A strategy for the measurement of CO 2 distribution in the stratosphere

In this study we introduce a new strategy for the measurement of CO2 distribution in the stratosphere. The proposed experiment is based on an orbiting limb sounder that measures the atmospheric emission within both the thermal infrared (TIR) and far-infrared (FIR) regions. The idea is to exploit the contribution of the pure rotational transitions of molecular oxygen in the FIR to determine the atmospheric fields of temperature and pressure that are necessary to retrieve the distribution of CO2 from its rovibrational transitions in the TIR. The instrument envisaged to test the new strategy is a Fourier transform spectrometer with two output ports hosting a FIR detector devoted to measuring the O2 transitions and a TIR detector devoted to measure the CO2 transitions. Instrumental and observational parameters of the proposed experiment have been defined by exploiting the heritage of both previous studies and operational limb sounders. The performance of the experiment has been assessed with two-dimensional (2-D) retrievals on simulated observations along a full orbit. For this purpose, optimal spectral intervals have been defined using a validated selection algorithm. Both precision and spatial resolution of the obtained CO2 distributions have been taken into account in the results–evaluation process. We show that the O2 spectral features significantly contribute to the performance of CO2 retrievals and that the proposed experiment can determine 2-D distributions of the CO2 volume mixing ratio with precisions of the order of 1 ppmv in the 10–50 km altitude range. The error budget, estimated for the test case of an ideal instrument and neglecting the spectroscopic errors, indicates that, in the 10–50 km altitude range, the total error of the CO2 fields is set by the random component. This is also the case at higher altitudes, provided the retrieval system is able to model the non-local thermal equilibrium conditions of the atmosphere. The best performance is obtained at altitudes between 20 and 50 km, where the vertical resolution ranges from 3 to 5 km, and the horizontal resolution is of the order of 300–350 km depending on latitude.

The role of CO 2 in the radiative budget of the biosphere is well established as well as the consequences of its growth on the Earth's climate.The knowledge of CO 2 abundance and distribution is then essential to monitor the evolution of global warming and to drive the appropriate political choices.The remote sensing techniques with instruments on board orbiting platforms have been demonstrated to represent a powerful strategy for monitoring the atmospheric state.In the case of CO 2 , it can be easily detected in the infrared spectral region (both near infrared, NIR, and thermal infrared, TIR), where the spectral features generated by its rovibrational transitions are clearly visible.A number of space missions have been designed to measure CO 2 from space (Chédin et al., 2003;Crevoisier et al., 2004Crevoisier et al., , 2009;;Kulawik et al., 2010;Buchwitz et al., 2005;Kuze et al., 2009;Boesch et al., 2011); most of them aim to measure tropospheric abundances and use the nadir observation geometry that is suitable for column measurements.Limb observations are more appropriate than nadir observations for discriminating the atmospheric behaviour at different altitudes.Actually, limb-scanning spectrometers have been widely used to obtain the distribution of Published by Copernicus Publications on behalf of the European Geosciences Union.physical and chemical parameters in the stratosphere and upper troposphere and to monitor their evolution over the years.Some limb experiments, such as the Michelson Interferometer for Passive Atmospheric Sounding, MIPAS (Fischer et al., 2008) and the High Resolution Dynamics Limb Sounder, HIRDLS (Gille et al., 2008), exploit the CO 2 rovibrational transitions to determine pressure (P ) and temperature (T ) fields that, besides their intrinsic importance, are necessary in the data analysis to retrieve any target molecule from its spectral features.In these experiments, the P , T retrieval process is based on the assumption that the CO 2 distribution is known.This assumption prevents the CO 2 spectral features from being used for the retrieval of its own atmospheric distributions.To date, only the Atmospheric Chemistry Experiment (ACE - Bernath et al., 2005) has provided daytime measurements of the CO 2 altitude distribution between 5 and 25 km and in the mesosphere (Foucher et al., 2011;Emmert et al., 2012;Sioris et al., 2014).In these measurements, the accuracy of a single profile is of the order of 10 ppmv (Foucher et al., 2009).The big challenge for ACE (which performs occultation measurements in the TIR region) was to retrieve P and T (and pointing information at low altitudes) from spectral signals that are generated by different sources from the CO 2 transitions.A thorough discussion about the strategies that space experiments have used to measure CO 2 in the atmosphere can be found in Foucher et al. (2011).
In this study we propose a new strategy for the measurement of CO 2 atmospheric distribution using a passive orbiting limb sounder.The starting idea is to exploit the pure rotational transitions of molecular oxygen (O 2 ) in the far-infrared (FIR) region for the retrieval of the P and T fields, which are necessary to retrieve the CO 2 distribution from its transitions in the TIR region.This strategy is expected to uncouple the strong correlation between T and CO 2 volume mixing ratio (VMR), which is intrinsic to the analysis of the TIR spectral features of CO 2 .As the O 2 rotational transitions originate from a magnetic dipole moment, their line strengths are very low; nevertheless, due to both the huge abundance of O 2 and the long optical path of limb observation geometries, they are among the most prominent features of the atmospheric spectrum below 200 cm −1 (Carli and Carlotti, 1992).This is illustrated in Fig. 1, where the upper panel shows the simulated FIR atmospheric emission at 20 km tangent altitude (as observed by the spectrometer described in Sect.2) and the lower panel shows the FIR emission when only O 2 is included in the simulation.The compressed scale of Fig. 1 prevents the identification of O 2 lines in the upper panel.However, the comparison of the two panels shows that, below 170 cm −1 , the intensity of the O 2 lines matches the maximum emission of the atmospheric spectrum.On the basis of these considerations we suggest using an instrument capable of measuring simultaneously the atmospheric FIR O 2 lines and the TIR CO 2 lines to determine the VMR of CO 2 stratospheric fields with the target precision of 1 ppmv.For this purpose, we propose to exploit a Fourier transform (FT) spectrometer with two output ports hosting a FIR detector to measure O 2 pure rotational transitions and a TIR detector to measure CO 2 rovibrational transitions in the 700-900 cm −1 region (roughly corresponding to MIPAS band A).The preliminary question to answer in our study is about the real contribution that the O 2 transitions can provide to the precision of the retrieved CO 2 VMRs.Actually, at stratospheric temperatures, the Planck function and its sensitivity to T are much lower in the FIR region than in the TIR region where CO 2 features will have to be analysed (the steep growth of the Planck function can be appreciated in the upper panel of Fig. 1).This suggests that the precision of the T values retrieved only from FIR transitions could not be sufficient to satisfactorily determine the CO 2 distribution from its TIR transitions.Moreover, a crucial step is to define experimental and retrieval conditions that are suitable to determine CO 2 VMRs with the required precision in the stratosphere and upper troposphere (extending as low as possible the altitude range at which these precisions are achievable).
This academic study is directed to assess the intrinsic capability of the proposed approach irrespective of some technological aspects that need to be evaluated when an operational experiment is considered.
In Sect. 2 of this paper we define the instrumental and observational parameters of the proposed experiment (which we will call OXYCO2), and we discuss the selection of the optimal spectral intervals to be used for the analysis on simulated observations.Section 3 is devoted to describe the strategy adopted for the retrievals on simulated observations while the results of the retrieval tests are shown and discussed in Sect. 4. Finally, in Sect. 5 we draw the conclusions of our study.

Instrumental and observational parameters
The instrumental and observational parameters for OXYCO2 have been defined by exploiting the heritage of previous experiments, whether implemented (MIPAS and SAFIRE, Spectroscopy of the Atmosphere by Far-Infrared Emission, Bianchini et al., 2004) or designed for future missions with the support of documented studies (PIRAMHYD, Passive Infrared Atmospheric Measurements of Hydroxyl, 1998 and IRLS, Infrared Limb Sounder, of the PREMIER, Process Exploration through Measurements of Infrared and millimetrewave Emitted Radiation, experiment) (Kraft et al., 2011).The OXYCO2 experiment is based on a limb-scanning FT orbiting spectrometer with two output ports hosting a FIR detector (to measure the O 2 emission) and a TIR detector (to measure the CO 2 emission).The SAFIRE spectral resolution of 0.004 cm −1 (FT) has been adopted for OXYCO2.This relatively high value is meant to avoid, as much as possible, systematic errors due to transitions of interfering species (see Sect. 4.4) and to enhance the sensitivity of the measurements.A similar spectral resolution, which mainly helps at high altitudes, was also suggested for the FT spectrometer in the PIRAMHYD study aimed at the measurement of the hydroxyl radical (OH) in the atmosphere.
The space platform is assumed in the same orbit of ESA/ENVISAT (Environmental Satellite), which is a sunsynchronous nearly-polar orbit at an altitude of about 700 km; the orbit period is about 101 min.The spectrometer measures the limb emission looking backwards along the orbit track (similarity to MIPAS and IRLS), which enables the use of the GMTR (geo-fit multi-target retrieval model) 2-D algorithm (Carlotti et al., 2006) in the data analysis stage (see Sect. 3).For OXYCO2, we envisage the use of onedimensional (1-D) vertical-array detectors, which allow the measurement of a whole limb scan with a single stroke of the moving mirror (2-D array detectors were proposed for the IRLS spectrometer, which was designed to measure out of the orbit plane).A further advantage of array detectors is that the pointing accuracy translates into an angular bias which is common to all of the limb geometries and does not affect the altitude sampling step (i.e. the separation between adjacent tangent points).We assume the time of 15 s for each interferogram recording cycle.It follows that in a full orbit we can collect about 400 limb scans roughly separated by 100 km.The field of view (FOV) of about 3 km at tangent point, and the limb geometries are the same as MIPAS full resolution (FR) nominal mode (Fischer et al., 2008); tangent altitudes are separated by 3 km from 6 to 42 km and then 47, 52, 60, and 68 km.Finally, the noise equivalent spectral ra- diance (NESR) is assumed to be 5 nW/(cm 2 sr cm −1 ), consistent with the values proposed for IRLS and PIRAMHYD.
The values of the spectral resolution, interferogram recording time, horizontal atmospheric sampling, and NESR are strictly interconnected.Actually, the recording time, which defines the atmospheric sampling step, is proportional to the spectral resolution and is conditioned by the detector integration time, which, in turn, affects the NESR.The NESR requirement assumed for this study is consistent with the outlined spectral measurement, which is performed with a detector-noise-limited spectrometer with an optical layout similar to the one used in SAFIRE, with an optical throughput of 0.015 cm 2 sr and uses 4.2 K cooled detectors.For the estimation of the NESR, the specifications of unstressed Ge : Ga detectors have been used.These detectors have already been deployed in space missions such as the ESA Infrared Space Observatory (ISO) (Lim et al., 1998) and require low temperatures that can be reached with a cryocooler system (Rogalski, 2012).The main instrumental and observational parameters of OXYCO2 are reported in Table 1.

Selection of optimal spectral intervals
The retrieval targets need to be defined before the selection of an optimal set of spectral intervals (MicroWindows, MWs) to be analysed in the inversion process.In our case, the spectral features of O 2 and CO 2 will have to be inverted to retrieve the atmospheric distributions of P , T , and CO 2 VMR (which is the ultimate goal).The visual inspection of observations simulated with different spectral resolutions and for different tangent altitudes shows that the main interference to the CO 2 spectral features is due to the O 3 ν 2 band rovibrational transitions.The high spectral resolution adopted for OXYCO2 reduces the O 3 interference to a level that suggests O 3 VMRs may not be included in the retrieval state vector (see discussion about this in Sect.4.4).Nevertheless, the observations simulated in the FIR show that the interference of H 2 O transitions with the O 2 spectral features cannot be avoided by acting on the spectral resolution alone.The size of the systematic errors induced by the water variability in www.atmos-meas-tech.net/9/5853/2016/Atmos.Meas.Tech., 9, 5853-5867, 2016 the inversion process is such that we decided to include H 2 O VMRs in the state vector of our retrieval tests (see Sect. 4.1).For this reason, the MWs selection system was operated for the retrieval of P , T , and H 2 O VMR from FIR transitions.The MWs used for the retrievals described in this paper were selected by adapting the algorithm described in Dudhia et al. (2002) to our specific needs.For this purpose a set of error sources must be defined and quantified in order to evaluate the uncertainty associated with each spectral point.Here we have considered errors deriving from the VMR uncertainty of all of the atmospheric constituents and the error deriving from the non-local thermal equilibrium (NLTE) conditions when they are not modelled in the retrieval system.Instrumental and spectroscopic errors have been omitted in this academic study by assuming that, in the case of operational implementation, they will have to be assessed on the basis of the existing technology.The MWs selection algorithm also requires the definition of the uncertainties associated with an a priori knowledge of the state vector.The MW growing process starts from a "seed" consisting of a 2-D set of spectral points (the two dimensions being frequency and retrieval altitude) which (when inverted) carry an error budget with respect to the retrieval targets.In this process, seeds are identified by scanning the full spectral band in search of the 2-D set that carries the highest information gain.The scanning procedure is repeated in search of a new seed after each new MW has been generated.Any improvement of the errors with respect to the a priori uncertainties is quantified by the "information gain" scalar (Dudhia et al., 2002).The initial set of spectral points is left to expand in both dimensions and the information gain is calculated for the extended set (including the additional spectral point or retrieval altitude).The new set is retained in the MW if this yields an increase of information.The MW growth process continues until either the inclusion of additional elements does not yield an increase of information or the maximum MW size has been reached (we allowed for a maximum of 0.5 cm −1 which corresponds to 125 spectral points).The next MW will grow around a new seed and its information gain will be evaluated with respect to the error budget of the previously selected MW.The search for new MWs continues until the information gain obtained introducing the new MW assumes values below a fixed threshold.A consequence of this MWs selection strategy is that different combinations of MWs may be used at each altitude of the vertical retrieval grid.We remark that the MWs selection process evaluates the error budget (random and systematic) by simulating retrievals with an algorithm that operates on individual limb scans (1-D algorithm), whereas the performance of OXYCO2 will be evaluated using the 2-D GMTR algorithm (see Sects. 3, 4 and Appendix A), which operates on the whole set of limb scans of the orbit.
Within the FIR measurements, 14 MWs were generated for the retrieval of P , T , and H 2 O VMR while 15 MWs were generated in the TIR for the retrieval of the CO 2 VMR.In order to limit the number of observations to be analysed, a fur-ther selection was performed among the initial set of 29 MWs using the information load (IL) analysis (Carlotti and Magnani, 2009;Carlotti et al., 2013).The basic concepts of the IL analysis are summarised in Appendix A. Both intensity and altitude coverage of the IL (with respect to the target quantities) were the criteria used to evaluate the performance of each MW.In this way, the final set of observations used in the retrieval tests reported in Sect. 4 was reduced to 15 MWs: 6 of them in the FIR and 9 in the TIR.The frequency intervals of these MWs are reported in Table 2 together with the corresponding number of spectral points and the altitude interval where the MW is used.Figure 2 shows maps of the overall IL of the selected sets of MWs along a whole orbit.The IL values in Fig. 2 (as well as all of the tests reported in this paper) have been calculated for the April 2004 atmosphere taken from the IG2 climatological database (Remedios et al., 2007); the altitude profiles of this atmosphere will be called "reference profiles".In all of the maps in this paper (Figs. 2, 4, and 8), the ground projection of the orbit track is represented by the orbital coordinate (OC), the angular polar coordinate originating at the North Pole (OC = 0 and 360 • ) and spanning the whole orbit plane (the South Pole has OC = 180 • while the Equator has OC = 90 and 270 • at the descending and ascending nodes respectively).In Fig. 2 the upper-left panel shows the IL with respect to T of the 6 FIR MWs while the upper-right panel shows the IL with respect to T of the 9 TIR MWs.In the same figure the lower-left panel shows the IL with respect to the CO 2 VMR of the TIR MWs.In order to interpret these results, the lower-right panel of the Fig. 2 reports the T field of the atmospheric model used in our computations.The comparison of the maps of the IL with respect to T for the TIR and FIR MWs (note the factor of two between the scale expansions of the two upper panels of Fig. 2) highlights how much larger the information on T is in the TIR region (the IL with respect to the T of the combined set of 15 MWs is not shown because, due to the different magnitudes, the quadratic-summation combination law makes this map quite similar to the upper-right panel).Furthermore, by comparing the maps of the temperature IL of the TIR MWs to the T field we can appreciate the correlation between their minima.On the contrary, the temperature IL of the FIR MWs does not show clear correlations with the T field; this is a consequence of the T dependence of the Planck function (P f ).Actually, at a temperature of 250 K (average T in the stratosphere) dP f /dT is of the order of 25 nW/(cm 2 sr cm −1 K) in the FIR while the numerical value of the derivative rises to about 115 in the TIR (around 800 cm −1 ).Finally, in Fig. 2 we can notice that the IL with respect to T in the set of analysed observations is much higher than the IL with respect to the CO 2 VMR.
The MWs resulting from our selection are sufficient to demonstrate the measurement approach; however, we note that this selection should be revisited once improved spectroscopic error estimates are available and that inclusion of CO 2  lines with less temperature dependence could potentially improve the retrieval performance.
3 Layout of the retrievals on simulated observations Retrieval tests have been carried out on simulated observations using the 2-D GMTR analysis system (Carlotti et al., 2006), which was specifically developed for MIPAS measurements.The GMTR implements the geo-fit algorithm M. Carlotti et al.: CO 2 distribution in the stratosphere (Carlotti et al., 2001a) in which the atmospheric state is allowed to vary in both the vertical and horizontal (latitudinal) dimensions.With this approach, the horizontal gradients are properly modelled by the simultaneous fit of measurements from a complete orbit.A further feature of GMTR is the multi-target retrieval (MTR) capability (Dinelli et al., 2004), which allows the simultaneous retrieval of atmospheric targets.The MTR strategy avoids the propagation of systematic errors induced by the overlapping of the spectral features of different retrieval targets when they are retrieved sequentially.We remark that when the MTR is applied to our case study (see Sect. 4.1), the information on P and T is gathered from all of the spectral features and not only from those of O 2 and/or CO 2 .For the reader's convenience, Appendix A reports the basic physical and mathematical concepts behind the GMTR analysis system.
The steps of our retrievals on simulated observations are as follows.
1. Generate simulated observations with the GMTR standalone forward model.A random Gaussian noise with standard deviation equal to the instrument NESR requirement (as defined above) is added to the synthetic spectra.Reference profiles are used at this stage to provide the atmospheric model.
2. Retrieve geophysical targets from the simulated observations starting from an initial guess obtained by applying random perturbations to the reference profiles.Auxiliary data are the same as used in step 1 so that their uncertainties are not taken into account in the systematic error budget.
3. Assess the retrieval precision by comparing the retrieved values with the reference values used to generate the simulated observations.In our tests the retrieval precision includes the propagation of spectral noise and the effect of ray tracing.Actually, the 2-D ray tracing is different in the retrieval step with respect to step 1 because the perturbed values of P and T (see Sect. 4.3) cause a different behaviour of the refraction index and, therefore, slightly different paths of the lines of sight of the instrument.
4. Assess the spatial resolution of the retrieval products through the 2-D averaging kernel of the inversion process.
In simulated retrievals a crucial issue is the strategy adopted (step 2) for the generation of the initial guess of the target quantities, that is, the algorithm used for the perturbation of the reference profiles.A possible approach consists of applying a random percent perturbation (with predefined maximum amplitude) to each altitude of the reference profiles.This strategy often leads to unphysical atmospheric fields.This drawback is amplified in the 2-D approach in which the initial guess of the retrieval is a set of adjacent vertical profiles and the random perturbations at each altitude may also produce unphysical horizontal gradients of the target quantities.On the other hand, if a constant perturbation is applied to the elements of a whole profile, unphysical shapes are avoided in the vertical domain but unphysical horizontal gradients may still be present.For these reasons we have implemented the following perturbation strategy.
Once the maximum amplitude of the perturbation (B) is defined, we apply a percent perturbation (pert) to each altitude (h) of the reference profiles given by the sinusoidal function: where Z = 80 km.
For each perturbation profile along the orbit, a random value of A ≤ B and a random value of φ between 0 and 2π is assigned.
This recipe generates a sinusoid of random amplitude (A) and phase (φ), which introduces a smooth variation (with altitude) of the perturbation.Since the reference (climatological) profiles are generally smooth, our perturbation strategy leads to physical shapes of the perturbed profiles.However, depending on the random value of φ, a perturbed profile will have altitudes that are affected by the maximum perturbation (A) and others with null perturbation.In order to assess the behaviour of our simulated retrievals with respect to a balanced altitude distribution of the perturbations, we then need to perform a number of retrieval tests which are large enough to make the perturbation almost constant with altitude when averaging the results.Figure 3 shows average altitude distributions of pert(h) (generated with Eq. 1) after a set of 20 GMTR-simulated retrievals.The five black lines of Fig. 3 correspond to perturbations calculated for B = 1, 2, 3, 4, 5 (values that will be used in our tests described in Sect.4).Due to the strong variability of the tropospheric H 2 O VMR, a value of B, constant at all altitudes, is not appropriate to perturb the H 2 O VMR profiles.Therefore, for this target we have used B = 30 at the lowest altitude.This value is linearly decreased up to 15 km, at which point it becomes 10 and remains constant.The red line in Fig. 3 (measured by the upper scale) refers to the average perturbations applied to H 2 O VMR profiles.In Fig. 3 we see that the average over 20 GMTR runs generates a satisfactory approximation of the constant percent values of 0.3, 0.7, 1.0, 1.3, and 1.5, which correspond to B = 1, 2, 3, 4, 5.The results of 20 runs have been averaged in all of the retrieval tests reported in Sect.4.3.

Retrieval strategy
Preliminary tests were carried out with the "sequential retrievals" strategy previously used in the MIPAS data analysis  1).The value of B is reported on top of each line except for the red line that refers to H 2 O perturbations and is measured by the upper scale.In this case B is 30 at 0 km and linearly decreases up to 15 km where it assumes the constant value of 10.
system (Ridolfi et al., 2000).In the case of MIPAS, T and P (at tangent points) are first retrieved from the CO 2 spectral features; the VMR of atmospheric species is then derived in subsequent steps by using the previously retrieved parameters as known quantities in the forward model.In our tests P , T , and H 2 O VMR were first derived from the FIR MWs, then used as known quantities in the retrieval of the CO 2 VMR from its TIR spectral features.This strategy led to CO 2 VMR precisions that did not approach the target value.We have verified that the insufficient precision of the retrieved T fields is responsible for the bad performance of the sequential strategy.The evidence is given by the precision of the CO 2 VMRs, which becomes satisfactory when, in the CO 2 analyses, we use the retrieved values of P and H 2 O VMR but the reference values of T .
The retrieval strategy was then changed to fully exploit the potentiality of both the measurement strategy and the MTR functionality of the retrieval system: FIR and TIR observations were simultaneously analysed to retrieve P , T , and the VMRs of H 2 O and CO 2 in a single step.As mentioned in Sect.3, the advantage of MTR is that the information on P and T is gathered from all of the analysed spectral features (including O 2 , CO 2 , and H 2 O).With this approach, the T dependence on the O 2 transitions reduces the correlation between T and CO 2 VMR, which is intrinsic in the analysis of CO 2 spectral features alone.Furthermore, if we look at the FIR and TIR spectra as two independent measurements sampling the Planck function in separated frequency regions, the retrieval of T is better constrained when simultaneously analysing both regions.On the other hand, the drawback of the MTR strategy is in the additional set of retrieval parameters introduced in the state vector by H 2 O VMRs and atmo-spheric continuum parameters associated with the FIR MWs (see Sect. 4.2).

Retrieval set-up
In all of the retrieval tests performed with the abovementioned strategy, the state vector contains the parameters describing the atmospheric field of the geophysical targets, including the atmospheric continuum (below 27 km) at the frequency of each MW.The whole set of 15 MWs (see Table 1) was used in all tests reported in the next subsection.By considering that not all of the MWs are used at each of the 17 tangent altitudes, a total of 18 904 spectral points are analysed for each limb scan; this corresponds to a total of 7 580 504 spectral points simultaneously fitted in the GMTR analysis of a full orbit.Our retrievals were carried out on a vertical grid coincident with the nominal tangent altitudes of the limb scans (see Sect. 2.1) except for the 9 km altitude, which was removed in order to improve the precision in the upper troposphere; this exclusion degrades the vertical resolution at tropospheric altitudes (see Sect. 4.3).The horizontal grid was defined with altitude profiles separated by 2 latitudinal degrees; this choice implies 180 profiles in the whole orbit and corresponds to a horizontal retrieval sampling which is about twice as wide as the measurement sampling.This set-up leads to 2880 retrieval parameters for each of the four main targets.The number of continuum parameters is 13 320 (this number does not match the product between number of retrieval altitudes, number of MWs, and number of limb scans because not the all MWs are used at all altitudes).The total number of retrieval parameters is then 24 840.
The GMTR analysis system makes use of the maximum a posteriori likelihood technique (optimal estimation).Since the initial guess of the retrieval parameters was obtained by applying "strong" perturbations (see Sect. 3), the reference profiles were used as a priori estimates of the state vector.However, large uncorrelated uncertainties (diagonal elements of the a priori variance-covariance matrix, VCM) were associated with the a priori values; they are 30 % for P , a constant error of 7 K for T , 80 % for the VMR, and 100 % for atmospheric continuum parameters.A constant term was also added to the H 2 O and atmospheric continuum a priori errors to prevent strong constraints where the a priori fields assume very small values.Dedicated tests have shown that the retrieval results barely depend on the errors assigned to the a priori estimates.

Retrieval results
The IL analysis reported in Sect.2.2 shows that the analysed observations are far more sensitive to T than to the CO 2 VMR.Therefore, in order to assess the stability of the retrieval system and its ability to sort out the information between T and CO 2 VMR in unfavourable conditions, we have also considered strongly perturbed atmospheric fields www.atmos-meas-tech.net/9/5853/2016/Atmos.Meas.Tech., 9, 5853-5867, 2016 as the initial guess for our retrieval tests (even if the perturbations might appear unrealistic).The initial guess atmospheric fields were generated by applying the perturbations generated by Eq. ( 1) to the reference profiles, wherein values of B = 2, 3, 4, 5 were used for CO 2 VMR profiles while different combinations of B values were used for the perturbation of P and T profiles.The impact of the perturbation adopted for the H 2 O profiles (see Sect. 3) has been evaluated by performing retrievals keeping the perturbation of P , T , and CO 2 fixed but using different magnitudes of B for the H 2 O perturbations.The difference between the retrieved CO 2 fields was negligible; therefore the perturbation reported in Fig. 3 for H 2 O was kept constant in all of our tests.
In order to provide a typical example of the retrieval results, Fig. 4 shows the 2-D distribution of the average difference (absolute value) between retrieved and reference CO 2 VMRs in the case of P , T , and CO 2 initial guess profiles obtained with B = 2, 1, 4 respectively (this example will be called "sample test" in the following part of the paper).An overall picture representing the results shown in Fig. 4 can be given in the form of a plot (which we call the overall plot) reporting, at each altitude, the average absolute value of the differences in Fig. 4 along the whole orbit.Retrieval results in term of overall plots will be shown for the case of "weak" and "severe" perturbations of P and T profiles.The results obtained in the case of weak P , T perturbations (B = 2 for P and B = 1 for T ) are reported in Fig. 5.In the four panels of this figure, the red lines show the overall plot for the CO 2 retrievals obtained by perturbing its reference profiles with values of B equal to 2 (top left), 3 (top right), 4 (bottom left), and 5 (bottom right).Figure 6 reports the same quantities of Fig. 5 but in the case of severe P , T perturbations (B = 3 for P and B = 4 for T ).The red curves in Figs. 5 and 6 suggest that the retrieval is rather stable with respect to the size of the applied perturbations and that the CO 2 VMRs can be retrieved by the OXYCO2 experiment with precisions that are of the order of 1 ppmv between 10 and 50 km.To fully understand the importance of measuring FIR and TIR spectral regions together, we need to assess whether and to what extent the FIR observations are necessary and how they contribute to the precision obtained in our retrieval tests.In order to answer this question we have excluded the FIR MWs from the set of the analysed observations.Since the H 2 O contribution is negligible in the TIR MWs, we have also excluded the H 2 O parameters (and the atmospheric continuum of FIR MWs) from the state vector of the retrieval.In these conditions the total number of observations analysed along the full orbit is reduced to 2 663 041 while the total number of retrieval parameters is 15 660.The green lines in Figs. 5 and 6 report the overall plots of this modified retrieval set-up in the case of weak and severe P , T perturbations (H 2 O profiles are unperturbed in this case).The results obtained without the FIR measurements indicate that the contribution of FIR observations to the retrieval precision is important and increases by increasing the size of the CO 2 perturbations.The uncertainty of the average differences plotted in Figs. 5 and 6 ranges between 1.2 and 2.3 %.
Another important issue is the spatial resolution of the retrieval products of the OXYCO2 experiment.The upper panels of Fig. 7 report, for the sample test, the vertical (left) and horizontal (right) resolutions of the retrieved CO 2 VMR field.They are calculated using the 2-D averaging kernel matrix as described in Carlotti et al. (2007).Since the correlation between CO 2 VMR and T is a major issue in our study, the same quantities as in the upper panels are reported in the lower panels of Fig. 7 for the retrieved T field.It can be seen in Fig. 7 that the vertical resolution of the retrieved CO 2 field is consistent with the geometrical sampling of the atmosphere (3 km) between 20 and 50 km.The horizontal resolution mostly exceeds the geometrical sampling of the retrieval grid with values lower than 350 km in the 20-50 km altitude range (to be compared with the 220 km of the retrieval grid).The complex interdependence between the many variables of the MTR inversion makes it difficult to interpret the 2-D distributions in the maps of Fig. 7.However, by looking at the IL maps of Fig. 2 and at the maps of Fig. 7, we can identify some correlation between the distributions of information load and those of spatial resolution.Finally, we remark that, since clouds represent an unquantified source of error, the results of our tests are only valid for cloud-free conditions.

Error budget
In this subsection we provide an estimate of the systematic error components that are associated with the choices implemented in the OXYCO2 experiment.As stated in Sect.2.2, the MWs selection algorithm provides an estimate of the total error budget together with the contribution of the individual sources.This budget indicates that, among the considered error sources (see Sect.N 2 O, and HNO 3 , and the approximation introduced by the NLTE which is not modelled by the retrieval system.Since the 1-D retrieval algorithm used for the MWs selection differs in several aspects from GMTR, we have estimated the systematic errors introduced by the interfering species by repeating the sample test with the VMR profiles of O 3 , N 2 O, and HNO 3 perturbed according to their climatological variability.In the case of NLTE, the sample test was carried out on observations simulated by using the vibrational temperature profiles of the target species along the full orbit (that is irrespective of day-night conditions).We remark that, since we do not impose hydrostatic equilibrium in our retrievals, a possible constant pointing bias (due to the use of 1-D array detector; see Sect.2.1) translates into the retrieval of an ef-fective value of P at the vertical retrieval grid points.Therefore, if the resulting CO 2 VMR values are represented as a function of P , any bias in the pointing will be corrected.On the other hand, the errors on P have a negligible impact on the possibility of measuring the CO 2 VMRs (see Sect. 4.1).About the errors induced by the uncertainty of spectroscopic parameters (in particular those of CO 2 and O 2 ) we note that, if such an experiment will be considered for operational implementation, there will be dedicated measurements of them.Figure 8 reports the main components and the total systematic error associated with the full set of 15 MWs used in our retrieval tests.The red curves of Figs. 5 and 6 compared to the total systematic error of Fig. 8 show that, below 50 km, the contribution of the systematic component of  the total error is negligible with respect to the random component.However, the interference of the O 3 transitions turns out to be the most important source of systematic error below 40 km.This outcome justifies the relatively high spectral resolution adopted for OXYCO2 (see Sect. 2.1). Figure 8 also shows that above 50 km the capability of modelling NLTE is a limiting condition for the retrieval accuracy.

