Synergetic use of IASI and TROPOMI space borne sensors for generating a tropospheric methane profile product

The thermal infrared nadir spectra of IASI (Infrared Atmospheric Sounding Interferometer) are successfully used for retrievals of different atmospheric trace gas profiles. However, these retrievals offer generally reduced information about the lowermost tropospheric layer due to the lack of thermal contrast close to the surface. Spectra of scattered solar radiation observed in the near and/or short wave infrared, for instance by TROPOMI (TROPOspheric Monitoring Instrument) offer higher sensitivity near ground and are used for the retrieval of total column averaged mixing ratios of a variety of atmospheric 5 trace gases. Here we present a method for the synergetic use of IASI profile and TROPOMI total column data. Our method uses the output of the individual retrievals and consists of linear algebra a posteriori calculations (i.e. calculation after the individual retrievals). We show that this approach is largely equivalent to applying the spectra of the different sensors together in a single retrieval procedure, but with the substantial advantage of being applicable to data generated with different individual retrieval processors, of being very time efficient, and of directly benefiting from the high quality and most recent improvements of the 10 individual retrieval processors. 1 https://doi.org/10.5194/amt-2021-31 Preprint. Discussion started: 23 February 2021 c © Author(s) 2021. CC BY 4.0 License.


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Measurements from different ground-or satellite-based sensors target at the observations of the same atmospheric parameters (e.g. the same trace gases), but with different characteristics (e.g. sensitivities for different vertical regions). Often the different sensors use different observation geometries (limb scanning, nadir, solar light reflected at the Earth's surface) and/or different spectral regions (e.g. UV/vis, near infrared, thermal infrared, microwave). Dedicated experts and efforts are needed to develop retrieval techniques that are specifically optimized for an individual sensor. An algorithm that uses coincident measurements 30 of all the different sensors for a multispectral approach for the optimal estimation of the atmospheric state would well exploit the synergies of the different observation geometries and spectral regions and thus allows for detecting the atmospheric state in more detail than achievable by individual optimal estimation retrievals. Cuesta et al. (2013) present such 'super retrieval', which performs an optimal estimation of atmospheric ozone (O 3 ) applying the spectra measured by the thermal nadir sensor IASI (Infrared Atmospheric Sounding Interferometer) and the UV/vis sensor 35 GOME (Global Ozone Monitoring Experiment). Their publication shows that using the multispectral approach allows the detection of lower tropospheric O 3 , which is not possible by an individual usage of the IASI and GOME spectra. Costantino et al. (2017) showed that the quality of this multispectral lower tropospheric O 3 product can be further improved with improved thermal nadir and UV/vis sensors.
The development of these 'super retrievals' requires experts in different remote sensing techniques to work closely together. 40 Furthermore, as soon as measurements from a new sensor become available (or as soon as sensors are modified/improved) such super retrieval processors have to be adapted accordingly, i.e. continuous collaborative retrieval developments are required.
While this is certainly possible, it might be not the most efficient way. The optimal exploitation of the already available individual retrieval results would be much less computationally expensive than running dedicated combined retrievals.
transferring logarithmic scale differentials into linear scale differentials. Appendix C presents the operators used for converting vertical profile data into total and partial column data. In this section we present the method for combining CH 4 profiles derived from IASI thermal nadir spectra and XCH 4 data obtained from the analysis of the near and short wave infrared spectra measured by TROPOMI.
The TROPOMI XCH 4 data used in this study are generated by the RemoTeC algorithm (Butz et al., 2011), which is used for the operational processing of Sentinel 5 Precursor/TROPOMI XCH 4 data (Hu et al., 2016;Hasekamp et al., 2019;Landgraf 85 et al., 2019). The current operational processing algorithm version is 1.2.0. Here we use data from version 1.3.0 with the improvements as presented and validated in Lorente et al. (2020). It is foreseen to become the operational processing version with the operational processor update in April 2021. The TROPOMI output files provide the XCH 4 data together with the used a priori data (constructed from simulations of the global chemistry-transport model TM5, Krol et al., 2005), the column averaging kernels, and the error values. In order to filter out data with reduced quality, here we only use TROPOMI data, for 90 which the variable qa_value has values larger than 0.5. This filter is consistent to the filtering as suggested in Table A1 of Lorente et al. (2020).
As IASI CH 4 data product we use the data generated by the retrieval processor MUSICA (MUlti-platform remote Sensing of Isotopologues for investigating the Cycle of Atmospheric water, a European Research Council project between 2011 and 2016).
The MUSICA IASI data full retrieval product encompasses trace gas profiles of H 2 O, the HDO/H 2 O ratio, N 2 O, CH 4 , and 95 HNO 3 . The data have been validated in several previous studies (Schneider et al., 2016;Borger et al., 2018;García et al., 2018), and it has been shown that the CH 4 product can very well detect the CH 4 signals originating in the upper troposphere/lower stratosphere. MUSICA IASI data using processor versions 3.2.1 and 3.3.0 are currently available for the 2014 to 2020 time period and are presented in Schneider et al. (2021). This MUSICA IASI data set is best suited for a posteriori data reusage (e.g. Diekmann et al., 2021), because in addition to the retrieved trace gas profiles it contains full information on retrieval settings (a 100 priori states and constraints) and on averaging kernel and error covariance matrices. In oder to ensure highest MUSICA IASI data quality, here we require the flag variable musica_fit_quality_flag to be set to 3 (the spectral fit of the MUSICA IASI retrieval has a good quality and the spectral rediduals are close to the instrumental noise level). Furthermore, we only use MUSICA IASI data for which the flag variable eumetsat_cloud_summary_flag is set to 1, which guarantees that the IASI instrumental field of view is cloud-free. 105 A particularity of the MUSICA IASI processor is that the trace gas inversions are performed on a logarithmic scale. In Appendix B of Schneider et al. (2021) it is shown that the MUSICA IASI retrieval can be considered as a moderately nonlinear problem, in particular if the differentials (averaging kernels and covariances) are used on the logarithmic scale. In the following equations we take special care on the correct usage of the corresponding logarithmic scale differentials. Nevertheless, all equations are also applicable for retrievals made on linear scale by replacing in the following the operator L by the identity

