Incremental 4D-Var requires forward nonlinear (NLM), adjoint (ADM), and tangent linear (TLM) models. The NLM is nearly identical to WRF-Chem , with the addition of emissions scaling factors. The GOCART option facilitates 19 species, including 4 gas and aerosol species for sulfate chemistry, hydrophobic and hydrophilic BC and organic carbon, 5 size bins for dust, 4 bins for sea salt, and 2 diagnostic species for PM2.5 and PM10. While we use GOCART, the results presented are limited to BC. The model configuration is the same as was used in , and is summarized as follows: ACM2 PBL mixing , the Pleim–Xiu land surface model and surface layer mechanisms without soil moisture and temperature nudging, Wesely dry deposition velocities , GSFC shortwave and Goddard longwave radiation, and microphysics turned off. Microphysical and radiative responses to online aerosols are not taken into account for GOCART aerosols in WRF-Chem.

We utilize the recently developed WRFPLUS-Chem , which contains ADM and TLM code extending the original WRFPLUS software . WRFPLUS-Chem describes chemical tracers in the context of planetary boundary layer (PBL) mixing, emissions, dry deposition, and GOCART aerosols. ADM and TLM gradients have been verified against finite difference approximations. Second-order checkpointing reduces the memory footprint to a feasible level for ADM and TLM simulations over longer durations (>∼ 6 h) and/or that use many chemical tracers (>∼ 10). applied the ADM in calculating sensitivities relevant to the emission inversion carried out here. Section  includes a comparison of the results of that study with the posterior emissions here.

The model domain is similar to that used by . The spatial extent encompasses California and other southwestern US states. We conduct two emission inversions, the first on 22 June with a focus on biomass burning sources, and the second on 23–24 June with a focus on anthropogenic sources. We generated chemical initial conditions by running WRF-Chem from 15 June 2008, 00:00:00 up until the beginning of each inversion period. We used the default WRF-Chem boundary condition for a BC concentration of 0.02 µgkg-1, which was found to be consistent with observations with an upwind flight on 22 June. Meteorological initial and boundary conditions are interpolated from 3 h, 32 km North American Regional Reanalysis (NARR) fields. The horizontal resolution is 18 km throughout 80 × 80 columns, and there are 42 vertical levels between the surface and model top at 100 hPa.

Our horizontal grid spacing was chosen to balance the wall-time and memory requirements of 4D-Var with model accuracy, and the ACM2 PBL option was chosen to reduce ADM and TLM development efforts. recommend that the complex terrain in California demands fine tuning of the WRF horizontal grid spacing, PBL, LSM, and reanalysis initialization. Among other conclusions, those authors found that at six surface sites near the land–ocean boundary, 4 and 12 km simulations with similar settings had mean wind speed biases of (0.15 to 1.5) ms-1 and (-0.38 to 1.9) ms-1, respectively. Supporting that conclusion, used a 36 km resolution chemical transport model (CTM), with offline meteorology, and found significant negative mean fractional bias (MFB) in modeled PM2.5 relative to surface observations of fires within narrow northern California valleys in July 2008 (MFB = -34.95 %) and during autumn 2007 Santa Ana winds (MFB = -110.22 %). During the July 2008 episode, their CTM predictions had a smaller positive bias (MFB = +21.88 %). Therefore, we would expect similar wind and concentration biases at 18 km resolution, which may or may not be improved by online meteorology. Incremental 4D-Var provides an opportunity to utilize a different model configuration (e.g., resolution) for the NLM comparisons of model to observations than that used for the ADM and TLM simulations. The adaptation of that capability from meteorological i.e., to chemical simulations and the subsequent testing is reserved for future WRFDA-Chem developments.

The prior includes sources of BC from anthropogenic activity and natural wildfires. Anthropogenic emissions are taken from the US EPA's 2005 National Emissions Inventory (NEI05) for mobile and point sources, including for example diesel on-road and power production from coal. The individual sectors are lumped together for each grid cell. We represent BB emissions using three different wildfire inventories, FINNv1.0 and v1.5, both at 1 km × 1 km resolution , and QFEDv2.4r8 at 0.1∘ × 0.1∘ resolution . FINNv1.5 is readily available through NCAR (http://bai.acom.ucar.edu/Data/fire/) to WRF-Chem users, while FINNv1.0 is no longer supported. However, we include FINNv1.0 in this study, because it shows the equivalent value as a prior. FINN and QFED fall into the first (bottom–up) and third (top–down constraint with AOD) categories of BB inventories described in Sect. , respectively.

