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2 - MODEL
The model proposed here starts from the radiative transfer equation applied to a thermal scanning sensor on board
a satellite. The radiance measured by the channel i, which is observing the land surface at zenith angle 0, is given
by:
B i Cr i ) = B i (T 1 *)x j (e)+L i t( 0 ) (!)
where B; is the Planck’s function averaged for channel i, T, is the brightness temperature measured by channel i,
and Tj and L^ are, respectively, the transmittance and upwelling radiance of the atmosphere. Tj* is the brightness
temperature at surface level, corresponding to the radiance that would be measured at surface level. This radiance
is composed of the radiance emitted by the surface, characterized by both the surface temperature and the
emissivity, and the reflected part of the downwelling atmospheric radiance. Assuming Lambertian reflection, it is
possible to write
B,(T,*) = £iBj(T) + Lj^(hem) ( 2 )
where T is the ground surface temperature; £; is the channel i emissivity of the surface; and Lj^(hem) is the
downwelling hemispheric radiance emitted by the atmosphere within the bandpass of channel i. Thus, Tj*
represents an atmospheric corrected surface temperature, but still containing the emissivity effects. It depends on
the ground surface temperature, the surface emissivity, and, to a lesser extent, on the atmosphere type, due to the
reflection term. Moreover, Tj* depends on the channel, mainly due to the spectral variation of surface emissivity.
Starting from equations (1) and (2) we have developed a split-window model for LST, and a procedure for
determining and mapping the surface emissivity spectral difference.
2.1. Operational Split-Window Model
From equations (1) and (2), a theoretical split-window model for LST has been derived, which is shown in Coll
et al. (1993a) in detail. For AVHRR channels 4 and 5 (i=4 and 5) it is obtained:
T = T 4 + A(T 4 -T 5 ) + A + B(e) (3)
with A=[1 -t 4 (0)]/[t 4 (0)-t 5 (0)]; A=-[l-X 5 ( 0 )] 8 T a T, where 8 T a ^ is the atmospheric temperature difference between
the two channels (Coll and Caselles, 1994); and B(e)=a(l-e)-PAe, where e=(e 4 +£ 5)/2 is the mean emissivity,
Ae =£ 4 -£ 5 is the spectral emissivity difference, and a and p are coefficients with dimensions of temperature,
which depend mainly on the atmospheric moisture and, to a lesser extent, on the surface temperature. The splitwindow
splitwindow coefficents A and A depend only on the atmospheric type and are independent on surface emissivity.
These coefficients have been calculated from a set of NOAA 11-AVHRR measurements over the sea surface and
coincident "in situ" measurements of temperature. The data set used cointains measurements performed over
world-wide oceans, then an atmospheric dependent coefficient A has been proposed in order to account for the
atmospheric variability in a global basis. Thus we have obtained (Coll et al., 1993a):
A = 1.0 +0.58 (T 4 -T 5 ) (4a)
A = 0.5 IK (4b)
where A is written as a linear function of T 4 -T 5 , which in turns is a measure of the atmospheric water vapor
content, whereas A is a constant. The accuracy obtained in surface temperature with these coefficients is about
O. 7 K, calculated from comparison of satellite derived temperatures with "in situ" measurements. It should be
noted that the sea surface is nearly a blackbody, and the emissivity effect in the split-window is close to zero.
However, for land surfaces, the emissivity effect must be taken into account through coefficient B(e). Using the
equations given in Coll et al. (1993a), we have calculated the parameters a and P as a function of the water vapor
content of the atmosphere, for a number of atmospheres used in Sobrino et al. (1994) (see Figure 1). From these
calculations, a is seen to be approximately constant, a=40±10 K, while P decreases perceptibly with the
atmospheric humidity, due to the larger contribution of the reflected atmospheric radiance. The coefficient P is
found to decrease exponentially with the water vapor content in a vertical column, W. For the data shown in
Figure 1 we have obtained the relationship P=284cxp[-0.621 W] (see Figure 2). Another relationship is found for
P, that is P=0.168exp[7.190R], where R is the ratio R=xs/x 4 , that can be obtained from the covariance and
variance of the brightness temperatures Tj (Sobrino et al., 1994). More simply, climatologic mean values can be
used for p; e. g. p= 150 K for midlatitude in winter, p=75 K for midlatitude in summer, and P=50 K for tropical
atmospheres.
