German Remote Sensing Data Center (DFD)}

German Aerospace Research Establishment (DLR)}

P.O. Box 1116} D-82234 Oberpfaffenhofen, Germany

Remote sounding from earth satellites using Fourier spectroscopy has become the main tool to monitor global changes in the atmosphere, like the formation of the ozone hole or the green house effect. The limb sounding observation technique consists of the measurement of emission spectra for different scan angles from the satellite through the atmosphere. By a least-squares fit procedure, temperature profiles and density distribution profiles of the major trace gases, are retrieved. The fit procedure is based on a forward model for relation between the unknown temperature and density distribution profiles and the measured emission spectra. Mathematically, the forward model is given by the radiative transfer equation, a nonlinear Fredholm integral equation of the first kind. Therefore, for the inversion of the radiative transfer equation a non-linear ill-posed problem has to be solved.

Due to the noise sensitivity of the inverse problem, regularization techniques
that incorporate additional information, have to be applied in order to
compute meaningful solutions. For Tikhonov regularization, which aims at
smooth solutions, an additional term consisting of a Sobolev norm of certain
degree of the solution is added to the least-squares functional. On the other
hand, the optimal estimation technique intends to obtain a solution close to a
known a priori profile. As additional constraint the difference between the
solution and the a priori profile measured in the L_{2} norm, weighted
by the variance-covariance matrix of the a priori information, is added to the
least-squares functional.

For the minimization of the resulting functional a nonlinear multigrid method in the form of the full approximation scheme (FAS) is applied. The coarse grid problems are formed by restricting the regularized least-squares functional to coarse grid subspaces. As error smoother the so called onion-peeling method, a nonlinear Gauss relaxation starting at the uppermost altitude grid point downwards to the lowest altitudes point, is used.

The performance (number of forward calculations) of the multigrid method is compared with standard retrieval methods (e.g., Levenberg-Marquardt iteration).