The measurement space solution (MSS) is a new type of representation of the information retrieved about the vertical profile of an atmospheric constituent. In this new representation the profile is not given, as in classical retrievals, by a sequence of values as a function of altitude, but as the combination of a set of functions each weighted with a measured amplitude. The set of functions are those that belong to the functional space in which the measurement is performed (the so called "measurement space"). The profile obtained in this new way does not directly provide, as classical retrievals do, a useful graphical representation (as seen in Fig. 1.1 and discussed in section 5, some "maquillage" is needed for a graphical representation), but has other important advantages that make it a tool for the full extraction of the information that the observations contain about the profile that is being retrieved.

Fig. 1.1 Comparison between the graphical representations of the MSS and of a classical retrieval of an ozone volume mixing ratio vertical profile. The MSS does not directly provide a useful graphical representation.

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.

We represent the observations (radiances) with a vector y of m elements and the vertical profile of the unknown atmospheric parameter with a vector x of n elements corresponding to a predefined altitude grid. The relationship between the vectors x and y is:

y = F (x) + ɛ , (3.1)

We expand F(x) up to the first order around a specific value of x, identified by x₀ and referred to as the linearization point, and, after some rearrangement, Eq. (3.1) becomes equal to:

y - F(x₀) + K X₀ = KX + ɛ , (3.2)

where K is the Jacobian matrix (that is the partial derivatives of F(x) with respect to the elements of x) calculated at x₀. Eq. (3.2) implies that the elements of  y-F(_X0) + K X₀ are the scalar products between x and the rows of K plus the errors.  It follows that the knowledge of  y-F(_X0) + K X₀ determines the knowledge of the component of x that lies in the space generated by the rows of K. This space is referred to as measurement space and its orthogonal complement in Rⁿ is referred to as null space.
In order to weigh the observations with their errors and avoid the complication of correlated errors it is useful to consider the quantity S_y ^-1/2 (which is characterized by a VCM that is the unity matrix) instead of the observations y. Multiplying both terms of Eq. (3.2) on the left by S_y ^-1/2 we obtain:

S_y ^-1/2 [ y - F(x₀) + K X₀] = S_y ^-1/2 Kx + S_y ^-1/2 ɛ , (3.3)

Since the space Rⁿ can be split into the direct sum of the measurement space and of the null space, we can write:

x_a and x_b can be expressed as:

X_a = Va , (3.5)

X_b = Wb , (3.6)

where V is a matrix whose columns are an orthonormal basis of the measurement space, W is a matrix whose columns are an orthonormal basis of the null space and a and b are the projections of x on these orthornormal bases:

a = V^T x ,  (3.7)

b = W^T x , (3.8)

S_y^-1/2 K = U Λ V^T ,  (3.9)

The columns of V are an orthonormal basis of the space generated by the rows of S_y^-1/2K. Since S_y^-1/2 is a nonsingular matrix the space generated by the rows of S_y^-1/2K coincides with the space generated by the rows of K, therefore, the columns of V are an orthonormal basis of the measurement space and, among all the possible orthonormal bases of the measurement space, it can be chosen for representing x_a with Eq. (3.5).  We can now determine the components of  x_a relative to this orthonormal basis. Substituting Eq. (3.9) in Eq. (3.3), multiplying both terms on the left by Λ^-1U^T and using Eq. (3.7), after some rearrangements, we obtain that â, i.e. the estimation of a deduced from the observations, is given by:

â = a + ɛ_a =  V^T x + ɛ_a = V^T x₀ + Λ^-1 U^T S_y ^-1/2 ( y - F(x₀) ) , (3.10)

is the error that we make taking â as the estimation of a and is characterized by the diagonal VCM:

S_a = Λ^-2 ,   (3.12)

The possibility to split the profile into a component retrieved from the observations and a component obtained from a priori information is particularly suitable to perform data fusion among independent measurements. Indeed the usual approach to data fusion implies the merging of products retrieved from the individual measurements, thus leading to a direct transfer into the fused data of any a priori information contained in a product. However, when a component of the profile is determined by a measurement there is no need for constraining this component in the other measurement by means of a priori information. This consideration suggests that the MSS is the optimal quantity to be used as input for data fusion.

In this section we analyze how the MSS can be used for the fusion of two independent indirect measurements of the same profile x. The results of this section can be easily extended to an arbitrary number of independent measurements.

