The measurement site is the St. Louis–Midwest supersite which was funded by the United States Environmental Protection Agency (EPA). It is located in East St. Louis, approximately 3 km east of the central business district of St. Louis, on the other side of the Mississippi river in a low-density, mixed-use neighborhood impacted by industrial point sources nearby. Elemental carbon (EC) and organic carbon (OC) concentrations were measured using two Sunset Laboratory semicontinuous ECOC field analyzers. By having two instruments operating in tandem, it was possible to obtain continuous hourly measurements with one instrument in the collection phase while the other instrument was in the analysis phase. The measurements were validated against 24 h samples and are described in detail in . This study is based on 7091 valid data points measured for the duration of 2002. Fig.  shows the location of the measurement site.

Hourly meteorological observations were obtained from Lambert–St. Louis International Airport (KSTL) and St. Louis Downtown Airport (KCPS) in Cahokia, IL from the Integrated Surface Hourly Data available from the National Climatic Data Center. KSTL is across the Mississippi river 24 km northwest of the measurement site, and KCPS is 5 km south of the measurement site on the same side of the river. Meteorological data were also available at the supersite. These data were in agreement with the KCPS data, but the latter was more complete and was therefore selected for the analysis.

The Lake Michigan Air Directors Consortium (LADCO) emissions inventory for 2007 for the Midwest was used as a prior for the inverse model . It is calculated on a 12 km grid with diurnal and monthly profiles and emissions separated by source category for on-road, non-road, marine/aircraft/railroad (MAR), “other”, biogenic and point sources. Point source emissions were specified using 2007 CEMS (continuous emission monitoring systems) data with updated temporal profiles to include adjustments for weekend/weekday emissions while still providing a solid platform for future projections . Mobile emissions were estimated using the MOVES2010a model . Non-road emissions were updated to reflect higher agricultural equipment emissions during the spring and fall season rather than the default of a single summer maximum based on midwest crop calendars and tilling, planting, pesticide application and harvesting cycles . For EC and OC, “other” sources consist mainly of residential wood and waste combustion with smaller contributions from unpaved roads, food preparation and construction.

Figure  shows the emissions for EC in metric tonnes per year (tpy). OC emissions have similar patterns with the following average OC to EC ratios: 0.62 for on-road, 0.64 for non-road, 0.49 for MAR, 6.7 for “other” and 2.5 for point sources. Table  presents the emission totals for the Regional domain shown in Fig. .

Biogenic emissions in were calculated using the Model of Emissions of Gases and Aerosols from Nature (MEGAN) version 2.03a . As an example, Fig.  shows the spatial map of the biogenic emissions of condensable gases, category “CG5” in non-dimensional units. These will be used as a tracer of biogenic emissions in the Eulerian simulations, and the concentrations will be normalized before being included in the inversion algorithm. As will be discussed in Sect. , the inverse results therefore do not represent an estimate of actual biogenic emissions, but rather an estimate of the fraction of OC that could be ascribed to aerosol formation due to these emissions.

In order to have an additional comparison to the LADCO prior emissions and the inverse model results, the 2008 National Emissions Inventory (NEI) version 3 was obtained from the US Environmental Protection Agency. EC and OC emissions were available in speciated files for PM2.5. The on-road emissions in the NEI were calculated using the MOVES2010b model . The data were provided as annual totals by Federal Information Processing Standards (FIPS) codes. These were mapped to the Regional model grid in order to compare NEI emissions with the emissions in the LADCO prior and with the inverse model posterior.

EC and OC have experienced a downward trend in the US, with around 1 to 2 % decreases per year . This means that emissions calculated based on 2002 measurements could be expected to be 5 to 10 % higher than an emissions inventory for 2007. Although emission inventories existed for 2002, it was felt that the considerable improvements and developments that went into the LADCO 2007 inventory meant that this would be a better choice for the prior, and that consequently the 2008 NEI was the most appropriate comparison point to the prior. Nonetheless, the temporal discrepancy should be borne in mind when interpreting the results.

