radiative transfer in nonuniformly refracting layered ... · the radiative-transfer model we...

$: Radiative transfer in nonuniformly refracting layered ... · The radiative-transfer model we present is designed for application to the atmosphere-ocean system, but it can be applied$
Radiative transfer in nonuniformly refractinglayered media: atmosphere-ocean system

Zhonghai Jin and Knut Stamnes

We have applied the discrete-ordinate method to solve the radiative-transfer problem pertaining to asystem consisting of two strata with different indices of refraction. The refraction and reflection at theinterface are taken into account. The relevant changes (as compared with the standard problem with aconstant index of refraction throughout the medium) in formulation and solution of the radiative-transferequation,' including the proper application of interface and boundary conditions, are described.Appropriate quadrature points (streams) and weights are chosen for the interface-continuity relations.Examples of radiative transfer in the coupled atmosphere-ocean system are provided. To take intoaccount the region of total reflection in the ocean, additional angular quadrature points are required,compared with those used in the atmosphere and in the refractive region of the ocean that communicatesdirectly with the atmosphere. To verify the model we have tested for energy conservation. We alsodiscuss the effect of the number of streams assigned to the refractive region and the total reflecting regionon the convergence. Our results show that the change in the index of refraction between the two stratasignificantly affects the radiation field. The radiative-transfer model we present is designed forapplication to the atmosphere-ocean system, but it can be applied to other systems that need to considerthe change in the index of refraction between two strata.

1. IntroductionThe discrete-ordinate method has been satisfactorilyused to solve the radiative-transfer problem in verti-cally inhomogeneous media. For example, it hasbeen applied to multilayered media with anisotropicscattering,' but there has been no need to considerchanges in refractive properties because opticallyonly one medium with a constant index of refractionwas considered. Today, however, more and moreapplications in oceanography and climate-change stud-ies have involved two or more subsystems withdifferent indices of refraction. For such situationsthe change in the index of refraction across theinterface between the two media has to be taken intoaccount. Examples are radiative transfer within theatmosphere-ocean system and the atmosphere-seaice-ocean system.

A number of models, most of them based on theMonte Carlo technique, have been developed and areused for various studies of radiative transfer in the

The authors are with the Geophysical Institute and the Depart-ment of Physics, University of Alaska, Fairbanks, Alaska 99775-0800.

Received 14 January 1993; revised manuscript received 28 May1993.

0003-6935/94/030431-12$06.00/0.3 1994 Optical Society of America.

atmosphere ocean system.2-7 The main purpose ofthis paper is to extend the discrete-ordinate method,which has proved to be a very efficient method forsolving radiative-transfer problems in the atmo-sphere,81 0 to a coupled system with a discontinuousinterface of the refractive index. The principal diffi-culty encountered in attempts to model radiativetransfer throughout such a system with the discrete-ordinate method originates from the bending orrefraction of radiation across the interface betweenthe media of different refractive properties. TheFresnel refraction and reflection will affect the formof the radiative-transfer equation and the particularsolutions, and the continuity relations at the inter-face are totally different from the nonrefractive case.We will take the atmosphere-ocean system as anexample and assume a flat air-ocean interface pres-ently. Also, the atmosphere and the ocean are bothassumed to be vertically stratified so that the opticalproperties depend only on the vertical coordinate.To account for the vertical inhomogeneity, the atmo-sphere and the ocean can be divided into any suitablenumber of horizontal layers, as required to resolvethe vertical structure of the optical properties of eachmedium.

In the following section, we first derive the formula-tion of the radiative-transfer equation, which is differ-ent from that for the uniformly refracting medium,

20 January 1994 / Vol. 33, No. 3 / APPLIED OPTICS 431

for a coupled system. We then select an appropriatequadrature and apply the discrete-ordinate method tofind a general solution which is suitable for everylayer. Finally we discuss the formulation and proce-dure required to apply the interface and boundaryconditions to such a coupled system. In Section 3,some model-consistency tests are performed, theimportance of proper stream distributions in theFresnel cone and the total reflection region in theocean to achieving fast convergence is discussed, andthe effect of scattering asymmetry on convergence isstudied. Following that, some results from applyingthis formalism to the atmosphere-ocean system arepresented, including comparisons performed withand without the inclusion of the effects arising fromthe change in the index of refraction at the air-oceaninterface. Those comparisons demonstrate the im-portance of including this change in the refractiveindex.

2. Equations and SolutionsFormulation and solution of the radiative-transferequation for the atmosphere-ocean system have a lotin common with that for the atmosphere only. Inthis paper we emphasize the differences. Therefore,only solar radiation is considered, because it is stronglyaffected by the refractive-index change and exhibits amuch different transfer process in the coupled system.We will neglect the thermal emission and give thehomogeneous solution directly, as it is basically thesame as that which is obtained when considering onlythe atmosphere.

