As widely recognized at the international level, there is a need for reduction in anthropogenic emissions (IPCC, 2014). This however implies the necessity for reliable climate predictions from atmospheric models in order to allow policymakers to make informed decisions. Unfortunately, current climate predictions are hampered by excessive uncertainties; for example intercomparisons of different models show important differences in their predictions as shown in Friedlingstein et al. (2006). This makes it difficult to assess the better environmental policies to implement. Because most biogenic fluxes in Europe are influenced by human activities, with 22 % of Europe's land dedicated to agriculture (FAO, 2013) and 45 % covered by forests, of which 80 % is managed for wood supply (UNECE, FAO, 2011), understanding and managing these biogenic fluxes must also be a component of any policy to reduce anthropogenic emissions.

A commonly used approach to estimate carbon budgets by teasing apart sources and sinks in a given spatial domain is the atmospheric Bayesian inversion. Atmospheric inversions combine prior knowledge from emission inventories with atmospheric observations acting as a top-down constraint to produce better posterior knowledge. As the main goal of this study is to assess the benefit of inter-species correlations in reducing the uncertainty of the posterior state space, we are particularly interested in the effects of such correlations on the uncertainty reduction, defined as the difference between prior and posterior uncertainty normalized by the prior. The vast majority of published papers on atmospheric inversions investigate the budget of a single species, usually a long-lived greenhouse gas like CO2 (e.g. Rödenbeck et al., 2003) or CH4 (e.g. Hein et al., 1997; Bousquet et al., 2006), but the technique can also be applied to active species like CO (Bergamaschi et al., 2000). Note that carbon dioxide is a special case as atmospheric CO2 mixing ratios result from a combination of strong anthropogenic sources with strong sources and sinks from biospheric processes, calling for a separation of anthropogenic fluxes from biospheric fluxes. One way to achieve such a separation is to measure CO alongside CO2, and use CO as a proxy for CO2 anthropogenic emissions. Several studies have made use the correlations among different species. One of the first examples is the work from Enting et al. (1995) on CO2 and 13CO2, while Brioude et al. (2012) attempted to derive a CO2 emission inventory without a prior emission estimate, instead using inventories of CO, NOy, and SO2. Similarly, Peischl et al. (2013) made use of CO and CO2 inventories to help quantifying sources of CH4 in the Los Angeles basin. The ability to measure multiple species has proved useful, also in remote sensing. For example, Pandey et al. (2015) made use of simultaneously retrieved CO2 and CH4 total column to reduce scattering effect. Further examples of studies making use of co-emitted species can be found in atmospheric chemistry (Konovalov et al., 2014; Berezin et al., 2013; Pison et al., 2009). More focused on exploiting inter-species correlation to reduce uncertainty in Bayesian inversion, Palmer et al. (2006) made use of CO2–CO correlations to improve inversion using data from the TRACE-P aircraft mission, while Wang et al. (2009) employed a similar method using satellite data, obtaining a reduction in the flux error of a CO2 inversion.

So far the lion's share of the studies investigating atmospheric inversions make use of both continuous in situ and flask measurements from ground-based observational networks of tall towers (e.g. Kadygrov et al., 2015; Sasakawa et al., 2010). However, as profiles collected from aircraft easily exceed the height of towers, airborne data may also offer an interesting option. This alternative was tested in some recent studies that made use of aircraft profiles alone or in combination with other data sources (e.g. Brioude et al., 2013; Gourdji et al., 2012). Methods to maximize the cost-effectiveness of airborne data are the use of unmanned aircraft (drones) and commercial airliners. The latter, in particular, allow for collecting data on a regular basis without requiring a particularly small or light sensor. The most important projects making use of commercial airliners are CONTRAIL (Comprehensive Observation Network for Trace Gases; Machida et al., 2008) and MOZAIC/IAGOS (Measurements of Ozone and water vapour by in-service AIrbus aircraft/In-service Aircraft for a Global Observing System; Marenco et al., 1998; Petzold et al., 2015). Both projects have been running for more than 2 decades and have produced extensive datasets that have proven to be important in the fields of atmospheric modelling and satellite calibration and validation (Zbinden et al., 2013; Sawa et al., 2012). Regarding carbonaceous species, CONTRAIL has so far collected CO2 mixing ratio measurement, while IAGOS is focused on CO. In the next years the IAGOS fleet will simultaneously provide CO, CO2, and CH4 atmospheric concentration measurements (Filges et al., 2015), enabling the use of multi-species synergy in modelling applications. This synergy follows the fact that the collocated measurements share the same atmospheric transport and have partially correlated emission uncertainties.

