- Email: [email protected]
A simplified formulation of photon transport in leaf canopies with scatterers of finite dimensions
J. Quwf.
Specrrosc. Radiar. Transfer Vol. 46, No. 3, pp. 13S140,
0022-4073/91 $3.00+ 0.00 PergamonPressplc
1991
Printedin GreatBritain
A SIMPLIFIED FORMULATION OF PHOTON TRANSPORT IN LEAF CANOPIES WITH SCATTERERS OF FINITE DIMENSIONS RANGA B. MYNENI
and BARRY D. GANAPOL~
Hydrological Sciences Branch, Mail Code 974, NASA-Goddard Space Flight Center, Greenbelt, MD 20771, U.S.A. (Received
30 November
1990)
Abstract-A formalism for photon transport in leaf canopies with finite-dimensional scattering centers that mutually cross-shade was recently proposed resulting in an equation set which is rather complicated. In this paper, we propose a simplification of the correction factors that describe the correlation in photon fates. These simplified correction factors permit the derivation of a modified integro-differential transfer equation with two additional terms that arise from considerations of finite size scatterers. A numerical procedure for the solution of this transfer equation in the framework of a successive collisions approach is also presented.
INTRODUCTION
When media such as planetary surfaces or plant canopies, which contain finite-dimensional scatterers’ that mutually cross-shade are illuminated by a monodirectional source, the emergent intensity distribution shows a sharp peak around the direction of incidence. This phenomenon is variously termed the hot-spot effect, Heiligenschein, opposition brightening, or backscatter enhancement and is due to the absence of shadows in the retro-illumination direction.’ In a recent paper,’ a formalism for photon transport in leaf canopies with finite-dimensional scattering centers was proposed. The inclusion of the scatterer dimension in the canopy problem leads to a correlation in photon fates since photons traveling in parallel or nearly-parallel directions, separated by an infinitesimally small spatial extent, may pass through the same gap in the canopy or may be captured by the same leaf. This means that if, after a scattering event, a photon were to trace back its trajectory, it would experience its recent history with unit probability. To imbue this property to the cross sections, dependence on the previous phase-space coordinates of the photons should be introduced. In this fashion, a quantitative link can be established between the dimensions of the scattering centers and the correlation in photon fates. At the canopy level, this nexus accounts for the hot-spot effect that accords well with experimental observations.3*4 The resulting transport process cannot be described as a Markov chain because of the dependence on the previous phase-space coordinates of the interacting photon stream. Consequently, transport between successive phase space points could only be described by a multipleintegral equality in the framework of a successive collision approach. It is, however, possible to group successive phase-space points into assemblies such that particle transport between successive assemblies in Markovian. Thus, an integral equation of transfer could be formulated. However, this equation relates not intensities but partial intensities, i.e., the energy density of those photons which have had a common history. Finally, it was not possible to reduce the general threedimensional transport problem to the slab geometry problem because of the dimensionality of the scattering centers. The specified formulation with all its ramifications leads to a rather complicated equation set and some simplification is highly desirable. It is our objective here to develop a simplified description of the transport process in leaf canopies with finite-dimensional leaves. tperrnanent address: Department of Nuclear and Energy Engineering, University of Arizona, Tucson, AZ 85721, U.S.A. 135
RANGA B. MYNENIand BARRY D. GANAPOL
136 THE
INTERACTION
CROSS
SECTIONS
We consider a spatially-homogeneous leaf canopy in which the random variables uL, the leaf size, and Qr, the orientation of the normal to the upper surface of the leaf, are independently distributed. Let (271) ’ gr(&) be the probability density of the leaf normal orientation [12r E (0, 2n)]. Also, let ~~(a,) be the probability density of the leaf-size distribution. Without loss of generality, we assume that ~~(a~) = ii(a, - ao). If nL(uL) is a function relating the number of leaves in the volume element dr to the leaf size uL, then ~,(a,)6 (uL - u,,) da, dr is the number of leaves in dr. all of size ~1,). The shadow cast by these leaves on a plane perpendicular to a monodirectional incidence along Q is ii)
~n(u,,,~~~4)d~Ldr=~L(~~1)~yL(Rr)aill1ZL.~I~(u,,.41.~~)d!)Ldr.