Calculation of the combined state vector
For this study we use the CH 4 a priori profile as provided by the TROPOMI product as the common a priori for all products (these are simulations of the global chemistry-transport model TM5, Krol et al., 2005). For this purpose we modify the MUSICA IASI product and bring it in line with the TROPOMI a priori profile choice by applying Eq. (A16). 115 For updating the IASI CH 4 profile product using the TROPOMI XCH 4 observation we apply a Kalman filter and obtain the combined CH 4 state as : Here the vectorx I and scalarx T are the MUSICA IASI CH 4 profile and the TROPOMI XCH 4 column averaged products.
The row vector a * T T is the total column averaged mixing ratio kernel of the TROPOMI product interpolated to the vertical 120 grid used by the MUSICA IASI processor (for details on the interpolation see Appendix C). The state vectorx l C represents the combined CH 4 profile product in logarithmic scale (i.e. the MUSICA IASI CH 4 data updated with the TROPOMI XCH 4 observation). The superscript ' l ' used withx l C andx l I indicates the use of the logarithmic scale. Here and in the following we will mark scalars, vectors or matrix operators that are in logarithmic scale by the superscript ' l '. The matrix L is the operator for the transformation of differentials or small changes (as given by averaging kernels or error covariances) from the logarithmic 125 to the linear scale (for more details see Appendix B).
The column vector m is the Kalman gain operator and it is given by: with the matrix S lx I and the scalar Sx T ,n being the logarithmic scale retrieval noise error covariance of the MUSICA IASI CH 4 product and the noise error variance of the TROPOMI XCH 4 product, respectively. The vector operator a * T is the transpose of 130 the TROPOMI column averaging kernel, i.e. a * T = (a * T T ) T .
Except for the logarithmic scale transformation, the Eqs. (1) and (2) are analogous to Eqs. (A9) and (A10). As demonstrated in Appendix A2 this kind of Kalman filter application is equivalent to an optimal estimation retrieval that uses a combined IASI and TROPOMI measurement vector. The application of this Kalman filter is possible because the MUSICA IASI data are provided with full information on a priori states, constraints, error covariances, and averaging kernels (Schneider et al.,135 2021), and because the TROPOMI data are provided together with their a priori state, averaging kernel, and retrieval noise error (Lorente et al., 2020).

Collocation of TROPOMI and IASI observations
As temporal collocation criterion we use four hours, for a valid horizontal collocation the centres of the TROPOMI and IASI ground pixels must be closer than 50 km, and the difference between the ground pressure at the TROPOMI and IASI ground 145 pixels must be within 50 hPa. Generally multiple TROPOMI/IASI ground pixel pairs fulfill the aforementioned criteria. In such case we use the pair with the smallest spatial distance, which we define as the Euclidean distance that considers a norm of 40 km for the horizontal distance and a norm of 5 hPa for the vertical distance. TROPOMI and IASI observations already belonging to a valid collocation pair are disregarded for further collocations. This ensures that an individual IASI or TROPOMI observation can only belong to a single collocation pair. The possible small difference in the TROPOMI and IASI ground pixel 150 pressures is taken into account by correcting the TROPOMI XCH 4 values, respectively.

Sensitivity and vertical resolution
In this section we compare the vertical representativeness of the individual retrieval products with those achieved when combining the two retrieval products. According to Eq. 1 the averaging kernels for the combined data product can be calculated as: Here A l I and A l C are the logarithmic scale averaging kernels of the MUSICA IASI CH 4 product and of the combined product (the MUSICA IASI CH 4 product after being updated with the information provided by the TROPOMI XCH 4 data product), respectively. These are the kernels for the profile products represented in nal (nal: number of atmospheric levels) levels, i.e. they are matrices of dimension nal × nal. Logarithmic scale kernels are also called fractional or relative averaging kernels 160 (e.g. Keppens et al., 2015). Figure 2 depicts the rows of typical averaging kernels for the MUSICA IASI product (panel a) and the combined data product (panel b). Adding the information provided by TROPOMI clearly improves the sensitivity in the lower troposphere: for the MUSICA IASI product the lower tropospheric kernels generally peak at the upper limit of the lower troposphere (at about 5 km a.s.l.). For the combined product these peak values are obtained at significantly lower altitudes (at about 2.5 km a.s.l.). In the 165 upper troposphere/lower stratosphere (UTLS) we see no significant difference between the kernels.
In this work we focus on the total column and the partial columns between the surface and 6 km a.s.l. (the tropospheric partial column) and between 6 km a.s.l. and 20 km a.s.l. (the UTLS partial column). The total and partial column kernels are calculated from A l I and A l C by their transformation on linear scale (see Appendix B) and the vertical resampling as explained in Appendix C. Figure 3 depicts the total and partial column kernels corresponding to the row kernels of Fig. 2.

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Total column amount kernels are available for all three products (see Fig. 3a): the TROPOMI, the MUSICA IASI, and the combined product. The TROPOMI kernel is close to unity for all altitudes, documenting the good sensitivity for CH 4 at all altitudes. The combined total column amount kernel is very similar to the respective TROPOMI kernel (even correcting the overshoot at 4-6 km) and means that the combined retrieval product does also well reflect the actual atmospheric total column amounts. The MUSICA IASI kernel has relatively low values in the lower troposphere and above 15 km, only in the UTLS 175 region the kernel values are between 0.75 and 1.25. This means that MUSICA IASI can actually not well detect total column amounts, because it lacks sensitivity in the lower troposphere. The altitude regions where the MUSICA IASI product has reduced sensitivities are the regions where TROPOMI's total column information has the strongest impact on the combined product (see Fig. 1).  Partial column amount kernels are only available for profile products, i.e. the MUSICA IASI and the combined product 180 (MUSICA IASI updated with information from TROPOMI). Figure 3b shows tropospheric partial column amount kernels. For the MUSICA IASI product we observe values that are generally lower than 0.5. The highest values are achieved around 6 km a.s.l., i.e. at the upper boundary of the vertical layer we defined as the tropospheric partial column. The kernel of the combined product shows a good sensitivity with peak values of almost 0.95 at 2.5 km a.s.l. and values above 0.75 for almost all altitudes between the surface and 6 km a.s.l. UTLS partial column amount kernels are depicted in Figure 3c. The values are close to unity for most of the altitudes that we attributed to the UTLS layer (altitudes between 6 km and 20 km a.s.l.). There is almost no difference between the MUSICA IASI and the combined kernels, meaning that the information provided by TROPOMI has almost no effect on the UTLS partial column, which is because the MUSICA IASI product is already very sensitive to this altitude region.
According to Eqs. (A1) and (A3) for the MUSICA IASI and combined retrieval data we can write with I being the identity operator and x l the actual atmospheric state in logarithmnic scale. Equation (4) reveals that the term (I − A l )x l a captures the relative contribution of the a priori to the retrieved product. The resampling of this term on total and partial columns is made according to Eq. (C6). For the TROPOMI total column averaged mixing ratios we can calculate the apriori contribution by (w * T − a * T T )x a . For more details see Appendix C.
195 Figure 4 depicts the a priori contribution relative to the retrieved values for the total column, the tropospheric and UTLS partial columns. Shown are time series for measurements over Central Europe, which confirm the observations made in the context of the example kernels of Fig. 3: for the total column (Fig. 4a) the a priori contribution on the TROPOMI and the combined products are rather small and can be neglected, i.e. both products can detect total column signals. In contrast the MUSICA IASI total column product is significantly affected by the a priori data, i.e. provides no independent observation of 200 the total column. Concerning partial column products ( Fig. 4b and c) we can compare the MUSICA IASI and the combined product (the TROPOMI product has no information on the vertical distribution). The tropospheric MUSICA IASI partial column is significantly affected by the a priori, but the combined product is largely independent on the a priori data. In the UTLS both the MUSICA IASI and combined products are largely independent on the apriori data. In summary, with IASI alone we can well detect signals in the UTLS, but not in the lower troposphere. The detection of signals in both altitude regions 205 independently from the a priori information is only possible by using the combined product.

