We use observations of column-averaged dry-air mole fraction (denoted Xgas) to tie model abundances to fluxes. Column averages are calculated by dividing the retrieved amount of the gas of interest (molecules cm-2) by the retrieved total column of dry air (molecules cm-2). Xgas values are less sensitive to changes in surface pressure and water vapor than total column amounts in units of molecules per square centimeter .

Data are obtained from the TCCON and OCO-2. We use TCCON data from the California Institute of Technology (Caltech) site in Pasadena, California , and from the NASA Armstrong Flight Research Center (AFRC) site near Lancaster, California . Values of XCO2, XCO, and XCH4 were generated using the operational GGG2014 algorithm . The Caltech site (lat 34.136, long -118.127, 240 m a.s.l.) is located in an urban environment within the SoCAB. As the name implies, the SoCAB is a basin surrounded by mountains, except towards the southwest, which boarders the Pacific Ocean. AFRC (lat 34.960, long -117.881, 700 m a.s.l.) is located outside the basin ∼100 km to the north in a much more sparsely populated area. Because of the lower population density, the AFRC is often considered a “background” site. However, depending on airflow patterns, recent emissions from the SoCAB may be observed at the AFRC so we use the term background loosely to indicate where lower concentrations are typically observed. Coincident data from both sites are available from July 2013 to August 2016, after which the AFRC instrument was relocated. In total, there are 5355 paired hourly averaged observations on 783 days.

OCO-2 data are available starting September 2014 when the instrument began its nominal operational mission . Here, we use XCO2 data generated using the NASA Atmospheric CO2 Observations from Space (ACOS) version 8r algorithm . We also do a partial analysis on v7r data for comparison with past studies that used these data with a focus on the SoCAB . Because OCO-2 is in a sun-synchronous orbit with an equatorial crossing time of around 13:00 local solar time, all observations are in the early afternoon. OCO-2 has 8 longitudinal pixels, with a footprint of ∼3 km2 each. To reduce over-weighting target mode observations, OCO-2 data are gridded to 0.01∘×0.01∘. Before filtering there are 6098 pre-averaged OCO-2 observations on 29 different overpass days when the AFRC TCCON site also collected background observations.

In Appendix  we describe filtering, background subtraction, boundary conditions, and our accounting for averaging kernels. In short, we determine enhancements of various gases (ΔXgas) by finding the difference between observations within the basin (either the Caltech TCCON or OCO-2) compared with the AFRC TCCON site.

Our flux estimate involves scaling the a priori spatial inventory, or subregions of the prior up or down to reduce the measurement–model mismatch. More important than the total prior absolute flux is the distribution of sources. EDGAR (Emissions Database for Global Atmospheric Research; ) and FFDAS v2.0 (Fossil Fuel Data Assimilation System; ) are available globally at a 0.1∘ resolution. We use the year 2016 version of the Open-source Data Inventory for Anthropogenic CO2 (ODIAC2016), which is available globally at a resolution of 30 arcsec from 2000 to 2015 . We also compare total SoCAB emissions from the 2015 version of ODIAC (ODIAC2015), which is based on a projection of the Carbon Dioxide Information Analysis Center (CDIAC) country total emissions. ODIAC has a monthly variation and compared to the annual average seasonal flux ratios are 1.06 (DJF), 0.97 (MAM), 1.00 (JJA), and 0.97 (SON). We assume that 2015 emissions are identical to those in 2016. A generic temporal hourly scaling factor product (TIMES – Temporal Improvements for Modeling Emissions by Scaling) available at a 0.25∘×0.25∘ resolution can be applied to spatial inventories such as ODIAC to improve temporal emissions . However, TIMES has a single peak for midday emissions, which is inconsistent with morning and afternoon rush hour periods in the SoCAB. We instead use the Hestia-LA v1.0 weekly profile reported by Hedelius et al. (2017a, Fig. 2 therein), which has both morning and afternoon rush hour peaks. We use ODIAC over the domain longitude -121.5 to -114.5 and latitude 30.5 to 37.5. Hestia-LA v2.5 is expected to be an even more accurate spatiotemporal inventory for the SoCAB the Hestia fossil fuel CO2 emissions data product for the Los Angeles Basin, submitted to Earth System Science Data. As a sensitivity test we also derive a flux based on Hestia-LA 2.5 over the region in which it is available, and Vulcan 3.0 is used for the rest of the area within the US. These were gridded to the same scale as the ODIAC.

This same prior is used for CO, but total emissions are 1 % of CO2 emissions on a molar basis (0.6 % of mass) based on the results of . Figure  shows the ODIAC2016 prior for 1 month.

