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The application of the principles of invariance to the radiative transfer equation in plant canopies
J. Quanr. Sptwrosc. Radiar. Transfer Vol. 48, No. 3, pp. 321-339, 1992 Printed in Great Britain. All rights reserved
0022-4073/92 $5.00 + 0.00 Copyright 0 1992 Pergamon Press Ltd
THE APPLICATION OF THE PRINCIPLES OF INVARIANCE TO THE RADIATIVE TRANSFER EQUATION IN PLANT CANOPIES B. D.
GANAPOL?
and R. B.
MYNENI
Hydrological Sciences Branch, Mail Code 974, NASA-Goddard
Space Flight Center, Greenbelt,
MD 20771, U.S.A. (Received
28 October
1991; received for publicution
27 February
1992)
Abstract-Solutions of the radiative transfer equation describing photon interactions with vegetation canopies are important in remote sensing since they provide the canopy reflectance distribution required in the interpretation of satellite acquired information. The general one-dimensional two-angle transport problem for a finite copy of arbitrary leaf angle distribution is considered. Analytical solutions are obtained in terms of generalized Chandrasekhar’s X- and Y-functions by invoking the principles of invariance. A critical step in the formulation involves the decomposition of the integral of the scattering phase function into a product of known functions of the incident and scattered photon directions. Several simplified cases previously considered in the literature are derived from the generalized solution, Various symmetries obeyed by the scattering operator and reciprocity relations are formally proved.
1. INTRODUCTION
The interaction of photons from the sun with the light harvesting elements of plant canopies is one of the most complex biophysical processes in nature. The complexity stems from the photon-atom scattering interactions which determine the photon direction as dictated by the canopy architecture and phytoelement morphology. These photon interactions are central to photosynthesis where radiant energy is transformed into chemical energy by leaves within the canopy. A reliable characterization of how photons penetrate into the canopy is paramount in developing an understanding of which attributes of the canopy (architecture, species, etc.) significantly influence photosynthesis. For example, how influential is the canopy architecture, which governs phytoelement shadowing and orientation, in determining the extent of photon penetration into the canopy. Currently, there exist several models of the radiative transfer process within a canopy’ including geometrical optics, the turbid medium approximation, the hybrid turbid-geometrical optics approach and stochastic simulation. Here, we will be exclusively concerned with the turbid medium approximation, however, from a different perspective than is usually taken. During the past years, an intensive effort has been mounted to solve the radiative transfer (RT) equation for the one-dimensional turbid medium approximation in a plant canopy using analytical methods previously applied to the conventional RT equation. The motivation behind this effort is primarily to develop a model that allows the isolation of particular biophysical phenomena and the determination of the influencing factors. An additional motivation is to provide a highly accurate determination of the bidirectional reflectance factor (BDRF) characterizing the canopy response which will also serve as a benchmark to compare to other more approximate numerical solutions of the RT equation. An essential aspect in specifying the analytical solution to the RT equation is in forming the RT equation such that it is mathematically tractable yet physically descriptive. For this reason, only bi-Lambertian leaf scattering with equal or unequal leaf reflectance and transmittance* without a specular component is assumed in the analytical investigation to be described. tpermanent address: Department of Nuclear and Energy Engineering, University of Arizona, Tucson, AZ 85721, U.S.A. 321
B. D. GANAPOL and
322
R. B.
MYNENI
The initial phase of the analytical effort concentrated on the azimuthally-integrated problem or, as commonly referred to in the literature, * the one-angle problem. The first model was a dense canopy with soil not participating, and where all leaves were assumed to be oriented at the same inclination, but azimuthally symmetric. The method of Chandrasekhar,3 invoking the principles of invariance, provided an efficient solution in terms of functions similar to the well known H-functions.4 Then, employing a quadrature approximation for the area scattering phase function, the RT equation for a canopy with a general leaf angle distribution (LAD) and for multiple species was solved by generalization of the H-functions.5 The BDRF for a single leaf angle canopy (SLAC) model with a Lambertian reflecting soil was also found using the principles of invariance again yielding expressions in terms of functions similar to Chandrasekhar’s X- and Y-functions.6 Recently,’ a major advance was made for solving the RT equation with a general scattering phase function by application of the FN method. In the second phase of the effort,’ the canopy with azimuthal variation of the radiance resulting from the scattering phase function and/or the LAD was considered. Here, as in the one-angle investigation, the first problem attempted was for a dense SLAC with equal leaf reflectance and transmittance again using Chandrasekhar’s formulation.a The purpose of this investigation is to extend Chandrasekhar’s solution theoretically to the general one-dimensional two-angle problem for a general LAD in a finite canopy. The solution hinges on the decomposition of the integral of the leaf area scattering phase function into a product of known functions of the incoming and scattered photon directions. Then, from the principles of invariance as presented by Busbridge,’ a solution in terms of generalized Chandrasekhar’s X- and Y-functions is obtained. Particular cases previously considered in the literature are then derived from this comprehensive solution. The analytical solutions developed will be of use in numerically verifying S, codes operational in the optical remote sensing arena” as has been done previously for the one-angle problem.7 2. SOLUTION
WITH
A LAMBERTIAN
REFLECTING
BOUNDARY
The radiative transfer equation for a plant canopy with leaf area index (total optical path; A) bordering a Lambertian soil with reflectance rs is
with the boundary
conditions
Z(A, 8) = ;
dQ’p’Z(A, Q’), s 277
p < 0.
Here, z is the optical depth, Q the solid angle [Q( CL,4)] and Z the leaf-area scattering phase function defined by2 Z(Q’, 0) = ]ri, Z -(a’, Q) + t, Z +(Q’, Q)l with f ‘(Q’,Q)=
&
s
dS_2,g,tO,)trC1’.n,)tsz.n,>,
(2’4
*n(c)
where the + indicates that the integration is over that part of the range where the integrand is positive (+) or negative (-) only. In Eq. (2b), g, is the probability density of the leaf normal orientation and, rL, tL are the leaf reflectance and transmittance respectively; the single scatter albedo w, is rL + tL. The area intercept function G is defined by (3)
Applying princibles of invariance to the RT equation in plant canopies
323
In Sec. 4, it is shown that Z obeys the following symmetry relations required for the principles of invariance:
T(Q’, Q) = r(Q’,
rtn, Q’>,
Q) = r( -@,
(4a) VW
-0).
From Eq. (3), one can easily see that G(Q) =
G(4).
(5)
For mathematical tractability, the LAD gL is assumed spatially homogeneous and FL is a known surface illumination. By superposition, the RT problem (1) can be decomposed into the following two problems: Problem 1
J
w?)=~
6’1,
(64
4n
4 (0, 0) = FL@), Z,(A,Q)=O,
P >
0,
.
(6b)
p
(6c)
Problem 2
J
Z2(A,Q)=S,,z~
dSZ’p‘Z(A, a’),
2n
(74
p < 0.
When Eqs. (6) and (7) are added, Eq. (1) results and, therefore, from its uniqueness, Z(T, Q) = Z,(r, 0) + Z,(7, Q).
(8)
If we assume the (Green’s function) solution to [p;+G(Z))]I(r,n;g)=i
J4~dn.~(~/,~)Z(r,n.;~),
Z(O,Q; 4) = &(Q - Qo), Z(A, Q a)
= 0,
P > 0,
(9a) (9b)
P < 0,
(9c)
is known where & is the delta function specifying the point (p,,, do) on the unit sphere, then multiplying Eq. (6) by FL, integrating the upper hemisphere of directions (Qzo,with & replaced by Q”) and applying the uniqueness of the RT equation, we have for the solution of problem 1 Z,(7, n) =
J
dQ”F,(Q”)Z(z, Q; n”).
(10)
2n
If T, Q and Q, are replaced by A - 7, -Q (cf. Ref. 5), there results r 1
QSRT 4.8/3-a
a pat+G(Q)
1
Z(A-7,
-Q; -Q,,)=!
and -Q,
A
Z(A,
-Q2;
-Q,)
=
&(O
Z(0,
-Q;
-QJ)
=
0,
J
in Eq. (9) and noting G is invariant
dQ’T(-Q’,
-Q)Z(A - T, -Q’; -Q,)i
(1 la)
0,
(1 lb)
4n
-Q,),
p
<
P
0.
>
(1
w
B. D. GANAPOL and
324
R. B. MYNENI
Multiplying by S, and integrating over the lower replaced by -&I”), we again find by uniqueness I,(r, Q) = & Therefore,
the solution
dQ”Z(A - z, -8; s Zn
in terms of the Green’s dQ”FL(Q”)Z(r,
I(? Q) =
hemisphere
function
Q, (- s_Z,, with Q,
a”).