Any inverse modeling study that depends on an initial guess should start in a region of high probability. In a Bayesian inversion, the first guess should be unbiased. Here we address several known errors in our prior inventories that we either fix or are unable to fix. All changes are consistent with either observations or the intended physical descriptions of the inventories.

QFED scales global aerosol emissions from four biome types through multiple linear regression between observed MODIS aerosol optical depth (AOD) and modeled GEOS-5 AOD during the years 2004–2009. For temperate forests QFED scales aerosols by ×4.5 throughout the world. That vegetation category accounts for 80 % of the wildfire BC in California during 22–30 June 2008. The global scaling is problematic for the California fires, because the GEOS-5 AOD is biased high in the western US during the summer fire seasons of 2006–2008 Fig. C14 of. In order to match the regional climatological AOD scaling factors for the western US, we scale all QFED BC sources by ×1/3. This scaling is already taken into account in the prior emissions shown in Sect. , and without it FINNv1.0 and QFED would differ by ×10 during the ARCTAS-CARB campaign.

The WRF preprocessor distributed with the FINN inventory is used to distribute ASCII formatted lists of both FINN and QFED daily speciated fire emissions to hourly netcdf files readable by WRF. The diurnal profile follows the Western Regional Air Partnership profile – – and is defined by a flux peak from 13:00 to 14:00 Local Time (LT), and flat fluxes equal to 2.5 % of the peak value between 19:00 and 09:00 LT. Through modeling experience, we found two bugs with how the FINN preprocessor interprets the WRAP profile and have fixed them for this case study. The total FINNv1.0 emissions across the model domain before and after fixing these bugs are plotted in Fig.  along with MODIS active fire counts . The first bug relates to how the time zone of a particular fire is calculated from longitude. The preprocessor converts a decimal longitude to integer time-zone bins; this allows a fire at 120.1∘ W to be an hour earlier in the diurnal profile than a fire at 119.9∘ W, even though they should be at nearly identical positions in the WRAP profile. Such behavior might apply to anthropogenic emissions, where cities near time-zone borders follow different daily cycles of activity, but not to natural activity related to the 15∘ per hour cycle of the Sun.

The second bug, and the one most visible in Fig. , is in the redistribution of UTC fire detections into LT emissions. MODIS Terra and Aqua overpass times are distributed around noon and midnight LT globally, with some adjustment as the image capture location moves farther from the Equator. The fire hotspots are detected on UTC days, and their emissions are profiled according to LT periods corresponding to the same UTC day as the detection. In California, where the LT is UTC minus 8 h, the noon overpass corresponds to 20:00 UTC, and 00:00 UTC corresponds to 16:00 LT on the previous day (Sun cycle). Therefore, when a fire is detected during nearly peak heat and emission fluxes at noon, a large fraction of the flux is apportioned to the previous afternoon. For locations east of the International Date Line, the LT reallocation is in the opposite direction. In either case, some portion of the profile is shifted by 24 h. This error is apparent as a temporal discontinuity in the case of transient fires that vary significantly in magnitude from one day to the next, especially after a recent ignition. Since the domain used here is nearly confined to a single time zone, we simply move the emissions forward 1 day for times between 16:00 and 23:00 LT (00:00–07:00 UTC). A more robust fix will need to be implemented in a future preprocessor.

Another error in the prior BB emissions is less easily resolved. Figure  shows where the MODIS active fires are located relative to the inventory fire locations. Since QFED fires are provided on a lat–lon grid, the fire centers do not coincide with its grid centers. When the inventory is distributed to the 18 km model grid, some emissions are shifted over by one column relative to the FINN locations. There are several additional spurious emission locations in QFED, where no active fires were detected on either 21 or 22 June. In a month long simulation, differences in fire gridding between several inventories can be averaged out. In the shorter-term inversions over California presented in Sect. , the locational differences do affect the results.

From late May until 20 June 2008, the southwestern US experienced a very dry period with little to no cloud cover appearing in MODIS true color imagery, and no recorded rainfall for most of California. On 21 June, the Aqua and Terra satellites recorded cloud cover for much of northern California, south of San Francisco, and along the Sierra Nevada mountain range, and there were widespread lightning strikes overnight. As is shown in Fig. , there was a spike in fire detections during the night between 21 and 22 June. Thus, from the morning through evening of 22 June, California experienced a transient fire initiation event. The wildfires burned well into July, exacerbating poor air quality throughout the state. The 20 June flight of ARCTAS-CARB characterized northern California anthropogenic sources, but was not influenced by fires. The 22 June flight embarked from Los Angeles, transited the offshore Pacific inflow, flew directly through smoke from forest fires in northern California, and then returned down the coastline. That flight encountered anthropogenic sources of BC in the morning, and BB sources for the remainder after returning to land. The 24 June flight passed back and forth in the downwind region between Los Angeles and San Diego, measuring the outflow from those cities and the transportation between them, and 1-day old diluted BB outflow from the north. A fourth flight on 26 June flew in the free troposphere from Los Angeles, north over the fires, and exited the model domain to the east.

