The inversion scheme for optimizing CO2 surface fluxes over Australia involves a Bayesian four-dimensional variational assimilation system. The system is a generalized minimization-based inverse-modelling framework, which can be applied to several potential models. We refer to it hereafter as “py4dvar”. py4dvar finds an optimal estimate of the CO2 surface fluxes (xa) that fits both observations (y) and the prior fluxes (xb) . Assuming Gaussian PDFs, finding this maximum a posteriori estimate is equivalent to minimizing the cost function J(x) shown in Eq.  .

1J(x)=12x-xbTB-1x-xb+12H(x)-yTR-1H(x)-y

The first term in Eq. () represents the sum of squared differences between the control variable (x) and its prior or background state (xb). The second term measures the sum-of-squared difference between the model simulation, H(x), and observations (y) during the time window of the assimilation. The term H(x) is the function composition of an atmospheric transport operator and an observation operator. Both terms in Eq. () are weighted by their respective error covariance matrices (B and R), and the errors are assumed to be Gaussian and bias free. As mentioned in the previous paragraph, the minimum of J(x) is found by an iterative process rather than by an analytical expression. The minimization inside py4dvar is performed using the limited-memory BFGS (L-BFGS-B) algorithm, as implemented in the scipy python module . The minimization algorithm, L-BFGS-B, requires values of the cost function and its gradient, which are calculated using the CMAQ forward model and the adjoint model, as shown in the third step in Fig. . 2∇xJ=B-1x-xb+HTR-1H(x)-y

The gradient of the cost function in Eq. () is calculated using the adjoint of the CMAQ model version 4.5.1;. We can observe that in the second term in Eq. (), the adjoint model (HT) is applied to the vector R-1H(x)-y, which is often called the “adjoint forcing”, or simply the “forcing”, and represents the error-weighted differences between the forward model and the observed concentrations. Applying the adjoint model to the forcing, running backward in time from ti-1 to t0, allows us to construct the gradient of the cost function, ∇xJ(x).

Our underlying physical variables are the monthly averaged fluxes at the spatial resolution of CMAQ (≈81 km). We do not split fluxes by day and night, consistent with only using daytime satellite observations, which are not subject to much influence by diurnal cycles in CO2 fluxes e.g.. Like most previous studies e.g. we use spatially correlated prior uncertainties to account for systematic errors in flux estimates. The variables exposed to the minimizer are not the fluxes themselves but rather multipliers for the principal eigenvectors of B. We truncate the eigenspectrum at 99 % of the total variance; doing this significantly reduces the size of the control vector x (relative to if the control vector was comprised of the fluxes at each grid cell). This requires a different number of eigenvectors for different months (Table ). The length of the control variables for our sensitivity experiments are defined in Table . Similar to , and because our inversion assimilation window is short, we also include (in the state vector for the inversion) a perturbation to the initial conditions (ICONs) of the CO2 concentration field. Because we are not interested in the analysis of this field, and in order not to significantly increase the size of the control vector, we added a scaling factor for the ICONs to our control variables x=i0,e0,e1,…,en, where i0 is the factor we solve for ICONs and en is the number of eigenvectors. The scaling factor was applied to the full three-dimensional concentration field. Some freedom in the initial condition avoids fluxes being unduly influenced by a mismatch in the initial concentrations. We assumed 1 % (≈4 ppm) uncertainties for the scaling factor.

We used OCO-2 level 2 satellite data (Lite File Version 9) distributed by the National Aeronautics and Space Administration (NASA) (available for download from https://oco2.gesdisc.eosdis.nasa.gov/data/s4pa/OCO2_DATA/, last access: 18 January 2020). We used the column-averaged dry air mole fraction of CO2, referred to as XCO2. We selected bias-corrected data, as described by . We used nadir and glint soundings over land that were flagged as good quality, except in some of our sensitivity experiments (described in Sect. ) in which we excluded glint mode data. We computed a weighted average for all OCO-2 measurements using a two-step process similar to . The first step is to average all the soundings into 1 s intervals and the second is to average these 1 s averages into the CMAQ vertical columns (81 km × 81 km) for each satellite pass, where the transit time over the CMAQ grid cell is about 11 s. For the 1 s averaging process, the weighted averaging is defined in Eq. (). 3x^CO2=∑i=1nwi×xCO2,i∑i=1nwi, where wi=1σi2 is the squared reciprocal of the OCO-2 uncertainties (σi). To get the uncertainties of these averaged soundings, we considered three different forms of uncertainty calculation (similar to ). First if we assumed that all errors are entirely correlated in a 1 s span, we can define the uncertainties as shown in Eq. (). 4σs2=1N∑i=1Nσi2

