Diffuse reflectance of oceanicwaters. I1 Bidirectional aspects
Andr6 Morel and Bernard Gentili
For visible wavelengths and for most of the oceanic waters, the albedo for single scattering () is not highenough to generate within the upper layers of the ocean a completely diffuse regime, so that the upwellingradiances below the surface, as well as the water-leaving radiances, generally do not form an isotropicradiant field. The nonisotropic character and the resulting bidirectional reflectance are convenientlyexpressed by the Q factor, which relates a given upwelling radiance Lo (', p) to the upwelling irradianceE, (' is the nadir angle, cp is the azimuth angle, and Q = E,/L,); in addition the Q function is alsodependent on the Sun's position. Another factor, denoted f, controls the magnitude of the globalreflectance, R (= E, lEd, where Ed is the downwelling irradiance below the surface); f relates R to thebackscattering and absorption coefficients of the water body (bb and a, respectively), according to R =f(bb/a). This f factor is also Sun angle dependent. By operating an azimuth-dependent Monte Carlocode, both these quantities, as well as their ratio (f/Q) have been studied as a function of the water opticalcharacteristics, namely si and -q; ' is the ratio of the molecular scattering to the total (molecular +particles) scattering. Realistic cases (including oceanic waters, with varying chlorophyll concentrations;several wavelengths involved in the remote sensing of ocean color and variable atmospheric turbidity)have been considered. Emphasis has been put on the geometrical conditions that would be typical of asatellite-based ocean color sensor, to derive and interpret the possible variations of the signal emergingfrom various oceanic waters, when seen from space under various angles and solar illuminationconditions.
Introduction
In a previous paper devoted to the diffuse reflectanceof oceanic waters (Morel and Gentilil), the magnitudeof this bulk process was examined as a function of thesolar elevation and for a variety of waters differing bytheir single-scattering albedo and by the relativeproportions of molecular and particle scattering.These proportions change with wavelength and, at agiven wavelength, also vary with the particle concen-tration. The partial scattering coefficient resultingfrom the presence of particulate materials can berelated to C, the chlorophyllous pigment concentra-tion in oceanic case 1 waters (Gordon and Morel2).Therefore the optical parameters regulating the radia-tive transfer in such waters can be derived from C andused when modeling the submarine light field.
The amount of the radiant flux returned towardthe atmosphere is governed by the value (just be-neath the surface) of the reflectance or irradianceratio, defined by
R(O-) =E.(-)1Ed(0-), (la)
where Ed and E are the downward and upwardirradiances at null depth (denoted O-). The Ed andE, are the integrals of the radiances L(O, p), weightedby the cosine of the zenith angle (0), over the upperhemisphere d (downward directions) or the lowerhemisphere E, (upwards directions),
EdU =-ad .- a
L(0, up) I cos 0 dfl. (lb)
The authors are with the Laboratorie de Physique et ChimieMarines, Universit6 Pierre et Marie Curie (Centre National de laRecherche Scientifique 353) B.P. 08, F 06230 Villefranche sur Mer,France.
In Eq. (lb) p is the azimuthal angle and 0 is the solidangle. It was previously foundl 3 that R alwaysincreases with diminishing solar elevation; R, how-ever, has different responses for various naturalwaters. The differences are predictable from (a) theactual shape of the volume scattering function (VSF)of the water body and (b) the average number of
collisions experienced by photons before they escapefrom water, (n).
The VSF of a given water results from an additionof the molecular VSF and the particle VSF. At agiven wavelength (), the first function is constant inmagnitude, whereas the particle VSF depends on theparticle concentration. Under the assumption thatthe shape of the particle VSF is stable, the shape ofthe resulting VSF, denoted (X), is fully determinedby the relative contributions of the two components,conveniently expressed by the ratio Ar, defined as
'9 = bw/(b. + bp), (2a)
where be and bp are the scattering coefficients for purewater and for particles, respectively, and their sum, b(=bw + bp), is the (total) scattering coefficient of theoceanic water. The resulting VSF for such a two-component water is therefore expressed as the follow-ing linear combination of PB and p, the molecularand particle-scattering phase functions, respectively(units sr-'):
(+3 = b[w3q(P ) + (1 - q)POM] (2b)
The average number of scattering events is uniquely(see Ref. 1) governed by the probability of photonsurvival (or single-scattering albedo), defined as theratio
= b/(b + a) = b/c, (3)
where a and c are the absorption and attenuationcoefficients, respectively. The radiative transferequation can be solved (here through Monte Carlosimulations) once the dimensionless numbers or func-tions, namely no, , and the two phase functionsabove, are fixed and, for practical scaling, when one ofthe three optical coefficients (a, b, or c) is given anabsolute value (as m'1).