EC and OC emissions from open burning were calculated using the Fire Inventory from NCAR (FINN) version 1 . FINN calculates daily emissions from fires identified by fire counts from the Terra Moderate Resolution Imaging Spectroradiometer (MODIS) fire and thermal anomalies data provided from the official NASA MCD14ML product, Collection 5, version 1 . Land cover and vegetation density needed to calculate the emissions were determined with the MODIS Land Cover Type product and the MODIS Vegetation Continuous Fields product (Collection 3 for 2001) , and fuel loadings from and . Ecosystem-specific emission factors for EC and OC emissions were compiled from existing literature Table 1,. Ratios of OC to EC emission factors range from 4.8 for fires in croplands, to 39 for fires in boreal forests. Daily emission totals were distributed evenly throughout the day as input to CAMx (The Comprehensive Air Quality Model with Extensions) simulations.

In FINN, open burning includes the fires which are detected by Terra MODIS. These are a combination of forest fires, prescribed burns and larger agricultural fires, with a minimum burn area of 1 km2. analyzed the detection rate of MODIS compared to a set of reference fires. The rates were high when both Terra and Aqua were used, but dropped to 60 % in the Great Plains and 39 % in the eastern US when only Terra was used. Because we only have Terra data for 2002, this is an added source of uncertainty in the emission estimates.

Figure  shows the total gridded open burning emissions for 2002 on the Large model domain, and Table  shows the total emissions by sector. The inverse model calculated posterior emissions independently for the following six geographical sectors: local emissions within 100 km of the measurement site followed by the northeast, southeast, southwest, west and northwest as shown in Fig. . The largest emissions are in the southeast and southwest sector.

The meteorological simulations were performed with the Weather Research and Forecasting (WRF) model version 3.5.1 . The North American Regional Reanalysis (NARR) was used for the initial and boundary conditions. The simulations used 3 domains with 27, 9 and 3 km horizontal resolution and 40 vertical levels. Figure  shows a map of the 3 domains, which will be referred to as the Large, the Regional and the Local domains.

The model was run with two-way nesting, with the Yonsei University (YSU) boundary layer scheme, the Kain–Fritsch convective parameterization, the NOAH land surface scheme, the WSM 3-class simple ice microphysics scheme, the Dudhia shortwave scheme and the Rapid Radiation Transfer Model longwave scheme. Individual simulations were performed lasting 162 h, of which the first 42 h were considered spin-up time and the remaining 5 days were used for analysis. The simulations are similar to those described in , where it was shown that the model accurately represents the statistical distribution of temperature, humidity, wind speed and wind direction at the surface see Fig. 3 in .

Particle back trajectories were calculated from the supersite with FLEXPART , using FLEXPART-WRF for a duration of 4 days starting every hour of the year using the WRF simulated wind fields. 1000 particles were released per hour between 0 and 100 m above the ground and were allowed to disperse in three dimensions using the WRF mixing heights and surface friction velocity. The particles were treated as passive tracers with neither wet nor dry deposition. Sensitivity tests presented in found that 1000 particles were sufficient to ensure that the results did not depend on the number of particles for inversions on a regional scale. The particle positions were converted to polar grids to provide a residence time analysis RTA, . This represents the amount of time that an air mass has spent in different grid cells before arriving at the measurement location and can be rescaled to yield the impact that a source in each grid cell would have at the receptor site .

Concentration field analysis CFA, was used as a preliminary method to evaluate possible source regions suggested by the residence time analysis and the hourly concentrations. Concentration field analysis is based on scaling the residence time analysis at each time step with the concentration at the measurement site. The sum over the entire measurement period is then normalized with the residence time analysis. This highlights air flow patterns that are associated with high receptor concentrations. As described in Sect.  below, standard CFA is sensitive to peak concentrations, and so we apply the method to an estimate of the column amount of pollutant. This “column CFA” is shown below to give a more reliable estimate of potential source regions than using CFA based on surface concentrations alone.

The Comprehensive Air Quality Model with Extensions CAMx v6.00, , an Eulerian 3-D grid model, was used to obtain hourly concentrations of EC and OC at the measurement site based on the prior emissions inventory. Dry deposition was calculated using the scheme, and wet deposition using the standard scheme in CAMx. For the LADCO inventory, CAMx was run with two nested domains: the Regional and the Local domains from the WRF simulations (shown in Fig. ), whereas for open burning, CAMx was run with the Large and the Regional domains.

