analytical solution of radiative transfer in the coupled ... · analytical solution of radiative...

Analytical solution of radiative transfer in the coupledatmosphere–ocean system with a rough surface

Zhonghai Jin, Thomas P. Charlock, Ken Rutledge, Knut Stamnes, and Yingjian Wang

Using the computationally efficient discrete-ordinate method, we present an analytical solution forradiative transfer in the coupled atmosphere–ocean system with a rough air–water interface. Thetheoretical formulations of the radiative transfer equation and solution are described. The effects ofsurface roughness on the radiation field in the atmosphere and ocean are studied and compared withsatellite and surface measurements. The results show that ocean surface roughness has significant effectson the upwelling radiation in the atmosphere and the downwelling radiation in the ocean. As wind speedincreases, the angular domain of sunglint broadens, the surface albedo decreases, and the transmissionto the ocean increases. The downward radiance field in the upper ocean is highly anisotropic, but thisanisotropy decreases rapidly as surface wind increases and as ocean depth increases. The effects ofsurface roughness on radiation also depend greatly on both wavelength and angle of incidence (i.e., solarelevation); these effects are significantly smaller throughout the spectrum at high Sun. The model-observation discrepancies may indicate that the Cox–Munk surface roughness model is not sufficient forhigh wind conditions. © 2006 Optical Society of America

OCIS codes: 010.1290, 010.4450, 030.5620.

1. Introduction

Cox and Munk1 described the statistical character-sitics of reflection by wind-blown ocean waves bymodeling the sea surface as a collection of individualmirror facets. They presented the probability distri-bution for the slopes of surface facets as a wind-speed-dependent Gaussian function. Based on thisCox and Munk formulation, several researchers in-corporated the ocean surface roughness in their ra-diative transfer models.2–8 Most of these models usedthe ray-tracing method or the Monte Carlo techniqueto treat the surface roughness. The Monte Carlo ap-proach consists of using probabilistic concepts andhas the advantage for geometries other than the

plane-parallel. Implementation of the statisticalsurface roughness by the Monte Carlo method isrelatively straightforward. The discrete-ordinatetechnique, on the other hand, can be more computa-tionally efficient and accurate, because it solves theradiative transfer equation analytically without theenormous statistical sample required to close aMonte Carlo solution and without the statistical fluc-tuation error. However, because of its analytical na-ture, implementation of the surface roughness in adiscrete-ordinate radiative transfer code is more com-plicated; a rigorous solution involves an additionalparameter that results in a different analytical solu-tion from the flat surface case.

To extend applications of the Discrete-OrdinateRadiative Transfer (DISORT) code for systems in-cluding two media (atmosphere and ocean, atmo-sphere and ice, etc.), Jin and Stamnes9 developed acoupled DISORT (CDISORT). The CDISORT code ac-counts for change in the refractive index at theboundary of the two media. For radiative transfer insuch a coupled system, CDISORT treats the ocean orice the same as atmospheric layers but with differentoptical properties, particularly, different refractiveindices. However, the interface between two stratawith different refractive indices was considered flat.This flat surface assumption limits the applicationsof CDISORT; the aforementioned wind-blown oceansurface is hardly flat. In addition to affecting reflec-

Z. Jin ([email protected]) and K. Rutledge are with AS&M,Incorporated and NASA Langley Research Center, One EnterpriseParkway, Suite 300, Hampton, Virginia 23666. T. P. Charlock iswith the Division of Atmospheric Sciences, NASA Langley Re-search Center, Mail Stop 420, Hampton, Virginia 23681-2199. K.Stamnes is with the Department of Physics and Engineering Phys-ics, Stevens Institute of Technology, Hoboken, New Jersey07030. Y. Wang is with the Anhui Institute of Optics and FineMechanics, Chinese Academy of Sciences, Hefei, Anhui 230031,China.

Received 4 January 2006; revised 6 March 2006; accepted 25April 2006; posted 19 May 2006 (Doc. ID 67000).

0003-6935/06/287443-13$15.00/0© 2006 Optical Society of America

1 October 2006 � Vol. 45, No. 28 � APPLIED OPTICS 7443

tion, the surface roughness itself significantly affectsthe directional character of the beam transmitted be-neath the air–water interface. Gjerstad and co-workers10 proposed an ad hoc method to consider thesurface roughness in the discrete-ordinate method.They mimic the irradiances from a Monte Carlomodel by adjusting the refractive index in CDISORT.This method has a number of limitations; for exam-ple, it calculates irradiances only in the ocean. Herewe present a more consistent and widely applicablesolution of the DISORT problem in the coupledatmosphere–ocean system with a rough surface.

2. Equation and Solution of Radiative Transfer

To incorporate ocean surface roughness into a radia-tive transfer equation and obtain an analytical solu-tion through the discrete-ordinate method, we needto make the following assumptions:

Y The rough surface can be resolved as a series ofsmall planar facets, and the orientations (slopes) ofthese facets follow a certain statistical distribution,for example, the Gaussian distribution described byCox and Munk.1

Y The dimensions of the elemental facets and sur-face undulations are large compared with the wave-length of light, so geometrical optics can be applied tocalculate the reflection and refraction at the surface.

