X BRDF and Canopy Reflectance Modelling
X BRDF and canopy reflectance modelling
In section XX the directional nature of the radiation field reflected from most natural surfaces was introduced, and sensors designed to exploit such reflectance anisotropy were discussed. As has been established (in this thesis) directional effects cannot be ignored in remotely sensed observations. They may be removed if undesirable, for example in comparisons of observations made at different times of day or from different viewing angles (Roujean et al., 1992; Cihlar et al., 1994), or they may be exploited as a source of information regarding surface structure (Hapke, 1981; Gerstl, 1990; Pinty and Verstraete, 1991). This chapter is intended to provide a theoretical framework for studying surface reflectance characteristics in addition to reviewing the specific area of interest of this thesis: canopy reflectance modelling.
X.1 Bidirectional Reflectance Distribution Function – the BRDF
It is relevant now to provide a more formal definition of surface scattered reflectance in order to better understand the problems associated with its measurement and analysis. In order to allow quantitative statements regarding the reflectance of any surface to be made, various definitions are required regarding the system under observation. The bidirectional reflectance of the Earth’s surface, that is, the surface reflectance as a function of viewing and illumination direction, is described by the bidirectional reflectance distribution function (BRDF) (Nicodemus, 1970; Hapke, 1981). The BRDF of a surface (at a particular wavelength and polarisation of illuminating radiance) is defined as the ratio of the incremental radiance leaving a surface through a solid angle in the direction q (usually the viewing direction) to the incremental irradiance from direction q’ (the illumination direction) (Nicodemus, 1970) i.e.
where dLe is the incremental radiance exitant from a surface (Wm-2sr-1nm-1); dEi is the incremental incident irradiance (Wm-2nm-1); q, q’ are the viewing and illumination vectors respectively. It should be noted that this quantity is defined only for infinitesimal viewing and illumination directions, and for an infinitesimal wavelength interval. In this form it cannot therefore be measured and is of little practical use. In practice illumination is typically over a hemisphere with possibly direct and diffuse sources (including the sky radiance); viewing is typically over some finite instantaneous field-of-view (IFOV) of a sensor with a spectral response over a continuum of wavelengths, rather than at some discrete value of l. In this case we can define a bidirectional reflectance factor (BRF) r, which is the ratio of radiance leaving the surface in a finite solid angle in the viewing direction q to the radiance from a perfect Lambertian reflector under the same illumination conditions as the target, into the same finite solid angle i.e.
where Le = radiance exitant from the surface (Wm-2sr-1nm-1); LLambertian = radiance from perfect Lambertian reflector (Wm-2sr-1nm-1), a constant; Lsky,sun = sky and sun radiance distributions (Wm-2sr-1nm-1). It can be seen that the BRF is unitless.
FIGURE HERE OF VIEW/ILLUM GEOM and schematic of surface reflectance envelope
As a result of the incident irradiance and exitant radiance being defined as through a finite but infinitesimal solid angle, the BRF, the ratio of the total exitant radiance from a surface in relation to that from a perfect Lambertian reflector, must be an integrated property over the viewing hemisphere i.e. the numerator in eqn. X.2 becomes
where rs is the surface reflectance function (BRDF). The factor |q*q’| describes the available reflecting surface area projected into the viewing direction. It can be seen from X.3 that in order to derive the exitant radiance, the viewing and illumination vectors are integrated over a hemisphere. In the same way, the reflectance from the Lambertian surface is defined as
where the integral is purely over the illumination direction as the observed reflectance, by definition, is the same regardless of the observation direction.
The BRDF can be considered as an intrinsic property of a surface, whereas BRF is not - it may change with sky radiance and other assorted environmental conditions. Approximations can be made to allow this formulation for BRF to be directly related to BRDF. If a point illumination source is considered i.e. no sky irradiance, then
It is important to note that BRDF cannot be directly measured for a number of reasons:
i) From X.1 it can be seen that BRDF is defined as the ratio of two derivatives i.e. reflectance relative to that of a perfect Lambertian reflector).
ii) From X.2 and X.3 it is clear that BRDF is an integrated property i.e. it is the integrated spectral directional hemispherical reflectance. In addition, no sensor has a perfectly discrete spectral response. Measurement of the BRDF therefore is inevitably a convolution of the sensor spectral response function with the signal over a range of wavelengths.
iii) In practice the Earth’s surface can only be observed through the atmosphere, so that the signal measured at a sensor is also a function of the atmospheric absorption, reflectance and transmittance. To retrieve measures of surface reflectance through the atmosphere, the scattering behaviour of the atmosphere (and in particular the scattering phase function) must be characterised accurately.