The dependence of R on Sun angle is a global effect,to the extent that the two irradiances involved in thedefinition of R are integrated quantities over twoopposite hemispheres (Eq. lb). More complicatedpatterns are expected if, instead of irradiance, up-welling radiances originating from various directionsare considered, and when the bidirectional aspect ofthe ocean reflectance is to be analyzed. This is theaim of the present work, which is an extension of theprevious paper' and relies on the same (azimuth-dependent) Monte Carlo simulation.
This topic is of practical interest in the remotesensing of ocean color. Until the present and whenprocessing the only data available (those delivered bythe Coastal Zone Color Scanner [CZCS] instrument),it has been asserted that the in-water upwellingradiances, denoted L>1, and consequently the water-leaving radiances, L, would be nearly isotropic, atleast for the directions comprising the swath of thesensor (390 for the CZCS). Note that the resultspresented by Smith 4 and recently by Voss5 actually donot support such a simplifying assumption. In con-
tradiction to this commonly adopted assumption, itwas nevertheless acknowledged that the ocean is nota true Lambertian diffuse reflector, because the (sup-posedly equal) upwelling radiances beneath the sur-face have been related to upwelling irradiance through
(4)
where Q is not Tr, as it should be. for a Lambertianreflector, and instead, was generally given a valueclose to 4.5 (an experimental value adopted fromAustin6). This geometrical Q factor theoreticallydepends on the wavelength even if, in absence ofsystematic data, such a dependence has been inpractice ignored.
With regard to other uncertainties arising whenprocessing CZCS data, the two above assumptionswere perhaps justified, even if the value assigned to Qremained largely uncertain. The validity of theseassumptions (constant L and Q • r), however,deserves critical examination, especially when con-sidering that future sensors will have improved ac-curacies and capabilities. One specific capabilityis directly related to the birdirectional reflectancedistribution function. Thanks to a wide-field-of-viewpolarization and directionality of the Earth's reflec-tances (POLDER) or a fore-and-aft nadir-pointablesetup or fixed tilts [moderate-resolution image spec-trometer-tilt (MODIS-T)], it will become possibleto observe an oceanic pixel under different viewingangles. The interpretation and the potential of addi-tional information that could be gained in using suchtechniques remain unexplored.
Basic Equations for Water-Leaving Radiances Lw(O, tp)
The water-leaving radiance just above the surface,Lw(0, p), is expressed as a function of the upwellingradiance just beneath the surface, L, (0', p), and thenof the reflectance, according to (Gordon and Morel,2Gordon et al.
where 0 is the zenith angle (in air); 0' is the correspond-ing refracted (nadir) angle in water, so that ' =sin-'[(sin 0)/n]; p is the azimuth angle, where p(0', 0)is the internal Fresnel reflectance for upwelling radi-ance and for the directions (0', 0); r is the water-airFresnel reflectance for the whole diffuse upwellingirradiance; p is the air-water Fresnel reflection at theinterface for the whole (Sun and sky) downwellingirradiance above the surface, denoted Ed(O+); andfinally n is the refractive index of the water (= 1.34).
In Eq. (6), the first fraction exclusively describesthe processes originating from the presence of aninterface between two media with different indices(Fresnel specular reflections in both directions andrefraction effect), whereas the second fraction de-
scribes the process of the diffuse reflectance by theupper oceanic layer. This second term, which is theobject of the present analysis, can be related to theinherent optical properties (as defined by Preisendor-fer8); namely to a, the absorption coefficient, and tobb, the backscattering coefficient of the water body,through a functional expression involving their ratio
R = f(bb/a),
where the number f is not a constant and actuallydepends on both the zenith-Sun angle 00 and theoptical parameters -o and of the water, as alreadydiscussed in Ref. 1. From Eq. (7) it follows that thesecond term in Eq. (6) can be written as
R/Q = (f/Q)(bb/a), (8)
where the symbol (0-) is abandoned for brevity. Agermane formalism was used by Gordon et al.,7 whoexpressed the same ratio as
R/Q = 0.1l(0.0 22 )(bb/Kd) (9)
where Kd, which replaces a in Eq. (8), is the attenua-tion coefficient for downwelling irradiance. The nu-merical coefficient (with its rms error) results fromextensive calculations with various optical propertiesand solar angles (Gordon9). To the extent that Kd(an apparent property, as defined by Preisendorfer8 )is always greater than the inherent absorption coeffi-cient, the numerical value 0.11 in Eq. (9) is anoverestimate of the ratio f/Q appearing in Eq. (8).Note that Kd being approximately equal to a/iiL,where i is the average cosine of the radiative field,the expected overestimation is of the order of 15-50%(JI from 0.85 down to 0.65).