Y The optical depth of either the ocean or theatmosphere is independent of the surface roughnessor horizontal position, because statistically there isno difference between any two points on the surface.

Under these assumptions, the time-averaged radi-ative effects at any two points on the surface are thesame, and a patch of surface area at an instant intime in which every possible slope occurs can repre-sent the surface as a whole. Therefore radiativetransfer in a coupled system with a horizontally ho-mogeneous atmosphere and ocean and with a roughocean surface is still in the one-dimensional category,as long as the calculated radiation is considered to betime averaged (statistically averaged) for a point or tobe relevant to a surface area larger than the patchaforementioned for an instant. We also treat the ra-diance and the reflection and refraction at the oceansurface as scalar. Therefore the model presented hereshould not be applied to problems where polarizationis important.

Jin and Stamnes9 (hereafter referred to as JS94)presented in detail the solution for coupled (i.e.,air–sea) radiative transfer by the discrete-ordinatemethod for the flat ocean case. Here we will follow thesame conventions defined in JS94 to describe thediscrete-ordinate radiative transfer equation and so-lution for the rough surface case. Because the formu-lations have a lot in common between these twocases, we will omit most common derivations andemphasize the differences here.

For the flat surface case, the ocean was divided intoa totally reflecting angular domain where upwellingphotons cannot return directly to the atmosphere,

and a refracting domain where upwelling photonscan pass the interface directly to the atmosphere.However, once the surface roughness is introduced,there are no such distinct angular domains. Becauseof the possible ranges of orientations for the surfacefacets, photons in the atmosphere may pass the in-terface and direct to any angle downward, and viceversa for photons from the ocean to the atmosphere.This difference results in different radiative transfersolutions.

The radiative transfer equation to be solved for aplane-parallel medium with one dimension can bewritten as

�dI��, �, ��

d�� I��, �, ��

��4� �

0

2�

d��

��1

1

p��, �, �, ��, ��I��, ��, ��

� d�� Q��, �, ��, (1)

where I��, �, �� is the radiance at vertical opticaldepth � (measured downward from the upper bound-ary) and in direction ��, ��; � is the cosine of thezenith angle (positive for upward directions); � is theazimuth angle; � is the single scattering albedo; andp��, �, �, ��, �� and Q��, �, �� are the phase functionand source term, respectively. We consider only thesolar radiation (i.e., not terrestrially emitted ther-mal infrared or microwave). The solar-beam source,Q��, �, ��, is different from the case of a flat ocean. Inthe flat surface case, part of the downwelling solarbeam is reflected specularly back to the atmosphere,and the rest is refracted into the ocean at an angle,which depends on the refractive index; this results intwo terms (downwelling and reflected) in the solarsource function for the atmosphere [Eq. (3) in JS94]and one refractive index dependent term for theocean [Eq. (4) in JS94]. However, for the case of arough ocean, the solar beam is diffused to variousdirections when it hits the surface. Therefore there isno beam source term in the ocean and only one ex-pression in the atmosphere for the rough ocean case,which is

Q��, �, ��

��4�F0p��, �, �, ��0, �0�exp��0�, � �a,0, � � �a,

(2)

where �a is the total optical depth of the atmosphere,�0 is the cosine of the solar zenith angle, �0 is thesolar azimuth angle, and F0 is the solar-beam inten-sity at the top of the atmosphere.

Expanding the radiance I��, �, �� into a Fouriercosine series of 2N and the phase function p��, �,�, ��, �� into a series of 2N Legendre polynomials,the discrete-ordinate method converts Eq. (1) into asystem of azimuthally independent, coupled differen-

7444 APPLIED OPTICS � Vol. 45, No. 28 � 1 October 2006

tial equations for each of the Fourier components.Detailed derivations of these equations were given inJS94 and will not be repeated here. Following thesame procedure, the equations for each azimuth ra-diance component (here we omit the index denotingthe order of Fourier series) can be derived, which arein the atmosphere,

�ia

dI��, �ia�

d�� I��, �i

a� � �j��N1j�0

N1

�jaD��, �i

a, �ja�

� I��, �ja� X0��, �i

a�exp��0�,i � 1, 2, . . . , N1; � �a (3a)

and in the ocean,

�io

dI��, �io�

d�� I��, �i

o� � �j��N2j�0

N2

�joD��, �i

o, �jo�

� I��, �jo�, i � 1, 2, . . . , N2;

� � �a. (3b)

Equations (3a) and (3b) are analogous to Eqs. (7) and(8) in JS94, but with different source terms. Here 2N1and 2N2 are the numbers of quadrature points (i.e.,stream numbers) applied in the atmosphere andocean, respectively. D��, �i, �j� and X0��, �i� were alsodefined in JS94. The ��i

a, �ia� and ��i

o, �io� are quad-

rature points and weights for the atmosphere andocean, respectively, with ��i � ��i and ��i � �i. Theyhave the following relationships:

�io � �1 � <1 � ��i

a�2=��nw�na�2, i � 1, 2, . . . , N1,(4)

�io �

na2�i

a

nw2 �i

o �ia, i � 1, 2, . . . , N1. (5)

