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Vegetation Science – MSc Remote Sensing UCL
Lewis & Saich
Radiative Transfer Theory
at Optical and Microwave wavelengths
applied to vegetation canopies: Part 2
P. Lewis & P. Saich, RSU, Dept. Geography, University College London, 26 Bedford
Way, London WC1H 0AP, UK.
Email: [email protected], [email protected]
3. The Radiative Transfer Equation
3.1 The Radiative Transfer Equation
Radiative transfer models have been used extensively since the 1960s to model
scattering from canopies at optical wavelengths (Ross, 1981). This approach first
exploited in the microwave scattering context during the 1980s.
The models take as a starting point consideration of energy balance across an
elemental volume. This links energy into the volume (either energy incident in the
propagation direction, or energy that is scattered from other directions) and energy
losses from the volume (either scattering out of the propagation direction, or
absorption losses). Whilst optical modelling generally exploits a scalar radiative
transfer equation, in microwave scattering, we deal usually with a vector of intensities
(typically using the modified Stokes vector defined in section 1.4). The reason for this
is that the propagating waves can have well-defined polarisation1, and orthogonal
1
http://www.its.bldrdoc.gov/fs-1037/dir-028/_4059.htm
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Lewis & Saich
polarisations are coupled by depolarising processes2 - therefore, we cannot consider
radiative transfer equations for polarised waves separately from one another. Note that
the intensities we are using are not themselves vectors - the introduction of a vector of
intensities is only a mathematical convenience.
Figure 3.1 Plane Parallel Medium geometry
3.1.1 The Scalar Radiative Transfer Equation
The (one-dimensional) scalar radiative transfer (SRT) equation for a plane parallel
medium (air) embedded with a low density of small scatterers defines the change in
specific Intensity (Radiance) I(z,) at depth z in direction  at any given wavelength
with respect to z through:

I z ,  
  e I (, z )  J s , z 
z
J S , z  
 P( z,   ; )I (, )d 
(3.1)
(3.2)
4
where  is the cosine of the direction vector (with the local normal (the ‘viewing
zenith angle) used to account for path length through the canopy (figure 3.1), e is the
volume extinction coefficient (section 3.2), Je is an emission source term, and P() is
“a process in which a beam of polarized light is reflected in all directions perpendicular to its axis so
that its vibrations no long occur along a single plane.”
(http://www.harcourt.com/dictionary/def/2/8/9/0/2890800.html)
2
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the volume scattering phase function (section 3.2.2). The terms on the RHS of
equation 3.1 account for radiation transfer by extinction in direction , and scattering
from all directions within an elemental volume in the canopy into direction  by the
embedded objects. We can also add an emission source term J e , z  on the RHS of
equation 3.1 for wavelengths where this is significant, though for the optical (and
active microwave) case, the emission source term is zero. Parameterising the radiative
transfer equation requires us to define e and s in terms of canopy biophysical
parameters, and to solve for some viewing direction , for given boundary conditions.
3.1.2 The Vector Radiative Transfer Equation
The vector form of the vector radiative transfer (VRT) equation expresses the transfer
of radiation through an elemental volume in the same way as the scalar form. Vectors
are used to express polarisation coupling, as noted above. Depolarisation of incident
horizontal and vertically polarised waves is an important part of the remote sensing
signal. For a linearly polarised wave, the cross-polarised response from a large
conducting sphere is zero (Ulaby and Elachi, 1990; p. 34). In atmospheric LiDAR
sensing, for instance, a measure of the degree of depolarisation3 allows liquid and
solid phases of water in the atmosphere to be distinguished. Spherical particles, such
as wet haze, fog, cloud droplets, and small raindrops do not significantly depolarise,
whereas non-spherical particles such as ice crystals, snow flakes or dust particles have
a much higher degree of depolarisation.
The vector form of the radiative transfer equation is most conveniently expressed
using Stokes vectors. In one dimension (after Ulaby and Elachi, 1990; p. 136):