Interestingly the relative stability (20%) of thecoefficient found by Gordon,9 implies roughly paralleldependencies of f and Q on Sun angle and thus hasallowed the remotely sensed ocean color data to besuccessfully processed. Although the changes in f,which are substantial,' have not been accounted forin the processing, they were in an unknown fashionimplicitly counterbalanced by correlative changes inQ. In addition, Q cannot be a constant for a givenwater and, a priori, depends on the direction (0', p) aswell as on the illumination conditions. This remainsto be quantitatively assessed.
The Bidirectional Aspect
If we abandon the handy, albeit incorrect, assump-tion of an isotropic upwelling radiance, Q in Eq. (4)must be more accurately rewritten:
Q(00, 0', Aq) = E.(0-)/L(0O, 0', Aq), (10)
where the geometrical structure of the upward radi-ance field is accounted for, as is its dependence on theillumination above the surface, namely on the Sunposition (00, uo); Ap in Eq. (10) represents the azi-muth difference (Ap = p - ypo) between the verticalhalf-plane containing the Sun and the vertical half-
plane containing the radiance L1 under consideration.The Lu field is also controlled by the optical propertiesof the water itself, which are summarized by the set ofparameters XW and m. Instead of A, a more adequateparameter is 'lb, a similar dimensionless quantity,except that only backscattering is concerned, so that
'lb = bbw/(bbw + bbp) (11)
where the additional subscript b stands for backscat-tering and the two others are as in Eq. (2a).
As previously shown,1 f is also a complex function of00, the zenith-Sun angle (denoted in the previousstudy), and XW and 'lb. Consequently when all thesedependencies are made explicit, the ratio f/Q can bewritten as
Q f[00 , (b, PI)]
Q[(0', 0, AY)(rb W)](12)
This variable ratio is important because it relates anyupwelling radiance (when normalized by Ed) to thebb/a ratio through Eqs. (la) and (4), which, whenmerged, lead to
L.(0o, O', Aq) f bb
Ed(0-) Q a (13)
The same remark holds true for the remote sensingequation and the water-leaving radiances Lw [Eq. (6)],which obviously involve the same ratio.
Simulation
The inherent optical properties of oceanic case 1waters are modeled as before' and the same two setsof simulations are analyzed here in terms of bidirec-tional properties. These sets (see Table 1) have beenformed in such a way that the first series includeswaters for which -W is approximately constant (about0.65), whereas decreases from 13.9% to 0.077% (ibdecreasing from 81% to 2%); the second series in-cludes waters for which remains close to 4.8% (bapproximately 57%), whereas -o increases from 0.07to 0.79. To these real situations occurring for givenwavelengths and for given chlorophyll concentrations(denoted C and given in Table 1), nonrealistic watershave been added. These nonrealistic waters are thepure molecular water ( = 100%, case 1 in the firstseries) and waters with oi values (0.90 and 0.95, cases16 and 17 in the second series) exceeding thoseobservable in oceanic waters but possibly encoun-tered in turbid coastal zones. The rationale for thisselection is that we want to understand better and toassess separately the influences of the two key param-eters ( and ) on the radiative regime by decouplingtheir variations. In addition, 20 cases have beenstudied by considering oceanic waters with increasingchlorophyllous pigment concentrations and selectinga nominal set of wavelengths that are crucial in theocean color interpretation (see practical applicationbelow). The Monte Carlo simulation has been de-scribed previously in detail,' and it suffices to add that
Table 1. Relevant Information for Selected Case Studies (1-7 and 11-17, as in Ref. 1) and for Oceanic Waters With Increasing ChlorophyllConcentration (Lower Part of the Table)
X C b, b b a I 'rb Q (nadir;Case Studies (nm) (mg m-3) (m-) (m-1 ) (m-l) (m-l) (%) (%) fdiff.Unfa diff)b
aEq. (7), fvalues for a uniform sky radiance distribution.bEq. (10), Q values for a uniform sky and for 0' = 0.