Figure 1 illustrates the radiative-transfer model forthe atmosphere-ocean system schematically. In theocean, region I is the total-reflection region. RegionII is the refraction region. The width of each regiondepends on the relative index of refraction of the twomedia. The downward radiation distributed over 27rsteradians in the atmosphere will be restricted to acone (less than 2 steradians) after being refracted

sun

Top /o -L0

Atmosphere

S1/ ~~~~~I

Ocean II

7r = T

.=,,

Fig. 1. Schematic diagram of the coupled-radiative-transfer modelfor the atmosphere-ocean system.

across the interface into the ocean. Photons inregion II of the ocean may be scattered into region I.Note, however, that photons in region I of the oceancannot reach the atmosphere directly and vice versa.Communication between the atmosphere and regionI must occur through the scattering process betweenregion I and region II in the ocean. All these charac-teristics for the coupled system can be described witha properly formulated radiative-transfer equationand appropriate implementation.

A. Basic EquationsThe equation describing the radiation transferthrough a plane-parallel medium is given by9 "'

id =T 1 -L,) u)1 J = I(T, J.L, 4~))- S(T, P, 4)), (1)

where I(T, , 4)) is the specific intensity (radiance) atvertical optical depth T (measured downward from theupper boundary) in direction (, 4)) (. is the cosine ofthe polar angle, which is positive with respect to theupward normal, and 4) is the azimuthal angle). Thesource function is

S(, ) = Xr) d+' f P(, I, A, p H,47 -1

x I(T, p.', 4)')dp.' + Q(T, A., 4)), (2)

where w(Or) is the single-scattering albedo,p(T, ,, p.', 4)') is the phase function, and Q(T, P, ))

represents the actual internal source. The solar-beam source in the atmosphere can be expressed as

Qair(Tr, P, () = -OT Fop(r, 1, 4), o, o)exp(-T/l10)

03(T)+ - FoR(- o, n)p-(T, a, 4), o, 4)o)

x exp[-(2Ta -)/LO], (3)where [Lo is the cosine of the solar zenith angle and ispositive, 4)o is the azimuthal angle for the incidentsolar beam, and F0 is the solar-beam intensity at thetop of the atmosphere. Here n is the index ofrefraction of the ocean relative to the atmosphere,and T is the total optical depth of the atmosphere.The first term in Eq. (3) represents the contributionfrom the downward, incident beam source, while thesecond term represents the contribution from theupward beam source reflected at the atmosphere-ocean interface because of the Fresnel reflectioncaused by the change in the refractive index betweenair and sea water. R(- Ro, n) is the ocean-surfacereflectance for the solar beam. In the ocean, thesource term is

Qocn(,, R 03(T) pLo--FT-Rn4Tr R.n(vLo n)

X p(T, , 4), - , 4o)exp(-Ta/./0)

x exp[-(T - Ta)/On],

432 APPLIED OPTICS / Vol. 33, No. 3 / 20 January 1994

(4a)

AT

where T(- [to, n) is the transmittance through theinterface, and Ron is the cosine of the solar zenithangle in the ocean, which is related to [Lo by the Snelllaw:

The source term in ocean can be expressed as

Qocnm(T) ) = X02m(T, p.)exp(-T/VLon),

where

lOn(p 0, n) = Vi - (1 - p2)/n2. (4b)

Expansion of the phase functionp(T, cos 0) in a seriesof 2N Legendre polynomials and the intensity in aFourier cosine series"ll

2N-1

I(T, p., ) = I ImQ(,, p.)cos MO( - 4 )o)m=O

p(T; p, 4), , 4)') p(T; Cos 0)

2N-1

= 1, (21 + 1)g1(T)P(cos 0),1=0

(5a)

X02 m(T, L)

- 4 R T|(-[o, n)Fo exp- Ta I 14 prIOn(pLO, n) P~O pon/J

2N- 1

X (2 - 8mo) I ( 1)l+m(21 + 1)gr(T)1=0

(1 - )!x (I + m)! plm(p)pm(p.n)

(5b)

which leads to the replacement of Eq. (1) with 2Nindependent equations (one for each Fourier compo-nent):

dlm(T, .)

p. dT= Im(T, p.) - Dm(T, p., p')Im(T, p.')dp.'

-1

(6h)

B. Quadrature Rule And Discrete-Ordinate Approximation

The discrete-ordinate approximation to Eq. (6a) isobtained by replacing the integral in Eq. (6a) by aquadrature sum, thus transforming the integro-differential Eq. (6a) into a system of coupled differen-tial equations. Thus, for each layer in the atmo-sphere, we obtain'

- Qm(T, p.), m = 1, ... ,2N- 1,

(6a)

W(T) 2N-1-3T (21 + 1)g1(T)2

( - )!X - pjm(p.)pjm(p.W),

(1y + in)!(6b)

where 0 is the scattering angle, Plm(p.) is the associ-ated Legendre polynomial, g(T) is the expansioncoefficient, and Qm(T, p) is the mth Fourier compo-nent of the beam source. In the atmosphere it is

a djm(T, pa)

a dT= I (T, pa) -

N1z ~Dm(T Ja }Jj)

j= -Njeo

X Im(T, p.la) -Qm(T pa),

(7)

and similarly, for layers in the ocean, we find

dI-(Qr, p.i°)

A o dT

N2

Im(T, PM o)- z WjoDm(T, Ip.1 O, j°)

i= -N2j•o

X I(T, pjo) - Qocn(T, pio),

i = + N2,

Qairm(T, p.) = Xom(T, P.)exp(-T/po)

+ XOlm(T, P.)eXp(T/p"o), (6c

where

03(T) .2N-1Xom(T, p.) = - F(2 - mo) E: (-1)l+m(21 + 1)gl(T)

1=0

( - )!x (I + m)! m(p.)plm(p. 0), (6d

XO (T , p) = r FOR(- RO, n)exp(- 2Ta/Lo)(2 - 8m0)

2N-1 ( -n)!x E (21 + 1)gl(T) (l + m)!