However, and because the average shown in Eq.  is sometimes low, we also considered the standard deviation of the XCO2 measurements (here referred to as the spread, or σr, of the OCO-2 measurements). In other words, if the spread (σr) of the XCO2 measurements was higher than the XCO2 uncertainty (σi), we used the spread value as shown in Eq. (). We did this because the spread in OCO-2 measurements may reflect real differences across the field within a 1 s time span. 5σr2=1N∑i=1Nx‾CO2-xCO2,i2

Third, we also considered a baseline uncertainty (σb), based on an error floor (ϵ) over land and ocean, as shown in Eq. (). We did this because sometimes we did not have enough OCO-2 soundings to compute a realistic spread. The values for our baseline uncertainties were taken to be 0.8 and 0.5 ppm over land and ocean, respectively. Finally, and after defining the uncertainties for the 1 s averages, we choose the maximum value between σs, σr and σb. 6σb2=ϵbase2N

The second step was to take these 1 s averages and average them within the CMAQ vertical columns using Eq. . 7x‾CO2=∑j=1Jwj×x^CO2∑j=1Jwj, where wj=1σj2 represents the squared reciprocal of the uncertainties average in the 1 s span (σj) and J is the number of those 1 s values. The average uncertainty over the CMAQ domain (Eq. ) was similar to the procedure outlined for 1 s average in Eq. (). However, we also added a term to represent the contribution of the model uncertainty (σm). We assumed that the model had a uncertainty of about 0.5 ppm. The observational error covariance matrix R was assumed to be diagonal. 8σ‾2=1J∑j=1Jσj2

After averaging the OCO-2 sounding over the CMAQ domain, we generated a set of pseudo-observations as described in step 1 of Fig. . In this process, we run the CMAQ model forward. We start with an assumed set of CMAQ inputs, which includes fossil fuel emissions, fires, land and ocean fluxes (see Sect.  for a description of these fluxes). Our py4dvar system takes in a vector x representing perturbations to the assumed emission profile, which is set to all be zeros in the “true case” and converts it into a format accessible to the CMAQ model (e.g. copying the monthly-average values into the hourly-resolution that the CMAQ model is configured to run with). These perturbations to the emissions (zero values in the true case) are then added to the assumed emission profile for CMAQ before the model is run to produce a four-dimensional CO2 concentration field, as is in step 2 of Fig. . Fourth, this modelled CO2 concentration field is then transformed using the OCO-2 observation space. Once it is transformed, we perturbed the “true observations” with Gaussian random noise to generate pseudo-observations as follows. 9y′=ysim+R1/2⋅p

The first term of Eq. (), ysim, represents the OCO-2- simulated observations using the true fluxes. The second term of Eq. (), p, is a vector with the same size as ysim and contains normally distributed random numbers with mean zero and variance of one. Scaling p by the square root of R ensures that the resulting realization has the assumed error distribution.

As is stated in Sect. , the CMAQ model needs hourly emissions to run forward in time. We use the atmospheric convention that a negative flux value indicates an uptake by the surface and a positive value means a release of carbon to the atmosphere. Our total fluxes were comprised of four datasets representing elements of the CO2 fluxes: terrestrial biospheric exchange, fossil fuels, fires and air–sea exchange. Hourly biosphere CO2 fluxes were calculated by combining two datasets: the net ecosystem exchange (NEE) at 0.5∘×0.5∘ and daily resolution and the gross primary production (GPP) at 0.5∘×0.5∘ and 3-hourly resolution from the Community Atmosphere Biosphere Land Exchange (CABLE) model (Vanessa Harverd, personal communication, 2018).

The post-processing of 3-hourly NEE data involved four steps. First, we calculated daily GPP. Then we used daily GPP to estimate the daily ecosystem respiration (ER); in terms of carbon balance, the ER can be calculated as ER=GPP-NEE. Finally, daily ER was assumed equal throughout the day and subtracted from 3-hourly GPP to obtain 3-hourly NEE. These 3-hourly NEE fluxes were interpolated to hourly resolution. Recall that for our OSSEs, only the uncertainties, not the values themselves, are used. Given that the optimization was performed to optimize monthly fluxes, the uncertainties were computed with monthly resolution. We assumed that the biosphere flux uncertainties were equal to the net primary production (NPP) simulated by CABLE, with a ceiling of 3 g C m-1 d-1 following .

Fossil-fuel CO2 emissions were obtained from the Fossil Fuel Data Assimilation System (FFDAS) . For this study, we used the 2015 FFDAS dataset (Kevin Robert Gurney, personal communication, 2018). The FFDAS uncertainty estimates were created by multiplying the FFDAS emissions dataset with a factor of 0.44. This factor was calculated by linear regression between the mean fluxes and the spread of an ensemble of 25 realizations of posterior CO2 fluxes, following . We did not directly use those realizations to get the posterior FFDAS uncertainties, because the realizations only contained emissions over land (i.e. excluding domestic, aviation, and maritime emissions). These “missing” emissions were taken from the Emissions Database for Global Atmospheric Research (EDGAR) . The highest value of FFDAS uncertainty over land was 2.3 g C m-2 d-1 and over ocean was 0.5 g C m-2 d-1. This surprisingly large value over the ocean was a coastal point coinciding with Perth (Western Australia), where one of the largest and busiest general cargo ports in Australia is located.

Fire emissions were taken from the Global Fire Emission Database, version 4 (GFEDv4). This version of GFEDv4 provides gridded monthly fire emissions at 0.25∘ . The GFEDv4 product combines four satellite datasets: the Moderate Resolution Imaging Spectroradiometer (MODIS) burned area data product with active fires, data from the Tropical Rainfall Measuring Mission (TRMM) Visible and Infrared Scanner (VIRS) and the Along-Track Scanning Radiometer (ATSR). We used biomass-burning carbon emissions, a product based on GFEDv4 and the Carnegie Ames Stanford Approach (CASA) biosphere model . Within the CASA model, fire carbon losses are calculated for each grid cell and month, based on fire carbon emissions based on burned area from the GFED dataset. We assumed uncertainties for GFEDv4 corresponding to 20 % of the biomass-burning carbon emissions.

Ocean CO2 fluxes were derived from the Copernicus Atmospheric Monitoring Service (CAMS) version 15r2 . The CAMS dataset is a global retrieval product, with a horizontal resolution of 3.75∘ in longitude and 1.875∘ in latitude at 3-hourly temporal resolution. Prior ocean fluxes estimated by CAMS were based on . We assumed that the error statistics were uniform, 0.2 g C m-2 d-1, over ocean, as in .

After defining the emission profiles and their uncertainties, we incorporated spatial correlations into our prior error covariance matrix B. We assume no temporal correlations. This differs from , who used a temporal correlation length of 4 weeks; though, this would only introduce weak correlations among our monthly averaged fluxes. Following Sect. 3.1.1, the spatial correlation between grid points r1 and r2 was defined as 10C(r1,r2)=exp⁡-d(r1,r2)/L, where d(r1,r2) is the distance (in km) between the two grid points, and L, the correlation length, was assumed to be 500 km over land and 1000 km over ocean following .

After defining B, we performed an eigendecomposition, B=WTwW, where W is a matrix of eigenvectors and w is a diagonal matrix of corresponding eigenvalues. Figure a shows the cumulative percentage variance and demonstrates that 20 eigenvectors account for about 60 % of the variance in B. We truncate the eigenspectrum to retain 99 % of the overall variance. The number required varied each month but was at most 400, compared to approximately 6700 grid points. The main reason for this strong truncation is the large correlation length relative to the CMAQ grid resolution. We will test and discuss this later.

We solve the minimization with a change of variable xb. Given that our control vector x depends on the size of the multipliers of the principal eigenvectors of B, our vector xb was reconstructed (as is given in Eq. ). This reconstruction includes a new vector q, which is normalized by the square root of the eigenvalues of B; this transformation involves minimization with respect to q, rather than xp.

