On the choice of retrieval variables in the inversion of remotely sensed atmospheric measurements

Marco Ridolfi; Luca Sgheri

doi:10.1364/OE.21.011465

1. Introduction

For the retrieval of atmospheric constituents and state variables such as pressure and temperature from remote sensing spectroscopic measurements it is necessary to model the radiative transfer through an inhomogeneous medium. The inhomogeneouos atmosphere is usually discretized in small elements which are assumed to be homogeneous. The law governing absorption in an homogeneous medium is the Beer-Lambert law

\frac{d}{d x} I (x) = - K_{σ} I (x),

where I(x) is the intensity at spatial coordinate x, and K_σ is a frequency-dependent constant. Equation (1) has the general solution I(x) = cexp(−K_σx). As a consequence the radiance reaching the instrument is modeled as a linear combination of exponential terms. Disregarding scattering, which is justified in the mid-to-far infrared spectral region in clear sky conditions, the radiance S(σ) at frequency σ reaching the instrument for a fixed observation geometry is described by the radiative transfer equation [1]

S (σ) = B_{0} (σ) τ (σ, 0, s_{obs}) + \int_{0}^{s_{obs}} B (σ, T (s)) \frac{\partial}{\partial s} τ (σ, s, s_{obs}) d s .

Here s is the coordinate along the line of sight, s_obs is the coordinate of the observing instrument, τ(σ, s₁, s₂) is the spectral transmittance along the line of sight from position s₁ to position s₂, B₀(σ) is the background radiance and B(σ, T(s)) is the source function which, under the assumption of local thermodynamic equilibrium, becomes the Planck function, which depends on the kinetic temperature T(s) and the frequency σ. In the limb scanning geometry, the term B₀(σ) is assumed to be zero. A common choice is to discretize the atmosphere with a series of spherical homogeneous layers. In this case, let l = 1,...,N be the layers crossed by a given line of sight, and

p_{l}^{e}

and

T_{l}^{e}

the equivalent pressure and temperature of layer l[2, 3]. Equation (2) may then be rewritten as

S (σ) = \sum_{l = 1}^{N} B (σ, T_{l}^{e}) (1 - τ_{σ, l}) \prod_{j = 1}^{l - 1} τ_{σ, j},

where τ_σ,l is the transmittance through layer l at frequency σ. The transmittance in the layer l has the following expression [4, 5]

τ_{σ, l} = exp (- \sum_{m} k_{σ, m} (p_{l}^{e}, T_{l}^{e}) c_{l, m}) = \prod_{m} exp (- k_{σ, m} (p_{l}^{e}, T_{l}^{e}) c_{l, m}) \equiv \prod_{m} τ_{σ, l, m},

where m is the gas index, k_σ,m is the cross-section of that gas at frequency σ, which depends only on

p_{l}^{e}

and

T_{l}^{e}

. Finally c_l,m is the gas column in the layer along the line of sight, which depends on the gas Volume Mixing Ratio (VMR) x_m(z) in the following way:

c_{l, m} = \int_{l} x_{m} (z) η (z) \frac{d s}{d z} (z) d z,

where z is the altitude, η(z) is the number density and ds/dz is the linear element along the line of sight. Let z_i be the retrieval grid. We may use the values x_i,m = x_m(z_i) as retrieval variables, and model the VMR with a piecewise linear curve with nodes z_i. It follows that c_l,m is a linear function of x_i,m so that the entire spectrum will be a linear combination of exponential terms e_j of the form exp(∑_i_∈_I_(j)d_ix_i,m) for some constants d_i, and the sum is extended on subsets I(j) of indexes.

The inversion of the measured spectra is obtained by optimizing the values of the retrieval variables. In the weighted least-squares approach the cost function χ² is defined as ℓ₂ norm of the differences between observed and simulated radiances (the so-called residuals) weighted with the inverse of the error covariance matrix of the measurements.

While the x_i,m have a physical meaning, their selection as retrieval variables makes the forward model a sloppy model, as described in [6]. The main drawback of such a model is that the model manifold includes wide regions where the residuals of the reconstructed spectrum are large, and long, narrow and tortuous valleys where the residuals are small. As a consequence many different sets of retrieval variables may lead to similar χ² values close to the minimum.

There are many methods to deal with sloppy models. However, most of them are based on a line search substep to calculate the increment along a descent direction [6]. Those methods are not suitable for atmospheric retrievals, because the evaluation of the simulated spectrum (the forward model) is a time-consuming operation. Our solution is to modify the retrieval variables to obtain a better dependency in the forward model and, ultimately, a better convergence towards the minimum of the χ². In the theory Section 2 we present the theoretical background of the modification. In Section 3 we present the results of some test retrievals applied to MI-PAS (Michelson Interferometer for Passive Atmospheric Sounding) observations. Finally in Section 4 we draw the conclusions.