This paper is focused on investigating the benefits of such a multi-species inversion on uncertainty reduction in comparison with a single-species inversion. To achieve this goal, we set up a synthetic experiment utilizing the measurement times and locations collected from the IAGOS projects in the year 2011. The present paper is intended to pave the way for future studies making use of multi-species IAGOS datasets when they become available. A receptor-oriented framework was set up to derive flux interactions between the atmosphere and the biosphere using IAGOS data. The modelling framework is composed of a Lagrangian particle dispersion model (LPDM, specifically the STILT model), a diagnostic biosphere–atmosphere exchange model (the VPRM model), gridded emission inventories, global tracer transport model output that provides the tracer boundary conditions for the regional domain, and a Bayesian inversion scheme. The present work is based on the modelling framework used in Boschetti et al. (2015) and builds upon that by adding other species, and using a formal Bayesian inversion. A multi-species inversion was carried out in order to exploit the correlations in uncertainties between CO2, CO, and CH4, specifically in their respective uncertainties in a priori anthropogenic emissions and in model representation error. The aim of this multi-species inversion is to provide better estimates of anthropogenic emissions, and, in the case of CO2, to better separate the biospheric from anthropogenic contributions. This paper is structured as follows: a short description of the different components of the modelling framework is given in Sect. 2; in Sect. 3 we present and discuss our results; Sect. 4 gives the conclusions.

Before evaluating the performance of the inversion scheme in reducing the uncertainty of the state space, a closer look at the ability of the modelling framework to reproduce the enhancements is necessary. Unfortunately, this can be done only for CO as actual measurements are not available for the other species. Figure 5 shows the mean daily enhancement of the three fossil fuel species for both observations and model outputs using prior emissions. A common feature of the three trace gases is that lower values tend to occur during summer time due to better mixing of the atmosphere. Conversely, enhancement values tend to be higher during winter, reflecting the more stratified atmosphere of the cold months.

In Fig. 5 the modelled CO plot was multiplied by a factor of 2.8, corresponding to the mean ratio between observed and modelled CO enhancements, similar to what was found in Boschetti et al. (2015). Mixing ratio values are highly variable, but the model provides a good indication of the temporal variation of the ML enhancement; the squared correlation coefficient between observed and modelled CO enhancements is 0.62, while the standard deviation of corrected model and observation residuals is 85 ppb; note that by not accounting for the zi correction, such values would be 0.56 and 87 ppb respectively. The median of the mixing ratio enhancement for the three trace gases is 2.8 ppm for CO2, 18.6 ppb for CO, and 26.6 ppb for CH4. For CO2 this seasonal difference is enhanced due to the simultaneous presence of both anthropogenic and biogenic emissions. During summer, values are slightly negative due to strong photosynthesis fluxes from growing vegetation from the active combined with deeper vertical mixing. Negative values arise in 31 % of the cases predominantly during the warmer months, implying that during the growing period uptake by photosynthesis dominates over release from combustion and respiration. Both CO and CH4 experience higher values during winter due to the shallow mixed layer usually associated with cold temperatures, and lower values during summer as higher temperatures cause the mixed layer to reach higher altitudes; differences related to seasonal domestic heating and transportation may also play a role. In addition, enhancement for both species is occasionally negative, most likely owing to advection of polluted air masses in the free troposphere. An alternative explanation is that strong winds at lower heights can disperse the emissions in the boundary layer and create a situation in which the mixing ratio in the FT is higher than in the ML.

Figure 6 shows the prior and posterior error covariance matrices for the base multi-species inversion. Note that CO2 from anthropogenic emissions is assumed to be independent from biogenic emissions; therefore prior error correlation between these categories is zero. The posterior error covariance matrix for the multi-species inversion (Fig. 6b) shows lower values corresponding to an average uncertainty reduction of 23 % across all state vector elements, while the posterior error covariance matrix for the single-species inversion (not shown) is characterized by a mean uncertainty reduction of 20 %. This result implies that the multi-species inversion improves the uncertainty reduction by roughly 15 %. Negative values in the posterior error correlation matrix are to be expected because different categories are bound together by correlations and therefore are not free to vary independently.

Figures 7 and 8 show a priori, a posteriori, and “true” fluxes related to different aggregated fuel types and to different emission categories as described in Tables 1 and 2 for the months of July and December. Figure 8 also shows the biospheric contribution (as absolute values) scaled down by a factor of 10. As is to be expected, the biospheric contributions show strong differences according to the seasonal cycle, while anthropogenic emissions remain rather stable. However, it is worth pointing out that while the fossil fuel prior is similar for both months, the assumed truth can be rather different due the random assignment of the prior error realization. In most cases, the posterior adapts and is therefore closer to the truth than the prior; the posterior uncertainty is also visibly reduced, as expected. Regarding the different tracers, CO2 and CO show a somewhat similar pattern indicating a partial overlap in dominating emission categories while CH4 is dominated by different contributions in both fuel types and emission categories.

Our modelling framework is currently not well suited to account for unreported sources of CH4 due to the lack of information about natural gas and oil production operations, or from recent and old mining areas. Many recent studies have discussed the problem, mainly referring to shale basins exploited via hydraulic fracturing in the USA (e.g. Kort et al., 2016; Karion et al., 2015; Lyon et al., 2015). For example, Karion et al. (2015) concludes that EDGAR underestimates methane emissions associated with oil and gas industry by a factor of 5 in the USA. However, the situation over the European continent may be quite different. In a review about risk assessment of shale gas development in the UK, Prpich et al. (2015) reports that the European Union is generally much more cautious about unconventional oil and gas sources, while a recent study on a methane plume over the North Sea (Cain et al., 2017) concluded that the bulk signature of said plume originated from on-shore coal mines and power stations in the Yorkshire area.

In general, the absence of some emission sources in an inventory is equivalent to the assumption of having point sources not included in the emission map, but still contributing to the measurements. The inversion scheme would typically react to this by assigning such point sources in some other sectors to another fuel type. As a result, the posterior enhancements would be biased low in proximity of those point sources, and (slightly) biased high for influences from other regions with the same sector or fuel type. This issue should definitely be considered in future study making use of actual CO, CO2, and CH4 observations from IAGOS but has limited effects on this paper, as our main focus is on the benefits of inter-species correlation on the posterior uncertainty in a synthetic experiment.

Note that our modelling framework does not allow for simulating CO biogenic fluxes during the growing season. Warm days in summer correspond to large amount of biogenic volatile organic compounds (VOCs) being emitted from vegetation, producing CO at non-negligible levels. According to Hudman et al. (2008), anthropogenic emissions account for only 31 % of CO emissions in the USA during summer. Conversely, according to estimates from EDGAR, CO anthropogenic emissions during summer are about 18 % of the annual anthropogenic emissions. Combining these two results, one could conclude that CO production from biogenic sources accounts for roughly 42 % of total annual CO emissions.

CO2 and CO are dominated by combustion sectors (Fig. 8). The most important emission sectors for CO2 are energy, industry, transport, and building, each contributing 7–10 in July and 6–14 MtC month-1 in December. Dominant fuels (Fig. 7) for CO2 are coal, gas, and oil, whose prior fluxes (pseudo-data) have a magnitude of 6–11 Megatons of carbon per year (MtC month-1) in July and 8–14 MtC month-1 in December. For CO the most important emission sector is heating of buildings during winter, contributing a flux of 0.19 MtC month-1. Contributions from industry and transport dominate during summer, with magnitudes of 0.04 and 0.05 MtC month-1, respectively. The dominant fuel for CO is biofuel with 0.19 MtC month-1 emissions during winter. The secondary industrial and transport contributions originate in summer from oil and biofuels with a magnitude of 0.06–0.08 MtC month-1 and from agricultural waste burning with a magnitude of 0.06–0.11 MtC month-1.

Contrary to CO2 and CO, CH4 is determined by non-combustion sectors, more specifically by a contribution of 0.15 MtC month-1 flux from agriculture (manure management and rice cultivation) in July with secondary contributions from waste and energy with a magnitude of roughly 0.06–0.08 MtC month-1 in both July and December. Other non-combustion sectors, in particular wastewater treatment and landfills contribute to a total of 0.16–0.24 MtC month-1 of emissions. These non-combustion sectors contribute to less than 20 % of total CO2 emissions, with 1.13 MtC month-1 from the cement and lime industry and less than 20 % of the total CO emissions (0.03 MtC month-1 from the metal industry).

The contribution to CO2 from biospheric primary production (a sink for atmospheric CO2) is about 100 MtC month-1 in July, which drops to almost zero in December, while respiration values are 50 MtC month-1 in July and roughly 15 MtC month-1 in December.

As further assessment of the inversion performance, we tested the ability of the inversion scheme to capture the truth compared with a perturbed version of the prior. Such perturbed version is obtained by adding a random distribution with mean and standard deviation equal one to the prior state space, similar to how the truth is obtained. For each simulated species we calculated the total annual fluxes for prior, posterior, truth, and perturbed prior. From these total fluxes we then derive the overall residual between prior and truth, posterior and truth, and perturbed prior and truth. It is clear from Table 4 that while the overall bias between posterior and truth is lower than the prior–truth bias, the bias between perturbed prior and truth is much higher, implying that the performance of the inversion is not an artifact of the pseudo-data generation. In addition, it was found that the truth–posterior bias of the multi-species inversion is mostly slightly lower compared to the single-species inversion. The difference is between -2.2 and 7.6 %, according to the simulated species, with an overall value of 0.3 %.

Improper characterization of the error correlation may result in systematic bias in the posterior estimate. As mentioned in Sect. 2.1.6, inter-species correlation, the correlation between different fuel types and the correlation between different emission sectors in Sprior, is assumed equal to 0.7 (Sect. 2.1.4). To assess how well the system will reproduce the true fluxes with incorrectly specified correlations, a series of experiments was performed in which the inter-species correlation in Sε remains equal to 0.7, while the three correlation coefficients in Sprior assume different values ranging from 0.1 to 0.9. Table 5 shows the residuals between total annual posterior fluxes and total annual true fluxes for the five simulated species, derived similarly as for Table 4. We found that for all species the uncertainty reduction increases with correlation. More precisely, from correlation 0.1 to 0.9, the annual uncertainty reduction for anthropogenic CO2 increases from 26.6 to 51.7 %, while the increase is lower for GEE (from 72.4 to 73.1 %) and respiration (from 39.3 to 41.3 %) because the biospheric fluxes are independent from other species. For CO, the uncertainty reduction increases from 60.7 (with correlation 0.1) to 66.4 % (with correlation 0.9). The annual uncertainty reduction for CH4 increases from 60.5 to 67.5 %.

In addition, the posterior–truth biases are always lower than the prior–truth biases. The posterior uncertainty values (1σ) are usually larger then the corresponding bias values as expected, except for CO and for CH4 with prior correlations equal to 0.9. Thus the posterior is not significantly different from the truth. Conversely, the prior (not shown) is significantly different from the prior in the majority of cases for fossil fuel fluxes, and in some cases also for biogenic fluxes. The effect of assuming the incorrect error correlations appears to be in general small, possibly implying a relative robustness of our methods. Following this result, the fact that CH4 is only partially co-emitted with CO2 and CO should not affect the inversion in a strong way. For all of the experiments, the residuals between true and posterior fluxes are lower than residuals between true and prior fluxes for each of the simulated species; the difference between the cases with maximum and minimum residuals is around 4.2 %. In addition, we found that the posterior aggregated fluxes in the nine experiments are not significantly different from each other, implying that the system is fairly robust against errors in the assumed inter-species correlation.

Before investigating the benefits of correlations between different tracers, it is worthwhile to evaluate the uncertainty reduction in the monthly budgets for all five modelled species (Fig. 9, based on targeted spatial domain in Fig. 1). The first thing to note is that for all of the five trace gases the posterior uncertainty is lower than the prior one, as it should be. In addition, prior uncertainty varies through the year, reflecting modulation in emission fluxes obtained by adding activity factors to describe the seasonal, weekly, and daily cycle.