This step (often called pre-conditioning) accelerates convergence. It also simplifies the system since, all target variables have unit standard deviation. In our case, where we solve for perturbations around a background state, they also have a true value of zero. Generating our prior flux for the inversion is achieved by defining a vector of normally distributed random numbers with unit standard deviation and zero mean. The process to generate the pseudo prior is represented in Eq. (). 11xb=xp+WTw1/2q

We used the CMAQ modelling system and its adjoint version 4.5.1; to conduct numerical simulation of the atmospheric CO2 concentration over the Australian region. The CMAQ modelling system is an Eulerian (gridded) mesoscale chemical transport model (CTM), initially created for air quality studies. It has been previously used to characterize the variability of CO2 at fine spatial and temporal scales . The choice of an older version of the CMAQ modelling system (cf. the latest version, v5.3) relates to the requirement of the model adjoint (needed to calculate the gradient of the cost function in the inversion).

We treat CO2 as an inert tracer, neglecting its chemical production . Thus modelled concentrations are determined only by emissions, the atmospheric transport (horizontal and vertical advection and diffusion), and initial and boundary conditions. Initial and boundary conditions were interpolated from atmospheric CO2 concentration data from the Copernicus Atmospheric Monitoring Service (CAMS) global CO2 atmospheric flux inversions . These data have a resolution of 3.75∘ in longitude and 1.875∘ in latitude with 39 vertical layers in the atmosphere; this dataset was also the basis for the oceanic fluxes used in the prior. The CMAQ chemical transport model (or CCTM) also requires 24-hourly three-dimensional emission data (recall that in our py4dvar system we solve for a perturbation around these background CO2 fluxes). Here our background CO2 fluxes were generated by adding the four CO2 flux fields described in Sect. : carbon exchange between biosphere and atmosphere, carbon exchange between ocean and atmosphere, fossil-fuel emissions, and biomass-burning emissions.

The CMAQ model is an offline model, and thus requires three-dimensional meteorological fields as inputs for the transport calculations. We simulated meteorological data using the Weather Research and Forecasting model (WRF) Advance Research Dynamical Core WRF-ARW (henceforth, WRF) version 3.7.1 . Details on the physics schemes used in our WRF configuration are shown in Table . Our domain has a horizontal resolution of 81 km and 32 vertical layers from the surface up to 50 hPa. The numerical simulation was carried out on a single domain (i.e. non-nested) of 89×99 grid cells.

The meteorological initial conditions were based on the ERA-Interim global atmospheric reanalysis , which has a resolution of approximately 80 km on 60 vertical levels from the surface up to 0.1 hPa. Sea surface temperatures were obtained from the National Centers for Environmental Prediction/Marine Modeling and Analysis Branch (NCEP/MMAB). The WRF model was run with a spin-up period of 12 h. The initial spin-up period stabilizes the model, that is, the inconsistencies between the initial and boundary conditions diminish in this period.

The WRF modelled meteorology was nudged towards the global analysis fields above the boundary layer. The default grid-nudging configuration was used; that is, nudging coefficients were assumed to be 10-4 s-1 for wind and temperature and 10-5 s-1 for moisture, as suggested by . Nudging has been widely used in mesoscale modelling as an effective and efficient method to reduce model errors . It relaxes the model simulations of wind, temperature and moisture towards driving conditions, preventing model drift over a long-term integration.

The WRF model output was post-processed by the Meteorology-Chemistry Interface Processor (MCIP) version 4.2 . MCIP prepares the meteorological fields in a form required by CMAQ and performs horizontal and vertical coordinate transformation. In this process, we removed the outermost six rows and columns from each edge of the WRF model domain, so the horizontal CMAQ domain was set up (with 77×87 grid cells). This was done to prevent numerical instabilities in the “relaxation zone” (the exterior rows and columns of the horizontal domain), where the lateral meteorological boundary conditions and the WRF model's internal physical processes both contribute.

As is seen in Eq. (), we need to compare the CMAQ simulated CO2 concentration with OCO-2 satellite retrievals. As outlined in Sect. , we averaged observations to approximate the observed XCO2 for any CMAQ grid cell observed by OCO-2. To compare modelled and observed concentrations, we used the Eq. () to convolve the simulated CO2 concentration with the relevant averaging kernels, as follows: 12xCO2m=xCO2a-∑jhjaCO2,jxa,j+∑jhjaCO2,jxjm, where xa is the OCO-2 a priori, h is a vector of pressure weights, hj is the mass of dry air in layer j divided by the mass of dry air in the total column, aCO2 is the averaging kernel of OCO-2, xa,j is the OCO-2 a priori profile, and xm is the simulated profile from the CMAQ model. In our py4dvar system, the first and second terms in Eq. () represent an “offset term”. The OCO-2 averaging kernel is defined on 20 pressure levels and we interpolate these to the CMAQ vertical levels.

One interesting diagnostic of the convergence is to compare the cost function at the end of the optimization to its expected theoretical value. In a consistent system, the theoretical value of the cost function at its minimum should be close to half the number of assimilated observations, assuming all error statistics are correctly specified p. 211. Table  shows the mean (across our five realizations) of the cost function J(x) and its gradient norm ∇xJ. For example, with 842 observations, the theoretical value should be 421. We see that the theoretical value is reached to within a few percent for all months. We see a corresponding decrease in the gradient norm by about 99 %.

The number of degrees of freedom for signal (DFS) in our OSSEs is another useful diagnostic of the inversion Eq. 2.46. The DFS quantifies the number of independent pieces of information that the OCO-2 measurements can provide given the prior information. In our experimental framework, we computed the DFS following Sect. 3.4.: 13J(xa)=xa-xbTB-1xa-xb, where xa represents our posterior estimates. Table  shows that on average the DFS in the prior for our four months is about 30. This value is consistent with Fig. a and b, which shows that only about 20 eigenvalues account for 60 % of the variance in our prior error covariance matrix. The inversion cannot add much information to other components, limiting the DFS. Australia is a special case in this respect since most of the continent comprises semi-arid and arid regions. We assumed that land flux uncertainties are driven by NPP, as simulated by CABLE. Thus, the prior uncertainty will be small in arid and semi-arid regions.

The uncertainty reduction between the posterior and prior fluxes is a useful way to evaluate the potential of satellite data to constrain CO2 fluxes. We calculated the percentage uncertainty reduction following Sect. 3.5., as follows: 14U=1-σaσb×100%, where σa and σb are the posterior and prior standard deviations, respectively. Figure  displays the monthly uncertainty reduction in CO2 fluxes for (a) March, (b) June, (c) September and (d) December 2015. We have masked areas with σb<10-7 mol m-2 s-2. We also mask areas with negative uncertainty reduction. Such uncertainty increase is simply a result of the small number of realizations. We will now describe the magnitude and spatial patterns in the uncertainty reduction, and in Sect.  we will discuss the uncertainty reduction aggregated by land cover class.

In March, the largest uncertainty reductions (Fig. a) are located in the north of Australia. In this area, the uncertainty reduction is greater than 30 %, reaching values up to 80 %. We note that the regions with the largest reduction in uncertainty coincide with the locations with high prior uncertainty (Fig. ). In June 2015 (Fig. b), for instance, the largest uncertainty reduction was found in the north, north-east, east and south-east of Australia, where values range between 70 % and 80 %. Uncertainty reductions in September (Fig. c) are higher compared to June in the south-east of the country, ranging between 70 % and 80 %. This is consistent with the fact that September is in the middle of the growing season in this part of Australia and our prior uncertainties are driven by NPP. Also, more satellite soundings are available for this region in September compared to other months. The uncertainty reduction in December (Fig. d) decreases in the north of Australia to a range of 20 %–30 %. This is likely due to the fact that relatively few OCO-2 soundings are available in that month (Fig. ), due to increased cloud coverage during the wet season in northern Australia. This is discussed further in the next section.

