . . . . . . . . . . . . .
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-80
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Perpendicular Plane .... Wavelength = 670 nm ....................................................................
g
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o
..........~ - ~ ~ ~ . i '
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'
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Figure 4. Same as F i g u r e 3 but with an aerosol layer of
Figure 5. Same as Figure 4 but for a w a v e l e n g t h of 670
optical thickness 0.15, a solar zenith angle of 35 °, a w i n d speed of 5 m s- l, and a w a v e l e n g t h of 500 nm.
nm. At this w a v e l e n g t h , the aerosol optical thickness is 0.097.
ical. The accuracy achieved with the analytical form is sufficient to correctly reproduce the reflectance angular variations. If greater accuracy is needed, one must perform a more precise estimate of double Rayleigh scattering. Note that, as Rayleigh optical thickness is constant, the precise
computation could be performed once and for all. A greater accuracy could therefore be achieved with a limited computational burden. We now add the aerosol layer to our atmospheric model. The aerosol layer optical thickness raer~ is 0.15, and its Angstr6m coefficient is 1.5.
Analytical Model of Atmosphere~Ocean Reflectance 189
The aerosol phase function is given by the Henyey-Greenstein model with an asymmetry parameter G equal to 0.7. The surface wind is 5 m s -1 and the wind direction is unchanged. The ocean reflectance is 1.5% at 500 nm and 0.2% at 670 nm. The aerosol optical thickness is small compared with typical values over the oceans. However, the Henyey-Greenstein phase function is similar to a truncated "real" phase function. Thus, the product optical thickness-phase function has the correct order of magnitude, Figures 4 and 5 show the comparison between MC and the analytical model estimates for wavelengths of 500 nm and 670 nm, respectively. We present the reflectance angular variations in the principal plane, the 30-210" azimuth plane, and the perpendicular plane. These comparisons show main characteristics similar to the "moleculesonly~ case: The main angular variations are correctly reproduced. The reflectance is overestimated by the analytical model in the glitter direction and slightly underestimated in the other directions, The aerosol layer smooths the angular variations of the reflectance. It is especially obvious in the glitter direction (Figs. 4a and 5a). The glitter pattern is not correctly reproduced by the analytical model. This is because the estimate of reflectance resulting from specular reflection and aerosol scattering assumes a fiat surface. As mentioned before, this hypothesis leads to accurate results when considering molecular scattering because the Rayleigh phase function is close to isotropy. However, an accurate estimate of the aerosol contribution would require a convolution of the glitter pattern at the surface and the Rayleigh phase function is close to isotropy. However, an accurate estimate of the aerosol contribution would require a convolution of the glitter pattern at the surface and the Rayleigh phase function, Note that the model approximation of a fiat surface would be more accurate for lower wind speeds. The small bias between the MC and the analytical model estimates is troublesome. If the measurement objective is the retrieval of ocean color, an accuracy greater than the analytical model uncertainty is needed. A more precise estimate of double scattering or an empirical correction for the analytical model bias would then be necessary, For other purposes, such as the aerosol optical
thickness and Angst6m coefficient estimate, the model accuracy is sufficient.