photons travelling upward are collected when theyreach the surface in 1296 submarine detectors corre-sponding to 18 polar directions (50 increments in 0',from nadir to horizon) and 72 azimuthal directions(2.50 increments in Ap, from 0 to 180°). Detectorsare also arranged just above the surface to allow thewater-leaving radiances to be checked against Eq. (5).As discussed in Ref. 1, the Sun disk is given anangular diameter of 5° to simulate the wavy interface,and the sky radiance distribution is considered asisotropic. The upwelling irradiance E (0-) isstraightforwardly determined by integration [Eq. (lb)].Then, using Eq. (10) with the LU(0o, 0', Ap) valuesresulting from the simulation, the bidirectional Qvalues can be derived for each water.
Results
Geometrical Structure of the Upwelling Light Field
An example of typical results is given in Fig. (la),which is also intended to represent the geometry and
summarize the symbols used. The upwelling radi-ance field, L(O', 00, Ap), depends on to and q in a(qualitatively) predictable manner. For this pur-pose, it is necessary to consider first the shape of theVSF (which actually varies with the value) and,second, the average number of scattering events thatthe photons have experienced before escaping. Thisnumber, denoted ii, is directly related to the _io valuethrough [see Eq. (19) of Ref. 1].
n = (1 -1
As long as XW remains sufficiently small, W is not muchgreater than 1 and single scattering predominates(see Fig. 11 in Ref. 1). In such circumstances thesubmarine radiance pattern tends to resemble theVSF, once the VSF has been properly oriented alongthe path corresponding to the refracted Sun direction[Fig. l(b)]. In such a case, the upwelling L, distribu-tion represents a variable portion of the VSF (see alsothe discussion in Ref. 10). The pattern, however, is
Fig. 1. (a) Polar plot in the vertical plane containing the Sun ofthe upwelling radiances L (0') (nadir angle within the water) and ofthe water-leaving radiances L.(0). The hatched portion, limitedby the critical angle 0,' corresponds to upwelling radiances that aretotally and internally reflected. The L(0 ') and L(0) distributionsare those computed for the water 13 in Table 1 and when thezenith-Sun angle 00 is 30°. (b) Downward radiation originatingfrom the Sun, undergoing scattering and generating an upwardradiance field. Even in the case of single scattering, the forwardscattering lobe can contribute to the formation of the upward field(Shaded portion of the VSF) and accounts for the elongation of theLu pattern (a) in the half-plane containing the Sun and forquasi-horizontal directions. Total (internal) reflection and mul-tiple scattering (involving the whole VSF) also enhance the radi-ances outside of the Snell cone (see a). The VSF shown is that ofPetzold for particles used in this simulation (as in Rof. 1).
1 2 3 45 6 7
00 "2/7T
Fig. 2. Polar plot (in a vertical plane) of the upwelling radiancefield when the Sun is at zenith and for the waters 1 to 7 (see Table1). For a meaningful comparison, the La's are divided by thecorresponding upwelling irradiance Eu, so that the plot actuallyshows the values of 1/Q(O'). The azimuth (Ap) does not interferein this axially symmetric configuration. The critical angle isindicated. The mean number of scattering events, 11, is 2.9 for allcases. The dotted curves are for 1/Q = 1/7r or 1/Q = 2/r.
