The theoretical framework of the inversion system developed in this study is the same as the traditional atmospheric inversions. The inversion derives a statistical estimate for a set of control variables x in a model x→y=Mx that simulates the satellite XCO2 measurements yo. The model M linking x and y is a combination of flux and atmospheric transport models (detailed in Sect. 2.4) and is called observation operator hereafter. As explained below, we do not have a constant term added to Mx in the observation operator of the PMIF that would gather the atmospheric CO2 signature of the fluxes not controlled by the inversion (like non-fossil fluxes and the background XCO2 field) since the uncertainty in such fluxes is ignored. The inversion follows a Bayesian statistical framework, updating the statistical prior estimate of x based on the statistical information from the assimilation of XCO2 measurements y into the observation operator. The distributions of the prior estimate and of the misfits between the actual observations yo and simulated ones due to errors in the observations and in the observation operator (called the “observation errors”) are assumed to be unbiased and to have the Gaussian forms N(xb,B) and N(0,R), where B and R are the prior and observation error covariance matrices. The statistical distribution of the posterior estimate of x, given the observation operator, xb and yo, also follows a Gaussian distribution N(xa,A), with xa being the mean and A being the error covariance matrix characterizing the posterior uncertainty. The problem is solved by deriving the following: 1A=B-1+MTR-1M-1,2xa=xb+AMTR-1yo-Mxb-yfixed, where T and -1 denote the transpose and inverse of a given matrix.

Equation (1) shows that A only depends on prior and observation error covariance matrices, on the observation operator, and implicitly on the structure of the observation vector (i.e., on the time, location and representation of the observations in M), while Eq. (2) shows that xa also depends on the actual value of xb and yo. PMIF is an analytical inversion system that solves for Eq. (1) or for an approximation of this equation (when accounting for temporal correlations in B) by building the different matrices involved in this equation.

We characterize B, R and A by the corresponding standard deviations (σ) of uncertainty in individual control parameters or aggregations of control parameters and by the temporal auto-correlations of the uncertainties (Sect. 2.6). In the following, the “uncertainty reduction” for a given control variable or for an aggregation of control variables (like emission budgets over larger timescales than that of the control vector) refers to the relative difference between its prior and posterior uncertainty: 1-σa/σb.

We use a Gaussian plume model (Sect. 2.4) to simulate the atmospheric transport at a spatial resolution consistent with that of the XCO2 measurements from the planned CO2 imager and with the highly heterogeneous distribution of emissions. Compared with complex 3-D atmospheric transport models, Gaussian plume models have a very low computational cost, making the global assessment of posterior uncertainty and uncertainty reduction at the scale of emission clumps from the assimilation of high-resolution data feasible. However, since a Gaussian plume model provides a highly simplified approximation of the atmospheric transport from emission clumps, we need to verify that its use in the PMIF yields estimates of the uncertainties in the inverted emissions that are consistent with those that would be based on more complex models. Therefore, we first compare the results for Paris from PMIF against those acquired based on a 3-D Eulerian atmospheric transport model by Broquet et al. (2018), the latter also accounting for uncertainties in diffuse CO2 fluxes. On the one hand, the signals from these diffuse and natural CO2 fluxes cannot be modeled effectively by a Gaussian plume model. On the other hand, the diffuse and natural CO2 fluxes in Paris were shown to have only a weak impact on the inversion of whole-city CO2 emissions (Staufer et al., 2016). For this comparison, we use the same simulation of the XCO2 sampling by CarbonSat (Sect. 2.2) and a similar control vector to Broquet et al. (2018). The corresponding inversion with the PMIF is called PMIF-Paris hereafter. Then we apply the system to all the emission clumps over the globe and over 1 year using a different control vector and a simulation of the XCO2 sampling by a single CO2M satellite (Sect. 2.2). The inversions for all emission clumps over the globe are called PMIF-Globe. In PMIF-Globe, we first investigate the potential of satellite observations in constraining emissions from individual time windows (ExpNoCor in Sect. 2.6). Then we assess the ability of satellite observations to constrain emissions at annual scale by accounting for the temporal auto-correlation of the prior uncertainties (other experiments in Sect. 2.6). Tables 1 and 2 summarize the different options for the configuration of the system and of the OSSEs. One distinction between PMIF-Paris and PMIF-Globe is that PMIF-Paris relates XCO2 signals with the mean emissions 6 h before overpasses, while it is assumed that in PMIF-Globe that the XCO2 signals only provide effective constraints on 3 h mean emissions before individual overpasses. The 6 h period corresponds to the period of emissions from Paris whose signature in the XCO2 field can still be detected by the satellite despite the atmospheric diffusion (Broquet et al., 2018). While Broquet et al. (2018) indicated that the period of “detectable” emissions from a large megacity like Paris could last up to 6 h, most of the clumps across the globe have smaller emission rates than Paris or are located in more complex environment close to other major emission areas where XCO2 signals can be attributed to multiple sources, making the detection of the XCO2 signature of emissions a few hours before the satellite overpass even more difficult. For the PMIF-Globe experiments, we thus conservatively assume that the XCO2 signals can only provide effective constraints on 3 h mean emissions before individual overpasses in general.