Here na and nw represent the refractive indices of airand water, respectively. Following the same proce-dure as in JS94, the solutions for Eqs. (3a) and (3b)can be obtained as

I��, �ia� � �

j�1

N1

�C�jG�j��ia�exp�kj

a�� CjGj��ia�

� exp��kja�� Z0��i

a�exp��0�,i � 1, 2, . . . , N1; � �a, (6a)

I��, �io� � �

j�1

N2

�C�jG�j��io�exp�kj

o�� CjGj��io�

� exp��kjo��, i � 1, 2, . . . , N2;

� � �a, (6b)

where Cj and C�j are constants of integration to bedetermined by the application of boundary and con-tinuity conditions as discussed below. kj, and Gj are

eigenvalues and eigenvectors, respectively, deter-mined by solving an algebraic eigenvalue problem asdescribed by Stamnes et al.11 Z0��i

a� is defined andobtained by solving Eq. (12b) in JS94. The solutionsrepresented by Eqs. (6a) and (6b) are seemingly sim-pler than those in JS94 for the flat ocean case. How-ever, the solution is not complete yet, becauseconstants C j in Eqs. (6a) and (6b) are still unknowns,which differ from layer to layer in the atmosphereand the ocean (for simplicity, we omitted the indexdenoting layers here). These constants will be deter-mined by boundary and interface conditions for radi-ances (intensities). The conditions for the top andbottom boundaries, for the interfaces among atmo-spheric layers, and for the interfaces among oceaniclayers are the same as those for the flat ocean case,which were given by Eqs. (16a), (16b), (16f), and (16g)in JS94. However, the continuity conditions for radi-ances at the interface between the atmosphere and theocean are very different from those for the flat oceancase. If we denote �a

� as the optical depth just abovethe ocean surface (i.e., the mean sea level) and �a

asthat just below the surface, these conditions for therough surface case can be expressed as

I��a�, �i

a� � �j�1

N1

I��a�, ��j

a�R��ia, �j

a, nw�na�

�j�1

N2

I��a, �j

o�T��ia, �j

o, na�nw�

1�

�0F0 exp��a��0�R��ia, �0, nw�na�,

i � 1, 2, . . . , N1, (7)

I��a, � �i

o� � �j�1

N1

I��a�, ��j

a�T��io, �j

a, nw�na�

�j�1

N2

I��a, �j

o�R��io, �j

o, na�nw�

1�

�0F0 exp�

��a��0�T��io, �0, nw�na�,

i � 1, 2, . . . , N2, (8)

in which reflection R and transmission T matricesappear without indices denoting the Fourier order.Equations (7) and (8) show that the emerging radi-ance at any direction at the air–water interface de-pends on incidences from all directions from both theatmosphere and the ocean for the rough ocean case.This contrasts with the simple one-to-one correspon-dence (pairing of each �i

a and �io) as presented by Eqs.

(16c)–16(e) in JS94 for the radiances across the air–water interface of a flat ocean. One asset of the roughocean case is the term accounting for the diffusion ofthe solar beam [the last term in Eqs. (7) and (8)]; itmakes a simpler formulation of the particular solu-tion [Eqs. (6a) and (6b)] than for the flat ocean case inJS94. The reflection and transmission matrices arecalculated as


R��, ��, n� �2 � �m0

2� �0

2�

R̃��, �, ��, ��, n�

� cos m�� d�� , (9)

T��, ��, n� �2 � �m0

2� �0

2�

T̃��, �, ��, ��, n�

� cos m�� d�� ,

�m0 ��1, if m � 0,0, otherwise. (10)

Here R̃��, �, ��, ��, n� and T̃��, �, ��, ��, n� repre-sent the reflection and transmission functions at therough surface, respectively (see Appendix A). ��, ��and ��, �� are the incident and exit light directions,respectively. Note that, in these functions, n is therelative refractive index, which equals nw�na if theincident light is from the air (na�nw if the incidence isfrom the ocean). The reflectance and transmittance atthe rough surface is closely related to the slope dis-tribution of the surface facets. This distribution isusually expressed as a Gaussian function as1

P��n� �1

��2 exp�1 � �n

2

�2�n2 �, (11)

where �n is the cosine of the normal to the surfacefacet. The � is the mean slope distribution width and,based on Cox and Munk,1 it is related to the wind-speed U �m�s� as

�2 � 0.003 0.00512U. (12)

The shadowing effect and multiple scattering (re-flection) among the surface wave facets are alsotaken into account in the reflection and transmis-sion functions.12,13 More details about functions[R̃��, �, ��, ��, n� and T̃��, �, ��, ��, n�] are providedin Appendix A. Substituting Eqs. (6a) and (6b) intothe boundary and interface conditions [i.e., Eqs.(16a), (16b), (16f), and (16g) in JS94 and Eqs. (7) and(8) here], we obtain a system of linear algebraic equa-tions for the unknown coefficients Cj. The method tosolve the equations and obtain the unknown coeffi-cients was described by Stamnes et al.11 and is notrepeated here. The implementation of these solutionsinto the CDISORT code is not trivial, however.