MODEL INVERSION We designed our bidirectional reflectance model to be simple and analytical so that it is easily inverted against airborne measurements such as POLDER's. We will now describe the inversion procedure: Let us define Nob~as the number of available measurement directions and Na the number of spectral bands. There are Nobs+ N~ + 2 unknowns, whereas the number of observations is NaNobs. Moreover, the parameters are bounded by physically possible values: Far from the glitter viewing direction, the specular reflectance tends to zero. Similarly, the ocean diffuse reflectance for the longer wavelengths can reasonably be set to very small values, therefore reducing the number of unknowns. However, these include reflectance from white caps and, therefore, they cannot be neglected in the case of a rough sea. A first guess for the model parameters can be obtained as follows: • Far from the glitter direction, we neglect the direct specular reflection. We also neglect the longer wavelength ocean reflectances, or their differences if white cap reflectance is suspected. These wavelength reflectances measurements are then used to obtain raer and y. Note that most theoretical aerosol phase functions show a local maximum in the backscattering direction. The aerosol signal is, therefore, larger when the measurement direction is antispecular. This viewing direction is preferred for this step of the method. • Using the same measurement direction, we can then obtain a first guess for the shorter wavelength ocean reflectances. • Using the aerosol parameters' first guesses and the 850 nm reflectance measurements, a first guess for the surface specular reflectance can be derived for each viewing direction. From this first guess set, the reflectance model
190
Brdon
is inverted with an iterative method; a function is defined as a normalized squared difference between the model and the measurements: F(r .... y, Sk, Pt~e~ 3~) f7~/~ i 2| 1 / Nob -- s -N 2- / O b %i - Moda~ - " 2_J 2 . J -,L--S-_i ~ ] . (23) N~NobsL,=lj~l~ Obs~j
has been developed in order to take full advantage of POLDER multidirectional reflectance measurements. Its input parameters are the aerosol optical thickness, the aerosol /kngstr6m coefficient, the spectral ocean albedo, and the ocean surface directional reflectance. The model can be inverted against directional spectral measurements in order to retrieve the above parameters
This function is numerically computed for the parameter set, and then repeated with each parameter increased and reduced by a given increment. The F-function partial derivatives are thus obtained, and the model parameters are modified accordingly in order to reduce F. The parameter increments are progressively reduced, and this process is repeated until all second-order derivatives are positive. Because an accurate first guess was obtained, this iterative method rapidly converges to the sought parameters. A product of the inversion is the ocean specular reflectance in many directions. These are corrected for the atmospheric effects. The Appendix shows that the specular reflectance is related to the wave slope distribution as in Eq. (22). The final step of the inversion procedure is therefore a fit of the slope distribution as retrieved by the inversion to the Cox and Munk (1954) model. The two model parameters are the wind speed and the wind direction relative to the sun. A crude estimate of the wind speed is easily obtained assuming that the wave slope distribution is isotropic and follows a Gaussian distribution as in Eq. (A9). An estimate of the wind direction and a better retrieval of the wind speed are then sought: For discrete values of the wind direction and the wind speed around our first estimate, we compute the correlation between the retrieved wave slope distribution and the model values. The wind speed and direction that correspond to the best correlation then become our estimates,
as well as the wind speed and direction. The model expresses the measured reflectance as a sum of nine terms that account for the most significant scattering/reflectance processes. It is then very well suited to estimate the relative importance of these processes. The model accounts neither for successive scattering from molecules and aerosols, nor for all third and higher orders of scattering/reflectance. It does account, however, for all firstorder processes and all significant second-order processes. We have shown, through comparison with a Monte Carlo model that accounts for all scattering/reflectance orders, that the analytical model accurately reproduces the spectral and angular reflectance variations. The largest errors are found in the glitter direction where an accurate reflectance estimate would necessitate a convolution of the aerosol phase function and the glitter pattern at the surface. It is clear that, as the atmopsheric optical thickness increases, the neglect of the higher scattering orders becomes more critical. Similarly, the analytical model would fail to accurately estimate the atmosphere/ocean reflectance for large solar and / or viewing angles. As said before, our model has been specifically designed to take full advantage of midatmosphere airborne POLDER observations. This instrument provides a full description of the reflectance angular distribuiton at wavelengths suitable for ocean color monitoring and aerosol observation. Scheduled to be launched aboard the ADEOS platform in 1996, several airborne campaigns over the ocean have been carried out or are planned. These include Medimar, whose measurements are used in the companion article (Brdon and Deschamps, 1993), RACER in the Antarctic ocean in December 1991, and SOFIA-ASTEX over the Atlantic during summer 1992. These will provide opportunities to validate and use the analytical algorithm and inversion procedure.
