An atmospheric inversion was performed for the city of Cape Town for the period of March 2012 to June 2013, making use of in situ measurements of

The inversion was shown to be sensitive to the spatial error correlation length in the biogenic fluxes – even a short correlation length – influencing the spatial distribution of the posterior fluxes, the size of the aggregated flux across the domain, and the uncertainty reduction achieved by the inversion. Taking advantage of expected spatial correlations in the fluxes is key to maximizing the use of a limited observation network. Changes to the temporal correlations in the observation errors had a very minor effect on the inversion.

The control vector in the original version consisted of separate daytime and night-time weekly fluxes for fossil fuel and biogenic fluxes over a 4-week inversion period. When we considered solving for mean weekly fluxes over each 4-week period – i.e. assuming the flux remained constant over the month – larger changes to the prior fossil fuel and biogenic fluxes were possible, as well as further changes to the spatial distribution of the fluxes compared with the reference. The uncertainty reduction achieved in the estimation of the overall flux increased from 25.6 % for the reference inversion to 47.2 % for the mean weekly flux inversion. This demonstrates that if flux components that change slowly can be solved for separately in the inversion, where these fluxes are assumed to be constant over long periods of time, the posterior estimates of these fluxes substantially benefit from the additional observational constraint.

In summary, estimates of Cape Town fluxes can be improved by using better and multiple prior information sources, and particularly on biogenic fluxes. Fossil fuel and biogenic fluxes should be broken down into components, building in knowledge of spatial and temporal consistency in these components into the control vector and uncertainties specified for the sources for the inversion. This would allow the limited observations to provide maximum constraint on the flux estimates.

Bayesian inverse modelling provides a top-down technique for verifying emissions and uptake of carbon dioxide (

Estimates of city-level

Verifying the accuracy of inventory-based estimates of emissions has become essential

Atmospheric inversions at the city scale are limited by available

Atmospheric monitoring sites targeting CT air masses were not available; therefore, temporary measurement sites were installed at Robben Island and Hangklip lighthouses, located to the north-west and south-east of the metropolis

One way that CT differs from the mega cities that previous inversions have targeted

We adopted the approach usually used from regional inversions, where the inversion modelled the concentrations at the measurement sites

The specification of the uncertainty covariance matrices substantially influences the inversion result

Additionally, we were interested in the composition of the control vector, also referred to as the state vector, which specifies the surface fluxes and domain boundary concentrations to be solved for by the inversion. The composition of this vector is determined by the size of the source pixels and the time length over which we assume the fluxes are homogeneous. This in turn impacts the assigned uncertainty covariance matrix. For the reference inversion we carried out 13 four-week inversions, which solved for weekly fluxes from each of the 101

The purpose of this paper is to present the results of these sensitivity tests in comparison with the CT reference inversion presented in

The Bayesian synthesis inversion method, as described by

Minimizing this cost function leads to the following solution:

The total

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

The reference inversion made use of two

CCAM is a variable-resolution global atmospheric model developed by the Commonwealth Scientific and Industrial Research Organisation (CSIRO)

The Jacobian matrix,

Previously, we modified the approach of

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

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

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

CCAM was dynamically coupled to the land surface model CABLE

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

The presence of the Cape Point GAW station provided a source of background

The Cape Point measurements of the background

The boundaries of the domain were deliberately set to be far from the measurement sites so that contributions to the

Error propagation techniques, as described in

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

To estimate spatial uncertainty covariances in the NEE fluxes, we assumed an isotropic Balgovind correlation model as used in

The observation uncertainties represented in

We added additional error estimates to these minimum observation errors. We assumed errors in modelled

The off-diagonal elements of

In order to assess the appropriateness of the uncertainty covariance matrices

The squared residuals from the inversion (squared differences between observed and modelled concentrations) should follow the