2. Theory

2.1. The continuum variables

Conventionally the cross-sections k_σ,m appearing in Eq. (4) are computed including only the contributions of transitions not farther than 25 cm⁻¹ from σ[4, 5]. In restricted spectral intervals, such as the microwindows (MWs) used in our retrievals (less than 3 cm⁻¹) [7], the contribution of the lines beyond the 25 cm⁻¹ threshold is modeled with a frequency-independent term, the so-called continuum, which is normally MW and altitude dependent. This term may also account for unmodeled contributions from aerosols or residual instrument calibration errors. In the forward model this term is usually implemented as an additional gas, with VMR equal to 1. As a consequence the columns of the continuum depend only on the observation geometry and the density of the layers. The retrieval variables model the continuum cross-section vertical profile. This profile can be approximated with a curve with nodes z_i (the retrieval grid) and values k_i = k(z_i). The discretization used for the radiative transfer calculation (3) may be finer than the retrieval grid. However, for each layer l the continuum k_l in that layer will depend on at most two values k_i and k_i₋₁ with consecutive indeces. Thus we have

k_{l, c} = a_{l, i} k_{i} + b_{l, i} k_{i - 1},

where 0 ≤ a_l,i, b_l,i ≤ 1 are constants determined by the interpolation law, for instance a linear interpolation in pressure.

The air columns can be explicitly calculated, but again we obtain a sloppy model because the contribution of the continuum in the productory (4) is

τ_{l, c} \equiv exp (- k_{l, c} c_{l, air}) = exp (- (a_{l, i} k_{i} + b_{l, i} k_{i - 1}) c_{l, air}) .

We can uniformly bound the air columns with a constant C_air ≥ c_l,air for any l and any line of sight. In our implementation we choose C_air = 10²⁵ cm⁻². Then we define the new variables:

ξ_{i} = exp (- k_{i} C_{air}),

so that

τ_{l, c} = ξ_{i}^{a_{l, i} c_{l, air} / C_{air}} ξ_{i - 1}^{b_{l, i} c_{l, air} / C_{air}} .

Note that while the acceptable values for the k_i are k_i ≥ 0, the range for the new variables is 0 ≤ ξ_i ≤ 1. This is clearly an improvement because the sensitivity of the forward model to k_i vanishes for large enough values of k_i because of the exponential dependence. The new variables ξ_i are polynomially connected with τ_l,c, that represents the optical transparency of layer l due to the continuum. Thus they are tightly linked to the measured spectra.

For the minimization required by the inversion, most techniques, such as the widely used Gauss-Newton method, require the derivatives of the simulated spectrum with respect to the retrieval variables. The derivatives with respect to the k_i continuum variables are calculated as

\frac{\partial S (σ)}{\partial k_{j}} = \sum_{l} \frac{\partial S (σ)}{\partial τ_{l, c}} \frac{\partial τ_{l, c}}{\partial k_{j}},

where the summation is extended to all the layers depending on the value k_j. Because of (6), for each l there are only two non vanishing derivatives,

\frac{\partial τ_{l, c}}{\partial k_{i}}

and

\frac{\partial τ_{l, c}}{\partial k_{i - 1}}

. With the new approach the derivatives become

\frac{\partial S (σ)}{\partial ξ_{j}} = \sum_{j} \frac{\partial S (σ)}{\partial τ_{l, c}} \frac{\partial τ_{l, c}}{\partial ξ_{j}} .

and from (9) the only non vanishing derivatives are

\frac{\partial τ_{l, c}}{\partial ξ_{i}} = \frac{a_{l, i} c_{l, air}}{C_{air}} \frac{τ_{l, c}}{ξ_{i}}, \frac{\partial τ_{l, c}}{\partial ξ_{i - 1}} = \frac{b_{l, i} c_{l, air}}{C_{air}} \frac{τ_{l, c}}{ξ_{i - 1}} .

2.2. The VMR variables

In principle this approach can be extended also to the VMR variables, however there is one important difference. In Eq. (4) the quantity holding the dependence on the continuum retrieval variables is the cross-section k_l. On the other hand, the cross-sections $k_{l, m} (σ) \equiv k_{σ, m} (p_{l}^{e}, T_{l}^{e})$ do not depend on the VMR retrieval variables, the dependence being via the gas columns c_l,m.

If the VMR profile is modeled with a piecewise-linear function with nodes z_i and values x_i = x(z_i), then there are constants

A_{l, i} = \int_{l} \frac{z_{i - 1} - z}{z_{i - 1} - z_{i}} η (z) \frac{d s}{d z} (z) d z, B_{l, i} = \int_{l} \frac{z - z_{i}}{z_{i - 1} - z_{i}} η (z) \frac{d s}{d z} (z) d z,

such that

c_{l, m} = A_{l, i} x_{i} + B_{l, i} x_{i - 1} .