Here, x is a dimensionless function that accounts for the overlap between the areas projected by the various leaves in the volume element with respect to Q. The total interaction cross section is obtained by integrating m over all orientations Q, and dividing by dr, viz. c?(Q) =
jl
13)
dSZ,.m(u,,~L,R)=u,G(a,.R)
where uL = u,,II~(u~,) is the leaf area density
function
and
1 ,.
G(u,,. Q) =
&
J dn,gL(nL)IR.n,I;I(u,,.n,,n, 2,”
is the non-overlapped area per unit leaf area that is projected on a plane perpendicular to the direction Q by leaves of all orientations. The cross section a’ accounts for mutual shading when viewed along Q of finite-size leaves in the volume element dr through the overlap function 1. However. it does not account for the fact that photons traveling in parallel or nearly-parallel directions, separated by an infinitesimally small spatial extent, tend to pass unhindered through the same gap or may be both intercepted by the same leaf. This property can be imbued to the total interaction cross section by including dependence on the previous phase space coordinates of the interacting photon stream through a correction factor [. This correction factor is, in general, a function of the current (r, Q and previous (r’, Q’) phase-space coordinates of the interacting photons and also of uL and 3.’ We propose a simplified model for the correction factor <. It is assumed to be independent of RL, which may be rigorously justified if the leaf normals are isotropically distributed. i can be made independent of uL by defining the characteristic dimension of the leaf I, from the mean leaf size a,; the leaf size distribution in plant canopies tends to be rather narrow. Further. if the leaf canopy is spatially homogeneous, as is assumed here, the dependence of < on the absolute spatial coordinates of two successive interaction centers can be replaced by the lineal distance separating the same, i.e., Ir - r’/ and, by averaging over these distances, < can be made independent of the spatial coordinates. Then, [ is a function of the characteristic dimension of the mean leaf size, physical depth of the canopy H, and of the incident @‘) and scattered (Q) photon directions only. Such a simplification will necessarily have its limitations in describing the physical process, but it is justifiable considering that the resulting transport process is Markovian and can be described by an integro-differential equation of transfer (see the next section for details). The actual form of the correction factor will depend on the model chosen to approximate it. It suffices here to specify the conditions that it should satisfy. We propose that it should be of the form 11 - r@, Q’.
4/H)l,
(G.Q’) < 0,
i(Q. Q’) =
(4) 1.
@.Q’)
> 0.
Photon transport in leaf canopies
The correlation
137
function 3”satisfies 1, @,9’)<0,
cl< r@,n,,IJH)2
rIg,n’,
ILlH) =
1,
Q=
0,
otherwise.
-g,
(5)
The modified total interaction cross section can now be written as c@,g)=iqQJ~@,4’).
(6)
It should be noted that cr also depends on a0 and the ratio I,/H but since these are constants, we shall not denote them as such explicitly. Also, if Q = -Q’, then cr = 0 and it is such absence of interaction that results in backscattering enhancement. Considering our earlier arguments leading to Eq. (2), we may define the differential scattering cross section as
where yL is the single-leaf scattering phase function. For a leaf with outward normal &, this phase function is the fraction of the intercepted energy, from photons initially traveling in the direction Q’, that is scattered into a unit solid angle about Q. 5.6The interaction of radiation with leaves enters the transport equation through yL. The relevant mechanisms and models can be found elsewhere.’ As with the total interaction cross section, we need to define the modified differential scattering cross section a, that accounts for correlation in photon fates. Thus, o,~-+Q;n”)=ri,(@+QJ[@,n”),
(8)
where Q’ is the direction of the incident photon, Q” is its previous direction of flight, and Q is the direction of travel of the scattered photon. THE
SLAB
GEOMETRY
PROBLEM
We consider a leaf canopy confined to the depth interval z = 0 (top of the canopy) and z = H, with H being the physical depth of the canopy. Also, we consider wavelength independent interactions and ignore emission. Let Z,(z, 2, Q’) be the partial intensity of those photons currently at the phase-space location (z, Q) whose previous direction of travel was Q’. Clearly,
I(& $2)=
s
dQ’ J, (z,a,2’ 1,
(9)
4a
where I is the standard intensity distribution. Using familiar arguments,’ integro-differential balance equation for the partial intensity ZPas
we may derive an
s
dQ” a,@‘+&& Q“)Z,(z, Q’, Q”),
(10)
4n
where p = cos 8 and p E (- 1,O) denote directions along which photons travel downwards. Let Z,(z,SZ,R’)=Z”,(z,~,~)+10(z,512,~), where Z; and I0 denote collided partial and uncollided full intensities, respectively. The transfer equation for the collided partial intensity is then
s
dQ”o,(Q’-+Q;Q”)Z;(z,n,,n”) 4X +
s
dQ”d,@‘-Q;Q”)Z”(z,Q’,q”),
4a
(Ila)
B. MYNENI and BARRY D. GANAPOL
RANGA
138
The equation
of transfer
for uncollided
intensity
is
-~~Zo(‘,“,“.‘+“(“,“/)Z”(z,n,n,=~~.