As part of a project assessing the carbon sinks of South Africa

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

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

The homogeneity of the biogenic

Spatial distribution of the prior daytime NEE fluxes produced by CABLE

As an alternative to the inventory analysis of the fossil fuel fluxes, we used current estimates of anthropogenic fossil fuel emissions from the 1

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

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

Spatial distribution of the prior fossil fuel fluxes produced from the Cape Town inventory analysis

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

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

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

We considered an inversion where the uncertainties in

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

When solving for only 1 week, or a mean weekly flux for a particular month, the number of surface sources reduced to 10 201

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:

Residuals in the posterior modelled concentrations were calculated as follows:

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

Description of sensitivity tests performed on the Cape Town inversion. Only those aspects that are changed for the sensitivity test are indicated. Other fields are the same as those for the reference inversion.

The results of the reference inversion (S0) are explained in detail in

The direction of the adjustments to the prior biogenic fluxes indicated that the CABLE model was overestimating the amount of biogenic carbon uptake over natural areas. Dynamic vegetation models have not been able to simulate fluxes over the fynbos biome well

Large uncertainty reductions were made over the natural areas bordering on the CBD, particularly over the Table Mountain National Park, and to natural areas near the Hangklip measurement site, where the uncertainty was lowered by over 50 %. Large uncertainty reductions also occurred over agricultural areas to the north of the CBD region. Uncertainty reductions of up to 60 % occurred over a few central CBD pixels but were generally smaller compared with the uncertainty reductions over natural areas, which reached as high as 92 %. When aggregating the fluxes over the domain, uncertainties in the prior aggregated fossil fuel fluxes ranged between 1.3 and 1.5

By assigning spatial correlation between biogenic flux uncertainties of neighbouring pixels and assuming independent fossil fuel flux uncertainties, we attempted to provide the inversion with additional information to allow it to better distinguish between these fluxes. The inversion-induced negative correlation between fossil fuel and biogenic flux uncertainties in the same pixel. We demonstrated that the posterior uncertainty of any linear combination of terms from the control vector of the fluxes (including the difference between fluxes from the same pixel and the sum of fluxes from the same pixel) will always be unchanged or smaller compared with the prior uncertainty of the same linear combination of elements

Clearly, the inversion result was strongly dependent on the assumptions regarding the prior fluxes and their uncertainties. The results of the sensitivity tests in subsequent sections explore to what degree these assumptions affected the inversion solution.

To assess the sensitivity of the inversion, we have calculated the aggregated posterior flux across the study period and over the full spatial domain, together with the posterior uncertainty and uncertainty reduction for each of the sensitivity tests, which are presented in Fig.

The aggregated fluxes were strongly sensitive to the uncertainty spatial correlations specified between the biogenic fluxes. Uncertainty correlations in the biogenic fluxes had a large impact on the spatial distribution of the resulting fluxes, and on the degree to which the inversion was able to make changes across the full domain (Fig.

A short temporal correlation length in the observation uncertainties did not have a large impact on the inversion. Increasing these to 7 h led to greater DFS (see Supplement Sect. S1.1 Fig. S1) but without having an impact on the flux solution or uncertainty reduction. The statistical consistency also fluctuated much more strongly from month to month when the temporal observation error correlation was larger compared to a 1 h correlation length or assuming independent observation uncertainties. With a correlation length of 1 h, non-zero off-diagonal elements persisted for approximately 7 h, whereas these off-diagonal elements persisted for much longer when the correlation was set at 7 h. Long correlation lengths are likely not realistic as wind fields observed at the measurement station during the day may be very different to those observed in the evening, reducing the chance of consistent errors in concentration.

The sensitivity test with the smoothed prior biogenic flux over the full domain produced the only posterior flux solution that was corrected to be further from the reference inversion posterior. This inversion did not assume any knowledge about the spatial variability in the surface fluxes, but it appears that providing at least some prior knowledge of where biogenic fluxes are likely to occur – at least separating the ocean and terrestrial fluxes – was important for a sensible posterior flux solution. The domain is not fully or representatively sampled by the observations. By providing a blanket biogenic flux prior across the domain, areas with large expected biogenic fluxes, which were well sampled by the observation network, had priors that were too carbon neutral, and thus biogenic fluxes were made more negative, which was propagated through to neighbouring biogenic fluxes, resulting in a posterior aggregated flux solution that was more negative than the prior. A blanket uncertainty estimate was also used, which meant that the uncertainty associated with the ocean fluxes was much larger compared with the reference inversion, allowing the inversion to make relatively large changes to oceanic pixel fluxes close to the measurement sites.

While all the sensitivity test inversions produced prior modelled concentrations that did not track the observations well (see Supplement Sect. S1.3 Figs. S10 to S27), the carbon assessment and ODIAC prior product inversions (S1 and S2) produced prior modelled concentrations that were on average too large compared with the observed concentrations at both sites, whereas the reference inversion (S0) underestimated the concentrations at Hangklip and overestimated the concentrations at Robben Island (Figs.

The carbon assessment total prior fluxes were notably different to those from ODIAC or the reference inversion. There was little seasonal variation, with fluxes remaining net positive throughout the study period. The uncertainty bands were very narrow, based on the carbon assessment NPP. The mean

The reference inversion generally made fluxes more positive, except for a few winter months when the innovations made fluxes more negative. The S2 inversion had innovations that made the fluxes more negative compared to the priors, except for September 2012. S1's innovation was to make the fluxes more negative for each month. The magnitudes of the innovations were smaller compared to those made to S0 and S2 prior fluxes, limited by the uncertainty placed on the prior biogenic fluxes. For the S1 inversion, both the biogenic flux uncertainties and the correlation lengths were smaller compared to those for S0, and therefore the posterior fluxes were not allowed to differ much from the prior, leaving the modelled concentration residuals before and after the inversion to be very similar, and posterior fluxes almost as uncertain as the prior fluxes.

The spatial pattern in the fluxes (Supplement Sect. S1.6 Figs. S56 and S57), as reflected in the time series pattern in the weekly fluxes (Fig.

With regards to the uncertainty reduction, the S0 inversion was able to obtain higher reductions than either S1 or S2 (Figs.

Altering the domestic fossil fuel emissions to be the same over time in S11 had little impact on the inversion results when compared with the wholesale change in the prior product. On the other hand, smoothing the biogenic emissions over space in the extreme manner where it was assumed NEE fluxes were the same throughout the domain (S12) had a large impact on the inversion. This resulted in the only inversion where the aggregated fluxes became more negative. The uncertainty reduction was also small (Fig.

Due to the small number of observations relative to the number of sources solved for in the inversion, it is unsurprising that the posterior solution is strongly dependent on the prior information. The results do show that the inversion brings these different prior estimates closer to each other, and therefore the inversion does assist in taking any selected prior closer to the true state, but this is limited by the assumed uncertainty limits placed on the priors, as demonstrated in the S1 inversion.

Prior and posterior modelled concentrations for the Hangklip site for the month of May 2012 for the reference inversion

Prior and posterior modelled concentrations for the Robben Island site for the month of May 2012 for the reference inversion

Prior and posterior aggregated weekly fluxes over the inversion domain from March 2012 to June 2013 for the reference, carbon assessment and ODIAC inversions. The dashed line represents prior flux estimates and the solid line represents posterior flux estimates.

The inversion solution was sensitive to the uncertainty spatial correlations assigned to the prior biogenic fluxes. This impacted on the spatial distribution of the fluxes, the magnitude of the total aggregated flux, and the uncertainty reduction achieved by the inversion. By not accounting for the spatial correlations in the biogenic flux uncertainties, this led to uncertainties that were too small, illustrated by average

In comparison, the removal of the temporal correlation in the observation errors in S3 only had a small penalty in the

As the flux uncertainties had already been scaled for the reference inversion to improve the statistical consistency of the uncertainty covariance matrices, it was expected that the

The spatial distributions of the posterior fluxes in this group of sensitivity tests (S7 to S10) were similar to that of the reference inversion S0. A notable feature in the September 2012 posterior fluxes is that when NEE uncertainties were doubled the inversion was able to reduce the aggregated flux with respect to the priors by creating a region of negative flux in an area close to the oil refinery point source to the north of the CBD region (see Supplement Sect. S1.6 Fig. S73).

Spatial distribution in the pixel-level uncertainty reduction achieved by the inversion to the prior fluxes in May 2012 for the reference inversion (S0)

S0 and S17, where separate weekly inversions were performed, had similar aggregated fluxes (Fig.

The spatial distribution of the posterior fluxes was very similar for S0 and S17 (see Supplement Sect. S1.6 Fig. S89) but was distinctly different for S16. Notably, the area around the oil refinery pixel was adjusted to negative fluxes for the month of September (Fig.

Consequently, the aggregated fluxes for S16 had uncertainty reductions that were twice as large as those for S0 and uncertainties in the aggregated fluxes were much smaller. For the aggregated flux over the full period, the posterior uncertainty was 66

Spatial distribution of the posterior fluxes and uncertainty reductions achieved by the reference inversion S0 and mean monthly flux inversion S16 for September 2012.

As Robben Island is dominated by fossil fuel influence from the Cape Town metropolitan area, and Hangklip is dominated by biogenic sources from natural and agricultural areas in its vicinity, the discrepancy in the modelled concentrations relative to the observations suggested that the fossil fuel fluxes provided by the prior products are too large in magnitude, and CABLE estimated too much carbon uptake by the biota around the Hangklip site. In the case of the carbon assessment inversion, the bias in the prior modelled concentrations was positive compared with the negative bias of the reference inversion, indicating that the carbon assessment product was underestimating the uptake by the biota. The direction of the correction to the prior fluxes made by the inversion using NEE fluxes from the carbon assessment product suggested that the amount of carbon uptake was insufficient. The NEE fluxes were also smaller compared to those from CABLE, leading to uncertainties that were too small and thus an ill-specified inversion. The inversion could not correct the fluxes sufficiently so that modelled concentrations could match better with observed concentrations, and therefore certain localized events (i.e. spikes in the

The comparison of inversion results using different prior products provides useful information regarding which direction the true flux estimates are likely to be. A pixel within the CBD limits had similar fossil fuel flux estimates from the ODIAC product compared with the reference inventory product, but the ODIAC product had emissions that were more widespread across the domain away from the CBD. This led to aggregated estimates that were larger under the ODIAC inversion than the reference inversion. Compared to the reference, the ODIAC inversion attempted to reduce the aggregated flux for most months – and to a greater degree – to better match the observations, indicating that compared with the reference inventory, the ODIAC prior was most likely overestimating the amount of fossil fuel emissions from Cape Town to a greater extent for most parts of the study period. When the two prior information products provide divergent prior flux estimates, such that the inversion reduced the flux for one product but increased the flux for the other, it suggests that the true flux lies somewhere between the posterior flux estimates from these two inversions. When the posterior aggregated flux was made smaller than the ODIAC prior but larger than the reference prior aggregated flux, such as during February and March 2013, the true aggregated flux should lie between these two posterior estimates. When the posterior flux was made smaller than the prior for both inversions, we could deduce that the true aggregated flux must be below the minimum of these two posterior estimates, and if we have accurate uncertainty estimates, the true flux should be no smaller than the lower uncertainty limit. Making use of the posterior uncertainties and the direction away from the prior in which the inversions made corrections, a region is suggested where the true flux is most likely to lie (Fig.

Using the posterior estimates of the reference and ODIAC inversions (S0 and S2) and the direction of change from the prior estimate, a region is inferred showing where exactly the true aggregated flux is expected to lie (indicated by the shaded pink area).