rendered more complicated by the effect of internalreflection. For instance, by assuming that singlescattering prevails and that the Sun rays are vertical(see Fig. 2), the L> (0') pattern will be similar to thebackward lobe of the VSF, but only within the Snellcone limited by the critical angle 0,. Outside of thiscone (for 0' > 0,'), the upwelling radiations remaintotally trapped within the medium, and thereforethese internally reflected radiations, now throughmultiple (including forward) scattering, contribute tothe enhancement of the upwelling radiances in theangular domain from O, to Ir/2. This enhancementis particularly intense for quasi-horizontal directions,with the consequence that the greatest departure ofthe Lu pattern from the VSF pattern is to be expectedfor these directions. A demonstrative example isprovided by water 1 in Fig. 2 (pure molecular scatter-ing), for which the minimum in the VSF pattern(occurring at a right angle) is definitely not reflectedby the L (0') pattern that does not exhibit anyminimum in the horizontal direction. Note that, forthis pure water, n is only 1.38, and therefore singlebackscattered photons largely prevail inside the Snellcone (but not outside). For all other examples in Fig.2, (waters 2 to 7), with n approximately 2.9, theresemblance to the VSF is largely degraded; nonethe-less the minimum in the particle VSF, at approxi-mately 145°, remains detectable in the radiance pat-tern at the corresponding angle (0' 350), at least forthe waters 5, 6, and 7. For nearly horizontal direc-tions ( < ' < ar/2), the Lo values are regularlyenhanced when the relative contribution of the par-ticle scattering is raised (waters from 2 to 7).
When the zenith-Sun angle increases, the forwardlobe of the VSF is increasingly involved [Figs. (lb) and3] and the L (0') pattern in the half-plane containingthe Sun is accordingly elongated. Now the water
Fig. 3. Polar plot of the upwelling radiances as in Fig. 2 but forthree particular waters (2, 5, and 17) and for a zenith-Sun anglevarying by steps from 0 to 80°. The normalized radiances, i.e., the1/Q (', Asp) values that are plotted, are those computed in thevertical plane containing the Sun. The Sun is in the right-handside of the figure (Ap = 0). The two dashed circular curves,correspond to the values 1/Q = 1 and 1/Q = 2.
VSF can be shaped either by molecular scattering orby particle scattering, depending on the q value.When X is high, i.e., when the round-shaped molecu-lar VSF governs the backscattering process, the influ-ence of the solar angle is reduced [Fig. 3(a)], comparedwith that occurring for low -q values, when thestrongly asymmetric VSF of particles dominates [Fig.3(b)].
When X approaches 1, the mean number of scatter-ing asymptotically raises and entails the formation ofa highly diffuse field. In such a regime, with mul-tiple collisions, the typical shape of the VSF com-pletely vanishes; the Lo spatial distribution tends tobecome isotropic and consequently less sensitive tothe illumination geometry. This is illustrated bywater 17, for which in is approximately 19 [Fig. 3(c)].
Q Factor
Following its definition [Eq. (10)], the Q(00, 0', Acp)factor, in an opposite way, reflects the spatialL, (0o, 0', Ap) distribution. In addition to the direc-tional variables, Q also depends on the water charac-teristics (oW and n). To analyze its variations case bycase (for the waters 1 to 7 and 11 to 17), a first surveycan be made by considering only the vertical planecontaining the Sun [Figs. 4(a) and (b)]. In this planeof symmetry Q actually takes both its minimal andmaximal values; the former, Qmin, is always found in ahorizontal direction (0' = Tr/2, whereas Ap may be 0or rr); the latter, Qmax, occurs in a certain directionthat is never far from the vertical one (0' close to 0).The entire structure of the Q field will be describedbelow (Fig. 5). These extreme values can be plottedfor each water as a function of the Sun angle 00 [solidlines in Figs. 4(a) and (b)] and also for an isotropic skyillumination. From the examination of these fig-ures, several points emerge:
0 30 60 SKY
Fig. 4. Solid curves represent extreme values of Q (Qmin and Qmx)as a function of the zenith angle of the Sun (in a black sky) and for asky with an isotropic radiance distribution (right part of eachpanel). The dashed curves represent minimal Q values in theremote sensing conditions (see text). Panel a shows waters 1 to 7;panel b shows waters 11 to 17.
1. When all waters and all solar angles are consid-ered, the Q values span a wide range, from approxi-mately 0.3 to 6.5.
2. For any water the difference between Qmin andQmax is regularly increasing for increasing zenith-Sun angle, 00.
3. For a given 0o value, this difference is wideningfor waters going from case 1 to case 7 (i.e., when X
decreases with constant -o value).4. For a given 00 value, the difference is diminish-
ing from case 11 to case 17, mainly because of theslight increase in Qmin, whereas the Qma, values are(for each 00) similar whatever the water (except forcases 16 and 17); the Qmax dependency on 00 appearsto be essentially insensitive to the to values (at qconstant) at least if X0 < 0.8.
differences and their variations with the solar angleare predominantly governed by - rather than by theo value; this remark meets the conclusions alreadyarrived at concerning the Sun angle dependence ofthe f factor [Figs. 7(b) and 8(b) in Ref. 1].
