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Biomass burning emissions of atmospheric aerosols, including black carbon, are growing due to increased global drought, and comprise a large source of uncertainty in regional climate and air quality studies. We develop and apply new incremental four-dimensional variational (4D-Var) capabilities in WRFDA-Chem to find optimal spatially and temporally distributed biomass burning (BB) and anthropogenic black carbon (BC) aerosol emissions. The constraints are provided by aircraft BC concentrations from the Arctic Research of the Composition of the Troposphere from Aircraft and Satellites in collaboration with the California Air Resources Board (ARCTAS-CARB) field campaign and surface BC concentrations from the Interagency Monitoring of PROtected Visual Environment (IMPROVE) network on 22, 23, and 24 June 2008. We consider three BB inventories, including Fire INventory from NCAR (FINN) v1.0 and v1.5 and Quick Fire Emissions Database (QFED) v2.4r8. On 22 June, aircraft observations are able to reduce the spread between a customized QFED inventory and FINNv1.0 from a factor of 3.5 (x3.5) to only x2.1. On 23 and 24 June, the spread is reduced from x3.4 to x1.4. The posterior corrections to emissions are heterogeneous in time and space, and exhibit similar spatial patterns of sign for both inventories. The posterior diurnal BB patterns indicate that multiple daily emission peaks might be warranted in specific regions of California. The US EPA's 2005 National Emissions Inventory (NEI05) is used as the anthropogenic prior. On 23 and 24 June, the coastal California posterior is reduced by x2, where highway sources dominate, while inland sources are increased near Barstow by x5. Relative BB emission variances are reduced from the prior by up to 35% in grid cells close to aircraft flight paths and by up to 60% for fires near surface measurements. Anthropogenic variance reduction is as high as 40% and is similarly limited to sources close to observations. We find that the 22 June aircraft observations are able to constrain approximately 14 degrees of freedom of signal (DOF), while surface and aircraft observations together on 23/24 June constrain 23 DOF. Improving hourly- to daily-scale concentration predictions of BC and other aerosols during BB events will require more comprehensive and/or targeted measurements and a more complete accounting of sources of error besides the emissions.