X P1m(p.)pjm(G.O),

1, ifm = 0;

bmo =lo, otherwise.

where pa, Wia and pio, wio are quadrature points andc) weights for the atmosphere and the ocean, respec-

tively, and pij = - p.j, w- = wj. Note that instead ofusing a constant number of streams for each layer asis usual, we have used different numbers of streamsfor the atmosphere and the ocean (2N, and 2N2,respectively). In region II of the ocean, which com-municates directly with the atmosphere, we use thesame number of streams (2Nj) as in the atmosphere.This properly accounts for the shrinking caused byrefraction of the angular domain in the ocean. Inregion I of the ocean, where total reflection of photonsmoving in the upward direction occurs at the ocean-atmosphere interface, we invoke additional streams(2N2 - 2N,) to accommodate the scattering interac-tion between regions I and II in the ocean. Althoughthere are many options for choice of quadrature, thischoice will strongly affect the application of interface-continuity conditions and the accuracy of the solution.The quadrature used here is essentially the same asthat adopted by Tanaka and Nakajima.12 The double


(6g)

D m(T, p, p.') =

(8)

Guass quadrature rule is used to determine thequadrature points and weights, pa and Wia (i = 1, . . ..NI), in the atmosphere, as well as the quadraturepoints and weights, pio and wio (i = N, + 1, . . , N2),in the total-reflection region of the ocean. Thequadrature points in the Fresnel cone of the ocean areobtained by simply refracting the downward streamsin the atmosphere (pla, .. , p.NIa), into the ocean.7,' 3

Thus, in this region, pio is related to ,1a by the Snelllaw,

pij0 = S(p,1 a) = V1 - [1 - (pa)2]/n2,

where k and G. are eigenvalues and eigenvectors,respectively, determined by solving an algebraic eigen-value problem, and Cj and C < are unknown constantsof integration, to be determined by the application ofboundary and continuity conditions as discussed be-low.

As for the particular solution, in the atmosphere itcan be expressed as

U(T, ,Wa) = ZO(ga)eXp(-T/ltO) + ZO1(p,a)eX(T/tto),

(12a)

(9) where i = +1, +2, . . ., N1. The coefficientsZO(p,1a)and Zo0,(pa) are determined by the following system of

and from this relation, the weights for this region can linear algebraic equations:be derived as

p.1a= t

2S(pa)

i = 1, 2,..., Ni. (10)

The advantage of this choice of quadrature is that thepoints are clustered toward p = 0 both in theatmosphere and in the ocean, and in addition, towardthe critical-angle direction in the ocean. This cluster-ing gives superior results near these directions wherethe intensities vary rapidly. Also, this choice ofquadrature will simplify the application of the inter-face-continuity condition and avoid the loss of accu-racy incurred by the interpolation necessitated byadopting the same quadrature (i.e., the same numberof streams) for the atmosphere and the ocean.

Finally, it is easy to show that the chosen quadra-ture points and weights make phase-function renor-malization unnecessary, so that energy conservationis satisfied automatically, as pointed out first byWiscombe.14

C. SolutionAn accurate, reliable, and efficient method of obtain-ing the solution of the homogeneous version of Eqs.(7) or (8) was presented by Stamnes and Swanson.10Following the same procedure, we give the homoge-neous solution here.' In the atmosphere, (omittinghereafter the superscript m denoting the Fouriercomponents), it is

Ni

I(T, p.ia) = E [C jGj(pa)exp(kjaT)i=1

+ CGj(.1 a)exp(-kjaT)],

i= +1, ... , N, (11a)

and similarly, in the ocean

N2

I(T, pi) = E [CjGj(Li0)exp(kjfT)j=1

+ CGj(pi")exp(-kjfT)],

i = +1, . . . N 2, (lib)

[ ( 1 +-aj - WjaD(T, p.a, Fja)jZo(1j a)

Nj

j=-Nij;-O

[(1 + Po)j - WJaD(T, p.,a,

= Xo(T, p1a),

p.3a) Zol(Aj a)

(12b)

N1

j= -Nj~e

= X0 1(T, p.sa). (12c)

The particular solution in the ocean can be expressedas

U(T, P1 ) = Z02(p.i)eXP[-T/p0.(p 0, n)],

where i =mined byequations:

(13a)

+1, +2, ... , N2, and Zo2(pio) is deter-the following system of linear algebraic

2 1 + 8i- WjOD(T, pio, Pajo)]Z02 (1jo)

j;O

= X02(T, .Llio)- (13b)

The general solution is just the sum of the homoge-neous solution and the particular solution.

