Why to use the MSS
The MSS solution provides the optimal exploitation of the information retrieved from level 2 analyses since it fully satisfies both requirements of providing the profile in a retrieval grid as fine as needed and of using no a priori information. As discussed below in a classical retrieval a compromise must be made between these two requirements.
In the retrieval of the vertical distribution of an atmospheric constituent the observations do not provide enough information to retrieve the continuous function of the vertical profile, therefore, the problem arises of how to represent, in terms of which and how many parameters, the finite number of independent pieces of information provided by the observations. The first choice is that of the vertical grid on which to represent the profile. This vertical grid should be as fine as possible in order both to make sure that all the information included in the observations is adequately represented by the profile and to limit the need for interpolation in the subsequent applications of the data. Indeed, operations of interpolation do introduce a loss of information [1]. However, whenever the information obtained from the observations is not sufficient to determine as many parameters as the grid points, a too fine grid makes the inversion problem ill-posed or ill-conditioned. This problem is generally overcome using some external information. In the optimal estimation method [2], according to the Bayesian approach, an a priori probability distribution function of the atmospheric state is used to fill the gaps present in the observations. In this case the solution is represented by the state corresponding to the maximum of the a posteriori probability distribution function, which combines the a priori information with that of the observations. An alternative method consists in applying a smoothness constraint to the profile. This can be done adding to the chi-square function minimized by the retrieval a term that increases when the profile oscillates [3-8]. In this case the information not coming from the observations is in the smoothness constraint. A further method that allows to make the problem well conditioned consists in writing the profile as a linear combination of a finite number of continuous functions (generally triangular functions, boxcars, polynomials, sine and cosine functions etc...) and to determine the coefficients of these functions fitting the observations [2]. In this case the information not coming from the observations consists in the number and in the kind of the chosen functions. In the following discussion the external information not coming from the observations is referred to as "a priori information" in all cases, even when it is applied in the form of a constraint.
A priori information makes it possible to obtain a vertical profile on a fine grid, but generates other types of problems in utilizing the data. In validation activities, when the compared profiles use different a priori information, it is necessary to assess the error due to this difference [9]. The estimation of the error component due to the use of different a priori information is a difficult task because it requires a reliable knowledge of the climatological variability of the quantity to be compared, which often is not available. Also in data assimilation, which is the combining of diverse data sampled at different times and different locations into atmospheric models, the measurements containing a priori information pose some problems [2]. Profiles in different geolocations obtained using the same a priori information are correlated among themselves as well as profiles obtained using nearby retrieved profiles as a priori. If not taken into account, these correlations can produce a bias in the products of the assimilation. Also in the case of data fusion, which consists in the synergetic use of more than one measurement of the same atmospheric state obtained with different instruments, the presence of a priori information in the original measurements can produce a bias in the final product.
The above considerations suggest that the optimal exploitation of the information retrieved from level 2 analyses requires data represented on a retrieval grid as fine as needed without any a priori information. These two requirements are not simultaneously satisfied by the products currently provided by level 2 analyses, and are met in the case of the MSS solution.