(lib)
Let the leaf canopy be illuminated by a monodirectional beam of intensity Z, incident along Q, (p,, < 0) at z = 0. We assume that the leaf canopy is bounded by an absorbing soil surface at I = H. Thus.
co.
z”(0,2)=z”6(Q-R,),p
(123)
ZO(H,cl) = 0, p > 0. The solution
of transfer
equation
(1 lb) subject
(12b)
to Eqs. (12) is
~“~~,52,RI)=~oe~p[-~(;Zo,R’~~il~Il~~-~~~~-~~’~, =(~o~~p[-~(~o)~iI~IIO~-~~~~-~‘~)~~R-Ro~ where 0 is a step function. Inserting the expressions for CJand cr, [Eqs. (6) and (S)] in Eq. (1 la) and integrating Eq. (9)] gives
over Q’ [cf.
J
dCJ rFY (Q’ -+ 9) Z’(z, Q’ )
4n
+
ss da’
dQ” (i,(Q’ -+QZ”(,_.
4n
-
ss dQ’
4n
Q’, Q”,
4n
dQ”C,(Q’+R)
Y(Q’,Q”. IL::H)Z;(:,Q’,~“,.
(13)
4n
In Eq. (13) I’ is the collided full intensity. The last two terms on the right-hand side of Eq. (13) can be interpreted as the correction factors to the standard radiative transfer equation that account for the correlation in photon fates due to the finiteness of leaf size. These terms effectively increase the permittivity of the medium for backscattered-photon travel. We now develop an algorithm for the solution of the transfer equation (13) subject to the boundary conditions given by Eq. (12) in the framework of a successive collisions approach. First collision problem Let 2, denote the directions of photon travel after one collision in the leaf canopy; Q, E (47~). Then, the direction of photon travel before the first collision Q’ is the direction of the incident beam a,. The governing equation for first collision intensity is
since
Y(Q’, Q”, IL/H)6 a’, Z’(z,Q,)=
a,)6
a”
-a,)
= 0. The solution
s
of Eq. (14) for p, < 0 is
~~ZO(z’,~,)d,~,-t~,)exp
0
IPII
[
where Zo(z,Qo) = I,exp[
-21.
-(zmz’)~(f”no), I
1
Photon transport in leaf canopies
139
Similarly, for CL,> 0, we obtain Z’(z,R,) =
PI I*
(z’--)~@,>n,)
“~z~(zl,Ro)d,~o~R,)exp s * PI
(16)
Second collision problem
Let 2, and 2, be the directions of photon travel after two collisions and one collision, respectively, in the leaf canopy; both Q, and 2, E (47~). Further, let S(z,R,,R,) J(z,n,,n,)=
= &@, -I,)Z,(z,Q,),
&@,+n,)r@,,Q,,
(17a)
ZJH)Z’(z92,).
(17b)
The governing transport equation for the second collision problem is
-p,~z~(z~n*,n,)+a~,,g,)z:(z,n,,n,)=s(z,2*,1,)-J(z,R,,B,) (18) The solution of Eq. (18), for ,uz> 0, is Z2(z, Q,)
=
dZ,Z;(z,R,,R,),
(19a)
s 4n
where Zi is z;(Z,RzA)
:
“~[S(z’,~,,R,)-J(z’,n,,~,)]
= s
x
exp -(Z’-z)~2@z’n,)
P2
1. (19b)
And similarly for p2 < 0. (N > 3) collisions problem f2 and We can generalize the indicated method for third- and higher-order collisions. Let _n, R _m a, denote the directions of photon travel after n (n 2 3), m = (n - 1) and k = (n - 2) collisions, respectively. Further, let
S(z,n,,Q,)
= &@,-+Q,)Z,(z,4,),
The governing transfer equation for the n th order collided partial intensity is -P.~Z~(z,n_“,n,)+a~,,n,)z:(z,n.,~,)=s(l,n,,n,)-J(Z,B.,n,).