D. Conditions for Boundary, Continuity, andAtmosphere-Ocean InterfacesThe vertically inhomogeneous medium is representedby multiple, adjacent homogeneous layers in theatmosphere and the ocean, respectively. The solu-tions derived previously will be used in each layer.We assume that the system consists of L, layers ofatmosphere and L2 layers of ocean. Then we maywrite the solution for the pth layer as

Ni

IP(T, pia) = I [CjpGjp(pa)exp(kjpaT)j=1

+ C GPp~a)eXp(_kjpaT)] + UP(,T, [La),

i=1 ±N..,±Nandp<L1,


(14)

i = 1 2..., N1,

Wo = Wa dVLa ,a=,,,a

N2

IP(T, p.i') = E [CjpGjp(l.Li)exp(kjpT)j=l

+ CjpGjp(pi)exp(-kjp°T)] + Up(T, p.io),

-i-N2 andLj <p <L,+L 2 .

(15)

There are 2N, x L, + 2N2 x L2 unknown coefficientsCjp in Eqs. (14) and (15). They are determined bythree factors: the boundary conditions to be appliedat the top of the atmosphere and the bottom of theocean; the continuity conditions at each interfacebetween layers in the atmosphere and ocean; andfinally the reflection and refraction occurring at theatmosphere-ocean interface, where we require Fres-nel's equations to be satisfied.

These conditions are implemented as follows: Atthe top we require

at the interfaces between atmospheric layers werequire

I , (T1, p.a) = IP(T P,.a),

(16b)

at the interface between the atmosphere and theocean we require

1L(Ta, p a) = ILJ(Ta, -,,.a)R(-.a, n)

+ [ILi+1(Ta, pio)/n 2]T(+piO, n),

(16c)

ILi+l(Ta, -piO)/n 2= [ILi+1(Ta, p.i)/n2]R(+i, n)

+ IL,(Ta, -,,a)T(-.a, n),

Il (0, -,,a) = I(- La), i = l,...,V,

Optical Depth = 0.0 (Top) Just Above The Interface

Conserv. Case:

'Total Down__

Total Up ........ 2

Net Irradi. - - -

OL1.0

Absorb. Case:

Total Down .... .

Total Up -

Net Irradi. - -

1.5 2.0 1.0 1.REFRACTIVE INDEX n

2 LZ

z0rf

of

. .. o

5 2.0

2

011.1,

Just Below The Interface

1.5 2.0 1.0 1.5REFRACTIVE INDEX n

Optical Depth = 2.00 (Bottom)

Ud

0

2 2 C.)Ldz

0(f(i]

OL0

2

1.0 1.5 2.0 1.0 1.5 2.0 1.0 1.5 2.0 1.0 1.5 2.0REFRACTIVE INDEX n REFRACTIVE INDEX n

Fig. 2. Variation of irradiances with the relative index of refraction at several locations for isotropic scattering, incident flux Fo = 1.0, withincident zenith angle 00 = 300, bottom-surface albedo = 1.0, Ta = 1.0, and T* = 2.0. For the absorption case, the only difference from theconserevative case is in the single-scattering albedo, o = 0.9, in the lower medium.


(16a) i = 1, 2,..., N1,

2ILL0z0

fl

(16d)

2

_----- - =. .,

. . . n . . . . .

2

Jo.02

oL o

a w w F | w E w w | | w - - - |. . . . . . .. . , . . .

. - - - - - -. -- .. - - --.. - - - - - - ... --- - - - -

i = 1 2..., N1,

I

IL,+1(Ta, -pi) = IL,+1(Ta, pi),

i = N + 1, . . ., N2; (16e)

at the interfaces between ocean layers we require

I , (T1,,, pi.) = I ,+,( , , io),

i = +1, . . ., ±N 2 , p = L + 1, . . . ,L1 + L2 - 1;

(16f)

and finally at the bottom boundary we require

IL,+L2 (T, pi ) = Ig(pio), i = 1, 2,..., N 2. (16g)

Formulas for R and T can be derived from the basicFresnel equations. The results are

[\- p. - n[L?) 2 + io - n [__a 2R(- _ a, n' 1 [{}J~ia-n~i°a + p.i-+ lp.~L2 Lyda npu° Rio + na

(17a)

(17b)

T(- .a, n) = 2n~aia[a (pa + npo) + (po + np a)2j

(17c)

We define R( pi, n) and T(±pi, n) as the specularreflectance and transmittance, respectively, of theinvariant intensity I/nabs2, where nabs is the absoluteindex of refraction at the location where I is measured.The minus sign applies for the downward intensity,and the positive sign for the upward intensity.

1.00

0.90

0.80

0.70

0.60

0.50

1.00

0.90

0.80

0.70

0.60

0.50

.0 1.2 1.4 1.6 1.8 2.REFRACTIVE INDEX n

) ......... ~~~............

8/10 _.8/8 - - - -8/6 ...........Benchmark_

T(+ p.O, n) = T(- ,,a, n). (17d)

Equations (16c) and (16d) ensure that, by satisfyingFresnel's equations, the radiation fields in the atmo-sphere and the ocean are properly coupled throughthe interface, whereas Eq. (16e) represents the total

1.00

0.90

id

0

0.8C

0.7C

0.60

0.500

1.00

0.90

id0z0

0.80

0.70

0.60

0.50

6/8 -----6/6 - --6/4Benchmark_D :,.,,,,,,,..,....

.0 1.2 1.4 1.6 1.8 2.REFRACTIVE INDEX n

16/18 -------16/16 - - - -16/14 ..........Benchmark_

)......

