X Canopy reflectance modelling
X.1 Linear kernel-driven BRDF models
The modelling of scattering from vegetated surfaces poses a number of problems, some of which have already been discussed (XX). A suitable model that is chosen to describe the BRDF of a vegetation canopy needs to satisfy a number of criteria, some of which are mutually exclusive. The following discussion outlines these problems and examines the way they have been approached before moving on to a specific investigation of linear BRDF modelling methods.
X.1 Difficulties of BRDF modelling
X.1.1 Physical accuracy
A major difficulty in deciding how to model scattering from real surfaces is the problem of modelling the mixed signal from non-homogenous surfaces. The signal measured by a remote sensing instrument will, in most cases, comprise an almost infinitely complex combination of signals of varying intensities, by no means all of which will originate from within the target area. It has already been shown (XX) that radiation can be scattered into to the path of the sensor from both the surface and atmosphere outside the target area. To further complicate matters, the signal registered at the sensor is a discrete measure of the sensor’s response to incoming radiation (at specific wavelengths), and it cannot fully represent the continuum of signals which make up the

incident radiation. Within the IFOV of a digital sensor, each finitely-sized area on the ground can only cause one discrete DN value to be registered (per spectral band). Hence if the imaged surface is not perfectly homogenous the signal must be a combination of the spectral responses of the various surface types within that pixel. In addition, scattered radiation from areas adjacent to the target can be scattered into the path of the sensor.
Considering the signal from a vegetation canopy for example, the signal arriving at the sensor (ignoring atmospheric effects) will not only be a function of the target canopy that is within that pixel, but also of any other vegetation that may be contained in the pixel
(secondary crops, understory, weeds etc.) as well as the soil at the base of the canopy.
Further, if there are different types of vegetation canopy surrounding the target area, radiation from these may also be scattered from into the path of the sensor, contributing to the measured signal. In this way the desired target signal will be combined with all the other signals, which may be considered noise.
As a consequence of all these factors, some way of separating the target signal from the noise must be found. This can be done using a physically realistic BRDF model.
However, because of the problems of scale and adjacency described above, any attempt at modelling surface BRDF in a physically accurate manner will be specific to a certain scale, the scale at which the modelled physical attributes can be considered to be dominant. Clearly, the physical parameters governing the scattering measured from a vegetation canopy at km-scale resolution are likely to be different from those controlling scattering at metre-scale resolution. Physical attributes of the canopy at the finer scale may be completely swamped by others at the coarser scale. Ideally we would like to be

able to model the canopy at arbitrary resolution, and not be limited in the choice of model by the scale of the data to which the model is to be applied. This type of approach is essential where comparisons of parameters derived from different instruments at different scales is desired.
This requirement that an ‘ideal’ BRDF model be applicable over as wide a range of scales as possible provides one of the biggest obstacles to the development of realistic physical BRDF models. If a highly accurate physical model of scattering is required it is clear that we must have detailed a priori knowledge of the canopy type and structure to enable such detailed physical models to be constructed, particularly if we wish to keep the number of model parameters down to a manageable level. But these strucrural characteristics are definitely particular to a certain scale: as the scale changes from fine to coarse, factors influencing scattering at one scale will be replaced by different factors.
This immediately contradicts the requirement for generality of scale outlined above. In addition the relatively specific nature of such a model would limit its use to canopies identical or very similar to the one for which it was formulated.
X 1.2 Complexity of inversion
As described in section XX, one of the major aims in developing BRDF models is to allow inversion against measured reflectance data in order to derive biophysical parameter information. If we wish to derive accurate information regarding the physical nature of the canopy, then ideally the BRDF model used should have some basis in

physical reality: that is, be a function of the physical parameters describing the radiation field within the canopy (or realistic approximations to these). In this case, inversion of the model against reflectance data may allow estimates of the model parameters to be made.
If these parameters are related to physical quantities, information regarding the physical state of the canopy can then be derived. In addition, a physically accurate BRDF model will require relatively few samples of reflectance to allow inversion. The derivation of useful surface biohysical parameter information is one of the primary goals that has driven the development of canopy reflectance modelling.
There are several difficulties inherent in the physically accurate approach to canopy reflectance modelling. Firstly, mathematical treatments of the scattering of radiation within vegetation canopies are not straightforwardly tractable. Realistic physical models generally utilise some form of solution of the radiative transfer equation within the canopy (SHOW) requiring solutions to a complex integro-differential equation. Even then, to find a unique solution, assumptions must be made regarding the horizontal extent, and azimuthal homogeneity of the canopy (typically the canopy is assumed to be infinite in horizontal extent, and azimuthally isotropic – REF Pinty & Verstrate, Asrar).
Secondly, due to their complex nature, these functions are not generally analytically invertible, and require some form of numerical inversion technique. Such techniques are often computationally intensive and time consuming. In the context of remote sensing data, this is not necessarily an insurmountable problem if the area to be studied is small. However it has become increasingly important to be able to derive