It is apparent from X.2-X.5 how BRDF can be used in the derivation of albedo. Albedo was defined in section XX as the ratio of incoming to exitant radiation from a surface. The BRDF explicitly describes the directional nature of exitant radiation from a surface, and, as a result, albedo is simply an integrated measure of the directional reflectance over the viewing/illumination hemisphere. Albedo is thus a function of the quantities of diffuse and direct illumination arriving at the surface, and in further approximations are made in order to derive measures of surface albedo (Wanner et al. 1996; Lucht et al., 1999??). A ‘black-sky’ albedo is defined as the directional hemispherical reflectance (integrated BRDF – i.e. albedo) assuming no diffuse component of illumination and is clearly dependent on the solar zenith angle i.e.
where rs is the surface reflectance. In addition, a ‘white-sky’ albedo can be defined as the albedo in the absence of a direct illumination component i.e. under conditions of isotropic illumination. This diffuse-to-diffuse irradiance (or bihemispherical reflectance) is defined as
where rhobar is the diffuse irradiance that is directly scattered into the viewing direction (mv, f) under illumination from ms.
The properties defined above are all approximations, but they may be measurable in practice, and can therefore prove extremely useful in deriving albedo from remote sensing measurements. It is clear, however, that these approximations are not sufficient on their own. In each case, in order to accurately characterise the up and downwellling radiation fluxes a description of the directioanl scattering nature of the surface under scrutiny is required. Albedo is a very important quantity in considering the Earth’s radiation budget. As has been described in section XX, vegetated surfaces can account for a great deal of variation in surface albedo and are also responsible for large fluxes of moisture, the uptake of CO2 and the interception of solar energy. Many models describing the scattering of radiation from vegetation have been developed in an effort to derive information regarding canopy structure and radiometric properties from remote sensing data. Such models can provide estimates of albedo over vegetated areas at the resolution of spacebeorn instruments also, through the use of the expressions detailed above. The subsequent section provides a review of the many varaious methods that have been developed to model canopy reflectace.
X.2 Canopy reflectance modelling
X.2.1 Reflectance anisotropy and surface structure
The importance of being able to describe and monitor land surface processes in a timely, continuous manner via measurement of biophysical properties such as LAI fAPAR/NPP and albedo was established in section XX (eg. Goel, 1988; Dickinson et al., 1990; Myneni et al., 1995). As a result, the ability to accurately describe the structure of vegetation canopies and their interaction with incoming solar radiation using remotely sensed measurements is a vital aspect of the monitoring process (Dickinson, 1983; Hall et al., 1995). Canopy reflectance modelling is the tool through which such monitoring is accomplished, and various techniques have developed in conjunction with the advancement of EO capabilities.
Whilst often appearing complex in practice, canopy reflectance modelling has relatively simple aims. It is impossible to measure the vegetation-related parameters of interest (e.g. structural parameters such as LAI/LAD, or radiation interception parameters controlled by vegetation amount such as fAPAR and NPP) accurately over large areas simply because vegetation is inherently difficult to characterise: it has many varieties with many different characteristic spatial (and to a certain extent temporal) scales. If an attempt is made to characterise vegetation structure accurately, the amount of effort required will necessarily restrict such measurements to very limited areas. Remote sensing can provide a solution to these problems. If the spectral directional reflectance of a vegetation canopy is controlled by the canopy structure (e.g. size and orientation of leaves, leaf reflectance, transmittance and absorption properties, understory behaviour (soil roughness and reflectance)) the radiation reflected from the canopy will contain information relating to the canopy structure (Ross (1981) provides an extremely detailed analysis of the effects of canopy structure on radiation interception). If an appropriate mathematical model of canopy reflectance can be formulated it may then be possible to use the model to (infer?) derive information regarding canopy structure from remote observations of canopy reflectance characteristics (Twomey, 1977; Goel, 1988; Asrar, 1989; Pint and Verstraete, 1991 etc.). This technique is common to many other disciplines where parameters controlling the system under observation cannot be measured directly, but may be estimated from the observations of surrogate properties of the system through the application of appropriate models. This process (so-called forward and inverse modelling) will be described in detail in subsequent sections. Twomey (1977) provides an excellent comprehensive introduction to computational inversion methods for a variety of remote observation systems.