Figure 5(b) now shows the dependence of Q on theazimuth Ap [see also Fig. 5(a) for the geometryadopted for this two-dimensional projection). Theresults are displayed for selected water cases (11, 12,and 13 omitted) and only for two 00 values 800 and 300(and for an overcast isotropic sky). As expected fromthe previous graph, the widest range of variations inQ is observed when the Sun angle is 800 and when theparticle VSF dominates (water 7). With respect totheir q values, all waters, from 11 to 17, are similar towater 3. The Q patterns for waters from 11 to 14,
Fig. 5. a, Geometry for the plates in b and c. The center of thecircle represents the nadir-zenith axis, with ' = 0. The twoinner circles successively represent 0' = 350 (corresponding to 0 500 in air) and ' = 48° (the critical angle); the projection iscylindrical and the external circle is for 0' = 90°. The Sun is in theright-hand side (p = 0), and the azimuth difference varies from 0to 'r (counterclockwise), up to the antisolar direction. b, accordingto the geometry shown in panel a, spatial distribution of the Qfactor for various waters as indicated (1 to 7 and 14 to 17) and for azenith angle of the Sun equal to 800 and 30°, or for a uniform sky(lower rows). The Q values are provided according to the graycoding shown in the inset. c, Ratio f/Q as in b for geometry,waters, and Sun angles.
practically confounded, are similar to that for water3; they are not shown in Fig. 5. A tangible evolutionof the Q patterns can only be seen when consideringthe waters from 14 to 17. With X tending toward 1for these waters, the increasing diffuseness in theradiant field reduces the overall Q variation, and theQ pattern tends to remain symmetrical with respectto the nadir-zenith direction, even in the case of lowsolar elevation.
In conclusion, the bidirectional Lo or Q patterns areincreasingly featured when the zenith-Sun distanceis increasing; at least for oceanic waters, where ingeneral -o is below 0.8, the bidirectional reflectancedistribution is essentially controlled by q, (actuallyvia the shape of the VSF); X plays a role when itexceeds 0.8, as in turbid waters, with the effect ofreducing the Sun angle dependency of the Q pattern.
Finally in the case of an axially symmetrical illumina-tion [ = 0, not shown; or for uniform sky, shown inFig. 5(b)], the symmetrical Q pattern always reachesits maximum along the nadir-zenith direction. Therange of Q variations with 0', reduced when molecu-lar scattering predominates, is widened when particlescattering is becoming preponderant.
f/Q Ratio
Before we examine the behavior of this ratio [Eq.(12)], it is worth recalling the variations of the f factor[Eq. (7)], already analyzed in Ref. 1. To a firstapproximation f is nearly independent from a, as longas o is < 0.8. Within this approximation, a param-eterization has been proposed [Eq. (14) in Ref. 1],allowing f to be expressed only as a function of pLo(= cos 00) and of qb; a more accurate parameteriza-tion including the effect of oW is given in Ref. 20. Thef function increases monotonically (almost linearly)when Lo decreases, with a slope that is governed bythe qb value. Consider, for instance, a Sun anglevarying from 0 to 80 (o varying from 1 to 0.17):f is increased by a factor of 1.77 when particlescattering dominates (water 7, with qb = 1.99%), orby a factor of 1.29 when molecular scattering predomi-nates (water 2, with 'qb = 81%). These values havebeen computed for a Sun in a black sky (no skyradiation).
When the Sun is sinking, the overall enhancementof the Q values is therefore partly (but in a variablemanner) compensated by the simultaneous increaseinf, in such a way that their ratio, f/Q, turns out to beless sensitive to the Sun elevation. This compensa-tion is more effective when 1b is small and lesseffective in pure water with high 9b values.
The polar plots in Fig. 5(c) show the f/Q patternsfor the same waters and Sun positions as in Fig. 5(b),and when the entire submarine field is considered(O < ' < 7r/2). In all situations, the maximalvalues are observed for quasihorizontal directions(0' = 'r/2) in the solar or antisolar half-plane. Thef/Q ratio experiences its largest variations, fromabout 0.08 to more than 0.24, at low solar elevation(0 = 800).
