Appendix 1: constraint of linear model parameters for inversion
The inversion of linear kernel-driven BRDF models against measured
reflectance data can, in practice, result in model inverted parameter values that are
physically unrealisable (e.g. negative isotropic parameter)(REF). This tends to occur
when a chosen kernel combination cannot describe surface reflectance adequately.
There are two reasons for this failure. Firstly, the selected kernel combination may
simply be inappropriate for the cover type under observation. This typically occurs in
areas where the surface may be at the extremes of what was envisioned in the kernel
formulation e.g. sparse canopies over very bright/rough soils; canopies with extreme
LAD (departure from spherical assumption of volumetric kernels); dense forest
canopy where the assumptions of discrete crown and ground areas break down.
Secondly there may be insufficient information in the signal (variation related to
surface structure in the reflectance data) for a full linear model comprising an
isotropic, volumetric and GO kernel, to be inverted. In this case, a full model may
have too many degrees of freedom to fit reflectance data using physically realisable
values, and hence the best-fit solution (lowest RMSE) may be physically unrealistic.
In order to overcome the problem of physically impossible values we must
consider what the parameters are to be used for. If the parameters are to be indirectly
used e.g. as an additional channel of information for classification (beyond the spatial
and spectral) then the absolute values are not important, only the spatial variations. As
was demonstrated in XXX, inverted linear model parameters do contain biophysical
information related to the volumetric and GO scattering properties of a surface, and
this is the important issue for indirect use. If parameters are to be used directly
however e.g. to extrapolate/interpolate directional reflectance values, or to derive
associated surface properties such as directional hemispherical reflectance or spectral
albedo, then clearly it is sensible to place physical constraints on the derived
properties. These constraints can then be used to direct the inversion procedure.
Assuming a linear system with a constant term (isotropic parameter) and two
variables (k1 and k2 representing the volumetric and GO kernels) canopy can be
represented as:
 canopy  f 0  f1 k1  f 2 k 2
Recall that the unconstrained inversion of such a model against a set of observations
over varying k1, k2 is simply the minimisation of e2, the sum squared error between
modelled and measured reflectances i.e.
e2 
1
N
i N
    f
i
i 1
0
 f1 k1  f 2 k 2 
2
which results in a series of linear equations, one for each observation. This can be
simply solved using matrix algebra (see section XX, after Lewis, 1995) i.e.
P  M 1V
where
V 
and
1
N
iN
  ,  k
i 1
i
i 1i
,  i k 2i 
T
1
1 iN 
M    k1i
N i 1
k 2i
k1i
2
1i
k
k 2i k1i
k 2i 
k1i k 2i 
k 22i 
Lagrange multipliers can be applied to such a system to impose arbitrary constraints
under which inversion may be carried out. Lagrange multipliers allow the minimum
(or maximum) of a function f(f0, f1, f2) to be found by using a relationship between the
function parameters and some constant i.e. (f0, f1, f2) = constant (e.g. Boas, 1983). In
this case, the function (f0, f1, f2) is the constraint equation, based on some physical
  af 0  bf1  cf 2  d  0
limits. e2 can now be minimised based on the original model in addition to the
constraint equation i.e. we wish also to minimise
or
CP  d
where C is the transpose of the matrix containing the Lagrange multipliers i.e.
C  a, b, c 
T
and the minimisation is now a function of the constraint equation in addition to the
original expression i.e.
1 i N
 i   f 0  f1k1  f 2 k 2 2  2 af 0  bf1  cf 2  d 

N i 1
Minimisation therefore leads to
e2 
V  MP  C
V  C  MP
and multiplying by the inverse of M
M 1V  M 1C  P
or
P  C   P
where
P   M 1V
C   M 1C
and P’ is the parameter values obtained from unconstrained minimisation. Now, from
XX above
P  C  C   C  P  C
and we have already seen that C.P=d, so that we can derive an expression for  i.e.

P  C  d
C  C
and since we know that M-1V - M-1C = P, then
 P  C  d  
P  
C  P
 C  C 
i.e. if M-1 and P’ are calculated in an unconstrained manner then the inverse matrix
can be multiplied by the constraint vector C to obtain C’, and hence P, the set of
constrained parameters, can be calculated.
The theory above can be applied to linear BRDF model inversion in a number
of ways. In each case, the practical difficulty is to find a constraint equation that
relates the model parameters to some constant. The most obvious physical constraint
is that the surface reflectance should be  0. In this case, we can say that f0, f1, f2  0
i.e.
f 0  f1  f 2  0
and
f 0  f1k1  f 2 k 2  0
Furthermore, it seems sensible to constrain the directional hemispherical reflectance
and for both the maximum and minimum kernel values, to lie between 0 and 1 i.e.
0  f 0  f1 k1 0  f 2 k 2 0  1
 
 
0  f 0  f 1 k1    f 2 k 2    1
2
2
In which case it should also be reasonable to constrain the bihemispherical reflectance
to lie within the same limits i.e.
0  f 0  f1k1  f 2 k 2  1
Constraint requires the calculation of the unconstrained parameters followed
by the a test of the various products against each constraint in turn. If any constraint is
not met, all constraints should be applied1. Following re-calculation of the constrained
model parameter sets, RMSE can be used to select a single, final set of values.
Although this may seem arbitrary, it requires no a priori assumptions to be made
about which constraints the model “should” and “should not” adhere to. This is not
dissimilar to the mechanism by which the original AMBRALS model ‘chose’
between a variety of kernel combinations for various surfaces. (It was initially
planned that a number (up to 15!) of kernel combinations would be inverted for every
land pixel and the combination with the lowest RMSE of model fit would be used to
derive parameter values. This has since been refined to a much smaller subset of the
Ross and Li kernels, along with the modified Walthall model, to reduce processing
time and storage space requirements.) The requirement for speed may be what limits
the application of constraints in practice, although the number of extra calculations
required for each constraint is not prohibitive.
1
Although individual constraints could be used, this is probably not physically very sound. If just one
of the constraints is not met, we can must conclude that the inversion is misbehaving.