 I z,  
  e I (, z )  J s , z 
z
J S , z  
 P( z,   ; ) I (, )d 
(3.3)
(3.4)
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which has the same form as the SRT equation, with I replaced by a modified Stokes
vector, e becoming a 4x4 extinction matrix, and the phase function replaced by a
(4x4) phase matrix. The resulting four equations are coupled (principally) through the
matrix multiplication by the phase matrix.
In order to fully specify our scattering problem, we therefore need to define the forms
of the extinction and phase matrices that appear in the VRT equation and solve the
coupled integro-differential equations. The extinction and phase matrices are formed
for an ensemble of canopy elements by a summation of extinction and Mueller
matrices for different canopy element types (potentially of different size, shape and
orientation). This can be otherwise expressed as the product of the number density
and a proportion-weighted average of the terms in the extinction and Mueller
matrices.
3
http://lidar.ssec.wisc.edu/papers/pp_thes/node20.htm
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This is where the solution to Maxwell's equations reappears - in defining the
scattering properties of individual particles for inclusion within the 'harness' of the
radiative transfer formulation. As we have seen earlier, it is usual to encounter
particular approximations in the course of this. Examples of this include far-field
scattering, or small, thin or planar scatterers. The phase matrix is closely related to the
scattering cross-sections of the 'average particles'.
3.1.3 Intrinsic Canopy Properties
In modelling canopy scattering, we typically wish to state the scattered quantity as an
intrinsic property of the canopy, rather than stating a scattered intensity as a function
of incident intensity. This allows us, for instance, to compare measurements made
under differing illumination intensities.
The fundamental intrinsic scattering quantity at optical wavelengths is known as the
Bidirectional Reflectance Distribution Function (BRDF) (sr-1):
BRDF  r , pr ,  i , pi ;   
dI r  r , pr , Fi 
dFi  i , pi ;  
(3.5)
where: p x represents polarisation of the receive/transmit wave (x=r or i, px=h,v); Fi is
the irradiance (Wm-2) on the surface; and Ir is the radiance (Wm-2sr-1) (Tomiyasu,
1988). The BRDF of an ideal diffuse (Lambertian) surface is 1/  (for an unpolarised
reflector) and is independent of viewing and illumination angles (= 1/ 2 for a
polarised detector). As this is defined for an infinitessimal soild angle, it is more usual
to use the Bidirectional Reflectance Factor (BRF)  c  r ,  i  (with implicit
wavelength etc. depoendence). This can be defined as the ratio of radiance leaving the
surface around direction  r I  r   due to irradiance Fi  i  to the radiance on a flat
totally reflective Lambertian surface under the same illumination conditions, i.e.:
 c  r ,  i  
Fi  r BRDF  r ,  i 
 BRDF  r ,  i 
Fi  r 1  
(3.6)
for an equivalent infinitessimal solid angle definition. Note however, that as it is
defined as a ratio of two radiances, it is a directly measurable quantity (i.e. allows for
model predictions to be compared with measurements), albeit over instrument finite
solid angles.
The fundamental intrinsic property for the microwave case is the differential
scattering coefficient  0  r , p r ,  i , pi ;   (Tomiyasu, 1988), a dimensionless term.
This is generally used as  0  r , p r ,  r , pi ;   , i.e. the radar backscatter coefficient,
often denoted  0pq for p=pr, q=pi. This is typically specfied in dB (log power ratio)
and is defined as for v or h transmit/receive as (Ulaby and Elachi, 1990; p. 138):
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 0pq
s
4r 2 E p

A Ei
q
Lewis & Saich
2
2
(3.7)
for an area A at distance r (in the far field). It is more conveniently calculated from the
first two components of the Stokes vector as:

0
pq
 4 cos  s
I ps
I qi
(3.8)
where  s is the angle between the surface normal and the transmit/scattering direction.
Tomiyasu (1988) notes a formal relationship between the differential scattering
coefficient and BRDF for a diffuse surface as:
 0  r , p r ,  r , pi ;    4BRDF  r , p r ,  r , pi ;   cos i cos r
3.2 Extinction Coefficient and Beer’s Law
The volume extinction coefficient,  e (the ‘total interaction cross section’, ‘extinction
loss’ or ‘number of interactions’ per unit length) is a measure of attenuation of
radiation in a canopy (or other medium). For a scalar radiation Intensity I (Radiance
or Brightness, in Wm-2sr-1) travelling in a homogeneous medium of randomly located
scatterers, the Intensity is exponentially attenuated over a distance l:
I l   I 0e  e l
(3.9)
where I0 is the Intensity at l=0. Equation 3.9 is known as Beer’s Law (also the BeerLambert Law). From it, we can see that:
dI
dl
   e I 0 e  e l

