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Deriving Specific Humidity from Relative Humidity
Soden and Bretherton
[1993] derived the following primary result:
where T6.7
is the cloud-free 6.7 µm brightness temperature,
r
is the layer-averaged upper-tropospheric humidity, p0 is the
ratio between the pressure of the 240K isotherm and 350 hPa, q
is the satellite zenith angle, and a and b are empirically derived constants
31.2 and 0.115 K-1 respectively.
In
order to derive specific humidity q (or, analogously, mixing ratio, w), we begin
with
the definition for
r
in terms of mixing ratio:
Following the
example of Soden and Bretherton [1993]
we assume an atmosphere in which r
is independent of height over the range of the upper troposphere, and we assume
the vapor pressure is much less than the pressure. The
Clausius-Clapyeron equation is approximated as an exponential function of
temperature and substituted into the equation above to give:
The saturation vapor pressure at 240K, es(To),
is 38.1 Pa. The quantity
T'
describes the deviation of the upper-troposphere layer-averaged
temperature from a reference value (
T'=T
- To), where To
and Po
are an upper-tropospheric reference level temperature and pressure, respectively
(taken as 240K and 400 hPa).
Taking the
natural logarithm of both sides, we have another expression describing the
layer-averaged upper-tropospheric humidity, this time as a function of two
terms; one depends on the absolute amount of water vapor in the layer and the
other is a function of layer temperature:
the
coefficients l and c
are constants (-23.1, 12.04) based on the reference values chosen. Now, we can
substitute the first equation for ln r
into the left-hand-side above, and rearranging terms we can show that the
remotely sensed quantity T6.7 can be regarded as the sum of
four independent terms plus a constant.
We can
simplify this conceptually by subtracting the reference level temperature of
240K from both sides of equation 8, and then rearranging:
Now, we can
readily see the remotely sensed radiative temperature, the equivalent brightness
temperature at 6.7mm, is comprised of a reference level temperature and three independent
terms, one a function of the layer-average water vapor mixing ratio (or specific
humidity), one a function of the layer-average temperature, and one a function
of the satellite viewing angle.
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