We use WRFDA-Chem 4D-Var to constrain BB and anthropogenic aerosols on 3 days during ARCTAS-CARB using aircraft and IMPROVE surface observations. We utilize aircraft measurements of absorbing carbonaceous aerosol at 10 s intervals from the single-particle soot photometer (SP2) on 22, 24, and 26 June . For this study, we assume equivalency between the SP2 measurement and modeled BC, and re-average to the 90 s model time step using the revision 3 product, a process described in . We also use 24 h average surface observations of light absorbing carbon (LAC) on 23 and 26 June , assuming an equivalence with modeled BC, and ignoring the 7 % high bias relative to the SP2 found by . All treatments of observations are identical to those described in , including an analysis of model–observation BC mismatch that feeds into the inverse modeling study.

Using measurements from 22, 23, and 24 June, the 4D-Var system constrains anthropogenic and BB sources simultaneously. Data collected between 07:00:00 and 16:00:00 LT on 22 June are used in an inversion from 22 June, 00:00:00 UTC to 23 June, 00:00:00 UTC, during which time WRF-Chem is run freely, without nudging. The emission scaling factors for this 24 h time period for both source types are applied to subsequent days from 23 to 26 June in a cross-validation experiment. The 24 and 26 June aircraft and 23 and 26 June surface observations are used to analyze the utility of observationally constrained scaling factors found on 1 day to fix source errors on subsequent days. The 23 and 24 June surface and aircraft data are used in a 48 h inversion from 23 June, 00:00:00 UTC to 25 June, 00:00:00 UTC, also without nudging. Cross-validation is performed for these source estimates using 26 June surface and aircraft data.

Through preliminary testing, we found that horizontal correlation length scales on the order of the grid spacing provide the lowest posterior cost function. For both time periods, this length scale is set to twice the grid scale, Lh=36 km. The emission scaling factors are aggregated in each hour, which coincides with the emission file reading interval for both source types. The correlation scale is set to Lt=4 h, following . In addition to spreading error information across adjacent grid cells, the correlation scales reduce the effective number of CVs. Through sensitivity tests where we considered the smoothness of the posterior and the stationary posterior cost function value, and after consulting published values for regional emission uncertainties (see Sect. ) in different global settings, we use a relative grid-scale BB uncertainty of ×3.8. The BB uncertainty might also be approximated from the ratio of prior domain-wide total emissions between FINNv1.0 and QFED, which is given in Table  as ×3.5. If the median emission strength lies in the middle of QFED and FINNv1.0, then the prior domain-wide relative uncertainty is ×3.5=×1.8. The uncertainty would then need to be inflated further to account for spatial and temporal disaggregation and the possibility that grid-scale sources from the two inventories do not bound the true value (see Sect. ). The prior anthropogenic grid-scale relative uncertainty is set to ×2, which is within the reasonable bounds discussed in Sect. .

In addition to these standard settings, several sensitivity scenarios are used to gauge the sensitivity of the posteriors during two time periods to alternative inversion settings. The full set of scenarios are summarized in Table , and are as follows. FINNv1.0 is used as the default BB inventory in a scenario called FINN_STD for both inversion periods. QFED_STD uses the QFEDv2.4r8 BB inventory. Both FINN_L18 and QFED_L18 use Lh=18 km. FINN_V1.5 utilizes the FINNv1.5 BB inventory. For the 23/24 June inversion, we show results for both QFED_STD and FINN_STD, the latter of which includes variations where either surface or aircraft observations are excluded. The number of aircraft observations is Nobs=241 on 22 June and Nobs=302 on 24 June. There were Nobs=35 active surface sites on 23 June, 13 of them within California. We use six outer iterations consisting of 10 inner iterations each. Given the number of inner iterations used, and the wall-time of the tangent linear plus the adjoint (×10 longer than the nonlinear model), the cost of 4D-Var is approximately ×600 more than that of a single forward simulation, which is much cheaper than using finite difference methods to approximate derivatives instead of the linearized models when n∼105.