This study is focused on estimating source contributions from specific source groups based on atmospheric transport and therefore does not use the aerosol module in CAMx. Both EC and OC are simulated as passive tracers with wet and dry deposition. This is adequate for EC, and so the inverse model results can be straightforwardly compared to the emissions inventories. In contrast to EC, there is extensive formation of OC in the atmosphere which is not simulated in our model. This means that the inversion will not distinguish between primary and secondary OC, and that results are therefore best interpreted as impacts at the measurement site rather than as emissions at the source location. It also means that we are not able to evaluate the non-linear interactions of different plumes together.

The least squares inverse model used in the present study was developed in , where it was used to evaluate emissions inventories of elemental and reactive mercury. The inverse model estimates emissions that contribute to measured concentrations at a receptor site. This is done by using both the passive transport from prior sources and the contribution of unknown sources using gridded back trajectories.

Inverse models based on back trajectories alone include . This work combines back trajectories with Eulerian simulations, and in this respect is similar to the methods presented in . The purpose of combining the Lagrangian and Eulerian simulations for was to combine global transport of inert species with higher definition impacts from specific locations. In our case, the background levels of EC and OC are very low (see Fig. ), and we expect minimal impacts from sources outside the study area. The purpose of combining Eulerian with Lagrangian simulations is therefore to estimate adjustments to known emission inventories with the Eulerian simulations, and to estimate impacts from unknown area sources in an overlapping domain with the Lagrangian simulations.

Hourly Eulerian simulations with CAMx were performed for the five different source groups in the LADCO inventory: on-road, non-road, MAR, “other” and point sources. Because we are interested in evaluating the temporal profiles of the sources, we carry out separate simulations for emissions during different times of the day and different days of the week. The time slots were selected based on the diurnal profile used in the emissions inventory: 11:00 p.m. to 5:00 a.m., 5:00 to 8:00 a.m., 8:00 a.m. to 2:00 p.m., 2:00 to 6:00 p.m., and 6:00 to 11:00 p.m. Days of the week were split into a weekday group and a group containing Saturdays, Sundays and holidays. As an example, an hourly time series of concentrations was obtained from a CAMx simulation with on-road emissions only between 5:00 to 8:00 a.m. on weekdays. With 5 source groups, 5 time slots and 2 day types, this means that there were 50 CAMx simulations. We are also interested in the annual profile of the emissions, and so we divide the 50 resulting concentration time series into 12 months for a total of 600 input time series into the inverse model. With this method of resolving temporal profiles, individual time series are used for each temporal interval of interest. This is in contrast with who use a Kalman filter to identify seasonal changes in emissions.

The open burning emissions are included in the inversion as six time series simulated by CAMx for the entire year for the six geographic sectors shown in Fig. . We also include a CAMx time series representing impacts from biogenic emissions, as discussed in Sect. .

In addition to the forward Eulerian simulations, we perform backward Lagrangian simulations of particle back trajectories for each hour of the measurement campaign. These are mapped onto a polar grid surrounding the measurement site. The time series from each grid cell gives an estimate of the concentration at the measurement site that would be caused by a constant area emission in that cell. We divided these gridded time series into impacts due to weekdays and weekends, and also into four time slots during the day: 3:00 to 9:00 a.m., 9:00 a.m. to 3:00 p.m., 3:00 to 9:00 p.m., and 9:00 p.m. to 3:00 a.m. These were selected to capture the morning and afternoon rush hours in the middle of two of the slots, and to differentiate the daytime and nighttime emissions between those. The polar grid was chosen to have eighteen 20∘ segments, in 20 radial bands extending to 1000 km from the measurement site. There were therefore 360 time series from 8 time slots, for a total of 2880 time series to be used as input into the inverse model.

The inverse model derives a posterior estimate of emissions based on the Eulerian simulations that used the emissions inventory as a prior. In addition, the inverse model uses the Lagrangian simulations to derive an estimate of sources that may be missing from the inventory. This is done by using the polar grids of residence time analysis that represent the impact that an emission in a given grid cell would have at the measurement site. As all the known sources were already included in the CAMx simulations with the emissions inventory, we use a field of zero prior emissions for the polar grids from the Lagrangian simulations.