(12)
is dQ”Z(A - 5, -Q; s 277
Q; Q”) + S,
s and we find from the definition
of directions
a”)
(13)
of S,,,
-r, de”&_ (0”) dQ’p’I(A. 71 Zn 2n____ :=--j s n 1-s 71 s ?n
dQ’p ’
9’; Q”) (14)
dQ”I(0, Q’; Q”) s
2n
Thus, from Eqs. (13) and (14) if I(0, -Q; Q,) and Z(A, Q; Q,) can be determined, then, the radiances I(0, Q) (or the BDRF) and Z(A, Q) as well as the reflection and transmission coefficients can be obtained. In the next section, the explicit determination of Z(0, -Q; 9”) and Z(A, Q; f&) is presented. 3. DETERMINATION
OF I(0, PRINCIPLES
3.1. Integro-d@erential (I) Integral
transport
-Q; Q,,) AND I(A, OF INVARIANCE
Equation for the Scattering
Q; Qr,)
FROM
THE
Source Function
equation
Tracking the photon along integral form for p > 0:
its trajectory
allows the reformulation
‘ds’exp[-(r !J I 0
I(r,R;n,)=fiz(S1-~,)exp(-z/~,)+l
s
of Eq. (9) into the following
-~‘)/5(sz)]J(t’,-n;~~),
(15a)
A
dT’ exp[-(7’
-
(15b)
~)/5(Q)lJ(z’, $2;%),
T
where
s 4n
In deriving Eq. (15) A, it follows that
(16)
dQ’r ( - n’, @l(T, n’; a,)
use was made of the symmetries
of r given by Eq. (4). Thus,
at t = 0 and
(174 0Adt ’ exp[ - (A -
Z(A, Q; &I,) = a,(!2 - !&)exp( - A/&) h s These relations
will be used extensively
in what
follows.
z’)/{]J(t ‘,
-Q;
Q,)
(I7b)
RT equation
Applying principles of invariance to the
325
in plant canopies
By forming J from Eq. (15) as
Jb, Q; Qd =
~~(-4,Q)exp(-T/to)
s 2x
s A
WW?‘, Q)+
dr’exp[-(r’bdr’exp[-(r
dQ’T(-Q’,Q);f; s
with t’ s r(Q)
t)/<‘]J(r’,
-Q’; Q,)
t
27
s
-
qlr’lw,
-0’; Qo)
(184
and upon defining
otherwise, which implies K(-L
(18~)
-Q) = r&t, Q,
Eq. (18a) becomes 11 - &JJ(r’, The operator
Sz’;r30) = i W-a,
(19)
Q)exp(-r/L).
L,* has been defined as A
L.,o=; and obeys several important
s In
(jsz’rK2’9
dr’K(t’ - r, Q)(a)
n)
(20)
s0
symmetry properties outlined in Appendix 1.
(2) Derivative of Eq. (19) with respect to r Taking a derivative of the operator ;
L,,J(G
n;
Ml = L&W, +; --
s tn 1
As Thus,
operating on J, we find (after some manipulation)
L,,
Q’;4) dO’T(-O’, RM, Q’W, -0’; 00) d@T(Q’, O)K(A - 7,Q’)J(A, Q’; Q,).
2n
taking a derivative of both sides of Eq. (19) results in
dQ’r(--n,, ??)K(?@MO, -n’; @I) --
I
n
s
dQ’T(Q’, Q)K(A - z,o’jJ(A, Q’; a).
(21)
2n
(3) Alternative representation of the rhs of Eq. (21) The first term on the rhs of Eq. (21) is T, =i and incorporating
lr I 2x
the definition of
dQ’r( -3’, @K(r, @)J(O, -n’; a); K
given by Eq. (18b), T, becomes
1 .
T, = s 2x
(22)
326
B. D. GANAPOL
After substitution o’-+U=U(PL,~)l
and R. B. MYNENI
of Eq. (19) for the term in the square
s s
bracket,
there
results [letting
dU ; J(r ‘, a’; U)J(O, -U; Q-,).
T, = [l - J&l
(23)
2n
Similarly, the second term on the rhs of Eq. (21) is given by dU;J(A-r’,
7’2= [l - ~,,I
-Q’; U)J(A, U; Q,).
(24)
2n
(4) Integro-differential
equation for J
The desired integro-differential into Eq. (21) to give
t1 -&,,I
i
equation for J is obtained when Eqs. (23) and (24) are introduced
~J(r’,n’;p,)+~J(r~,R’;R,)-
s 2n
0
dU ; J(z’, a’; U)J(O, -U; Q,,)
dU;J(A-r’.
+
-Q’; U)J(A, U; Q,)
s 2n
= 0.
(25)
I
Thus, in order for this equation to be satisfied in general, the term in curly brackets must vanish9
[
1
&+;0
J(T,
Q; &lo)=
s$ da’
Jtz,
Q Q’)J(O,-a’; C&d
2n
-
s
da’ + J(A - r, -Q; Q’)J(A, 0’; 4).
(26)
2TI
3.2. Exiting Radiance Distributions Equation (26) can be straightforwardly integrated to obtain the exciting radiance distributions at r = 0 and A directly. To accomplish this, Eq. (26) is first multiplied by exp(-r/l) and then integrated over r on [0, A] with liberal use of Eq. (17) to give JtO,
$2;Qo)
+
s
dQ’J(0, -Q’; Q,)Z(O, - 0’; Q)
2n
-
s
1
dQ’J(A, a’; Q)I(A, g; Qo) .
2a
(27)
Similarly for Z(A, Q Q,), Eq. (26) is multiplied by exp[ -(A - r)/<] and integrated over r on [0, A] using Eq. (17) to give
x ZtA, 8’; Q) -
JtA, --Sri 4) +
s
dQ’J(A, Q’; &))I(O, -Q’; Q)
2n
11 .
(28)
In Eqs. (27) and (28), the reciprocity relations, PZ(O, -n;
Qo) = PoZ(0, -Qo; n),
P~(A, Q; slzo)
= PHI@,
Qo; Q,
(29a) Pb)
derived in Appendix 2 have been employed to simplify the resultant integrals in Eqs. (27) and (28). No further simplification can be made until the specific form of the leaf area scattering phase function is specified.
Applying principles of invariance to the RT equation in plant canopies 4.
FORM
OF
THE
AREA
SCATTERING
PHASE
327
FUNCTION
For bi-Lambertian leaf scattering model, the area scattering phase function is given by Eq. (2) with Z’* written in a more revealing form f*(Q’,R)=$
d%gt_(RL)lR’.QLI In.n,lo[+(n’.sr,)(~.R,)l (30) s 2n where 0 is the unit step function. From this form, the symmetry relations given by Eq. (4) are easily realized. Upon application of the identity O(f.~~)=O(+x)O(y)+O(Tx)O(-y), and Eq. (2a) becomes Z(Q’, n) = & where
a@,QL) = IQ~QLI[fL@tR-QL)+ rL@(-O.OL)l,
Wa)
Pa
WW
QL)
= In.QLl@(sz.nL).
Then, noting that a@,
QL)
= @C-Q,
-RL),
BeA
a,)
= PC-Q,
-8L),
Eq. (31) becomes f (Q’, Q) =
s 4n
where g,_ is
dQLgL(QL)a(a’, QLV (Q, BL>
EL (QL)
= &TL@L
(334
Wb)
)/27c
with i?L(fJL)
= kL(RL)@(PL)
(33c)
+gL(-.RL)@(-PL)l.
As will now be shown, a theoretical advantage of this form for Z is the decomposition integrand into a product of functions of Q’ and Q. 5. EXPRESSIONS
FOR
EXCITING
of the
RADIANCES
From Eq. (16) evaluated at z = 0 and A and using the symmetry of Z, we have
s
J(O,R;Ql)=+?&)+~ dn,r K?‘,WtO, 277
J@,
Q; Q,)
= ;
d$?‘r
(Q’,
- QW,
Q’;
4
-a’; f&J,
PW
),
s 2R
and substitution
(344
of Eq. (33a) shows that J(0, Q Q,) and J(A, Q a,) are of the form J(o,nW=;
s 4n
J@, Q; Q,) =;
s 4n
dQ,&(~,)P(Q,
Q,)~(Q,,
d~&.(~2,M(-rlr,
OLPWO,
R,_), OL),
(354 WW
with
YtSr, QL) = 4-G @@,QL)
QL)
+ s 2x
dR’a (Q’, QL)Z(O, - a’; R),
dQ’a(R’, Q,)Z(A, Q’; a).
= s 2n
(W t36b)
328
B. D. GANAFQL and R. B. MYNENI
Substitution
40,
of Eqs. (35) and (36) into Eqs. (27) and (28) gives
-o;w=&-&) 0
-S
dQd,(Qr,)
x Pf’(Qo, QLLW,
478
G_) - @(Qo, Q,)WL
I(A,R;~)=exp(-A/5,)62(~--~)+~~
4n
.S
x [‘YaO,
QL)W,
QL)
-
w&l,
QLMQ9
Q,ll,
(37)
QLifL(LIL)
(38)
RLII,
where
Jw,
rlr,)
Y(Q, a,)
= /m
QL)
dn’B(-W,
+
(39a)
Q,)l(O, -Q’; Q),
f 2n
=
@‘PC-n,,
QLW,
S?‘; Qr).
W’b)
s 2n
To obtain integral equations for the auxiliary functions Y, @, X and Y; Eqs. (37) and (38) are substituted into the Eqs. (36) and (39): wQL)=.(-c?,Q,)+; s 4n
x [Y(Q, f&)Y(nl, n;) - @(Q,n;)qr, dW,_(Rt) x(n,n,)=B(Q,8,)+~
Y&t Q,) = exp(-AlOB(-R,
QL) + i
x [Y(Q, &)Y(Q’, By direct substitution following:
f 2n
&.)I,
W -
r
B(-~,t?,)
5’+5
W?‘)
s s 4n
dW, OX,)
rlrl) - w2, ntbw’,
2n
dn, -
5
t’--t
8(-=9QL) G(Q’)
n;)l.
into Eq. (40), the auxiliary functions can be shown to be related by the
VQ, 0,) = ev( - A/t M -Q, QL),
VW
WQ, QL) = w(--A/Wu(--R,
@lb)
rlL).