As in the continuum case we may define new variables ζ_i = exp (−x_iC_m) for some gas-dependent constant C_m, so that:

τ_{σ, l, m} = ζ_{i}^{k_{l, m} (σ) A_{l, i} / C_{m}} ζ_{i - 1}^{k_{l, m} (σ) B_{l, i} / C_{m}} .

The derivatives of the spectrum with respect to these new variables are:

\frac{\partial S (σ)}{\partial ζ_{j}} = \sum_{l} \frac{\partial S (σ)}{\partial τ_{σ, l, m}} \frac{\partial τ_{σ, l, m}}{\partial ζ_{j}} .

and from (15) the only non vanishing terms are

\frac{\partial τ_{σ, l, m}}{\partial ζ_{i}} = \frac{A_{l, i} k_{l, m} (σ)}{C_{m}} \frac{τ_{σ, l, m}}{ζ_{i}}, \frac{\partial τ_{σ, l, m}}{\partial ζ_{i - 1}} = \frac{B_{l, i} k_{l, m} (σ)}{C_{m}} \frac{τ_{σ, l, m}}{ζ_{i - 1}} .

3. Test of the method on MIPAS measurements

Starting from March 2002, MIPAS [8] has been sounding for ten years the mid-infrared atmospheric limb-emission spectrum from the polar satellite ENVISAT, developed by the European Space Agency (ESA). The measured spectral region contains the signatures of many atmospheric constituents. ESA set up a retrieval algorithm [4, 5, 7] to determine pressure at tangent points and geolocated profiles of temperature and of VMR of some atmospheric constituents. The retrieved constituents in version 6.0 [7] of the ESA operational processor are H₂O, O₃, HNO₃, CH₄, N₂O, NO₂, CFC-11, CFC-12, ClONO₂ and N₂O₅. The ESA operational retrieval algorithm V6.0 is based on the Gauss-Newton method, modified with the regularizing Levenberg-Marquardt (LM) technique (see [7] and the references therein). Due to the computational cost of the forward model, no search methods are viable for the LM constant. The strategy adopted for the evolution of the LM constant is the damping-undamping strategy (see [9]).

For the tests presented in this paper we used more conservative convergence thresholds with respect to those selected in V6.0. The V6.0 thresholds were optimized for near real-time processing of MIPAS data, taking into consideration the stringent runtime requirements. Since April 2012 the instrument is no longer operational, and computing time is no longer a primary issue, provided it remains into reasonable bounds. In the next reprocessing of the whole MIPAS mission convergence thresholds similar to those used for our tests will be adopted, in order to decrease the convergence error at the expenses of an increase in computing time. Moreover, we implemented the following selectable options.

The Iterative Variable Strength (IVS) regularization technique [10, 11], which is already scheduled for implementation in the next release of the ESA processor.
In addition to the nominal (NOM) version, we implemented the change of retrieval variables (as explained in Section 2), either for the continuum variables only (CONT version) or for the continuum and VMR variables (POLY version). The better conditioning of the inversion obtained in the CONT and POLY approaches permits to start the retrieval with a smaller LM damping parameter for the new retrieval variables. The advantages of this modification will be discussed in Section 3.1.

We introduce the following scalar quantifiers to characterize the performance of the retrieval.

The average (on the set of processed limb scans) number of retrieval iterations $\bar{NIt}$ .
The consistency of the final simulated spectra with the observations is measured by ${\bar{χ}}_{R}^{2}$ . This is the arithmetic mean (on the set of processed scans) of the normalized chi-square $χ_{R}^{2}$ (see [12]) related to individual profile retrievals.
We consider the number of degrees of freedom (DoF) of the retrieval, calculated as the trace of the averaging kernel matrix [13]. This quantifier represents the overall reduction of the vertical resolution as a consequence of the applied constraints. We divide this number by the number n of points of the retrieved profile, since this latter can vary from scan to scan due to cloud contamination. We then take the arithmetic mean $\bar{DoF / n}$ on the set of processed scans. Note that $1 - \bar{DoF / n}$ measures the fraction of the vertical resolution lost because of the combined effect of the LM technique and the a-posteriori regularization.
Finally, to evaluate the profile smoothness we use the oscillation quantifier Ω₂. For any profile x_i = x(z_i), i = 1,..., n, Ω₂ is defined as $Ω_{2} = 100 \cdot \sqrt{\frac{1}{n - 2} \sum_{i = 2}^{n - 1} {[x_{i} - x_{i - 1} - \frac{x_{i + 1} - x_{i - 1}}{z_{i + 1} - z_{i - 1}} (z_{i} - z_{i - 1})]}^{2} .}$ The quantity Ω₂ is proportional to the root mean square distance between each profile point x_i and the linear interpolation at z_i from the two adjacent points x_i₋₁ and x_i₊₁. When the z_i are equispaced, Ω₂ is proportional to the ℓ₂ norm of the discrete second derivative of the profile. We then take the arithmetic mean Ω̄₂ (on the set of processed scans) of the Ω₂ related to individual profiles.