For a linear model operator and Gaussian distributed errors, the cost function can be used to evaluate the consistency of the statistics in B and R. The χ2 criteria states that the posterior cost function should be equal to 12NOBS e.g.,. The convergence properties of the 22 and 23/24 June inversion scenarios are shown in the outer loop cost function progression in Fig. . All of the 22 June scenarios led to comparable cost function values at numerical convergence, as shown in Fig. . The gradient norms are also reduced by nearly two orders of magnitude in all cases. In all of the scenarios, J converges to approximately NOBS, indicating that a portion of the model errors are not fully spanned by prior emission errors. For the 23/24 June inversion, QFED_STD reaches a lower cost function value, and both scenarios achieve similar χ2 values as the 22 June cases. Scrutinizing other sources of error (e.g., initial and boundary conditions for BC and meteorological variables, transport, BB plume rise, and model discretization) either independent from source strengths or simultaneously in the inversion framework should elicit further cost function reductions. Considering the top subplot in Fig. , the non-emission sources of error on 22 June are evident when the prior and posterior predictions are on top of each other, and remain on the edges of the low probability uncertainty region. For example, the observations before 08:00 LT after takeoff from Edwards are likely sensitive to the Pacific inflow, but not early morning emissions. Since these locations have relatively small uncertainties, the posterior cost function will never be reduced there.

Figure  also shows that during the inversion period the posterior is within the combined model–observation uncertainty (see Sect. ) much more often than the prior. In the afternoon, when the DC-8 passed over the wildfires, an increase in posterior emissions captures several of the observed BC peaks. The posterior is able to match the high-resolution variability of the observations at 13:30 LT, which may support the validity of the temporal averaging scheme. The only time during the inversion when the forecast degrades is for an observed peak at 22 June, 08:00 LT. The larger absolute observation uncertainty at that time relative to that at 08:30 LT enforces a weaker constraint in the inversion. Coupled with the assumed relative emission error correlation length scale and the close proximity of these two measurements, that stronger constraint at 08:30 LT dominates the morning anthropogenic emission analysis increment.

The R2 coefficients and slopes for linear fits between the prior and posterior and both aircraft and surface observations are summarized in Tables  and . Those results include cross-validation data on non-inversion days, which is discussed in Sect. . For both inversion periods, there are considerable model performance improvements for observations that are used in the inversion. FINN_STD improves R2 from 0.11 to 0.82 and slope from 0.26 to 0.8 on 22 June. QFED_STD improves R2 from 0.03 to 0.73 and slope from 0.34 to 0.71. Similar improvements occur for 23 June surface observations during the 23/24 June inversion. The posterior match to 24 June aircraft observations is improved, but not nearly as much as the other two data sets. The 22 June inversion results are also shown in Fig. , where the progression of the fit parameters is shown for the multiple scenarios. While all scenarios show similar improvements, the FINN_STD and QFED_STD results indicate the posteriors are still underpredicting many low and high concentrations. A similar phenomenon occurs for the 24 June observations in Fig.  in the inversion that uses both surface and aircraft observations. On both 22 and 24 June, the remaining low bias is either due to large prior observation and model error (diagonal of R) or due to the prior errors not being sensitive to emission increments.

For the appreciable measured BC concentrations (> 0.25 µgm-3), which are likely caused by a source within the model domain and simulation period, the lack of a source–receptor relationship is likely caused by low resolution. Changing a point source to a grid-scale area source changes its effective location. Temporal averaging of the observations will not necessarily solve that problem since perfectly modeled transport could still send a mislocated source in an entirely different direction than the truthfully located source. This effect is evident for valley fires , since placing the sources in the basin or spreading them throughout the basin and the peaks will result in different “downwind” concentrations. Downwind might be a very different direction if the convective-scale winds contribute more information than the mesoscale winds to the true source–receptor relationship. Since the emissions are smoothed in the model and not in reality, the mislocation is more likely to cause underprediction than overprediction.