In addition, moments relations can be shown to apply using the reciprocity radiance, if the moments are defined by Qa?L)
=
d&(+3,
dRZgL(W I 4n
s 2n
WWQ,
RL),
relation for the
(42)
Applying principles of invariance to the RT equation in plant canopies
329
where K = Ij/, @, Xor Y and y = a or /?. If y = P(N), then - Q( +Q) is specified for the first argument of y in the integrand. By applying reciprocity to Eqs. (36a) and (39a), adding and manipulating, the following moments relations can be derived:
-2Yy,(QL) + 2
X
s 3
d%‘gi. (CC’) s 4R s 2n
[a?,
sz;>pa
nt>
+
dR c@, QL) - G(R)
P -Q
&M-Q
WI
= 0. (43c)
2n W3
The BDRF for the original problem is obtained from Eq. (13) with 7 = 0 and 0 is replaced with -Q. Thus for a beam source impinging on the canopy upper surface [FL(a) = k5,(Q- a)], we have
where
dQ,&(Q,)
SN=~~exp(-A/&,)i-~
da’ 5’50
s 4n
5’-5
s 2n
x [Y(Qcl, QL)Y(R’, Q,) - @(Qoo,Q,MQ’, d&g,(&) s 4n
(44c)
(“5 de’ ___
dQ” s 2n
WI,
s 2n
5’+
5”
x W’@“, Q, P-W’, 0, >- WX Qt.>W’, G )I. (444 The determination of the BDRF therefore has been reduced to the solution of the integral equations for Y, @, X and Y [Eq. (40)]. While these equations seem far too complicated for application of analytical solution techniques such as the Wiener-Hopf approach,’ they can be solved numerically which will be the subject of a future effort. In any event, several simplified cases previously reported in the literature can be obtained. 6.
SPECIAL
CASES
For the following simplified cases, only the BDRF for the Green’s function, Eq. (37), will be obtained except in the last case. The distribution at the soil can be obtained as well using the auxiliary function for the particular problem. 6.1. Equal Leaf Rejectance
and Transmittance
For the case, rL=tL-Ws/2
(45)
and from Eq. (31)
s 2n
dRLgL(QL)IR’.RLl IQ+Q,l.
(46)
330
and R. B. MYNENI
B. D. GANAPOL
Thus
Wb) and therefore
Y@z,$2,) =
T wn, RL),
WW
W,Q,)=~ w&Q,).
WW
From Eq. (37)
Z(O,-am=-
to 50+
dnLgL(nL)[X(~O,RL)X(I_Z,S_Z,)s 277
m&43
Q,)l
Wo>Q,)W,
(49)
with
s
r
da
2n
x pm, nt)Y(Q’,
w - un, n;Mw,
-
t-‘-t
KQ’,QL>
G(R’)
nt11.
(51)
For this case, the moments relation Eq. (43a), becomes
s
dR;g,(W
We
2n
-
Y#X.)l& - 2Xo(R;) +2
Note the similarity to the conventional
dQkL(&) s 277
WW,
Q3-W
QWVQ)
s 2n
radiative transfer case.’
6.2. Azimuthally
Ungorm LAD
For this case, =gL(PL)r
(53)
G(Q) = G(P),
(54a)
r(Q) = 5(P)*
(54b)
sL(nL>
which implies
and Eq. (37) is then somewhat simplified to Z(0, -!a, Q-J =
where
50 t-o-
1
’ d~LgL(~L)[R,,(RO,R,~,)-R,,(~,n,
s -1
c1dl
(55)
331
Applying principles of invariance to the RT equation in plant canopies
with M and N representing
Y, #, X or Y. An integral equation for Rlx is obtained from Eq. (40~)
bd?o~ Q,PL)= cl&ot Ckc1d+;
1
s s ’
dfi;g,_(d)
-I
2n
da’
where $9v(Qo9 n,
PL)
(574
=
can be found from Eq. (40a) to be
x r;+(Q’, 3’3 PL)&X@O,Q/4) - &,@o, 9’3 &)I
Wb)
with 257 d4La(f%
~$9*Q,n4L)=
Q,)B(+O,
a,).
10 The additional transformations
functions ReY, RvY and R,, are obtained from Eqs. (56) and (57) using given by Eq. (41) and therefore will not be derived here. 6.3. Single Leaf Angle Canopy (IL = t‘)
Here all leaves are oriented along the same direction such as would be true for a heliotropic canopy. Then irL@L)
=
62@L
-
(58)
a*)
and Eq. (37) becomes (59) where
Jm
z$K(Q,Q*), K=Xor
Y.
(60)
8 and P are obtained from Eqs. (4Oc) and (40d) along with the following relations from Eq. (3)
‘39 =
&B(Q,Q*).
(61)
Therefore,
d(Q)= G(Q) + o,
s& dQ
mw(~)
-
m?m?')l,
(624
dfJ &
tRQ)P(o’)
- RQ)RQ’)l.
(62b)
2n
P(Q)= G(Q) + co,
s 2n
B. D. GANAPOLand R. B. MYNENI
332
For the moments
with K = X or Y,
(63)
1 =o
(W
we have c@;-
Pi]-2&+
where the normalization
s
dQG(Q) = f
2n
has been used. It should be noted how closely the results for the single leaf canopy resemble the conventional radiative transfer equation with rotationally invariant scattering. For this case, there also exists a transformation of the nonlinear form of the integral equations for .%?and P [Eq. (62)] to a linear form containing principal values. 6.4. Reduction If the LAD is azimuthally
to the One-angle
symmetric g,(R,)
it is shown by proposition
and by proposition
Problem
(65)
= gL(c(L)v
(A30) of Appendix 3 that
(A31)
Z(r, n;Qo)= From this relation and azimuthal periodicity,
IO,
I.c; PO)=
;
w, P, 4 -
40; Pd.
(67)
the one-angle radiance is
s 2n
W(t, Q; Qo,).
W)
In addition,
and integration
G(O) = G(P),
(@a)
5(Q) = 5(P),
Wb)
of Eq. (37) over 4 gives
where 2n s0
dWW,
QL)
V’W
and K is !I’, @, X or Y. To obtain Eq. (70b), the symmetry of K K(Q,
QL) = m4
PLY 4 -
$L)
(71)
Applying principles of invariance to the RT equation in plant canopies
333
proved by proposition (A32) of Appendix 3 was introduced. The integral equations for K(p, ,u~) are found from Eq. (40) by integrating over C$and noting Eq. (69):
x P”(P, P;)~(P’, ~3 - @(P,/~M/J’, ~ut)l, (72b)
x [Y(P, P~)x(P’,
it) - Q(P, ~ut)y(k,
put)],
(72~)
with
C(P,14_)= kH(,4 A) + rLW-h
14
(73)
and H is* if(-cotBcotO,)< if(-cotOcotfJ,)>
H(P, C(L)=
-1, 1,
(74)
otherwise,
where c#l,(p) = cos-’
WL
[ &G&K-a.
1
6.5, Conventional Radiative Transfer with Isotropic Scattering To recover the solution tL and
to the conventional
transfer equation
with isotropic
scattering let
rL =
&TL(crL) = m.
- P*x
(75)
Thus,
and
C’W
B. D. GANAPOLand R. B. MYNENI
334
If in addition, G(p) = 1 and o, = o/2, then Eq. (59) becomes the conventional Z(0, -p,
/A)) =
w--k2n
~
+
~
result
W/Jo)X(P) - mh)y(P)I
and from Eq. (38)
(77b) with the usual X and Y functions defined by x(/4=1+;, s
0’s
[X(/J)X(p’) -
VPL)W’)l~
W)=eM-A/r)+:r I.
SUMMARY
The general one-dimensional two-angle photon transport problem for a finite vegetation canopy of arbitrary leaf angle distribution is considered. Analytical solutions are obtained in terms of generalized Chandrasekhar’s X- and Y-functions by invoking the principles of invariance. The solution hinges on the decomposition of the leaf area scattering phase function into a product of known functions of the incident and scattered photon directions. Several simplified cases previously considered in the literature are then derived from the generalized solution. Iterative algorithms will be developed to evaluate the defining integral equations for the Chandrasekhar’s functions in a future effort. Acknowledgemenrs-This work was made possible with the support of G. Asrar of NASA Headquarters. BDG warmly acknowledges his support and that of the NASA-ASEE Summer Faculty Fellowship Program and H. Boroson, the program coordinator. Participation of RBM was made possible through NASA grant NASS-30442.
REFERENCES N. Goel, Remote St ns. Revs 4, 1 (1988). J. K. Shultis and R. B. Myneni, JQSRT 39, 115 (1988). S. Chandrasekhar, Radiative Transfer, Dover, New York, NY (1960). B. Ganapol, TTSP 18, 475 (1990). B. Ganapol, NASA Ames Research Center Report (1992). B. Ganapol, TTSP, 19, 56 (1990). B. Ganapol and R. B. Myneni, Remote Sens. Envir. 39, 213 (1992). B. Ganapol, Proc. 11th Transport Theory Conf., Virginia Polytechnic and State University, Blacksburg, VA (1989). 9. I. Busbridge, The Mathematics of Radiative Transfer, Cambridge, Univ. Press, UK (1960). 10. R. B. Myneni, G. Asrar, and S. A. W. Gerstl, TTSP 19, 1 (1990). 1. 2. 3. 4. 5. 6. 7. 8.
APPENDIX 1 Symmetries of the Operator L,, (i) Proposition
(Al) integral symmetry:
Adr@(r)L,o{~(Q’, s0 Proof: we consider the integral
-Q4W}
=
dW~)L,,(WX,
and introduce the definition of L,,
AdrX(W,,~{W’, s0
-Q)@(r’)}.