3.1. Tests on a set of 12 MIPAS orbits

To assess the improvements of the change of variables described in Section 2, we first tested the retrieval on a limited set of real observations. We selected 12 orbits of MIPAS measurements from the year 2007. The measurements cover the four seasons to ensure that the test includes sufficient atmospheric variability. For this test we compared the NOM, CONT and POLY approaches, both without regularization and with the IVS regularization (NOM_R, CONT_R, POLY_R).

Figure 1 shows the impact of the proposed change of the retrieval variables on the algorithm. We calculated the spectral condition number of the GN approximation of the Hessian of the cost function, modified with the LM method. This is the matrix which needs to be inverted for the calculation of the next iterate of the GN sequence. In the figure we report the probability density of this condition number calculated from our NOM_R and CONT_R samples at the first and the final iteration. We noticed that the new approach generally led to a significant reduction of the condition number of the matrix. For some weak species like CFC-11 (cfr. right panel of Fig. 1) this reduction also solved the numerical problems of the inversion caused by a condition number greater than ∼ 10¹⁷ (the inverse of the machine precision). Furthermore we observed that this conditioning was less sensitive to the LM damping parameter. Hence we were able to decrease by a factor of 10 the initial value of this parameter for the new retrieval variables. This reduction is possible also thanks to the reduced non-linearity of the forward model with respect to the new continuum variables. The smaller value of the LM damping parameter leads to larger steps in the GN sequence. As a consequence we are able to get closer to the true minimum of the cost function, with fewer iterations. We found that the results of the retrieval do not depend critically on the initial value of the LM parameter for the new variables. Thus a fine tuning of this parameter on a per-target basis would not lead to significant improvements.

Fig. 1 Probability density of the log₁₀ of the spectral condition number of the GN approximation of the Hessian of the cost function, at first and last iteration. Retrieval of temperature (left panel) and CFC-11 (right panel)

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The results are summarized in Table 1. For each test and each target parameter, the table reports the quantifiers described above. The last block of the table contains the percentage variation of the quantifiers with respect to the NOM approach, averaged over the retrieval targets.

Table 1. Average retrieval performances in a sample of 12 MIPAS orbits.

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Generally, all the retrieval targets exhibit similar behaviors, with two notable exceptions. The NO₂ VMR is retrieved only above 24 km, while the continuum parameters are fitted only below 30 km. Thus, only a small number of continuum parameters are retrieved, in an altitude range where the opacity of the atmosphere is small. Hence the sensitivity of the retrieval to continuum parameters is good also in the NOM approach. In the retrieval of N₂O₅, the selected MWs span a limited spectral range (approximately 27 cm⁻¹), therefore a single continuum profile is retrieved to make the inversion better conditioned. Hence the effect of the CONT approach on these two species is small.

From the last block of Table 1 we see that the CONT approach is able to reduce the final ${\bar{χ}}_{R}^{2}$ of about 3% and the number of iterations of about 15% with respect to the NOM approach, at the expenses of a 5% increase of the Ω̄₂. The regularizing effect of the LM method in this case is weaker due to the smaller initial value of the LM damping parameter. This is confirmed by the 5% increase in $\bar{DoF / n}$ . In the CONT_R approach the regularization is able to reduce the amplitude of the oscillations to the level of the NOM_R approach, while maintaining the 3% reduction of the ${\bar{χ}}_{R}^{2}$ .

We tested the statistical significance of the reduction in the number of iterations and in the ${\bar{χ}}_{R}^{2}$ with a per-target Welch’s t-test. Given the relatively large sample size, all the improvements of the CONT_R with respect to the NOM_R method are significant with a probability threshold of 0.01, except for the aforementioned NO₂ and N₂O₅ species.

As pointed out in Section 2, the new approach changes the search domain for the continuum variables from the semi-unbounded interval [0, +∞) to the bounded domain [0, 1]. The LM method does not constrain the retrieval variables within their domain, thus a check on their value is implemented in the program, similarly to the approach already adopted for the 0 bound in V6.0 [7]. This check however intervenes only in a very marginal number of retrievals, usually corresponding to difficult observational conditions.