By limiting the input of the model to passive tracers and individual time series, we can use a least squares simplification developed in to the Bayesian formulation used in . This hybrid least squares method derives an estimate of the emissions vector x that minimizes the cost function J given by the sum of the observation cost function and the emissions cost function: J=(Hx′-y′)2+α2x′2. Where x′=x-xo is the vector of emission corrections given prior emissions estimates xo. The individual entries in x can take different forms: they can be actual emissions in units of mass per time, or they can be non-dimensional scaling factors. H is the sensitivity matrix that converts emissions parameters x into simulated concentrations. Vector y′=y-Hxo is the residual between the vector of concentration measurements y and the time series produced by the prior emissions estimates Hxo. α is the regularization parameter that balances the two parts of the cost function. In practice, α can be replaced by a vector of parameters s that scales each term in x within the L2 norm. In this way, the method was shown to be equivalent to a Bayesian derivation when diagonal error covariance matrices are used . In these cases, the regularization parameter is equal to the ratio of the uncertainty of the measurements to the uncertainty of the emissions parameter, as described in .

The columns of H contain the 606 input time series from the forward Eulerian simulations (in the same units as the measurements) as well as the 2880 time series from the back-trajectory grids in units relating area emissions to measurement concentrations, see, all of which are hourly time series for the whole of 2002. The rows of H correspond to the impact of the different sources for each of the 7091 h with valid data, which are contained in vector y. The vector x contains (606 + 2880) entries which yield the posterior emissions estimate for the source groups and for the gridded area sources represented by the back trajectories. For the CAMx time series, the entries in x are scaling factors on the LADCO emissions that went into the CAMx simulations. For the FLEXPART polar grids, the entries in x represent emissions.

The system of equations can be solved with a single step of least squares using J=s⋅(H′′x′-y′′)2, where H′′=(H,I) and y′′=(y′,xzero)T are the augmented versions of H and y′. I is the identity matrix the size of x, and xzero is a vector of zero values. Hence, the first part of H′′ and of y′′ correspond to the observation cost function and the second part to the emissions vector cost function. H′′ has dimensions of (7091 + 3486) by (3486), and y′′ has dimensions of (7091 + 3486). The vector s contains scaling factors on the parts of the cost function: these are taken to be unit values for the observation cost function and contain the regularization parameter α for the emissions cost function. A strength of the method is that boundaries can be straightforwardly applied to the vector x′ during the least squares solution to prevent nonphysical negative emissions.

An iteratively reweighted least squares (IRLS) scheme is used to reduce the sensitivity of the method to outliers in the data: after solving for x, observation times that have a residual larger than 3 times the standard deviation of the residual values are removed from the analysis. This is performed iteratively to converge on a stable set of times to include in the inversion. EC and OC simulations were evaluated separately using the inverse model.

In a Bayesian framework, uncertainty estimates are required to obtain the error covariance matrices on the two parts of the cost function. In the absence of detailed prior information, recommends using empirical Bayes methods where the prior information is obtained from the data set itself. If this is insufficient, then using frequentist methods is recommended as a check on the Bayesian simulations. In this context, the current method can be understood as a frequentist method where the inversion is performed multiple times using bootstrapping, and where the regularization parameter is obtained from the data itself.

The inverse model therefore does not need prior error estimates, but rather relies on an optimization routine to determine the values of the regularization parameters in the vector s that minimize the total error following . While in principle we can ascribe different values for each entry in the sensitivity matrix, we decided to use common values by source groups. The values of s were therefore determined separately for the emissions inventory sources, for the open burning sources and for the emissions based on back trajectories. The regularization parameter for the gridded emissions is scaled by the cell area to account for the increase in uncertainty with increasing distance from the measurement site. This yields values for the EC inversion of 0.025 for gridded emissions, 1 for emissions inventory sources and 0.03 for open burning. The corresponding parameters for OC are 0.015, 1, 0.25 and an additional parameter of 0.5 for the biogenic contribution. Taking the uncertainty of the measurements to be 1 µgm-3, this corresponds to an uncertainty of 100 % for the emissions inventory sources, and to an uncertainty factor for open burning of 33 for EC and 4 for OC.

review different methods to enforce positive emissions in the inversion, and show that some of these may bias the results. In the inverse model, the inversion is performed by the function lsqlin in Matlab. This uses a trust-region reflective Newton method to solve the least squares problem and enforce positive constraints on the results. This does not prevent the model from estimating uncertainties, as we derive a regularization parameter from the data and obtain the uncertainty estimates using bootstrapping.