3. Examples of Model Simulations

A. Brief Description of the Model

The CDISORT just described has been used as theradiative transfer solver by our coupled ocean–atmosphere radiative transfer (COART) model14,15

(http://www-cave.larc.nasa.gov/cave/). CDISORT ac-counts for the change in refractive index at theair–sea interface9 and now includes the interfaceroughness into the analytic solution of the radiative

transfer equation. Hence COART considers the atmo-sphere and ocean as one system and treats the oceanstrata just as additional atmospheric layers with dif-ferent optical properties. Similar to the atmosphere,the ocean can also be divided into an arbitrary num-ber of layers required to resolve the vertical varia-tions of the water properties. COART models theabsorption and scattering processes in the atmo-sphere and ocean explicitly. These include the scat-tering and absorption by molecules, aerosols, andclouds in the atmosphere, and by liquid water mole-cules, dissolved and particulate matter in the ocean.

COART calculates radiances and irradiances atany level of the atmosphere and ocean in bothnarrowband (spectral) and broadband. For the nar-rowband scheme, users can specify both the band(wavelength) limits and computational resolution ar-bitrarily. In this scheme, COART employs theLOWTRAN 7 band model (spectral resolution of 20cm�1) and the molecular absorption database for theatmosphere. This corresponds to a wavelength reso-lution of approximately 0.5 nm at 500 nm and 8 nmat 2000 nm. For efficient broadband calculations ofradiance and irradiance, COART divides the solarspectrum �0.20–4.0 �m� into 26 fixed wavelength in-tervals; in each spectral interval, the k-distributiontechnique parameterizes molecular absorption in theatmosphere using the HITRAN 2000 database.16

The most prominent effects of ocean surface rough-ness on solar radiation are on upwelling fields in theatmosphere and downwelling fields in the ocean. Itseffects on the downwelling radiation in the atmo-sphere and the upwelling radiation in the ocean aresignificantly smaller. While COART can simulate avariety of quantities, including the water-leaving ra-diance, we show here mainly the types of calculationsthat pertain to surface roughness.

B. Effects of Surface Roughness on Radiance

The COART calculations in Fig. 1, which use aMcClatchey mid-latitude summer atmosphere17 witha marine aerosol optical depth of 0.1 (at 500 nm) andcase 1 water for the ocean18 with a chlorophyll con-centration of 0.1 mg�m3, span the upwelling radiancedistribution at the top of the atmosphere (TOA) andthe downwelling radiance distribution at four depthsin the ocean (0, 10, 100, and 200 m) for three differentwind speeds. The average Petzold19 phase functionfor marine particle scattering is used in the calcula-tions. Figure 1 uses polar coordinates, with view ze-nith angle (�) on the radial axis and relative azimuthangle (�) as the azimuthal coordinate. To facilitatecomparisons of different wind speeds, wavelengths,and levels, the radiance in Fig. 1 is normalized bythe upwelling or downwelling irradiance (E) at thesame level to obtain the anisotropic radiance function(ARF) as

ARF��, �� I��, ��

E , (13)


where E is the upwelling irradiance if I��, �� is theupwelling radiance (� is positive in this case in Fig. 1)and otherwise, E is the downwelling irradiance(� is negative in this case). The ARF here is, in fact,the ratio of the actual radiance, I��, ��, and theimagined isotropic radiance, E��, with the same ir-radiance. Therefore the gradient in the ARF repre-sents the departure of the radiance field from theisotropic case �ARF � 1.0�.

The solar zenith angle (SZA) in Fig. 1 is 40°. Be-cause the slope distribution in Eq. (11) is independentof the wind direction, the ARF (and the radiance field

itself) of Fig. 1 is symmetric with the principal plane(the vertical plane containing the Sun, the surfacetarget, and the nadir). So only the ARF for the azi-muth from 0° to 180° is presented. The entire princi-pal plane is covered by the horizontal axis of eachpanel in Fig. 1, and the Sun (observer) is on the left(right). The top three rows of Fig. 1 show upwellingARF for the broadband shortwave �0.20–4.0 �m�,531 nm [the central wavelength of the Moderate Res-olution Imaging Spectroradiometer (MODIS) channel11], and 865 nm (MODIS channel 16), respectively.The bottom four rows show the downwelling ARF at

Fig. 1. Model-simulated upwelling radiance field at the TOA and the downwelling radiance field at depths of 0, 10, 100, and 200 m in theocean for three different wind speeds and for three wavelength sets (broadband, narrowband at 531 nm, and narrowband at 865 nm). TheSZA is 40°.


four ocean depths for 531 nm only (865 nm is notshown because of strong absorption by liquid water,and the broadband is not shown because it is similarto 531 nm).