Retrieval noise error
In this section we compare the retrieval noise errors of the individual retrieval products with those achieved when combining the two retrieval products. According to Eq. (1) we can calculate the retrieval noise covariance matrix for the combined data product by Here S l x I ,n is the retrieval noise covariance matrix of the MUSICA IASI retrieval. The error covariances resampled to the total and partial columns are then determined according to Appendix C. Figure 5 shows the retrieval noise errors (which are the square root values of the error variances) relative to the retrieved values for the total column and the tropospheric and UTLS partial columns.

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The errors for the total columns ( Fig. 5a) are generally below 0.2% for the TROPOMI product. For the MUSICA IASI product they are rather stable at about 0.6%. Concerning the combined product the retrieval noise error is very similar to the retrieval noise error of the TROPOMI data. For the tropospheric partial columns (Fig. 5b) the error is in general above 1% for the MUSICA IASI product and below 1% for the combined product. For the UTLS partial columns (Fig. 5c) we observe an error of generally below 1% and no 220 significant difference between the MUSICA IASI and the combined data products. This suggests that the error in the combined product is dominated by the error in the MUSICA IASI data, which reveals the very limited impact of the TROPOMI data on the combined UTLS data product.

Validation
In this section we compare the TROPOMI, MUSICA IASI, and combined products to different reference data products. As 225 reference for the total column averaged mixing ratio (XCH 4 ) we use TCCON (Total Carbon Column Observing Network, Wunch et al., 2011) ground-based remote sensing data from six sites located in different climate zones. As reference for the total and the partial columns we use in-situ profiles measured by the AirCore system (Karion et al., 2010) at two geophysically different European locations. Furthermore, we use in-situ data measured at two nearby Central European Global Atmospheric Watch (GAW) mountain stations. 230 Figure 6 depicts the geographical location of the European reference observations. We consider the European TCCON stations at Sodankylä (Finland) and Karlsruhe (Germany). They are indicated as red crosses together with a circle around the stations with a radius of 150 km indicating the spatial collocation criteria: only satellite observations with ground pixels inside this circle are compared to the TCCON data. Blue crosses and circles represent the locations of AirCore measurements (at Trainou, France, and Sodankylä, Finland) and their spatial collocation criteria, respectively. Here we relax the radius of 235 the collocation circle to 500 km in order to achieve a sufficient high number of coincidences between AirCore and satellite observations. The two grey dots indicate the locations of the two GAW stations (Jungfraujoch in Switzerland and Schauinsland in South-western Germany) and the respective grey spatial collocations circle around the middle distance point of the two stations has a radius of 150 km.  We use TCCON ground-based remote sensing data from six exemplary sites located in different climate zones representative for high, mid and low latitudes. The Sodankylä site is located at high latitudes, Karlsruhe and Lauder are located in the northern and southern hemispheric mid-latitudes, Wollongong is located in the subtropics, and Burgos and Darwin are located in the tropics. More details on the locations of these sites and references and amount of the used data sets are given in Table 1. sensing products. For this purpose the TCCON product is adjusted to the TROPOMI a priori settings according to Eq. (A16), which ensures the usage of the same a priori data for all the remote sensing products. As spatial collocation criteria we require that the ground pixels of the TROPOMI and the IASI measurement fall within a circle with a radius of 150 km around the TCCON sites. For collocation with respect to time, TCCON, TROPOMI, and IASI observations have to be made within at least 6 hours. Furthermore, we require that the altitude differences between the TCCON stations and the satellite ground pixels 250 are within 250 m.
We estimate the reliability of the TCCON data as reference for the satellite observations. For this estimation we consider the TCCON retrieval noise errors, the incomparableness of TCCON and satellite data caused by their different averaging kernels, and the collocation mismatch. The total column uncertainty variance (the scalar S ref ) for using the TCCON data as reference for the satellite data can be estimated by: The first term (the scalar S ∆TC ) is the TCCON retrieval error covariance (the TCCON error is provided with the TCCON data is typically 1‰). The second term accounts for the different averaging kernels. The row vectors a * T and a * TC T are the total column averaged mixing ratio kernels of the satellite and the TCCON retrievals, respectively (calculated according to Appendix C). The matrix S ∆a describes the uncertainty covariances of the used a priori data. We determine these uncertainty 260 covariances from the TM5 CH 4 simulations (Krol et al., 2005), which are provided in the TROPOMI data set as the a priori data. For this purpose we assume a hypothetical out-of-phase of the model of 24 hours and in addition a horizontal mismatch of the modeled CH 4 fields of 500 km. The covariances obtained for the differences between the original TM5 model fields and the TM5 fields with the hypothetical model deficits are then used as the uncertainty covariances. We found an a priori uncertainty covariance S ∆a having largest values close to the surface but even there, the uncertainty variance is smaller than (4‰) 2 . Due 265 to this good a priori knowledge the effect of the different averaging kernels on the comparison is less than 0.5‰ (even for the comprison between the TCCON and the MUSICA IASI products, where the difference in the averaging kernels is most significant). The third term takes into account that TCCON and the satellites might detect different air masses. The respective uncertainty covariances are again estimated with the TM5 CH 4 simulations. We determine the covariances between out-ofphase model fields and the correct model fields for different out-of-phase time intervals. Similarly we calculate the covariances 270 between model fields that have a horizontal mismatch and the correct model fields for different horizontal mismatch intervals.
The temporal collocation uncertainty covariance (S ∆t ) and the horizontal collocation uncertainty covariance (S ∆h ) are then the covariances interpolated to the actual temporal and horizontal mismatch of the satellite and the TCCON measurements. The effect of this collocation mismatch on the comparison of the total columns is estimated to be smaller than 0.5‰. In summary, we estimate the reliability of the TCCON data as reference for the satellite total column observations to be within 2‰.

