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The Rayspread Canopy Reflectance Model 223
Lateral Boundary Conditions
Within Rayspread, the computations associated with the physical interac-
tions of a primary ray are pursued until 1) an absorption event occurs, 2)
the primary ray exits the world object upward, or 3) a user-specified max-
imum event number is reached. Contrary to a pure MC scheme, primary
rays are used only to sample the scene’s geometric and scattering properties
and thus do not contribute to any BRF measurements. To simulate the hor-
izontal exchange of primary (and secondary) rays with neighbouring areas
cyclic boundary conditions may be applied, that is, laterally exiting rays are
re-ingested into the world object through the side opposite the one of their
departure without affecting their direction of travel. This assumes that the
simulated scene is part of a larger and similarly structured domain.
2.2 Benchmarking of the Model
The 3-D virtual laboratory of [GV98] was evaluated both against actual labo-
ratory BRF measurements and with respect to a panoply of other RT models
in the context of the RAdiation transfer Model Intercomparison (RAMI)
exercise [PWT04]. Among others, RAMI compares the total, single and mul-
tiple collided BRF components generated by its participating RT models
over a variety of structurally homogeneous and heterogeneous canopy rep-
resentations, similar to those presented in Fig. 2. The results displayed on
the web site http://rami-benchmark.jrc.it/ indicate that the Raytran
model of [GV98] is in close agreement with other 3-D RT models in the het-
erogeneous cases, and was deemed accurate enough to contribute towards
the “most credible” BRF solutions for the various homogeneous test cases.
Given these credentials, and since the Rayspread model utilizes many of the
geometrical routines and radiation transfer formulations implemented within
the model of [GV98], a direct comparison of their various BRF components is
a first step in the validation process of the Rayspread model. The two panels
of Fig. 3 show that the implementation of the photon spreading acceleration
technique did not introduce any bias between the Raytran and Rayspread
generated BRF values for the discrete homogeneous (left) and heterogeneous
(right) test cases of the second phase of RAMI. Included in these graphs
are BRF simulations along different planes of observation, a wide range of
illumination conditions as well as spectral surface properties.
Since these simulations do not permit an evaluation of the Rayspread
model in absolute terms, the black canopy BRF component for a vertically
and spatially homogeneous turbid medium canopy with spherical LND and
Lambertian soil was generated (using different numbers of primary rays), and
subsequently compared against the analytical true solution, which is:
BRF(θ
v
)=exp
−0.5 LAI
|µ
0
|
α
soil
exp
−0.5 LAI
|µ
v
|
224 J.-L. Widlowski et al.
RAMI Phase 2 HOM DIS
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
BRF
Rayspread
-0.010
-0.005
0.000
0.005
0.010
D = (BRF
Raytran
- BRF
Rayspread
)
D = -5.47 10
-5
RAMI Phase 2 HET DIS
0.0 0.1 0.2 0.3 0.4 0.5 0.6
BRF
Rayspread
-0.004
-0.002
0.000
0.002
0.004
D = (BRF
Raytran
- BRF
Rayspread
)
D = 4.40 10
-5
Fig. 3. The difference, D between BRF simulations of the Raytran and Rayspread
models over the entire set of homogeneous (left panel) and heterogeneous “floating
spheres” (right panel) test cases of the RAMI (phase 2) exercise. The mean deviation
between the two model outputs,
D is also indicated.
where µ =cosθ, LAI = Λ(z) H is the leaf area index, α
soil
is the albedo of
the soil, and Beer-Bouger’s exponential law is applied to both the downward
and the upward transmission probabilities within a canopy with G(θ)=0.5.