The three columns of Fig. 1 cover wind speeds of 3,6, and 10 m�s for the atmospheric ARF and windspeeds of 3, 9, and 18 m�s for the ocean ARF. The hotspot in each panel represents the specular reflection(the sunglint in the atmosphere) or transmission (inthe ocean) of the solar beam at the rough surface. Thesunglint is conspicuous at the right of each TOApanel (top three rows), but the sunglint region widensas wind speed increases and is much more prominentfor the 865 nm case because of less atmospheric scat-tering. Because the downwelling radiation in the up-per ocean is sharply focused around the refractedsolar beam, a larger wind variation than for the at-mosphere is required to show the hot spot variationwith wind (i.e., the widening as wind increases). Asthe depth in the ocean increases, however, the sharpradiance peak around the refracted solar beam rap-idly decreases, and the position of the maximum ra-diance gradually shifts from the refracted solarzenith direction to the nadir. At deep ocean levels, thediffusion by ocean water and particles becomes moreimportant, and the wind effect on the ARF or theradiance anisotropy diminishes. Eventually, the ra-diance distribution in deep water will approach anasymptotic shape with a maximum at the nadir, andthe directional and depth dependences of the radi-ance distribution decouple. The asymptotic distribu-tion is independent of the surface roughness or windspeed but dependent only on the inherent opticalproperties (IOPs) of the ocean. However, the speedwith which this approaches an asymptotic distribu-tion depends on both the ocean IOPs and the surfaceroughness. The radiance distribution approaches theasymptotic shape faster for a high-scattering waterthan for a high-absorption water, and faster for ahigh wind than for a low wind.

For the same atmospheric and ocean inputs, Fig. 2further shows the radiance distribution at 531 nmonly; and just around the hot spots in Fig. 1 in theprincipal plane, where radiance varies most sharplyand wind effect is most glaring. Figure 2 highlightsthe different impacts of wind on the radiance fields atvarious levels in the atmosphere and ocean. Threedifferent wind speeds (3, 6, and 9 m�s) are used here.Note, the radiance in the sunglint region at the TOAcould be larger than at the surface when wind isweak, because the reflected solar radiances at thesurface in this particular small region are muchlarger than the radiances outside; and the Rayleighand aerosol radiances at the TOA are not enough tocompensate for the attenuation of the strong reflectedsolar radiance at the surface. While the color scale ofrow 5 in Fig. 1 revealed virtually no effect of surfaceroughness on downwelling radiance broadly over thehemisphere, rows 2 and 3 of Fig. 2 show that wind,indeed, has an impact on the radiance distributionaround the forward-scattering direction in the ocean.Figure 2 further delineates how the wind effect di-

minishes, and the radiance anisotropy rapidly de-creases, as the depth in the ocean increases.

Figure 3 shows a model-observation comparison ofthe shortwave radiances at the TOA. The measure-ment data were from NASA’s Cloud and Earth Radi-ation Energy System (CERES) instrument20 during aspecial field program at the CERES Ocean ValidationExperiment (COVE) site.21 The CERES concerns ra-diation energy budget over the global ocean. TheCOVE site may not be strictly the case 1 water. How-ever, in situ measured ocean optical properties [ab-sorptions for phytoplankton and nonpigmentparticles, and for colored dissolved organic matter(CDOM)] instead of the parameterization for case 1water were directly used in the model. The CERESwas programmed to a special mode for intense obser-vation at the COVE, and only the measurements inthose clear days during the field experiment are pre-sented here. In this experiment, comprehensive mea-surements on a variety of physical and opticalproperties of the atmosphere, surface, and oceanwere also available for the model input here. Thehorizontal coordinate in Fig. 3 is the sunglint angle,defined as the angle between the view direction andthe specular solar reflecting direction for an imaginedflat surface. The nine numbers in the lower portion ofFig. 3 are the mean model-observation biases for thenine glint-angle intervals (10° each) from 0° to 90°,respectively. Though the aerosol loadings and thesurface and ocean properties were different for dif-ferent days, the model and observation agree fairlywell away from the sunglint region. The difference issomewhat larger near the sunglint center (smallerglint angles), probably due to the error in the surfaceroughness treatment in the calculations, for example,the uncertainties in the Cox–Munk model. The SZAis approximately 20° when the CERES took the mea-surements, and so a large glint angle (larger than 75in Fig. 3) also represents a large-view zenith angle,where the view path is longer and the surface foot-print is larger, and therefore, the possible horizontalvariations of the aerosol and the surface have largereffects than at a small-view angle. This might beresponsible for the increased biases in the large-angleregime.

C. Effects of Surface Roughness on Irradianceand Albedo

The effects of ocean surface roughness on irradiancesare shown in Fig. 4, which has upwelling irradiancesin the atmosphere (linear scale) and downwelling ir-radiances (logarithmic scale) in the ocean for 531 and865 nm, and the broadband shortwave (the three col-umns) at four different levels (the four rows). Themodel inputs for the atmosphere and ocean are iden-tical in Figs. 1 and 4. In each panel of Fig. 4, theirradiances for different wind speeds are presented asa function of the cosine of the SZA. Results for a flatocean case �wind � 0 m�s� are plotted as solid curvesin each panel, and thus the difference of irradiancesbetween a rough ocean case (represented by a non-zero wind) and the flat ocean case represents the


surface roughness effect quantitatively. Figure 4shows that the effect of surface roughness is smallerfor high Sun than for low Sun; and the upwellingirradiance just above the surface (row 2) is the fieldwith the most action. Note that for upwelling irradi-ance at low Sun, roughness has a larger effect at865 nm (where much of the downwelling irradiance isdirect, striking the surface at a glancing angle,thereby obtaining strong Fresnel reflection) than at531 nm (where the downwelling irradiance is morediffuse and has a component that is closer to normal).However, we find the opposite for high Sun: rough-ness has a larger effect on above-surface upwellingirradiance at 531 nm than at 865 nm. The turningpoint is at approximately the SZA of 60 (cos SZA of0.5), where the direct and diffuse reflectances aresimilar, and this angle can be considered as an effec-tive angle for diffuse radiation. For each rough sur-face case, the variation with respect to the flat surface