In this study, we consider the samplings from two different virtual CO2 imagers.

The first sampling used in PMIF-Paris (Table 1 and Sect. 2.7.1) is the simulation of the sampling for CarbonSat by Buchwitz et al. (2013) exactly as in Broquet et al. (2018). XCO2 is sampled by a 240 km swath instrument with 2 km spatial resolution. Given the presence of cloud and aerosol and their impacts on the precision of XCO2 retrievals, only “good” XCO2 observations, for which the sum of the retrieved aerosol optical depth (AOD) at near-infrared (NIR) wavelength and atmosphere cirrus optical depth (COD) is less than 0.3, are used in the inversions. The preferable condition, AOD(NIR)+COD<0.3, for a good XCO2 observation is referred to as “clear sky” hereafter. The CarbonSat sampling was simulated over the whole globe and for a full year by Buchwitz et al. (2013), but it is used here for the inversion of the emission of Paris only. Thus, only the passes with at least one good XCO2 measurement in the 100 km radius circle centered on Paris are used, as in Broquet et al. (2018).

The second sampling is global and is used for all the other experiments of PMIF-Globe (Table 2 and Sect. 2.7.2). It corresponds to that of a single CO2M satellite with a 300 km swath and 2 km spatial resolution. CO2M is similar to CarbonSat for sampling but has a larger swath and better precision (Sect. 2.5). The simulation is based on the method and model described by Buchwitz et al. (2013) but uses different values for the parameters in the model.

In the PMIF-Paris inversion, the satellite observations are sampled at 11:00 local time, in line with the experiments from Broquet et al. (2018). The inversion solves for the mean emissions for the 6 h before 11:00 local time. Broquet et al. (2018) solved for the hourly emissions during this 6 h period but PMIF can only solve for the mean emissions during the 6 h period due to the fact that the Gaussian plume model cannot be used to compute the signatures in the XCO2 field of individual hourly emissions during that period. The control vector in PMIF-Paris thus consists of a set of scaling factors for the mean emission between 05:00 and 11:00 for all individual overpasses near Paris (Sect. 2.7.1). The prior and posterior scaling factors are used to rescale the 1 h and ∼1 km resolution emission fields from an emission map and its temporal profile which are parts of the observation operator (Sect. 2.4).

In the PMIF-Globe inversion, the satellite observations are sampled at a local time of approximately 11:30 over all the clumps. The inversion solves for a scaling factor for 3 h mean emissions between 08:30 and 11:30 and a scaling factor for the emissions during of the rest of the day (00:00–08:30 plus 11:30–24:00) for each day over 1 year and for all the clumps over the globe: 3x=[λclump1day1,morning,λclump1day1,rest,λclump1day2,morning,λclump1day2,rest,…,λclump1day366,morning,λclump1day366,rest,λclump2day1,morning,λclump2day1,rest,…,λclumpNday366,morning,λclumpNday366,rest].