(20)
The solution of Eq. (20) for the nth order collided intensity can be found as discussed earlier. Finally, the total collided intensity is given by ZC(z,Q = 2 zn(Z, QJ. n=l CONCLUDING
REMARKS
A formalism for photon transport in leaf canopies with finite dimensional leaves was recently developed by Myneni et al,’ who report that the transport process in such media is not Markovian and the derived equation set is numerically awkward to deal with. In this paper, we have simplified their formalism while retaining the physical mechanism of backscatter enhancement. In particular, the correction factors can be simplified under certain restrictive conditions, which can then be approximated by models or with empirical relations. These simplifications reduce the transport problem to one resembling the standard radiative transfer problem with dimensionless scatterers. In particular, the governing transfer equation resembles the standard integro-differential transfer
140
RANGA
B. MYKENIand BARRYD.
GANAHII
equation but contains two additional terms that account for the correlation in photon fates. This equation can be solved by using a successive collisions approach; an aigorithm for such a calculation is also given. Acknowledgements--RBM is funded by a grant from NASA through contract NASS-30432. BDG was a GSFC-ASEE Summer Faculty Fellow at the Hydrological Sciences Branch during the course of this investigation. We gratefully acknowledge the support, financial and otherwise. REFERENCES 1. 1. 3. 4. 5.
6. 7. 8.
R. B. Myneni. A. L. Marshak, and Y. V. Knyazikhin. JQSR7’. in pws (IYYI I A. Kuusk, .Soz%t J. Remote Sens, 3, 645 (1985). R. B. Myncni and E. T. Kanemasu. JQSRT 40, l65 (1988). D. W. Deering, in Theor), and Applications of’ Opticul Rcmotr Seruing, pp. 14 .hS. G. Asrad ed., Wiley, New York, NY (1988). J. Ross, The Radiation Regime and Arrhirectuw of Piunr Stund~. W. Junk. The Hague ( 198 I). J. K. Shuitis and R. B. Myneni, JQSRT 39. 115 {1%X). A. L. Marshak, JQSRT 42, 615 (1989). G. C. Pomraning, The Equations of’ Rrrdiation N?,rhod~,r2trnli(,s. Pergamon Press. Oxford ( 1973)
Specrrosc. Radiar. Transfer Vol. 46, No. 3, pp. 13S140,
0022-4073/91 $3.00+ 0.00 PergamonPressplc
1991
Printedin GreatBritain
A SIMPLIFIED FORMULATION OF PHOTON TRANSPORT IN LEAF CANOPIES WITH SCATTERERS OF FINITE DIMENSIONS RANGA B. MYNENI
and BARRY D. GANAPOL~
Hydrological Sciences Branch, Mail Code 974, NASA-Goddard Space Flight Center, Greenbelt, MD 20771, U.S.A. (Received
30 November
1990)
Abstract-A formalism for photon transport in leaf canopies with finite-dimensional scattering centers that mutually cross-shade was recently proposed resulting in an equation set which is rather complicated. In this paper, we propose a simplification of the correction factors that describe the correlation in photon fates. These simplified correction factors permit the derivation of a modified integro-differential transfer equation with two additional terms that arise from considerations of finite size scatterers. A numerical procedure for the solution of this transfer equation in the framework of a successive collisions approach is also presented.
