Some exact solutions for coupled atmosphere-surface radiation models

Pergamon

J. Quant. Spectrosc. Radiat. Transfer Vol. 56, No. I, pp. 47-55, 1996 Couvriaht 0 1996 Elsevier Science Ltd Printed’k &ea;Britain. All rights reserved SOO22-4073(%)00023-4 0022-4073196 $15.00 + 0.00

SOME EXACT SOLUTIONS FOR COUPLED ATMOSPHERE-SURFACE RADIATION MODELS J. J. SETTLE Environmental System ScienceCentre, The University of Reading, Whiteknights, Reading RG6 6AB, U.K. (Received 8 November 1995)

Abstract-The problem of a plane parallel atmosphere bounded below by a reflecting surface is considered. It is well known that when that surface reflects isotropically, the radiation distribution emerging from the top of the atmosphere may be expressed in terms of the scattering and transmission functions of that atmosphere. For more general non-Lambertian surfaces this is not usually the case. However, it will be shown that for a certain class of surfaces this can still be done, and the problem reduced to evaluating certain integrals; the integrands of which are the product of ground reflectance terms and either the scattering or transmission functions of the atmosphere. The models described in this note were derived in order to provide “closed form” solutions with which to validate coupled atmospheric-surface radiation code, but they have a certain intrinsic interest beyond that, and may find application in sensitivity studies of retrieval algorithms for data from multi-view satellite instruments. Copyright 0 1996 Elsevier Science Ltd

INTRODUCTION

The study of the propagation of radiation through the Earth’s atmosphere has a long pedigree, but the study of the reflection properties of the Earth’s surface is less well advanced. In consequence, computer models that attempted to predict the intensity of reflected sunlight at the top of the atmosphere have, until recently, consisted of a fairly sophisticated atmospheric code coupled with poor representations of surface reflectance. A recent model, 654’ has improved on earlier representations with the inclusion of a number of more realistic surface reflection sub-models. These include the models of Minnaert,* Hapke,3 Verstraete et aL4 and Iaquinta and Pinty’ as well as the empirical function of Walthall et al6 which was claimed by its authors to be a fair approximation to the reflectance distribution of a range of cover types. Future satellite missions proposed for studying the land surface will aim to take account of the bi-directional nature of land surface reflectance and perhaps even to take advantage of it. Coupled atmosphere-surface radiation models will form an important element in the exploitation of the data that these instruments will generate. However, the validation of those models may present a problem. Mathematical models, as opposed to numerical models of the coupled system, usually involve simplifications for some part of the system (e.g., the CSAR model of Raman et al7 parameterizes some of the interactions) and cannot be used to validate numerical methods. The top-of-atmosphere (TOA) radiances can be predicted accurately when the land surface is Lambertian, for Chandrasekhar (Ref. 8, Sec. 72.1) has shown how that problem can be solved in terms of the “standard problem”, that is, the problem for an atmosphere bounded below by a non-reflecting surface. His approach appears to fail for an arbitrary surface reflection function, and it seems that his solution for the Lambertian surface is the only known solution in closed form for the coupled model. In this paper a new family of closed-form solutions is presented which are not much more complex than the Chandrasekhar solution, but which take into account a wide variety of surface reflection functions. These are not intended to be especially realistic realisations of a coupled system, but simple models which can be used to help verify the complicated computer programs that are required to investigate more complicated, more realistic models. 47

48

J. J. Settle SCATTERING

AND

TRANSMISSION

FUNCTIONS

The basic co-ordinate system used is polar, with f3(=cos-‘(p)) denoting the zenith angle, and 4 the azimuth angle. The shorthand o will be used to denote the direction (13,4) and do the element of solid angle, dp d4. To avoid ambiguity over the sign of p for upwards- and downwards-directed radiation quantities, p will be restricted to lie in the range [O,l] and the direction indicated by a superscripted arrow. Integrals over the hemisphere will be denoted thus (1) Consider a one-dimensional atmosphere of finite optical thickness. If details of the radiance distribution within that slab are not needed, but just those at top and bottom, then it is usual to refer to the scattering and transmission functions (S and T) of the atmosphere. There is a certain lack of uniformity in the literature of the definitions of these functions, and Chandrasekhar’s (Ref. 8, Sec. 13) will be followed in this note. Thus, if light of intensity Zinc(o) is incident on one side of a slab, the distribution of reflected light on that same side occasioned by scattering within the atmosphere is S(W; o')Z~~,(O') dw’, and the distribution