In both types of experiments, we do not include the diffuse emissions outside the selected clumps and the natural fluxes (more generally, any parameter of the “background concentrations”, Kuhlmann et al., 2019) in the control vector. The setup of the R matrix also ignores uncertainties in the background concentrations (Sect. 2.5). This is another divergence from the inversion configuration of Broquet et al. (2018), who accounted for such uncertainties.

The observation operator in PMIF (which is used in Eq. 1) is composed of two sub-operators. The first operator (Minventory) describes the spatial distribution (within the clumps) and temporal variations of the emissions whose budgets are controlled by the inversion during 08:30–11:30 and during the remaining 21 h for each clump: x→E=Minventoryx. The spatial distribution of the emissions are based on estimates from ODIAC (Oda et al., 2018) for the year 2016. ODIAC provides the monthly mean emissions for 12 months through a year at a 0.0083∘×0.0083∘ (approximately 1km×1km) spatial resolution. The weekly and diurnal (at hourly resolution) profiles from the Temporal Improvements for Modeling Emissions by Scaling (TIMES) product (Nassar et al., 2013) are applied to the monthly emission maps of ODIAC to generate the hourly emission fields. The second operator (Mplume) simulates the plumes of XCO2 enhancement above the background at and downwind of the emission clumps at 11:30: E→y=MplumeE. We assume that the plume of XCO2 enhancement related to a given emitting pixel within a clump of the ODIAC map has a Gaussian shape and the plume from a clump is a sum of multiple Gaussian plumes from all the ODIAC pixels within that clump. For a given emitting pixel, the Gaussian plume model writes the following: 4yi,j=αE2πσjue-j22σj2, where y is the XCO2 enhancement (in ppm) downwind of the emitting pixel. The i direction is parallel to the wind direction, and the j direction is perpendicular to the wind direction. y depends on the mean emission rate during 08:30–11:30 at local time (E, in g s-1), the wind speed (u, in m s-1), the cross-wind distance (j) and the parameter σj (see below). The wind direction and speed is taken from the Cross-Calibrated Multi-Platform (CCMP) gridded surface wind fields for the year 2008 (Atlas et al., 2011). The CCMP product uses a variational analysis method (VAM) to combine the data from Version-7 RSS radiometer wind speeds, QuikSCAT and ASCAT scatterometer wind vectors, moored buoy wind data, and ERA-Interim model wind fields. The σj is a function of downwind distance i and atmospheric stability parameter σj=βj/(1+γj)-1/2, where α is a coefficient that converts the computed XCO2 enhancement in the unit of parts per million, and β and γ are coefficients depending on the atmospheric Pasquill stability category, which is a function of the wind speed and solar radiation (Turner, 1970). The values for β and γ can be found in Bowers et al. (1980). The original Gaussian plume model generates a stationary plume of an infinite length and width downwind of the emissions. Because we assume that the XCO2 plumes sampled from a satellite overpass is only related to the emissions 3 h before, the Gaussian plume corresponding to each emitting pixel is cut off at the downwind distance equaling the wind speed multiplied by 3 h. The width of the plume is also cut off beyond 3 times the value of σj in the cross-wind direction. The observation operator is null for emission of the remaining 21 h (00:00–08:30 plus 11:30–24:00).

The size of the full theoretical control vector corresponds to 11 314 emission clumps times 2 time windows for each day times 366 d. The size of this full theoretical observation vector over the year is thus more than 30 000 000. Building matrices and applying Eq. (1) with such spaces is, in practice, not computationally affordable. Therefore, we divide the globe into 5400 spatial inversion windows (from 180∘ W to 180∘ E and from 90∘ N to 60∘ S), each inversion window covering an area of 10∘×10∘ and being extended on the four boundaries with margins of 500 km to ensure that the plumes from the clumps near the boundary of inversion windows are fully simulated and accounted for in the corresponding inversions. Mplume is constructed with a set of block matrices, each block representing a single spatial inversion window and a single day. When an emission clump and its plume are comprised within more than one inversion window on a single day, only the results obtained in the window that covers the full plume is used in Mplume.