0

1.0 1.2 1.4 1.6 1.8 2.0REFRACTIVE INDEX n

1.0 1.2 1.4 1.6REFRACTIVE INDEX

1.8 2.0n

Fig. 3. The effect of stream combinations within the refractive region and the total reflective region (represented by the number ratio) onthe convergence for the Henyey-Greenstein scattering-phase function, with asymmetry factor g = 0.7. Other input parameters are thesame as those for the conservative case in Fig. 2. Shown in the panels are the upward irradiances at the tops of the slabs and theircomparison with the Benchmark irradiance incident on the system.


Li

0ixcx

- / *--.

/

4/64/4

.4/2 ............Benchmark__

C-)

0

............... ----------------

R (+ piO, n = R (- Ra , n),

. . . . I . . . . . . . I . . . :

reflection in region I in the ocean. The total opticaldepth of the atmosphere and ocean is denoted by T inEq. (16g). I.(- pa) is the intensity incident at thetop of the atmosphere, and Ig(pijo) is determined by thebidirectional reflectance of the underlying surface atthe bottom of the ocean.' Substitution of Eqs. (14)and (15) into Eqs. (16a)-(16g) leads to a system of2N x L + 2N2 x L2 linear algebraic equations forthe same number of unknown coefficients, the Cjp.Matrix inversion of this system of equations yieldsthe desired coefficients and thereby completes thesolution for the coupled atmosphere-ocean system.Once we have obtained the solutions for each (and all)Fourier components using Eqs. (14) and (15), we maycompute the intensity (radiance) at the quadraturedirections from Eq. (5). Irradiances (fluxes) andmean intensity can now be easily computed from thezero-order Fourier component of the intensity givenabove by using the same quadrature rule to convertintegrals into simple summations.9

1.00

0.90

0.80

0.70

0.60

0.50l

1.00

0.90

0.80

0.70

0.60

0.50

g=0.0

4/6 -------

4/4 - - - -4 /2 ............Benchmark

.0 1.2 1.4 1.6 1.8 2.0REFRACTIVE INDEX n

g=0.6

4/6 _-

4/4 ____4/2 .........Benchmark_

3. Some Results and Discussion

A. Model Test

Before considering an actual application, we shalldiscuss a number of consistency and convergencetests aimed at checking the basic soundness of thesolution. First we consider a conservative situation,in which case there is no absorption at all in the wholesystem. In another words, we replace the atmo-sphere and the ocean with two strata consisting ofmedia with different refractive indices, but withoutabsorption. At the bottom of the lower stratum, weassume that the surface is totally reflecting. Thevariation in irradiances with the relative index ofrefraction n at four particular levels is shown in Fig.2. The effect of adding some absorption is alsoillustrated. The net irradiance is defined as thedifference between the total downward irradianceand the total upward irradiance. In the conservativecase, the net irradiance is zero everywhere, consistentwith the energy-conservation requirement, so that

1.00

0.90

LId

z0DfO]

0.80 .

0.70

0.6(

0.5(

1 .0(

0.9(

UI

z0Z3

r"I

0.8(

0.71

0.6'

0.511.0 1.2 1.4 1.6 1.8 2.0

REFRACTIVE INDEX nFig. 4. Similar to Fig. 3 but showing the effect of scattering asymmetrycombinations).

1.0 1.2 1.4 1.6 1.8 2.(REFRACTIVE INDEX n

.

i g=0.9

/ . ... .

/ il ,:-'~

4/6_.__._.4/4

0 4/2 *,Benchmark____

0 . . . . I . . .

1.0 1.2 1.4 1.6 1.8 2.'REFRACTIVE INDEX n

0

on the irradiance computation (for only one group of stream


Id

0

g=0.3

./- -_ --- - - I T_ : a 0 0 0 � .

. . .. . . . . . . . . . . . .|

. . . . . . .

4/64/4 - - - -4/2 ...........Benchmark

O _

Ui0Z

0

a-,af

the total upward irradiance overlaps with the totaldownward irradiance everywhere. The total down-ward and upward irradiances increase rapidly withincreasing n in the lower stratum, but not in theupper stratum because of energy trapping, as dis-cussed by Stavn et al. 15and by Plass et al. 13 Figure 2also shows that the variation in the irradiances with nis very sensitive to the absorption in the lowermedium. A single-scattering albedo of 0.9 in just thelower medium leads to a drastic reduction of theirradiance changes versus n, as compared with theconservative case. At the bottom, the total upwardirradiance overlaps with the total downward irradi-ance for this absorptive case and owes to the assumedbottom-surface albedo of unity. The results shownin Fig. 2 pertain to isotropic scattering, but we haveverified by computations that the choice of phasefunction and optical depth does not significantlyaffect the general behavior of the irradiances indi-cated above, for the case of an absorbing lowermedium, and there is no effect for the conservativecase.

As just discussed, in the conservative case with asurface albedo of unity, the upward irradiance shouldequal the total downward irradiance at the top. Thissuggests that for the conservative case, the incidentdownward irradiance, which is equivalent to p.OF 0,can be taken as a benchmark for the total-upward-irradiance computation at the top. Figure 3 showsthe computed upward irradiances at the top by usingdifferent streams and their comparison with thebenchmark with the same conditions as in Fig. 2, butwith an asymmetry factor of 0.7 for the Henyey-Greenstein phase function used here. The number

2000

1500

:.