SUMMARY AND CONCLUSION In this article, we have first developed an analytical radiative transfer model that predicts the directional reflectance as measured by an airborne or spaceborne directional radiometer. The model
Analytical Model of Atmosphere / Ocean Reflectance 191
APPENDIX: OCEAN SPECULAR REFI,ECTANCE AND WAVE SLOPE DISTRIBUTION
within d[~. The Jacobean of Eq. (A2) and simple trigonometry show that
Pws(Zx,Zy) $Zx 6Zy
Let Pws(Z~,Zy) describe the wave slope distribution. Z~ and Zv are the two components of the surface slope: aZ
Pw~(Z~,Z~) 6fi' = Pw~(Z~,Z¢) 6 ~ eos3(fl) 4 cos3(fl) cos(x)"
The total reflected irradiance measured perpendicular to the reflected rays is then
aZ
(A1)
Z,~= O---x-' ZY=O---y"
6F=psp(X)]O S Pws(Z~,Zy) ~n 4 cos'(#)
where Z is the surface height. Pws(Z~,Zy)6Z~ 6Zy is the probability for the wave surface slope to be such that its two components are in the interval (Z~ -1-~Zx / 2,Zy ± 6Zy / 2). Simple trigonometry shows that Z~ and Z~ are related to the incident and reflected rays, defined by the solar and viewing direction, by
(A7)
cos(0s) '
where psp(x) is the water specular reflectance as given by the Fresnel laws. The directional reflectance, dependent on the wave slope distribution, is therefore
R(O~,Or, ~a,WS) nPw~(Z~,Zy)psp(x)
=4 cos'(#) eos(0s) eos(Ov)" Z~ = sin(0v)cos(fp) - sin(Os) cos(0v) + cos(0~) ' sin(0v)sin(fp) Zy = cos(0o) + cos(0s)
(A2)
(the x axis is chosen in the direction of the sun). Let us consider a large horizontal surface S over the ocean. We consider the portion of the corresponding ocean surface whose slope lies in the interval Z~ ± gZ~ / 2,Zy ± 6Zy / 2). Its actual surface that is not projected in the horizontal is
OS = Pws(Z~,Zy)S6Z~gZy cos(fl)
Let J0 be the solar irradiance at the surface. The total flux intercepted by the surface portion is then cos(0~)
'
(A8)
Cox and Munk (1954) measurements showed that P~ is related to the wind speed and the wind direction. Regardless of the wind direction, it is shown that
Pws(Zx,Zy)-~ !7[0-2 exp
(
O-2
]
(A9)
where a 2 depends linearly on the wind speed (WS) following tr2 = 0.003 + 5.12 x 10 -3 WS
(m
s -~)
(A10)
One can also account for the wind direction posing:
'
where fl is the wave tilt, related to Z~ and Zy through tan(t) = ~ + ~ . (A3)
OF_Jo cos(x) 6S
(A6)
(A4)
where X is the incidence angle. Z is related to the incident and refracted rays directions through cos(2x) = eos(0s) cos(0v) - sin(O~) sin(0o) cos((a). (A5) As the two components of the wave slope vary within (Z~ ± ~SZ~/ 2,Z u± ~SZy/ 2), the normal to the surface varies within 6~', and the reflected ray
~=Z~/a~,
tl=Zu/a,,
(All) where Zc and Z, are the wave slope components crosswind and upwind, respectively. The wave slope probability can then be modeled as a GramCharlier series: exp[
ews(Z~,Z,)=
('~ +2 r/2)]
2nactr, C21 2 C03 3 1 --~-(~ - 1)t/---~-(t/ - 3r/)
+ C4o(~,_ 6~2 + 3) + Co4(q4_ 6q2 + 3) 24 24 +-~--. - 1)(r/z - 1) + " • (A12)
192
Br~on
where a~, a~, and the cij coefficients depend linearly on the wind speed [see Cox and Munk (1954) for t h e analytical expressions].
This research was conducted while the author was at the Meteorological Research Institute, Japan, under an STA (Science and Technology Agency of Japan)fellowship. We thank the Centre National dT~tudes Spatiales for funding assistance, We are indepted to two anonymous reviewers whose editorial comments greatly improved this article's readability.
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