3.2.2 Analytic BRDF Models
Assuming the atmospheric correction can be done (and the surface interaction
needs to be flagged here as the coupling means the atmospheric correction
and surface BRDF estimation are not independent) the surface BRDF can sometimes
be defined by an analytic model.
Among the many models for the volume effect are the Suits and Sail models
as well as many more sophisticated ones such as the hotspot based model
described (as an example) in Qin and Jupp (1993). The literature is vast
(Myneni and Ross, 1990 provides a very good review although it is becoming
dated!).
The hotspot effect is a geometric or occlusion effect and from among the
many papers that exist describing it the following text has been editied
and extracted from Jupp and Walker (1996).
"A simple model for the remote sensing of a canopy is the Geometric
Optical (GO) model. The simple GO (or hotspot) model for scenes which describe
open forest or woodland areas is based on the one described in Jupp et al.
(1986), Strahler and Jupp (1991a&b) and Li and Strahler (1992). In this
model, there are four kinds of ground cover 'visible' from a given direction.
These are referred to as scene components and consist of sunlit canopy (symbol
sc), shaded canopy (shc), sunlit background (sb), and shaded background
(shb). Each component is assumed to have a characteristic radiance and the
radiance of a pixel is modelled as the area weighted combination (or linear
mixture) of the characteristic component radiances. That is, the observed
radiance of a single pixel (rs) is modelled as:
where the subscripts sc, shc, sb, and shb indicate the radiances of the
four components as named above, Rj represents the (mean) radiance of component
'j' and k indicates the sensed proportion of each component within the pixel
from the given view direction.
The mean radiance over the scene (Rs), assuming the view and sun directions
are constant, can be written as:
where, capital Kj represents the mean or expected value of the varying
proportions kj over the scene for j as the components sc, shc, sb or shb.
The mean value (Rs), as a function of sun and observer position, defines
the BRDF of the scene.
In order for the scene BRDF model to be computed, a description of the size
and shapes of the objects, their density and how they are distributed over
the background is needed and the geometrical relationships between the objects
and the expected values of the four components must be established. Jupp
et al. (1986), Strahler and Jupp (1991a&b) and Li and Strahler (1992)
describe such a model for spheroidal crown (not necessarily opaque) volumes
which is valid for any view or illumination angles using the 'Boolean' model
of Serra (1982). In the Boolean model, the object 'centres' are assumed
to be randomly distributed in a 'Poisson' distribution. By defining the
geometry and the distributions, expressions for Kj may be derived. Strahler
and Jupp (1991a&b) use a simple model for spheroids which is adequate
for moderate sun and view zenith angles and Li and Strahler (1992) provide
some more general alternative models for resolving the Kj. These basic scene
BRDF models are quite simple and are easily implemented in various forms
such as mathematical packages or spreadsheets.
In the woodlands and open forest areas typical of the area of Australia
where the model studies have been made, the pixel to pixel behaviour of
the image is conveniently (if not as accurately) described by a simpler
form of the model in which the shaded background, sunlit (but still relatively
dark) tree and shaded tree components are combined into one so that:
where X is a composite component combining sunlit and shaded tree and
shaded background and RX. is computed as:
The simpler model has the advantage for this discussion that it shows
clearly how, in many woodlands, the image pixel to pixel variation is driven
primarily by the variation in the proportion of sunlit background which
is visible in the pixels and the contrast between this sunlit background
and the other components. It also provides a simple estimate for ksb from
images where Rsb and RX are known for an appropriate image channel, or channel
combination, as:
For such a model, the mean radiance (ie BRDF) over all pixels in a patch
with the same basic underlying type of cover and structure is therefore:
where Ksb is the mean value of ksb, or the expected proportion of visible
sunlit background for the particular sun and view positions.
Figure 2. DATM Hotspot effect and models
This simple model has been found to be adequate to describe the data
obtained by a Daedalus scanner over woodlands, Figure 2 shows the data averaged
over many scanlines for the aircraft scanning in and across the principal
plane. The models shown vary the ratio of tree crown diameter to tree height.
As can be seen, the models are very sensitive to this ratio which has been
independently confirmed at the site by ground measurements.
