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