I

.s

'2

1:

C,

1 CCC

500

0.30 0.40 0.50 0.60 0.70 0.80

ratio represents 2N,/(2N2 - 2N,), the number ofstreams adopted within the refractive region dividedby those adopted for the total reflective region. Theresults indicate that although an increase in thenumber of streams will eventually increase the accu-racy, the convergence speed depends strongly on n fora given combination of streams in the two regions.For example, the upper left panel of Fig. 3 indicatesthat the combination of 4/2 (four streams for regionII and two streams for region I) will produce the bestaccuracy for smaller values of n (approximately 1.1 to1.2). The combination of 4/4 is best for n between1.3 and 1.5, while the combination of 4/6 is better forlarger n. Furthermore, as demonstrated in the otherthree panels, when the number of streams is in-creased in both regions, the overall accuracy in-creases as it should. Our computations also showthat the convergence behavior indicated above is trueat any level. Generally, for larger n, to obtain opti-mal results we need to assign more streams to thetotal reflective region because it becomes wider. Thequantitative relation between this optimum streamratio and n is unknown.

In addition to the distribution of quadrature pointsbetween the two regions, the convergence speed isalso strongly dependent on the asymmetry of thephase function. For the Henyey-Greenstein phasefunction, which depends only on the asymmetryfactor g, Fig. 4 shows that for smaller asymmetryfactors (i.e., less anisotropy), fewer quadrature pointsare needed to attain good accuracy. We have adoptedthe delta-M transformation,'2 which has been shownto optimize the performance of the model and toimprove the accuracy for strongly forward-peakedscattering.

E-1

0

I0.L.

EI

Wavelength(pm)

Fig. 5. Spectral distributions of downward irradiances at the topof the atmosphere, just above the ocean surface (+0 m), just belowthe ocean surface (-0 m), and at several depths in the ocean for amodel of a clear midlatitude atmosphere and pure sea water, with asolar zenith angle e0 = 300.

400

Top

+0m ...............

-Om _ _

300 25m

50m

200

100

\.' ; - " ......>'X'.-

... . .. . .. . .

0.30 0.40 0.50 0.60Wavelength(,m)

0.70 0.80

Fig. 6. Similar to Fig. 5 but showing the upward irradiances.


I ! ..'.

| ,. , / \ \, 50m

I N m.I , \ '

, . . . . .

. . . . . . . . . . . .. . . . . . . . . . . . . , 1r. . . . . . . . . . . . . . . .

1\ ......

n

B. Application for the Atmosphere-Ocean SystemThe comprehensive model described above was ap-plied to the atmosphere-ocean system. We adoptedthe profile of the midlatitude summer atmospheredeveloped by McClatchey et al.'6 and divided theatmosphere into 24 layers. For simplicity, only pureatmosphere and pure sea water are considered here.Therefore, we may adopt the Rayleigh scattering-phase function for both the atmosphere and theocean. The optical properties of the clearest oceanwater are taken from Smith and Baker.' 7 Because ofthe homogeneity of pure sea water, it is not necessaryto use multiple layers in the ocean even though ourmodel can easily accommodate an arbitrary numberof layers in the ocean. By assuming a solar zenithangle of 30° and a relative refractive index for theocean of 1.33 (neglecting for simplicity the wave-

00= 1 0°

length dependence here), we computed downwardand upward irradiances at the top of atmosphere, justabove and just below the ocean surface, as well as atseveral depths in the ocean, as shown in Figs. 5 and 6.

In Fig. 7 we show the computed reflectance of theocean surface for several solar zenith angles andcompare it with the result obtained when the effectsof refraction are ignored. The differences in theupward and downward irradiances at the ocean sur-face for having included and neglected refraction arealso shown. The results indicate that underestima-tion of the reflectance would occur for most wave-lengths if the change in index of refraction across theinterface between the ocean and the atmosphere isignored. As the solar zenith angle increases, thespecular reflection, caused by the difference in theindices of refraction between the atmosphere and the

0o=30

0.40 0.50 0.60 0.70Wavelength(Um)

0,=600

E40

20

0

-200.10

0C0U

aW.

0.80

0.05

0.00 .............0.30 0.40 0.50 0.60 0.70 0.80

Wavelength(Im)

0o=80

0.40 0.50 0.60Wavelength(4m)

E

.E

50

25

00.50

a,UC0C.a,

a,

0.70 0.80

0.25

0.00F..................-. ....

0.30 0.40 0.50 0.60Wavelength(rm)

0.70 0.80

Spectral distribution of ocean-surface reflectance for considering refraction and neglecting refraction, and for several solar zenithAlso shown are the downward and upward irradiance differences at the ocean surface when refraction is considered and neglected,

FRefri - FNoref ,AF1 = FRefrt -FNoref t. The same atmospheric and oceanic models used in Fig. 5 are used here.


40

20

E:3-

1_.

E1-13:

0

-200.10

a,UC0Ua,

a,

. § l ,... .. .. .. ..............

: AFt

.. ........................

Refract.No Refract-_

/ \

.. ~.. ... .

0.05

0.0010.; 50

50

25

E

E.