Since the predetermined grid can be chosen freely and be common to the two measurements, we can assume that the two measurement spaces are subspaces of the same space Rⁿ. The two MSSs are characterized by the matrices V₁ and V₂ that identify the measurement spaces (of dimensions p₁ and p₂ respectively) and by the two vectors â₁ and â₂ (with their VCM S_{a1 and} S_a2). The relationships between these MSSs and the profile x are given by Eq. (3.10):

â_{2 =} V₂^Tx + ɛ_{a2     ,} (4.2)

which can be written in the compact form:

,       (4.3)

where the notation:

means the matrix (vector) obtained arranging the rows of the matrix (vector) Q below the rows of the matrix (vector) P and ɛ_a1 and ɛ_a2 contain the errors with which a₁=V₁^Tx and a₂=V₂^Tx are stimated by â₁ and â₂. Eq. (4.3) implies that the elements of the vector  are the scalar products between x and the columns of V₁ and of V₂ plus the errors. It follows that the knowledge of determines the knowledge of the component of x that lies in the space generated by the columns of V₁ and of V₂ . This space is the union space of the measurement spaces of the two individual measurements. We are now in a situation similar to that encountered in the case of Eq. (3.2) where the vector  plays the role of y-F(x₀)+Kx₀ and the matrix   plays the role of K. Following the procedure described in the section "Theory of MSS calculation" we can calculate the MSS in the union space of the measurement spaces of the two individual measurements.

We can represent the profile x in the form of the summation of a vector of the union space of the two measurement spaces and of a vector of the orthogonal complement space to this space (which coincides with the intersection space of the two null spaces of the original measurements). The vector in the union space is the MSS of the data fusion problem and its identification implies the determination of the elements of the following relationship:

â₁₂ = V₁₂^T x + ɛ_a12 , (4.4)

where â₁₂ is the estimation of the components of x in the basis of the union space represented by the matrix V₁₂, and ɛ_a12 is its error.

As in section theory we transform the vector  in such a way that its VCM becomes the unity matrix. This can be obtained multiplying Eq. (4.3) by the matrix  , obtaining:

,  (4.5)

If we perform the SVD of the kernel of Eq. (4.5) we obtain:

,   (4.6)

where U₁₂ is a matrix of dimension (p₁+p₂)xp₁₂, with p₁₂≤(p₁+p₂), Λ₁₂ is a nonsingular diagonal matrix of dimension p₁₂x p₁₂ and V₁₂ is a matrix of dimension nx p₁₂ whose columns are an orthonormal basis of the space generated by the rows of . Since S_a1^-1/2 and S_a2^-1/2 are nonsingular matrices, the space generated by the rows of  coincides with the space generated by the columns of V₁ and of V₂. Consequently the columns of V₁₂ are an orthonormal basis of the union space of the two measurement spaces.

The knowledge of V₁₂ can be used to determine the vector â_12. Substituting Eq. (4.6) in Eq. (4.5) and multiplying both terms on the left by Λ₁₂ ^-1U₁₂^T we obtain:

,  (4.7)

where ε_a12 is given by:

,   (4.8)

and is characterized by the VCM:

_{S a12} =  Λ_{12 ^-2} ,    (4.9)

â₁₂ obtained from Eq. (4.7) and V₁₂ obtained from Eq. (4.6) provide the MSS of x in the union space of the two measurements. Since the space characterized by V₁₂ exclusively includes the two original measurement spaces, the MSS in the union space has been obtained by utilizing all the information provided by the two sets of observations and without any a priori information. This characteristic makes this method for data fusion optimal and equivalent to the simultaneous analysis of the two sets of observations.

This method of data fusion has been applied to the fusion of IASI and MIPAS ozone data to retrieve the tropospheric and stratospheric columns [11] and the vertical profile [12]. In Fig. 4.1 and example of the improvement in retrieval errors obtained using data fusion with respect to using single measurements is reported.

Fig. 4.1 Example of the improvement obtained using data fusion with respect to using single measurements. Percentage retrieval errors on ozone volume mixing ratio [12] for the profiles retrieved from simulated measurements using only the IASI measurement (green line), only the MIPAS measurement (blue line) and the IASI-MIPAS data fusion (black line). The standard deviation of the climatological profile (red line) is also reported. Panel (b) shows a blow up of the panel (a) in the low altitude region.