We evaluate the projection of the measurement noise of the satellite observation and ignore uncertainties in the observation operator. The measurement noise is derived from the simulations of random measurement errors from Buchwitz et al. (2013), and the impact of the systematic measurement errors is ignored. The random measurement errors are simulated as a function of geographic location (e.g., solar zenith angle, SZA), surface (e.g., albedo) and atmosphere characteristics (e.g., AOD). The random measurement error is 1.4 ppm for vegetation albedo and SZA 50∘ in the CS sampling, and it is 0.7 ppm in the CO2M sampling, thus being 2 times smaller for the latter. The random measurement errors are uncorrelated from one XCO2 data to the other, and the R matrix is thus a diagonal matrix as generally done in atmospheric inversion.

Two configurations for the prior uncertainty are used in the OSSEs (Sect. 2.7). In the PMIF-Paris inversion, the prior uncertainty is 22.4 % for the each of the scaling factors for 6 h mean emission, the choice of this value being consistent with the configuration used by Broquet et al. (2018).

In the PMIF-Globe inversions, the prior uncertainty is downscaled from its estimate for the annual budget of emissions of each clump. A prior uncertainty in annual emission of 30 % is assumed for all clumps. This value is chosen to be of the same order of magnitude as the typical difference between emission inventories for a single point source and city. For example, Gurney et al. (2016) found that one-fifth of the power plants had monthly emission differences larger than 13 % between the estimates by two different US agencies. Gurney et al. (2019) compared the emission maps from ODIAC and Hestia for four US cities and found the whole-city differences are between -1.5 % and +20.8 %. Gately and Hutyra (2017) compared the inventories reported by local authorities and bottom-up fossil fuel CO2 emission maps for 11 US cities and found the differences range from 33 % to 78 %. Then, the downscaling of the uncertainty in annual emissions into uncertainties at the sub-daily scale of the control variables (i.e., 3 h mean emission over 08:30–11:30 and 21 h mean emission during the rest of the day; Sect. 2.3) follows a decomposition of the total uncertainty into components with different temporal auto-correlations.

The hourly emissions in inventories are usually derived from the periodic typical temporal profiles to annual emissions (Andres et al., 2011; Nassar et al., 2013). There are large variations in actual emissions from hour to hour and from day to day, resulting in large differences between the emission estimates derived based on typical temporal profiles and actual emissions. These differences are sources of uncertainties in the emission inventories which are used in the inversion as prior information. However, there is no consensus regarding the uncertainty in emission inventories and their error structures (Gurney et al., 2019). We compare the typical temporal profiles of transport emissions and energy sector from the TIMES product with the TOMTOM traffic index (https://www.tomtom.com/en_gb/, last access: 20 September 2020, which provides indications of the level of variability in the traffic, though not of that of the CO2 emissions themselves), respectively, and with the actual hourly CO2 emissions from electricity production in France (https://www.services-rte.com/en/home.html, last access: 17 July 2017). Although these comparisons are only made for two sectors, the results already show that it is challenging to describe the temporal auto-correlations of the uncertainty in emissions with simple exponentially decaying functions (Figs. S1 and S2 in the Supplement) like what is usually done in traditional atmospheric inversions (Chevallier et al., 2010; Kountouris et al., 2015). We thus make several assumptions regarding the decomposition of the prior uncertainty into components with different modes of auto-correlation.

