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The calculation of a brightness temperature within a spectral bandpass is inconsistent with the true temperature; where the error increases with larger bandpass. This is a result of solving
based on a single wavenumber representation of the given bandpass.
A correction can be applied by assuming a linear representation of the temperature in the Planck function, such that
where a and b are the y-intercept and slope, respectively, determined with a least squares fit over an expected temperature range using the central bandpass wavenumber, .
For ground-based measurements of downwelling radiance in the atmospheric window, a typical brightness temperature range is between 150 and 270 K; dependent upon cloud base altitude and optical depth. The theoretical radiance, integrated over the spectral bandpass, is calculated in 1 K increments in the given temperature domain. Brightness temperature, , is determined using the theoretical radiance and . A least squares fit to these values is performed to yield the necessary adjustment; where the corrected form is represented as
Figure 37 illustrates the relative error (upper figure) and percent error (lower figure) in deriving the brightness temperature from radiance data in a spectral bandpass. Corrected and uncorrected values are represented by the solid and dashed curves, respectively, using a bandpass of 909 to 1000 cm (10 to 11 m) over the temperature domain 150 to 270 K.
Figure 37: A comparison of broadband brightness temperature using parameterized Planck function linear in temperature (solid line) and standard Planck function (dashed line); each using the mean wavenumber, relative to the theoretical value. The upper figure represents temperature difference from theoretical, while the lower gives percent error.