INTRODUCTION
When media such as planetary surfaces or plant canopies, which contain finite-dimensional scatterers’ that mutually cross-shade are illuminated by a monodirectional source, the emergent intensity distribution shows a sharp peak around the direction of incidence. This phenomenon is variously termed the hot-spot effect, Heiligenschein, opposition brightening, or backscatter enhancement and is due to the absence of shadows in the retro-illumination direction.’ In a recent paper,’ a formalism for photon transport in leaf canopies with finite-dimensional scattering centers was proposed. The inclusion of the scatterer dimension in the canopy problem leads to a correlation in photon fates since photons traveling in parallel or nearly-parallel directions, separated by an infinitesimally small spatial extent, may pass through the same gap in the canopy or may be captured by the same leaf. This means that if, after a scattering event, a photon were to trace back its trajectory, it would experience its recent history with unit probability. To imbue this property to the cross sections, dependence on the previous phase-space coordinates of the photons should be introduced. In this fashion, a quantitative link can be established between the dimensions of the scattering centers and the correlation in photon fates. At the canopy level, this nexus accounts for the hot-spot effect that accords well with experimental observations.3*4 The resulting transport process cannot be described as a Markov chain because of the dependence on the previous phase-space coordinates of the interacting photon stream. Consequently, transport between successive phase space points could only be described by a multipleintegral equality in the framework of a successive collision approach. It is, however, possible to group successive phase-space points into assemblies such that particle transport between successive assemblies in Markovian. Thus, an integral equation of transfer could be formulated. However, this equation relates not intensities but partial intensities, i.e., the energy density of those photons which have had a common history. Finally, it was not possible to reduce the general threedimensional transport problem to the slab geometry problem because of the dimensionality of the scattering centers. The specified formulation with all its ramifications leads to a rather complicated equation set and some simplification is highly desirable. It is our objective here to develop a simplified description of the transport process in leaf canopies with finite-dimensional leaves. tperrnanent address: Department of Nuclear and Energy Engineering, University of Arizona, Tucson, AZ 85721, U.S.A. 135
RANGA B. MYNENIand BARRY D. GANAPOL
136 THE
INTERACTION
CROSS
SECTIONS
We consider a spatially-homogeneous leaf canopy in which the random variables uL, the leaf size, and Qr, the orientation of the normal to the upper surface of the leaf, are independently distributed. Let (271) ’ gr(&) be the probability density of the leaf normal orientation [12r E (0, 2n)]. Also, let ~~(a,) be the probability density of the leaf-size distribution. Without loss of generality, we assume that ~~(a~) = ii(a, - ao). If nL(uL) is a function relating the number of leaves in the volume element dr to the leaf size uL, then ~,(a,)6 (uL - u,,) da, dr is the number of leaves in dr. all of size ~1,). The shadow cast by these leaves on a plane perpendicular to a monodirectional incidence along Q is ii)
~n(u,,,~~~4)d~Ldr=~L(~~1)~yL(Rr)aill1ZL.~I~(u,,.41.~~)d!)Ldr.
Here, x is a dimensionless function that accounts for the overlap between the areas projected by the various leaves in the volume element with respect to Q. The total interaction cross section is obtained by integrating m over all orientations Q, and dividing by dr, viz. c?(Q) =
jl
13)
dSZ,.m(u,,~L,R)=u,G(a,.R)
where uL = u,,II~(u~,) is the leaf area density
function
and
1 ,.
G(u,,. Q) =
&
J dn,gL(nL)IR.n,I;I(u,,.n,,n, 2,”
is the non-overlapped area per unit leaf area that is projected on a plane perpendicular to the direction Q by leaves of all orientations. The cross section a’ accounts for mutual shading when viewed along Q of finite-size leaves in the volume element dr through the overlap function 1. However. it does not account for the fact that photons traveling in parallel or nearly-parallel directions, separated by an infinitesimally small spatial extent, tend to pass unhindered through the same gap or may be both intercepted by the same leaf. This property can be imbued to the total interaction cross section by including dependence on the previous phase space coordinates of the interacting photon stream through a correction factor [. This correction factor is, in general, a function of the current (r, Q and previous (r’, Q’) phase-space coordinates of the interacting photons and also of uL and 3.’ We propose a simplified model for the correction factor <. It is assumed to be independent of RL, which may be rigorously justified if the leaf normals are isotropically distributed. i can be made independent of uL by defining the characteristic dimension of the leaf I, from the mean leaf size a,; the leaf size distribution in plant canopies tends to be rather narrow. Further. if the leaf canopy is spatially homogeneous, as is assumed here, the dependence of < on the absolute spatial coordinates of two successive interaction centers can be replaced by the lineal distance separating the same, i.e., Ir - r’/ and, by averaging over these distances, < can be made independent of the spatial coordinates. Then, [ is a function of the characteristic dimension of the mean leaf size, physical depth of the canopy H, and of the incident @‘) and scattered (Q) photon directions only. Such a simplification will necessarily have its limitations in describing the physical process, but it is justifiable considering that the resulting transport process is Markovian and can be described by an integro-differential equation of transfer (see the next section for details). The actual form of the correction factor will depend on the model chosen to approximate it. It suffices here to specify the conditions that it should satisfy. We propose that it should be of the form 11 - r@, Q’.
4/H)l,
(G.Q’) < 0,
i(Q. Q’) =
(4) 1.
@.Q’)
> 0.