We estimate uncertainties in the inverse model by two different methods. The first is to use expert judgment to determine an uncertainty on the measurements (y) and on the model sensitivities (H) and to use Monte Carlo error propagation. We perform 100 realizations of the inversion with randomized scaling of the entries in y and H in order to estimate the uncertainties in x. In practice, we assume that entries in y vary by plus or minus 20 % and those in H by plus or minus 50 %.

An alternative method is to assume that by randomly sampling the data included in the inversion we are randomly sampling both the measurement errors and the simulation errors at the same time. This can be done with the bootstrap algorithm. Although measurement errors are assumed to be uncorrelated in time, meteorological events vary on the order of hours to days. In order to obtain samples that have different meteorological conditions, we perform block-bootstrapping with a block length of 24 h. We therefore perform 100 inversions with random selection with replacement of the days included in the analysis. In this way, the bootstrapping yields an estimate of the combined uncertainty due to measurement errors and due to transport modeling errors.

In outline, we first perform the optimization of the regularization parameters without bootstrapping for each set of sources in turn: for the RTA grids, for the LADCO emissions, for the open burning emissions and for the biogenic tracer. This is repeated to make sure the values are stable. We then use the set of regularization parameters to obtain inverse results with the full data set, and for 100 realizations with block-bootstrapping.

Before presenting the results of the inverse model, this section presents the results of analyzing wind roses and back trajectories from the measurement site. Winds come from all directions at the Lambert–St. Louis International Airport with a predominance for westerly flow, as shown in the wind roses in Fig. . Nearer to the supersite at the Downtown St. Louis airport, however, there is a clear peak of southeasterly flow and a much larger proportion of calm hours (17 % compared with 7 % at KSTL). As a first cut analysis, Fig.  shows the wind rose for the hours in the top 10 percentile of EC concentrations. 54 % of these have calm winds that occur between midnight and 9:00 a.m. As for the non-calm hours, they are most frequently from the southeast. This suggests that high EC concentrations are associated with calm conditions and hence with local sources. It also suggests that significant sources could be found southeast of the site, which is at odds with known inventories.

Figure  shows the probability density function for both the measurements and the simulations at KCPS. The distributions are very similar, and all variables passed the Kolmogorov–Smirnov test to much lower than the 1 % significance level, showing that the model does not suffer from significant systematic biases. The auto-correlation times of the meteorological variables also shows that by using block-bootstrapping with 24 h intervals we will be sampling independent events.

We use residence time analysis to display the spatial pattern of wind transport to the measurement site over the course of 2002, see Fig. . On the Regional domain, this shows that air masses from all directions impact the site but that there is a predominant signature from the southwest, in agreement with the wind rose at KSTL. On the Local domain, we see again impacts from all directions, but in addition there is a very clear river valley effect. Simulated particles from the south follow the Mississippi river going north towards the measurement site.

Concentration field analysis of EC and OC (Fig. ) shows that peak concentrations are associated with transport from the southeast. This is in agreement with the pollution rose shown in Fig.  but is puzzling given that southern Illinois does not stand out as a large source region in Fig. . To resolve this conundrum, we consider the influence of mixing heights and stable atmospheric conditions at the supersite: the last rose in Fig.  shows the wind direction for hours with the lowest 10 percentile of mixing heights in the WRF simulations. This shows a picture similar to the EC pollution rose with nearly half of the hours experiencing calm winds, and the remaining having winds predominantly from the southeast. found an episode where high levels of cadmium, antimony, barium and selenium were associated with a very clear southeast signature. This could be due to a power station 53 km away in that direction, although simulations with CAMx did not support such high impacts from this source. Further analysis found that peak EC concentrations are associated with these hours with very stable vertical mixing conditions which themselves are associated with weak southeasterly transport. They appear to be linked to occurrences of the low-level jet. This suggests that micrometeorology needs to be taken into account when analyzing high pollution events in St. Louis.