To get a better understanding of the constraint on CO2 surface fluxes provided by OCO-2, we aggregated the prior and posterior fluxes into six categories over Australia: grasses and cereal (GC), shrubs (SH), evergreen needleleaf forest (ENF), savannah (SAV), evergreen broadleaf forest (EBF), and unvegetated land (UN). We used the MODIS Land Cover Type Product (MCD12C1) Version 6 data product. The distribution is shown in Fig. . After aggregating fluxes for each realization we calculated standard deviations and uncertainty reductions following Eq. ().

The bar chart in Fig.  shows the prior (green bar) and the posterior (orange bar) uncertainties of our five realizations (in Pg C yr-1) split into six land-use classes for (a) March, (b) June, (c) September and (d) December 2015. The uncertainty reduction for each land-use class and each month is represented by circles. Also shown is a second estimate of the prior uncertainties, comprising 100 realizations (purple bar). The prior of 100 realizations is plotted to assess the representativity of the five random prior realizations of the prior uncertainties. We see clearly in each figure that with only five realizations we can represent quite well our assumed prior uncertainties (we should also note that, due to computational limitation, the uncertainty reduction is based only on these five realizations).

The largest uncertainty reduction in March is over SH (81 %). The large uncertainty reduction is likely due to the large number of OCO-2 soundings in this region (464 observations). The next largest uncertainty reductions are over GC (78 %) and ENF forest (68 %) likely due to the relatively large NPP in these regions (Fig. a).

June shows less uncertainty reduction for GC (51 %) compared to March, likely due to the smaller number (one third as many) of OCO-2 soundings (Fig. ) in southern Australia. similarly, uncertainty reduction over the SH ecotype decreases. Due to the small number of realizations, however, this percentage of reduction might not be representative of this region. For this category, we can see that the prior uncertainty with five realizations is about 0.1 Pg C yr-1, whereas with 100 realizations it is about 0.25 Pg C yr-1. Uncertainty reduction over SAV is about 31 %, similar to the percentage of reduction found in March. Even though we found relatively few soundings over EBF and ENF in June, uncertainty reductions for these regions are 47 % and 7 %, respectively. The reduction over UN areas is about 39 %, again demonstrating the potential of OCO-2 data to constrain fluxes.

In September the most significant uncertainty reduction was found over EBF (74 %) and GC (68 %) compared with all other months, associated with the peak of the growing season in much of Australia. Uncertainty reductions in these categories are much larger due to the increase of OCO-2 soundings in south-eastern Australia (see Fig. c). The uncertainty reduction over areas designated as SAV and ENF is about 53 % and 30 %, respectively. Over areas classified as SH and UN, we see a weaker uncertainty reduction of 22 % and 33 %, respectively.

Similar to September, in December we found the largest uncertainty reductions over EBF (72 %) in line with the structure of the uncertainties seen in south-eastern of Australia in (Fig. c). The percentage of uncertainty reduction found over GC (77 %) may not represent the precise percentage for this category (given the small number of realizations used). For this category, we see that the prior uncertainties of 100 realizations is about 0.17 Pg C yr-1, whereas with five realizations it is about 0.28 Pg C yr-1. We would expect to have a smaller uncertainty reduction for this category due to scarcity of soundings available in the north and north-eastern Australia for this month, likely due to cloudiness associated with the wet season. Uncertainty reductions found over areas classified as SH, SAV and ENF were 56 %, 62 % and 36 %, respectively. Different to other months, the uncertainty reduction over UN is about 58 %.

Table  shows the standard deviation of the total CO2 flux uncertainty over Australia for the four months in which inversions were run. Months with the largest uncertainty reductions are found in December (80 %), March (76 %) and September (70 %). In contrast with these results, the smallest reduction is found in June (31 %). The last of these results is not surprising, since June is the month with the smallest number of OCO-2 soundings (for this month we only find 694 observations compared to September and March, with 1002 and 842 soundings, respectively).

Differences in the uncertainty reduction between months not only depend on the number of soundings and the structure of the uncertainty but also other variables (e.g. wind direction). Coastal grid points present a problem for our inversion when the wind direction comes from the ocean because our system only assimilates data over land). Prevailing winds in this coastal zone restrict the ability of OCO-2 to constrain surface fluxes (Figs. S1–S3 in the Supplement).

Table  shows the number of retained eigenvalues from B and the DFS for sensitivity experiments S1, S2, S3 and control cases. Experiment S1 shows that merely reducing correlation lengths does not lead to extra information being resolved by the observations. S2 shows that, as expected, subtracting observations from our inversion resolves less information on fluxes. Experiment S3 (in which we reduce correlation lengths but also increase the uncertainty on many grid points) demonstrates an increase in the number of components resolved by the observations. The comparison of S1 and S3 suggests it is the low uncertainty rather than the smoothness imposed by the uncertainty correlations that limits the DFS.

Figure  shows the spatial distribution of the uncertainty reduction at grid-scale over Australia for sensitivity experiments S1, S2 and S3. These figures should be compared to Fig. a (control case). Experiment S1 shown in Fig. a demonstrates that the correlation length plays a significant role in the uncertainty reduction. A lower correlation length yields a lower reduction of the uncertainties. For example, the error reduction over the productive areas in northern and north-eastern Australia is between 0 % and 20 % compared to the control experiment's 40 %–80 %. This implies that longer correlation length scales allow for information to be effectively “transferred” in space, thus pooling data over a wider region and magnifying the benefit from the assimilation.

Experiment S2 (Fig. b) illustrates that decreasing the number of observations also reduces the percentage of reduction per grid cell. The uncertainty reduction (40 %–60 %) is much weaker than the control experiment. These results complement Table , where the DFS decrease from 38.66 (control experiment) to 35.32 (S2).

Experiment S3 (Fig. c) shows how the structure and magnitude of the prior uncertainty influence uncertainty reduction. The uncertainty reductions are distributed almost uniformly across Australia, and their values range between 0 % and 20 %. Our assumption of a linear relationship between uncertainty and NPP means much of Australia has negligible impact on the prior uncertainty in the control case. This result shows the importance of that assumption. Assuming equal uncertainty across Australia may have a significant impact on the final total flux estimate, because most of the continent is largely composed of arid and semi-arid land. The small percentage of uncertainty reduction is due to the negligible correlation length assumed in the prior error covariance matrix.

Figure  shows the uncertainty reduction for the sensitivity cases S1, S2, and S3 aggregated by ecotype. There is good consistency between the geographical distribution (Fig. ) and these spatial aggregates. Thus for case S1, the uncertainty reductions were found to be small compared to the results in the control experiment (Fig. a). For example, the sensitivity case S1 in Fig. a shows uncertainty reductions over GC and UN are about 30 % and 1 %, respectively. No uncertainty reductions are observed over SH, SAV, EBF and ENF. Because of an insufficient number of realizations, for these particular categories, we found a negative error reduction. In these land-use classes, we display the posterior to be equal to the prior uncertainty.

Similarly, case S2 (Fig. b) displays significantly weaker uncertainty reductions for some of the six land-use classifications compared to the control experiments (Fig. a). For instance, the fractional uncertainty reductions over GC and SH reach values of about 51 % and 57 %, respectively. In the control experiment in (Fig. a) these values were 78 % and 81 %, respectively. As mentioned in the previous section, the stronger posterior reduction is due to the correlation length in the prior covariance and an increase of the OCO-2 soundings over Australia. Findings in the sensitivity case S3 (Fig. c) show similar results to those found in sensitivity case S1: the smaller the correlation length, the less efficient the inversion.

Uncertainty reduction of the total Australian CO2 flux for sensitivity experiments S1, S2 and S3 are shown in Table . Experiment S1 shows that the regional flux uncertainty in Australia was only reduced by ∼9 % compared to the control case (which was 76 %). In this test, we can see again the importance of the choices of the correlation length in B. We saw in Table  that by decreasing the spatial correlation to 5 km over land, we increase the number of principal components. Given the small number of realizations and an increase in the number of components in the prior, we expect that this estimate of the uncertainty reduction may be less representative using our randomization approach.

Experiment S2 shows an uncertainty reduction over Australia from 73 % compared to 76 % (control case). This small shift in the percentage of reduction is related to the number of soundings found in the northern region of Australia. By removing glint land data from our observations, we are reducing the coverage of surface flux footprints.

Experiment S3 demonstrates the same artefact as S1, though the generally higher prior uncertainties in S3 result in a higher uncertainty reduction for the total Australian flux. In this case, the assimilation reduces the total uncertainty to 34 %.