Conclusions
In this paper we have studied the possibility of measuring the CO 2 distributions in the stratosphere by exploiting the synergism between FIR and TIR limb-sounding measurements from space.The pure rotational transitions of O 2 in the FIR region are exploited to circumvent the strong correlation between T and CO 2 VMR, which is intrinsic in the analysis of CO 2 spectral features in the TIR region.In order to assess the performance of this concept, we have considered a hypothetical experiment (OXYCO2) based on a FT spectrometer for which we have defined instrumental and observational parameters.The proposed experiment is inspired by both previous experiments and previous studies for space limb sounders.The ability to measure CO 2 distributions was established through retrievals of OXYCO2 observations simulated at optimal spectral intervals.
We have shown that accurate T fields are necessary to retrieve CO 2 VMRs from the rovibrational spectra of this molecule with precisions of the order of 1 ppmv.For the analysis of FIR observations, the VMR of H 2 O must be included in the state vector in order to avoid the systematic errors due to the interference of its spectral features.The best performance of the simulated retrievals is obtained when FIR and TIR observations are simultaneously analysed to derive CO 2 fields together with P , T , and H 2 O fields.
A number of retrieval tests were carried out with the 2-D GMTR algorithm starting from initial guess profiles of the retrieval targets, obtained with different perturbations of the reference profiles.The outcome of these tests shows that a precision of the order of 1 ppmv can be achieved for the CO 2 VMRs in the altitude range between 10 and 50 km.These results have been obtained along a full orbit onto a vertical grid defined in steps of 3 km up to 42 km and 5 km above and onto a horizontal grid of profiles separated by about 220 km.The vertical resolution of the retrieved CO 2 fields is consistent with the retrieval grid between 20 and 50 km.The horizontal resolution depends on altitude and latitude and mostly exceeds the size of the retrieval grid; the best performance is in the 20-50 km altitude range where the horizontal resolution does not exceed 350 km.
If the FIR observations are excluded from the retrieval process, the precision of the CO 2 fields worsen by a factor which depends on altitude but may exceed 2 in the 20-40 km altitude range.This outcome demonstrates the starting hypothesis that FIR observations contribute to the CO 2 retrieval by introducing the information on T contained in the pure rotational spectrum of O 2 .
The assessment of the systematic errors considered in this study (VMR of the atmospheric constituents and NLTE conditions) shows that below 50 km their contribution to the total error is negligible with respect to the random component.However, below 40 km the interference of O 3 transitions tends to be a meaningful source of systematic errors; therefore the spectral resolution needs to be high enough to ensure the retrieval accuracy to be driven by the random component of the error.Above 50 km, the accuracy of the CO 2 fields mainly depends on the capability of the retrieval system to model NLTE conditions.
Finally, we remark that the target of this study was the retrieval of CO 2 VMR fields.However, a number of side products can be identified within the spectral ranges of the OXYCO2 experiment, first of all the hydroxyl radical (OH, of fundamental importance in atmospheric chemistry), which displays unique spectral features around 118 cm −1 (Carlotti et al., 2001b).For the analysis of these features, FIR spectroscopy was identified as the best measurement strategy (PI-RAMHYD, 1998).

Data availability
Data used in this study are available from the authors upon request.
Figure 1.(a) Far-infrared atmospheric spectrum simulated at 20 km tangent altitude, (b) same as upper panel but only O 2 is included in the simulation.

Figure 2 .
Figure 2. (a) IL with respect to T of the FIR MWs.(b): IL with respect to T of the TIR MWs.(c) IL with respect to CO 2 VMR of the TIR MWs, (d) T field of the reference atmosphere.

Figure 3 .
Figure 3. Average % values (over 20 GMTR runs) of pert (h) corresponding to different values of B in Eq. (1).The value of B is reported on top of each line except for the red line that refers to H 2 O perturbations and is measured by the upper scale.In this case B is 30 at 0 km and linearly decreases up to 15 km where it assumes the constant value of 10.

Figure 4 .
Figure 4. 2-D distribution of the averaged absolute difference between reference and retrieved CO 2 VMRs resulting from the sample test (see text).

Figure 5 .Figure 6 .
Figure 5. Red lines: overall plots for CO 2 retrievals in the case of B =2 for P -and B =1 for T -perturbations in Eq. (1); each panel contains the B value used for the CO 2 VMR perturbations.Green lines: same as red lines but excluding FIR MWs and H 2 O parameters from the retrieval set-up.

Figure 7 .
Figure 7. Maps of the vertical (a and c) and horizontal (b and d) resolutions as a function of altitude and orbital coordinate obtained with the averaging kernel matrix of the sample test retrieval.Panels (a-b) refer to the retrieved CO 2 field and panels (c-d) refer to the retrieved T field.

Figure 8 .
Figure 8. Error components and total systematic error associated with the set of 15 MWs that were used in retrieval tests.Only the three atmospheric gases that represent the main sources of error are reported.NLTE indicates the error propagated by neglecting nonlocal thermal equilibrium conditions in the retrieval system.

Table 1 .
Instrumental and observational parameters of OXYCO2.

Table 2 .
Definition of the MWs selected for the retrieval tests.
FIR MW number Frequency interval (cm −1 ) No. of points Altitude coverage (