The Backus-Gilbert theory, originally developed for analysis of inversion problems associated with the physics of the solid earth, was applied to the problem of the vertical sounding of the atmosphere by means of remote radiation measurements. An application was made to spectral intervals 2.8/cm wide in the 667/cm band CO2, and tradeoff curves are presented which quantitatively define the relationship between intrinsic vertical resolution and random error in temperature profile estimates. It is found that for a 1-2 K random error with state-of-the-art instrumentation, the intrinsic vertical resolution ranges from approximately 0.5 locale scale height (l.s.h.) in the lower troposphere to greater than 2 l.s.h. in the upper stratosphere with approximately 1 l.s.h. resolution in the vicinity of the tropopause. These values are somewhat smaller than the widths of the radioactive transfer kernels at similar levels. Increasing the number of spectral intervals from 7 to 16 is found to produce only a marginal improvement in vertical resolution.