A detailed CH4 inventory is also available for the SoCAB, which we do not use because it would be difficult to scale . For the US the Harvard–EPA (Environmental Protection Agency) inventory is already available at 0.1∘×0.1∘ resolution , and globally the EDGAR inventory is available at 0.1∘×0.1∘ resolution . We make our own 30 arcsec × 30 arcsec methane prior using landfills, nightlights, expected total emissions, and the US Harvard–EPA inventory shown in Fig. . Due to a lack of information outside the US on point sources, such as landfills, our methane prior is also not scalable beyond a national level.

For our methane prior we first distribute emissions from landfills as point sources (available 2010–2015, https://ghgdata.epa.gov/ghgp/main.do, last access: 17 August 2017) and use 2015 emissions for 2016. Emissions from the Puente Hills landfill were doubled because the EPA estimate (average of 13.6 Gg CH4 yr-1) is low compared to previous estimates of 34 Gg CH4 yr-1 . After doubling Puente Hills emissions, EPA total SoCAB (144 Gg CH4 yr-1) and Olinda Alpha (13.5 Gg CH4 yr-1) landfill emissions are similar enough to other studies 164 Gg CH4 yr-1 and 12.5 Gg CH4 yr-1, respectively; that we do not double emissions from other landfills in the SoCAB. Chino dairy emissions were added in as a ∼0.1∘×0.1∘ source . Outside of the SoCAB, CH4 manure and enteric fermentation were added from the 0.1∘×0.1∘ Harvard–EPA inventory . SoCAB emissions are assumed to sum to 400 Gg CH4 yr-1 based on the work of , and the rest of the emissions were distributed based on population, which was assumed to correspond with the January 2017 Suomi NPP nightlights (15 arcsec). An average monthly trend was included based on results of , and emissions were assumed to be constant on a monthly timescale. Because the Aliso Canyon leak effectively doubled the SoCAB CH4 emissions for its duration from 23 October 2015 to 11 February 2016 , it was also added as a point source.

We use various publicly available statistics to get a sense of annual CO2 emissions from the SoCAB. Literature estimates range from 99 Tg CO2 yr-1 Vulcan; to 211 Tg CO2 yr-1 EDGAR v4.0; as reported by. Table lists statistics for the SoCAB. We assume the nonresidential natural gas (NG) use is for industry or power already accounted for in the EPA inventory. Because most of the food consumed in the SoCAB is grown outside the basin, such as in the Midwestern US and Central Valley (CV), there is a CO2 return flux to the croplands from both human respiration and food waste. In the US, 60 million metric tons (MMT) of food are lost annually at the retail and consumer levels compared with 129 MMT consumed , roughly one-third of all food calories (not counting inedible food-related biomass). Presumably, most food waste decomposition would be accounted for in EPA landfill emissions. However, CO2 emissions from food waste could be underestimated if food waste is composted, if there were unaccounted for methanotrophs, or if aerobic respiration is significantly underestimated (e.g., from rapid decomposition while still exposed to oxygen), which would decrease the CH4 : CO2 emission ratio commonly assumed to be unity for managed landfills on a per mole basis . Thus, we add 30 % to human respiration emissions of 917 g CO2 d-1 person-1 for food waste losses. We assume the flux from vegetation is balanced (i.e., no net change in plant biomass or soil carbon) within the basin. This choice is because of uncertainty as to whether there is a net uptake of CO2 by the biosphere in the SoCAB or if the excess CO2 in the atmosphere from the biosphere is due to more respiration than photosynthetic uptake. We estimate the uncertainty due to the biosphere is less than ±10 %. Based on these various statistics we estimate a BU net flux of the order of 110 Tg CO2 yr-1 from the SoCAB.

A dynamical model is needed in conjunction with the a priori flux estimates to generate forward model Xgas enhancements. Our model uses Lagrangian trajectories driven by existing archived forecast or reanalysis datasets. An advantage of archived model data is there is no need to run a Eulerian model first, and they are more accessible to a broader community. However, taking existing results without model evaluation may propagate hidden errors and biases, which could influence flux results. Archived data usually have coarser spatiotemporal resolutions than custom models and cover larger domains than the area of interest. Custom runs allow models to be parameterized differently and nudged to reduce the measurement–model mismatch for the regions of interest.