In section XX it was described that the reflectance signal of structurally complex surfaces tends to be anisotropic. Because reflectance anisotropy is generally caused by surface structural features (Hapke, 1981, 1993), the directional component of surface scattered reflectance may prove to be of greater use in determining vegetation structural charcteristics than the spectral and temporal signatures. This signal contains significantly different information from the other components and must be treated separately in any investigation of canopy structure (Privette et al., 1997). The modelling of anisotropic surface scattering has its origins in planetary astronomy, with the discovery of the bidirectionality of surface reflectance in images of the lunar surface (Minnaert, 1941). It was recognised that in order make inferences regarding the nature of the lunar surface from observations, and, in particular, to compare the scattering behaviour of different points observed at different times and under different conditions of illumination, all external sources of reflectance variation would need to be removed. Minnaert (1941) proposed a simple empirical function based on Kirchoff’s law of reciprocity to account for the variations in reflectance caused by surface anisotropy. This theory has been extended by others (e.g. Hapke, 1981; Pinty et al., 1989) to provide a greater degree of flexibility in describing different surface types and conditions. It has been recognised that radiation scattered from vegatation canopies also exhibits directionality, and that techniques applied to anisotropic reflectance from planetary surface might be extended and applied to canopy reflectance modelling (Ross, 1981; Goel, 1988, Asrar, 1989; Myneni et al., 1989; Verstraete et al., 1990). In particular, much effort has been directed towards the development of scattering models of vegetation (Ross, 1981, Goel, 1988). This effort has been driven by the importance of vegetation, both economically and due to its impact on the global climate (GARP report, 1975; Dickinson, 1983, 1992).
X.2.2 BRDF and canopy reflectance: a definition of the problem
As an extension of the purely macroscopic BRDF described above (a gross representation of the flux exitant from the surface), reflected radiation from the Earth’s surface can be described as a function of five separate signatures (Gerstl, 1990; Liang and Strahler, 1994)
1. l - spectral signatures - e.g. the reflectance, transmittance and absorptance behaviour of canopy element, manifested as the total spectral signature of a target.
2. (x,y) – spatial signatures - e.g. spatial structure, arrangement of scattering objects on a surface, appearance of target at different scales, adjacency, mixed pixels.
3. t – temporal signatures - e.g. seasonal change of vegetation growth profiles; inter-annual variability of parameters e.g. LAI.
4. (q, q’) – angular signatures - e.g. BRDF and reflectance anisotropy caused by surface reflectance; features such as hot-spot.
5. p (f) – polarization signature - e.g. polarization information contained in surface reflectance signal, atmospheric phase function.
It is the exploration of the first three signatures, the spectral, spatial and temporal components, that has formed the basis of conventional remote sensing (e.g. Goel, 1988; Gerstl, 1990). The fifth component contains separate information related to the polarization of incident and reflected radiation, and has only been exploited recently with the advent of the POLDER instrument (POLarization and Directionality of Earth’s Reflectance) (Leroy et al., 1996). The directional signal (4) is one of the least-well understood and exploited of these signatures (Goel, 1992; Liang and Strahler, 1994; Myneni et al., 1995; Privette et al., 1997), and because of its dependence on canopy struacture it is the one that we will be most concerned with in this thesis.
If we wish to describe the reflectance of a vegetation canopy there are a number of variables to take into account. Goel (1992) extends the description by Gerstl (1990) to propose the following functional description of the relationship, R, between the measured spectral directional signature S of a vegetation canopy, and the parameters controlling that signature: