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A simple analytical method is presented that shows some potential for application to the problem of extracting attenuation and backscatter coefficients in an inhomogeneous atmosphere from the return signal of a monostatic single-wavelength lidar system. The method assumes the validity of the single-scattering lidar equation and a power law relationship between backscatter and attenuation. For optical depths greater than unity the inversion method can be applied in principle using only information contained in the signal itself. In contrast to a well-known related analytical inversion solution, the new solution form is shown to be stable with respect to perturbations in the signal, the postulated relationship between backscatter and attenuation, and the assumed or estimated boundary value of attenuation.

The conventional approach to solving the single-scattering lidar equation makes use of the assumption of a power law relation between backscatter and extinction with a fixed exponent and constant of proportionality. An alternative formulation is given herein which assumes the proportionality factor in the power law relationship is itself a function of range or extinction. The resulting lidar equation is solvable as before, and examples are given to show how even an approximate description of deviations from the power law form can yield an improved inversion solution for the extinction. A further generalization is given which includes the effects of a background of Rayleigh scatterers.

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