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where represents the ratio of visible to infrared optical depths. Here, the `visible' optical depth is assumed to be represented by the HSRL measured 532 nm optical depth. This is a valid assumption because the visible extinction cross-section is due to scattering rather than absorption. The infrared cloud transmissivity, as a function of infrared cloud optical depth, can then be described in terms of the HSRL measured visible cloud optical depth,
Using this information, the HSRL integrated column optical depth can be transformed into an infrared value. This approach uses the cloudy RTE solution, Equation 6, in a forward calculation to derive the IR radiance,
|=||HSRL derived column integrated downwelling|
|radiance, mW (m sr cm);|
|=||radiosonde determined Planck radiance,|
|mW (m sr cm); and|
|=||HSRL differential visible cloud transmissivity.|
The first term in Equation 31 represents clear sky radiance below the cloud, calculated using the FASCOD3P atmospheric transmission model for a given radiosonde profile. The second term is the cloud contribution, attenuated by the clear sky transmittance below the cloud, , which is also determined by the model. The last term is the cloud reflected upwelling terrestrial and clear sky radiance from below the cloud. The radiance contribution from above the cloud was previously shown to be negligible. Multiple-scattering effects are also neglected for visible scattering.
The previous technique utilizes independent observations of AERI derived infrared and HSRL measured visible optical depth to determine a spectral optical depth ratio and HSRL derived downwelling brightness temperature. Equation 30 suggests an alternative method, where the HSRL measured visible optical depth is used to determine the column radiance. This is accomplished by iterating until the value agrees with the AERI measured value. The infrared cloud optical depth follows immediately from Equation 30 after is determined. It is expected that the iteration of will yield better results because the cloud is effectively weighted by the HSRL measured visible optical depth. The former solution assumes a uniform cloud, governed by HSRL measured cloud boundaries, to determine the infrared optical depth and optical depth ratio.