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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.

Sun Aug 11 10:02:40 CDT 1996