4.2.3 Multi-parameter regularization

If the inversion deals with the joint retrieval of several gas species (and optionally of some
auxiliary parameters), the state vectorxis a concatenation of the vectors x_{p} corresponding to
the concentration profile of thepth molecule. Moreover, all individual regularization termsL^{p}
are assembled into a global matrix Lwith a block-diagonal structure based on the assumption
that the components of the state vector are independent:

L =

√λ_{1}L^{1} 0 · · · 0

0 √

λ_{2}L^{2} · · · 0
... ... . .. ...
0 0 · · · √

λ_{N}L^{N}

, (4.32)

whereNis the number of target molecules. Note that the regularization parameters in Eq. (4.32) are included in the expression of the global regularization matrix. Multi-parameter regulariza-tion schemes can be classified into two types according to the objective of the inverse problem:

• complete multi-parameter regularization scheme, in which the regularized solution corre-sponding to the entire state vector is computed;

• partial multi-parameter regularization scheme, in which only some components of the state vector are retrieved with a sufficient accuracy, e.g. the joint fit of one molecule considered as a main target and an auxiliary atmospheric profile or instrument parameter considered as a contamination.

In this regard, the regularization parametersλ_{k}can be selected by minimizing certain objective
functions with respect to the entire state vector or corresponding to the main target of the state
vector, respectively [Doicu et al., 2010].

Figure 4.2: Data flow diagram for illustrating the different type of errors.

In this context, we can say that the instrument is sensitive over the “entire” spectral domain
to a ±ε-variation in the kth component of the state vector about the a priori, if|[∆F_{k}]_{i}|> σ
for all i= 1, . . . , m, whereσ^{2} is the noise variance.

By using these diagnostic techniques, the sensitivity of the limb radiances to the unknowns of the inverse problems (i.e. vertical distributions of the molecular concentration or the temper-ature) can be studied. For a finer altitude grid, the unfavorable situation is that the retrieval of these quantities is essentially based on information coming from the a priori knowledge and not from the measurement itself. To remediate this deficiency, we can choose a coarser retrieval grid or employ higher-order interpolation schemes.

4.3.2 Error analysis and characterization

When the iterative process converges, it is important to assess the reliability of the retrieval products through an error analysis and characterization. An elaborate discussion covering the topic of error analysis and characterization can be found in Rodgers [2000] and Doicu et al.

[2010]. To explain the different type of errors, we illustrate the data flow diagram in Fig. 4.2.

The state vector x and the forward model parametersb are mapped by the radiative transfer
model into the forward model functionF(x,b). Similarly, the signal measured by the instrument
s^{δ} is mapped by the instrument model together with the instrument model parameters c into
the noise data vector y^{δ}. The noisy data vector y^{δ} sums the contribution of the instrument
model function and the measurement (or the instrument) noiseδ, wheresis the signal measured
by the instrument in the noise-free case. Finally, in the inversion model, the residual vector
f(x) =F(x,b)−y^{δ} is used to compute the Tikhonov function.

Given the regularized generalized inverseK^{†}_{λ}, the averaging kernel (or the model resolution)
matrix is defined by

A = K^{†}_{λ}K. (4.36)

Providing that the higher-order terms at the solution are neglected, the retrieval error is straight-forwardly represented by

e_{λ} = x_{λ}−x_{t} = e_{s}+e_{y}+e_{b}+e_{c} , (4.37)
where the smoothing (or null-space) error

es = (A−In) (xt−xa) (4.38)

quantifies the loss of information due to the regularization in the inversion model,

e_{y} = K^{†}_{λ}δ (4.39)

is the noise error quantifying the loss of information due to the measurement noise δ,

e_{b} = K^{†}_{λ}δ_{b} (4.40)

is the forward model error in the state space resulting from the forward model error in the data
space δ_{b}, and

e_{c} = K^{†}_{λ}δ_{c} (4.41)

is the instrument model error in the state space resulting from the instrument model error in
the data spaceδ_{c}.

Although in practice the smoothing error is not a computable quantity, it can be alternatively approximated by

e_{s} ≈ (A−I_{n}) (x_{λ}−x_{a}) . (4.42)

Evidently, keeping the smoothing error small requires a small regularization parameter λ, whereas the stability of the inversion process requires a reasonably large regularization pa-rameter. Thus, the optimal value of λ has to be determined through a compromise between stability and accuracy.

As can be inferred from Eq. (4.38), the deviation of the averaging kernel matrix A from
the identity matrix I_{n} characterizes the smoothing error. In atmospheric remote sensing, the
averaging kernel matrix provides more information than just a characterization of the smoothing
error. The rows of A, namely the averaging kernels, theoretically tend to peak at the diagonal
value with a width which is a measure of the vertical resolution of the instrument. In the
ideal case, A would be a unit matrix, but in practice, the regularization degrades the vertical
resolution, that is reflected by the peak value and the full width at half maximum (FWHM) of
this peak. The measurement response is given by the sum of the elements of each averaging
kernel row:

M_{i} =

n

X

j=1

[A]_{ij} , i= 1, . . . , n , (4.43)
and high values around unity assure that the retrieved information comes mostly from the
measurement and the contribution of the a priori profile is almost negligible. Furthermore, the
trace of the averaging kernel matrix yields the degree of freedom (DOF) for the signal, that is
interpreted as the number of useful independent quantities in a measurement, or a measure of
information for brevity.

