Application of the MSS to data fusion

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.