From the analysis of the reference inversion

The impact of the inversion on the posterior fluxes and their uncertainties strongly depended on the specification of the correlation between the uncertainties in the NEE fluxes. In particular, the aggregated fluxes were distinctly different between the reference and test cases ignoring covariances between NEE flux uncertainties, which tended to have aggregated fluxes closer to the priors and uncertainty reductions achieved by the inversion that were much lower (7.6 % compared with 26.6 % on average by the reference inversion). This indicates that advantage should be taken of knowledge related to the correlation induced by homogeneity of biogenic productivity in subregions of the domain. If this correlation is correctly specified in

Specification of the uncertainties in the prior flux estimates is one of the most challenging tasks in an atmospheric inversion exercise. There is little consensus on the correct approach to follow, and it is difficult to ensure that the most important sources of uncertainty are accounted for. The

An inversion will nudge the flux solution closer to the truth and will always result in reduced uncertainty compared to that which was placed on the prior. If the prior estimates for the fluxes are far from the truth and the uncertainties are made small, the modelled concentration residuals will be similar before and after the inversion, and uncertainty reduction will be small. Therefore, the uncertainties need to be correctly specified to allow the inversion to correct the fluxes as close as possible to the true fluxes. Ideally, large enough to give the inversion the freedom to correct the fluxes towards the truth but small enough so that the posterior uncertainty is within the required limits. This motivates for the hierarchical Bayesian approach where a distribution is assigned to the uncertainty estimates. It can be shown that in the absence of observation error, doubling or halving the prior uncertainty in the fluxes results in a respective doubling or halving of the posterior uncertainty (see Supplement Sect. S1.7). Therefore, it us unsurprising that if a prior uncertainty is made larger with respect to a reference inversion specification that the posterior uncertainty of this inversion will be larger than the posterior uncertainty of the reference.

Normally when an inversion framework is assessed, we are interested in how much uncertainty reduction can be achieved by the available observation network. The uncertainty reduction is dependent on the influence of the observations and on how well the prior information is specified. This set of sensitivity tests demonstrated that if we wish to ensure that the uncertainty bounds around the posterior fluxes are within a prespecified margin, say 10 % of the aggregated flux estimate, then we have to ensure that we know enough about the sources such that the prior uncertainty we begin with is sufficiently small. Assuming no large shifts in the mean estimate, it can be shown that if we wish to obtain an uncertainty estimate that is within 10 % of the aggregated flux estimate, and we are able to reduce the uncertainty by 25 % through the inversion as we have achieved in the Cape Town inversion, then the prior uncertainty estimate would need to be within 13.3 % of the prior aggregated flux estimate.

Simplifying the

The separate weekly inversions obtained similar results to those of the reference inversion. Therefore, if necessary, for example due to computational costs, the separate weekly inversions could have been performed in place of the monthly inversions used in the reference case. In addition to the reduction in computation resources required, this allows additional features of the inversion to be tested more easily.

The large uncertainty reduction achieved by the solving for a mean weekly flux is expected, as a mean weekly flux estimate over 4 weeks has 4 times as many observations to constrain this estimate as separate weekly estimates. The estimates from the inversion solving for a mean weekly flux were consistent with those from the reference inversion, except in the summer months. During these months observations were often missing. We would expect smaller discrepancies between mean weekly and separate weekly fluxes if data were complete during these periods.

An alternative control vector, which could improve on all three of the alternative control vectors used in this study, would be to solve for separate components of fossil fuel and NEE fluxes. For example, if fossil fuel fluxes were split into those fluxes from sectors that change slowly and those that change more quickly, the inversion could solve for a mean weekly flux over the month for the slow fluxes, and for sectors with faster changes, the inversion could solve for individual weekly fluxes. This would allow greater uncertainty reductions for those fluxes for which a mean weekly flux could be solved, which would in turn reduce the overall uncertainty in the aggregated fossil fuel flux. The NEE flux could also potentially be split into a slow and fast component. The fast component responds to local climate conditions and this component could be tightly constrained by the available climate data. The inversion could solve for the slower component, which is much harder to model, allowing this estimate to be constant for a relatively long period, thereby allowing for stronger constraint from the observations.

