Use of the MSS for the graphical representation of the vertical profile
In general the information provided by the observations is not sufficient to identify a single profile, indeed it selects a set of possible profiles given by Eq. (3.4), where xa is determined by the observations and xb is given by Eq. (3.6) with an arbitrary value of the components b. When further processing of the data is planned it is adequate to provide the measurement information in a sub-space and, taking advantage of this opportunity, it is better to avoid any choice about xb. Indeed, the further processing may provide some estimation about the null-space components and any constraint (even the condition b=0) does introduce a bias. However, if a graphical representation of the profile is requested different considerations apply. In a graphical representation the result is to be presented in a complete space. The representation of the x profile with the MSS corresponds to selecting the value zero for b. This is a particular choice among the infinitive possible ones, which does not necessarily provide the best graphical representation.
Various ways exist for selecting b values which provide a good graphical representation of the vertical profile.
Null-space regularization
Since we do not expect that real profiles show unphysical oscillations, a good graphical representation can be obtained selecting the smoothest profile compatible with the observations. This can be done choosing a well conditioned MSS (obtained selecting only the well measured components of a) and determining in the null space (that now includes the null space enlarged with the poorly measured components rejected from the measurement space) the value of b that minimizes the oscillations. To this purpose, we minimize the quantity:
as a function of b, where L1 is the first-derivative matrix with respect to altitude and Eqs. (3.4)-(3.6) have been used for the definition of x. In this equation the matrices V and W do not necessarily coincide with the matrices used in section "Theory of MSS calculation"; they are now, respectively, the matrices containing the bases of the well measured subspace of the measurement space and of the new complementary null-space. Setting equal to zero the derivatives of the quantity in Eq. (5.1) with respect the components of b and substituting for a our estimation â, it results that the estimation of b is given by:
, (5.2)
where R=L1TL1. This calculation of the components of x in the new null space allows obtaining the smoothest profile compatible with the observations and corresponds to a regularization of the profile; accordingly, it is referred to as null- space regularization (NSR). The profile obtained as the sum of the MSS and of the null-space component calculated with the NSR is referred to as regularized measurement-space solution (RMSS). This approach has been used in the retrieval of the ozone volume mixing ratio profile from MIPAS measurements in [10] and the results are shown in Fig. 5.1.
When a reliable estimation of the climatological profile and of its VCM are known, it is possible to exploit the climatological profile to determine the null-space component. Also in this case is recommendable to choose a well conditioned MSS (obtained selecting only the well-measured components of a) and determining in the null space (that now includes the null space enlarged with the poorly measured components rejected from the measurement space) the value of b as the projection of the climatological profile on the base W. This approach has been used in the retrieval of ozone partial columns from data fusion of IASI and MIPAS measurements [11].
Another way to represent the vertical profile in a complete space using the climatological profile is to perform the weighted mean between the profile obtained from the MSS and the climatological profile, the weights being the inverses of the VCMs. In this way in the obtained profile the components well measured are practically equal to those of the MSS while the components not measured or poorly measured are practically equal to those of the climatological profile. This choice removes the need to choose the number of well-measured components to represent the MSS, because the weighted mean can be done using all the components and associating infinity error to the components that are not measured. This approach has been used in the retrieval of ozone vertical profiles from data fusion of IASI and MIPAS measurements [12].