The outcome of the POLY and POLY_R tests shows no advantage with respect to the CONT and CONT_R approaches. We attribute this result to the fact that the cross-sections k_l,m(σ) of the retrieved gas depend on the frequency within each MW. On the other hand the cross-sections k_l,c for the continuum are constant within each MW. The same argument also applies to the transparencies τ_σ,l,m and τ_l,c. In the NOM approach a sudden large increase of a continuum parameter may easily lead to a fully opaque layer τ_l,c ≈ 0 for all the frequencies within the involved MW. As a consequence, any emission from a layer beneath l is blocked by the fully opaque layer. On the other hand, a large increase in the VMR parameter hardly leads to a full MW with τ_σ,l,m ≈ 0, due to the dependence on σ. Also in the CONT approach we can get an opaque layer with τ_l,c ≈ 0. However, while in the NOM case we have $\frac{\partial τ_{l, c}}{\partial k_{j}} \to 0$ when k_j → ∞, in the CONT case $\frac{\partial τ_{l, c}}{\partial ξ_{j}} \to \infty$ when ξ_j → 0, because the exponents in Eq. (9) are less than 1. Consequently, in this case the sensitivity to the continuum parameters is not lost in the CONT approach.

3.2. Tests on a whole month of orbits

To make sure that the results of the previous test are not biased by the relatively small sample of orbits, we extended the test to a whole month of MIPAS orbits. We selected the measurements acquired in October 2007, consisting of 279 orbits, for a total of approximately 21400 scans. In order to limit the necessary computations, we compared only the most promising CONT_R approach to the standard NOM_R approach. We collected the results in Table 2, which has the same format of Table 1. The numbers of Table 2 fully confirm the conclusions of the previous test, i.e. reduced number of iterations, smaller ${\bar{χ}}_{R}^{2}$ and slightly smaller Ω̄₂ for the CONT_R over the NOM_R approach.

Table 2. Average retrieval performances in a sample of 1 month of orbits.

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In Fig. 2 we report the behavior of the $χ_{R}^{2}$ versus the latitude of the limb scans. The graph is obtained by binning the $χ_{R}^{2}$ in 5 deg. latitude intervals, and then averaging the data within each bin. The four panels refer to different retrieval targets. We do not report all target species, because they all show very similar behaviors. We note that the reduction of $χ_{R}^{2}$ obtained by the CONT_R approach is visible at all latitudes. For the reasons explained in Section 3.1, the change of variable shows no advantage in the retrieval of N₂O₅, see bottom right panel of the figure.

Fig. 2 Behavior of normalized χ² versus latitude for the NOM_R and CONT_R approaches for the October 2007 orbits. Each panel refers to the target indicated in the plot.

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Despite the smaller $χ_{R}^{2}$ achieved, we found that the average differences between the NOM_R and CONT_R profiles show no evident systematic features above the noise error.

4. Conclusions

Normally in the inversion of remote sensing atmospheric measurements, the atmospheric continuum is retrieved as an additional molecular cross-section, constant in limited spectral intervals. The selection of such retrieval variables however leads to a so called sloppy model. In this case the numerical search for the minimum of the cost function may be difficult.

In this paper we propose a different set of retrieval variables for the atmospheric continuum. Tests with real MIPAS observations show that a more stable retrieval is obtained with the new variables. As a consequence we obtain a reduction in the number of iterations, a smaller minimum of the cost function, and slightly less oscillating profiles. The same modification can be applied also to the VMR retrieval variables. Our tests show however that – at least in the present formulation – we get no real advantage connected with this additional change.

The technique can be applied to any inverse problem whenever the forward model depends exponentially on the inversion variables. Its effectiveness however should be evaluated case by case on the basis of numerical tests.

Acknowledgments

We thank the IFAC-CNR institute in Firenze for making available computing and technical facilities through associate contract to M.R., in the frame of the research activity TA.P06.002. This study was supported by the ESA-ESRIN contract 21719/08/I-OL.

References and links

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		NOM	NOM_R	CONT	CONT_R	POLY	POLY_R
T	$\bar{NIt}$	4.694	4.700	4.392	4.454	4.413	4.444
	${\bar{χ}}_{R}^{2}$	2.026	2.048	1.862	1.903	1.863	1.904
	$\bar{DoF / n}$	0.688	0.557	0.760	0.590	0.759	0.590
	Ω̄₂	427.711	339.341	448.269	344.410	448.275	344.021

H₂O	$\bar{NIt}$	3.455	3.395	3.179	3.154	3.635	3.598
	${\bar{χ}}_{R}^{2}$	1.295	1.304	1.271	1.280	1.272	1.284
	$\bar{DoF / n}$	0.819	0.450	0.856	0.460	0.905	0.474
	Ω̄₂	443.762	361.493	455.129	365.349	467.102	370.936

O₃	$\bar{NIt}$	6.225	6.242	3.526	3.402	3.772	3.601
	${\bar{χ}}_{R}^{2}$	2.345	2.344	2.269	2.273	2.272	2.278
	$\bar{DoF / n}$	0.756	0.605	0.888	0.659	0.908	0.671
	Ω̄₂	58.915	40.924	63.661	39.474	67.668	40.424