We use the North American Mesoscale Forecast System (NAM) at 12 km resolution (3 h temporal) from the NOAA data archive as the primary model source. NAM is run with a non-hydrostatic version of the WRF at its core with a Mellor–Yamada–Janjić planetary boundary layer (PBL) scheme . Estimates of model error are described in Appendix . Though NAM data are only available over North America, other archived models are available at lower resolution with global coverage (e.g., the Global Data Assimilation System (GDAS) 0.5 ∘, 3 h product). The NOAA ESRL recently began publicly releasing 3 km, 1 h archived data from the High-Resolution Rapid Refresh (HRRR) model that covers the US . This product holds the potential to improve flux estimates at smaller scales.

We use HYSPLIT-4 Hybrid Single Particle Lagrangian Integrated Trajectory-4; with the three archived NOAA data products described above. Our base method is to use mean 48 h back trajectories with NAM 12 km for the lowest 20 % of the atmosphere, which we assume is the only part of the atmosphere enhanced with local emissions at the measurement site. Trajectories are equally spaced in pressure every 0.3 % of the column. By comparison, the GDAS model takes 0.71±0.18 (1σ) times as long to run, and the HRRR model takes 33.3±7.1 (1σ) times as long. Because HRRR takes substantially longer, we only run it for a subset of months – July and October 2015 and January and April 2016. Other studies e.g., used multiple particles released at each level. We assume that over the multiyear time series the ensemble of mean trajectories is, on average, representative of the upwind influences on the receptor sites without the additional turbulence term.

Figure  shows back trajectories for one layer that end at the observation sites at 14:00 (UTC-7). Trajectories from multiple vertical levels are combined to determine residence times or footprints as described in Appendix . There are three major origins for air at the Caltech site. The primary source is from over the ocean and over downtown Los Angeles (southwest). The second major source is from the Mojave Desert (northeast), and the third source is from the CV (northwest; see Fig.  in Appendix).

Others interested in undertaking similar studies may also consider using the recently developed X-STILT (X-Stochastic Time-Inverted Lagrangian Transport) model to obtain footprints for column observations .

Different schemes can be applied to reduce the measured–model mismatch. One of the simplest is to find the ratio between the average enhancements in the observations and the forward model and then to scale the prior based on this ratio. Bayesian inversions are more complex, but can also improve information on the spatial distribution and intensity of fluxes e.g.,; they can be solved by analytical or adjoint methods . Different cost functions can be used, which might change the results. Here we test and compare three different methods. The first is a Kalman filter (described in Appendix ), which is computationally cheap but has only 1 degree of freedom. For scaling retrievals, using too few degrees of freedom can cause the results to be heavily weighted by the largest model results relative to the observations (Appendix ). We also use Bayesian inversions based on the methods of (described in Appendix ). One Bayesian inversion is based on a nonlinear forward model with 40 different scaling factors (Eq.  in Appendix), and the other is a linear forward model with up to nearly 35 000 scaling factors (Eq. ), though only a fraction (<1000) of these are used. Because of potential bias in the first two methods, we focus on the linear forward model. Uncertainty estimates are stated for the linear forward model while disregarding the other methods.

In summary, we have four sets of observations of Xgas differences: Caltech TCCON – AFRC TCCON (CO2, CH4, and CO) and OCO-2 – AFRC TCCON (CO2). We use one gridded spatiotemporal inventory for both CO2 and CO (ODIAC2016, with a weekly pattern for hourly emissions), one for sensitivity tests for CO2 (Hestia-LA v2.5) and one gridded spatiotemporal inventory for CH4 (Sect. ). HYSPLIT is run with three dynamical models for the Caltech TCCON – AFRC TCCON differences (GDAS 0.5∘, NAM 12 km, and HRRR 3 km for a subset) and is run with NAM 12 km for the OCO-2 – AFRC TCCON differences. Three different inversion techniques are used, including a Bayesian inversion with a linear forward model, a Bayesian inversion with a nonlinear forward model, and a Kalman filter. Unless specified, values reported are from the Caltech TCCON – AFRC TCCON difference with the NAM 12 km model and the Bayesian inversion with the linear forward model.