  eI
(3.10)
We can see equation 3.2 as a no-source statement of the SRT equation – zero order
scattering solution.  e can be defined as a function of travel of the direction of the
radiation,  (Fung, 1994):
 e   N v Qe 
(3.11)
where Qe() is the extinction cross section for a particle (units of m2).  e can be
defined for a specific polarisation state, in which case we give it a further subscript p
for p-polarisation (  ep ).
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 e can also be expressed in terms of volume absorption and scattering coefficients,
 a and  s respectively through:
e  a  s
(3.12)
where dependence on  is implicit and on polarisation allowed, for all terms. These
two terms represent loss due to absorption by the particles (leaves) and scattering by
the particles away from the direction of propagation (Fung, 1994; p11). They are
related to number density through particle absorption and scattering cross sections
similarly to above.
We also note the related term ‘optical thickness’,  l  (Fung, 1994; p.16):
t l
 l     e dt
(3.13)
t 0
which is often used in radiative transfer formulations.
A further term, the single scattering albedo of a particle,  0 , can be defined as the
probability of radiation being scattered rather than ‘extinguished’:
0 
s
e
(3.14)
3.2.1 Optical Extinction Coefficient for Oriented Leaves
For optical wavelengths comprising oriented objects (leaves) which are large
compared to the wavelength of the radiation, we can define an effective ‘particle’
extinction cross section Qe() in terms of leaf area as:
Qe   Al Gl 
(3.15)
ignoring any dependence of canopy properties on z, for constant leaf area Al. Gl   is
the foliage area orientation function (the ‘G-function’), a dimensionless geometry
factor equal to the projection of a unit area of foliage on a plane perpendicular to the
direction  , averaged over elements of all orientations:
Gl  
1
2
 g     d 
l
l
l
l
(3.16)
2 
For a spherical leaf angle distribution ( gl l   1), this is simply 0.5. It can be
similarly shown that for an azimuthally independent distribution, for the special case
of a completely horizontal distribution ( m   / 2 ,   1 in an elliptical description),
Gl  cos . For a completely vertical distribution, Gl  2   1  cos 2  (Ross,
1981).
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Myneni et al. (1989; p. 29) show G-functions for a variety of measured canopies.
After Ross (1981), they comment that:



the range of G-functions for the measured canopies is relatively small (0.3-0.8)
and is considerably smoother than the measured leaf inclination distributions;
for near planophile canopies, the G-function is high (>0.5) for low  and low
(<0.5) for high  ; the converse is true for near erectophile canopies;
the G-function is always close to 0.5 between 50 and 60 degrees, being
essentially invariant at 0.5 over different leaf angle distributions at   57.5o .
0.8
0.7
0.6
G _l(theta)
0.5
0.4
0.3
0.2
0.1
0.0
0
10
20
30
40
50
60
70
80
90
z enith angle / degrees
s pheric al
plagiophile
planophile
ex trem ophile
erec tophile
Figure 3.2 Leaf Projection Functions ('G-functions')
for Archetype Leaf Angle Distributions
Equation 3.16 expresses the G-function as a simple geometric attenuation factor for
‘blocking’ of unpolarised radiation. Note that it does not vary with wavelength, but
that it will generally vary with  . From equations 2.1, 3.11 and 3.15, we can write:
 e  ul G
(3.17)
which is the more usual way of expressing the extinction coefficient for canopies at
optical wavelengths (Ross, 1981). Inserting this into equation 3.9, we can see that the
attenuation of radiation is exponential and controlled by path length l, leaf area
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density, and the normalised leaf projection function. In a plane parallel canopy, l can
be expressed as:
l
z
(3.18)

where   cos  (figure 3.1). Note that z is defined from 0 at the top of the canopy to
–H at the soil layer in these notes (figure 3.1). The optical depth at the bottom of the
canopy (z=-H) is then:
t l
 H     e dt 
t 0
z  H

z 0

ul G

dz 
LG

i.e. the radiation at the bottom of the canopy is I 0 e
(3.19)

LG  

 I 0e

0.5 L

for a spherical leaf
L
angle distribution,  I 0 e for a horizontal distribution.
Note that for the optical case, we typically define the single scattering albedo of a
particle (leaf), (), which varies as a function of wavelength, through reflectance,
l() and transmittance, l(), so
()=l()+l()
(3.20)
3.2.2 Extinction and Scattering in a Rayleigh Medium
The optical case is ‘straightforward’ and relatively intuitive, as we can use geometric
optics principles to consider blocking (interception) of the radiation in defining the
extinction coefficient or optical depth. For a more general case, we turn to the
example of a so-called Rayleigh medium within which the particles are assumed
spherical, of radius a. It is further assumed that the particle radius is small relative to
the wavelength  in the dielectric material of the particle. This condition can be stated
as:
ka 
2a

 1
where k is the wavenumber ( 2 /  ), with  in the same linear units as a (Ulaby and
Elachi, 1990; p.141). This condition holds for scattering by gasses in the atmosphere
(Fröhlich and Shaw, 1980) and for long wavelength microwave (P-band) extinction
by vegetation canopies. Where this is not the case, we must apply modifications to the
extinction and scattering coefficients, as shown for discs and needles in section 1.3
(equations 1.7 and 1.8). We do not attempt to derive the extinction coefficient here,
rather we state it and examine its properties. A Rayleigh scattering layer can be
characterised by two parameters: the single scattering albedo    s /  e (equation
3.14); and the optical depth    e d (equation 3.13) for a path length d for constant
e.
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For a canopy in air, the scattering coefficient for a random medium of spherical
Rayleigh scatterers is (Fung, 1994; p. 122-124; Ulaby and Elachi, 1990; p. 142):
2
 1
 1
8
s 
Nvk 4a6 s
 2 fk 4 a 3 s
3
s  2
s  2
2
(3.21)
where f is the volume fraction of scatterers ( (4 / 3)a 3 N v ),  S (=  s  i s ) is the
dielectric constant of the sphere material (leaves) and k is the wavenumber in air. The
absorption coefficient is:
3
 a  fk s
s  2
2
(3.22)
Figure 3.3 Single scattering albedo for 70% leaf GMC as a function of leaf radius
From this we see that:

1
9 s
1 3 3
2
2
2k a  s  1   s

(3.23)

We can see that the single scattering albedo increases for increasing leaf radius (figure
3.3) and decreases with increasing wavelength. Equation 3.23 shows that it is
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effectively a function of the ratio of leaf linear dimension to wavelength, modulated
by leaf moisture (dielectric constant). As the leaf dielectric constant is a function
wavelength, the relationship of single scattering albedo with normalised leaf radius
(leaf radius over wavelength) is not obvious, but figure 3.4 (with dielectric constants
for 70% GMC) demonstrates that the impact of the variation in dielectric constant
with wavelength is typically small compared to the overall behaviour.
Figure 3.4 Single scattering albedo for 70% leaf GMC
as a function of leaf radius to wavelength ratio
Note that when the leaf radius is more than about 0.04 times the wavelength, the
single scattering albedo is no longer small (‘small’ being conidered as 0.2 or less
here). The condition for validity of the model given above implies a/ << 0.16, so a
threshold of 0.04 would seem generally reasonable. We can therefore state that for
leaf dimensions and wavelengths within the region of validity of the Rayleigh
scatterer assumption, the single scattering albedo is low.
The optical depth is given by:

fHk
 ( s  2
2
9   2k a    1
   )
3
2
s
2
3
s
2
  s

(3.24)
s
at the bottom of the canopy (z=H), where  is the cosine of the zenith angle (figure
3.1). We see that the optical thickness of the canopy depends on wavenumber, the size
of the scatterers, the volume of scatterers per unit area (fH) and the real and imaginary
parts of the leaf dielectric constant.
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Figure 3.5 Normalised optical depth for 70% leaf GMC as a function of leaf radius
Figure 3.5 shows the dependence of normalised optical depth ( /(fH)) on leaf radius.
The general trend is an increase in optical depth with increasing leaf radius.
Figure 3.6 shows the same information, on a linear scale, for leaf sizes limited to
within the assumed range of validity of the model ( <=0.2). Note that the rate of
increase in optical depth increases significantly with decreasing wavelength, and that
it is essentially flat for L- and P-band over the range of validity of the model.
For spherical scatterers, the extinction matrix  e is equal to a 4x4 identity matrix
multiplied by  e   s   a  . Since the extinction term is effectively scalar, we can
use Beer’s Law to express extinction within the Rayleigh medium (see equation 3.9):
I z   I 0 e

 I 0e

fzk
 ( s  2 ) 2
9   2k a  1 
2
3 3
s
s
(3.25)
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Figure 3.6 Normalised optical depth for 70% leaf GMC as a function of leaf radius for
range of validity of model (threshold at =0.2)
The phase matrix P , of a Rayleigh medium is defined as a function of incident and
scattering polar angles (  ,  ) with subscripts i and s respectively is (Ulaby and
Elachi, 1990):
 P11
P
P  s ,  s ,i ,  i    21
 P31

0
P12
P13
P22
P23
P32
P33
0
0
0 
0 
0 

P44 
(3.26)
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where

P11


w sin 2  s sin 2 i  cos 2  s cos 2 i cos 2  s   i 
P21
 2 sin  s sin i cos  s cos i cos s   i 

w cos 2  s sin 2  s   i 
wsin  s cos  s sin i sin  s   i 

 cos 2  s cos i sin  s   i  cos s   i 

w cos 2 is sin 2  s   i 
P22

P23

P31

P32

P33

P44

w

P12
P13

w cos 2  s   i 
 w cos  s sin  s   i  cos s   i 
w 2 sin  s sin i cos i sin  s   i 