Photon transport in leaf canopies
The correlation
137
function 3”satisfies 1, @,9’)<0,
cl< r@,n,,IJH)2
rIg,n’,
ILlH) =
1,
Q=
0,
otherwise.
-g,
(5)
The modified total interaction cross section can now be written as c@,g)=iqQJ~@,4’).
(6)
It should be noted that cr also depends on a0 and the ratio I,/H but since these are constants, we shall not denote them as such explicitly. Also, if Q = -Q’, then cr = 0 and it is such absence of interaction that results in backscattering enhancement. Considering our earlier arguments leading to Eq. (2), we may define the differential scattering cross section as
where yL is the single-leaf scattering phase function. For a leaf with outward normal &, this phase function is the fraction of the intercepted energy, from photons initially traveling in the direction Q’, that is scattered into a unit solid angle about Q. 5.6The interaction of radiation with leaves enters the transport equation through yL. The relevant mechanisms and models can be found elsewhere.’ As with the total interaction cross section, we need to define the modified differential scattering cross section a, that accounts for correlation in photon fates. Thus, o,~-+Q;n”)=ri,(@+QJ[@,n”),
(8)
where Q’ is the direction of the incident photon, Q” is its previous direction of flight, and Q is the direction of travel of the scattered photon. THE
SLAB
GEOMETRY
PROBLEM
We consider a leaf canopy confined to the depth interval z = 0 (top of the canopy) and z = H, with H being the physical depth of the canopy. Also, we consider wavelength independent interactions and ignore emission. Let Z,(z, 2, Q’) be the partial intensity of those photons currently at the phase-space location (z, Q) whose previous direction of travel was Q’. Clearly,
I(& $2)=
s
dQ’ J, (z,a,2’ 1,
(9)
4a
where I is the standard intensity distribution. Using familiar arguments,’ integro-differential balance equation for the partial intensity ZPas
we may derive an
s
dQ” a,@‘+&& Q“)Z,(z, Q’, Q”),
(10)
4n
where p = cos 8 and p E (- 1,O) denote directions along which photons travel downwards. Let Z,(z,SZ,R’)=Z”,(z,~,~)+10(z,512,~), where Z; and I0 denote collided partial and uncollided full intensities, respectively. The transfer equation for the collided partial intensity is then
s
dQ”o,(Q’-+Q;Q”)Z;(z,n,,n”) 4X +
s
dQ”d,@‘-Q;Q”)Z”(z,Q’,q”),
4a
(Ila)
B. MYNENI and BARRY D. GANAPOL
RANGA
138
The equation
of transfer
for uncollided
intensity
is
-~~Zo(‘,“,“.‘+“(“,“/)Z”(z,n,n,=~~.
(lib)
Let the leaf canopy be illuminated by a monodirectional beam of intensity Z, incident along Q, (p,, < 0) at z = 0. We assume that the leaf canopy is bounded by an absorbing soil surface at I = H. Thus.
co.
z”(0,2)=z”6(Q-R,),p
(123)
ZO(H,cl) = 0, p > 0. The solution
of transfer
equation
(1 lb) subject
(12b)
to Eqs. (12) is
~“~~,52,RI)=~oe~p[-~(;Zo,R’~~il~Il~~-~~~~-~~’~, =(~o~~p[-~(~o)~iI~IIO~-~~~~-~‘~)~~R-Ro~ where 0 is a step function. Inserting the expressions for CJand cr, [Eqs. (6) and (S)] in Eq. (1 la) and integrating Eq. (9)] gives
over Q’ [cf.
J
dCJ rFY (Q’ -+ 9) Z’(z, Q’ )
4n
+
ss da’
dQ” (i,(Q’ -+QZ”(,_.
4n
-
ss dQ’
4n
Q’, Q”,
4n
dQ”C,(Q’+R)
Y(Q’,Q”. IL::H)Z;(:,Q’,~“,.
(13)
4n
In Eq. (13) I’ is the collided full intensity. The last two terms on the right-hand side of Eq. (13) can be interpreted as the correction factors to the standard radiative transfer equation that account for the correlation in photon fates due to the finiteness of leaf size. These terms effectively increase the permittivity of the medium for backscattered-photon travel. We now develop an algorithm for the solution of the transfer equation (13) subject to the boundary conditions given by Eq. (12) in the framework of a successive collisions approach. First collision problem Let 2, denote the directions of photon travel after one collision in the leaf canopy; Q, E (47~). Then, the direction of photon travel before the first collision Q’ is the direction of the incident beam a,. The governing equation for first collision intensity is
since
Y(Q’, Q”, IL/H)6 a’, Z’(z,Q,)=
a,)6
a”
-a,)
= 0. The solution
s
of Eq. (14) for p, < 0 is
~~ZO(z’,~,)d,~,-t~,)exp
0
IPII
[
where Zo(z,Qo) = I,exp[
-21.
