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Given a temperature profile and vertical distribution of gaseous constituents in a clear atmosphere, one can derive the radiance if the spectroscopic properties of the gas are also known. For monochromatic radiation, the differential transmissivity through the atmosphere is determined by
where
 | = | atmospheric transmission from  to p; |
 | = | absorption of
atmospheric constituent x, m kg
; |
 | = | wavenumber, cm; |
| g | = | gravitational acceleration, m s ; |
 | = | mixing ratio of
constituent x, g kg; and |
| p | = | pressure at given level, hPa. |
Atmospheric pressure units are often expressed in mb, and are equivalent to hPa.
The general radiative transfer equation (RTE) for downwelling spectral radiance in a clear atmosphere is described by the relation
where
represents the Planck radiance, mW (m sr cm);
is the differential transmission over the pressure layers;
 | = | AERI measured downwelling column radiance, |
| mW (m sr cm ) ; | ||
| T(p) | = | temperature at pressure level p, K; |
| h | = | Planck's constant, J s; |
| c | = | speed of light, m s ; |
| k | = | Boltzmann's constant, J K ;
|
and the integral limits, 
and 0, specify surface layer and
top of atmosphere, respectively.
The radiance is often described in terms of a temperature
because it removes the spectral dependence,
normalizing the data with respect to the Planck curve.
This `effective temperature' is referred to as the
brightness temperature, 
, and is the solution of
Equation 3 for T(p), in terms of the measured radiance,
where the measured downwelling column radiance, ,
has been substituted for the Planck radiance, 
.
Equation 4 will yield small errors when used for a spectral
bandpass rather than a single wavenumber; where the bandpass would be
represented by the mean wavenumber, 
. A correction, based on a
least-squares fit of the measured radiance in a given bandpass over a
typical temperature domain, can be applied to Equation 4,
such that
where a and b are the least-squares fit y-intercept and slope, respectively. Appendix B summarizes this approach and illustrates the associated errors.
Additional absorption and radiative feedback must be accounted for if a cloud is introduced to the atmosphere. When this occurs, the atmosphere can be partitioned into layers: clear sky below the cloud, cloud layer, and clear sky above the cloud; assuming a single cloud layer. Equation 2, the clear sky RTE, would be adjusted as
where
 | = | cloud transmissivity, from cloud base to p; |
 | = | angular integrated cloud reflectivity; |
 | = | cloud base pressure, hPa; and |
 | = | cloud top pressure, hPa. |
The first three terms of
Equation 6 are an expansion of Equation 2,
whereas the last term accounts for the reflection of upwelling
terrestrial and atmospheric radiation from below the cloud.
A cloud particle size distribution outside the 50 
m radius
reflectance parameterization yields a very small (see
Appendix D) change in the radiance. This
accounts for the approximation in Equation 6.
The following sections discuss the individual terms of the cloudy RTE, where:
an atmospheric transmission model is used to calculate the clear
sky values; an evaluation of cloud optical properties will lead to a
cloud reflectivity term; and the RTE can be inverted to derive an
optical depth for the cloud. The final section describes a technique to
determine an estimate of the optical depth using the 9.6 m ozone band.