Figure  shows the prior and posterior BB emissions for FINN_STD and QFED_STD during both simulation periods. In that figure there are several outlined emission areas (EAs); each EA was chosen to identify regions where a subset of the grid-scale analysis increment (δxEAX⊂δx) from both prior inventories is of a similar sign. The coordinates of the EAs are listed in Table . The two inversions do not reach identical total posterior BC emissions, but they do converge in certain aspects. Table  gives the emission subtotals for the EAs. During both inversions, each EA has emission increments of the same sign for both scenarios. Therefore, while domain-wide sources seem to be bounded by the two priors (as evidenced by their convergence), the same might not be true within the individual EAs. EA3, which accounts for the smallest average posterior total, is the only region where the magnitude of the log ratio between QFED and FINN is smaller in the posterior on 22 June. The ratio is reduced in EA2, but there the FINN posterior is ×2 larger than that for QFED. On 23/24 June, the two scenarios have less posterior spread in all of the EAs. Although Table  indicates large changes in source strengths across the EAs, Fig.  reveals that a majority of the absolute emission increment (posterior minus prior) in both FINN_STD and QFED_STD arose in only a few grid cells, often where the prior has the largest magnitude. The linear scaling factor pattern is similar between the two scenarios, with those for QFED_STD shifted toward decreases due to the high prior bias.

The temporal distributions of prior and posterior BB emissions within the four EAs are shown in Fig.  across all inversion scenarios on 22 June. The FINNv1.5 prior is an extreme outlier on the local afternoon of 21 June for EA1, EA2, and EA4. The same is true all day on 22 June for EA2, where the posteriors from other scenarios adjust toward the FINNv1.5 prior. Meanwhile, at other times when FINNv1.5 appears to converge toward the posteriors found using the other two priors, the prior relative uncertainty of ×3.8 is too restrictive to allow full convergence, since the priors differ by ×10. EA1 is characterized by decreases for all scenarios at all times. EA2, EA3, and EA4 exhibit early morning peaks between 03:00 and 06:00 LT that were not captured in the prior. In separate sensitivity tests, these peaks only appear when Lt>1 h, and become more prevalent as Lt is increased. attributed similar behavior in posterior estimates of the 2013 Rim Fire to persistent large-scale burning. found similar, less pronounced bimodal behavior for all of North America, which could be more noticeable in a regional inversion. Another possibility on 22 June 2008 is that the early morning burning is caused by the transient fire initiation event, which would explain the ramping of emissions for the QFED and FINNv1.5 posteriors in EA2. For both QFED and FINNv1.0, reducing the correlation length to Lh=18 km reduces the analysis increment in all EAs. This is especially apparent in EA4 for FINN_L18, where the increment is negligible.

The differing diurnal patterns in EA2 across scenarios could be attributed to variation in plume heights, QFED regridding errors, and the regularization term of the cost function. The observations most sensitive to EA2 sources were captured within or very near fire plumes. Plume heights are calculated hourly in an online 1-D vertical mixing scheme in WRF-Chem , which depends strongly on burned areas. With FINN, the areas are provided for each fire independently, while for QFED the areas use a default value of 0.25 km2 per fire. In both cases, the maximum area burned per grid cell per day is 2 km2. The regridding error discussed in Sect.  introduces fire locational errors, especially in EA2. A small error in vertical or horizontal mapping of a discrete point source on the model grid could hinder the optimization in distinguishing it from others. The uniform relative uncertainty in the prior inhibits consolidation of multiple posteriors when the prior spread is heterogeneous and sometimes very large. Quantifying the heterogeneity of uncertainty could contribute to posterior agreement between inversions using different priors, as well as to reducing the cost function.

The spread of local emissions provides some sense of that heterogeneity. Each EA covers a region approximately the size of a grid box in a global simulation with a chemical transport model. Due to the nature of variance aggregation, uncertainty grows as the grid scale gets smaller. In individual EAs, the spread between FINNv1.0 and QFED priors is ×2–×6 for both hourly (Fig. ) and daily (Table ) strength on 22 June. If the median emission strength lies in the middle, then a proxy for prior EA relative uncertainty is ×2-×6=×1.4-×2.4. Since the two inventories use identical diurnal patterns, the hourly estimate is missing information about uncertainties in daily emission timing. Using the posterior spread in a similar way gives approximate EA uncertainties of ×3-×10=×1.7-×3.2 on hourly scales and ×2-×7=×1.4-×2.6 on daily scales. This posterior estimate accounts for contributions in the prior definitions, including regridding, plume rise, and diurnal patterns. These ranges provide much more detail estimates than simply taking the domain-wide ratio of total emissions for the campaign period. However, the spread is itself missing information about uncertainty that could be found through carrying out similar inversions across an ensemble of model configurations and meteorological initial and boundary conditions e.g.,, or by comparing many more inventory priors e.g., and posteriors. All this is to say that the BB inventories used in this study are not provided with analytical estimates of uncertainty, and a lack of information for deriving such values at hourly grid scales is a topic for future research.

Figure  shows the total prior and posterior anthropogenic emissions and Fig.  displays the analysis increment and linear scaling factor for FINN_STD on 22 June and separately on 23–24 June. The only difference in QFED_STD, not shown here, is that anthropogenic scaling factors are shifted in the negative direction in the posterior, likely due to the higher bias in that BB prior. The increments found in a new set of EAs are presented in Table .

The 23 and 24 June observations provide much more detailed information about anthropogenic sources. The analysis increment reveals potentially misrepresented city-level emissions in the NEI05 prior. Posterior BC near Barstow, Victorville/Hesperia, Fresno, Edwards Air Force Base, and El Centro/Calexico are increased, while sources near the three coastal cities are decreased. Since Barstow is a crossroads for the BNSF and the Union Pacific railroads, and since Fresno, Victorville/Hesperia, and El Centro/Calexico lie at switching locations for major rail lines, the inversion results may suggest that the prior is missing diesel rail sources of BC. However, for locations where the prior magnitude of BB and anthropogenic emissions are of similar magnitude, their posteriors are subject to projection from one sector to another. It is more likely that the low bias fire emissions north of Fresno are responsible for the prior underpredictions of 23 June surface concentration measurements exceeding 2 µgm-3 see Fig. 5 of. This is corroborated by the posterior BB emissions being scaled up near Fresno on 23 and 24 June, and by the much smaller model bias for IMPROVE on 22 June before the fires started.

There are also small negative increments near Los Angeles (EA6) and San Francisco (EA7) during both the 22 June and 23/24 June inversions, which are likely attributable to on-road mobile sources. These results are consistent with model bias in surface and aircraft observations on 20 June near both of those cities . found a decreasing trend in ambient measurements of BC and in a fuel-based bottom–up inventory for both Los Angeles and San Francisco from 1990 to 2010 that might not be captured for the 2008 model year by the snapshot in NEI05. Using a similar fuel-based approach, derived 2010 CO emissions in the South Coast Air Basin surrounding Los Angeles that are ×1/2 the magnitude of NEI05. On-road and other mobile sources make up 36 and 62 % of that difference, respectively, and their bottom–up inventory matches more closely with NEI 2011. While not a perfect comparison to BC in 2008, the sign of error in NEI05 relative to the coastal posterior and that study is consistent. An inventory with sector-specific breakdowns of BC emissions, additional inversions with more thorough speciated local observations, and higher resolution would all be required to investigate sector-specific anthropogenic pollution.

Analysis of posterior emissions uncertainties is useful for understanding the value of the posterior emissions themselves. The diagonal terms of Pa are the posterior variances, σxa, which are always smaller than prior variances. The variance reduction could instead be presented in β space, by utilizing Eq. (). However, σβk,i2<σβ0,i2 is not guaranteed when xa,i>xb,i=0, because the posterior relative emission uncertainty depends on xa,i. For this work, the reductions in variance are presented in CV space. The low-rank estimate of Pa is only valid for linear perturbations away from xa. The final outer loop estimate of Pa is the most accurate, since it is linearized around the state preceeding xa. A quantitative measure of error reduction in the koth outer loop in the ith CV is ρi,ko=1-Pi,iaBi,iko∈[0,1). Values of ρi,ko closer to 1 reflect locations where the observations provide a stronger constraint than the prior. This estimate may not reflect the entire error reduction, since it does not capture potential reductions in previous outer loops. Without propagating updated estimates for B to subsequent outer loops e.g.,, we also define ρagg, a qualitative metric that accounts for increases in curvature (decreases in error) in all outer loops: ρi,agg=1-∏ko=1kPi,iaBi,iko∈[0,1). ρi,agg reveals additional information about observation footprints not shown by ρi,ko=6. The nonlinear nature of the problem means ρi,agg is not quantitative.

Both (ρko=6) and ρagg are presented in Figs.  and  for the BB and anthropogenic members of xa, respectively. Fifty inner loop iterations were taken in the final outer loop to improve ρ estimates. ρko=6 is <45 % across all scenarios, except for QFED_STD BB sources near the IMPROVE sites on 23/24 June. If the inner loop were halted at 10 iterations, the error reduction estimates would be reduced by up to ∼ 10 % (i.e., 35 instead of 45 %) in the darkest grid cells. The BB error reduction shown in Fig.  has similar spatial distributions for FINN_STD and QFED_STD scenarios, but differs significantly between the two time periods due to the different spatial coverage of the observations. The reductions in the north on 22 June are more disperse for QFED_STD, which could be caused by the same regridding errors and plume rise differences that influence the posterior emissions. There is also more error reduction in the south for the QFED_STD emissions. In general, the grid-scale uncertainty improvement is confined to sources close to the observations.

