Abel Integral Inversion in Occultation Measurements

Abstract

Occultation geometry under a spherical symmetry assumption leads to models described by the Abel integral equations. Analyzing general properties of the Abel transform, this work derives practical rules for discretization and for solution of the inverse problems, containing Abel-type integral equations. Two applications in remote sensing are considered: reconstruction of local densities from horizontal column densities (vertical inversion) in absorptive stellar occultation measurements and reconstruction of air density from refractive angle measurements. In the case of continuous functions, it is shown that the vertical inversion problem is ill-posed: small errors in measurements may cause errors of arbitrary size in retrieved quantities. The refractivity reconstruction problem is well posed: a noise in measurements is smoothed in inversion. In the reality of a finite number of measurements, the inverse problems can be made even-determined by discretization. The difficulties in discretization of the Abel-type integrals are the weak singularity at the lower limit and the upper limit initialization. Possible solutions to these problems are discussed together with different discretization schemes. The amplification of error coefficient is used as a criterion of ill- or well-posedness of the problems. Together with the averaging kernel, it also characterizes quality of the discretization schemes. For the vertical inversion problem, three matrix inversions: standard onion peeling, onion peeling with quadratic interpolation and discretization by trapezoidal rule in pole formulation are compared with discretization of inverse Abel transform. For refractivity inversion, the discretized inverse Abel transform is compared with two matrix inversions. Necessity of regularization for the considered inverse problems is also discussed.