An iterative regularization method with B-spline approximation for atmospheric temperature and concentration retrievals

Abstract

A retrieval algorithm using B-spline approximation for solving ill-posed inverse problems arising in atmospheric remote sensing is presented. The nonlinear ill-posed inversion is achieved by means of a hybrid approach combining higher-order B-spline functions and the iteratively regularized Gauss–Newton method. The performance of the inversion algorithm is studied by means of simulations for atmospheric temperature retrieval from far infrared airborne observations and molecular concentration retrievals from infrared limb emission spectra.

Introduction

Ill-posed inverse problems are occurring widely in the environmental sciences (e.g. Parker, 1994, Maier et al., 2004) and are usually solved by regularization methods. The optimal estimation method (otherwise known as the Bayesian approach) is a stochastic regularization method with a dominating role in atmospheric remote sensing (Rodgers, 2000). When statistical information about atmospheric variability is poor, semi-stochastic regularization methods, as for instance Tikhonov regularization and the iteratively regularized Gauss–Newton method, are pleasant alternatives. For nonlinear ill-posed problems, the iteratively regularized Gauss–Newton method has been studied for the first time by Bakushinskii (1992). Convergence results have been given by Blaschke et al., 1997, Hohage, 1997, and Deuflhard et al. (1998). The performance of this method (with respect to the selection of regularization matrices and sequences of regularization parameters) for atmospheric remote sensing has been discussed by Doicu et al. (2002). The extension to bound-constraint ill-posed problems has been addressed in Doicu et al. (2003), while a comparison with other iteratively regularized methods has been presented in Doicu et al. (2004).

The construction of the stabilizing term strongly depends on the particular discretization employed. The atmospheric profiles to be recovered can be approximated by means of interpolating polynomials (usually over adjacent intervals) or by expansions in basis functions with global or local support. The problem is first regularized in a continuous setting by controlling the magnitude and smoothness of the solution and then discretized, that is, the strategy “first regularize, then discretize” is advocated. This is in sharp contrast to the quadrature approach, where the discretization takes place first (without any regularization) and then an additional regularization is performed in a discrete setting. Various simulations suggested that basis functions of small support, e.g., B-splines, are more adequate for solving integral equations than basis functions with global support, e.g., Legendre or Chebyshev polynomials (Böckmann, 2001).

In the present work we extend our regularization method with interpolating polynomials (Doicu et al., 2002) to a hybrid regularization approach combining B-spline approximation and the iteratively regularized Gauss–Newton method. In fact the inversion process consists of a discretization part and a regularization part using B-splines. The second part itself has regularization properties. Note that due to their excellent approximation properties, B-spline functions were extensively used for the solution of ill-posed problems. O'Sullivan and Wahba (1985) used first-order B-spline approximation and Tikhonov regularization with an extended form of the generalized cross validation for vertical temperature retrieval. Böckmann (2001) employed a hybrid regularization method combining various high-order B-splines and a truncated singular value decomposition for ill-posed inversion of multi-wavelength lidar data in the retrieval of aerosol size distributions.

The organization of our paper is as follows. In Section 2 we formulate the forward and inverse problem arising in atmospheric remote sensing. The peculiarities of the iteratively regularized Gauss–Newton method with B-spline approximation are presented in Section 3. Two applications of the algorithm are considered in Section 4. Section 5 summarizes our results.