(Al)
(W
-rlzoFW’)}
with a change of variable (Q’ + -Q’), we have drx(r - T’; Q’)r(Q’)J-(Q’,
-Q)@(r)
Applying principles of invariance to the RT equation in plant canopies
335
where Eqs. (18~) and (4b) have been used. Thus, again from the definition of L,, , A
K=
dN~)L,,,(W’,
-O)@(O)
643)
f0 and the proposition is proved. (ii) Proposition (A2), translational
symmetry:
L,+,@(A - T‘, -0’)
= LA_,-Q@(“,
@).
644)
Proof: consider the relation U 3 L,,@(A
- t’, --a’)
and from Eq. (20) A U=
dr’@’
dfJ'UQ',Q)
(A9
- r, Q’)@(A - r’, -Q’).
s0
s 4n With the change of variables
Q’-, -Q’, r”+A-T’, Eq. (A5) becomes [cf. Eq. (18c)] A
dr”rc[t” - (A - T), &I’]@(?‘,a’). s 4n s0 From Eqs. (20) and (4b) the proposition follows. (iii) Proposition (A3), repeated application of integral symmetry: d@f (-&I’, Q)
U=
646)
A
dz@(z)L;,,[l-(a’,
-Q,)X(r’)]
=
s0
A d7X(r)L:,00U2’, s0
-OP(~?l,
647)
where v = 1,2,. . . , L::,,=L,,
Lr,.,,
J&O? . _ ...
648)
L,!&_,* ” .
Proof: the proof will follow by induction. From Proposition Next assume the conjecture true for v - 1
(Al) the conjecture is true for v = 1.
A
dt@(t)L:,‘[f
(a’, -Q,)X(r’)J
s0
=
AdtX(r)L;.$[T(Q’, s0
-Q@(r’)].
(A9)
Now consider A
dr@(r)L:,o[r(O’, -GW’(~‘)l= ‘dr~(r)L~,,‘L,,,~,[r(~‘, --GM~‘)1. s0 f0 From the definition of L,,,,, and with some rearrangement, change of variable (0 + -0’) application of Eq. (18~) K=
K = JoAdr’J’(r’) 6. dCl’f(Q’, 4) JoAdr@(r)LZ,‘[r(Q’, Upon application
, -Q)lc(t,
(AJO) and
- T’, Q’)].
of (AS), we have
K = JoAdz’X(t’) 6. dQ’r(@, Q,) JoAdvc(z - r’, Q’>L:,![l-(Q,
-Q)@(r,)].
(Al 1)
Finally, from the definition of L,, A
K= s0
A
d7sX(z)L,,,L;;-d.[T(~‘,,
-QP(~,11=
dWr)L:,c$W’, s0
-8M~‘)l.
(A121
336
(iv) Proposition
B. D. GANAPOL
and R. B. MYNENI
(A4), repeated application of translational L:,o[@(A - r’, --&I’>]= Li-,,
Proof: again the proof is inductive. From proposition the proposition true for v - 1 L;,;‘[@(A - t’, -!J)]
symmetry:
-Q[@(T’, Q’)].
(A13)
(A2), the above is true for v = 1. Assume
= LLl1,. _&P(t’, q.
(A14)
Consider K z L;.,@(A - t’, -a’)
or K = L;,<’ {L,.,J@(A - z”, -a”)]} and apply proposition
(Al3
(A2) to the term in the curly brackets to give K = L;,;’ {LA-_r., _g[@(T”, @‘)I>.
6416)
K = L;:‘,,
(Al7)
Then from Eq. (A14)
and the proposition
-&_,,,
_n~[@(z”, n”)]
is proved. APPENDIX
2
Reciprocity Relations
(i) Upper surface (z = 0) The Neuman series representation
for J can be obtained directly from Eq. (19) as
J(t, Q; f20)= f &Jr@‘, I =
-Gdexp(--~‘/to)l
(AIf
0
provided the series converges. To show this, we first assume that r is bounded F(O’, sz) < w. One can then show
Q’kv(-~‘&)I
L,o]F(-QoY
G WL,dIl
(Al9)
and since [cf. Eq. (18b)], K(Z, Q) < 1,
LT,,[~lG
dQ’r(Q’, Q) = w,G(Q) < o,
s 4n
where from Eq. (3), G(Q) < 1. Thus,
L.
n
[r (Q’, -
Q. >exp(- t ‘Ito )I s 0, W
WO)
and
&Jr@‘,
-%)exp(-r’/tO)l
G Wo;
implying that the series for J is bounded by
WV and is therefore convergent for 0, < 1. Since the exiting radiance at t = 0 is
soA
dt’exp(-t’/S)J(t’,
$2 Qd,
6422)
Applying principles of invariance to the RT equation in plant canopies
337
the convergent series for Jean be substituted into Eq. (A22) with the integral and sum interchanged to give M(O, -n; Therefore
00) = c
* dr exp(-rlW:o[r(S?‘9
from proposition
(~23)
-%)exp(--r/Ml.
Y s0
(A3)
MO, -Q; Qo) = c
-O)ex~(-r’/t11
* dr exp(-rl~&u,,[F(Q’,
Y 10
which is identical to the sum in Eq. (A18) with Q and Q, interchanged. MO, -n;
(A24)
Thus (A25)
Qo) = /.klJ(O, -Qo; Q)
(ii) Lower surface (T = A) At t =A, Z(A,R;W)=s*(B-Q)exp(-Alt,)+t
and upon substitution
s
oAdrexp[-(A-r)irlJ(r,
-3; Qd
(A26)
of Eq. (Al8), we have
~Z(A,~_2;~)=~062(~-~)exp(-Alto)
s
+c Y
A
dt ev[--(A - WIG-,PW’,
-Q&W-~‘/to)l
WV
0
or letting r be A - T in the integral MA, Q; C&,) = clo4 (Q - Qo)exp( - No) A +c
From proposition
dr
I I0
--C?&x~(--~‘/t~11. WW
exr{-~lr)L~-,-~[T(~,
(A4), the sum becomes - O/tol)
*dr ,x~(-~l~)~~.,{~(~‘,~)ex~[-(A ?IY 0 and from proposition
(A3) A
” 0
d7
exp[--(A
-
~Y~ol~: +,F(&?‘, -Qhp(-r’/Ol
?I
and comparing to Eq. (A26) with Q and Q, interchanged /W,
gives
Q; G,) = iMA, GG,zo; Q).
(A291
Note that the delta functions cancel. APPENDIX
3
Propositions for Reduction to the One-angle Problem (i) Proposition
(AS): if gl_(G_) = 8L(k)
(A30)
then F(@,O)=W’,/&$
-4’).
(A3L)
B. D. GANAPOL and R. B. MYNENI
338
Proof: since’
J~J~
(a *QL ) = WL +
CM4 - 4L 1,
from Eq. (32)
a@,QL) = ah4 PL, 4 -
4L)9
(A32a)
B(Q,
4L)r
(A32b)
QL)
=
Bh
PL,
4
-
and in addition a and /_Iare azimuthally periodic. Then from Eq. (33) with a change of variables I r(n’,
Q) =
d‘ ch.&(d
s -I
Because of the periodicity
s @- 2n
duab’,
L(~, u)LG
clLI u + 4 - 9’).
(A33)
of the integrand of the inner integral, Eq. (A33) can be written as I
rcO’,
n)
=
2’dua(~‘.~L,u)B(~,~L,u+~ f 0
d~Lh,(pL)
s -I
-4’)
and the proposition follows. (ii) Proposition (A6): if tcL@L)
=
tTL(PL)
then (A34a) and W, Q; Qo) Proof: consider the representation
= I(&
~134
-
90;
/.+,I.
(A34b)
for J given by Eq. (A18)
J(r, Q; G) = exp(-r/
-t f
1’= I
L:,[T(-~2,,n’)exp(-r’/ro)l.
(A35)
4 - 40 - n)
(A36)
Since
J-(-$20, Q) = from proposition
u-p,,
P,
(A5) and Eq. (20), it can be shown that xl = &J~(-s_Z~
B’)ev(-~‘/50)1 = 4b.b P, 4 - 40).
(A37)
Then assuming (A38) we have x, =
LI,[f(-~,R’)exp(-r’/r,)l
=&J~“-l(PQ,P’~4
-4011
where it can again be shown from the definition of L,, [cf. Eq. (20)] that L,o]r(-&,
@)exp(-r’/M
= k(cco, k 4 - 40).
(A39)
Thus from Eqs. (A36) and (A39) J(r, Q; Qo) = J(r, /Ju,4 - $0; I4I) and the proposition
follows from Eqs. (A22) and (A26)
(A40)
Applying
(3) Proposition
principles
of invariance
to the RT equation
in plant
canopies
339
(A7): if &_(QL.)= gI,(,%)
then K(QZ,Q,) = K(K
PL, 4
(A41)
- 4~)
where K = Y, @, X or Y. Proof: the proof will be performed for Y and Qi since with only minor modifications the proof follows for X and Y. From the defining equation for Y with explicit dependence on azimuthal angle indicated, Eq. (36a) becomes Y(O, OL) = a(-pu,
PLY 4
- 4L + x> 2n +
wwhL,
G' S'0
4'-
4m
-PW-4
+m.
