The proper weighting function for retrieving temperatures from satellite measured radiances

One class of methods for converting satellite measured radiances into atmospheric temperature profiles, involves a linearization of the radiative transfer equation: delta r = the sum of (W sub i) (delta T sub i) where (i=1...s) and where delta T sub i is the deviation of the temperature in layer i from that of a reference atmosphere, delta R is the difference in the radiance at satellite altitude from the corresponding radiance for the reference atmosphere, and W sub i is the discrete (or vector) form of the T-weighting (i.e., temperature weighting) function W(P), where P is pressure. The top layer of the atmosphere corresponds to i = 1, the bottom layer to i = s - 1, and i = s refers to the surface. Linearization in temperature (or some function of temperature) is at the heart of all linear or matrix methods. The weighting function that should be used is developed.