In some scenarios, we consider an “annual component” that is fully correlated in time over 1 year. We also consider “uncorrelated” components whose temporal auto-correlations are null and “sub-annual” components whose temporal auto-correlations follow the exponential decaying model with a correlation length smaller than 1 year. Specifically, we assume that the correlation between two instants of the sub-annual component at hourly scale is described by the following: 5r=exp⁡-Δh/τ1×exp⁡-Δd/τ2, where Δh is the time lag (in hours) between the two times of the day that are considered, and Δd is the time lag (in days) between the two dates that are considered. The parameters τ1 and τ2 follow the fit of the misfits between the TIMES profiles and the TOMTOM and electricity production indices to the exponential functions at the hourly scale and at the daily scale, respectively (Figs. S1 and S2). The temporal auto-correlations between the emissions during the aggregated time windows (08:30–11:30 and the remaining 21 h) are computed by re-aggregating the uncertainties at the hourly scale, accounting for temporal auto-correlation.

The detailed configuration of the different scenarios for the decomposition of the prior uncertainty are listed below: 1.

Annual component and moderately correlated sub-annual component (AMS). This is composed of an annual component and a sub-annual component. The temporal auto-correlation of the sub-annual component follows Eq. (5) with τ1=12 h and τ2=7 d. The ratio of the uncertainty in the annual component to that in the sub-annual component for 3 h emissions is assumed to be 3:5. This leads to an annual uncertainty component ∼N(0, 29 %) and a sub-annual component ∼N(0, 49 %) for 3 h emissions and ∼N(0, 38 %) for 21 h emissions.

The prior uncertainty in the 3 h mean emissions between 08:30 and 11:30 is close to or larger than 100 % in scenarios SCS and MCS, and it even reaches an abnormally huge value of 1623 % in NoCor. Andres et al. (2016) estimated the uncertainty in the widely used emission map CDIAC (Carbon Dioxide Information Analysis Center). They found that the average uncertainty in monthly emissions for one 1∘×1∘ grid cell is 120 % and further suspected that the uncertainties in hourly and daily emissions at urban scale could be even larger (from a few percent to 1000 %). But these large values challenge the assumption that the uncertainty in anthropogenic emissions is normally distributed (Gurney et al., 2019). In this study, we follow the traditional assumption used in atmospheric inversions that the prior uncertainty follows a Gaussian distribution, allowing the prior uncertainty to exceed 100 % in some scenarios. This assumption ensures that the system is analytically solvable using Eqs. (1) and (2). In addition, we focus our analysis on 08:30–11:30 time windows or days for which the posterior uncertainties of underlying emissions are smaller than 20 % (Sect. 2.7.2), a value that is significantly smaller than the prior uncertainty in any scenario. In these cases, Eq. (1) ensures that the posterior uncertainty is almost driven by the projection of the observation error on the control space and is not sensitive to the level of prior uncertainty.

Two sets of OSSEs are conducted under different configurations adapted to different purposes, as described below. Tables 1 and 2 summarize the different configurations of the OSSEs.

The comparison of the results from the PMIF-Paris experiment to that of Broquet et al. (2018) is used to demonstrate that the PMIF produces meaningful statistics for other clumps despite its relative simplicity at the local scale (its complexity being linked to its global and annual coverage). Figure 1 shows the theoretical uncertainty reduction for the 6 h mean emissions obtained in PMIF-Paris inversions with the 1st, 5th, 10th, 15th, 19th and 25th best observation sampling from CarbonSat over 31 inversion days (Sect. 2.7.1), each day being characterized by the average wind speed over Paris. We compare these results with the Fig. 6 from Broquet et al. (2018). Like Broquet et al. (2018), Fig. 1 illustrates the strong correlation between the uncertainty reduction and the average wind speed, indicating that lower wind speed results in a larger signal close to the city that is easier to assimilate than elongated plumes under large wind speeds. For the best observation sampling, the uncertainty reduction remains smaller than 40 % when the wind speed is larger than 13 m s-1, and this value is generally twice as low as the values obtained when the wind speed is smaller than 5 m s-1.