(~42)
s0
Then with the change of variable, u = 4’ - c$~, 2n 'Y(~,OL)=cr(-/l,~L,~-~L+n)+
&'
S’0
dMp',~L,uYP,
-P',u
-(do
-4~)+n;
s0 =Y(P,PL,~-~L)
where the periodicity of a and Z have been noted. Similarly, for CDfrom Eq. (36b)
QSRT 48,3--H
PI (A43)
0022-4073/92 $5.00 + 0.00 Copyright 0 1992 Pergamon Press Ltd
THE APPLICATION OF THE PRINCIPLES OF INVARIANCE TO THE RADIATIVE TRANSFER EQUATION IN PLANT CANOPIES B. D.
GANAPOL?
and R. B.
MYNENI
Hydrological Sciences Branch, Mail Code 974, NASA-Goddard
Space Flight Center, Greenbelt,
MD 20771, U.S.A. (Received
28 October
1991; received for publicution
27 February
1992)
Abstract-Solutions of the radiative transfer equation describing photon interactions with vegetation canopies are important in remote sensing since they provide the canopy reflectance distribution required in the interpretation of satellite acquired information. The general one-dimensional two-angle transport problem for a finite copy of arbitrary leaf angle distribution is considered. Analytical solutions are obtained in terms of generalized Chandrasekhar’s X- and Y-functions by invoking the principles of invariance. A critical step in the formulation involves the decomposition of the integral of the scattering phase function into a product of known functions of the incident and scattered photon directions. Several simplified cases previously considered in the literature are derived from the generalized solution, Various symmetries obeyed by the scattering operator and reciprocity relations are formally proved.
1. INTRODUCTION
The interaction of photons from the sun with the light harvesting elements of plant canopies is one of the most complex biophysical processes in nature. The complexity stems from the photon-atom scattering interactions which determine the photon direction as dictated by the canopy architecture and phytoelement morphology. These photon interactions are central to photosynthesis where radiant energy is transformed into chemical energy by leaves within the canopy. A reliable characterization of how photons penetrate into the canopy is paramount in developing an understanding of which attributes of the canopy (architecture, species, etc.) significantly influence photosynthesis. For example, how influential is the canopy architecture, which governs phytoelement shadowing and orientation, in determining the extent of photon penetration into the canopy. Currently, there exist several models of the radiative transfer process within a canopy’ including geometrical optics, the turbid medium approximation, the hybrid turbid-geometrical optics approach and stochastic simulation. Here, we will be exclusively concerned with the turbid medium approximation, however, from a different perspective than is usually taken. During the past years, an intensive effort has been mounted to solve the radiative transfer (RT) equation for the one-dimensional turbid medium approximation in a plant canopy using analytical methods previously applied to the conventional RT equation. The motivation behind this effort is primarily to develop a model that allows the isolation of particular biophysical phenomena and the determination of the influencing factors. An additional motivation is to provide a highly accurate determination of the bidirectional reflectance factor (BDRF) characterizing the canopy response which will also serve as a benchmark to compare to other more approximate numerical solutions of the RT equation. An essential aspect in specifying the analytical solution to the RT equation is in forming the RT equation such that it is mathematically tractable yet physically descriptive. For this reason, only bi-Lambertian leaf scattering with equal or unequal leaf reflectance and transmittance* without a specular component is assumed in the analytical investigation to be described. tpermanent address: Department of Nuclear and Energy Engineering, University of Arizona, Tucson, AZ 85721, U.S.A. 321
B. D. GANAPOL and
322
R. B.
MYNENI
The initial phase of the analytical effort concentrated on the azimuthally-integrated problem or, as commonly referred to in the literature, * the one-angle problem. The first model was a dense canopy with soil not participating, and where all leaves were assumed to be oriented at the same inclination, but azimuthally symmetric. The method of Chandrasekhar,3 invoking the principles of invariance, provided an efficient solution in terms of functions similar to the well known H-functions.4 Then, employing a quadrature approximation for the area scattering phase function, the RT equation for a canopy with a general leaf angle distribution (LAD) and for multiple species was solved by generalization of the H-functions.5 The BDRF for a single leaf angle canopy (SLAC) model with a Lambertian reflecting soil was also found using the principles of invariance again yielding expressions in terms of functions similar to Chandrasekhar’s X- and Y-functions.6 Recently,’ a major advance was made for solving the RT equation with a general scattering phase function by application of the FN method. In the second phase of the effort,’ the canopy with azimuthal variation of the radiance resulting from the scattering phase function and/or the LAD was considered. Here, as in the one-angle investigation, the first problem attempted was for a dense SLAC with equal leaf reflectance and transmittance again using Chandrasekhar’s formulation.a The purpose of this investigation is to extend Chandrasekhar’s solution theoretically to the general one-dimensional two-angle problem for a general LAD in a finite canopy. The solution hinges on the decomposition of the integral of the leaf area scattering phase function into a product of known functions of the incoming and scattered photon directions. Then, from the principles of invariance as presented by Busbridge,’ a solution in terms of generalized Chandrasekhar’s X- and Y-functions is obtained. Particular cases previously considered in the literature are then derived from this comprehensive solution. The analytical solutions developed will be of use in numerically verifying S, codes operational in the optical remote sensing arena” as has been done previously for the one-angle problem.7 2. SOLUTION
WITH
A LAMBERTIAN
REFLECTING
BOUNDARY
The radiative transfer equation for a plant canopy with leaf area index (total optical path; A) bordering a Lambertian soil with reflectance rs is
with the boundary
conditions
Z(A, 8) = ;
dQ’p’Z(A, Q’), s 277
p < 0.
Here, z is the optical depth, Q the solid angle [Q( CL,4)] and Z the leaf-area scattering phase function defined by2 Z(Q’, 0) = ]ri, Z -(a’, Q) + t, Z +(Q’, Q)l with f ‘(Q’,Q)=
&
s
dS_2,g,tO,)trC1’.n,)tsz.n,>,
(2’4
*n(c)
where the + indicates that the integration is over that part of the range where the integrand is positive (+) or negative (-) only. In Eq. (2b), g, is the probability density of the leaf normal orientation and, rL, tL are the leaf reflectance and transmittance respectively; the single scatter albedo w, is rL + tL. The area intercept function G is defined by (3)
Applying princibles of invariance to the RT equation in plant canopies
323
In Sec. 4, it is shown that Z obeys the following symmetry relations required for the principles of invariance:
T(Q’, Q) = r(Q’,
rtn, Q’>,
Q) = r( -@,
(4a) VW
-0).
From Eq. (3), one can easily see that G(Q) =
G(4).
(5)
For mathematical tractability, the LAD gL is assumed spatially homogeneous and FL is a known surface illumination. By superposition, the RT problem (1) can be decomposed into the following two problems: Problem 1
J
w?)=~
6’1,
(64
4n
4 (0, 0) = FL@), Z,(A,Q)=O,
P >
0,
.
(6b)
p
(6c)
Problem 2
J
Z2(A,Q)=S,,z~
dSZ’p‘Z(A, a’),
2n
(74
p < 0.
When Eqs. (6) and (7) are added, Eq. (1) results and, therefore, from its uniqueness, Z(T, Q) = Z,(r, 0) + Z,(7, Q).
(8)
If we assume the (Green’s function) solution to [p;+G(Z))]I(r,n;g)=i
J4~dn.~(~/,~)Z(r,n.;~),
Z(O,Q; 4) = &(Q - Qo), Z(A, Q a)
= 0,
P > 0,
(9a) (9b)
P < 0,
(9c)
is known where & is the delta function specifying the point (p,,, do) on the unit sphere, then multiplying Eq. (6) by FL, integrating the upper hemisphere of directions (Qzo,with & replaced by Q”) and applying the uniqueness of the RT equation, we have for the solution of problem 1 Z,(7, n) =
J
dQ”F,(Q”)Z(z, Q; n”).
(10)
2n
If T, Q and Q, are replaced by A - 7, -Q (cf. Ref. 5), there results r 1
QSRT 4.8/3-a
a pat+G(Q)
1
Z(A-7,
-Q; -Q,,)=!
and -Q,
A
Z(A,
-Q2;
-Q,)
=
&(O
Z(0,
-Q;
-QJ)
=
0,
J
in Eq. (9) and noting G is invariant
dQ’T(-Q’,
-Q)Z(A - T, -Q’; -Q,)i
(1 la)
0,
(1 lb)
4n
-Q,),
p
<
P
0.
>
(1
w
B. D. GANAPOL and
324
R. B. MYNENI
Multiplying by S, and integrating over the lower replaced by -&I”), we again find by uniqueness I,(r, Q) = & Therefore,
the solution
dQ”Z(A - z, -8; s Zn
in terms of the Green’s dQ”FL(Q”)Z(r,
I(? Q) =
hemisphere
function
Q, (- s_Z,, with Q,
a”).