 2 cos  s cos 2 i sin  s   i  cos s   i 
2 w cos  s sin  s   i  cos s   i 
wsin  s sin i cos s   i 


 cos  s cos  cos 2  s   i   sin 2  s   i 
wsin  s sin i i cos s   i   cos  s cos i 
3
s
8
(3.27)
The defined phase matrix is dependent on the incident and scattering geometry (but
not on the orientation of the scatterers as they are spherical) and on  s . This depends
in turn on  S (thus essentially on leaf water content), on k4 (so -4 – scattering
decreases as wavelength increases), and on the total volume of scatterers (product of
fractional volume and a3).
3.2.3 A Solution to the Vector Radiative Transfer Equation
For a medium of low scattering (single scattering albedo, , is low) we can use the
iterative method to solve the RT equation (vector or scalar). At visible wavelengths, 
is generally low, but it increases dramatically to close to 1.0 in the near infrared (see
figure 2.6). In the microwave, (see section 3.2.2 for Rayleigh scatterers) can often
be small, particularly for low leaf moisture, small leaves, or long wavelengths.
Under these conditions, a solution to the VRT (or SRT) equation can be obtained by
first computing a zero-order solution, then plugging the resultant terms in as source
terms to compute first order scattering (Ulaby and Elachi, 1990; pp. 146-151).
Second+ order scattering terms are then calculated by inserting the results of the
previous order as source terms. Under the condition that is low, the terms rapidly
converge to zero for increasing scattering order. Generally, only up to second order
terms are used with this method.
Considering a two-stream model concerning the propagation of upward- and
downward-going intensities, I+ and I- respectively, we can write the VRT equation as:
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d 
I  s , z 
dz


Lewis & Saich
e 

I  s , z   F  s , z 
s
(3.28)

 
d 

I   s , z    e I   s , z   F   s , z 
dz
s
with source functions:
F  s , z 


1 
  

P  s ,   I   , z d  


2


 

s 
  P 
2
s
 


,  I    , z d  

 

(3.29)
F   s , z  

1 
  

P   s ,   I   , z d  

 

 s  2 
  P  
2
s
 


,  I    , z d  

 

subject to boundary conditions (figure 3.9):
I   s , z  0  

I 0  s   0 
I  s , z   H   R 23 I   s , z   H 

(3.30)

where R 23 is a specular reflectivity matrix for a ‘flat’ soil, describing scattering at the
interface between medium 2 (air) and medium 3 (soil), and (x) is a delta function
which is unity for x=0 and zero elsewhere. It is obtained by exchanging subscripts 23
for 12 in equation 2.6.
Following Ulaby and Elachi (1990; p.148) we rephrase the differential equations in
3.28 as integral equations:
I  s , z 

   z   H   
 I  s , z   H  
T   e
s



z z
  e  z  z   
z H T    s  F  s , z dz 
(3.31)
I   s , z 

  z
T   e

  s
 
 I   s , z  0  

z  0    z  z   
e
z z T     s  F   s , z dz 
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Lewis & Saich
where T(-x)=e-x is an attenuation (extinction) term due to Beer’s Law.
Figure 3.7 Upward scattering terms
Figure 3.8 Downward scattering terms
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In the upper equation we see (figure 3.7) the upward intensity defined by an
attenuation of the upward intensity at the canopy bottom over distance z-(-H), along
with an integral (sum) from the canopy base to height z of the upward source term,
attenuated over vertical distance z - z’.
In the lower equation we see the downward intensity at the canopy top is attenuated
over the distance z, with the addition of the integral of a source term from canopy top
to height z attenuated over a distance z-z’ (figure 3.8).
Inserting the boundary conditions (figure 3.9):
I  s , z 

  z   H  
   H  
 R  0 T   e
 I 0   s   0  
T   e
23
s
  0 



z0
  z   H  
  e  z  z   
 R  s  

 F   s , z dz 
T   e
T

23
z   H 



s
s




z z
   z  z   
 F  s , z dz 

T   e
(3.32)
z  H
s



I   s , z  
  z
T   e  I 0   s   0  
  0 
z   0    z  z   
e
z z T     s  F   s , z dz 
we see the upward intensity now to consist of:
(i)
(ii)
(iii)
a double attenuation of the incident intensity (over distance z-(-H) at
angle s and distance –H at angle 0);
an integral of the downward intensity (attenuated over distance –z-z’
within the integral) over the canopy height, reflected at the canopy
base an attenuated over distance z-(-H);
an integral from the canopy base to height z of the upward intensity
attenuated over distance z-z’.
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Figure 3.9 Boundary conditions
3.2.3.1. Zero-Order Solution
The zero order solution is arrived at by setting the source terms in equation 3.28 to
zero (Ulaby and Elachi, 1990; p.148):
I
 0 
I
 , z 
 0 
, z 

  z
T   e  I 0  s   0 
 0 
  z  H  
  H
 R 23  0 T   e  I 0  s   0 
 T   e
0


 0 
(3.33)
The downward intensity is the incident intensity attenuated by a Beer’s Law term over
height z. The upward intensity comprises the incident intensity modulated by a
downward attenuation over height H, reflectance at the ground interface, and
attenuated over height z-(-H).
3.2.3.2. First-Order Solution
Using the iterative method, we obtain the solution to first-order scattering by inserting
the zero-order solution as the source terms in equation 3.29.
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F
 0 
 s , z
I0

e
s
  e z 
 

 0 
Lewis & Saich
  e 2H 

 

