Backus-Gilbert formulation

A retrieval formulation developed by Backus Gilbert (1970) and used in geophysical inverse problems of the solid Earth has also found application in remote sensing of atmospheres. Although this formulation is developed from a somewhat different point of view, it is formally related to the constrained linear inversion as well as other techniques. We include a brief description here because of the additional physical insight it can provide into inversion theory. 

For radiance measurements at m discrete frequencies, the linearized radiative transfer equation can be written 

To address this problem let the estimated average property of f z) be written 

The extent to which f(z) is useful depends on the properties of A z, z ), such as its width relative to the individual A, (z)s. 

To pursue this line of reasoning further, it is necessary to define some measure of the width or shape of A(z, z ). Two quadratic measures of iitterest are 

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