1 Spatial Modelling of the Terrestrial Environment - Georeferencial

64 Spatial Modelling of the Terrestrial Environment
(1993), or they can be simplified by assuming that the scattering effects (and hence
the phase functions) are negligible, as they are in the ‘simple’ model of Ulaby et al.
(1986) and the ‘extended Wilheit’ (1978) model (Lee et al., 2002a). Both the simple
model and the extended Wilheit (1978) model are only valid at longer wavelengths
where the dimensions of portions of the canopy are of the same order of magnitude as
the wavelength of the radiation detected. These three approaches are discussed further
below.
Discrete Model. Discrete models estimate the scattering phase functions and the absorption
and scattering coefficients from measurements of the vegetation characteristics. Discs
are used to represent leaves and finite cylinders the stems of vegetation. Numerical solutions
of equations (2) and (3) can then be calculated and the microwave brightness temperature
above the canopy estimated. Wigneron et al. (1993) and Ferrazzoli et al. (2000) discussed
two different versions of a discrete model in detail. An example of the application of the
discrete model is given by Burke et al. (1999) who used the Wigneron et al. (1993) model
to successfully predict the time series of brightness temperatures measured over a soybean
canopy using a truck-mounted L-band radiometer.
Simple Model. The simple model assumes that the scattering phase functions and, thus,
the final term in equations 2 and 3 above are negligible. The brightness temperature above
the vegetation canopy is then given by the sum of the emission from the soil (the first term
in equation (4) below), the upward emission from the canopy (the second term in equation
(4)), and the downward emission from the canopy that is reflected by the soil (the third
term in equation (4)), the microwave emission being then attenuated by any vegetation it
passes through. The simple model therefore gives:
with:
T Bp = ƔT Bpsoil + (1 − Ɣ)(1 − )T can + (1 − Ɣ)(1 − )ƔT can r sp (4)
Ɣ = e −τ sec θ (5)
τ = K e d (6)
= K s
(7)
K e
where T Bpsoil (K) is the brightness temperature of the soil for polarization p, r s is the
reflectivity of the soil surface, Ɣ is the transmissivity of the canopy, τ is the optical depth, d
is the canopy height (m), is the single scattering albedo, K e is the extinction coefficient
of the canopy, and K s is the scattering coefficient of the canopy. The single scattering
albedo () determines the relative importance of absorption and scattering by the canopy.
For L-band radiation, K s and, therefore, are close to zero (equation (7) – Jackson and
Schmugge, 1991). However, has rarely been estimated (Kerr and Wigneron, 1994) and
its dependence on vegetation characteristics is unknown. The optical depth, τ, represents
the amount of absorption and emission by the canopy and is often approximated by:
τ = βθ veg (8)