(2)

of diffusely transmitted light on the other side of the slab is given by (3)

This assumes that there is no reflection at the lower boundary; it is with the modification to (2) in the presence of a proper surface reflection that this paper is concerned. If the atmosphere is vertically homogeneous, i.e., if its single scattering albedo and scattering phase function are independent of optical depth, then it does not matter which side of the slab is illuminated; the same scattering and transmission functions apply for both directions. In this case, these quantities, which are functions of four variables, can often be expressed in terms of functions of a single variable only (Chandrasekhar’s X and Y functions of p and trigonometric functions of azimuth angle). However, if the scattering properties vary with depth in the atmosphere then the usual two functions are insufficient, and different scattering functions are needed for “top” (St(o, ; wz)) and “bottom” (Sl(o, ; q)) of the atmosphere. Similarly, the transmission function from the top to the bottom (Tl(o, ; q)) and bottom to the top (T f(o, ; q)) are different. However, they are related through the reciprocity principle, which gives us S’(0, ; 02) = Sf(co*; co,);

9(0, ; 02) =

sqw,;0,);

Tf(w,; 02) = Tl(o*; 0,).

(4)

The T functions only account for diffuse transmission, and light is also directly transmitted. For the problem at hand, the T functions are sometimes less convenient than functions Gf and GL, which account for all transmission and are defined thus 1 GJ(,; w’) = G1@L,#J; CL’,4 ‘) = 4nC1T’(P, 4; P’, 4 ‘) +exP(-z/C1)6b-p’)6(4

-~'>~'G'(o';o)=~~Gl(o;w'), (5)

where 6 (.) is Dirac’s delta-function and T the optical depth. G defines the intensity of all transmitted light, both diffuse and direct.

Coupled atmospheresurface TOP-OF-ATMOSPHERE

AND

49

radiation models SURFACE

RADIANCES

I assume that the atmosphere is illuminated at the top by a collimated beam of radiance I., from the direction o,, = (p,,, do). The incident and reflected radiance distributions at the ground are denoted by Z:(w) and Zb(w) respectively. These are related by the Bidirectional Reflectance Distribution Function (BRDF) f, a function of four angles. As defined by Nicodemus et al,9 the BRDF is a differential quantity, but its most intuitive expression is possibly in the following form, where it specifically links the radiance distributions of the incoming and reflected radiation:

Zi(~r)=

s

f(wr; ~i)Z~(~i)Pid~i.

(6)

4

Note that in the definitions of all these functions of two directions (S, T, G, f), the first argument is the direction of the emergent light, and the second that of the incident beam. The reciprocity principle requires that f satisfy:

fCwr; =f(wi;or).

(7)

Oi>

Following Chandrasekhar, the emergent radiation at the top of the atmosphere is expressed as the sum of three components. These are (i) the path radiance, consisting of light that has been scattered in the atmosphere and has not reached the ground, (ii) light reflected from the ground transmitted directly towards the sensor, and (iii) light that leaves the ground and is scattered in the atmosphere on its way to the sensor. These are given by the first, second and third terms respectively on the right hand side of the equation: ZT~,(O; coo> = LZeST(w; 47W

oo) + Zi(o)e-“p

+

J47cp

Tf(o;

w')Z#o') do’,

s H'

which can be more compactly written as:

s

GT(o; w’)Z

5(co’) do’.

H'

The first term on the right hand side of (9) is the black-ground, or “standard” solution; the second term gives the surface related contribution. S and T (and hence G) may be considered known functions for a given atmosphere, and can be obtained in a closed form for simpler atmospheres. For these simpler atmospheres there will therefore be a closed form solution for the top-of-atmosphere radiance if a closed-form expression for the ground-leaving radiance can be found. This quantity is given in part by an expression similar to parts of (8), but taking account of multiple atmosphere-surface interactions means that the expression for ZLcan only be defined in terms of itself; an integral equation must be solved. AN

INTEGRAL

EQUATION

FOR

THE

GROUND

LEAVING

RADIANCE

The light that arrives at the ground has three main components. These are (i) the radiance transmitted directly to the ground from the extra-atmospheric source, (ii) light scattered by the atmosphere and transmitted diffusely towards the ground, and (iii) ground leaving radiance, reflected back down by the atmosphere. They are given by the respective terms on the right hand side of the following equation: Z/(0’) = Z.e-“Po6(p’

+-

-

po) 6(fj ’ - qbo> +

&

TLb’, 4 ‘3PO, 40)

1 S ‘(0 ‘; o “)Z;(w “) dw ” 471p’s H”

= Z.G$o’;

1 $(w’; w”)Z;(o”) dw”. 47rp’ s H”

co,,) + -

(10)

50

J. J. Settle

If this is multiplied by the product of the BRDF and the cosine of the incidence angle, and the result integrated over all incoming directions, the following integral equation satisfied by the ground leaving radiance is obtained: Z;(w) = I.

f(o; w’)G@‘; s H'

o&’

f(o; o’)S(o’;

do’ o”)Z@“)

do” do’.

(11)

If we define functions 0 and K by @(co; CD,,)=

f(w;

w’)G@‘;

o&u’ do’

s H' K(o;

co”)

=A jH,f(co; o’)Sl(o’;

co”) do’,

(12)

then it is seen that (11) is equivalent to finding a solution of: Z(0) = Z*Q(o; 00) +

K(o;w")Z(w") do”, s H”

(13)

which is formally an integral equation of the second kind, with kernel function K. In practice, a series of separate integral equations will have to be solved if the functions are expanded as Fourier series in the azimuth angle. The last term on the right hand side of (13) corresponds to light which has been reflected from the ground and then reflected back towards the surface by the atmosphere. It will frequently be small, especially for fairly dark targets beneath an optically thin atmosphere. If it is ignored there will often be little error in just feeding the function @ into the TOA radiance equation; with the BRDF and atmosphere known, this will give a good approximation, although a systematic underestimate, to the TOA radiance for a good many cases. Other simple approximations are possible. However, for precise calculations this integral equation has to be solved, and this is not usually possible in finite terms. A FAMILY There are a number of and the results can be Chandrasekhar’s solution dent of direction and may here we have

OF SOLUTIONS

cases in which the integral equation (11) can be solved in closed form, used to give closed form TOA solutions. One obvious example is for a Lambertian surface; the ground leaving radiance is then indepenbe found directly from (6). The Minnaert reflectance function* is another:

fCwi;

we>

=f(Pi9

4i;

CLe5 4,)

CC P?-‘Pt-‘Y

(14)

where K, the Minnaert constant, >O (K = 1 for a Lambertian surface). K w 0.25 has been found to represent adequately some vegetated surfaces in the visible and near infra-red portions of the spectrum. ‘w* It follows from (6) that the emergent radiation must be of the form z;(o) cc /Y’,

(15)

irrespective of the precise form of the surface irradiance distribution Zi. The integral equation is used merely to determine the constant of proportionality. The appropriate solution is substituted in (9) to give the TOA radiance in terms of the various S and G functions, and the function (p”- ‘). The solution is straightforward because of the simple factorization of the BRDF; the details are given later. Similar solutions arise whenever the kernel function K(o,co') is degenerate, that is when it can be expressed as a finite sum of terms each of which is a product of a function of w and a function of 0’. This will clearly be the case whenever the BRDF can itself be expressed as such a sum; the Minnaert surface is a particularly simple example. Not all reflectance distributions we might want to consider can be so expressed. Specular reflection is one such case, and the physically

Coupled atmosphere-surface

51

radiation models

based models of, for example, Hapke3 and Verstraete et al4 are others which cannot be decomposed in that way. The skeleton of the method consists in, first of all, expressing the BRDF thus:

5

_/X0;w’> =

(16)

xj(O>Y,(w').

j=l

Substituting this expression into (6) shows that the upwelling radiance can be expressed as the finite sum:

II(o) = I. f kj(O,)Xj(0). j= I

(17)

The ks are found as the solutions of the following sets of equations: k, - 2 A,,k, =

/=I

yj(o’)GL(w’;

(18)

co&’ do’ = ~+Q~li(o~),

s H'

say, where Aj,=&

yj(o’)SL(o’; ssH"

w”)x,(w”)

dw’ do”.

(19)

H'

The formal solution is simply: k = (I - A))‘$, giving Z;(o) =

Z.xT(u)(I - A)-'t&q,),

(20)

where x and + are vector functions, thejth components of which are respectively xj(o) and IJ?~(w,,). Thus the ground-leaving radiance is obtained in closed form, and the closed form TOA solution follows, being: Z,,,(o;

co,,) = &Z.ST(w;

q,) + ZsJl’(o,,)(I - A)-'

RECIPROCITY

AND

THE

TOA

s

G’(o, o’)x(w’)

do’.