Fig. 2. Effects of surface rough-ness on radiance distributions at531 nm in the components of theprincipal plane containing mostof the reflected solar beam in theatmosphere (top row), and mostof the refracted solar beam in theocean (rows 2–4). The SZA is 40°.

Fig. 3. Comparison of modeled and measured broadband radi-ances as a function of sunglint angle. The nine numbers are themean model-observation biases for the nine glint-angle intervals(10° each) from 0° to 90°, respectively.


case (the solid curve) in the downwelling irradiancejust below the surface (row 3) is equivalent to the var-iation in the upwelling irradiance just above the sur-face (row 2) but with opposite sign. However, becausethe downwelling irradiance just below the surface ismuch larger than the upwelling irradiance just abovethe surface, the relative variation in the downwellingirradiance in the ocean from the flat ocean case to arough surface case is smaller and is less obvious thanin the upwelling in the atmosphere in Fig. 4, espe-cially for high Sun conditions. At a depth of 10 m (row4), the effects of internal ocean optics on irradiance

outweigh the effects of surface roughness and theirradiance at 865 nm is none due to strong waterabsorption. At great depths, where the radiance dis-tribution does not change, the irradiance attenuationalso approaches a constant, dependent only on theocean IOPs.

Because the downwelling irradiance in the atmo-sphere has little dependence on the surface conditionof an ice-free ocean, the large effect of surface rough-ness on upwelling energy (top half of Fig. 4) will havea signal in the surface albedo. The left panel of Fig. 5shows the multifilter rotating shawdowband radiom-

Fig. 4. Modeled irradiances versus cos(SZA) with upwelling irradiance in the atmosphere and downwelling irradiance in the ocean, fordifferent wind speeds and different wavelengths.


eter (MFRSR) measured �670 nm� (the dots) andmodeled (the solid curves) surface albedo for threeclear afternoons with quite different wind regimes(right panel) at the COVE; the aerosol loadings werelow. Aerosol optical properties used in the model weremeasured from the same platform by NASA’s Aero-net Cimel instrument.22 The Cimel Sun photometermade periodic scans in the almucantar and in thesolar principal plane; inversions of these data yieldedaerosol phase functions and particle size distribu-tions.23 The wind data were from the National Oce-anic and Atmospheric Administration meteorologystation also at the COVE. The ocean optical proper-ties and chlorophyll concentration were also from insitu measurements,15 but ocean optics have little ef-fect on the total surface albedo at 670 nm. To removethe relative difference between the two surface-basedMFRSR instruments and obtain accurate ocean albe-dos, the instruments subsequently used for thedownwelling and upwelling spectral irradiance mea-surements were calibrated relative to each other inadvance, by observing the same target at the sametime.15,21 Results in Fig. 5 show the significant effectsof wind on ocean surface albedo, especially for a largeSZA. The dependences of albedos on the SZA andwind are consistent between model and measure-ments.

When light is incident on the rough surface at agrazing direction, the photons are more likely to un-dergo multiple scattering or reflection among the sur-face wave facets. There is also a shadowing effect ofone wave facet blocking rays from getting to anotherfacet (occultation).13 Figure 6 shows the effects ofmultiple reflection and shadowing among the surfacewave facets on albedo simulation. In each panel, thesolid curve is the same modeled albedo as shown inFig. 5, with both shadowing and multireflection con-sidered. The short dashed curve is the calculationwithout shadowing but with multireflection, whilethe long dashed curve represents the results withoutany multireflection but with shadowing considered.The dotted curve is the calculation without roughness(flat surface). The error bar shows the range of mea-

Fig. 5. Effects of wind speed on ocean surface albedo at 670 nm.The left panel shows the modeled and measured surface albedoduring three afternoons. The right panel shows the observed windspeed for each afternoon. Different colors are for different days.

Fig. 6. Effects of multiple scat-tering (reflection) and shadowingamong surface wave facets onocean albedo simulation. Thethree panels are for the three se-lected days as in Fig. 5. The longdashed curve is the albedo com-puted without multireflection butwith shadowing; the short dashedcurve is the albedo computedwithout shadowing but with mul-tireflection; the dotted curve is thealbedo calculated with a flat sur-face. The error bar represents themeasured albedo range to within3° of the SZA.


sured albedo to within 3° of the SZA centered by theerror bar. As expected, the calculated albedo is re-duced after including the shadowing effect and isincreased after including the multireflections. Theeffect of multireflection is larger on day 1 when thewind was high, and the model-observation agreementis improved after the effects of shadowing and multi-reflection are taken into account. However, the effectsof shadowing and multireflection are small for highSun (small SZA). The small differences between theflat ocean albedos (the dotted curves) are mainly dueto the slightly different aerosol loadings for the threeselected days.