 P s , 0 e  0  R 23  0   P s ,0 


(3.34)
F
 0 
  s , z 
I0
s
e
  e z 
   
0 


 P  s , 0 e


  e 2H 
  

0


R 23  0   P  s ,0 
The upward intensity for first order scattering (Ulaby and Elachi, 1990: p. 149) is:
e  e
I
(1) 
 s , z  
R 23  s e
 e
( z 2 H )
s
 z 2 H 
0
R 23  0 I 0  s   0  
H

 e  


e  0   1H
1  e  2H  I 0
e  1  P  s , 0 

 R 23  0 P  s ,  0 
1
 2   s







 2H 
  e  z 
 2H

 e 
 e 2 z
0 
  e

e
R 23  0 P s ,  0 e  0 
 P s , 0  e1z  e1H
2





 I
0


s
where 1=e(1/0+1/s); 2=e(1/0-1/s). The solution for upward intensity at z=0,
for s=0, s=+0 is:

e
I
(1) 
0 , z  0  R 23  0 e

2
0
2
0
R 23  0 I 0  0   0  



0
 0e

 R 23  0 P  0 ,  0  2


 2 
 e 0  1 P  0 , 0 H  I 0 
(3.35)


0


2
2



 I
0

 R 23  0 P 0 ,  0 e  P 0 , 0  1  e 0  H 0

  s





Noting e 2 H  1   2 H f or small  2 , so e 2 H  1 /  2  H for  2  0 . We can
recognise the various terms in this result as:

1. e
2
0
R 23 0 I 0 0  0 
direct intensity reflected at the canopy base, doubly attenuated through the
canopy;
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Figure 3.10 First-order microwave scattering mechanisms
2. R 23  0 e

2
0

R 23  0 P 0 , 0 
e

0
 2

 e 0  1 I 0

2 

A ‘double bounce’ term involving a ground interaction, (downward)
volumetric scattering by the canopy, and an additional ground interaction.
The term includes a double attenuation (on the way to the ground the first
time & back from the ground at the end) and attenuation as part of the
volume scattering. This term is generally very small and is often ignored.
3. R 23 0 e

2
0
P 0 ,0 H
I0
0
A downward volumetric scattering term, followed by a soil interaction,
including a double attenuation on the upward and downward paths.
4. R 23 0 P0 , 0 e

2
0
H
I0
s
A ground interaction followed by volumetric scattering by the canopy in
the upward direction, again including double attenuation.
2

 I
0 

5. P0 ,0  1  e H 0

 s


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Pure volumetric scattering by the canopy. Note that term expresses the
equivalent of ‘path radiance’ in atmospheric applications.
Remembering that the phase function depends on  s (equation 3.27) and that this is a
function of the volume fraction (sphere volume per unit volume) of spheres (equation
3.21) we can interpret the canopy height factor, H, which always (other than
interaction 2) appears with the phase function as expressing a dependence on the
sphere volume per unit area.
The phase function for spherical scatterers is given in equations 3.26 and 3.27. We
noted that the extinction coefficient for spherical scatterers was essentially scalar,
allowing us to treat it simply in the formulation above. In the general case both the
extinction terms and the phase matrix will also be functions of scatterer orientation,
and may include cross-polarised components.
The backscatter coefficient is given by Ulaby and Elachi (1990; p. 150):
 0pq 0  
40 I ps 0 ,  0   , z  0
I 0q
(3.36)
The second order solution is similarly obtained by setting the first-order solutions as
the source terms. Note that whilst there are no cross-polarisation terms for spherical
(Rayleigh) scatterers, they do occur in second+ order scattering, i.e. cross-polarisation
for spherical scatterers is a result of multiple scattering. The second order solution is
given in Ulaby and Elachi (1990; pp. 150-151).
3.2.4 A Solution to the Scalar Radiative Transfer Equation
It is interesting to note at this point the term ‘first-order interactions’ used in the
microwave literature to describe these five terms. Essentially, it is used to express
terms which have a single interaction with the vegetation canopy. In the optical
canopy modelling literature, only terms 1 and 5 would be included in a first-order
scattering model (the other terms having multiple interactions with soil and
vegetation).
Other than this, we can proceed for the scalar approach for optical wavelengths in
much the same way as section 3.2.3. One other issue we must be aware of is that we
are not (generally) interested in a backscattering term – we need to keep a dependence
on incident and scattered intensities in the fomulation.
We have already dealt with the extinction coefficient for the optical case (section
3.2.1), where we noted  e   ul G (equation 3.17). We will therefore generally
also need to allow this extinction term to vary with the angle of the incident radiation.
The phase function at optical wavelengths is often expressed as:
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P   
Lewis & Saich
1
u   
 l
(3.37)
where ul is the leaf area density, ’ the cosine of the incident illumination zenith
angle, and  the area scattering phase function.
   