Prior uncertainty assumes values around 0.4–0.6 MtC month-1 for CO2, 5–15 ktC month-1 for CO, and 15 ktC month-1 for CH4. For GEE the prior uncertainty is between 0.3 and 46.7 MtC month-1, and for respiration it is 5.1–19.0 MtC month-1. Posterior uncertainty for CO2 is 0.24–0.38 MtC month-1 for fossil fuel emissions, 0.3–9.9 MtC month-1 for GEE, and 3.1–10.4 MtC month-1 for respiration, while it has a range of 3.3–4.7 for CO and 2.7–7.0 ktC month-1 for CH4. Mean uncertainty reduction of the monthly values is 38 % for fossil fuel emissions of CO2, 41 % for GEE, and roughly 45 % for respiration, 63 % for CO, and about 67 % for CH4. It is worth pointing out that such values are higher than the mean uncertainty reduction in the scaling factors (23 %); this happens because the most representative emission sectors are those influencing the observations the most and thus are also the most constrained.

In addition, note that in this case, the posterior uncertainties for single- and multi-species inversions are similar for the modelled species, with the exception of the CO2 anthropogenic contributions. To generalize this last result, we tested the benefit of a multi-species inversion for the different cases of prior uncertainty values shown in Table 3. As an indicator for the benefit of including correlation between different species, we use the ratio between posterior uncertainty of the multi-species inversion and the posterior uncertainty of the corresponding single-species inversion. A value of 1 means that there is no benefit from adding an inter-species correlation to the inversion, while values greater than 1 mean that a multi-species inversion is even less constrained than a single-species one. We expect this indicator to be less than 1, meaning that inter-species correlations actually improve the constraint power of the inversion. As before, we consider here the uncertainties of the retrieved budgets for the 50 % footprint, where the surface influence is strongest (Fig. 1). Values of this uncertainty ratio for the different trace gases as function of month are shown in Fig. 10 for the different cases listed in Table 3. The benefit of including inter-species correlations shown in Fig. 10 does not depend on different manifestations of the true fluxes, but only on the posterior uncertainty of the multi- and single-species inversions.

All of the species experience a reduction in the posterior uncertainty ratio due to the addition of inter-species correlation; said reduction is up to 20 % for fossil fuel CO2 and up to 10 % for the other species; in addition, anthropogenic CO2 is more sensitive to the prior relative error values than CO and CH4. As the uncertainty of GEE and respiration is not modified, they show little to no variation for different cases (Fig. 10). There is a dependence of the benefit of the multi-species inversion over the single-species inversion on the prior uncertainty values (differences between Cases 1–3), with the largest difference for fossil fuel emissions of CO2. Interestingly for Case 2 with reduced prior uncertainty for fossil fuel CO2 emissions the benefit nearly doubles over the default case (Case 1). Also reducing the prior uncertainties of CO and CH4 emissions (Case 3) more or less compensates for this increase in benefit. The reason for both of these results is probably to be found in Eq. (8). In fact, changing the prior uncertainty in CO2 emissions means to also change the off-diagonal blocks linking the different species together. However, by reducing the anthropogenic CO2 uncertainty from 20 to 10 % (Case 2), the diagonal block for CO2 in the prior uncertainty changes by a factor of 4, while the off-diagonal blocks change only by a factor of 2. This effectively ties the emissions of CO2 tighter to the emissions of the other species, resulting in greater benefit from a multi-species inversion over a single-species inversion. Conversely, when all prior uncertainties are reduced by a factor of 2 (Case 3), both diagonal and off-diagonal blocks are reduced by a factor of 4. This explains why Case 1 and Case 3 show similar benefit values. Note that the assumed prior uncertainties for the default case (Case 1) are quite conservative; therefore lower uncertainties were chosen for Cases 2 and 3. While the absolute benefit of adding inter-species correlation is not a game-changer, it is worth pointing out that such improvement also comes with only slightly greater computational effort than multiple independent single-species inversions.

