
Optimal use of the information provided by indirect measurements of atmospheric vertical profiles
Piera Raspollini^{(1)}, Simone Ceccherini^{(1)} and Bruno Carli^{(1)} ^{(1)} IFACCNR, Via Madonna del Piano, 10, 50019 Sesto Fiorentino (FI), Italy
Abstract
A procedure for the optimal exploitation and representation of the information on the vertical profile of an atmospheric constituent retrieved from remote sensing observations is presented. The optimal exploitation implies to extract all the information contained in the observations and to make it fully available to subsequent applications, such as data comparison, data assimilation and data fusion. To this purpose the retrieved profile must be without any apriori information and be represented on a vertical grid as fine as needed both for its exhaustive description and for the prevention of subsequent interpolation approximations. We show that this result can be obtained with the measurementspace solution (MSS) which gives the retrieved profile in terms of its components in the subspace defined by the rows of the Jacobian matrix of the forward model and leaves completely undetermined the components of the retrieved profile in the null space (that is the complement space to the measurement space).
The MSS, while being optimal for further processing because comprehensive, unbiased by apriori information and with a diagonal variancecovariance matrix, is not suitable for a representation of the retrieved profile in the form of a graph. Indeed, using simulations applied to a test case of ozone retrieval with the MIPAS instrument, we show that if the dimension of the vertical retrieval grid is chosen with the desirable redundancy, components of the MSS that correspond to small singular values are poorly determined and the retrieval problem becomes illconditioned. On the other hand, if the poorly determined components are removed, the size of the null space grows at the expenses of the measurement space and the retrieved profile acquires a rather unphysical shape. Therefore, when a representation of the profile in the form of a graph is made it is important to choose an adequate nullspace component. To this purpose we present the nullspace regularization (NSR) method for the calculation of the nullspace component which allows to obtain the smoothest profile compatible with the observations. This is a regularization and is different from the other regularizations since it does not affect the measured component and only determines the one that has not been measured. The fact that two components (the MSS and the NSR) are determined respectively in the measurement space and in the null space provides a clear and easily traceable distinction between the measurement and the constraint. Simulations show that summation MSS plus NSR, that is referred to as the regularized measurementspace solution (RMSS), provides a good graphical representation of the retrieved profile.
Finally, the procedure developed for the calculation of the RMSS is applied to data fusion problems. A method is presented that allows, from the MSS of two or more independent measurements, the calculation of the fused profile with the determination of its MSS in the union space (that is the space obtained merging the measurement spaces of the original measurements). The components of the profile in the orthogonal complement of the union space can be calculated with the NSR method. The sum of the MSS component with the NSR component provides the RMSS of the fused profile. This RMSS is the optimal data fusion in the sense that the result is the same as it would have been obtained with the simultaneous analysis of the measurements.
 