(12)
is dQ”Z(A - 5, -Q; s 277
Q; Q”) + S,
s and we find from the definition
of directions
a”)
(13)
of S,,,
-r, de”&_ (0”) dQ’p’I(A. 71 Zn 2n____ :=--j s n 1-s 71 s ?n
dQ’p ’
9’; Q”) (14)
dQ”I(0, Q’; Q”) s
2n
Thus, from Eqs. (13) and (14) if I(0, -Q; Q,) and Z(A, Q; Q,) can be determined, then, the radiances I(0, Q) (or the BDRF) and Z(A, Q) as well as the reflection and transmission coefficients can be obtained. In the next section, the explicit determination of Z(0, -Q; 9”) and Z(A, Q; f&) is presented. 3. DETERMINATION
OF I(0, PRINCIPLES
3.1. Integro-d@erential (I) Integral
transport
-Q; Q,,) AND I(A, OF INVARIANCE
Equation for the Scattering
Q; Qr,)
FROM
THE
Source Function
equation
Tracking the photon along integral form for p > 0:
its trajectory
allows the reformulation
‘ds’exp[-(r !J I 0
I(r,R;n,)=fiz(S1-~,)exp(-z/~,)+l
s
of Eq. (9) into the following
-~‘)/5(sz)]J(t’,-n;~~),
(15a)
A
dT’ exp[-(7’
-
(15b)
~)/5(Q)lJ(z’, $2;%),
T
where
s 4n
In deriving Eq. (15) A, it follows that
(16)
dQ’r ( - n’, @l(T, n’; a,)
use was made of the symmetries
of r given by Eq. (4). Thus,
at t = 0 and
(174 0Adt ’ exp[ - (A -
Z(A, Q; &I,) = a,(!2 - !&)exp( - A/&) h s These relations
will be used extensively
in what
follows.
z’)/{]J(t ‘,
-Q;
Q,)
(I7b)
RT equation
Applying principles of invariance to the
325
in plant canopies
By forming J from Eq. (15) as
Jb, Q; Qd =
~~(-4,Q)exp(-T/to)
s 2x
s A
WW?‘, Q)+
dr’exp[-(r’bdr’exp[-(r
dQ’T(-Q’,Q);f; s
with t’ s r(Q)
t)/<‘]J(r’,
-Q’; Q,)
t
27
s
-
qlr’lw,
-0’; Qo)
(184
and upon defining
otherwise, which implies K(-L
(18~)
-Q) = r&t, Q,
Eq. (18a) becomes 11 - &JJ(r’, The operator
Sz’;r30) = i W-a,
(19)
Q)exp(-r/L).
L,* has been defined as A
L.,o=; and obeys several important
s In
(jsz’rK2’9
dr’K(t’ - r, Q)(a)
n)
(20)
s0
symmetry properties outlined in Appendix 1.
(2) Derivative of Eq. (19) with respect to r Taking a derivative of the operator ;
L,,J(G
n;
Ml = L&W, +; --
s tn 1
As Thus,
operating on J, we find (after some manipulation)
L,,
Q’;4) dO’T(-O’, RM, Q’W, -0’; 00) d@T(Q’, O)K(A - 7,Q’)J(A, Q’; Q,).
2n
taking a derivative of both sides of Eq. (19) results in
dQ’r(--n,, ??)K(?@MO, -n’; @I) --
I
n
s
dQ’T(Q’, Q)K(A - z,o’jJ(A, Q’; a).
(21)
2n
(3) Alternative representation of the rhs of Eq. (21) The first term on the rhs of Eq. (21) is T, =i and incorporating
lr I 2x
the definition of
dQ’r( -3’, @K(r, @)J(O, -n’; a); K
given by Eq. (18b), T, becomes
1 .
T, = s 2x
(22)
326
B. D. GANAPOL
After substitution o’-+U=U(PL,~)l
and R. B. MYNENI
of Eq. (19) for the term in the square
s s
bracket,
there
results [letting
dU ; J(r ‘, a’; U)J(O, -U; Q-,).
T, = [l - J&l
(23)
2n
Similarly, the second term on the rhs of Eq. (21) is given by dU;J(A-r’,
7’2= [l - ~,,I
-Q’; U)J(A, U; Q,).
(24)
2n
(4) Integro-differential
equation for J
The desired integro-differential into Eq. (21) to give
t1 -&,,I
i
equation for J is obtained when Eqs. (23) and (24) are introduced
~J(r’,n’;p,)+~J(r~,R’;R,)-
s 2n
0
dU ; J(z’, a’; U)J(O, -U; Q,,)
dU;J(A-r’.
+
-Q’; U)J(A, U; Q,)
s 2n
= 0.
(25)
I
Thus, in order for this equation to be satisfied in general, the term in curly brackets must vanish9
[
1
&+;0
J(T,
Q; &lo)=
s$ da’
Jtz,
Q Q’)J(O,-a’; C&d
2n
-
s
da’ + J(A - r, -Q; Q’)J(A, 0’; 4).
(26)
2TI
3.2. Exiting Radiance Distributions Equation (26) can be straightforwardly integrated to obtain the exciting radiance distributions at r = 0 and A directly. To accomplish this, Eq. (26) is first multiplied by exp(-r/l) and then integrated over r on [0, A] with liberal use of Eq. (17) to give JtO,
$2;Qo)
+
s
dQ’J(0, -Q’; Q,)Z(O, - 0’; Q)
2n
-
s
1
dQ’J(A, a’; Q)I(A, g; Qo) .
2a
(27)
Similarly for Z(A, Q Q,), Eq. (26) is multiplied by exp[ -(A - r)/<] and integrated over r on [0, A] using Eq. (17) to give
x ZtA, 8’; Q) -
JtA, --Sri 4) +
s
dQ’J(A, Q’; &))I(O, -Q’; Q)
2n
11 .
(28)
In Eqs. (27) and (28), the reciprocity relations, PZ(O, -n;
Qo) = PoZ(0, -Qo; n),
P~(A, Q; slzo)
= PHI@,
Qo; Q,
(29a) Pb)
derived in Appendix 2 have been employed to simplify the resultant integrals in Eqs. (27) and (28). No further simplification can be made until the specific form of the leaf area scattering phase function is specified.
Applying principles of invariance to the RT equation in plant canopies 4.
FORM
OF
THE
AREA
SCATTERING
PHASE
327
FUNCTION
For bi-Lambertian leaf scattering model, the area scattering phase function is given by Eq. (2) with Z’* written in a more revealing form f*(Q’,R)=$
d%gt_(RL)lR’.QLI In.n,lo[+(n’.sr,)(~.R,)l (30) s 2n where 0 is the unit step function. From this form, the symmetry relations given by Eq. (4) are easily realized. Upon application of the identity O(f.~~)=O(+x)O(y)+O(Tx)O(-y), and Eq. (2a) becomes Z(Q’, n) = & where
a@,QL) = IQ~QLI[fL@tR-QL)+ rL@(-O.OL)l,
Wa)
Pa
WW
QL)
= In.QLl@(sz.nL).
Then, noting that a@,
QL)
= @C-Q,
-RL),
BeA
a,)
= PC-Q,
-8L),
Eq. (31) becomes f (Q’, Q) =
s 4n
where g,_ is
dQLgL(QL)a(a’, QLV (Q, BL>
EL (QL)
= &TL@L
(334
Wb)
)/27c
with i?L(fJL)
= kL(RL)@(PL)
(33c)
+gL(-.RL)@(-PL)l.
As will now be shown, a theoretical advantage of this form for Z is the decomposition integrand into a product of functions of Q’ and Q. 5. EXPRESSIONS
FOR
EXCITING
of the
RADIANCES
From Eq. (16) evaluated at z = 0 and A and using the symmetry of Z, we have
s
J(O,R;Ql)=+?&)+~ dn,r K?‘,WtO, 277
J@,
Q; Q,)
= ;
d$?‘r
(Q’,
- QW,
Q’;
4
-a’; f&J,
PW
),
s 2R
and substitution
(344
of Eq. (33a) shows that J(0, Q Q,) and J(A, Q a,) are of the form J(o,nW=;
s 4n
J@, Q; Q,) =;
s 4n
dQ,&(~,)P(Q,
Q,)~(Q,,
d~&.(~2,M(-rlr,
OLPWO,
R,_), OL),
(354 WW
with
YtSr, QL) = 4-G @@,QL)
QL)
+ s 2x
dR’a (Q’, QL)Z(O, - a’; R),
dQ’a(R’, Q,)Z(A, Q’; a).
= s 2n
(W t36b)
328
B. D. GANAFQL and R. B. MYNENI
Substitution
40,
of Eqs. (35) and (36) into Eqs. (27) and (28) gives
-o;w=&-&) 0
-S
dQd,(Qr,)
x Pf’(Qo, QLLW,
478
G_) - @(Qo, Q,)WL
I(A,R;~)=exp(-A/5,)62(~--~)+~~
4n
.S
x [‘YaO,
QL)W,
QL)
-
w&l,
QLMQ9
Q,ll,
(37)
QLifL(LIL)
(38)
RLII,
where
Jw,
rlr,)
Y(Q, a,)
= /m
QL)
dn’B(-W,
+
(39a)
Q,)l(O, -Q’; Q),
f 2n
=
@‘PC-n,,
QLW,
S?‘; Qr).
W’b)
s 2n
To obtain integral equations for the auxiliary functions Y, @, X and Y; Eqs. (37) and (38) are substituted into the Eqs. (36) and (39): wQL)=.(-c?,Q,)+; s 4n
x [Y(Q, f&)Y(nl, n;) - @(Q,n;)qr, dW,_(Rt) x(n,n,)=B(Q,8,)+~
Y&t Q,) = exp(-AlOB(-R,
QL) + i
x [Y(Q, &)Y(Q’, By direct substitution following:
f 2n
&.)I,
W -
r
B(-~,t?,)
5’+5
W?‘)
s s 4n
dW, OX,)
rlrl) - w2, ntbw’,
2n
dn, -
5
t’--t
8(-=9QL) G(Q’)
n;)l.
into Eq. (40), the auxiliary functions can be shown to be related by the
VQ, 0,) = ev( - A/t M -Q, QL),
VW
WQ, QL) = w(--A/Wu(--R,
@lb)
rlL).
In addition, moments relations can be shown to apply using the reciprocity radiance, if the moments are defined by Qa?L)
=
d&(+3,
dRZgL(W I 4n
s 2n
WWQ,
RL),
relation for the
(42)
Applying principles of invariance to the RT equation in plant canopies
329
where K = Ij/, @, Xor Y and y = a or /?. If y = P(N), then - Q( +Q) is specified for the first argument of y in the integrand. By applying reciprocity to Eqs. (36a) and (39a), adding and manipulating, the following moments relations can be derived:
-2Yy,(QL) + 2
X
s 3
d%‘gi. (CC’) s 4R s 2n
[a?,
sz;>pa
nt>
+
dR c@, QL) - G(R)
P -Q
&M-Q
WI
= 0. (43c)
2n W3
The BDRF for the original problem is obtained from Eq. (13) with 7 = 0 and 0 is replaced with -Q. Thus for a beam source impinging on the canopy upper surface [FL(a) = k5,(Q- a)], we have
where
dQ,&(Q,)
SN=~~exp(-A/&,)i-~
da’ 5’50
s 4n
5’-5
s 2n
x [Y(Qcl, QL)Y(R’, Q,) - @(Qoo,Q,MQ’, d&g,(&) s 4n
(44c)
(“5 de’ ___
dQ” s 2n
WI,
s 2n
5’+
5”
x W’@“, Q, P-W’, 0, >- WX Qt.>W’, G )I. (444 The determination of the BDRF therefore has been reduced to the solution of the integral equations for Y, @, X and Y [Eq. (40)]. While these equations seem far too complicated for application of analytical solution techniques such as the Wiener-Hopf approach,’ they can be solved numerically which will be the subject of a future effort. In any event, several simplified cases previously reported in the literature can be obtained. 6.
SPECIAL
CASES
For the following simplified cases, only the BDRF for the Green’s function, Eq. (37), will be obtained except in the last case. The distribution at the soil can be obtained as well using the auxiliary function for the particular problem. 6.1. Equal Leaf Rejectance
and Transmittance
For the case, rL=tL-Ws/2
(45)
and from Eq. (31)
s 2n
dRLgL(QL)IR’.RLl IQ+Q,l.
(46)
330
and R. B. MYNENI
B. D. GANAPOL
Thus
Wb) and therefore
Y@z,$2,) =
T wn, RL),
WW
W,Q,)=~ w&Q,).
WW
From Eq. (37)
Z(O,-am=-
to 50+
dnLgL(nL)[X(~O,RL)X(I_Z,S_Z,)s 277
m&43
Q,)l
Wo>Q,)W,
(49)
with
s
r
da
2n
x pm, nt)Y(Q’,
w - un, n;Mw,
-
t-‘-t
KQ’,QL>
G(R’)
nt11.
(51)
For this case, the moments relation Eq. (43a), becomes
s
dR;g,(W
We
2n
-
Y#X.)l& - 2Xo(R;) +2
Note the similarity to the conventional
dQkL(&) s 277
WW,
Q3-W
QWVQ)
s 2n
radiative transfer case.’
6.2. Azimuthally
Ungorm LAD
For this case, =gL(PL)r
(53)
G(Q) = G(P),
(54a)
r(Q) = 5(P)*
(54b)
sL(nL>
which implies
and Eq. (37) is then somewhat simplified to Z(0, -!a, Q-J =
where
50 t-o-
1
’ d~LgL(~L)[R,,(RO,R,~,)-R,,(~,n,
s -1
c1dl
(55)
331
Applying principles of invariance to the RT equation in plant canopies
with M and N representing
Y, #, X or Y. An integral equation for Rlx is obtained from Eq. (40~)
bd?o~ Q,PL)= cl&ot Ckc1d+;
1
s s ’
dfi;g,_(d)
-I
2n
da’
where $9v(Qo9 n,
PL)
(574
=
can be found from Eq. (40a) to be
x r;+(Q’, 3’3 PL)&X@O,Q/4) - &,@o, 9’3 &)I
Wb)
with 257 d4La(f%
~$9*Q,n4L)=
Q,)B(+O,
a,).
10 The additional transformations
functions ReY, RvY and R,, are obtained from Eqs. (56) and (57) using given by Eq. (41) and therefore will not be derived here. 6.3. Single Leaf Angle Canopy (IL = t‘)
Here all leaves are oriented along the same direction such as would be true for a heliotropic canopy. Then irL@L)
=
62@L
-
(58)
a*)
and Eq. (37) becomes (59) where
Jm
z$K(Q,Q*), K=Xor
Y.
(60)
8 and P are obtained from Eqs. (4Oc) and (40d) along with the following relations from Eq. (3)
‘39 =
&B(Q,Q*).
(61)
Therefore,
d(Q)= G(Q) + o,
s& dQ
mw(~)
-
m?m?')l,
(624
dfJ &
tRQ)P(o’)
- RQ)RQ’)l.
(62b)
2n
P(Q)= G(Q) + co,
s 2n
B. D. GANAPOLand R. B. MYNENI
332
For the moments
with K = X or Y,
(63)
1 =o
(W
we have c@;-
Pi]-2&+
where the normalization
s
dQG(Q) = f
2n
has been used. It should be noted how closely the results for the single leaf canopy resemble the conventional radiative transfer equation with rotationally invariant scattering. For this case, there also exists a transformation of the nonlinear form of the integral equations for .%?and P [Eq. (62)] to a linear form containing principal values. 6.4. Reduction If the LAD is azimuthally
to the One-angle
symmetric g,(R,)
it is shown by proposition
and by proposition
Problem
(65)
= gL(c(L)v
(A30) of Appendix 3 that
(A31)
Z(r, n;Qo)= From this relation and azimuthal periodicity,
IO,
I.c; PO)=
;
w, P, 4 -
40; Pd.
(67)
the one-angle radiance is
s 2n
W(t, Q; Qo,).
W)
In addition,
and integration
G(O) = G(P),
(@a)
5(Q) = 5(P),
Wb)
of Eq. (37) over 4 gives
where 2n s0
dWW,
QL)
V’W
and K is !I’, @, X or Y. To obtain Eq. (70b), the symmetry of K K(Q,
QL) = m4
PLY 4 -
$L)
(71)
Applying principles of invariance to the RT equation in plant canopies
333
proved by proposition (A32) of Appendix 3 was introduced. The integral equations for K(p, ,u~) are found from Eq. (40) by integrating over C$and noting Eq. (69):
x P”(P, P;)~(P’, ~3 - @(P,/~M/J’, ~ut)l, (72b)
x [Y(P, P~)x(P’,
it) - Q(P, ~ut)y(k,
put)],
(72~)
with
C(P,14_)= kH(,4 A) + rLW-h
14
(73)
and H is* if(-cotBcotO,)< if(-cotOcotfJ,)>
H(P, C(L)=
-1, 1,
(74)
otherwise,
where c#l,(p) = cos-’
WL
[ &G&K-a.
1
6.5, Conventional Radiative Transfer with Isotropic Scattering To recover the solution tL and
to the conventional
transfer equation
with isotropic
scattering let
rL =
&TL(crL) = m.
- P*x
(75)
Thus,
and
C’W
B. D. GANAPOLand R. B. MYNENI
334
If in addition, G(p) = 1 and o, = o/2, then Eq. (59) becomes the conventional Z(0, -p,
/A)) =
w--k2n
~
+
~
result
W/Jo)X(P) - mh)y(P)I
and from Eq. (38)
(77b) with the usual X and Y functions defined by x(/4=1+;, s
0’s
[X(/J)X(p’) -
VPL)W’)l~
W)=eM-A/r)+:r I.
SUMMARY
The general one-dimensional two-angle photon transport problem for a finite vegetation canopy of arbitrary leaf angle distribution is considered. Analytical solutions are obtained in terms of generalized Chandrasekhar’s X- and Y-functions by invoking the principles of invariance. The solution hinges on the decomposition of the leaf area scattering phase function into a product of known functions of the incident and scattered photon directions. Several simplified cases previously considered in the literature are then derived from the generalized solution. Iterative algorithms will be developed to evaluate the defining integral equations for the Chandrasekhar’s functions in a future effort. Acknowledgemenrs-This work was made possible with the support of G. Asrar of NASA Headquarters. BDG warmly acknowledges his support and that of the NASA-ASEE Summer Faculty Fellowship Program and H. Boroson, the program coordinator. Participation of RBM was made possible through NASA grant NASS-30442.
REFERENCES N. Goel, Remote St ns. Revs 4, 1 (1988). J. K. Shultis and R. B. Myneni, JQSRT 39, 115 (1988). S. Chandrasekhar, Radiative Transfer, Dover, New York, NY (1960). B. Ganapol, TTSP 18, 475 (1990). B. Ganapol, NASA Ames Research Center Report (1992). B. Ganapol, TTSP, 19, 56 (1990). B. Ganapol and R. B. Myneni, Remote Sens. Envir. 39, 213 (1992). B. Ganapol, Proc. 11th Transport Theory Conf., Virginia Polytechnic and State University, Blacksburg, VA (1989). 9. I. Busbridge, The Mathematics of Radiative Transfer, Cambridge, Univ. Press, UK (1960). 10. R. B. Myneni, G. Asrar, and S. A. W. Gerstl, TTSP 19, 1 (1990). 1. 2. 3. 4. 5. 6. 7. 8.
APPENDIX 1 Symmetries of the Operator L,, (i) Proposition
(Al) integral symmetry:
Adr@(r)L,o{~(Q’, s0 Proof: we consider the integral
-Q4W}
=
dW~)L,,(WX,
and introduce the definition of L,,
AdrX(W,,~{W’, s0
-Q)@(r’)}.