00.2

C0Ua,

a,

0.1

0.0L0.30

Fig. 7.angles.AF =

ocean, contributes more to the total reflectance of theocean surface, and consequetly more to the upwardirradiance at the ocean surface. Therefore, for largersolar zenith angles, the discrepancy between thereflectance computed with and without refractionbecomes larger.

Figure 8 shows the distribution of downward irradi-ance and upward irradiance with height in the atmo-sphere and with depth in ocean for the same atmo-sphere and ocean model used above and for awavelength of 500 nm. For comparison, correspond-ing results obtained by neglecting the refraction arealso displayed. There is considerable interest inenergy absorption as a function of altitude in theatmosphere or of depth in the ocean, because thisabsorbed energy drives the atmospheric and oceaniccirculation. Because the absorbed energy withineach layer is proportional to the mean intensity (thesame as the total scalar irradiance, defined by Moreland Smith,'8 divided by 4'r) in that layer, we show inFig. 9 the mean intensity versus both the height inthe atmosphere and the depth in the ocean. Alsoshown are the same results obtained by ignoringrefraction and the relative error, I (ref - inorefr)/Irefr I X100. We note that the relative error may increase upto 20% just below the ocean surface. Although theradiation field is large in the deep ocean, it is alreadysignificantly attenuated there.

Figure 10 shows the azimuthally averaged inten-sity distribution just above and just below the ocean

c

a

.21

II

To

1

.S

c1

,.

Deviation 0

0 20 30 50

50 100 150 200Mean Intensity (wott/m'/,um/Sr)

Fig. 9. Distributions of the total mean intensity (total scalarirradiance/47r) with height in the atmosphere and depth in theocean, the result of neglecting refraction, and the relative devia-tion. 00 = 30°, X = 500 nm.

I

.'

I

I

._

r0 500 1000

Irrodiance (Watt/m'/Am)

Z

E

E

.i,._

R

-

E

1500 200030 60 90

Polar Angle (degree)120 150 180

10 20 30Deviation ()

I. 1!E

2.

1SN

.1I

I7

'5

-1

c

40 50 30 60 90Polar Angle (degree)

120 150 180

Fig. 8. Distributions of the downward and the upward irradianceswith height in the atmosphere and depth in the ocean, the resultsof neglecting refraction, and the relative deviation. = 300, X =500 nm.

Fig. 10. Distributions of the azimuthally averaged intensity(radiance) for the refraction case (n = 1.33) and the no-refractioncase (n = 1.0); Oo = 300; = 600 nm. (a) Just below the oceansurface, (b) just above the ocean surface.


50km

2

I

so

Downward Irrad.

-_ | Upward Irrod. - _

- Atmosphere

- _ _ Ocean

*~~~~~~~~~~~~~~~~ . . . . .. .. . . . .. . . . .

25km I

Om

1 OOm I

C

. . . . . . . . . .. . . . . . . . . . . . . . . .

- 1.1 . . . . . . . . . . . . . . . . --._

0t "nn_

To

interface. Again results obtained by ignoring refrac-tion are also displayed. The results indicate that therefraction significantly alters the radiative-intensitydistribution. Just below the ocean surface, the down-ward-intensity discontinuity position shifts from thehorizontal direction for the case of no refraction, tothe critical-angle direction when refraction is included.The refraction also significantly changes the upwardradiation field just above the ocean surface. Knowl-edge of the intensity distribution here is importantfor correct interpretations of intensity measurementsin remote-sensing applications.

All of the quantities computed above for Figs. 5-10are calculated to be accurate to within 1%. Toachieve this accuracy, a stream ratio of 6/10 (6streams in the atmosphere, 10 streams in the ocean)was applied in the calculations for Figs. 5-9, while aratio of 40/70 is used for Fig. 10, but only for thepurpose of providing enough points to plot a smoothercurve.

The present model has recently been comparedwith six other models that have approached theradiative-transfer problem in the atmosphere-oceansystem with different methods.19 A typical oceanicphase function with strongly forward scattering wasadopted there. Our results calculated with thedelta-M transformation agreed very well with theresults from the other models. The computer re-sources required for calculation among the modelswere also compared, showing that our model isespecially efficient for the computations of irradi-ances and azimuthally averaged radiances.

4. ConclusionWe have developed a comprehensive methodology,based on the discrete-ordinate method, for solving theradiative-transfer equation pertinent for a systemconsisting of two strata with different indices ofrefraction. The method is well suited to providingconsistent solutions of the radiative-transfer problemfor the coupled atmosphere-ocean system. The re-fraction and total reflection at the interface of the twostrata have been taken into account by assigningdifferent numbers of angular quadrature points (dis-crete ordinates or streams) in the atmosphere and theocean. Thus the interpolation at the interface of twomedia is entirely avoided, and the radiation field inboth the atmosphere and the ocean can be efficientlysolved at once. The vertical inhomogeneity of theatmosphere and the ocean can be accounted for bydividing each stratum into a suitable number ofhomogeneous layers so that the optical propertiesmay be regarded as constant within each layer, butare permitted to vary from layer to layer.