We mentioned in Sect.  that potential biases in the observations prevent the inversions from converging on optimal fluxes. The results of experiment S4 confirms that biases in the observations do indeed affect the resulting posterior fluxes. After adding biases of about 3.3 ppm, our inversion produced a posterior flux, which was bias by approximately 5.0 Pg C yr-1 over Australia. This value indicates that in order to obtain an accuracy of 0.1 Pg C yr-1 in the total Australian flux, bias in the observation must be reduce roughly to 0.07 ppm. This sensitivity case shows us the importance of minimizing biases in the observations if the goal is to estimate accurately CO2 fluxes. Figure  illustrates the impact of the observational biases on the posterior mean fluxes in each of the six MODIS land-use categories. Significant biases are observed over SH (1.7 Pg C yr-1), GC (1.4 Pg C yr-1), and EBF (0.9 Pg C yr-1). For each category, the inversion system only generates positive flux biases, consistent with the direction of the bias in the observations. Our results are mainly due to large biases we prescribed in the observations. Finally, we found that uncertainty of prior and posterior were 0.68 and 0.25 Pg C yr-1, respectively. Given the magnitudes of the prior uncertainties (and hence biases in this case), this result is consistent with the control case.

Results of experiment S5 are illustrated in Fig. . This figure shows the monthly-mean biases (black diamonds) added to our prior true fluxes (assumed to be 0.0 Pg C yr-1) categorized by MODIS ecotype. In this figure, we can see that after performing the inversion we can recover successfully the mean of our true fluxes (dashed grey line). On average the total biases added to our Australian prior flux was about 0.21 Pg C yr-1 (using a conversion factor of 2.12 Pg C ppm-1, this value is equivalent to adding 0.1 ppm bias). After performing the inversion, the posterior mean bias was reduced to 0.024 PgC. The distribution of the fluxes across the different land-use classes (centred around zero; Fig. ) reflects the fact that biases added to our prior were randomly distributed. We added negative biases to GC and SH (-0.12, -0.05 Pg C yr-1) and positive bias to SAV and EBF (-0.05, 0.05 Pg C yr-1). We can see clearly in this figure that the inversion system is able to handle negative and positive biases.

Unlike global flux inversions, regional flux inversions are sensitive to lateral boundary conditions (BCs). To explore how sensitive our system is to biased BCs, we ran two further sensitivity experiments (collectively termed “S6”). In sensitivity experiment S6-A we increased the BCs by adding 0.5 ppm to each boundary grid cell. Findings of this experiment show that our system is indeed sensitive to the altered BCs. Adding an extra 0.5 ppm to the BCs yields a posterior bias in Australia of about -0.7 Pg C yr-1. These findings are in line with the values found in sensitivity case S4, but in a opposite direction. The negative value of the bias means the inversion system is trying to reduce the fluxes to compensate for the positive bias in the BCs. The mean posterior bias flux for each land category is shown in Fig. .

Experiment S6-B was designed to see if the inversion could correct for biases in the boundary conditions given additional parameters to optimize. After solving for BCs in the inversion, the biases introduced to BCs in S6-A were corrected. We analysed the corrections by looking at the bias of the posterior flux for each land-use category. Figure  shows that the decrease of biases over GC was significant. In this category biases were reduced from -0.11 to -0.019 Pg C yr-1. Similar results were found over SAV, EBF and ENF, where biases were also reduced. Biases over SH does not show much improvement. In this category biases decreased only from -0.30 to -0.20 Pg C yr-1. After a wind-rose analysis for 10 selected locations around the coast in west Australia (Figs. S1–S3), we found that the small reduction of the biases in this category is explained by the orientation of the wind in March. When winds come from ocean, the inversion loses the ability to correct the wrong BCs. The treatment of the bias in BCs is relatively simple, with a goal of introducing relatively few additional parameters into the control vector. The experimental design assumes that these biases are constant in time and across large areas of the domain. The biases in the BCs were generated with the same framework as was used to solve them (i.e. fully specified by eight parameters). In reality, error in BCs will vary in both space and time. Thus, the results here are indicative but suggest that biases (as opposed to fluctuations) at least can be accounted for in such a system.