Wind rose analysis and CFA are sensitive to peak concentrations occurring during situations with very shallow boundary layers and so we need to expand the methods to be more sensitive to the amount of pollutant rather than to the peak concentration. This can be done by calculating a “column CFA”: CFA is carried out with an estimate of the total column of EC rather than with the surface concentration of EC. To do this, we assume that EC and OC are mainly in the planetary boundary layer and that concentrations are well mixed throughout. The column amount is obtained by multiplying the surface concentration by the height of the boundary layer. Since we do not have measurements of the mixing height, we use simulated values from the WRF model. The two graphs on the right in Fig.  show the results of the Column CFA for EC and OC. For EC, we can now see a clear signature from the St. Louis metropolitan area as well as a smaller signature from the Illinois side of the urban zone. For OC, the St. Louis metropolitan area shows up but there is a stronger signal of general impacts from both the southeast and the southwest, which is consistent with regional atmospheric formation of OC compared with local transport of EC.

Figure  shows the EC and OC time series of the measurements and of the inverse model results. The time series from the inverse model are much improved compared with those simulated using the emissions prior, as shown by the statistical measures in Table . For the full time series, Pearson's correlation coefficient squared (r2) increases from around 0.1 to above 0.4. As described above, the inverse model uses Iteratively Reweighted Least Squares to reduce the impact of outliers on the results. The r2 statistics are also shown for this subset points, with an agreement of 0.53 for the EC inverse time series and of 0.56 for the OC time series.

The inverse model decomposes the measurement time series as the sum of the contributions from different source groups. If these are sufficiently well separated spatially and temporally it is possible to estimate the contribution of individual source groups to the average concentration at the site. In our current case, there is a certain level of overlap between the different source categories, as can be seen in Fig. . The closest time series are the impacts of on-road and those of “other” sources, with a correlation coefficient (r) of 0.82, and with non-road sources with r of 0.65. By impacts, we mean the surface concentration of EC or OC at the measurement site that are due to transport of particular emissions to the site. The most distinct time series are the point sources, which have an r of 0.5 with MAR emissions but small or negative r with the other categories. In practice, block-bootstrapping was used to determine uncertainties in the inverse model results, and these were found to be robust, as will be discussed below.

Figure  shows the contributions of different source groups to average EC and OC concentrations at the measurement site for both the prior and the posterior emissions inventory. The prior inventory overestimates the average EC concentration at the measurement site by 13 % and suggests that on-road emissions account for 36 % of the pollutant load, non-road for 20 %, MAR for 13.9 %, “other” for 23 %, point sources for 6 % and open burning for 1 %. The posterior emissions underestimate average impacts by 10 % (“Missing” on the graph), and attribute 33 % to area emissions from the polar RTA grids. This leaves on-road emissions with 13 %, non-road with 16 %, MAR with 10 %, “other” with 11 %, point sources with 5 % and open burning with 4 %.

Whereas EC behaves as a tracer species from source to receptor, OC is due to the combination of transport from source to receptor and formation in the atmosphere during transport. Because this paper only considers transport, we expect the model results to underestimate average concentrations: the prior time series represents 60 % of the average OC concentration. As discussed in Sect. , this suggests that 40 % of OC at the measurement site is from secondary formation, in line with the estimate provided in . The largest contributor in the prior is the “other” category with 68 % of simulated impacts, followed by point sources with 12 %, on-road with 9 %, non-road with 6 %, and MAR and open burning with 3 % each. The posterior accounts for 88 % of the average OC levels, mainly by reducing the impacts of the source groups and using the RTA grids to represent 46 % of the simulated impacts. Simulated impacts from open burning are increased in the posterior so that they make up 5 % of the total OC.

Normalized time series of biogenic precursor concentrations were included in the analysis. Because the units are non-dimensional, the results from the inverse model give an indication of the fraction of EC or OC that correlates with these emissions, without giving an estimate of the emissions themselves. As expected, none of the biogenic precursors contributed to the EC time series in the inversion, and these were therefore left out of the EC inversions. For OC, we tested different biogenic components and found that condensable gases category 5 “CG5” yielded the best inverse time series of OC compared to the measured time series. The model was therefore run just with this species as an input. The model estimated that 4 % of simulated OC at the measurement site is associated with emissions of CG5.