Results above show that the surface roughnesshas the largest effect on ocean albedo at low Sun(large SZA). Both Figs. 5 and 6 show a larger model-observation discrepancy for a large SZA (higher thanabout 80) for the day with the strongest wind (day 1).This may indicate a larger error tendency in theCox–Munk surface roughness model in high windconditions. An alternate distribution was recentlyproduced by Ebuchi and Kizu,24 based on approxi-mately 3 � 107 satellite observations over 5 years.The Ebuchi–Kizu function has a narrower slope dis-tribution, and less sensitivity to wind, than the Cox–Munk function. The calculated albedo (not shown)based on the Ebuchi–Kizu model is close to that basedon the Cox–Munk model for low winds, but higher forhigh winds and large SZAs. However, we cannot con-clude the superiority of either the surface roughnessparameterization, based on limited comparison. Val-idation of this parameterization is not a subject here.We simply indicate that the surface slope distributionfunction has a significant impact on the albedo calcu-lation too. On the other hand, the radiative transfermodel itself could also introduce errors for calculationswith large SZAs. For example, we have not accountedfor the orientation of the wave slope with the winddirection and the Earth curvature, both have largerimpacts on glancing incidences.

4. Conclusion

We have presented an analytical approach for radia-tive transfer in a coupled atmosphere–ocean systemhaving a rough surface between two media with dif-fering indices of refraction. The discrete-ordinate tech-nique is used in the formulation and solution. Thesolution is implemented in the radiative transfer codeCDISORT. Using CDISORT as the radiative transfersolver, a coupled ocean–atmosphere radiative transfermodel is now available to calculate various radiancesand irradiances at any altitude in the atmosphere anddepth in the ocean. This model is demonstrated on-line at http://www-cave.larc.nasa.gov/cave/ (search for“COART model” on Google).

Model simulations show that the ocean surfaceroughness has significant effects on the upwellingradiation in the atmosphere and the downwelling ra-diation in the ocean. As wind speed increases, theangular domain of sunglint broadens, the surface al-bedo under low Sun decreases, and transmissionthrough the air–water interface to the interior of the

ocean increases. The transmitted radiance just belowthe ocean surface is highly anisotropic, but this an-isotropy decreases rapidly as the surface wind in-creases. Deeper below the surface, as the opticalproperties of the ocean interior eventually overcomethe impact of surface roughness, the anisotropy de-creases and the radiance distribution gradually ap-proaches an asymptotic shape with a maximum atthe nadir. The effects of surface roughness on radia-tion depend greatly on both wavelength and angle ofincidence (i.e., solar elevation); these effects are sig-nificantly smaller throughout the spectrum at highSun.

The models and observations agree fairly well onthe effects of surface roughness. Some discrepanciesmay indicate that the original Cox–Munk surfaceroughness model is not sufficient for high wind con-ditions, or other errors exist in the treatment of thesurface roughness.

Appendix A: Reflection and Transmission at a RoughOcean Surface

The light incident at a flat water surface will bereflected or refracted directly. However, photons in-cident at the rough surface may scatter more thanonce among the surface wave facets before exit toair or water. The single-scattering reflectance at theair–water interface from ��, �� to ��, �� can bewritten as

R0��, �, ��, ��, n� � r�cos �r, n�� p��, �� → �, �, �n

r , �� s��, ��, ��, (A1)

where r�cos �r, n� is the Fresnel reflection coefficientfor relative refractive index n under incident angle �r.n � nw�na for air incidence and n � na�nw for waterincidence. The p��, �� → �, �, �n

r , �� is the fractionof the sea surface (i.e., the effective area of the wavefacets) with normal �n

r to reflect light from ��, �� to��, �� and is given by

p��, �� → �, �, �nr , ��

1

4��nr�4 P��n

r�, (A2)

where P��nr � is the surface slope distribution function

given by Eq. (11). The required surface normal, �nr , to

fulfill the specular reflection from ��, �� to ��, �� isdetermined by ��, ��, �, and �. Defining

cos � � �� 1 � �2 �1 � ��2 cos�� , (A3)

cos �r in Eq. (A1) can be derived as

cos �r � ��1 � cos ��2, (A4)

and �nr can be derived as


�nr �

� ��

�2�1 � cos ��. (A5)

In Eq. (A1), the shadowing effect, representing theprobability that the incident and the reflected lightsare intercepted by other surface waves, is corrected bythe function s��, ��, ��, which is based on Sancer13

and is widely used.5,7

Similarly, the single-scattering transmittance atthe air–water interface from ��, �� to ��, �� can bewritten as

T0��, �, ��, ��, n� � t�cos �t, n�� p��, �� → �, �, �n

t , �� s��, ��, ��, (A6)

where t�cos �t, n� is the Fresnel transmission forrelative refractive index n for incident angle �t.p��, �� → �, �, �n

t , �� is the fraction of the sea sur-face with the required orientation to refract lightfrom ��, �� to ��, �� and is given by

p��, �� → �, �, �nt , ��

n�n2 cos2 �t � 1

4��nt �4 cos �t

P��nt �.