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In Fig  series of the differences with respect to the TCCON references. The daily mean data have error bars, which is the 1σ standard deviation of the data used for calculating the daily mean.

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Statistics in form of mean difference and 1σ standard devation (scatter) around the mean difference are given in each panel (for statistics using daily mean data in black fonts and for statistics using all individual valid collocations in grey fonts).
Concerning TROPOMI (Fig. 7a)  the observed good agreement with the TCCON XCH 4 data can be partly explained by the reliable a priori data.  For the MUSICA IASI product, we observe a similar good correlation than for the TROPOMI and the combined products.
However, concerning the MUSICA IASI data part of the common signal might be due to the a priori on which the MUSICA IASI total column product is not independent (see Fig. 4a).
The different satellite XCH 4 data products show a very good agreement with the TCCON data (within 0.9%-1.1% concerning 295 the comparison of single satellite pixel data with individual reference data). Figure 8 shows the correlations between the theoretically predicted scatter -considering the reference uncertainty according to Eq. (6) and the satellite products' noise error (see Fig. 5) -and the actually observed scatter. For the comparison of the MUSICA IASI product with TCCON (red square) we observe a larger scatter than for the comparison of the TROPOMI and combined products with TCCON (black and blue squares). The relatively larger scatter for the MUSICA IASI comparison is also theoretically predicted and mainly due to 300 the increased noise error of this product (see Fig. 5a). Nevertheless, for all satellite products there is a systematic difference between the theoretically predicted and actually observed scatter values. Because it is consistently seen in all satellite products we assume that it is due to an underestimation of the reference uncertainty calculated according to Eq. (6).
We observe small systematic negative and positive differences for the TROPOMI versus TCCON and the MUSICA IASI versus TCCON comparisons, respectively. Although these systematic differences are not significant (within the observed standard 305 deviation), they might indicate to a bias between the TROPOMI and MUSICA IASI XCH 4 products of about 1%.

Air-Core in-situ CH 4 profiles
We use the AirCore balloon borne in-situ measurements (Karion et al., 2010) as the reference for CH 4 total columns as well as for the CH 4 vertical distribution. The AirCore system samples the vertical distribution of CH 4 with a much better vertical resolution than the satellite remote sensing systems. For this reason we can generate an AirCore profile (x AC ) that has the same 310 vertical sensitivity and resolution characteristics as the remote sensing data. According to Eqs. (A1) and (A3) for the MUSICA IASI and the combined retrieval data we can write: Here A l and x l a are the logarithmic scale averaging kernels and the logarithmic scale a priori state of the satellite retrieval, respectively, x l AC is the measured logarithmic scale AirCore profile regridded to the atmospheric model grid used for the 315 satellite retrievals. The resampling of these data on total and partial columns is made with the linear scale data according to Eq. (C6). For the TROPOMI total column averaged mixing ratios we calcuated the adjusted AirCore total column averaged . For more details see Appendix C.
As spatial collocation criteria we require that the ground pixels of the TROPOMI and the IASI measurement fall within a circle with a radius of 500 km around the mean horizontal location of the AirCore system when sampling between the 450 and 320 550 hPa pressure levels. The temporal collocation is 6 hours. AirCore data are typically not available close to the ground and above the burst altitude of the balloon (approximately 25 hPa). At low altitudes we extend the profile with the concentrations closest to the ground. At high altitudes we extend the profile with the TM5 model data, with a smooth transition between the measured values and the modelled data.
Similar to the TCCON data we estimate the reliability of the AirCore profile data as reference for the satellite observations. 325 For this estimation we consider an AirCore measurement noise covariance (S ∆AC,n ). It is calculated assuming an uncertainty for altitudes with AirCore CH 4 data of 0.3% (Karion et al., 2010) and the uncertainty according to S ∆a from Sect.3.1 for all other altitudes. The outer diagonal elements are determined by assuming the same vertical correlation as derived for S ∆a .
In addition, we consider uncertainties in the height attribution, which is according to Wagenhäuser et al. (2021) below 10 m close to ground, about 200 m at 20 km a.s.l. and about 1 km at 27 km a.s.l. For some AirCore soundings there was a problem 330 with the electronic board. For those measurements information on pressure, altitude and temperature had to be reconstructed from the radiosonde data and we use for all altitude levels an additional height attribution uncertainty value of 500 m. We construct a respective height attribution uncertainty covariance (S ∆AC,v ) by assuming a very strong correlation of the height attribution uncertainties between different altitude levels. The temporal and spatial collocation uncertainty covariance between the AirCore and the satellite observations (S ∆t and S ∆h , respectively) are calculated as described in Sect. 3.1.

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All the aforementioned uncertainties are independent and we can calculate the total uncertainty as: The reliability of the AirCore data -after its adjustment according to Eq. (7) -as reference for the MUSICA IASI and combined satellite data can then be estimated by:

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Here and in Eq. (8) the covariances are determined for the full vertical profile. Respective covariances for total or partial columns can be derived according to Appendix C. The reliability for the TROPOMI total column averaged mixing ratio data can be calculated by S ref = a * T T S ∆AC a * T . Table 2 gives an overview on the AirCore profiles measured at Trainou (France, 48.0 • N, 2.1 • E) and Sodankylä (Finland, 67.4 • N, 26.6 • E). In total we have 24 individual AirCore profiles with collocated satellite observations. The total number of 345 collocated single pixel satellite observations is 4993. We estimate that the AirCore data can serve as reliable references for the satellite data validation. The three columns on the right report the uncertainties determined according to Eq. (8). For the reliability -according to Eq. (9) -we get very similar values (exept for the total column and the partial tropospheric column of the MUSICA IASI product, because of the limited sensitivity). It is 3-6‰, 3-5‰, and 3-7‰, for the total column, the tropospheric partial column and the UTLS partial column, respectively. In the troposphere the reliability depends mainly on 350 the availability of AirCore data close to the ground and in the UTLS uncertainties of the altitude attribution have a dominating influence.
The comparison between the TROPOMI and the AirCore XCH 4 data is shown in Fig. 9. The differences of collocated measurements are shown in Fig. 9a. The agreement between TROPOMI and AirCore is very good and the mean difference and the 1σ sigma standard deviation (scatter) around the mean difference is similar to the comparison between TROPOMI and 355 TCCON. Considering the mean values for all coincidences corresponding to the same AirCore flight, we observe a coefficient of determinarion (R 2 ) of about 30%. This is lower than the R 2 value achieved for the correlation with TCCON data; however, we have to consider that the amplitude in the analysed total column signals is much smaller in the AirCore data set if compared to the TCCON data set. Figure 10 presents the comparison between the MUSICA IASI and AirCore total column and tropospheric and UTLS partial 360 column data. The differences between both data sets are depicted in Fig. 10a-c. We find a very good agreement for the UTLS partial column data (mean difference of about 0.6% and a scatter of about 1%). Because at this altitude region the MUSICA IASI product is almost independent from the a priori assumption (see Sect. 2.3), the a priori effect onx AC from Eq. 7 can also be neglected and we compare here two independent data sets. For the total column we also see a good agreement (Fig. 10a).
However according to Sect. 2.3 the MUSICA IASI total column products are significantly affected by the a priori data and so 365 isx AC from Eq. 7, i.e. here we actually do not compare two independent data sets and a significant part of the good agreement might be due to the common a priori effect. For the tropospheric partial columns (Fig. 10b) the agreement worsens a bit. We get a mean difference of about 2.4% and a scatter around the mean differences of about 1.3% (for data averaged per flight). The increased mean difference might indicate a systematic bias in the MUSICA IASI lower tropospheric partial columns, which might also explain the increased scatter: the bias will depend on the actual sensitivity of the MUSICA IASI product, which in Table 2. List with information about the AirCore flights. [Pmin,Pmax] is the pressure range covered by the AirCore measurements. N is the number of collocated satellite observations (one collocation of IASI and TROPOMI counts as one). Pmax = P Sat. GND − Pmax is the mean difference between AirCore maximum pressure value and the pressure values for the collocated satellite ground pixels. ∆h is the mean horizontal distance between the AirCore system (location for AirCore system at 450-550 hPa) and the locations of the satellite ground pixels. ∆t = t Sat. − t is the mean time difference between the AirCore observations (time for AirCore system at 450-550 hPa) and the satellite observations. ∆ACtot, ∆ACtro, and ∆AC utls are the square roots of the variances (determined according to Eq. (8) and the column calculations according to Appendix C). These are the estimated uncertainties for using the adjusted AirCore data as reference for the satellite data: for the total column (index: 'tot') and the tropospheric and UTLS partial columns (indices 'tro' and 'utls', respectively Table 2. turn varies with the conditions present during the observation (mainly the surface temperature and the vertical temperature and humidity profiles).
Figure 10d-f shows respective correlation plots. We get very high R 2 values for the UTLS partial column, where the two data sets are largly independent (almost not affected by the a priori data). This demonstrates that the MUSICA IASI product reliably captures the actual atmospheric CH 4 signals in the UTLS. Concerning the total column and the tropospheric partial  UTLS partial coloumns. This is due to a low amplitude of the respective signals (total column) and due to varying MUSICA IASI sensitivities, which causes a varying impact of a possible systematic bias (tropospheric partial column).
All combined products (total column and tropospheric and UTLS partial columns) are practically independent from the a priori assumptions (see Fig. 4). Figure 11a-c illustrates the differences between AirCore data and the combined products. For 380 the total column we achieve values for the scatter that are similar to the comparison of the respective TROPOMI product.
However, we observe a mean difference that is outside the 1σ scatter and also outside the uncertainty estimated for the AirCore references (see Table 2), which might indicate to a positive bias in the total columns of the combined data product. For the tropospheric partial column we observe a low scatter, but also mean difference of about 1.4% that is slightly outside the 1σ scatter of about 1.2% (for data averaged per flight). For the UTLS partial column the mean difference and scatter values are 385 similar to the comparison of the respective MUSICA IASI product.
The correlation plots (Fig. 11d-f) allow similar conclusions: the combined product can capture total column signals as reliable as the TROPOMI product (apart from a possible weak bias) and UTLS partial columns signals as reliable as the MUSICA IASI product. Concerning the tropospheric partial column we observe higher R 2 values than for the respective correlation with MUSICA IASI data; however, only when correlating the mean values for all coincidences corresponding to 390 Figure 11. Same as Fig. 10, but for comparisons with the combined data products.
the same AirCore flight. When correlating all individual coincidences the R 2 values are even lower than the already low R 2 values achieved for the respective correlation with MUSICA IASI data (compare Fig. 10e with Fig. 11e). The low values for R 2 are explained by the low CH 4 variability encountered during the 24 individual AirCore profiles. for TROPOMI and combined data products than for MUSICA IASI data product. Secondly, for XCH 4 as well as for the tropospheric partial column the observed scatter is significantly larger than the theoretically predicted scatter. For the UTLS 400 partial column we find a very good agreement in the absolut values of the theoretically predicted and actually obsereved scatter.
Because the observed scatter is consistently higher than the theoretically predicted scatter independent from the kind of XCH 4 or tropospheric partial column satellite data product and also independent from the reference data (AirCore or TCCON), it is very reasonable to assume that close to ground we underestimate the temporal collocation uncertainty covariance (S ∆t ) and the Similar to the TCCON comparisons, the AirCore study suggests a systematic bias between the TROPOMI and MUSICA IASI XCH 4 products of slightly above 1%, which seems to be mainly due to a positive bias of about 2% in the MUSICA IASI tropospheric data product. In the UTLS we observe no significant bias in the MUSICA IASI data. cycles) at both sites were below a certain threshold. Sepúlveda et al. (2014) showed that the common signals are well representative for a broader layer in the lower free troposphere. Here we follow this approach and use the mean of the Jungfraujoch and Schauinsland CH 4 mixing ratio -whenever identified as a common signal -as a validation reference for the remote sensing data in South-western Germany and Northern Switzerland (indicated by the grey circle in Fig. 6). We assume that the signals obtained from this GAW data filtering are well representative for the tropospheric partial column averaged mixing ratios (sur-420 face -6 km a.s.l.) and compare these data directly to different satellite products as a fully independent data set: we do not adjust the data to a common a priori data usage as in Sects. 3.1 and Sect. 3.2, because the in-situ data represent absolute measurements and do not depend on any a priori information. Furthermore, we do not adjust sensitivities as in Sect. 3.2 (see Eq. (7)), which means that we validate here also the sensitivities of the products. Figure 12 shows the comparison with the different satellite products. Concerning the comparison with TROPOMI XCH 4 425 data we observe a very large systematic difference and very low values for R 2 (Fig. 12a and d). This indicates that the total column (XCH 4 ) signals are not a good proxy for lower tropospheric CH 4 signals, instead the former are strongly affected by signals in the UTLS, where CH 4 values are strongly affected by shifts of the tropopause height.