Practical Applications
When presenting and discussing the above results,our intent is to assess the values and understand thevariations of the Q and f/Q quantities in response tochanging solar elevations. This was achieved byselecting various oceanic waters with specific XW and x(or 'lb) values (Table 1). For practical applications,however, another selection is now desirable, and onemust account for additional phenomena and atmo-spheric effects.
In relation to the ocean color interpretation, 20other realistic cases have been studied. The in-herent optical properties of oceanic case 1 waterswith increasing chlorophyll pigment concentrations(C = 0.03, 0.1, 0.3, 1, and 3 mg m- 3 ) have beenmodeled as in Ref. 1, and the main wavelengths
(X = 440, 500, 565, and 665 nm) involved in the oceancolor remote sensing have been considered for theMonte Carlo simulation. The corresponding (X)and Tb(X) value are displayed in Fig. 6. The validityof the qb values at high chlorophyll concentrations(above 1 mg m-3 in particular) is questionable andwill be discussed later. The f values, as they resultfrom the simulation for these specific wavelengthsand pigment concentrations (except 3 mg m-3 ), aredisplayed in Fig. 7, for two zenith-Sun angles and foran overcast sky. The continuous f curves that jointhe data are modeled from the spectral values of 'band _o (Fig. 6), using a parameterization slightlymodified with respect to Eq. (14) in Ref. 1 (whichdepended only on b). In the blue part of thespectrum and for C = 1 or 3 mg m-3, the XW values arein the vicinity of 0.8, and therefore the effect of thisparameter, which cannot be neglected, is accountedfor by the modified parameterization given in Ref. 20.Real situations combine Sun and sky and the range ofvariations in f with o are subsequently smoothed bythe contribution of diffuse radiation (note that the fvalues for an isotropic sky radiation are intermediatebetween the values derived for Oo = 150 and 60°).
The entire upwelling radiance field (over 2ir sr) hasto be examined for submarine optical problems, suchas visibility and shadowing effects. If only the up-welling radiances that are able to emerge are exam-ined, the extreme (low) Q values and extreme (high)f/Q values that were observed when 0' > 0,' can nowbe discarded. Note that the high Q values (low f/Qvalues), which are expected in the vicinity of thevertical direction, obviously still remain to be consid-ered. The in-water field involved in remote sensing
1 .0 I I I I I I I I I
1.8
.8 ~~~~~0.03 (mg m 3)81 440(nm)
- , < ~~~~~0.1.6
565 ~~~0.3
.4
.2 3.
.0 .2 .4 .6 .8 1.0
Fig.6. Spectral values of i and b for each water characterized byits chlorophyll pigment concentration from 0.03 to 3 mgm- 3. Theblack or white circles are for specific wavelengths as indicated,whereas the continuous solid curves are for all spectral valuesbetween 400 and 700 nm.
665nmFig. 10. For various wavelengths and pigment concentrations as indicated, Q values as a function of scan angle (abscissas) and latitude(ordinates, positive and negative values for northern and southern latitudes, respectively). These values derive from the geometricalconditions (Sun angle and viewing angle) that are those corresponding to a satelliteborne sensor in a quasi-polar, Sun-synchronous orbit(see Ref. 10). (b) Values of the ratio f/Q for the same A, C, and geometrical conditions as in (a).
M0
b
-VU '1-1-1 ... I... I'- -�R'elf% - 1? I A� I
L_
Fig. 13. Upper left panel: isopleths of 9b in the wavelength-chlorophyll concentration plane; the shaded area corresponds to i valuesexceeding 0.8. The other panels show the isopleths of the Q(O' = O) factor, i.e., those associated with the nadir radiance in the same (X - C)plane and for the zenith-Sun angle as indicated (00 = 15, 30, 600). Note that the illumination conditions combine the Sun and an isotropicsky with a visibility of 23 km [see also Table 1 for the same Q (' = 0) factor computed for an overcast sky and uniform incident radiancedistribution].
waters and sediment-dominated case 2 waters has tobe made, and this discrimination is based on thedetection of anomalously high reflectances;18 there-fore a separate knowledge of the Q factor is needed.The use of iterative schemes to partition the signalinto its atmospheric and marine components18"19 alsorequires an adequate value for this factor. Theparameterization of the f factor20 and the predicted(and tabulated) Q values for case 1 waters that resultfrom the present study certainly can improve theaccuracy of any processing of the remotely sensedocean color data.