N’

(21)

RADIANCE

We would expect that if the BRDF obeys the reciprocity principle then so will the TOA distribution for the combined atmosphere-surface model. This is not easy to show for the general case, but is straightforward when the BRDF takes the form (16). We first note that if the BRDF obeys reciprocity (7) and can be expressed as a finite sum of the form (16), then it is possible to re-write the expression as a finite sum of the form:

fCO;w’) =

2

(22)

X,(w)Xj(m'),

I

where N, < 3N. This follows because reciprocity implies:

./X0; 0’1=

5

xjCw>.Yj(w')

I

=

$

xj("'lYj(w)

Lx’Cw) + Vj(w)l[xj(w’)+YjCO’)l

:1{+Jq$)&$7i+,J~~

=-

I

(23)

52

J. J. Settle

Now we define a number of functions gj by: ~~(o’)G$o’;

gj(O) =

o)p’

dw’ = p

~~(w’)Gf(o; w’) do’.

(24)

JH’

Writing:

(25)

1: = kCkj(w,)Xj(w), we find the following equation satisfied by the functions k [equivalent to Eq. (18)] kj-IAuk,

=gj(Po)

or k = (I - A)-‘g,

(26)

where g is a vector function of o which ith component is g, (w), and the ijth element of the matrix A is: A,=

~,(w’)S1(o’; ssH” H’

o”)~j(w”) do’ dw”.

(27)

The TOA radiance is now easily shown to be: &A(m) = &ST@;

c-+,)+ Z*ClogT(W,)(I- A)-‘g(w).

(28)

The fact that A is symmetric shows immediately that pZToA(m;w,,) is symmetric in o and o,,, which is the appropriate expression of the reciprocity principle for the TOA radiances. THE

MINNAERT

MODEL

The Minnaert model was originally propounded to explain the observed variability of reflection from the lunar surface, but has been used in a number of studies of the Earth’s surface,‘S’3 usually to effect topographic correction of satellite imagery. It is a very simple model for which a Lambertian ground is a special case. The solution for a Minnaert surface will accordingly be couched in similar terms to those used by Chandrasekhar for the Lambertian surface solution. Let us specifically write: I f(Oi; W,) =-~u1-‘~L,K-‘, z;(W) = r/PI, (29) It where r is a constant to be determined, and 1 is a parameter related to the surface albedo (it is precisely the albedo when K = 1 and the surface is Lambertian). Let us also define the following functions: ” Sl(w; wo) d+.

S’O’(P,PO)= & s0

fcp= 2

’ /.Ps,(,u)d~

=.FPO

s0 2n

FO’(

p,

/Jo)

=

L 27t

TL(,;

00)

d4

s 0

I $0)

(~‘)V’(O’; 4rr s H’

T’O’(p’ 9,u)(p’)” dpLI= i.

=; s 0

s

o) dw’

Y,(P)= ; H,(II')PGl(W';W)dW'=~P-le-~+~.

(30)

Coupled atmosphere-surface

53

radiation models

Chandrasekhar’s functions t(p), s(p) y, (cl) and 5 (Ref. 8, Sec. 72) are equivalent to t,,( cc), so(p), y, (p) and &,,,in this nomenclature. Z.p,,y, (p,,) is the total irradiance at the bottom of the atmosphere above a black ground. Plugging the expressions (29) into (11) enables us to calculate the constant r; we find: (31) The TOA radiance is given by: I.

ZTOA(O)= G

41 ST@; 00) + (1 _ nsK_,,c_ ,) cLo~Y,(II)Y,(cL0)5

[

(32)

1

which reduces to Chandrasekhar’s It is a feature of non-Lambertian irradiation) depends on the surface isotropically illuminated with light reflected flux are given respectively

result for a Lambertian surfaces that the albedo illumination. If we can of intensity Zi,,, we find by:

surface when JC= 1. (the ratio of reflected flux to incident arrange for a Minnaert surface to be that the emergent radiation field and

s

4xJ.ZIso F’ = -2n4so p ~-1~ do _ K+l

(K + 1)”

H

(33)

while the downward flux, or irradiance is simple rcZiso.The “diffuse albedo” is therefore F;

41

(34)

adiff=E=(rc+

We can easily show that when the surface is illuminated with a collimated beam of intensity Zco,, from the direction (oO), the albedo (the “black sky albedo”) is then given by:

(35) For the problem at hand, with a planetary atmosphere, the irradiance is given by: Zf(o’)$

/~‘Gl(o’; o,J do’

do’ = I, sH

sH

ss

r +

4n:

=PJ*

S’(o’;w”)($‘)“-‘do’do”

H.

