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The ability of a band to
shape the temperature
profile of the whole atmosphere should tend to
be maximum at intermediate optical thicknesses (for a given band width), because at small optical thicknesses, the amounts of emission and absorption within any layer will
be small relative to what happens
in other bands, while at large optical thicknesses, the net fluxes will tend to go to zero (except near TOA and, absent convection, the surface) and will
be insensitive to changes
in the temperature
profile (except near TOA), thus allowing other bands
greater control over the temperature
profile (depending on wavelength —
greater influence for bands with larger bandwidths at wavelengths closer to the peak wavelength — which will depend on temperature and thus vary with height.