If the atmospheric microwave absorption coefficient γ varies with height z as γ = τ βexp (-βz), τ being the optical thickness of the entire atmosphere, show that the brightness temperature measured by a passive microwave radiometer looking vertically down through the atmosphere is given by

where the weighting function a(z) is given by

Show that a(z) takes its maximum value at (ln τ)/β, and that, for a fixed value of β, the only effect of changing τ is to shift a(z) along the z-axis without change of scale. Interpret this result.

| a(2)T(2)dz, T(0)exp(-t)+ al) p(-B-)ep(- xp(-B-)