Test results show that the solution conserves en-ergy and is both reliable and efficient. The accuracydepends on the number of streams utilized to makethe angular dependence discrete. Good accuracy forirradiance and mean intensity are obtained with justa few streams. Preliminary results of the applica-tion to the pure atmosphere-ocean system show that

the refraction significantly affects the radiation fieldand radiative-energy absorption in both the atmo-sphere and the ocean. A more realistic quantifica-tion of the radiation field in the atmosphere andocean environments can be simulated when actualoptical properties of the atmosphere, including cloudsand aerosols, and of the ocean, including particulates,versus depth are available. The radiative-transfermodel presented in this paper provides a means forestimating light reflection and transmission, as wellas rates of warming and cooling, and for studying theradiative interaction between the atmosphere and theocean. The model could, if so desired, be extended todeal with changes in the index of refraction betweenall layers instead of just the atmosphere-ocean inter-face. Such an extension would make it feasible tostudy radiative transfer within media in which theindex of refraction changes continuously throughoutthe medium. Possible future extensions of the dis-crete-ordinate method presented here include (1) thesimulation of ocean-surface roughness, (2) the compu-tation of inelastic-scattering effects to treat phenom-ena such as Raman scattering, and (3) the consider-ation of polarization.

Z. Jin thanks F6 Seymour for help with the Englishlanguage. This work was supported by the NationalScience Foundation through grant DPP92-00747and the Department of Energy through contract091574-A-Q1.

References1. K. Stamnes, S. C. Tsay, W. Wiscombe, and K. Jayaweera,

"Numerically stable algorithm for discrete-ordinate-methodradiative transfer in multiple scattering and emitting layeredmedia," Appl. Opt. 27, 2502-2509 (1988).

2. C. Mobley, "A numerical model for the computation of radi-ance distributions in natural waters with wind-roughenedsurfaces," Limnol. Oceanogr. 34, 1473-1483 (1989).

3. H. Gordon and M. Wang, "Surface-roughness considerationsfor atmospheric correction of ocean color sensor. I: TheRayleigh-scattering component," Appl. Opt. 31, 4247-4260(1992).

4. G. Kattawar and C. Adams, "Stokes vector calculations of thesubmarine light field in an atmosphere-ocean with scatteringaccording to a Rayleigh phase matrix: effect of interfacerefractive index on radiance and polarization," Limnol. Ocean-ogr. 34, 1453-1472 (1989).

5. A. Morel and B. Gentili, "Diffuse reflectance of oceanic waters:its dependence on Sun angle as influenced by the molecular-scattering contribution," Appl. Opt. 30, 4427-4438 (1991).

6. J. Kirk, "Monte Carlo procedure for simulating the penetra-tion of light into natural waters," Division of Plant IndustryTech. Paper 36 (Commonwealth Scientific and IndustrialResearch Organization, Canberra, Australia, 1981), p. 16.

7. T. Nakajima and M. Tananka, "Effect of wind-generatedwaves on the transfer of solar radiation in the atmosphere-ocean system," J. Quant. Spectrosc. Radiat. Transfer 29,521-537 (1983).

8. S. C. Tsay, K. Stamnes, and K. Jayaweera, "Radiative transferin stratified atmospheres: development and verification of aunified model," J. Quant. Spectrosc. Radiat. Transfer 43,133-148(1990).


9. K. Stamnes, "The theory of multiple scattering of radiation inplane-parallel atmospheres," Rev. Geophys. 24, 299-310(1986).

10. K. Stamnes and R. A. Swanson, "A new look at the discrete-ordinate method for radiative-transfer calculations in aniso-tropically scattering atmospheres," J. Atmos. Sci. 38, 387-399(1981).

11. S. Chandrasekhar, Radiative Transfer (Dover, New York,1960) p. 12.

12. M. Tanaka and T. Nakajima, "Effects of oceanic turbidity andindex of refraction of hydrosols on the flux of solar radiation inthe atmosphere-ocean system," J. Quant. Spectrosc. Radiat.Transfer 18, 93-111 (1977).

13. G. Plass, T. Humphreys, and G. Kattawar, "Ocean-atmo-sphere interface: its influence on radiation," Appl. Opt. 20,917-931 (1981).

14. W. J. Wiscombe, "The delta-M method: rapid yet accurate

radiative-flux calculations for strongly asymmetric phase func-tions," J. Atmos. Sci. 34, 1408-1422 (1977).

15. R. Stavn, R. Schiebe, and C. Gallegos, "Optical controls on theradiant-energy dynamics of the air/water interface: the aver-age cosine and the absorption coefficient," in Ocean Optics VII,M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 489,62-67 (1984).

16. R. A. McClatchey, R. W. Fenn, J. E. A. Selby, F. E. Volz, andJ. S. Garing, Rep. AFCRL-72-0497, (Air Force CambridgeResearch Laboratories, Bedford, Mass., 1972).

17. R. C. Smith and K. S. Baker, "Optical properties of the clearestnatural waters," Appl. Opt. 20, 177-184 (1981).

18. A. Morel and R. C. Smith, "Terminology and units in opticaloceanography," Mar. Geol. 5, 335-349 (1982).

19. C. Mobley, B. Gentili, H. Gordon, Z. Jin, G. Kattawar, A Morel,P. Reinersman, K. Stamnes, and R. Stavn, "Comparison ofnumerical models for computing underwater light fields,"Appl. Opt. (to be published).


radiative transfer in nonuniformly refracting layered ... · the radiative-transfer model we...

Documents