The Backus-Gilbert theory, originally developed for analysis of inversion problems associated with the physics of the solid earth, is applied to the problem of the vertical sounding of the atmosphere by means of remote radiation measurements. An application is made to spectral intervals 2.8 cm−1 wide in the 667 cm−1 band of 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 1-2K random error with state-of-the-art instrumentation, the intrinsic vertical resolution ranges from ˜0.5 local scale height (l.s.h.) in the lower troposphere to >2 l.s.h. in the upper stratosphere with ˜1 l.s.h. resolution in the vicinity of the tropopause. These values are somewhat smaller than the widths of the radiative 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.