surface biophysical parameter information over large (potentially global) areas. In these cases numerical model inversion is simply not feasible given current limits on computing power. Wanner et al. (1995) estimate that in order to derive parameter information for a global, 1km
-2
scale weekly product, for only a few spectral bands, and requiring only 15 forward model calls per inversion (realistically the number is likely to be far greater than this), would require 1.5x10
4
forward model calls per second.
Finally, the more physically accurate the model is required to be, the more complex it must be and the more parameters it will have. This will tend to further complicate the inversion process in addition to making it much slower. In theory it should be possible to describe the scattering from an arbitrary surface exactly, no matter how complex it may be – provided we can use a model with an arbitrary number of parameters. However to numerically invert such a model would be practically impossible due to the large number of free parameters. There may also be no unique solution to the inversion with the likelihood that many of the parameters are likely to be coupled.
In summary, providing a full physical description of the canopy for modelling purposes is difficult for small areas, requiring many assumptions to be made, and almost impossible for large areas. Added to this is the difficulty of finding solutions to such models at a variety of scales. For global-scale products, detailed physically accurate models are likely to remain impractical for some time.

X 1.3 Empirical Models
A second approach that has been applied to modelling surface BRDF is that of empirical (or statistical) modelling. In this approach, a function is sought that describes scattering simply on its ability to fit observed reflectance data. The empirical approach to modelling surface reflectance anisotropy is well established, and has provided many of the early developments in BRDF modelling (Minnaert, 1941; Hapke, 1963). Walthall et al. (1985) for example, proposed an empirical model based on a three-parameter combination of functions of the view zenith angle. This model is very flexible and has been widely used for correction of BRDF effects in multi-angular data but does not describe certain features, such as the canopy hot-spot. Empirical models are generally simple with relatively few parameters, and can often be inverted analytically. However they have a number of drawbacks. The most obvious of these is the fact that because they are not based on physical scattering behaviour, any parameters that may be inverted from them will be physically meaningless. This is a major disadvantage against physically based BRDF models. A second disadvantage is that they generally require a greater number of angular samples for inversion than the physical models. However, if all that is required is some form to correct for view-angle effects in remotely sensed data sets, then simple empirical functions may suffice.
X.1.4 Semi-empirical models and the linear kernel-driven approach

In response to the challenge of deriving biophysical parameter information rapidly from global data sets at a variety of moderate to low resolutions a new approach to BRDF modelling has been developed. Primarily in an effort to overcome the problems of scaling of BRDF model parameter information inherent in physically-based BRDF modelling and that of obtaining physically meaningful parameters from empirical models, a new class of BRDF models has been developed. These models are referred to as linear kerneldriven BRDF models and are based on a semi-empirical approach to modelling surface
BRDF (Wanner et al., 1995).
The semi-empirical approach postulates that if relatively simple linear functions, or kernels, can be constructed from approximations to more detailed and complete solutions of scattering of radiation from a surface, then these kernels may be combined in a linear fashion to describe scattering from an arbitrary surface. If the functions are flexible enough, then in principle any surface BRDF shape can be described in this manner if the correct weights of each kernel in the linear combination can be deduced. In effect, this type of model attempts to describe any BRDF shape using a combination of basic functions, each representing a different aspect of underlying surface scattering behaviour, namely geometric-optic and volume scattering. If the overall scattering behaviour of a surface region can be decomposed into these constituent components then functions describing these components might be combined to describe other surfaces whose scattering behaviour is also likely to be some combination of geometric-optic and volume scattering. This approach has many advantages: if adjacency effects are assumed small enough to be ignored and the kernels are designed to be linear functions of viewing

and illumination zenith angles only, the model parameter information will scale linearly with areal scale i.e. these models can be applied at all scales with only a simple linear transformation. In this way, products derived at one scale can be mapped to arbitrary scales.
Another important feature of this type of model is that because the BRDF is described by a sum of linear functions, they are ideally suited to the methods of linear inversion i.e. simple analytical matrix-based inversion, setting the derivative of the error function to zero (Lewis, 1995). Such inversions are easy and rapid to perform, making this type of model especially suitable for inversion against large data sets.
A further advantage of the linear kernel-based BRDF models is the ease with which they can be used to derive surface albedo. As discusssed in XX, surface albedo is an important boundary parameter for short-wave energy budget studies and global circulation modelling. Both the directional-hemispherical integral of BRDF (“black-sky” albedom i.e. no sky radiance considered) and the bihemispherical integral of BRDF
(“white sky” albedo i.e. uniformly radiant sky considered) can be pre-calculated for each kernel, for any given angular sampling regime. In this way, model albedo values can be calculated as a weighted sum of the integrated kernel values, with the weights being the same as those used to construct the BRDF. In this way, albedo can be rapidly calculated using a look-up-table (LUT) approach.
X. 1.5 Components of canopy reflectance