1
4
1
4
  , g     
l
l
l
l
  l d l 
2 
  , g     
l
l
l
(3.38)
l
  l d l
2 
where l and l are the leaf directional reflectance and transmittance factors
respectively (Ross, 1981). We can see this as a double projection of the leaf angle
distribution, modulated by reflectance for the upper hemisphere and transmittance for
the lower hemisphere, in much the same way as the G-function expresses a projection
in the direction of the radiation for the extinction coefficient.
A typical assumption used in building a canopy reflectance model is to let the leaves
be biLambertian scatterers. In this case, the angular dependence is removed from the
reflectance and transmittance, which can be taken outside the integral. This is
sometimes modified by the addition of a ‘rough surface’ specular component from the
leaves (Nilson and Kuusk, 1989). If the reflectance is assumed equal to transmittance
(or a linear function thereof) we can more simply express the spectral dependence in
terms of the single scattering albedo and a weighting of the upper and lower
hemisphere integrals.
To formulate for first order scattering in the optical (scalar) case, we consider firstorder interactions (as in only one interaction with canopy or soil elements) in a scalar
expression of equation 3.32:
I  s , z  

e
  e  s  z   H  
 

s


I0
s

z  z
z   H
e
 soil  s , 0 e
  e  s  z  z    e 0  z
 

s
0


e
  e 0   H  
 

0


I 0  s  0  
(3.39)
P 0   s dz
where soil is the directional reflectance factor of the soil. Inserting the phase function
defined in equation 3.37:
I  s , z  

e
  e  s  z   H  
 

s


e
  e 0   H  
 

 0


 soil  s , 0 I 0  s  0  
  e  s  z  z    e 0  z

s
0

I 0ul     z z  
z He
 s 0
e
(3.40)
dz
so:
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I  s , z  

e
        
    
 H  e s  e 0   z  e s 
0 
 s
 s 
e
Lewis & Saich
 soil  s , 0 I 0  s  0  
  e  s  z 
       

z e s  e 0 
 s  z z
0 
 s
z   H
I 0ul      
e
 s 0

e
(3.41)
dz
       
d  e s  e 0 
0 
 s
We refer to the term e
in the integral as the joint gap probability at
optical wavelengths, as it is the probability of incident radiation passing from the top
of the canopy to depth d and being able to pass unhindered in the scattered direction.
We shall consider this term further later, but can note for the present that it is
currently expressed as the product of two Beer’s Law attenuation terms.
Performing the integral to give the intensity at z=0:
e
I  s , z  

       
 H  e s  e 0 
0 
 s
 soil  s ,  0 I 0  s   0  
       

H  e s  e 0  
I 0ul    
1  e   s 0  

        

 s  0  e s  e 0  
0 
 s
so:
e  H 3  soil  s , 0 I 0  s  0  

I  s , z  
I 0   
1  e H 3
 e  s  0   e 0  s


(3.42)
 e  s  0   e 0  s
. Inserting the optical extinction coefficient
 s 0
(eqnation 3.17):
where  3 
 3  ul
G  s  0  G 0  s
(3.43)
 s 0
Since the LAI, L=ulH:
I  s , z  

e
 G   s  0  G   0  s 

 L 
 s 0


 soil  s ,  0 I 0  s   0  
 G  s  0  G   0  s 

 s 0

 L 
I 0     
1 e 
G  s  0  G  0  s 





(3.44)
The canopy directional reflectance factor is simply found by dividing terms in
equation 3.44 by I0. We can recognise the following two mechanisms for first-order
scattering at optical wavelengths from this (figure 3.11):
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Vegetation Science – MSc Remote Sensing UCL
1. e
 G s 0 G 0  s 