The biogenic tracer serves to highlight that the posterior estimate does not differentiate between direct emissions at the source and chemical formation inside a plume associated with those direct emissions. The biogenic emissions are in the gas phase, and the model obtains an estimate of OC concentrations that results from them. The same applies for the individual source categories. For example the 19 % of simulated impacts from the “other” category are the sum of both direct emissions and chemical formation resulting from those emissions. A finer grained study using an aerosol module would be required to deconvolve these two processes.

We used both Monte Carlo error propagation and bootstrapping to estimate the uncertainties in the emissions estimates. Figure  shows the histogram of total emissions for each of the main categories in the inversion, along with correlation scattergrams of the results for the bootstrapped simulations for EC. The standard deviation of the contributions is between 3 and 5 % of the mean contribution for all emission categories except for open burning where it is 20 %. There is little correlation in the emissions estimates from the different source groups. The highest r2 is 0.22 for realizations of the on-road and “other” emissions. Overall this suggests that our results are not excessively impacted by cross-correlation terms.

The results of the Monte Carlo error propagation are included in the Supplement. The uncertainties vary between 1.5 and 3 % except for open burning where they are 6 %. These are noticeably lower than the bootstrapping estimates as well as what we expect from knowing about emission inventories and from the values of the regularization parameters that were determined from the inversion themselves. These suggest that using block-bootstrapping provides a better estimate of the uncertainties.

The results for OC are included in the Supplement. The bootstrapped standard deviations are between 5 and 10 % of the mean contributions for all emission categories except for open burning where they are 18 %. This suggests that the emissions estimates are robust with respect to uncertainties in the model inputs.

As described in Sect. , we performed the inversion using separate time series for five different time periods during the day, for weekdays and weekends, and for each month of the year. This led to 5×2×12 entries in the inverse algorithm for each of 5 source types. We now present the monthly variation and the diurnal variation for emissions of EC and OC for each of the source types for weekdays and for weekends (which include Saturdays, Sundays and holidays, SSH).

Section  showed that using emissions from FINN as the prior for CAMx simulations of open burning led to impacts of 1 % of EC and 3 % of OC. The posterior impacts were increased to 4 % for EC and 5 % for OC. Table  shows the emission totals by geographic sector in metric tonnes per year for the prior and for the posterior. For the Local sector (within 100 km of the receptor), the northeast sector and the southeast sector, the inverse model increases the emissions by a factor of around 30 for EC and around 20 for OC. Emissions from the southwest sector are increased by a factor of 3 for EC and by a factor of 2 for OC. The open burning emissions from the west were kept at a similar level in the posterior as in the prior. The emissions from the northwest did not match the data and were set to 0 in the posterior by the inversion.

Table  also shows the posterior impact fractions by sector. The largest contributions are 1.4 % of EC and 2.5 % of OC from the southeast sector, followed by the southwest and the west sector. Local fires account for 0.7 % of EC and 0.5 % of OC in the posterior.

As discussed in Sect. , the emissions in FINN are based only on the Terra MODIS sensor, as the Aqua satellite was not yet in orbit in 2002. This means that the uncertainties in these emissions are greater than those following the launch of Aqua where there is twice as much satellite data available for fire detection . In addition to missing fires, there are uncertainties in the estimates of area burned and of the type and amount of vegetation burned. As shown in Fig. , uncertainty estimates based on bootstrapping are largest for open burning, with 20 %. However, adjustment factors of 20 to 30 suggest either that the uncertainties are underestimated, or that the inversion of these emissions are underconstrained. Overall, these results suggest that future work with more surface measurements and emissions estimates from more recent satellite sensors are needed to improve the inverse estimates, but that nonetheless emission factors in FINN should be revised upwards.

The inverse model combines emission estimates using Eulerian (CAMx) and Lagrangian (FLEXPART-WRF) simulations. Polar grids of residence time analysis calculated using back trajectories are used to estimate emission sources that could be missing in the LADCO emissions inventory. The polar gridded emissions have zero prior and represent a way of decomposing the residual between the CAMx posterior and the measurements into a spatial emission signal. The inverse model includes separate grids for 3:00 to 9:00 a.m., 9:00 a.m. to 3:00 p.m., 3:00 to 9:00 p.m. and 9:00 p.m. to 3:00 a.m., as well as for weekdays and weekends, for a total of eight grids.