This concept of describing the BRDF shape using simple linear functions representing the various scattering processes within the canopy was introduced indirectly by Suits (1972) who extended the model of Allen et al. (1970). Suits developed a model of scattering canopy components (vertical and horizontal projectioons?) which could be combined linearly. This approach was further developed by Roujean et al. (1992) who described variations in surface directional reflectance using a combination of a geometrical-optical scattering component, modelled by opaque rectangular protrusions on a surface, and a volumetric scattering component, an approximation to Ross’s theory of volumetric scattering in a horizontally infinite homogenous medium (1981) i.e. the canopy reflectance is described as
 canopy
  volumetric
  geometric
 optic
The justification that Roujean et al. (1992) give for this approach is evidence deduced from analysis of the many examples of measured bidirectional reflectance spectra available in the literature (e.g. Hapke and Van Horn, 1963; Kimes, 1983; Kimes et al.,
1985, 1986 REFS!!!!). They concluded that many of the observed features of these spectra appeared to be common, and might be characterised by a general set of functions or kernels, representing the behaviour of the major processes controlling scattering from non-Lambertian surfaces.

One of the observed features of the measured data was the strong presence of a peak in reflectance in the backscattering or hot-spot direction. This is due to a minimum of shadowed surface area being visible (zero) when the illumination source is directly behind the viewer, regardless of the viewing direction. This effect is clearly illustrated by
Norman et al. (1985 – from Goel in Asrar), reproduced in fig. XX. Leaf 2, nearly perpendicular to the direction of illumination, is well lit, whilst leaf 1 is dimly lit. Viewer a, with a good view of leaf 2 and a poor view of leaf 1, will see a relatively bright scene.
Viewer b however will see the dimly lit leaf 1 more clearly than leaf 2, and will hence see a dimmer scene. In the hot-spot direction, with the sun directly behind view a, it can be seen that the projected shadowing of canopy elements will be a minimum, and hence a peak in reflectance will be observed. The amount of mutual shadowing within the canopy will be controlled by the size, shape and angular distribution (leaf angle distribution,
LAD) of the scattering facets in the canopy.

view b specular reflection direct illumination view a shadow leaf 2 shadow leaf 1
The second clear feature of the measured data identified by Roujean et al. (1992) is the gradual increase in reflectance in all spectral bands away from nadir viewing and illumination angles (regardless of the relative azimuth angle between viewing and illumination directions), with the exception of the hot-spot direction. This is caused by the shadowing of lower canopy layers by the layers above them, and the varying quantities of scattered light viewed from each layer with changing view zenith angle. As view zenith increases, the chances of viewing light scattered from lower, shadowed layers reduces, implying that more scattered radiation will come from the better-lit upper layers
(MENTION GAP PROBABILITY HERE?). This is a volumetric effect.
Hence, Roujean et al. (1992) postulated that canopy reflectance could be represented by two main components, a geometric-optic component, and a volumetric component as in eqn. XX. If these components could be generalised to a linear form, depending only on the viewing and illumination angles, then they could be combined in a

weighted linear sum to represent the BRDF of any surface displaying elements of either geometric-optic or volume scattering, or both.
This approach can be interpreted in two ways. Firstly, considering a mixed pixel containing two predominant cover types such as woodland and grassland, it would seem reasonable to suppose that the total reflectance of the pixel would be some combination of the respective reflectances of the two cover types in proportion to their areal weighting within the pixel. In this case, the weightings of the kernel values combined in linear superposition are just that: the respective areal proportions of the cover types from which each component of scattering is presumed to dominate. (For this to be the case, adjacency effects must be ignored, which is a reasonable assumption presuming the view and illumination zenith angles are not high (and the respective areal extents of the two cover types do not vary spatially over too fine a scale)). The assumption of sub-pixel homogeneity typically made in physcially-based BRDF models is thus avoided, permitting linear models to be applied with a greater degree of confidence over areas likely to have a degree of heterogeneity. When considering the scales at which typical moderate resolution instruments such as MODIS operate (km-scale) this is a tremendous advantage over the direct phsyical approach to modelling BRDF.
A second interpretation is possible in the case where the cover type is homogenous at the observed scale. There are still likely to be elements of both volume and geometric-optic scattering present in the shape of the surface BRDF. For example, the BRDF of a forested area is likely to be composed of a geometric-optic component due