The left panel of Fig. 4 shows how the normalized absolute BRF difference ,
when averaged over 119 view directions (θ
v
≤ 75
◦
) and multiple model runs
with different starting seeds, decreases as a function of the number of pri-
mary rays. To assess the efficiency of the speed-up, results are shown for two
contrasting scenarios: 1) a sparse canopy with bright soil and a small solar
zenith angle (thick lines), and 2) a very dense canopy with dark soil and a
large solar zenith angle (thin lines). As can be expected for MC simulations,
the results of both cases improve as a power law: ∝1/
√
N
in
for both
Rayspread (solid lines) and Raytran (dashed lines) provided that the latter
receives sufficient numbers of primary rays into its angular BRF bins. This is
particularly visible in the dense canopy scenario where the true BRF(θ
v
=0)
≈ 0.000168, and it takes Raytran more than 10
6
primary rays to guaran-
tee sufficient BRF contributions. For case 1, any primary ray number in the
range 10
3
to 10
7
yields an improvement in the accuracy of Rayspread sim-
ulations that is constant with respect to that of Raytran and given by the
offset between the two (thick) lines:
rayspread
raytran
/ 50. By the same
token, in case 1 the Raytran model will need about 2500 times the number of
The Rayspread Canopy Reflectance Model 225
10
2
10
3
10
4
10
5
10
6
10
7
10
8
Number of primary rays
10
4
10
2
10
0
10
2
10
4
Average absolute normalized difference <ε> (%)
soil
α = 0.89
soil
α = 0.10
θ = 20
0
o
θ = 50
o
0
LAI=2.0
LAI=5.0
Homogeneous Turbid Scene: Black Canopy BRF
10
2
10
3
10
4
10
5
10
6
10
7
10
8
Number of primary rays
10
4
10
2
10
0
10
2
10
4
Average absolute normalized difference <ε> (%)
Homogeneous turbid canopy, LAI=3, red
Heterogeneous discrete canopy; red
RAMI scenes: Total BRF
Fig. 4. The mean value of the normalized absolute difference, that exists in
between a set of reference BRF values and simulations by the Rayspread (solid
lines) and Raytran (dashed line) models as a function of the number of primary
rays used by these models. Left panel: results for the black canopy BRF contribution
over two homogeneous turbid medium canopy scenarios (thick and thin lines) where
the reference BRFs are known theoretically. Right panel: results for the total BRF
over the homogeneous (thick) and heterogeneous “floating spheres” (thin) test cases
of RAMI. The reference BRFs were given by a Rayspread run with 10
9
primary
rays
primary rays of the Rayspread model to achieve a similar accuracy level (for
case 2 and N ≥ 10
6
the advantage of the Rayspread model is even larger).
It can be seen, for example, that the Rayspread model lies within 1% of the
true solution (of case 1) after only 10,000 primary rays, whereas Raytran
requires more than 10
7
rays to achieve a similar BRF accuracy.
The right panel of Fig. 4 displays the same uncertainty measure, i.e.,
the mean value of the normalized absolute difference , existing between
a reference set of total BRFs and the corresponding simulations carried out
by the Rayspread (solid lines) and Raytran (dashed lines) models over the
homogeneous (thick) and heterogeneous “floating spheres” (thin) test cases of
RAMI (these data were also part of Fig. 3). Based on the previous discussion
the reference BRFs were taken from a Rayspread reference run with 10
9
primary rays. Again, the slope of the various lines is close to –0.5, and the
offset between Rayspread and Raytran indicates that the latter will need
about 4500 times the number of primary rays of the former to achieve a
similar accuracy level . Figure 5 shows the quasi-linear speed-up that can
be achieved by running Rayspread in parallel mode on a small PC/Linux
cluster. [GV96] found that this linearity was maintained for at least 100
processors in the case of the Raytran model. When run on a single processor,
it took ∼17 minutes for a 2.4 GHz machine (2 Gb RAM) to compute the total
226 J.-L. Widlowski et al.
2 4 6 8 10
Number of processors, N
2
4
6
8
10
Speed Up
RAMI "Floating Spheres" scene
750,000 scatterers
Speed Up = time (1 proc) / time (N proc)
Fig. 5. Speed-up in computation time achieved when running Rayspread on in-
creasing numbers of processors using the Message Passing Interface (MPI) as a
communication layer between the various nodes
BRF using 100,000 primary rays in the near-infrared over the heterogeneous
“floating spheres” scenario with 750,000 individual scatterers.
