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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
|=||atmospheric transmission from to p;|
|=||absorption of atmospheric constituent x, m kg ;|
|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
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
|=||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.