(A7)

The surface normal ��nt � and the incident angle ��t�

required to fulfill the refraction are

cos �t ��n cos � � 1�

�n2 � 2n cos � 1, (A8)

�nt � �� cos �t sin �t�1 � ��2

��1 � �1 � �2�sin2�� sin �. (A9)

Due to the roughness nature, some photons after afirst scattering at the surface may not exit to the airor water directly but experience a second or evenhigher orders of scattering processes. The reflectanceand transmittance from these multiple scatteringscan be derived from the single-scattering values (i.e.,R0 and T0 represented by Eqs. (A1) and (A6) and theslope distribution function. For example, the secondscattering reflectance from ��, �� to ��, �� is

R1��, �, ��, ��, n� ��1

1

d�1�0

2�

d��1 � ��

� R0��1, �1, ��, ��, n�� R0��, �, �1, �1, n�, (A10)

the third scattering reflectance is

R2��, �, ��, ��, n� ��1

1

d�2�0

2�

d��2 � ��

� R1��2, �2, ��, ��, n�� R0��, �, �2, �2, n�, (A11)

the fourth scattering reflectance is

R3��, �, ��, ��, n� ��1

1

d�3�0

2�

d��3 � ��

� R2��3, �3, ��, ��, n�� R0��, �, �3, �3, n�, (A12)

and so on for higher orders of scattering reflectance.Finally, the total reflectance is

R̃��, �, ��, ��, n� � �i�0

N

Ri��, �, ��, ��, n�. (A13)

This is the reflection function used in Eq. (9). HereN 1 represents the highest order of multiple scat-tering to be considered. N � 0 is for single scatteringonly. In theory, N could be very large. But a large Nmeans more computation time. In reality, most pho-tons will be either reflected or refracted into the air orwater after a single interaction with the surface. Pho-tons could survive, for higher orders of successivemultiple scattering decrease rapidly as the scatteringorder increases. Test results indicate that includingthe second scattering �N � 1� is sufficient for virtuallyall conditions.

Similar to the reflectance, the formulations formultiple-scattering transmittance can be written as

T1��, �, ��, ��, n� ��1

1

d�1�0

2�

d��1 � ��

� R0��1, �1, ��, ��, n�� T0��, �, �1, �1, n�, (A14)

T2��, �, ��, ��, n� ��1

1

d�2�0

2�

d��2 � ��

� R1��2, �2, ��, ��, n�� T0��, �, �2, �2, n�, (A15)

T3��, �, ��, ��, n� ��1

1

d�3�0

2�

d��3 � ��

� R2��3, �3, ��, ��, n�� T0��, �, �3, �3, n�, (A16)

and then the total transmission used in Eq. (10) is


T̃��, �, ��, ��, n� � �i�0

N

Ti��, �, ��, ��, n�. (A17)

The effects of multiple scattering on the total reflec-tance for a beam incidence (direct albedo) and for adiffuse incidence (diffuse albedo) are presented inFig. 7. Albedo represents the irradiance reflectance.The direct albedo for a beam incidence from ��0, �0� is

RR��0� �1�0

�0

�1

d��0

2�

d�� 0�

� �R̃��, �, �0, �0, n�. (A18)

From the direct albedo, the diffuse albedo can beobtained by integrating RR weighted by the incidentradiances. For the uniform (isotropic) incidence, thisdiffuse albedo is simply

Rdf � 2�0

1

�RR��d�. (A19)

In Fig. 7, the upper two panels are for the directalbedo and the lower two panels are for the diffusealbedo. The left panels are for light incident from air�n � 1.34�, and the right panels are for light incidentfrom water �n � 1�1.34�. Here the water refractiveindex of 1.34 is used, which is a typical number forvisible wavelengths. The solid curves in Fig. 7 are forsingle scattering and the dashed curves are for mul-tiple scattering. In panels (a) and (b), the numbersby each pair of lines on the right edge represent theincident angles. These results show that themultiple-scattering effect is small for radiation withsmall incident angles and increases as wind speedincreases. Note, for a flat surface, the direct albedo is1.0 (i.e., total reflection) for water-incident light withan incident angle larger than the critical angle (whichis 48.2° for nw � 1.34). Figure 7 (panel b) indicatesthat this total reflection region disappears when thesurface is rough.

The total transmittance for a beam incidence or fora diffuse incidence simply equals one minus the totalrelevant albedo as defined in Eqs. (A18) and (A19).Therefore opposite to the albedo, the multiple scat-tering will decrease the transmittance across the air–water interface incident from either direction.

We thank the late Glenn Cota, Xiaojun Pan, DavidRubble at Old Dominion University for the measure-ments of ocean optical properties, B. N. Holben of theNASA Goddard Space Flight Center for the Aeronetdata, Fred Denn for the measurements at the COVE,and Fred Rose for help with the CERES data. Thisresearch was supported by NASA’s Earth Observa-tion System through the CERES. K. Stamnes is sup-ported by NASA grant NRA-02-OES-06.

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