760 
sky radiance is 
written, 
small, these approximations are reasonable. The atmospheric radiance may be 
U\B) = J‘" B4T(z)]j±(z,6)d. 
(3) 
where z is height, z, is the height of the surface, z D is the height of the sensor, T(z ) is the atmo 
spheric temperature profile and t\(z,6) is the atmospheric transmittance profile. Monochro 
matic quantities have been used in these equations because of the simplification gained. When 
making a real measurement it is necessary to integrate over a waveband, and the monochro 
matic quantities must be replaced with wavelength integrated values convolved with the filter 
response function, 5(A), of the instrument. Fortunately, the waveband required is generally 
narrow (< 1 /xm) and for most temperatures encountered on the earth it is possible to use an 
equivalent wavelength which when used in the Planck function gives radiances that are close to 
those obtained by integrating over the response function. It is also a reasonable approximation 
to replace the integrated transmittances and emissivities by equivalent values, that is, define, 
jjl s[mm 
it; s(x)i\ 
(4) 
where i/> is the quantity (e.g. e„(A)) and A e the equivalent wavelength. Other averaging schemes 
are possible and Sobrino et al. (1992) include a weighting with the Planck function and the 
atmospheric transmittance. The value of this approximation will not be discussed here and we 
will simply assume that it is good. 
The atmospheric transmittance between two heights, Z\, z 2 (z 2 > z\) is, 
r(z 1 ;z 2 ,0) = expj-y k\(z')p(z') sec Odz'^ , (5) 
where k\ is the absorption coefficient for ail gases affecting the radiation at wavelength A and 
p(z) is the water vapour density profile. 
The set of equations given so far represent a reasonable starting point for describing the 
basic physics of the measurement of land surface temperatures using infrared radiometry. How 
ever, as formulated they are not particularly useful for deriving surface temperatures because 
there are generally too many unknowns and too few measurements. Thus to obtain a method 
ology for using this basic framework, further approximations and constraints must be applied. 
These are described next. 
2. APPROXIMATIONS 
The parameters appearing in (l)-(5) all have some dependence on zenith angle. By making 
some simplifying approximations we will investigate the size and behaviour of these dependen 
cies. 
2.1 Transmittances. 
Assuming that the absorption coefficient is independent of height and defining the précipitable 
water amount (g cm -2 ) to be, 
U = f p(z)dz, 
Jz, 
( 5 ) can be integrated from the surface to the height of the sensor to give, 
t,(A, 6) = exp { — k\U sec0} .