GAW surface in-situ CH
For the MUSICA IASI tropospheric partial column averaged mixing ratio product ( Fig. 12b and e) we observe a smaller mean difference than for the TROPOMI XCH 4 comparison, but at the same time an increased 1σ standard deviation (scatter) 430 around the mean. The R 2 values are larger than for the correlation of TROPOMI data; however, we have to be careful, because the lower tropospheric MUSICA IASI CH 4 data are significantly affected by the a priori assumptions (see Fig. 4b). This means that the observed correlation might actually be due to a correlation with the a priori data. Furthermore, the slope of the linear regression line is significantly larger than unity.
The combined tropospheric partial column averaged mixing ratio product is practically independent from the a priori as-

435
sumptions (see Fig. 4b). The good agreement and correlation between the GAW data and the combined products as illustrated in Fig. 12c and f demonstrates that the combined product can reliably capture actual tropospheric CH 4 signals independently   According to Eq. (4) a varying error in the a priori state together with a poor sensitivity (i.e. an averaging kernel being very different from an identity matrix) can cause a varying bias. If the error in the a priori state is latitudinal dependent the bias will also be latitudinal dependent. Furthermore, a systematic error source (like an error in a spectroscopic parameter) can have a variable impact on the remote sensing product, if the sensitivity is variable. If the sensitivity has a dependency on latitude, a systematic error source can thus also cause a latitudinal dependent bias. In this context, variabilities (e.g. latitudinal 450 dependencies) of biases are likely for a low or variable sensitivity. In contrast, inconsistencies in the bias are unlikely in case of a high and constant sensitivity (as observed in Fig. 4 for the total column and tropospheric and UTLS partial column of the combined data product). Figure 13 depicts the overall mean total and partial column values obtained at the six TCCON and and two AirCore observation sites. For total column data (Fig. 13a)  can investigate possible latitudinal inconsitencies in the satellite data products. We find that the TROPOMI and the combined satellite data product capture practically the same latitudinal dependency as the TCCON data. Figures 3a and 4a reveal that for these data products the sensitivities are very high and stable, in contrast to the MUSICA IASI data product, which has a relatively weak and seasonally (and supposed latitudinally) varying sensitivity. This explains that in Fig. 13a the latitudinal dependency of the MUSICA IASI XCH 4 data is slightly different from the TCCON data. Table 3  MUSICA IASI data product shows poorer performance with regard to the values of standard deviation, R 2 , and regression line slope, which is in line with its weak and varying sensitivity.

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A similar study of the latititudinal consistency of the partial column data products is compromised by the lack of profile references for low latitudes and southern hemispheric sites (Fig. 13b). Nevertheless, because the combined product has a rather high and constant sensitivity for the tropospheric as well as the UTLS partial column (see Figs. 3 and 4), we expect -as for XCH 4 -a good latitudinal consistency, i.e. a bias at low and/or southern latitudes that is similar to the bias of about +1% as observed at middle and high northern latitudes.  In summary, by reducing the systematic inconsistency between the TROPOMI and MUSICA IASI XCH 4 data we might be able to further improve the quality of the combined data product. Assuming that this inconsistency is mainly due to a positive bias in the lower/middle tropospheric MUSICA IASI CH 4 data (see Sect. 3.2), follow-up studies can focus on removing the MUSICA IASI bias (for instance, by an approach as discussed in Sect. 3.4 of Kulawik et al., 2021) and in documenting the impact of the bias removal on the quality of the combined data product. However, such studies are currently compromised by 485 the lack of high quality CH 4 profile data at low latitudes over land for the period after October 2017 (period with operating TROPOMI and IASI instruments).
We present a method for a synergetic use of TROPOMI total column and IASI vertical profile retrieval products. The method is based on simple linear algebra calculations, i.e. the execution of computationally expensive dedicated combined retrievals 490 is not needed. Nevertheless, it approximates closely to a dedicated combined optimal estimation retrieval using the combined TROPOMI and IASI measurements (see Appendix A2). We apply the method to CH 4 data. By providing a compilation with all important equations we support the application of this method to other data products.
We theoretically examine the sensitivity, vertical resolution, and errors of the individual TROPOMI and IASI products and of the combined product. The TROPOMI product consists of reliable total column CH 4 data, but does not offer information on 495 the vertical distribution. The IASI product offers some information on the vertical distribution and has best sensitivity in the UTLS region, but lacks sensitivity in the lower troposphere, i.e. it is not well sensitive to the total column. We show that the combined product combines both strengths: it is a reliable reference for the total column and also for the UTLS partial column.
In addition, we found as a clear synergetic effect that the combined product is also a reliable reference for the tropospheric partial column. 500 We generate the combined CH 4 product for the time period between November 2017 and December 2019 and compare the individual and combined products to reference data of TCCON, AirCore and GAW. TCCON data are available for the different latitudes in the northern and southern hemisphere and offer good references for XCH 4 . We get an agreement of all satellite XCH 4 products with the TCCON data within 1%. This comparison reveals a good reliability of the TROPOMI and the combined XCH 4 products, because of their independency on the a priori data (the comparison of the IASI data is affected 505 by the a priori data and thus cannot be directly interpreted). We found that the AirCore data are a very good reference for the consistent validation of the CH 4 total column amounts and the CH 4 vertical distribution; however, they are limited to northern hemispheric high and middle latitudes. Concerning total column comparison we get a very low 1σ scatter between the satellite products and the AirCore reference data (within 1%, which is similar to the comparison with TCCON). For the UTLS partial columns the scatter is also within 1% and for the tropospheric partial columns it is 1.1% -1.7%. While the comparison to 510 TCCON shows no significant bias, the comparsion to AirCore reveals a significant positive bias in the MUSICA IASI XCH 4 and tropospheric partial column data (significant in the sense that the systematic difference is outside the 1σ scatter and that it can also be not explained by the uncertainty of the AirCore references). For the combined tropospheric partial column product we report a slightly significant positive bias of about +1.4%.
We have only 24 AirCore profiles measured in collocation to satellite observations. A statistically more robust validation 515 of the tropospheric partial column products can be achieved by using continuous CH 4 observations from two nearby GAW stations. The CH 4 signals that are common at both stations are a good validation reference for the troposphere. We get collocations between the GAW data and satellite observations for 100 individual days and the comparison to the tropospheric partial column averaged mixing ratios generated from the combined data product confirms and widens the conclusions based on the comparison with the AirCore data: for the comparison of the daily mean data we get a mean difference and 1σ scatter 520 of +1.2%±1.2%, which is in good agreement to the comparison with the AirCore data (i.e. the combined product agrees very well with reference data, but we find indications of a weak positive bias). The continuous GAW CH 4 reference data cover seasonal cycle signals and have a larger amplitude than the AirCore data. We demonstrate that the lower tropospheric partial column averaged mixing ratio generated from the combined data product is able to capture these signals much better than the respective IASI product or the TROPOMI total column averaged product.

