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IR Radiative Transfer

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


tex2html_wrap_inline2777 =atmospheric transmission from tex2html_wrap_inline2779 to p;
tex2html_wrap_inline2783 =absorption of atmospheric constituent x, mtex2html_wrap_inline2787 kgtex2html_wrap_inline2789 ;
tex2html_wrap_inline2791 =wavenumber, cmtex2html_wrap_inline2793;
g =gravitational acceleration, m stex2html_wrap_inline2797;
tex2html_wrap_inline2799 =mixing ratio of constituent x, g kgtex2html_wrap_inline2803; 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 (mtex2html_wrap_inline2807 sr cmtex2html_wrap_inline2809)tex2html_wrap_inline2811;


is the differential transmission over the pressure layers;

tex2html_wrap_inline2813 =AERI measured downwelling column radiance,
mW (m tex2html_wrap_inline2815 sr cm tex2html_wrap_inline2817) tex2html_wrap_inline2819;
T(p) =temperature at pressure level p, K;
h =Planck's constant, J s;
c =speed of light, m s tex2html_wrap_inline2827 ;
k =Boltzmann's constant, J K tex2html_wrap_inline2831 ;

and the integral limits, tex2html_wrap_inline2833 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, tex2html_wrap_inline2837 , and is the solution of Equation 3 for T(p), in terms of the measured radiance,


where the measured downwelling column radiance, tex2html_wrap_inline2841 , has been substituted for the Planck radiance, tex2html_wrap_inline2843 .

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, tex2html_wrap_inline2845 . 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



tex2html_wrap_inline2847 = cloud transmissivity, from cloud base to p;
tex2html_wrap_inline2851 =angular integrated cloud reflectivity;
tex2html_wrap_inline2853 =cloud base pressure, hPa; and
tex2html_wrap_inline2855 =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 tex2html_wrap_inline2857 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 tex2html_wrap_inline2859 m ozone band.

next up previous
Next: FASCOD3P Model Up: Infrared Spectrum Previous: Infrared Spectrum

Daniel DeSlover
Sun Aug 11 10:02:40 CDT 1996