-(zmz’)~(f”no), I
1
Photon transport in leaf canopies
139
Similarly, for CL,> 0, we obtain Z’(z,R,) =
PI I*
(z’--)~@,>n,)
“~z~(zl,Ro)d,~o~R,)exp s * PI
(16)
Second collision problem
Let 2, and 2, be the directions of photon travel after two collisions and one collision, respectively, in the leaf canopy; both Q, and 2, E (47~). Further, let S(z,R,,R,) J(z,n,,n,)=
= &@, -I,)Z,(z,Q,),
&@,+n,)r@,,Q,,
(17a)
ZJH)Z’(z92,).
(17b)
The governing transport equation for the second collision problem is
-p,~z~(z~n*,n,)+a~,,g,)z:(z,n,,n,)=s(z,2*,1,)-J(z,R,,B,) (18) The solution of Eq. (18), for ,uz> 0, is Z2(z, Q,)
=
dZ,Z;(z,R,,R,),
(19a)
s 4n
where Zi is z;(Z,RzA)
:
“~[S(z’,~,,R,)-J(z’,n,,~,)]
= s
x
exp -(Z’-z)~2@z’n,)
P2
1. (19b)
And similarly for p2 < 0. (N > 3) collisions problem f2 and We can generalize the indicated method for third- and higher-order collisions. Let _n, R _m a, denote the directions of photon travel after n (n 2 3), m = (n - 1) and k = (n - 2) collisions, respectively. Further, let
S(z,n,,Q,)
= &@,-+Q,)Z,(z,4,),
The governing transfer equation for the n th order collided partial intensity is -P.~Z~(z,n_“,n,)+a~,,n,)z:(z,n.,~,)=s(l,n,,n,)-J(Z,B.,n,).
(20)
The solution of Eq. (20) for the nth order collided intensity can be found as discussed earlier. Finally, the total collided intensity is given by ZC(z,Q = 2 zn(Z, QJ. n=l CONCLUDING
REMARKS
A formalism for photon transport in leaf canopies with finite dimensional leaves was recently developed by Myneni et al,’ who report that the transport process in such media is not Markovian and the derived equation set is numerically awkward to deal with. In this paper, we have simplified their formalism while retaining the physical mechanism of backscatter enhancement. In particular, the correction factors can be simplified under certain restrictive conditions, which can then be approximated by models or with empirical relations. These simplifications reduce the transport problem to one resembling the standard radiative transfer problem with dimensionless scatterers. In particular, the governing transfer equation resembles the standard integro-differential transfer
140
RANGA
B. MYKENIand BARRYD.
GANAHII
equation but contains two additional terms that account for the correlation in photon fates. This equation can be solved by using a successive collisions approach; an aigorithm for such a calculation is also given. Acknowledgements--RBM is funded by a grant from NASA through contract NASS-30432. BDG was a GSFC-ASEE Summer Faculty Fellow at the Hydrological Sciences Branch during the course of this investigation. We gratefully acknowledge the support, financial and otherwise. REFERENCES 1. 1. 3. 4. 5.
6. 7. 8.
R. B. Myneni. A. L. Marshak, and Y. V. Knyazikhin. JQSR7’. in pws (IYYI I A. Kuusk, .Soz%t J. Remote Sens, 3, 645 (1985). R. B. Myncni and E. T. Kanemasu. JQSRT 40, l65 (1988). D. W. Deering, in Theor), and Applications of’ Opticul Rcmotr Seruing, pp. 14 .hS. G. Asrad ed., Wiley, New York, NY (1988). J. Ross, The Radiation Regime and Arrhirectuw of Piunr Stund~. W. Junk. The Hague ( 198 I). J. K. Shuitis and R. B. Myneni, JQSRT 39. 115 {1%X). A. L. Marshak, JQSRT 42, 615 (1989). G. C. Pomraning, The Equations of’ Rrrdiation N?,rhod~,r2trnli(,s. Pergamon Press. Oxford ( 1973)