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There might be a chance to further improve the quality of the combined data product by performing detailed investigations on the inconsistency between the TROPOMI and the MUSICA IASI XCH 4 data. The availability of additional CH 4 profile reference data for low latitudes (e.g. obtained by the AirCore system) would be very beneficial for such purpose.
The proposed method takes benefit from the outputs generated by the dedicated individual TROPOMI and IASI retrievals, it needs no extra retrievals, and is thus computationally very efficient. This makes it ideal for an application at large scale, and 530 allows the combination of operational IASI and TROPOMI products in an efficient and sustained manner. This has a particular attraction, because IASI and TROPOMI successor instruments will be jointly aboard the upcoming Metop (Meteorological operational) Second Generation satellites (guaranteeing observations from the 2020s to the 2040s). There will be several 100,000 globally distributed and perfectly collocated observations (over land) of IASI and TROPOMI successor instruments per day, for which a combined product can be generated in a computationally very efficient way.

Appendix A: Theoretical considerations
In this appendix we give a brief overview on the theory of optimal estimation remote sensing methods and follow the notation as recommended by the TUNER activity (von Clarmann et al., 2020), which is closely in line with the notation used by Rodgers (2000). The overview focuses on the equations that are important for our work, i.e. the optimal a posteriori combination of 545 two independently retrieved optimal estimation remote sensing products. We show analytically that our method of combining two individually retrieved optimal estimation products by means of a posteriori calculations, is in most cases equivalent to a combined optimal estimation retrieval that uses a combined measurement vector.
For a more detailed and general insight into the theory of optimal estimation remote sensing methods we refer to Rodgers (2000) and for a general introduction on vector and matrix algebra dedicated textbooks are recommended.

A1 Basics on retrieval theory
If we assume a moderately non-linear problem (according to Chapter 5 of Rodgers, 2000), the retrieved optimal estimation product (the retrieved atmospheric state vectorx) can be written as: Here x and x a are the actual atmospheric state vector and the a priori atmospheric state vector, respectively. K is the Jacobian 555 matrix, i,e, derivatives that capture how the measurement vector (the measured radiances) will change for changes of the atmospheric state (the atmospheric state vector x). G is the gain matrix, i.e. derivatives that capture how the retrieved state vector will change for changes in the measurement vector: with S y,n and S a −1 being the retrieval's noise covariance and the constraint matrices, respectively. In a strict optimal estimation 560 sense, the constraint matrix is the inverse of the a priori covariance matrix S a .
The averaging kernel is an important component of a remote sensing retrieval, because according to Eq. (A1) it reveals how changes of the real atmospheric state vector x affect the retrieved atmospheric state vectorx.

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Very useful is also the a posteriori covariance matrix, which can be calculated as follows: The linearised formulation of the retrieval solution according to (A1) is very useful for the analytic characterisation of the product. The retrieval state's noise error covariance matrix for noise can be analytically calculated as:

570
where S y,n is the covariance matrix for noise on the measured radiances y.
Further very helpful equations are the relations between the a posteriori covariance, the averaging kernel, the constraint (or the a priori covariance), and the retrieval's state noise error covariance matrices: and 575 Sx ,n = ASx, with I being the identity matrix.

A2 Optimal combination of retrieval data products
In this subsection we discuss an optimal estimation retrieval that uses a combined measurement vector (two measurements from different instruments). Then we briefly introduce the Kalman filter and show that the Kalman filter formalism enables us 580 to combine two individually retrieved remote sensing data products in equivalence to the optimal estimation retrieval using the combined measurement vector.

A2.1 Optimal estimation using a combined measurement vector
According to Eqs. (A1), (A2), and (A4) the retrieval product obtained from a combined measurement vector {y 1 , y 2 } can be written as: where S y1,n and S y2,n are the respective measurement noise covariances, K 1 and K 2 the respective Jacobians and Sx 1 and Sx 2 the respective a posteriori covariances.