to the scattering caused by the crown shape and size, as well as the gaps between the crowns. A volumetric component may also be present due to scattering within the crown, goverened by the density and angular orientation of scattering elements within the envelope of the individual crowns. If multiple scattering interactions between these components is ignored (i.e. ignoring coupling between the two effects) then the weights of the kernels in linear superposition in this case represent the respective weights of the separate components of first and second-order scattering.
Wanner et al. (1995) refined the kernel-based approach and provided a formalism for developing new kernels, and arrived at a more general formulation of the linear kernel-driven approach – a suite of semi-empirical (and empirical) kernels describing the geometric-optic and volumetric components of surface scattering, combined with an isotropic brightness term (expressing nadir reflectance at nadir illumination), that can be combined arbitrarily to provide the most flexible description of BRDF possible. The capacity of the linear models to operate over a variety of cover types and to scale linearly with spatial scale, combined with their ability to be inverted analytically and rapidly, makes them the only realistic choice of model for producing surface biophysical parameter products such as albedo in near real-time from forthcoming global data sets.
As a result, this suite of models, termed the AMBRALS model (Algorithm for MODIS bidirectional reflectance anisotropy of the land surface), has been adopted for processing data from the moderate resolution imaging spectroradiometer (MODIS ) to be launched aboard NASA’s Earth Observing System (EOS) platform some time in early 1999??.

In summary, due to the fact that physically accurate BRDF models are impractical for rapid inversion against large data sets and cannot be operated over many different scales, and that empirical models, while flexible and often rapidly invertible, do not posses meaningful parameters, severe problems arise if we wish to derive biophysical parameter information from moderate resolution data over global areas. Linear semiempirical kernel-driven BRDF models have been developed to try and solve this problem. Different kernels, based on approximations to solutions for geometric-optic or volume scattering from a surface can be combined linearly to describe different components of the BRDF shape. Because of their properties they can be sacaled linearly and inverted analytically, whilst potentially maintaining relevant surface biophysical parameter information.
X 2 Kernel types
As has been decribed above (XX) the assumption behind linear kernel-driven models is that canopy reflectance can be satisfactorily represented by an isotropic, GO and volumetric component. If the simple expression of eqn. XX is extended (Wanner et al.) canopy reflectance can be expressed as follows
 canopy
 f istropic
 f geo k geo
 f vol k vol where the f values represent the kernel weights, and the k values are the kernels themselves.

Introduce kernels, and assumptions behind kernels i.e. roh=tau, high/low LAI assumptions – point out problem of non-reciprocity, solved arbitrarily, and the different b/h ratios which are arbitrary for the Li kernels.
X 2.1 Geometric-optic (GO) kernels
The GO kernels assume
 canopy
to be a weighted sum of the sunlit and shadowed crown and ground components of reflectance, the weights in each case being the relative areal proportions of each component, with the sunlit components assumed to be equally bright
(Li and Strahler, 1986). The sunlit and shaded areas are calculated by considering the projections of the spheroids onto the background under the assumptions of parallel-ray geometry.
. the changes made to the kernels in order to ensure reciprocity. Reciprocity is a desirable property of the kernels - in regards to energy conservation and therfore physical reality, it is essential. However, it has been enforced in the case of the shadowing (Li) kernels in a rather arbitrary manner simply by dividing the kernel by the cosine of the view zenith angle,

, and it may be that there are more theoretically justifiable ways of ensuring reciprocity in the formulation of the kernels (REF??).
Add on bits

in the case of the GO (Li) kernels, approximations are made regarding the density of scattering ellipsoids distributed on a surface: no mutual shadowing is permitted in the
LiSparse family of kernels whilst the LiDense kernels are formulated to allow mutual shadowing through the calculation of an overlap function (Li and Strahler, 1986, 1992).
The GO kernels are further complicated by containing two free parameters describing the crown shape and relative height which must be fixed in order to obtain strictly linear kernels (see section XX ) ( h / b = the height-to-base ratio where h is the distance of the centre of the spheroids to ground and 2 b = length of spheroids; b / r = the ratio of the semimajor to semi-minor axes of the spheroids – b is the minor axis of the spheroids). Strictly these kernels are functions of b and h also and so single kernels exist only for fixed values of b and h . Similarly, approximations for low LAI and high LAI cases are made to
Ross’s (1981) solution for volume scattering in a homogenous vegetation canopy in the
RossThin (LAI << 1) and RossThick (LAI >> 1) volumetric kernels in order to obtain linear models.
Volumetric
This component is controlled by the amount of light penetrating the canopy to a particular depth, undergoing a single scattering event, and then exiting the canopy towards the viewer without being further scattered. This behaviour will be controlled by the joint gap probability (the probability of finding a line-of-sight gap into the canopy to a certain depth and then out of the canopy to the viewer – see section XX ), the exponentially increasing likelihood of the incident beam being intercepted with depth in the canopy, the

single scattering phase function and the leaf projection functions in the direction of the viewer (G) and illumination (G’).