In order to assess the contribution of inter-species correlation in the prior uncertainty vs. that of model–data mismatch uncertainty, Fig. 11 also shows the resulting posterior uncertainty ratios for Case 1 (Table 3) from inversions only using prior or model–data mismatch correlation. For the anthropogenic component of CO2, the greatest constraint is given by the prior correlation, while for GEE, respiration, and CH4 the strongest contribution is from the model–data mismatch correlation. In the case of CO, the inter-species correlations for different components are dominant for different months of the year. What makes CO sensitive to different correlation structures during different seasons is CO enhancement showing a stronger seasonal cycle compared to, for example, the fossil fuel component of the CO2 enhancement, with average values of around 150 ppb (25 ppm for CO2) for January and around 9 ppb (4 ppm for CO2) for July. This results in a much weaker constraint on the CO emissions from the CO observations during summer, but there is still some constraint through the other species such as CO2 via the a priori correlation in the emissions.

Palmer et al. (2006; in the following referred to as P06) studied the importance of inter-species correlation to improve inverse analysis using airborne data from the TRACE-P mission conducted in March/April 2001 over the western region of the Pacific Ocean. P06 derived a prior error correlation lower than 0.2 by analysing the uncertainty of emission factors from an Asia-specific emission inventory (Streets, 2003), which is significantly smaller than the correlation of 0.7 assumed in the present study. P06 deemed CO2–CO prior correlation greater than 0.5 to be unrealistic for the emissions in Asia, which is mostly associated with the uncertainty in emission factors for CO of 67 % for fossil fuel emissions and 240 % for biofuel emissions in China (P06 Table 1). However, for the European region used in the present paper we argue that values around 0.7 are appropriate. The resulting uncertainty in the CO2–CO ratio, diagnosed from the prior error covariance matrix used in this study, is about 50 % for both biofuel and fossil fuel emissions in Europe, which we regard as reasonable. To compare results from P06 with those from the present study, ratios of posterior uncertainties resulting from inversions using correlations between CO2 and CO of 0.7 in the prior uncertainties and to those using no correlations have been extracted from Fig. 7 in P06 and are also shown as orange diamonds in Fig. 11. It is easy to see that for anthropogenic CO2, the value derived from P06 is higher than in our study, while the two values are very similar for CO. Similarly, posterior uncertainty ratios using model–data mismatch correlations of 0.7 between CO2 and CO are derived from Fig. 8 of P06 and are shown as red diamonds. In this case, the value derived from P06 is slightly lower than in our study for anthropogenic CO2, while the two are again very similar for CO.

From this comparison we can see that the estimates of the benefit of including inter-species correlation in atmospheric inversions in P06 and in this paper are of the same order of magnitude for anthropogenic CO2 and almost identical for CO, suggesting a general continuity of results.

The present paper presents a synthetic experiment aiming to evaluate the effects of exploiting correlations between different trace gases in an atmospheric inversion. We quantitatively described the capability of the modelling framework to reproduce observations, the performance of the inversion scheme in reducing the uncertainty of the different trace gases, and the benefits of multi-species inversions compared to corresponding single-species inversions. We also describe a method to re-scale different prior uncertainty covariance matrices so that the corresponding posterior uncertainties are actually comparable.

Where possible, we compared model outputs with available observations. Such comparison, possible only for CO, showed a good degree of agreement between the model and observations with an overall correlation of roughly 0.75; modelled values for CO enhancement underestimate the observed ones by a factor of roughly 2.8, compatible with what was found in Boschetti (2015). It is found that posterior uncertainty is much lower than the prior for all of the five simulated species. The mean uncertainty reduction for CO2 emissions from fossil fuels is roughly 38 %, for GEE it is around 41 %, while for respiration it is roughly 44 %. For CO and CH4 the uncertainty reduction is about 63 and 67 % respectively. Finally, we described quantitatively the benefit of using multi-species inversions by exploiting correlations in different chemical species. It is found that considering correlations between different trace gases can reduce the posterior uncertainty by up to about 20 % for monthly fluxes. These benefits are however dependent on the error structure of the prior uncertainty.

The present paper paves the way for future studies using simultaneous measurements of different trace gases. This will be especially important in the context of the upcoming routine measurements of CO2, CO, and CH4 vertical profiles within IAGOS. As IAGOS makes use of commercial airliners, such profiles will be collected in the vicinity of major international airports, and hence in the vicinity of major metropolitan areas, where many different human activities take place simultaneously. In such a context, any improvement in the constraint of atmospheric inversions will be particularly useful. A possible improvement in this analysis would be to evaluate the effects of different correlation factors specific to different pairs of anthropogenic species, fuels, and emission sectors.