offset 
T . /[(1 - 
air 
after which the slope equation may be solved for 
then the offset equation may be solved for 
4’ 
This method may also be used if one can 
identify adjacent regions of greatly different 
surface temperatures, e.g. the land surface at 
midday and a lake, with a surface temperature 
difference of 10 to 20 C. Of course one may not 
rely on this approach to produce values over a 
large area, as such local temperature variations 
are relatively uncommon. A procedure of this 
type has been used by Kleespies and McMillan 
(1987) . 
We do not pursue the correlation approach because 
it lacks generality and thus can provide 
incorrect results. The problem is that 
atmospheric water vapor may also be spatially 
nonuniform. For example, (Price, 1984) if a 
LOCAL area is spatially uniform in surface 
properties, while the atmosphere is variable, 
then one may relate the variation of temperatures 
in channels 4 and 5 to obtain an estimate for R. 
Qualitatively the issue is as follows: for a 
laterally uniform atmosphere, channel 4 is more 
responsive to surface temperature variations than 
is channel 5, while for a uniform surface, 
atmospheric moisture variations produce greater 
variability in channel 5 than in channel 4. Both 
sources of variation must be considered in 
interpreting the observations. 
The analysis in this paper is susceptible to 
error from many sources: radiometric calibration 
of the satellite data, variation of surface 
emissivity between the spectral intervals of 
channels 4 and 5, cloud effects, and nonlinearity 
of radiances with temperature in the thermal 
infrared. As we neglect these problems we desire 
a formalism which is as robust to errors as 
possible. We deal only with variances from a 
local average and take differentials from the 
averages of equations 1 and 2 
ST, 
ST r 
ST (1-<r,>) + St (T 
s 4 4 
5T (1-R<r.>) + RSt, 
s 4 4 
air - 
<T >) 
s 
(7) 
< T air 
- <T >) 
s 
(8) 
where 5x = x - <x>, with <x> a local average of a 
variable x. In eqs 7 and 8 we have acknowledged 
the variability of atmospheric transmittance r, 
but have dropped as second order the tendency for 
local convection of moisture to alter the mean 
temperature of the atmospheric column (product of 
St with the vertical displacement of moisture). 
We next square and average equation 7, and 
average the product of equations 7 and 8, 
assuming in both cases that the average 
< St^ ST^ > may be neglected. This implies that 
organized convection, such as in sea breeze 
circulations, is not present. Although such 
effects may occur over land, they are generally 
greatly reduced if large scale advection is 
present, as is usually the case. 
< ST. 2 > = (1 - < t. >) 2 < 6T 2 > 
+ (T . - <T >) 2 < St . 2 > (9) 
air s 4 
< ST. 5T C > = (1 - < t. >) (1 - R<r. >) < 5T > 
4 5 4 4 s 
+ R (T . - <T >) 2 < St, 2 > (10) 
air s 4 
Then taking the difference between the product of 
equation 9 with R and equation 10, we may solve 
for < t, >. 
4