The forward model error eb is caused by inaccurate knowledge of the forward model pa-rameters b (atmospheric and spectroscopic parameters in the forward model). If ∆b are the uncertainties inb, the forward model error can be approximated by

δ_{b} = K_{b}∆b ≈ F(xt,b+ ∆b)−F(xt,b), (4.44)

Figure 4.3: Instrument model error.

whereK_{b} is the Jacobian matrix with respect to the forward model parameters, ∂F/∂b.

The instrument model error ec is due to inaccurate knowledge of the instrument model
parameters c (Fig. 4.3). The signal delivered by the instrument is given by the inverse of the
instrument model functionR^{−1}acting on the exact data vectoryand the instrument parameters
c, plus the additive measurement noiseδ, i.e.s^{δ}=s+δ=R^{−1}(y,c) +δ. If in the Level-1 data
processing step, the instrument model parameters c0 match the true instrument parametersc,
the output of the instrument model read as

y^{δ} = R(s,c) +δ = R(R^{−1}(y,c),c) +δ = y+δ . (4.45)
If this is not the case, i.e. if c_{0} 6=c, we have

y^{δ} = R(s,c_{0}) +δ = R(R^{−1}(y,c),c_{0}) +δ = y+δ_{c}+δ, (4.46)
whereδc is the instrument model error in the data space defined by

δc = R(s,c0)−R(s,c) = R(R^{−1}(y,c),c0)−R(R^{−1}(y,c),c) . (4.47)
Thus, assuming that ∆care the uncertainties inc, the instrument model error in the data space
can be computed as

δc = R(s,c+ ∆c)−R(s,c) . (4.48)

Because in the Level-2 data processing, we deal only with the noisy data vectory^{δ}=y+δ_{c}+δ,
it is a common practice to employ an additional correction step to account for the instrument
model error δ_{c} = δ_{c}(c). This situation is depicted in Fig. 4.4, and note that the instrument
parameters can or cannot be included in the retrieval.

In a semi-stochastic framework, the smoothing and model parameter errors are deterministic,
whereas the noise error is stochastic with zero mean and covariance matrix S_{y}. The quality of
the regularized solution can be estimated through the mean square error matrix computed as

Sλ = Eh

(xλ−xt) (xλ−xt)^{T}
i

≈ Ss+Sy+Sb+Sc , (4.49)
where under the assumption that the measurement noise δ is a white noise with variance σ^{2},

Figure 4.4: Correction step for the instrument model error.

we have

S_{s} = e_{s}e^{T}_{s} , (4.50)

S_{y} = σ^{2}K^{†}_{λ}K^{†T}_{λ} , (4.51)

S_{b} = e_{b}e^{T}_{b} , (4.52)

S_{c} = e_{c}e^{T}_{c} . (4.53)

The square root of the diagonal elements of S_{λ} gives the retrieval error, while the off-diagonal
elements quantify the correlations between the components of the regularized solution x_{λ},

C_{λ}(i, j) = Sλ(i, j)

pS_{λ}(i, i)S_{λ}(j, j) , (4.54)

whereiand j are the row and column index, respectively.

Note that the sum of the square root of the diagonal elements of S_{λ} is the expected value
of the retrieval error, i.e.

Eh
ke_{λ}k^{2}i

= ke_{s}k^{2}+Eh
ke_{y}k^{2}i

+ke_{b}k^{2}+ke_{c}k^{2} , (4.55)
where E is the expected value operator. As stated previously, the model parameter errors are
deterministic. More precisely, if for example, ∆b=εbfor some scalar ε <1, we have

S_{b} = ε^{2}K^{†}_{λ}K_{b}bb^{T}K^{T}_{b}K^{†T}_{λ} . (4.56)
Defining the diagonal matrixB by [B]_{ii}= [b]_{i},i= 1, . . . , n, the matrix of all ones1 by

1 =

1 · · · 1 ... . .. ...

1 · · · 1

, (4.57)

and the symmetric and positive definite matrixC_{b} by

C_{b} = ε^{2}B1B^{T} , (4.58)

i.e.

[Cb]_{ij} = ε^{2}[b]_{i}[b]_{j} , i, j= 1, . . . , n . (4.59)
yields

S_{b} = K^{†}_{λ}K_{b}C_{b}K^{T}_{b}K^{†T}_{λ} . (4.60)
In a stochastic framework, when b is assumed to be a stochastic quantity, C_{b} defined by
Eqs. (4.58) and (4.59), is interpreted as the covariance matrix of b. Thus, in view of Eq. (4.22)
with l → ∞, Eqs. (4.57)–(4.59), a deterministic treatment of b is equivalent to a stochastic
treatment, in which εis the model parameter standard deviation, and all components of bare
perfectly correlated. If the components of b are not correlated, then the matrix of all ones 1
should be replaced by the identity matrixIn.