HNO₃	$\bar{NIt}$	3.446	3.414	2.993	2.965	3.023	3.010
	${\bar{χ}}_{R}^{2}$	1.369	1.368	1.333	1.329	1.337	1.341
	$\bar{DoF / n}$	0.731	0.555	0.800	0.583	0.852	0.599
	Ω̄₂	0.095	0.064	0.101	0.063	0.108	0.065

CH₄	$\bar{NIt}$	4.198	4.117	3.320	3.293	3.293	3.278
	${\bar{χ}}_{R}^{2}$	1.941	1.933	1.840	1.837	1.842	1.841
	$\bar{DoF / n}$	0.807	0.603	0.910	0.651	0.948	0.661
	Ω̄₂	18.553	10.658	19.328	10.141	18.994	9.671

N₂O	$\bar{NIt}$	3.881	3.883	3.208	3.191	3.169	3.163
	${\bar{χ}}_{R}^{2}$	1.888	1.889	1.802	1.805	1.802	1.807
	$\bar{DoF / n}$	0.777	0.554	0.835	0.579	0.898	0.597
	Ω̄₂	2.604	1.505	2.643	1.440	2.829	1.498

NO₂	$\bar{NIt}$	2.095	2.098	2.360	2.362	2.309	2.308
	${\bar{χ}}_{R}^{2}$	1.275	1.277	1.275	1.277	1.276	1.278
	$\bar{DoF / n}$	0.840	0.598	0.857	0.610	0.915	0.656
	Ω̄₂	0.380	0.311	0.397	0.328	0.370	0.302

CFC-11	$\bar{NIt}$	4.434	4.420	3.545	3.546	3.654	3.659
	${\bar{χ}}_{R}^{2}$	1.282	1.272	1.222	1.217	1.227	1.228
	$\bar{DoF / n}$	0.716	0.480	0.608	0.395	0.695	0.693
	Ω̄₂	0.004	0.002	0.004	0.001	0.004	0.004

CFC-12	$\bar{NIt}$	3.683	3.649	2.982	2.970	2.936	2.950
	${\bar{χ}}_{R}^{2}$	1.289	1.293	1.259	1.261	1.264	1.267
	$\bar{DoF / n}$	0.865	0.506	0.849	0.495	0.907	0.897
	Ω̄₂	0.010	0.003	0.011	0.003	0.010	0.010

N₂O₅	$\bar{NIt}$	3.421	3.417	3.018	3.019	3.110	3.122
	${\bar{χ}}_{R}^{2}$	1.360	1.351	1.364	1.352	1.371	1.373
	$\bar{DoF / n}$	0.925	0.474	0.922	0.467	0.955	0.653
	Ω̄₂	0.051	0.015	0.054	0.016	0.052	0.023

ClONO₂	$\bar{NIt}$	3.651	3.680	2.913	2.865	3.121	3.091
	${\bar{χ}}_{R}^{2}$	2.048	2.034	1.973	1.965	1.984	1.985
	$\bar{DoF / n}$	0.652	0.431	0.704	0.432	0.769	0.481
	Ω̄₂	0.027	0.011	0.037	0.012	0.032	0.012

Avg wrt NOM	$Δ \bar{NIt} (%)$		−0.422	−15.239	−15.675	−13.059	−13.430
	$Δ {\bar{χ}}_{R}^{2} (%)$		−0.029	−3.264	−3.148	−3.030	−2.631
	$Δ \bar{DoF / n} (%)$		−31.652	4.973	−30.398	11.136	−18.531
	ΔΩ̄₂(%)		−41.470	5.755	−43.347	4.679	−31.013

		NOM_R	CONT_R
T	$\bar{NIt}$	4.761	4.432
	${\bar{χ}}_{R}^{2}$	1.981	1.824
	$\bar{DoF / n}$	0.534	0.594
	Ω̄₂	316.132	329.808

H₂O	$\bar{NIt}$	3.381	3.137
	${\bar{χ}}_{R}^{2}$	1.300	1.274
	$\bar{DoF / n}$	0.445	0.455
	Ω̄₂	357.329	357.488

O₃	$\bar{NIt}$	6.018	3.527
	${\bar{χ}}_{R}^{2}$	2.382	2.308
	$\bar{DoF / n}$	0.599	0.650
	Ω̄₂	39.881	39.312

HNO₃	$\bar{NIt}$	3.377	2.962
	${\bar{χ}}_{R}^{2}$	1.344	1.310
	$\bar{DoF / n}$	0.554	0.582
	Ω̄₂	0.056	0.056

CH₄	$\bar{NIt}$	4.259	3.311
	${\bar{χ}}_{R}^{2}$	2.063	1.956
	$\bar{DoF / n}$	0.592	0.656
	Ω̄₂	12.050	11.623

