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Theory of MSS calculation

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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)

 

where the function F(x) is the forward model and ɛ is the vector containing the experimental errors of the observations, characterized by a variance-covariance matrix (VCM) Sy.

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 Sy -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 Sy -1/2 we obtain:

 

Sy -1/2 [ y - F(x0) + K X0] = Sy -1/2 Kx + Sy -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 = Xa + Xb ,   (3.4)

where xa and xb, respectively, belong to the measurement space and to the null space (see Fig. 3.1).


Fig._3.1

Fig. 3.1 Decomposition of the vertical profile in the measurement-space and null-space components.

 

 

xa and xb can be expressed as:

Xa = Va , (3.5)

Xb = 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 = VT x ,  (3.7)

b = WT x , (3.8)

 

 

where the superscript T denotes transposed matrices. The component xa is the only quantity that can be derived from the observations. In order to find xa from Eq. (3.5) we need to identify V and a. To this purpose we perform the singular value decomposition (SVD) of:

 

Sy-1/2 K = U Λ VT ,  (3.9)

 

The columns of V are an orthonormal basis of the space generated by the rows of Sy-1/2K. Since Sy-1/2 is a nonsingular matrix the space generated by the rows of Sy-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 xa with Eq. (3.5).  We can now determine the components of  xa relative to this orthonormal basis. Substituting Eq. (3.9) in Eq. (3.3), multiplying both terms on the left by Λ-1UT 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 + ɛa =  VT x + ɛa = VT x0 + Λ-1 UT Sy -1/2 ( y - F(x0) ) , (3.10)


where:

ɛa = Λ-1 UT Sy -1/2 ɛ , (3.11)

 

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

 

Sa = Λ-2 ,   (3.12)

 

From Eq. (3.12) we see that the components of the vector â are independent of each other and are characterized by variances given by the inverse of the squared singular values of Sy-1/2K. Therefore, components corresponding to large singular values are well determined while components corresponding to small singular values are poorly determined. The definition of xa in terms of V and â, determined respectively by Eq. (3.9) and by Eq. (3.10), is referred to as the measurement-space solution (MSS).