 L 
 s 0


Lewis & Saich
 soil s , 0 
radiation travelling through the canopy to the soil, being reflected, and
travelling out of the canopy in the scattering direction. The radiation is
subject to a double attenuation due to the two paths through the canopy.
The (Beer’s Law) attenuation depends on the (zenith) angles of viewing
and illumination and on the leaf projection functions. The exponential term
includes a dependency on LAI – one-sided leaf area per unit ground area,
so we can see the soil contribution as small for high LAI (or high zenith
angles of high G-functions), but small for low LAI.
 G   s  0  G   0  s 

s 0


 L
   
1  e 
2.
G  s  0  G 0  s 





volumetric scattering by the canopy, dependent on: the area scattering
phase function (a direct dependence on leaf single scattering albedo) and
on the double projection of the leaf angle distribution; an inverse
dependency on the viewing and illumination angles and G-functions; an
LAI (and G-function and zenith angle) dependency which is small for low
LAI and tends towards 1 for high LAI.
Figure 3.11 First-order optical scattering mechanisms
We note that for the special case of a spherical leaf angle distribution for biLamberian
scatterers:
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Vegetation Science – MSc Remote Sensing UCL
G  

Lewis & Saich
0.5
l  l
sin    cos    l cos 
3
3
    
(3.45)
where  is the phase angle between viewing and illumination angles ( cos      ).
Similar simple formulae can be obtained for purely horizontal or purely vertical
leaves (Ross, 1981; pp. 254-258).
Further, if we assume reflectance to be linearly related to transmittance
( k  1  l / l ):
    
l k 


 sin   (   ) cos  
3 
k

(3.45)
and we can write the first order scattering from the canopy as:
e

1
c

L   s
  0
2   s 0



 soil  s ,  0  
L     s   (3.46)


  0
2
l k 


2   
 sin   (   ) cos  1  e  s 0  
 s   0 3 
k



This simple analytical expression allows us to gain an insight into the factors affecting
first-order canopy scattering. We will investigate further uses of this expression in a
following lecture.
Note that the single scattering albedo for leaves is not always low. This is particularly
true in the near infrared. In this case, we still need to account for further orders of
scattering. The iterative method used above for the microwave case is not directly
appropriate, but a range of methods exist to solve for multiple scattering. In general
terms, we can state that the multiple-scattered component will be high for high LAI
and high single scattering albedo. As a multiple-scattered term, it is less dependent on
the specific leaf, viewing, and illumination geometries, and is typically an upward
shallow ‘bowl’ shape when plotted as a function of viewing zenith angle.
3.2.4 Modifications to a Simple radiative Transfer Approach
3.2.4.1 Optical Canopy Hotspot
Under the far field approximation, where scatterers are distant from one another, the
optical gap probabilities of incoming and outgoing radiation can be assumed
independent and can be described by Beer’s Law. Thus, the joint gap probability, Q
is:
Q(    )
-L
=
e
G(  )   + G(  ) 
 
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Vegetation Science – MSc Remote Sensing UCL
Lewis & Saich
We came across this term in equation 3.41, in considering the radiation reaching a
given level of a canopy and being able to escape from the canopy.
However, when the viewing vector is leaving in the same direction as the incident
solar radiation, using this equations we obtain:
-
Q(    )
=
2 L G(  )

e
In this direction, we must consider that the conditional probability of photons being
able to follow the same path back up through the canopy as they took on their way
down through the canopy is 1.0; thus we require:
-
Q(    )
=
L G(  )
e

Denoting the single (Beer’s Law) gap probability P(), we can state that we require
some compensation factor, C(Ω',Ω):
Q(    )
P(  ) P(  ) C( , )
=
where:
C( , )
=
1
P(  )
in the retro reflection direction. Away from the retro reflection direction, we require:
C( , )
=
1
From a mathematical standpoint then, we have the basis of a functional form for this
correction factor. The enhanced joint gap probability in the retro reflection direction
gives rise to a localized peak in reflectance. This feature is known as the hot spot. The
angular width of the joint gap probability (of the hot spot feature) can be shown to be
a function of the ratio of the average leaf size to canopy height - a dimensionless
'roughness' parameter.
In studying soils or other rough surfaces, the angular width is again found to be
directly related to a dimensionless measure of roughness.
There have been many formulations of a hot spot correction factor, both from an
empirical standpoint and from physically-based modelling. The model of Pinty et al.
(1989) is physically-based, being derived from considerations of the common volume
between two cylindrical voids in the canopy. The depth at which a cylinder of radius r
from the viewing direction will overlap with one from the illumination direction is
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Vegetation Science – MSc Remote Sensing UCL
Lewis & Saich
denoted Li. A more physically-consistent hot spot model, provided by Nilson and
Kuusk (1989), is based on considerations of random overlapping of horizontal circular
disc leaves.
Other hot spot formulations exist: the vast majority are simply variants on the themes
presented above, but many do not consider reciprocity of the joint gap probability
well, preferring to opt for empirical 'correction' terms.
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Lewis & Saich
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