Some differences are seen in Fig. S3, between the results obtained by PMIF and by Broquet et al. (2018). For example, the PMIF-Paris inversion slightly overestimates the uncertainty reduction under high wind speed (>15 m s-1) using the best observation sampling compared to Broquet et al. (2018). These differences reflect the impact of using the Gaussian plume model instead of a 3-D atmospheric transport model, and more importantly, the impact of accounting for more sources of uncertainties (in diffuse emissions and natural fluxes) in Broquet et al. (2018). Despite these differences, the general coherence in the ranges of uncertainty reductions (Fig. S3) under different wind speeds between the PMIF-Paris experiment and Broquet et al. (2018) is a strong indication that the PMIF generates the correct order of magnitude for the uncertainty reduction for a single clump. In addition, Nassar et al. (2017) used the Gaussian plume model to process actual XCO2 plumes generated from several power plants, which were sampled by OCO-2, adding the indication that Gaussian plume model can simulate the typical spread and amplitude of actual XCO2 plumes and thus support the application of PMIF to a large range of clumps.

Figure 1 shows that the uncertainty reduction on 6-hourly emissions from Paris before the satellite overpass can be up to 74 % under calm wind conditions (wind speed <1 m s-1) with the best observation sampling (in clear sky and with the satellite swath nearly centered on Paris), while it is systematically smaller than 45 % for the 25th best observation sampling, over a full year of CS simulation. In addition, the uncertainty reductions have a large variation for a narrow range of wind speeds, illustrating the strong impacts of the satellite track position with respect to the target and plume, together with the fraction of “clear sky” that modulates the sampling. In particular, the number of observations sampling the plume on the days when the wind direction is perpendicular to the satellite overpass tends to be lower than the number of days when the wind direction is parallel to the satellite overpass. This is illustrated in Fig. 1 by the uncertainty reductions on the days when the wind speeds are 1.73, 7.6 and 8.1 m s-1, which are lower than on the days with similar wind speeds.

Figure 2a shows the distribution of the number of 08:30–11:30 time windows per clump for which the posterior uncertainty of 3 h mean emissions is smaller than 20 % (this number is called N20) in Exp-NoCor. Clumps with small emission budgets tend to have lower N20 values than those with large budgets, due to the fact that the atmospheric plume generated by small emission clumps is difficult to distinguish from the measurement noise. Typically, N20 is smaller than 5 d for clumps emitting less than 2 MtC yr-1 (like the city of Aswan, Egypt). Conversely, N20 is larger than 10 d for clumps emitting more than 2 MtC yr-1 (like the cities of Manchester, UK; Boston, USA; and Chongqing, China). Note that clumps with emissions larger than 2 MtC, although representing less than 25 % of the total number of clumps, contribute more than 83 % of the total clump emissions. At regional scale (Figs. S4, S5), South America, North America, and Africa tend to have larger N20 values for same bin of clump annual emission than the other regions, while the Middle East and Asia have the lowest ones. In addition, there are large variations and spatial heterogeneity in the N20 values within each emission bins (Fig. S5), which will be further discussed in Sect. 4.

We also show the numbers of 08:30–11:30 time windows per clump being labeled as “well-constrained” when the posterior uncertainty of 3 h mean emission is smaller than other thresholds, e.g., 10 % and 30 % (Fig. 2b). In general, using a posterior uncertainty larger than 20 % as a threshold, we could expect more “well-constrained” cases. But for a given threshold, we still find the number of well-constrained cases increases with the emission budgets.