2.3 Case Study: Impact of Woody Structure on BRF Pattern
The woody structure of vegetation canopies is in general disregarded in
canopy reflectance simulations, despite the fact that land surface-atmosphere
interaction models do acknowledge their effect on surface albedo, canopy ab-
sorption, evaporation and run-off [ZSD02]. The central graph of Fig. 6 shows
variations of the total BRF, in the red spectral band and along the princi-
pal plane, obtained over 3-D canopy representations where first the branches
within the crowns, and then also the trunks themselves were omitted. Shown
are BRF simulations over a mixed coniferous – deciduous forest target in
late autumn. Only the Scots Pine trees still feature their crowns, which were
modelled to contain 1) a vertical LAI gradient, 2) a foliage free zone (of verti-
cally varying width) that surrounds the main stem, and 3) five branches per
whorl (as well as multiple second order branches) [SKK94] – as displayed in
the left panel of Fig. 6. All necessary spectral and structural tree characteris-
tics were compiled from the available scientific literature [WPG03]. Assuming
The Rayspread Canopy Reflectance Model 227
-60 -40 -20 0 20 40 60
Viewing angle
0.10
0.15
0.20
0.25
0.30
Bidirectional Reflectance Factor
0.381
Full canopy
0.418
0.303
Only crowns
Full canopyNo branchesOnly crowns
Canopy absorption
No branches
Red
vertical
volume
Fo iage−free
Branching pattern
Branching
levels
LAI profile
projected crown
Trunk
area
Fig. 6. The middle panel shows the total BRF simulated in the principal plane for
the three forest representations depicted in the right panel: Full tree representation –
as shown in the left panel – (solid line), no branches within crowns (dashed line), and
no woody elements in canopy (dotted line). Also indicated is the canopy absorption
in each case. Note that the woody structures were brightened for better contrast in
the right hand colour plates
Lambertian scattering properties, the reflectance (transmittance) values for
the foliage was set to 0.054 (0.014), whilst woody elements had a reflectance
of 0.142, and α
soil
= 0.125. The density of trees with (without) crowns was
106 (331) stem per hectare and their height was 24.0 (19.8) m, resulting in
an LAI of 1.50 for the entire scene (250 × 250 m
2
).
As expected, the presence or absence of woody elements does not affect
the overall width of the hot spot which is primarily governed by the size (and
shape) of the tree crowns. On the other hand, it was found that the omis-
sion of woody elements in 3-D canopy representations reduced the canopy
absorption (in this particular scenario by 27.4%), and increased the direc-
tional hemispherical albedo values of the canopy. More specifically, canopy
representations without the (mainly horizontal) branch structures lead to in-
creases in the surface leaving radiation field at predominantly small θ
v
values,
whereas the omission of the (mainly vertical) tree trunks showed its largest
impact at increased θ
v
values. In the latter case, BRF values in the forward
scattering direction (positive θ
v
) were mostly affected.
3 Conclusion
A concise overview of the impact of vegetation structure on the radiative
transfer within plant canopies was given. Furthermore, in the context of in-
verting 3-D RT models against satellite observations the need for speedy, yet
228 J.-L. Widlowski et al.
accurate and reliable, RT simulations over realistic 3-D canopy architectures
have been highlighted. To this effect, the Monte Carlo ray-tracing model
Rayspread makes use of a variance reduction technique, known as photon
spreading, and the capability to operate in a multi-processor environment.
Verification and benchmarking tests of the Rayspread model, as well as, a
small study regarding the impact that 3-D woody structures may have with
respect to the surface leaving BRF field are also provided.
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Part III
High Energy Density Physics