The most obvious application of ρ is to evaluate the footprint of a set of measurements. For example, the large relative BB emission increments in EA1–EA3 on 23/24 June indicate that distant observations can have a large impact on the posterior emissions magnitudes. However, ρi,ko=6 in Fig.  indicates there is nearly zero uncertainty reduction for those emissions. Also, upon considering the last two columns of Table , one might conclude that there is a missing weekend (22 June) to weekday (23/24 June) variation in BC emissions within EA8–10. However, Fig.  shows that the 22 June observations only weakly reduce uncertainty in emissions.

In a more tangible application, ρ can be used to assess existing and future observing strategies in a similar way to how used adjoint sensitivity information to plan future meteorological observing sites to improve forecasts of extreme dust events in the Korean Peninsula. Fig.  presents anthropogenic ρ for different combinations of surface and aircraft observations on 23/24 June. The surface observations primarily resolve sources near Fresno, and to a lesser extent near Los Angeles. Since the purpose of the IMPROVE network is to measure background concentrations, it is mostly successful on 23 June in not being influenced by anthropogenic sources of BC from the major cities. If the goal were to measure anthropogenic sources, inflows, or domain-wide concentrations on daily timescales, then ρ would suggest using a different surface network distribution. Such a conclusion does not conflict with the success of using IMPROVE observations to provide top–down constraints on both BB and anthropogenic emissions on monthly timescales e.g.,. That strategy is consistent with what is generally known: further decreasing uncertainty requires observing the same phenomena more thoroughly. For hourly to daily timescales, more observations are needed close to and downwind of chemical sources, and at high spatial and temporal resolution (e.g., from repeated aircraft overpasses, extra aircraft, hourly-average surface sites, or satellites).

Another piece of information useful for comparing observing configurations and inversion scenarios is the trace of the resolution matrix, or degrees of freedom for signal, i.e., DOF=TrIn-PaB-1, which is equal to the number of modes of variability in the emissions that are resolved by the observations . Substituting the approximation for Pa from Eq. (), DOF≈n-TrB+U∑ki=1lλki-1-1ν^kiν^ki⊤U⊤B-1≈-TrU∑ki=1lλki-1-1ν^kiν^ki⊤U⊤B-1. Since U is square, U⊤B-1U=In, and Trν^kiν^ki⊤=ν^ki⊤ν^ki=1, the expression simplifies as DOF=-Tr∑ki=1lλki-1-1ν^kiν^ki⊤U⊤B-1U≈∑ki=1l1-λki-1Trν^kiν^ki⊤≈∑ki=1l1-λki-1. Therefore, the only information needed to compute DOF is the eigenvalues of Tl. Each inner loop, ki, has the potential to constrain one additional mode of variability in the emission scaling factors. For all of our inversion scenarios, the leading eigenvalue is on the order of 102-103, which is equal to the condition number of the full-rank Hessian. As the Lanczos optimization proceeds, each subsequent λki is smaller, asymptotically approaching unity, and each eigenmode provides less information than the one preceding it about scaling factor variability.

Figure  gives three estimates of DOF at each level of truncation in the final outer loop, that is, if higher degrees of eigenvalues were ignored. In that figure, we plot eigenvalue spectra of the FINN_STD and QFED_STD scenarios on 22 June. Similar to ρ, we use a 50-iteration linear optimization to improve the bounds on DOF. The ki=l estimate of the eigenvalue spectrum at each iteration is represented by a single colored line. Each member of the eigenvalue spectrum, represented by vertical grid lines in Fig. , converges toward an upper bound as more iterations are taken. Initial guesses for the least dominant eigenvalues are less than 1 for ki≥8 for FINN_STD, but they exceed 1 after an additional iteration, consistent with the properties of the Lanczos sequence. The first DOF value in parentheses adheres to the philosophy that only converged eigenvalues should be used to estimate DOF; it excludes λki,…,λl such that λki is more than 5 % changed from the previous estimate. The second DOF value in parentheses uses all of the current estimates of the eigenvalues available in iteration l. This is still a conservative estimate of DOF, because the true eigenvalues of the full-rank Tn are always larger than their current numerical estimate. After enough iterations, the numerical growth in DOF is very small, and further computation is not warranted. As the eigenvalue spectra in Fig.  and the cost function reduction in Fig.  show, this is long after the cost function is converged enough for practical purposes. The posterior CVs, which are the primary result from inverse modeling, do not change significantly in the final outer loop. Finally, the best estimates of DOF in red brackets are evaluated at different truncations using the most-converged values of the eigenvalues found in the 22nd iteration.