Linear end-member analysis is similar to the estimation of components described
above. It has been the subject of useful research and application in Australia
(Pech et al. 1986, Pickup and Foran 1987) and at regional scales where all
pixels are mixtures of land covers of interest (Cross et al., 1991). End-member
analysis assumes each pixel to be a composition, or mixing, of a few base
components or 'end-members'. The pixel signature is assumed to be a linear
sum of reflectances from each of n end-members weighted in proportion to
its cover (kj) in the pixel:
End-member analysis seeks to invert this mixing by deriving the proportions (kj) of each component in the pixel signature. This can feasibly be derived from the remotely sensed data provided that if there are n components (trees, shrubs, grass etc) then there are at least (n-1) channels of data that separate the end-members spectrally. The key assumptions built into the end-member method are that:
a) The end-members (pure examples of total cover by trees, shrubs, grass and background) are spectrally consistent between sites and
b) Reflectance values for end members (Rj) are available from remotely sensed data or can be accurately derived by other means (such as field spectral measurements).
There has been considerable work aimed at deriving end-members from the data (a form of principal components analysis, see Boardman, 1990) and employing high spectral resolution data to effect separation of more than a few components (Adams et al., 1989). However, with a lack of available high resolution spectral data, this linear approach suffers from several significant limitations to its applicability in Australia:
1. Available broad band signatures of the tree and shrub crowns over much of Australia whilst different, are not markedly spectrally distinct.
2. Even if spectrally distinct crowns did exist for the available bands, their distinction is confounded by the effects of shadowing within crowns and cast shadow on the background (with bigger plants shading smaller plants). This makes the signature of the end-members difficult to estimate as the signature depends on the proportions of crowns and shadows present and variations in sun and look angles.
3. Relatively low covers of trees and shrubs, together with shadowing, introduce such high spectral variance into the data relative to the spectral contrasts between end-members that the numerical methods used in the end member analysis become highly unstable.
Shadow effects obviously depend primarily on the sun angle. Although the
crown cover is the same, lower sun angles clearly decrease image brightness.
Differences due to shadowing can be taken into account in end-member analysis,
provided the end-member values are recalculated for each temporal image
and one or more components labelled 'shade' are added to the list. However,
its successful application still depends on an assumption of linear scaling
along cover gradients due to sun positional and sensor view angle changes.
These assumptions in practice are erroneous in structured vegetation (e.g.
vegetation with discontinuous cover of trees or shrubs), and this limits
the application of such methods to general synoptic estimates of change
in cover.
It is therefore better to model vegetation cover directly as an assemblage
of various sizes and shapes of 3-dimensional objects (trees, shrubs, grass
tussocks, herbs, etc.) scattered on a background that may be uniform or
heterogeneous (Li and Strahler 1985, Jupp et al. 1986). The GO model may
then be used to model the bidirectional reflectance of the canopies. In
this approach, the effects due to shadowing on the overall reflectance (or
infrared temperature) from a scene become important and useful features
and the correlated interactions between shaded and sunlit components are
built into the analysis - although it now becomes nonlinear. The directional
radiance of the vegetation is then a mixture of four components (sunlit
and shaded tree crowns, and sunlit and shaded backgrounds) that is seen
from a given viewing angle. The areal proportions of these four components,
for given illumination and viewing directions (which can be off-nadir),
will be a function of the sizes, shapes, orientations and placements of
the objects (i.e. individual plants) within the scenes.
A GO model is most appropriate to woodlands or vegetation in which the vegetative
cover is discontinuous, that is, where tree and shadowed background interactions
account for a large proportion of the variance in the image. The further
advantage of these models is that they are also potentially invertible to
provide structural as well as cover information. The invertibility of GO
models was demonstrated by Strahler et al. (1988), Franklin and Strahler
(1988) and Wu and Strahler (1993) in which tree size and density were estimated
from reflectance data. When size, shape and orientation are fixed or characterised
by distributions of known parameters, and the object centres are randomly
distributed, then the proportions of the four components can be estimated
using the Boolean model of Serra (1982). This GO model is termed the Boolean
version (Strahler and Jupp, 1991a&b; Li and Strahler, 1992). It accounts
for the changes in proportions that occur with random overlapping objects
as the density of objects increases and can easily model scale effects and
changing sun and view directions. The GO aspect of the model implies that
multiple scattering of radiation in the vegetation layer is neglected. While
the evidence of our eyes supports this, there are wavelengths (in particular
the near infrared) where multiple scattering is very significant. This has
been recently addressed by Li et al. (1995)."