If we consider the aggregated posterior fluxes, the variability between flux estimates across those inversions that used the reference control vector is 1962

Exceptions are the inversions that changed the prior estimates of the fossil fuel fluxes. The fossil fuel fluxes were not assigned uncertainty correlations. Those inversions that altered the prior estimates of the fossil fuel fluxes also had aggregated fluxes that differed when compared with the reference inversion. This is due to the inversion having limited ability to make large changes to the fossil fuel fluxes. The ensemble of posterior fluxes obtained from inversions with alternative prior fluxes allowed us to determine in which direction the inversion was attempting to adjust these fluxes and provided us with an interval in which we could deduce the true aggregated flux would most likely be located. Changing the control vector also had a large influence on the aggregated flux, but this was largely due to periods with low data completeness.

Sensitivity tests have shown that to improve the inversion results for the Cape Town inversion, two important advancements should be made to the inversion framework. Firstly the NEE estimates need to be improved. The results from the reference inversion and from these sensitivity tests clearly indicate that CABLE is generally overestimating the amount of

Solving for mean weekly fluxes over a month produced much larger uncertainty reductions. Using an alternative control vector that solves for separate components of the fossil fuel and NEE fluxes that can be split into slow and fast components could take advantage of the larger uncertainty reduction achieved from solving for a mean weekly flux for each month. This could potentially allow the inversion to better distinguish between NEE and fossil fuel fluxes, allowing the inversion to apply corrections to the right flux component (fossil or biogenic) and at the same time obtain aggregated flux estimates with smaller uncertainties than those obtained for the reference inversion. The estimates of the aggregated fluxes were shown to be more reliable in the reference inversion than those for the individual fossil fuel and NEE fluxes

The posterior uncertainties are highly dependent on the prior uncertainties. Of more concern is the large impact that the uncertainty correlation assumed for the NEE fluxes had on the aggregated flux estimates and on the spatial distribution of the posterior fluxes. This has been observed in previous inversions

Approaches that allow the data to inform the estimates of the uncertainties and correlation lengths are likely to be more successful at obtaining estimates of the true uncertainty bounds around the inversion posterior flux estimates.

These sensitivity analyses performed for this paper did not consider alternative atmospheric transport models. Sensitivity tests on previous city-scale inversions have shown this to be an important source of variation between inversion results

The hourly

The supplement related to this article is available online at:

AN installed and maintained all the instrumentation at Robben Island and Hangklip, obtained the measurements and processed these into hourly concentrations, ran and processed the result of the LPDM in Fortran, produced all code and ran the inversion in Python, processed all the inversion results using R statistical software, produced all graphics and tables, designed the sensitivity tests, and was responsible for the development of the paper, which forms part of her PhD. PJR was the main scientific supervisor, oversaw all implementation of the inversion, and provided guidance on the presentation and interpretation of results. FE performed the coupled CCAM-CABLE simulations. BE provided guidance on statistical issues. RJS provided guidance on the location of the sites and provided input on the interpretation of the results. All authors commented on the manuscript.

The authors declare that they have no conflict of interest.

We would like to acknowledge and thank Casper Labuschagne, Ernst Brunke, and Danie van der Spuy of the South African Weather Service for their assistance in maintaining the instruments at Robben Island and Hangklip; Casper Labuschagne for his guidance on processing the instantaneous

This research has been supported by the Council for Scientific and Industrial Research (grant nos. EEGC030 and EEGC066).

This paper was edited by Neil Harris and reviewed by Neil Harris and one anonymous referee.