Figure 3 shows the posterior uncertainty in the clump emissions for the “well constrained” 08:30–11:30 time windows (identified in Exp-NoCor) from different OSSEs. It confirms that in all OSSEs, the posterior uncertainties for clumps with larger emissions are smaller than those with lower emissions. Within a given bin of clump annual emission, the posterior uncertainties from the various OSSEs are very similar, even though they are obtained with different hypotheses regarding the temporal auto-correlation in the prior uncertainty. The interpretation is that, for the inversion of the 3 h emissions before a given satellite overpass, most of the constraint is imposed by the direct satellite observations during this overpass. These observations are independent on different days, and the constraints on different days are not strongly crossed even when errors in the prior estimate are highly correlated in time. However, although small, the impact of temporal auto-correlations in the prior uncertainties can be seen. For example, the posterior uncertainties in ASS (SCS) are systematically smaller than those in AMS (MCS), which confirms that the capability of the inversion system to use the information from observations from previous or subsequent days to reduce the posterior uncertainties increases with the temporal auto-correlations. In SectCS, the posterior uncertainties are smaller than those in MCS and SCS in most regions (Fig. S5), due to the fact that the uncertainty in industrial emissions has a long temporal auto-correlation (τ2=180 d).

In previous sections, we analyzed the uncertainty reduction and the posterior uncertainty for the 3 h emissions that generate the atmospheric plume observed from space at 11:30. We now analyze the potential to monitor the daily emission, relying on the extrapolation of constraints on emissions between 08:30 and 11:30 using temporal auto-correlation of the prior uncertainties in step 2 of the inversion (Sect. 2.7.2). Figure 4 shows the distribution of the number of days when the posterior uncertainties in daily emissions are smaller than 20 % (D20) for the same bins of emission clumps as in the previous section. Similar to the distribution of N20, clumps with small emission budgets tend to have lower D20 values than those with large budgets, due to having smaller signal-to-noise ratios for clumps with smaller emissions. The D20 values also strongly depend on the temporal auto-correlation in the prior uncertainty. When no correlation (Exp-NoCor) or short correlation (MCS) are assumed, D20 remains zero even for the largest clumps, since most of the daily emissions are disconnected from the 3 h emissions that are constrained by the satellite observation and keep on bearing the large prior uncertainties associated with the Exp-NoCor and MCS scenarios. When significant temporal auto-correlations (e.g., in the case of AMS, ASS and SCS) are assumed, the results get better and the posterior uncertainties for the daily emissions become less than 20 % for more than 100 d for clumps emitting more than 5 MtC yr-1. At regional scale (Fig. S6), the distribution of D20 values shows a similar pattern to N20: North America, South America and Africa have larger D20 values than the Middle East and Asia for same bin of clump annual emission. But the distribution D20 values in SectCS have large regional variations, reflecting the regional differences in the share of emissions from different sectors.

We now analyze the results for the annual emissions, allowed again by the derivation of the posterior uncertainty covariance matrix A for individual clumps in step 2 of the inversion, and thus the aggregation of the posterior uncertainties in time. Figure 5 shows the posterior uncertainties in annual emissions from the OSSEs. When we assume that there is no temporal auto-correlations in the prior uncertainties, the uncertainties obtained from the inversions remain very close to the prior uncertainties (30 %) for all emission bins since the information from the few well-constrained 08:30–11:30 time windows within the year is not extrapolated to the huge unobserved fraction of the total annual emission over the year. The benefit of satellite observations becomes apparent when assuming that the prior uncertainties have temporal auto-correlations. Similar to the posterior uncertainties for 3 h emissions during 08:30–11:30, the posterior uncertainties in annual emissions are smaller in the OSSEs where the prior uncertainties have stronger temporal auto-correlation. This indicates that temporal auto-correlations help to extrapolate the information on the emissions from the satellite passes over a given clump to emissions during other hours and days when there are no direct observations. Small clumps tend to have a larger relative posterior uncertainty in annual emissions than large clumps even when temporal error correlations are accounted for. The posterior uncertainties in the annual emissions of large cities with annual emission >5 MtC yr-1 can be constrained to better than 20 % in AMS, SCS and SectCS, and to better than 10 % in ASS. On the other hand, the posterior uncertainties for small emission clumps with annual emissions <0.5 MtC yr-1 are always larger than 15 %, regardless of the temporal auto-correlations in prior uncertainties.