This research was jointly supported by the AgenceSpatiale Europ6enne (under contract ESTEC 9111/90/NL-PR) and by the Centre National d'EtudesSpatiales (under contract CNES/91/0262).
References and Notes1. A. Morel and B. Gentili, "Diffuse reflectance of oceanic waters:
its dependence on Sun angle as influenced by the molecularscattering contribution," Appl. Opt. 30, 4427-4438 (1991).
2. H. R. Gordon and A. Morel, Remote Assessment of Ocean Colorfor Interpretation of Satellite Visible Imagery: A Review(Springer-Verlag, New York, 1983), p. 114.
3. H. R. Gordon, "Dependence of the diffuse reflectance ofnatural waters on the Sun angle," Limnol. Oceanogr. 34,1484-1489 (1989).
4. R. C. Smith, "Structure of solar radiation in the upper layersof the sea," in Optical Aspects of Oceanography, N. G. Jerlovand E. Steemann Nielsen, eds. (Academic, New York, 1974),pp. 95-119.
5. K. J. Voss, "Use of the radiance distribution to measure theoptical absorption coefficient in the ocean," Limnol. Oceanogr.34, 1614-1622 (1989).
6. R. W. Austin, "Coastal Zone Color Scanner radiometry," inOcean Optics VI, S. Q. Duntley, ed., Proc. Soc. Photo. Opt.Instrum. Eng. 208, 170-177 (1979).
7. H. R. Gordon, 0. B. Brown, R. H. Evans, J. W. Brown, R. C.Smith, K. S. Baker, and D. K. Clark, "A semianalytic radiancemodel of ocean color," J. Geophys. Res. 93, 10909-10924(1988).
8. R. W. Preisendorfer, "Application of radiative transfer theoryto light measurements in the sea," Monogr. Int. Union Geod.Geophysics Paris 10, 11-30 (1961).
9. H. R. Gordon, "Ocean color remote sensing: influence of theparticle phase function and the solar zenith angle," in EOSTrans. Amer. Geophy. Union 14,1055 (1986).
10. R. H. Stavn and A. D. Weidemann, "Shape factors, two-flowmodels and the problem of irradiance inversion in estimatingoptical parameters," Limnol. Oceanogr. 34, 1426-1441 (1989).
11. D. Tanr6, M. Herman, P. Y. Deschamps, and A. de Leffe,"Atmospheric modeling for space measurements of groundreflectances including bidirectional properties," Appl. Opt. 18,3587-3594 (1979).
12. The orbital features are a semimajor axis of 7159.5 km, withan eccentricity of 0.001165, an inclination of 98.55, and anequator crossing time at 10 am in descending orbit. Theswath of the sensor is ±50°.
13. This probability is defined as the ratio of the backscatteringcoefficient to the (total) scattering coefficient and is denotedbb; the additional subscriptp stands for particle.
14. A. Morel, "Optical modelling of upper ocean in relation to itsbiogenous matter content (case 1 waters)," J. Geophys. Res.93, 10749-10768 (1988).
15. A. Morel and A. Y. Ahn, "Optics of heterotrophic nanoflagel-lates and ciliates: a tentative assessment of their scattering
role in oceanic waters compared to those of bacterial and algalcells," J. Mar. Res. 49, 177-202 (1991).
16. D. Stramski and D. A. Kiefer, "Light scattering by microorgan-isms in the open ocean," Prog. Oceanogr. 28, 343-383 (1991).
17. H. R. Gordon and W. R. McCluney, "Estimation of the depth ofSun light penetration in the sea for remote sensing," Appl.Opt. 14, 413-416 (1975).
18. A. Bricaud and A. Morel, "Atmospheric corrections and inter-pretation of marine radiances in CZCS imagery: use of areflectance model," Oceanol. Acta 7, 33-50 (1987).
19. J. M. Andr6 and A. Morel, "Atmospheric corrections andinterpretation of marine radiances in CZCS imagery, revis-ited," Oceanol. Acta 14, 3-22 (1991).
20. The modified parameterization of f as a function of rb, Xi, andcos 0 p= ois as follows: f= 0.5575 - 0.1 0 6 7 b + 0.1045i -