(Al)
(W
-rlzoFW’)}
with a change of variable (Q’ + -Q’), we have drx(r - T’; Q’)r(Q’)J-(Q’,
-Q)@(r)
Applying principles of invariance to the RT equation in plant canopies
335
where Eqs. (18~) and (4b) have been used. Thus, again from the definition of L,, , A
K=
dN~)L,,,(W’,
-O)@(O)
643)
f0 and the proposition is proved. (ii) Proposition (A2), translational
symmetry:
L,+,@(A - T‘, -0’)
= LA_,-Q@(“,
@).
644)
Proof: consider the relation U 3 L,,@(A
- t’, --a’)
and from Eq. (20) A U=
dr’@’
dfJ'UQ',Q)
(A9
- r, Q’)@(A - r’, -Q’).
s0
s 4n With the change of variables
Q’-, -Q’, r”+A-T’, Eq. (A5) becomes [cf. Eq. (18c)] A
dr”rc[t” - (A - T), &I’]@(?‘,a’). s 4n s0 From Eqs. (20) and (4b) the proposition follows. (iii) Proposition (A3), repeated application of integral symmetry: d@f (-&I’, Q)
U=
646)
A
dz@(z)L;,,[l-(a’,
-Q,)X(r’)]
=
s0
A d7X(r)L:,00U2’, s0
-OP(~?l,
647)
where v = 1,2,. . . , L::,,=L,,
Lr,.,,
J&O? . _ ...
648)
L,!&_,* ” .
Proof: the proof will follow by induction. From Proposition Next assume the conjecture true for v - 1
(Al) the conjecture is true for v = 1.
A
dt@(t)L:,‘[f
(a’, -Q,)X(r’)J
s0
=
AdtX(r)L;.$[T(Q’, s0
-Q@(r’)].
(A9)
Now consider A
dr@(r)L:,o[r(O’, -GW’(~‘)l= ‘dr~(r)L~,,‘L,,,~,[r(~‘, --GM~‘)1. s0 f0 From the definition of L,,,,, and with some rearrangement, change of variable (0 + -0’) application of Eq. (18~) K=
K = JoAdr’J’(r’) 6. dCl’f(Q’, 4) JoAdr@(r)LZ,‘[r(Q’, Upon application
, -Q)lc(t,
(AJO) and
- T’, Q’)].
of (AS), we have
K = JoAdz’X(t’) 6. dQ’r(@, Q,) JoAdvc(z - r’, Q’>L:,![l-(Q,
-Q)@(r,)].
(Al 1)
Finally, from the definition of L,, A
K= s0
A
d7sX(z)L,,,L;;-d.[T(~‘,,
-QP(~,11=
dWr)L:,c$W’, s0
-8M~‘)l.
(A121
336
(iv) Proposition
B. D. GANAPOL
and R. B. MYNENI
(A4), repeated application of translational L:,o[@(A - r’, --&I’>]= Li-,,
Proof: again the proof is inductive. From proposition the proposition true for v - 1 L;,;‘[@(A - t’, -!J)]
symmetry:
-Q[@(T’, Q’)].
(A13)
(A2), the above is true for v = 1. Assume
= LLl1,. _&P(t’, q.
(A14)
Consider K z L;.,@(A - t’, -a’)
or K = L;,<’ {L,.,J@(A - z”, -a”)]} and apply proposition
(Al3
(A2) to the term in the curly brackets to give K = L;,;’ {LA-_r., _g[@(T”, @‘)I>.
6416)
K = L;:‘,,
(Al7)
Then from Eq. (A14)
and the proposition
-&_,,,
_n~[@(z”, n”)]
is proved. APPENDIX
2
Reciprocity Relations
(i) Upper surface (z = 0) The Neuman series representation
for J can be obtained directly from Eq. (19) as
J(t, Q; f20)= f &Jr@‘, I =
-Gdexp(--~‘/to)l
(AIf
0
provided the series converges. To show this, we first assume that r is bounded F(O’, sz) < w. One can then show
Q’kv(-~‘&)I
L,o]F(-QoY
G WL,dIl
(Al9)
and since [cf. Eq. (18b)], K(Z, Q) < 1,
LT,,[~lG
dQ’r(Q’, Q) = w,G(Q) < o,
s 4n
where from Eq. (3), G(Q) < 1. Thus,
L.
n
[r (Q’, -
Q. >exp(- t ‘Ito )I s 0, W
WO)
and
&Jr@‘,
-%)exp(-r’/tO)l
G Wo;
implying that the series for J is bounded by
WV and is therefore convergent for 0, < 1. Since the exiting radiance at t = 0 is
soA
dt’exp(-t’/S)J(t’,
$2 Qd,
6422)
Applying principles of invariance to the RT equation in plant canopies
337
the convergent series for Jean be substituted into Eq. (A22) with the integral and sum interchanged to give M(O, -n; Therefore
00) = c
* dr exp(-rlW:o[r(S?‘9
from proposition
(~23)
-%)exp(--r/Ml.
Y s0
(A3)
MO, -Q; Qo) = c
-O)ex~(-r’/t11
* dr exp(-rl~&u,,[F(Q’,
Y 10
which is identical to the sum in Eq. (A18) with Q and Q, interchanged. MO, -n;
(A24)
Thus (A25)
Qo) = /.klJ(O, -Qo; Q)
(ii) Lower surface (T = A) At t =A, Z(A,R;W)=s*(B-Q)exp(-Alt,)+t
and upon substitution
s
oAdrexp[-(A-r)irlJ(r,
-3; Qd
(A26)
of Eq. (Al8), we have
~Z(A,~_2;~)=~062(~-~)exp(-Alto)
s
+c Y
A
dt ev[--(A - WIG-,PW’,
-Q&W-~‘/to)l
WV
0
or letting r be A - T in the integral MA, Q; C&,) = clo4 (Q - Qo)exp( - No) A +c
From proposition
dr
I I0
--C?&x~(--~‘/t~11. WW
exr{-~lr)L~-,-~[T(~,
(A4), the sum becomes - O/tol)
*dr ,x~(-~l~)~~.,{~(~‘,~)ex~[-(A ?IY 0 and from proposition
(A3) A
” 0
d7
exp[--(A
-
~Y~ol~: +,F(&?‘, -Qhp(-r’/Ol
?I
and comparing to Eq. (A26) with Q and Q, interchanged /W,
gives
Q; G,) = iMA, GG,zo; Q).
(A291
Note that the delta functions cancel. APPENDIX
3
Propositions for Reduction to the One-angle Problem (i) Proposition
(AS): if gl_(G_) = 8L(k)
(A30)
then F(@,O)=W’,/&$
-4’).
(A3L)
B. D. GANAPOL and R. B. MYNENI
338
Proof: since’
J~J~
(a *QL ) = WL +
CM4 - 4L 1,
from Eq. (32)
a@,QL) = ah4 PL, 4 -
4L)9
(A32a)
B(Q,
4L)r
(A32b)
QL)
=
Bh
PL,
4
-
and in addition a and /_Iare azimuthally periodic. Then from Eq. (33) with a change of variables I r(n’,
Q) =
d‘ ch.&(d
s -I
Because of the periodicity
s @- 2n
duab’,
L(~, u)LG
clLI u + 4 - 9’).
(A33)
of the integrand of the inner integral, Eq. (A33) can be written as I
rcO’,
n)
=
2’dua(~‘.~L,u)B(~,~L,u+~ f 0
d~Lh,(pL)
s -I
-4’)
and the proposition follows. (ii) Proposition (A6): if tcL@L)
=
tTL(PL)
then (A34a) and W, Q; Qo) Proof: consider the representation
= I(&
~134
-
90;
/.+,I.
(A34b)
for J given by Eq. (A18)
J(r, Q; G) = exp(-r/
-t f
1’= I
L:,[T(-~2,,n’)exp(-r’/ro)l.
(A35)
4 - 40 - n)
(A36)
Since
J-(-$20, Q) = from proposition
u-p,,
P,
(A5) and Eq. (20), it can be shown that xl = &J~(-s_Z~
B’)ev(-~‘/50)1 = 4b.b P, 4 - 40).
(A37)
Then assuming (A38) we have x, =
LI,[f(-~,R’)exp(-r’/r,)l
=&J~“-l(PQ,P’~4
-4011
where it can again be shown from the definition of L,, [cf. Eq. (20)] that L,o]r(-&,
@)exp(-r’/M
= k(cco, k 4 - 40).
(A39)
Thus from Eqs. (A36) and (A39) J(r, Q; Qo) = J(r, /Ju,4 - $0; I4I) and the proposition
follows from Eqs. (A22) and (A26)
(A40)
Applying
(3) Proposition
principles
of invariance
to the RT equation
in plant
canopies
339
(A7): if &_(QL.)= gI,(,%)
then K(QZ,Q,) = K(K
PL, 4
(A41)
- 4~)
where K = Y, @, X or Y. Proof: the proof will be performed for Y and Qi since with only minor modifications the proof follows for X and Y. From the defining equation for Y with explicit dependence on azimuthal angle indicated, Eq. (36a) becomes Y(O, OL) = a(-pu,
PLY 4
- 4L + x> 2n +
wwhL,
G' S'0
4'-
4m
-PW-4
+m.
(~42)
s0
Then with the change of variable, u = 4’ - c$~, 2n 'Y(~,OL)=cr(-/l,~L,~-~L+n)+
&'
S’0
dMp',~L,uYP,
-P',u
-(do
-4~)+n;
s0 =Y(P,PL,~-~L)
where the periodicity of a and Z have been noted. Similarly, for CDfrom Eq. (36b)
QSRT 48,3--H
PI (A43)