Similar to ρ, the quantitative application of DOF is limited to the final outer loop, when δxn is small enough that (Hδv)-1|xn-1≈(Hδv)-1|xn. Absent the need to estimate the posterior Hessian, the outer loop could be ended an iteration earlier. In the inner loop, truncated estimates of H-1 and its eigenvalue spectrum at earlier iterations will provide conservative values for both DOF and ρ. The actual DOF value is higher than any value shown in Figs.  (22 June) and  (23/24 June). Therefore, the 22 June observations constrain > 14 modes of hourly grid-scale variability through 4D-Var in both the FINN_STD and QFED_STD scenarios. Just like for ρ, the optimization constrains additional modes in the earlier outer loop iterations, but that quantification is not straightforward since DOF are defined for linear behavior. If all outer loops were similar, then the total DOF value for the entire nonlinear optimization is on the order of 30 to 40.

As shown in Fig. , the DOF values on 23 and 24 June after 50 iterations are 10, 17, and 23 for the SURF, ACFT, and FINN_STD(23/24) scenarios, respectively. The relative magnitudes show that using combined surface and aircraft observations provides an additional value over using either independently, although the two platforms might have some redundancy. This conclusion is consistent with the maps of BB and anthropogenic ρ in Fig. , where the footprints of SURF and ACFT have slight overlap near Los Angeles, but are otherwise independent. Additionally, the higher DOF value of ACFT is consistent with its more widespread and larger magnitude ρ values. The slower eigenvalue convergence when both observing types are utilized means that additional inner iterations could yield higher estimates for DOF in that case. What is even more clear, and intuitive, is that ρ and DOF estimates require more iterations as the number of constrained CVs increases, which is directly dependent on the number of observations. The sparse ρ map for SURF in Fig.  and the large spike near Fresno illustrate that while near-source surface measurements can be a powerful constraint, measurements of background concentrations provide relatively little constraints to characterize CA anthropogenic emissions on 1-day timescales.

As an additional evaluation of the robustness of the emission scaling factors, we apply them in cross-validation tests. In two separate evaluations, the 22 June scaling factors are applied to 23–26 June emissions, and the 23/24 June scaling factors are applied to 25–26 June emissions. The heterogeneous adjoint sensitivity signs and magnitudes for each source sector we found on each day of the campaign are an indication that corrective scaling factors in each day will be unique. In that work, we found that the 24 June observations were most sensitive to southern California anthropogenic sources on 24 June and to northern and southern California coastal sources of both sectors on 23 June. The 26 June observations were most sensitive to northern California fires, and the adjoint sensitivities were of opposite sign than on 23 and 24 June.

As shown in Fig. , the cross-validated 22 June scaling factors rarely generate improvements to model performance, when compared to 24 and 26 June aircraft observations. On 24 June, some of the high bias predictions are corrected, or even over-compensated, but the low bias prior locations are unaffected. Table  shows the R2 and slope of the linear trend lines. The scatter of the fit for QFED_STD on 24 June and the slope for FINN_STD on 26 June are slightly improved, but all other metrics degrade. The increase in slope for FINN_STD comes as a result of better fit to very large concentrations above the PBL associated with fire sources on multiple previous days. The posterior scaling factors generated from the 23/24 June inversion degrade the forecast of aircraft measurements on 26 June. Since the posterior primarily serves to reduce coastal anthropogenic and BB emissions, it is not surprising that it does not improve a low bias prior 2 days later.

Table  includes cross-validated surface measurements on 23 June and 26 June. There is very little change to the modeled surface concentrations as a result of posterior scaling factors derived from inversions that only use aircraft observations. Assimilating surface observations on 23 June (Monday) does improve model comparisons to surface observations on 26 June (Thursday). Those small improvements imply that errors are weakly correlated between weekdays. Although it is beyond the information content provided by the observations used in this work, future studies could compare the efficacy of using weak multiday correlation in B and the hard constraint of 24 h periodic scaling factors used herein. Aircraft and surface observations do not appear to be useful for cross-validation of each other over the short timescales and limited set of flights considered here. At least for this study period, when they are not collocated, each provides some unique information to the inversion.