N₂O	$\bar{NIt}$	3.952	3.202
	${\bar{χ}}_{R}^{2}$	1.900	1.807
	$\bar{DoF / n}$	0.539	0.582
	Ω̄₂	1.649	1.546

NO₂	$\bar{NIt}$	2.078	2.322
	${\bar{χ}}_{R}^{2}$	1.251	1.254
	$\bar{DoF / n}$	0.600	0.607
	Ω̄₂	0.265	0.282

CFC-11	$\bar{NIt}$	4.471	3.580
	${\bar{χ}}_{R}^{2}$	1.247	1.211
	$\bar{DoF / n}$	0.464	0.389
	Ω̄₂	0.002	0.001

CFC-12	$\bar{NIt}$	3.618	2.925
	${\bar{χ}}_{R}^{2}$	1.284	1.261
	$\bar{DoF / n}$	0.501	0.493
	Ω̄₂	0.003	0.003

N₂O₅	$\bar{NIt}$	3.212	2.938
	${\bar{χ}}_{R}^{2}$	1.355	1.357
	$\bar{DoF / n}$	0.464	0.463
	Ω̄₂	0.019	0.012

ClONO₂	$\bar{NIt}$	3.503	2.753
	${\bar{χ}}_{R}^{2}$	1.984	1.932
	$\bar{DoF / n}$	0.421	0.219
	Ω̄₂	0.010	0.012

Avg wrt NOM_R	$Δ \bar{NIt} (%)$		−15.119
	$Δ {\bar{χ}}_{R}^{2} (%)$		−2.957
	$Δ \bar{DoF / n} (%)$		−1.696
	ΔΩ̄₂(%)		−4.987

		NOM	NOM_R	CONT	CONT_R	POLY	POLY_R
T	$\bar{NIt}$	4.694	4.700	4.392	4.454	4.413	4.444
	${\bar{χ}}_{R}^{2}$	2.026	2.048	1.862	1.903	1.863	1.904
	$\bar{DoF / n}$	0.688	0.557	0.760	0.590	0.759	0.590
	Ω̄₂	427.711	339.341	448.269	344.410	448.275	344.021

H₂O	$\bar{NIt}$	3.455	3.395	3.179	3.154	3.635	3.598
	${\bar{χ}}_{R}^{2}$	1.295	1.304	1.271	1.280	1.272	1.284
	$\bar{DoF / n}$	0.819	0.450	0.856	0.460	0.905	0.474
	Ω̄₂	443.762	361.493	455.129	365.349	467.102	370.936

O₃	$\bar{NIt}$	6.225	6.242	3.526	3.402	3.772	3.601
	${\bar{χ}}_{R}^{2}$	2.345	2.344	2.269	2.273	2.272	2.278
	$\bar{DoF / n}$	0.756	0.605	0.888	0.659	0.908	0.671
	Ω̄₂	58.915	40.924	63.661	39.474	67.668	40.424

HNO₃	$\bar{NIt}$	3.446	3.414	2.993	2.965	3.023	3.010
	${\bar{χ}}_{R}^{2}$	1.369	1.368	1.333	1.329	1.337	1.341
	$\bar{DoF / n}$	0.731	0.555	0.800	0.583	0.852	0.599
	Ω̄₂	0.095	0.064	0.101	0.063	0.108	0.065

CH₄	$\bar{NIt}$	4.198	4.117	3.320	3.293	3.293	3.278
	${\bar{χ}}_{R}^{2}$	1.941	1.933	1.840	1.837	1.842	1.841
	$\bar{DoF / n}$	0.807	0.603	0.910	0.651	0.948	0.661
	Ω̄₂	18.553	10.658	19.328	10.141	18.994	9.671

N₂O	$\bar{NIt}$	3.881	3.883	3.208	3.191	3.169	3.163
	${\bar{χ}}_{R}^{2}$	1.888	1.889	1.802	1.805	1.802	1.807
	$\bar{DoF / n}$	0.777	0.554	0.835	0.579	0.898	0.597
	Ω̄₂	2.604	1.505	2.643	1.440	2.829	1.498

NO₂	$\bar{NIt}$	2.095	2.098	2.360	2.362	2.309	2.308
	${\bar{χ}}_{R}^{2}$	1.275	1.277	1.275	1.277	1.276	1.278
	$\bar{DoF / n}$	0.840	0.598	0.857	0.610	0.915	0.656
	Ω̄₂	0.380	0.311	0.397	0.328	0.370	0.302

CFC-11	$\bar{NIt}$	4.434	4.420	3.545	3.546	3.654	3.659
	${\bar{χ}}_{R}^{2}$	1.282	1.272	1.222	1.217	1.227	1.228
	$\bar{DoF / n}$	0.716	0.480	0.608	0.395	0.695	0.693
	Ω̄₂	0.004	0.002	0.004	0.001	0.004	0.004