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An important application of a Kalman filter (Kalman, 1960;Rodgers, 2000) is data assimilation in the context of atmospheric modelling. There the filter operates sequentially in different time steps. Kalman filter data assimilation methods determine the analysis state (x a ) by optimally combining the background (or forecast) state (x b ) with the information as provided by a new observation (x o ): Optimal means here that the uncertainties of both, the background state and the observation, are correctly taken into account by the Kalman gain matrix (M): with Sxb and Sx o ,n being the uncertainty covariances of background state and the new measurement, respectively. The matrix 600 H is the measurement forward operator, which maps the background domain into the measurement domain.
The similarity between Eqs. (A9) and (A10), on the one hand, and Eqs. (A1) and (A2), on the other hand, reveals that remote sensing optimal estimation and Kalman filter data assimilation methods use the same mathematical formalism.

A2.3 Optimal a posteriori combination of individually retrieved data products
We have a first estimation of the atmospheric state (the first retrieval productx 1 ) and we want to optimally improve this 605 estimation by using a second retrieval product (x 2 ). This is a typical data assimilation problem and we can use the Kalman filter formalism. We make the following settings: In Eqs. (A11) and (A12) we assume that the two individual retrievals use the same constraint (S a −1 ). This is generally not the case and we can a posteriori modify a constraint and its effect on state vectors and covariances by the formalism as presented in Chapter 10.4 of Rodgers (2000) or Sect. 4.2 of Rodgers and Connor (2003). For our problem here this is of secondary importance, because we assume that TROPOMI total column data products are almost independent on the constraint (as long as the constraint is reasonable).

630
In Eqs. (A14) and (A15) we assume the usage of the same a priori for the two individual retrievals. Since generally two indivudually performed retrievals use two different a priori settings we have to perform an a priori adjustment. Using the a priori of retrieval 2 as the reference (x 2,a = x a ) we can adjust the output of retrieval 1 by (see Eq. (10) of Rodgers and Connor, 2003): 635 where x 1,a is the a priori used by retrieval 1.
Substituting Eq. (A17) together with the settings from Eqs. (A14) and (A15) in Eq. (A9) finally yields: i.e. the analysis state is the same as the outputx of a retrieval with a combined measurement vector from Eq. (A8). This means that we can a posteriori calculate the result that would be obtained by an optimal estimation retrieval using a combined measurement vector.

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The optimal a posteriori combination of two remote sensing products is possible, whenever: (1) the two remote sensing observations are made at the same time and detect the same location, (2) the problem is moderately non-linear (according to Chapter 5 of Rodgers, 2000), and (3) the individual retrieval output as listed by Eqs. (A11) to (A15) is made available. This is for the first retrieval the aposteriori covariances (Sx, which might also be reconstracted from A and R according to Eq. (A6)), the averaging kernels (A), and the retrieved and a priori state vectors (x and x a , respectively). For the second retrieval we need 650 the noise covariances (Sx ,n ), the averaging kernels (A), and the retrieved and a priori state vectors (x and x a , respectively).

Appendix B: Operator for transformation between linear and logarithmic scales
Linear scale differentials and logarithmic scale differentials are related by ∆x = x∆ ln x. For transforming differentials or covariances of a state vector with dimension nal (nal: number of atmospheric levels) from logarithmic to linear scale we define the nal × nal diagonal matrix L: Herex i is the value of the ith element of the retrieved state vector (i.e. in case of an atmospheric CH 4 state vector the CH 4 mixing ratios retrieved at the ith model level).
A logarithmic scale averaging kernel matrix A l can then be expressed in the linear scale as: Similarly a logarithmnic scale covariance matrix S l can then be expressed in the linear scale as: our results.
For converting mixing ratio profiles into amount profiles we set up a pressure weighting operator Z, as a diagonal matrix with the following entries: Using the pressure p i at atmospheric grid level i we set ∆p 1 = p2−p1 for 1 < i < nal. Furthermore, g i is the gravitational acceleration at level i, m air and m H2O the molecular mass of dry air and water vapour, respectively, andx H2O i the retrieved or modelled water vapour mixing ratio at level i.
In analogy we can define a row vector w T (with the dimension 1 × nal) with all elements having the value '1', which allows the resampling for the total column amounts.

C1 Column amounts 680
The kernel that discribes how a change in the amount at a certain altitude affects the retrieved partial (or total) colunm amount can be calculated as: For the total column we replace W by w T and get the row vector a T (dimension 1 × nal). This is the total column kernel provided by the TROPOMI data and it is typically written as a T . Figure 3 shows examples of such total and partial columns 685 amount kernels. The total column amount kernel can be interpolated to different altitude grids. For the applications in Sects. 2 and 3 we interpolate the TROPOMI total column amount kernel to the vertical grid used by the MUSICA IASI retrieval.

C2 Column averaged mixing ratios
We can also combine the operators Z and W for the calculation of a pressure weighted resampling operator by: This operator resamples linear scale mixing ratio profiles into linear scale partial column averaged mixing ratio profiles. Its inverse is calculated as: with W −1 = (W T W) −1 W T . The respective total column operators w * T and (w * T ) −1 can be calculated in analogy by replacing W by w T .

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With operator W * we can calculate a coarse gridded partial column averaged statex * from the fine gridded linear mixing ratio statex by: The kernels matrix of the partial column averaged mixing ratio state can then be calculated from the fine gridded linear scale kernel matrix (A) by: This kernel discribes how a change in the mixing ratio at a certain altitude affects the retrieved partial colunm averaged mixing ratio. Covariances of the partial column averaged mixing ratio state can be calculated from the corresponding covariance matrices of the fine gridded linear scale (S) by: The respective calculations for total column averaged mixing ratios can be made by replacing W * by w * T . For the total column avereraged mixing ratios the covariance is a simple variance (the scalar S * ) and the kernel has the dimension 1 × nol, i.e. it is a row vector a * T .
The total column amount kernel (a T T ) provided with the TROPOMI data set can be converted into a total column averaged mixing ratio kernel a * T T by the following calculation: The total column averaged mixing ratio kernel a * T T used in Sects. 2 and 3 is valid for the vertical grid used by the MUSICA IASI retrieval. It is calculated according to Eq. (C9), but using a TROPOMI total column amount kernel (a T ) that is interpolated onto the MUSICA IASI grid (see also Appendix C1).
Author contributions. Matthias Schneider developed the idea for the optimal a posteriori combination of two remote sensing products We acknowledge the support by the Deutsche Forschungsgemeinschaft and the Open Access Publishing Fund of the Karlsruhe Institute of Technology.