Our flux estimate of CO2 using the TCCON sites and linear model (Eq. ) is 104±26 TgCO2 yr-1. An error assessment is described in Sect. . This estimate is shown, along with estimates from past studies, in Fig. . Our estimate is lower than those from Vulcan , Hestia-LA v1.0, Hestia v2.5, and the California Air Resources Board (CARB) 2017. Our result is also slightly lower than those of , who estimated emissions by comparing OCO-2 observations with forward model results from a WRF-Chem model. Our result differs significantly from previous TD estimates from aircraft flights, EDGAR v4.0 as reported by, and CARB 2011. Between 2011 and 2012 CARB changed how bunker fuels and aircraft emissions were reported for the state, which caused a significant decrease in reported emissions. Our posterior estimate is similar to that of EDGAR v4.2 and ODIAC2016, which is slightly less than ODIAC2015. The ODIAC2016 is based on disaggregation of CDIAC national total emissions. Thus, unlike locally developed emission inventories the interannual variations in subnational emissions are driven by the national emission trends. ODIAC could be low from incorrectly distributing too much of the emissions to rural areas due to blooming effects . Blooming effects refer to the tendency for nightlights to exaggerate settlement areas compared with actual extent due to coarse gridded spatial resolution and indirect or nonelectrical light.

Most of the estimates from previous studies include only emissions from fossil fuel use. We have not separately accounted for biospheric uptake (emissions) in the model, and if it is significant, the anthropogenic flux would be larger (smaller) than our net estimate. In the GEOS-Chem model described by the nearby ocean is a neutral to weak sink, likely from biological activity.

OCO-2 provides better spatial coverage than TCCON (Fig. ), and the orbit tracks can change longitudinally with season or when the spacecraft moves for collision avoidance. However, observations only occur at the same local solar time, and are days to weeks apart. The estimate using OCO-2 data is slightly larger at 120±30 TgCO2 yr-1, which is in better agreement with the results of . This value varies by up to 12 TgCO2 yr-1 depending on filtering methods (e.g., warn levels, Appendix ).

Using the same methodology we estimate a CH4 flux of 360±90 Gg CH4 yr-1. This is less than the estimate by but similar to estimates from and CARB (Fig. ). CARB-based CH4 fluxes for just the SoCAB were estimated by subtracting agriculture and forest emissions (53 %–61 % of total depending on version and year) and out-of-state electricity generation (0 %–0.1 %). The remaining flux was scaled by 42 % based on the population of the SoCAB, and 5 % of the agriculture and forestry emissions were added back in. Our estimate is slightly lower than previous estimates of CH4 fluxes using in situ (tower and aircraft) data .

We also estimate a CO flux of 487±122 Gg CO yr-1. This is significantly less than the estimates by of 1400±300 Gg CO yr-1 from August 2007 to June 2008 and the estimate of 1440±110 Gg CO yr-1 by for summer 2010. used a tracer–tracer relationship in which the assumed CO2 was likely too large (191 TgCO2 yr-1). When their results are scaled down based on our posterior CO2 fluxes (104–120 TgCO2 yr-1), the CO flux is 750–880 Gg CO yr-1, which is in better agreement with the CARB inventory. The CARB CO inventories, specific to the SoCAB, have decreasing CO emissions; part of the difference could be from different observation periods. CARB2017 emissions are 581 Gg CO yr-1 for 2015.

For a single estimate of the SoCAB flux, we have a sufficiently large sample that random uncertainty is small. This is supported by a bootstrap analysis in which we select a random subset of data equal in size to the original n=200 times . The random uncertainty estimate is 4 Tg CO2 yr-1 (2σ), or about 4 %. Persistent biases from a priori flux uncertainty, model errors, observation biases including boundary conditions, and poorly chosen initial values are more detrimental to our flux estimate.

Several variables (xa, Sϵ, Sa) need initial values (see Appendix ), and how these are chosen can affect the final flux calculated. We evaluate four sensitivity tests (Fig. ). For the first test, we filter out data for which the observations differ from the model above a threshold. We scale from the starting factor of 10× (Appendix ). We also adjust values of xa, Sϵ, and Sa by factors of 2-10 to 210. These results show the overall flux generally has low sensitivity to scaling Sϵ and Sa but has some sensitivity when filtering more data and about a 30 % sensitivity to the scaling of xa. The interannual variability, which we expect is less than about 25 %, increases for large Sa. Increasing Sa increases r and the degrees of freedom for the signal (dofs) with only a small effect on the overall flux but also increases the interannual range. Decreasing Sϵ increases r and dofs, but it also increases χ2 and the interannual range. We estimate an overall uncertainty of 10 % from these parameters.

We next test the sensitivity to different inversion and modeling schemes (Table ). The Kalman filter (Appendix ) and the nonlinear inversion (Eq. ) results are not unreasonable for CO2. However, their CH4 flux results are unreasonably low, likely from high model : measured values having unreasonably high weights in these particular schemes with few scaling factors (Appendix ). GDAS and HRRR results are within uncertainty.