CFC-12	$\bar{NIt}$	3.683	3.649	2.982	2.970	2.936	2.950
	${\bar{χ}}_{R}^{2}$	1.289	1.293	1.259	1.261	1.264	1.267
	$\bar{DoF / n}$	0.865	0.506	0.849	0.495	0.907	0.897
	Ω̄₂	0.010	0.003	0.011	0.003	0.010	0.010

N₂O₅	$\bar{NIt}$	3.421	3.417	3.018	3.019	3.110	3.122
	${\bar{χ}}_{R}^{2}$	1.360	1.351	1.364	1.352	1.371	1.373
	$\bar{DoF / n}$	0.925	0.474	0.922	0.467	0.955	0.653
	Ω̄₂	0.051	0.015	0.054	0.016	0.052	0.023

ClONO₂	$\bar{NIt}$	3.651	3.680	2.913	2.865	3.121	3.091
	${\bar{χ}}_{R}^{2}$	2.048	2.034	1.973	1.965	1.984	1.985
	$\bar{DoF / n}$	0.652	0.431	0.704	0.432	0.769	0.481
	Ω̄₂	0.027	0.011	0.037	0.012	0.032	0.012

Avg wrt NOM	$Δ \bar{NIt} (%)$		−0.422	−15.239	−15.675	−13.059	−13.430
	$Δ {\bar{χ}}_{R}^{2} (%)$		−0.029	−3.264	−3.148	−3.030	−2.631
	$Δ \bar{DoF / n} (%)$		−31.652	4.973	−30.398	11.136	−18.531
	ΔΩ̄₂(%)		−41.470	5.755	−43.347	4.679	−31.013

		NOM_R	CONT_R
T	$\bar{NIt}$	4.761	4.432
	${\bar{χ}}_{R}^{2}$	1.981	1.824
	$\bar{DoF / n}$	0.534	0.594
	Ω̄₂	316.132	329.808

H₂O	$\bar{NIt}$	3.381	3.137
	${\bar{χ}}_{R}^{2}$	1.300	1.274
	$\bar{DoF / n}$	0.445	0.455
	Ω̄₂	357.329	357.488

O₃	$\bar{NIt}$	6.018	3.527
	${\bar{χ}}_{R}^{2}$	2.382	2.308
	$\bar{DoF / n}$	0.599	0.650
	Ω̄₂	39.881	39.312

HNO₃	$\bar{NIt}$	3.377	2.962
	${\bar{χ}}_{R}^{2}$	1.344	1.310
	$\bar{DoF / n}$	0.554	0.582
	Ω̄₂	0.056	0.056

CH₄	$\bar{NIt}$	4.259	3.311
	${\bar{χ}}_{R}^{2}$	2.063	1.956
	$\bar{DoF / n}$	0.592	0.656
	Ω̄₂	12.050	11.623

N₂O	$\bar{NIt}$	3.952	3.202
	${\bar{χ}}_{R}^{2}$	1.900	1.807
	$\bar{DoF / n}$	0.539	0.582
	Ω̄₂	1.649	1.546

NO₂	$\bar{NIt}$	2.078	2.322
	${\bar{χ}}_{R}^{2}$	1.251	1.254
	$\bar{DoF / n}$	0.600	0.607
	Ω̄₂	0.265	0.282

CFC-11	$\bar{NIt}$	4.471	3.580
	${\bar{χ}}_{R}^{2}$	1.247	1.211
	$\bar{DoF / n}$	0.464	0.389
	Ω̄₂	0.002	0.001

CFC-12	$\bar{NIt}$	3.618	2.925
	${\bar{χ}}_{R}^{2}$	1.284	1.261
	$\bar{DoF / n}$	0.501	0.493
	Ω̄₂	0.003	0.003

N₂O₅	$\bar{NIt}$	3.212	2.938
	${\bar{χ}}_{R}^{2}$	1.355	1.357
	$\bar{DoF / n}$	0.464	0.463
	Ω̄₂	0.019	0.012

ClONO₂	$\bar{NIt}$	3.503	2.753
	${\bar{χ}}_{R}^{2}$	1.984	1.932
	$\bar{DoF / n}$	0.421	0.219
	Ω̄₂	0.010	0.012

Avg wrt NOM_R	$Δ \bar{NIt} (%)$		−15.119
	$Δ {\bar{χ}}_{R}^{2} (%)$		−2.957
	$Δ \bar{DoF / n} (%)$		−1.696
	ΔΩ̄₂(%)		−4.987

On the choice of retrieval variables in the inversion of remotely sensed atmospheric measurements

Abstract

1. Introduction

2. Theory

2.1. The continuum variables

2.2. The VMR variables

3. Test of the method on MIPAS measurements

3.1. Tests on a set of 12 MIPAS orbits

3.2. Tests on a whole month of orbits

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (2)

Tables (2)

Equations (18)

Optics Express