2 Multiple scattering experiment
For the first experiment we will examine in detail the behaviour of multiple scattered
radiation within the barley canopies. Using the 3-D geometric representation of the
measured barley data within the BPMS, it is possible to track the scattering behaviour of
photons ‘fired’ into the canopy through each interaction. In this way a picture of the
multiple scattering within the canopy can be built up.
In general when modelling canopy reflectance, multiple scattering is either ignored
(in the more primitive models, ref ??) particularly in the visible wavelengths where it
may be an order of magnitude smaller than the singly scattered radiation, or modelled as
an approximation to the solution for radiative transfer in a homogenous medium (such as
demonstrated in Hapke, 1). This has implications for the level of structural detail that
may be inverted from such models. However, if the scattered radiation field within the
canopy can be characterised at each sepearate soil/canopy interaction, the relative
importance of each level of scattering, and the contribution from each order of scattering
can be calculated. This enables the multiple scattering behaviour of the canopy under a
variety of conditions (such as varying row or solar azimuth angles) to be investigated. It
also allows the evaluation of the various assumptions made in the formulation of multiple
scattering terms in other canopy reflectance models.
The assumptions made in this analysis are that leaf reflectance is equal to leaf
transmittance, and that both these and the soil reflectance are Lambertian. This allows for
mulitple leaf-soil or leaf-leaf scattering events, but not soil-soil scattering events.
Contributions to the total multiple scattered reflectance of the canopy are calculated as a
function of scattering order i. Successively higher orders of scattering take increasingly
long times to compute, so this experimant is limited to 8 orders of scattering. This is
enough to contain the vast majority of canopy scattering information as the relative
magnitude of the scattering events decreases exponentially with i.
To provide sufficient information to allow the scattering at each order i to be
calculated, a set of twenty ‘pseudo-wavelengths’ are used. The soil and leaf reflectances
at each wavelength are chosen arbitrarily to provide a set of different soil and canopy
contrasts. Physcial reality of the simulations is not a requirement in this case, so numbers
are chosen for convenience. In this way the various orders of leaf and scattered
reflectance from the leaf and soil surfaces, s and l can be separated.
The experiment is repeated for the barley canopies at a variety of stages of growth,
for row orientations of 0o, 45o and 90o (relative to the viewing direction) and with solar
zenith angles of 0o, 30o and 60o.
As can be seen from the figure above, the contributions to the total scattered canopy
reflectance from each scattering order i are the sum of the contributions of the various
components lj sk (where j and k are the number of leaf and soil scattering events
respectively). The fact that are no soil-soil scattering events – only leaf-leaf and leaf-soil
(or soil-leaf) as explained above, leads to the increasing powers of l down the left-hand
branch, whilst there are no comparable s scattering events on the right-hand (obviously
ljsk where j=k is the same as sjlk). For the first 8 orders of scattering, it can be seen
that there are 2, 2, 3, 3, 4, 4, 5, 5 ljsk terms respectively, with j increasing from 1 to 8,
and k from 1 to 4.
Total canopy reflectance  can be expressed as a series of summated ljsk terms:
i 
i 
i 
j  i 


i 1
i 1
i 1
j 0
 i j

    l i a0i    s  l (i 1) a1i   s 2  l (i 2) a2i ....     s j  l (i  j ) a ji  a00 
where i is the order of canopy scattering events, and j that of soil scattering events. The aij
terms are the reflectance contribution terms (or weights) at each order of scattering. If
these terms can be determined, then this expression is an exact solution for the multiple
scattered canopy reflectance (in this simplified case) to order i. It also has the advantage
of being wavelength independent. . In this way, the reflectance field can be broken down
and analysed in terms of the number of individual soil or leaf scattering events, as a
function of scattering order.
[The method for analysing the multiple scattered radiation is similar to the approach
taken by Cooper and Smith (Cooper, K. D. and Smith, J.A. (1985) A Monte Carlo
reflectance model for soil surface with three-dimensional structure, IEEE Transactions on
Geoscience and Remote Sensing, GE-23, 5, p668-673) who developed a Monte Carlo
soil reflectance model for use in canopy reflectance modelling.]
For each of the chosen viewing and illumination conditions, reflectance is simulated
at the set of twenty psuedo-wavelengths with the chosen soil/leaf reflectances. The direct
and diffuse components of scattered radiation are collected for each scattering order, for
each band, resulting in a set of 160 measurements (20 bands * 8 orders of scatterng) for
both the direct and diffuse components. The reflectance contribution terms are calculated
both as a function of scattering order, but also as a function of the number of soil
interactions.
If the log of the normalised reflectance contributions are plotted as a function of
scattering order, for successively greater numbers of soil interactions, from 0 to 4 there
appears to be a linear decrease in the reflectance contribution with scattering order. The
single-scattered contribution is still by far the largest component, with the multiple
scattered contrbutions falling off exponentially with scattering order. Each line sj on the
graph shows the contribution to the total reflectance, at scattering order i, of j soil
interactions and (i-j) canopy interactions i.e. s0 is the terms a0i as a function of i, s1 is the
terms a1i etc.
The apparent linear nature of the reflectance contributions (particularly beyond the
first order of scattering, which we may have justification for ignoring in this analysis)
with scattering order suggests that
aij   ij  j
i.e. the reflectance contributions can be described by a linear combination of  and 
terms, where  is a canopy structural attenuation term, and  is a soil attenuation term. If
this linear relationship (empirical at the moment) is correct, then we may model the
canopy reflectance as follows:
j  i 


 i j

     j  sj  l(i  j ) ij   j 
j 0
In this way, modelled values of  and  can be calculated, based on the linear
assumption, and modelled reflectance contributions âij can be calculated, and compared
with experimentally obtained values for canopies under different conditions and at
different stages of growth.
The linear assumption appears true of initial results: the gradients for the different sj’s
are very similar, and the values of the aij terms at i=8 appear to increase monotonically.
This may or may not be general. This formulation, while of general interest in the
analysis of the multiple scattered radiation field, is only of real use if the relationship of 
and  to parameters such as row orientation, gap probability, LAI, and viewing and
illumination conditions can be established. If this is possible however, it potentially
provides another source of information for determining vegetation parameters from
reflectance data. The next stage of analysis is to examine the relationships of  and 
with varying row orientations, solar zenith angles, and at different stages of growth. It
should then be possible to establish whether there really is a functional (linear)
relationship between the reflectance contributions and  and , or whether this is a
feature of a particular canopy at a particular stage of growth. From there, it may be
possible to apply these generalised rules to other canopies.
Comparison with treatment of multiple scattering in linear and non-linear BRDF
models
The results of the experiment described above are compared with the treatment of
multiple scattering in linear BRDF models. In addition, the comparison is made with the
treatment of multiple scattering in the more physically accurate non-linear Kuusk canopy
reflectance model (Nilson, T. and Kuusk, A. etc. etc.). If there are significant
discrepancies (in either case) can these can these be explained in terms of the weakness
of the model formulation, or are they mainly due to errors in the geomteric (BPMS)
model?