H’

Ylbd+l_ds

[

and so the albedo is:

Fd a=c=

s

r

1

~YL(PO)~o,KI K

_

I,K

I

(36)

3

pK-‘p do

H

=

Z;(w’)p’ dw’ sH 21

=Ic+1

Ylc(~o) 1 Yl(PO)U

-~~,-l,,-,)+~Y,(~o)~o,,-l

1 .

1 (37)

The factor ~y,(&/(l - nS; _ ,,c _ ,) is common to the surface albedo and that part of the TOA radiance that derive from surface reflectance; unfortunately there does not appear to be a simple way to use this fact to estimate directly surface albedo from satellite radiances.

J. J. Settle

54

ANOTHER

SIMPLE

MODEL

For the second example we shall take the simple BRDF defined by: f(0;

1b +b(p

+p’)+cpp’+dJ1_Sli&pcos(cfJ -$‘)I,

0’) = -

(38)

where a, 6, c and d are constants. For all but the largest values of zenith angles this is very similar to the empirical BRDF of Walthall et aJ6 and for certain combinations of the constants (b = 0 = d + c) is a function of the angle between the incident and emerging beams. The expression (38) may be represented as a sum of the form (16) in a number of ways, but our approach will be slightly different. We expand the scattering and transmission functions as Fourier series in (4 - &), defining functions S(O)and S(I), T(O)and T(‘) by: S’ = S’O’(cL, PO)- 4pfioJ77 +

c cosU(4-

J7z!

~oW”(P,

cos(4 - bo)S”‘(U PO)

PO)

j=2 T1

=

T’“‘(K

+

PO)

4ppoJ77

+

c cdA4

JI7&osG$

- #o)T”‘(& /Jo)

- ~O~lWP~ PO).

(39)

j=2

Our solution for the ground leaving radiance will clearly be of the form (40) for some functions k, , k, and k,. Solving for k, gives: k

-3-

where:

drlbo)

1 +dcr’

I q(p)

= e-+

+

JJ I

6=

0

WP’;

PL)P’U

-

~3

&’

s0 I ~‘~20

-

P:)U

-

&WPI;

p2)G’

G2,

(41)

0

while k, and k, are linear combinations k, (1 - a& -

of yl and y2, found from:

bJo’)

-

-k’@k+cfo’)+k,(l

k2Wo’

+

6’)

-bfo’-~6’)

=

ay’boo)

+

by20.4

=by’hd+~~2hJ

(42)

where the various fV and rj were defined at (30). Substituting back in Eq. (4) gives us I:(o). We can once again obtain, if we wish, expressions for the surface albedo under different conditions. The TOA radiances are easily shown to be given by:

h*(W) =

& Sf(w;00) +Z.A~[~+(b~-uc)S,,ly,(~)~l(~) + ZaAF[b

-(b2--c)So,l[y,(~)Y,(~o)+Yz(~)YI(~CLo)l

+ 1. Aff 1~ + (b2 - ~c)hJr2(~>r2(/d (43)

Coupled

atmosphere-surface

radiation

models

55

where A=[l-(&,,+26&,,+cS,,)+(ac-b*)($,,,S,,-a?;,)]-’.

(44)

Finally, if the atmosphere is assumed to be characterized by conservative scattering with Raleigh’s phase function, then the TOA radiance can be expressed in terms of various X and Y functions. The expressions of y, and yz are given by Chandrasekhar (Ref. 8, Sec. 64.3) and the function q is simply the Y function corresponding to the characteristic polynomial &*(l - p’). CONCLUSIONS It has been shown that exact solutions can sometimes be found to the problem of determining the emergent radiance distribution from a plane parallel atmosphere, bounded below by a non-Lambertian reflecting surface. These solutions are obtained by first solving an integral equation for upward radiance at the lower boundary. This cannot usually be solved in closed form, but such solutions always exist when the integral equation is degenerate. The coupled problem then reduces to solving the corresponding problem for the atmosphere alone, and then evaluating certain straightforward integrals involving the scattering and transmission functions of the atmosphere, and functions defining the surface reflectance. Solutions were presented for the Minnaert surface and another of practical importance. Acknowledgemenf-The the United Kingdom.

author

is supported

under contract

F60/G6/12

of the Natural

Environment

Research

Council

of

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