There is some uncertainty due to the accuracy and resolution of the emission inventories. compared emission inventories over the northeastern US and noted inventory differences of 100 % for half of the 0.1∘ grid cells in the domain. and compared aggregate posteriori inversion results from different emission inventories and noted differences of only 5 %–8 % for the Indianapolis region despite large differences at the grid level. We make a similar comparison in which we use the more spatially and temporally accurate Hestia v2.5 fossil fuel inventory instead of ODIAC as the prior. We note that the correlation between the forward model data and TCCON is slightly higher with Hestia than ODIAC, and there are fewer outliers that differ by a factor of 10× or more. However, the flux estimate of 110±28 is similar to the posterior flux estimate using ODIAC.

Finally we consider the observation uncertainty. reported a 2σ measurement bias of less than ∼0.2 ppm XCO2 (central estimate, maximum range <0.5 ppm) between the AFRC and Caltech TCCON sites, but even a bias of 0.2–0.3 ppm XCO2 will produce an error of ∼ 10 % in the flux. This bias could also arise from improper boundary conditions or application of averaging kernels.

In summary, we estimate 5 % random uncertainty from the bootstrap analysis, 10 % from our choice of initial values, 5 % from the prior flux, 10 % from observations and the boundary condition, and 20 % from model winds (Appendix ). The sum in quadrature is 25 %. By comparison, uncertainty estimates from other inversions were 11 % (inner 50 percentile range from an ensemble) for Indianapolis and 5 % for the Bay Area using pseudo-observations . Both of these studies benefited from additional sites (nine and 34, respectively) and custom WRF model runs. estimated an uncertainty of 5 % for the SoCAB flux by using data from 10 OCO-2 tracks; however this is not directly comparable with our result because it does not include uncertainty from biases in the forward model, observations, and inversion scheme.

Emission ratios can help us evaluate the inversion for the SoCAB. In previous studies it was noted that the Pasadena area is a good receptor site for the basin, so tracer–tracer ratios observed there should approximately correlate with emission ratios . If the ratios are significantly different it could highlight an error in the inversion scheme or the a priori assumption of sources. However, errors in the model can be correlated for different tracers, which would obscure universal biases to all gases. For example, the CO : CO2 flux ratio of 7.5 ppb : ppm is in good agreement with past literature, despite the absolute fluxes being lower on average.

We estimate emission ratios using the solar zenith angle (SZA) anomaly method described by and using the average enhancement compared with AFRC or the Pasadena : Lancaster gradient ratio. Errors are assumed to equal the standard deviation of all the data, and a linear fit is made using the methods of on monthly timescales. We estimate the emission ratio from the work of using the weighted mean of the excess ratios from their five in-basin sites, with weights of 1σ2. Emission ratios from the SZA anomaly method and the differenced enhancement are in agreement with ratios from previous studies (Fig. ). The CH4 : CO2 from the inversion at 9.7 ppb : ppm is larger than past studies, suggesting either our CH4 flux is too large or our CO2 flux is too low (or some combination of both). If both were adjusted by 15 % the ratio would be 7.1 ppb : ppm. The CO : CO2 ratio is in good agreement with the ratios using the SZA anomaly method, but is lower than past estimates. Based on the CARB inventories, a decrease is expected because CO emissions have decreased almost exponentially over the past decades, whereas CO2 emissions have decreased only moderately (e.g., compare Figs.  and ).

In November 2015, the large CH4 : CO2 ratio is from additional methane emissions from the Aliso Canyon gas leak . Though this leak persisted until February 2016, different wind patterns caused less of the highly methane-enriched air to be transported and observed in Pasadena after the first 2 months. The large CO : CO2 ratios seen in summer 2016 are from wildfires. The San Gabriel Complex Fire was less than 25 km to the east and burned 22 km2 over a month. It was close enough for ash to be transported to Pasadena. Eight other major fires within 150 km burned an additional 400 km2 during June–August 2016 (http://cdfdata.fire.ca.gov/incidents/incidents_archived?archive_year=2016, last access: 12 November 2018).

The weekday-to-weekend (WD : WE) flux ratios are listed in Table . The uncertainty is estimated to be ±0.10 based on changes in the ratio from the Sa scaling test up to 16× (Fig. ). Weekday : weekend ratios are similar to those from previous studies for CO . The CO2 and CO ratios are scaled down compared with the prior, which puts CO in better agreement with Hestia-LA v2.5. Methane has a ratio that is slightly larger than unity. Methane is not expected to vary as much as CO2 or CO on weekdays compared to weekends because production from biogenic sources and fugitive losses from natural gas infrastructure are less time variant.