Retrieval of Vertical Temperature Profiles in the Atmosphere

In this paper a new regularization technique is introduced and applied to the problem of retrieval of vertical temperature profiles in the atmosphere from remote sensing data. This is a key issue in Meteorology since it provides an important input for weather forecasting models, mainly in the Southern Hemisphere, where there are large areas uncovered by data collecting ground stations. The new regularization technique is derived from the well known Maximum Entropy method, and is based on the maximization of the entropy of the vector of second-differences of the unknown parameters. Simulations using real satellite data achieved a good agreement with radiosonde measurements. Numerical simulations have also shown that the temperature profiles retrieved with the new technique are relatively independent on the choice of the initial guess. INTRODUCTION Basic to most regularization techniques is the idea of restoring the well-posedness of the original problem by restricting the class of admissible solutions with the help of suitable a priori information. Prior knowledge is normally exploited under the form of a stabilizing functional that impose constraints on the variations of the model parameters, bounding them to such a degree that the final solution looks physically reasonable. Generally, this rather vague notion of reasonable means in fact smoothness. First proposed as a general inference procedure by Jaynes (1957), on the basis of Shannon’s axiomatic characterization of the amount of information (Shannon and Weaver, 1949), the Address all correspondence to this author. maximum entropy (MaxEnt) principle emerged at the end of the 60’s as a highly successful regularization technique, mainly due to the pioneering contributions of Burg (1967), Frieden (1972), Wernecke and D’Addario (1977), and Gull and Daniel (1978). Since then, the MaxEnt principle has successfully been applied to a variety of fields, from computerized tomography (Smith et al., 1991) or non-destructive testing (Ramos and Giovannini, 1995), to pattern recognition (Fleisher et al., 1990) or crystallography (de Boissieu et al., 1991). As with others standard regularization techniques, such as Occam’s razor (Constable, 1987) or Tikhonov’s regularization (Tikhonov and Arsenin, 1977), MaxEnt searches for solutions that display global regularity. Thus, for a suitable choice of the penalty or regularization parameter, MaxEnt regularization yields the smoothest reconstructions which are consistent with the available data. However, in spite of being very effective in preventing the solutions to be contaminated by artifacts, many times explicit penalizing roughness during the inversion procedure may not be the best approach to be followed. If, for instance, it is realistic to expect spikiness in the reconstruction of an image, or if there is prior evidence on the smoothness of the, say, second-derivatives of the true model, imposing an isotropic smoothing directly on the entire solution may lead to an unnecessary loss of resolution or to an unacceptable bias. In other words, the solution so obtained may no longer reflect the physical reality. In this work, we describe a generalization of the standard MaxEnt regularization method which allows for a greater flexibility when introducing prior information about the expected structure of the true physical model, or of its derivatives, into 1 Copyright  1999 by ASME the inversion procedure. Also, we discuss a particular implementation of this generalization, called “second-order maximum entropy (MaxEnt-2) regularization”, and applied it to problem of retrieval of vertical temperature profiles in the atmosphere from remote sensing data. ENTROPIC REGULARIZATION OF HIGHER ORDER We assumed that the inverse problem to be solved is defined as follows (Ramos and Campos Velho, 1996; Campos Velho and Ramos, 1997): find x such that y = A(x) ; (1) where x2R n denotes the unknown parameters, y2R m the datavector and A : R n ! R m is an operator, linear or not, modeling the relation between x and y. A traditional approach for solving (1) is to determine x in the least square sense. Unfortunately, minimization of the distance between computed and experimental data alone does not provide a safe inversion technique, due to the presence of noise in y. A better approach, is to formulate the inverse problem as: