The Bayesian synthesis inversion method, as described by and , was used to solve for the fluxes in this study. The observed concentration (c) at a measurement station results from contributions from the surface in the form of fluxes, from the domain boundaries, and from the initial concentration at the site. Concentrations at the measurement site can be modelled as follows: 1cmod=Hs, where cmod is the modelled concentrations and s is a vector of source fluxes or concentrations. H is the Jacobian matrix representing the first derivative of the modelled concentration at the observational site and dated with respect to the coefficients of the source components . It provides the sensitivity of each observation to each of the sources, where the sources can be fluxes or concentrations of CO2. Estimates of the unknown sources can be obtained by minimizing the following cost-function with respect to s: 2J(s)=12(cmod-c)TCc-1(cmod-c)+(s-s0)TCs0-1(s-s0), where s is the control vector of unknown surface fluxes and boundary concentrations we wish to solve for, s0 is the vector of prior flux and boundary concentration estimates, Cc is the uncertainty covariance matrix of the observations, and Cs0 is the uncertainty covariance matrix of the fluxes and boundary concentrations .

Minimizing this cost function leads to the following solution: 3s=s0+Cs0HTHCs0HT+Cc-1(c-Hs0), with the following posterior covariance matrix: 4Cs=HTCc-1H+Cs0-1-1,5=Cs0-Cs0HTHCs0HT+Cc-1HCs0.

The total CO2 flux from a single surface pixel can be thought of as being made up of the following individual fluxes: ssf;i=sffweekday;i+sffweeknight;i+sffweekendday;i6+sffweekendnight;i+sNEEday;i+sNEEnight;i, where ssf;i is the total weekly surface flux from the ith pixel, sffweekday;i is the total fossil fuel flux during the working week day, sffweeknight;i is the total night-time fossil fuel flux during the working week, sffweekendday;i is the total weekend daytime fossil fuel flux, sffweekendnight;i is the total weekend night-time fossil fuel flux, and sNEEday;i and sNEEnight;i are the total daytime and night-time biogenic fluxes for the full week from the ith spatio-temporal pixel. The reference inversion solved for each of these separate fluxes for each week. There are 101×101=10201 surface pixels. Over the 16-month period from March 2012 to June 2013, separate monthly inversions were carried out for all months with sufficient valid concentration observations: a total of 13 inversions. Each monthly inversion solved for four weekly fluxes. Therefore, a monthly inversion solves for 10 201 × 6 × 4 = 244 824 surface fluxes.

The mean daytime and night-time concentrations at each of the four domain boundaries for each week are included in the control vector. The inversion solved for 4 × 2 × 4 = 32 boundary concentrations (four boundaries, day–night, 4 weeks). We solved for weekly concentrations at the boundaries, as we expected these concentrations to show small changes on synoptic time scales, particularly inflow from the ocean boundaries. We avoided solving for too short a period so that the percentile filtering technique (see Sect. ) would never discard all measurements for a period. The maximum standard deviation in the hourly background CO2 concentrations for a week was 0.8 ppm.

The reference inversion made use of two CO2 monitoring sites that were established at Robben Island and Hangklip lighthouses. Each site was equipped with a Picarro cavity ring-down spectroscopy (CRDS) (Picarro G2301) instrument. Sufficient data for 13 of the 16 months were available to perform monthly inversions. The Robben Island site predominantly viewed air influenced by the Cape Town city bowl, whereas Hangklip viewed air influenced by biogenic fluxes from nearby fynbos vegetation and agricultural areas. Details about these measurement sites are provided in . Rigorous calibration was performed on a regular basis, ensuring that these sites measured on the same scale as the Cape Point background site, which is calibrated to the WMO-X2007 scale. The high-frequency observations were processed into hourly concentrations, which provided the observed data for the inversion.

CCAM is a variable-resolution global atmospheric model developed by the Commonwealth Scientific and Industrial Research Organisation (CSIRO) and has been validated over South Africa . Full details are provided in . CCAM was applied in stretched-grid mode to function as a regional climate model. A multiple-nudging approach was followed to downscale the 250 km resolution National Centers for Environmental Prediction (NCEP) reanalysis data to a resolution of 60 km over southern Africa, 8 km over the south-western cape and subsequently to a 1 km resolution over the study area. The model produced hourly estimates on a 1 km × 1 km spatial grid, which extended from 34.5∘ to 33.5∘ S and from 18.2∘ to 19.2∘ E.

The Jacobian matrix, H, provides the sensitivities of the concentrations observed at the receptor sites to the surface fluxes and boundary inflows. To generate this matrix in our application the particle counts were processed from a Lagrangian particle dispersion model (LPDM) run in backward mode . The LPDM was driven by hourly three-dimensional fields of mean winds (u, v, w), potential temperature and turbulent kinetic energy (TKE), which were obtained from the CCAM model. LPDM simulates atmospheric transport by releasing particles from the observational sites and tracking these particles backward in time. These particle counts were used to derive the elements of the Jacobian matrix H as originally described by and subsequently used in several inversion studies .

Previously, we modified the approach of to use particle counts – as produced by our LPDM – instead of mass concentrations that were output by the atmospheric transport model FLEXPART in their study . The elements of the matrix H corresponding to the surface fluxes in s were calculated as follows: 7∂csf‾∂sin=ΔTgΔPNinNtot‾4412×103, where csf‾ is a volume mixing ratio (receptor) expressed in ppm and sin is a mass-flux density (source), Nin the number of particles in the receptor surface grid from source pixel i released at time interval n, ΔT is the length of the time interval, ΔP is the pressure difference in the surface layer, g is the acceleration due to gravity, and Ntot the total number of particles released during a given time interval.

The spatial resolution of the surface flux grid boxes was set to be the same as that of the high-resolution subregion of the atmospheric transport model, resulting in a gridded domain consisting of 101 × 101 grid boxes (a resolution of 1 km × 1 km). The units of the surface fluxes are given in kg CO2 m-2 week-1 and are transformed through H into contributions to the concentration at the measurement site in units of ppm. To solve for the concentrations at the boundary showed that the Jacobian can be calculated as follows: 8∂cb‾∂sB=NBNtot, where sB are the concentrations at the domain boundary, cb‾ is the volume mixing ratios, NB is the number of particles from the domain boundary, and B and Ntot are the total number of particles viewed at the receptor site from any of the domain boundaries. The contribution to the observed concentration at the receptor site can be written as follows: 9cb=HBsB, where HB is the Jacobian with respect to the domain boundary concentrations, sB is the domain boundary concentrations and cb is the contributions from the boundary to the observed concentration at the measurement site in units of ppm. The row elements of HB sum to one. Therefore, the elements of cb represent a weighted average of the concentrations at the domain boundaries and provide a basis concentration to which the contributions from the surface fluxes are added. Each inversion solves for weekly domain boundary concentrations at the northern, eastern, southern and western borders of the inversion domain box, separated by day and night.

The inventory analysis carried out for CT subdivided the anthropogenic emissions into road transport, airport and harbour, residential lighting and heating, and industrial point source emissions . Road transport emissions were derived from modelled values of vehicle kilometres for each section of the road network, based on observed vehicle count data. The vehicle kilometres were scaled for each hour of the day and separated into weekdays and weekend days, leading to distinctive vehicle emissions for the weekday–weekend and day–night periods. Airport emissions were derived from landing and takeoff cycles, as reported by Airports Company South Africa for each month. The IPCC average emission factors for domestic and international fleets were used to convert the airport activity data into emissions of CO2. Harbour emissions were derived from gross tonnage of vessels that docked at the CT port during each month published by the South African Ports Authority and emissions derived as described in . Residential emissions for lighting and heating were derived from population count data obtained for each of the municipal wards in 2011 . The South African government reports on the fuel used for domestic heating and lighting . This was divided between the total population and then allocated pro rata to each ward. It was assumed that 75 % of the annual energy consumed was used for heating, 20 % for cooking and 5 % for lighting. The majority (75 %) of the emissions for heating were allocated to the winter months. CT provided monthly fuel use for the largest industrial emitters. These were converted directly into CO2 emissions by multiplying the fuel amount with the DEFRA greenhouse gas emission factors . The fuel types that were considered included heavy fuel oil, coal, diesel, paraffin and fuel gas, which was divided into liquid petroleum gas and refinery fuel gas.

Based on this inventory analysis, the percentage contribution of industrial point sources to the total fossil fuel emission was 12.0 % for CT, 34.6 % from vehicle road transport, 51.0 % from the residential sector, and 2.4 % from airport and harbour transport. Residential emissions are a large contributor to the fossil fuel emission budget, as well as one of the largest contributors to the uncertainties in the fossil fuel flux. This is due to the dependency that many people living in CT have on raw fossil fuel burning for heating and lighting. Emissions from power stations are a small component of the total fossil fuel flux from CT, as the bulk of the direct emissions from power stations occur elsewhere in the country.

The total fossil fuel CO2 emissions for the domain were within the range of CO2 emissions reported in the EDGAR (Emission Database for Global Atmospheric Research) (v4.2) database . EDGAR is a global product on a 0.1∘ × 0.1∘ grid, which provides the total anthropogenic emissions of CO2 as estimated from proxy data such as population counts and information on the road transport network . The total emissions from the inventory for 2012 were 22 % higher than the EDGAR emissions reported for 2010. The emissions in the inventory tended to be concentrated over specific sources, such as over an oil refinery or along the road network, whereas the EDGAR emissions were smoothed over the city region.

CCAM was dynamically coupled to the land surface model CABLE , which allows for feedbacks between land surface and climate processes, such as leaf area feedback on maximal canopy conductance and latent heat fluxes . This also has the consequence that the spatial resolution of the biogenic fluxes were at the same spatial resolution of 1 km × 1 km as for the transport model. The model produces hourly estimates of net ecosystem exchange (NEE), which were aggregated into weekly (day and night) flux estimates in units of kg CO2 m-2 week-1 and used as the prior estimate of biogenic fluxes over the land surface.

The natural areas within the target domain of the inversion are dominated by the fynbos biome. This is a biodiverse biome with many endemic species, and it covers a relatively small area in South Africa but a large proportion of the area within the domain of the inversion. The fynbos biome is poorly represented by dynamic vegetation models and their ability to simulate biogenic fluxes in the fynbos region is largely untested. CABLE was selected as the land–atmosphere exchange model to couple with CCAM due to its development for regions in Australia that are similar to the savanna biome in South Africa. In addition to the natural vegetation, a large agricultural sector is within the proximity of CT, particularly vineyards and fruit orchards. The CT region experiences a Mediterranean climate with winter rainfall, hot and dry summers, and mild and wet winters. Significant NEE fluxes take place during both winter and summer periods, as biogenic activity in this region is limited by the amount of water availability, whereas temperatures are usually sufficiently high for plant production and respiration. The CO2 fluxes over the ocean were obtained from a study that characterized the seasonal cycle of air–sea fluxes of CO2 in the southern Benguela upwelling system off the South African west coast .

The presence of the Cape Point GAW station provided a source of background CO2 concentrations for the inversion. The Cape Point station is located approximately 60 km south of CT within a nature reserve situated on the southernmost tip of the Cape Peninsula at a latitude of 34∘21′12.0′′ S and longitude of 18∘29′25.2′′ E. The inlet is located on top of the 30 m measurement tower mounted on a cliff 230 m above sea level (m a.s.l.). The station observes background measurements of CO2 when observing maritime air advected directly from the south-western Atlantic Ocean – an extensive region stretching from 20∘ (sub-equatorial) to 80∘ S (Antarctic region) . Therefore, maritime measurements at Cape Point from the Southern Ocean are representative of the background CO2 signal influencing the Cape Peninsula, which are the concentrations expected at the boundary of the inversion domain. The background signal at Cape Point, represented by a subset of the measurements obtained from a percentile filtering technique , was used as the prior estimate of the concentrations at each of the four domain boundaries. The percentile filtering technique removes data influenced by the continent or anthropogenic emissions. When applied to the Cape Point CO2 measurements, approximately 75 % of the data are selected. The percentile-filtering technique has been shown to compare well with the more robust method of using contemporaneous radon (222Rn) measurements to differentiate between marine and continental air .

The Cape Point measurements of the background CO2 levels meant that we were not dependent on the atmospheric transport model to produce estimates of CO2 concentrations at the domain boundary, which are prone to large errors . The mean weekly background concentrations, separate for day and night, were determined from the percentile filtered measurements at the site and were used as the prior domain boundary concentrations for each of the four cardinal directions. The prior uncertainty assigned to the boundary concentrations was set at the standard deviation of the measured hourly concentrations for that period, which resulted in a tight constraint on the prior background CO2 concentrations. Large adjustments by the inversion to the domain boundary concentrations were not expected, including the terrestrial boundaries. The standard deviation in the hourly background CO2 concentrations ranged between 0.32 and 0.90 ppm, with a mean of 0.62 ppm.

The boundaries of the domain were deliberately set to be far from the measurement sites so that contributions to the CO2 concentration at a measurement site were dominated by the surface fluxes within the domain, rather than by the domain boundary concentrations.

Error propagation techniques, as described in and , were used to estimate the relative uncertainties for each of the sector-specific fossil fuel estimates. The relative uncertainties were scaled by a value of 2, in order to ensure that the elements of the covariance matrix were statistically consistent with the assumptions of the inversion . The resulting uncertainty estimates (expressed as standard deviations) ranged between 6.7 % to 71.7 % of the prior fossil fuel emission estimate, with a median percentage of 34.9 % to 38.4 % depending on the month. These values were more conservative compared with uncertainties of for the AirParif inventory, which were set at 20 % throughout. Since we solved for weekly, rather than daily fluxes, we used a strong assumption that fossil fuel fluxes within the same week were homogeneous over this time. To allow the inversion to react to local conditions within a given week, no temporal uncertainty correlation was assumed between weekly fluxes. Since fossil fuel emissions were expected to be localized in space, we also assumed no spatial uncertainty correlation between fossil fuel fluxes.

The uncertainty in the biogenic prior fluxes was set at the absolute value of the net primary productivity (NPP) as produced by CABLE. Therefore, the uncertainties assigned to the NEE estimates were large, but there is a great deal of uncertainty in both the productivity and respiration fluxes contributing to the NEE flux . The estimates of NEE are strongly dependent on the assumed model forms selected for different processes in the CABLE model. For example, the model forms used for the soil temperature–respiration function and the soil moisture-respiration function have large impacts on the NEE estimates, with resulting NEE estimates differing by over 100 % compared with eddy-covariance measurements . The approach of assigning either the productivity or respiration component of NEE as the uncertainty has been used by . We wished to avoid assigning fixed proportional uncertainty to the NEE estimates as, for semi-arid regions in particular, small NEE fluxes could occur as a result of both large productivity and respiration fluxes. Proportional uncertainties would lead to unrealistically low estimates of the uncertainty in NEE fluxes. This is different to the approach used by , where an uncertainty level of 70 % was assigned to biogenic fluxes, but in their case absolute NEE estimates were usually large in summer and expected to be small in winter. For the ocean fluxes, the standard deviations in the daily CO2 fluxes from were assigned as the uncertainties.

To estimate spatial uncertainty covariances in the NEE fluxes, we assumed an isotropic Balgovind correlation model as used in . The off-diagonal covariance elements for sNEE;i and sNEE;j were calculated as follows: Cs0;NEEsNEE;i,sNEE;j=Cs0;NEEsNEE;iCs0;NEEsNEE;j101+hLexp⁡-hL, where sNEE;i and sNEE;j are NEE fluxes in pixels i and j, Cs0;NEEsNEE;i and Cs0;NEEsNEE;j are the corresponding variances in the NEE flux uncertainties in pixels i and j, the characteristic correlation length L was assumed to be 1 km, and h is the spatial distance between the centres of pixels i and j.

The observation uncertainties represented in Cc contain both the measurement error and the error associated with modelling the concentrations. We assigned a minimum uncertainty variance of 4 ppm2 for daytime observations and 16 ppm2 for night-time observations. These values were assigned as baseline (i.e. minimum) errors and accounted for measurement errors, atmospheric transport modelling errors, aggregation errors and representation errors. These minimum errors are smaller than those for city-scale inversions conducted in the Northern Hemisphere. We justify the use of these values in our application since CT is a smaller city compared with the cities considered in the megacity applications, such as Paris or Indianapolis. Measurements of background CO2 in the Southern Hemisphere have smaller variability compared with measurements in the Northern Hemisphere. For example, for the years 2012 to 2013 the standard deviation between the monthly CO2 means for the Mauna Loa GAW station in the Northern Hemisphere was 2.3 ppm , whereas for the same time period at Cape Point the standard deviation between the monthly means was 1.6 ppm.

We added additional error estimates to these minimum observation errors. We assumed errors in modelled CO2 concentrations due to the transport model would be larger when the wind speed was lower , and this would be compounded at night when the planetary boundary layer height was shallower and more stable . Additional error, ranging between 0 and 1 ppm2, was added to the daytime uncertainty variance of 4 ppm2, linearly scaled depending on the wind speed, with 0 ppm2 added when wind speeds were high (20 m s-1 or higher) and 1 ppm2 when the wind speed was close to zero. At night the additional uncertainty ranged between 0 and 16 ppm2. We also accounted for the standard deviation of the measured CO2 concentrations during each hour. We assumed that variability within the instantaneous measurements at the site during an hour would be associated with larger errors in the atmospheric transport model. The variance of the observed instantaneous CO2 concentrations within an hour was added to the overall uncertainty. Therefore, each hour had a customized observation error dependent on the prevailing conditions at the measurement site. Therefore, the total observation uncertainty variance for hour k is given as follows: 11Cc(k,k)=Cc;base2+Cc;wind2+Cc;obs2, where Cc;base is the baseline observation error of 2 ppm during the day and 4 ppm during the night, Cc;wind is the additional error due to the wind speed conditions that ranged between 0 and 1, and Cc;obs is the standard deviation of the observed concentrations within that hour. The final observation uncertainties reached up to 15 ppm at night, reducing the weight of these measurements in the estimation of the prior fluxes.

The off-diagonal elements of Cc were calculated, based on the Balgovind correlation model as used in , as follows: 12Ccci,cj=CcciCccj1+hLexp⁡-hL, where ci and cj are the average concentrations during hours i and j, Cc(ci) and Cc(cj) are the corresponding error variances for the concentrations in hours i and j, the characteristic correlation length L was assumed to be 1 h, and h is the length in time between observations i and j. The impact of this (albeit short) correlation length was assessed in a sensitivity tests discussed in the next section. No consensus has yet been reached on how these observation uncertainty correlations should be treated in city-scale inversions .

In order to assess the appropriateness of the uncertainty covariance matrices Cc and Cs0, the χ2 statistic, as described in , was calculated as follows: 13χ2=(Hs0-c)T(HCs0HT+Cc)-1(Hs0-c), with degrees of freedom equal to ν, the dimension of the data space, in this case the length of observations in the inversion.

The squared residuals from the inversion (squared differences between observed and modelled concentrations) should follow the χ2 distribution with degrees of freedom equal to the number of observations . The expected value of χ2/ν is one. Values lower than one indicate that the uncertainty is too large, and values greater than one indicate that the uncertainty prescribed is lower than it should be. The error in the assignment of the uncertainty could be in either Cc or Cs0 (or both). In order to ensure the suitability of Cs0, the prior uncertainty variances were multiplied by a factor of 2. This ensured that the χ2/ν statistic was close to a value of one for almost all months of the inversion. These details are provided in . Due to the length of time it takes to run a single inversion, we did not calculate an individual scaling parameter for each month.

As part of a project assessing the carbon sinks of South Africa , monthly 1 km × 1 km estimates of terrestrial carbon stocks and fluxes were produced . To estimate these fluxes, a distinction was made between carbon stocks in natural to semi-natural areas and those on transformed land, such as annually cropped cultivated land, plantation forests and urban areas (which was based on the IPCC 2006 value for closed urban forests). As a sensitivity test, the NEE and NPP from CABLE estimates used for the biogenic flux priors and their uncertainties were replaced with NEE and NPP from the carbon assessment product and the inversion rerun with these priors (inversion S1).

To estimate gross primary productivity (GPP), 10 years (2001 to 2010) of monthly climatologies (temperature, rainfall, relative humidity) and satellite products for photosynthetically active radiation (PAR) and fraction of absorbed photosynthetically active radiation (FAPAR) were assimilated. Autotrophic respiration (Ra) was calculated based on the inputs for temperature, above-ground biomass, below-ground biomass and FAPAR. NPP could then be calculated as NPP = GPP–Ra. The heterotrophic component (Rh) of ecosystem respiration (Re) was based on estimates of soil organic carbon stocks and above-ground litter. The basic calculation to obtain NEE was NEE = GPP–Re, and additional losses of CO2 through biomass burning and export and import fluxes from harvest and trade-related activities were accounted for.

To disaggregate the monthly products into day and night fluxes, it was assumed that all GPP took place during the day, and that half of Re occurred during the day and half at night. Therefore, the weekly NEE and NPP estimates used for the prior information in the inversion were based on the GPP and respiration products from the assessment. The carbon assessment estimated the GPP flux for the year in the fynbos biome to be 521 g CO2 m-2 yr-1, with a standard deviation of 492 g CO2 m-2 yr-1 across pixels with 1 km2 resolution. Therefore, as for the CABLE estimates used in the reference inversion, we assign uncertainties to the prior NEE estimates equal to the NPP estimate. A map of the prior daytime NEE fluxes in May 2012 from the CABLE and carbon assessment products is provided in Fig. . The prior estimates from the two products are deliberately plotted on the same scale to illustrate how much more homogeneous the carbon assessment estimates are compared with those from CABLE. For the CABLE product, regions with the highest productivity are associated with the largest uncertainties.

The homogeneity of the biogenic CO2 fluxes across the domain from the carbon assessment product can be explained by the products used as inputs for the estimation of the carbon stock components, such as FAPAR. These products would not be expected to differ considerably from pixel to pixel in this domain. CABLE predicts greater CO2 uptake. The average CO2 flux over the course of the study period and across the domain was -41 g CO2 m-2 week-1 according to the carbon assessment and -172 g CO2 m-2 week-1 according to CABLE. Both these estimates can be considered small. The true flux is likely to be highly variable but close to carbon neutral over a long period of time (several years).

As an alternative to the inventory analysis of the fossil fuel fluxes, we used current estimates of anthropogenic fossil fuel emissions from the 1 km × 1 km ODIAC (Open-source Data Inventory for Anthropogenic CO2 product) product for the years 2012 and 2013 (ODIAC2017; ; inversion S2). The product provides monthly emissions of CO2 in kt of carbon. The original ODIAC product made use of global energy consumption statistics and distributed the emissions from these activities based on known point source emitters, such as power plants, and based on a global nightlight distribution satellite product. Emissions from point sources, such as those from power plants, were estimated separately from the diffuse emissions, for example those due to transport. These emissions were disaggregated onto to a 1 km × 1 km grid. The updated product has further disaggregated the diffuse emissions to a 30 m × 30 m grid by making use of global road network data, a satellite product of surface imperviousness, and population census data . This 30 m × 30 m diffuse emission product together with the point source emission product were aggregated back up to the 1 km × 1 km grid. ODIAC has been shown to give comparable flux estimates when used in an inversion as a prior product in place of the ultra-high-resolution inventory product Hestia , carried out for Indianapolis, Indiana .

The ODIAC monthly estimates were rescaled according to the day of the week and to the hour of day using scaling factors for South Africa as estimated by . These estimates were re-aggregated into day and night and working week and weekend fossil fuel fluxes in units of kg CO2 m-2week-1. These estimates for the fossil fuel fluxes were used as prior estimates for the inversion in place of the inventory-based estimates used for the reference inversion. The daytime fossil fuel fluxes produced by the inventory analysis and the ODIAC product are provided in Fig. .

The ODIAC product gave similar fossil fuel fluxes over pixels in the CBD area compared with the inventory estimates. The inventory estimates were concentrated over the road network, point sources and areas of high population density, whereas the ODIAC product dispersed emissions over the domain, with an area of high concentration over the CT metropolitan area and decreasing emissions away from this region. The average fossil fuel flux for the domain over the study period was 134 g CO2 m-2 week-1 according to the inventory and 274 g CO2 m-2 week-1 according to the ODIAC product.

The specification of the prior uncertainty covariance structures has been shown to have a substantial impact on the pixel-level flux estimates, the total flux estimate for the domain, and on the spatial distribution of the fluxes . For example, in the Indianapolis inversion, assuming correlation lengths of 4 or 12 km in the prior uncertainty covariance matrix of the fluxes resulted in total flux estimates for the city that were 17 % and 25 % larger than the total flux estimate, assuming no correlation . The effect of changing the correlation length had a larger impact on the total flux estimate than changing the prior emission product from Hestia to ODIAC.

To assess the sensitivity of the posterior flux estimates, their uncertainties and their distribution in space to the specification of the uncertainty correlations, we ran inversions where the non-zero off-diagonal elements of Cs0 and Cc in the reference inversion were systematically set to zero. We considered an inversion, which assumed no temporal observation uncertainty correlation in the specification of Cc (inversion S3), an inversion where no spatial uncertainty correlations were assumed for Cs0 (inversion S4), and an inversion, which assumed no uncertainty correlations in the specification of Cs0 and Cc (inversion S5).

We tested what would happen if observation error correlations were set at 7 h (inversion S6) instead of 1 h, as was set for the reference inversion. A 1 h observation error correlation length results in nonzero off-diagonal covariance terms for up to approximately 7 h from the observations. Assigning a 7 h correlation length resulted in non-zero covariances extending through to at least a day away from the observation.

We also considered inversions where the prior fossil fuel flux uncertainty was doubled (inversion S7) and where it was halved (inversion S8), and similarly for the NEE flux uncertainties (inversions S9 and S10). By doubling or halving the uncertainty of the fossil fuel or NEE component of the total flux, we changed the relative uncertainty contribution each of these made to the total uncertainty when compared with the reference inversion.

Due to the large impact that the estimation of the domestic fossil fuel emissions had on the temporal profile of the total fossil fuel fluxes, we considered a modification of the estimated domestic emissions in the inventory product. In the reference inversion, 75 % of the domestic emissions from heating were assumed to take place during the 6 winter months. We tested the impact of this assumption by altering the domestic emissions so that they were distributed uniformly through time but still spatially distributed according to the population size. This changed the prior estimates of the fossil fuel fluxes and their distribution through time, as well as their uncertainties, which were set at 60 % of the domestic emission estimate (inversion S11).

Due to the large uncertainty in the modelling of NEE , particularly over the fynbos biome, we considered that perhaps the average of the NEE estimates from CABLE over the domain may be a more reliable representation of the true flux compared with the pixel-level estimates. Therefore, we averaged the NEE and NPP estimates from CABLE over the inversion domain and assigned this average NEE (and NPP for its uncertainty) as the prior biogenic flux estimates (inversion S12).

We considered an inversion where the uncertainties in Cc were set at 2 ppm during the day and 4 ppm at night (inversion S13), excluding the additional components for the error due to wind speed and observation variability that were used in the reference inversion. In this case all the errors in the modelled concentrations are contained within these values, and we disregard the climatic conditions under which the measurements were taken. We tested the impact of increasing the night-time uncertainty in the observation errors to 10 ppm (inversion S14). We further simplified Cc by using the simplified uncertainties of 2 ppm during the day and 4 ppm at night and also set the temporal observation uncertainty correlation to zero (inversion S15).

As a sensitivity analysis we examined two alternative approaches to the control vector. If we assumed that neither the NEE nor fossil fuel flux would change very much from week to week, an option would be to solve for the mean of the six individual fluxes over the 4 weeks in a given month. We therefore considered a sensitivity test where the inversion solved for one average day and one average night NEE flux within each pixel and four fossil fuel mean weekly fluxes (day and night working week, day and night weekend) (inversion S16). We also considered performing a separate inversion for each week, i.e. four separate weekly inversions in place of each of the monthly inversions (inversion S17). In this case only the concentration measurements for 1 week were used and the individual weekly fluxes (two NEE and four fossil fuel) were solved for, and this was repeated for each of the 4 weeks in the month. The benefit of these two alternative control vectors is that for each individual inversion the resulting Cs0 matrix is much smaller compared with the reference case.

When solving for only 1 week, or a mean weekly flux for a particular month, the number of surface sources reduced to 10 201 × 6 = 61 206. Solving for individual weeks required 4 × 2 additional boundary concentrations to be added to the control vector, and when solving for the mean weekly flux for the month, we allowed the boundary concentrations to differ for each week, and therefore we still solved for the 32 boundary concentrations as in the reference case. Therefore, the Cs0 for these two alternative control vectors is 16 times smaller than that of the reference inversion.

The benefit of these two alternative approaches is a substantial reduction (at least a 75 % reduction) in the time taken to perform the inversion. If the results are similar to that of the reference inversion, this type of saving in the computational time and resources would allow more components of the inversion to be tested in a shorter period of time.

A description of the sensitivity tests is presented in Table .

The modelled concentrations from each inversion were compared with the observations by assessing the bias and standard deviation of the prior and posterior modelled concentration residuals. Residuals in the prior modelled concentrations were calculated as follows: 14cresprior=c-cmodprior.

Residuals in the posterior modelled concentrations were calculated as follows: 15crespost=c-cmodpost, where cmodprior are the CO2 concentrations modelled from s0, cmodpost are the CO2 concentrations modelled from the posterior estimate of s, and cresprior and crespost are the respective residuals in the modelled concentrations. The bias, calculated as the mean of these residuals, and standard deviation of these residuals were provided for each inversion. We plotted the time series of the observed and modelled concentrations to assess the skill of the inversion for reproducing the observed concentrations, particularly “local events”, which were periods of larger than normal spikes in the observed concentration signal. These are presented in the Supplement (Sect. S1.3) for all the sensitivity tests.

The posterior fluxes from each inversion were compared with those of the reference inversion in a number of ways. The posterior flux estimates and their spatial distribution were assessed for each inversion by mapping the mean total weekly flux within each pixel for 2 months (May and September 2012). We calculated the total flux over the domain and plotted these weekly total fluxes over time together with the uncertainty bounds. We also considered the total flux over the domain for each month. These total flux estimates are the net flux resulting from the fossil fuel and NEE flux estimates solved for by the inversion. The inversion induces negative correlations between the fossil fuel and NEE flux components from the same week and pixel. When the total flux is considered in a particular pixel, the uncertainty for the total flux will be lower than the sum of the uncertainties for the individual components due to the negative covariance terms. The size of these negative covariances will depend on the prior information specified in the inversion framework. The total estimate gives an indication of the central tendency, which we can compare between inversions, and allows us to assess, for example, if the inversion is predicting the region to be a net source or a net sink. The uncertainties of these posterior total estimates allow us to assess the confidence we can place around these totals, and how this compares to the estimate itself.

In order to assess the suitability of the prior uncertainty estimates contained in Cc and Cs0, the χ2 statistic as described in was calculated (see Eq. ). We compared these statistics between the different inversions to assess the suitability of the uncertainties prescribed to the prior fluxes. Due to the adjustments made, particularly in cases where the uncertainty covariance matrices were simplified, it was expected that some of the inversions would have χ2 statistics that deviated from 1. We chose not to make additional changes to the sensitivity test inversions to improve these statistics, as it would then not be possible to attribute the sensitivity of the inversion solution between the adjustment tested and the additional adjustment made to the covariance parameters to improve the statistical consistency of the inversion. The number of degrees of freedom of the χ2 statistic can be divided into the degrees of freedom for signal (DFS) and degrees of freedom for noise . The DFS describes the number of independent pieces of information provided by the measurements. The DFS were calculated for the first week of March 2012 for the reference and sensitivity test inversions. These statistics are provided in the Supplement Sect. S1.1 Fig. S1.