Abstract
In atmospheric science we are confronted with inverse problems arising in applications associated with retrievals of geophysical parameters. A nonlinear mapping from geophysical quantities (e.g., atmospheric properties) to spectral measurements can be represented by a forward model. An inversion often suffers from the lack of stability and its stabilization introduced by proper approaches, however, can be treated with sufficient generality. In principle, regularization can enforce uniqueness of the solution when additional information is incorporated into the inversion process. In this paper, we analyze different forms of the regularization matrix L in the framework of Tikhonov regularization: the identity matrix L0, discrete approximations of the first and second order derivative operators L1 and L2, respectively, and the Cholesky factor of the a priori profile covariance matrix LC. Each form of L has its intrinsic pro/cons and thus may lead to different performance of inverse algorithms. An extensive comparison of different matrices is conducted with two applications using synthetic data from airborne and satellite sensors: retrieving atmospheric temperature profiles from microwave spectral measurements, and deriving aerosol properties from near infrared spectral measurements. The regularized solution obtained with L0 possesses a reasonable magnitude, but its smoothness is not always assured. The retrieval using L1 and L2 produces a solution in favor of the smoothness, and the impact of the a priori knowledge is less critical on the retrieval using L1. The retrieval performance of LC is affected by the accuracy of the a priori knowledge.
Highlights
An inverse problem is the process of using a given set of data to infer the parameters in a model that can characterize the given data
The retrieval experiments were performed using synthetic measurements simulated with realistic measurement conditions and instrument parameters
The retrieval quality is essentially associated with the accuracy, the goodness of fit, and the useful information content extracted from measurements
Summary
An inverse problem is the process of using a given set of data to infer the parameters in a model that can characterize the given data. Rank-deficiency (insufficient information contained in the given data) and ill-posedness (no solution or a non-unique solution, and/or an unstable solution procedure) are often encountered when dealing with practical inverse problems in science and engineering, e.g., astronomy, geophysics, remote sensing, signal processing, etc. Both situations are characterized by a function which is ill-conditioned, i.e., the condition number of this function is very large. This leads to the primary difficulty that these inverse problems are practically underdetermined. It is essential to incorporate additional information about the desired solution in order to stabilize the problem-solving process and to obtain a meaningful solution.
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