Note that the FLEXPART-WRF simulations do not include deposition, and that secondary OC formation is not included either. Both of these limitations would impact the estimation of actual emission amounts from the inverse model. In this section, we therefore report only impacts of different source regions on concentrations at the measurement site, which are not affected by deposition and include estimated impacts of both primary emissions and in-plume secondary formation. As will be discussed in Sect. , deposition is estimated to account for a 4 % loss of EC, and secondary formation is estimated to account for around 40 % of OC.

Figure  shows the sum of impacts from the eight grids for both EC and OC. As shown in Fig. , these account for 33 % of the EC posterior time series and 46 % of the OC posterior time series. The main signal is from the south, and especially the southwest for both EC and OC, indicating that these could be areas to be explored for updating the spatial distribution of emissions.

Figure  shows the total contribution to the average concentration for EC and OC for each of the RTA grids. For EC, the contribution varies from a minimum of 0.05 ngm-3 to just above 0.25 ngm-3. The contribution from the early morning to afternoon (3:00 a.m. to 3:00 p.m.) are lower than those for the late afternoon and nighttime (3:00 p.m. to 3:00 a.m.). The weekdays and weekends have a similar trends, but the diurnal variation is more pronounced on weekends. For OC, there is a similar pattern with lower contributions from 3:00 a.m. to 3:00 p.m., of around 0.8 ngm-3 rising to around 1.5 ngm-3 in the nighttime. Weekdays and weekends RTA impacts are more similar for OC than they are for EC.

In this section we compare the emissions in metric tonnes per year of the different source types from the inverse model with the NEI 2008 and the LADCO inventory. Table  and Fig.  show the annual total emissions for the St. Louis Regional domain for the 2008 National Emissions Inventory, the 2007 LADCO inventory used as a prior, and for the posterior.

Overall, the LADCO inventory is slightly larger than the NEI for both EC and OC. For EC, the on-road emissions are 50 % larger, and the MAR emissions are 25 % larger while the remaining categories are similar. For OC, the largest category by far in both inventories are the “other” sources which are 17 % higher in the LADCO inventory. These include residential wood and waste combustion, non-vehicle road emissions and food cooking (estimates of agricultural burning are high in the NEI but low in the LADCO inventory). OC emissions from on-road, non-road, MAR and point sources are all increased by up to a factor of 2 in the LADCO inventory compared with the NEI.

The posterior emissions are calculated from the model as departures from the LADCO prior. As discussed in Sect. , the simulated EC concentrations were too high at the site, and so the inverse model has lower emissions for all categories. The EC emission estimates from both on-road sources and “other” sources are reduced by 50 % in the prior, whereas the remaining categories are only slightly reduced. For OC, there is only a slight reduction in the total emissions with a small shift in emissions from the “other” category into non-road emissions. This suggests that the inverse results are in agreement with the inventory, bearing in mind that the model does not distinguish between primary and secondary OC. As around 40 % of OC is estimated to be secondary (see Sect. ), this is a significant source of uncertainty. Nevertheless, comparison with Fig.  would suggest that the secondary OC is represented in the model more by the polar grid emissions or as missing carbon rather than as adjustments to the known sources.

Also shown in Fig.  are emissions for three time periods during the year that correspond to a natural grouping in the data: January to April, May to August and September to December. The emissions rates are annualized by multiplying the emissions in tonnes per 4 months by 3 in order to have emissions in tonnes per year. This yields the annual emission rate that would be obtained if the emissions of the 4 months continue for an entire year. Compared with the LADCO inventory, the emissions estimates are low for January–April, high for May–August and similar for September–December. This shows that there is uncertainty in the model results that depends on the time of year and that in particular simulations are in greater disagreement with the inventories for January to April. Although the variation exists for both EC and OC, it is stronger for OC because May–August is when there is the most secondary OC formation (see Sect. ). At this stage it is not possible to say what part of this is due to limitations in the inventories, what part to measurements and especially what part due to modeling errors. Further research with more sites and longer time series would be able to better constrain the estimates.