2.1 Inversion scheme

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 (x_a) that fits both observations (y) and the prior fluxes (x_b) (Ciais et al., 2010; Rayner et al., 2019). Assuming Gaussian PDFs, finding this maximum a posteriori estimate is equivalent to minimizing the cost function J(x) shown in Eq. 1 (Rayner et al., 2019).

The first term in Eq. (1) represents the sum of squared differences between the control variable (x) and its prior or background state (x_b). 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. (1) 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 (Byrd et al., 1995). 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. 1.

The gradient of the cost function in Eq. (2) is calculated using the adjoint of the CMAQ model (version 4.5.1; Hakami et al., 2007). We can observe that in the second term in Eq. (2), the adjoint model (H^T) is applied to the vector ${\mathbf{R}}^{-\mathrm{1}}\left(\mathbf{H}\left(\mathit{x}\right)-\mathit{y}\right)$, 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 t_i−1 to t₀, allows us to construct the gradient of the cost function, ∇_xJ(x).

2.2 Choice of control variables

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. Deng et al., 2014; Houweling et al., 2015). Like most previous studies (e.g. Chevallier et al., 2007; Baker et al., 2010; Basu et al., 2013; Crowell et al., 2019) 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 1). The length of the control variables for our sensitivity experiments are defined in Table 6. Similar to Chevallier et al. (2005a), 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 $\mathit{x}=\left\{i_{\mathrm{0}},e_{\mathrm{0}},e_{\mathrm{1}},\mathrm{\dots },e_n\right\}$, where i₀ is the factor we solve for ICONs and e_n 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.

Table 1Number of eigenvectors included in our control vector (x). Date format is YYYY-MM.

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2.3 Observations and their uncertainties

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 Kiel et al. (2019). We used nadir and glint soundings over land that were flagged as good quality, except in some of our sensitivity experiments (described in Sect. 4) in which we excluded glint mode data. We computed a weighted average for all OCO-2 measurements using a two-step process similar to Crowell et al. (2019). 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. (3).

where $w_i=\frac{\mathrm{1}}{{\mathit{\sigma }}_i^{\mathrm{2}}}$ is the squared reciprocal of the OCO-2 uncertainties (σ_i). To get the uncertainties of these averaged soundings, we considered three different forms of uncertainty calculation (similar to Crowell et al., 2019). First if we assumed that all errors are entirely correlated in a 1 s span, we can define the uncertainties as shown in Eq. (4).

However, and because the average shown in Eq. 4 is sometimes low, we also considered the standard deviation of the XCO₂ measurements (here referred to as the spread, or σ_r, of the OCO-2 measurements). In other words, if the spread (σ_r) of the XCO₂ measurements was higher than the XCO₂ uncertainty (σ_i), we used the spread value as shown in Eq. (5). We did this because the spread in OCO-2 measurements may reflect real differences across the field within a 1 s time span.

Third, we also considered a baseline uncertainty (σ_b), based on an error floor (ϵ) over land and ocean, as shown in Eq. (6). 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 σ_s, σ_r and σ_b.

The second step was to take these 1 s averages and average them within the CMAQ vertical columns using Eq. 7.

where $w_j=\frac{\mathrm{1}}{{\mathit{\sigma }}_j^{\mathrm{2}}}$ represents the squared reciprocal of the uncertainties average in the 1 s span (σ_j) and J is the number of those 1 s values. The average uncertainty over the CMAQ domain (Eq. 8) was similar to the procedure outlined for 1 s average in Eq. (4). However, we also added a term to represent the contribution of the model uncertainty (σ_m). We assumed that the model had a uncertainty of about 0.5 ppm. The observational error covariance matrix R was assumed to be diagonal.

After averaging the OCO-2 sounding over the CMAQ domain, we generated a set of pseudo-observations as described in step 1 of Fig. 1. 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. 2.4 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. 1. 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.

The first term of Eq. (9), y_sim, represents the OCO-2- simulated observations using the true fluxes. The second term of Eq. (9), p, is a vector with the same size as y_sim 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.

2.4 Prior CO₂ fluxes and their uncertainties

As is stated in Sect. 2.5, 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 $\mathrm{0.5}{}^{\circ }\times \mathrm{0.5}{}^{\circ }$ and daily resolution and the gross primary production (GPP) at $\mathrm{0.5}{}^{\circ }\times \mathrm{0.5}{}^{\circ }$ 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 $\mathrm{ER}=\mathrm{GPP}-\mathrm{NEE}$. 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 Chevallier et al. (2010a).

Fossil-fuel CO₂ emissions were obtained from the Fossil Fuel Data Assimilation System (FFDAS) (Rayner et al., 2010; Asefi-Najafabady et al., 2014). 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 Asefi-Najafabady et al. (2014). 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) (Olivier et al., 2005). 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^∘ (van der Werf et al., 2017). 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 (Randerson et al., 1996). 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 (Chevallier, 2016). 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 Takahashi et al. (2009). We assumed that the error statistics were uniform, 0.2 g C m⁻² d⁻¹, over ocean, as in Chevallier et al. (2010a).

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 Chevallier et al. (2010a), who used a temporal correlation length of 4 weeks; though, this would only introduce weak correlations among our monthly averaged fluxes. Following Basu et al. (2013, Sect. 3.1.1), the spatial correlation between grid points r₁ and r₂ was defined as

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 Basu et al. (2013).

After defining B, we performed an eigendecomposition, B=W^TwW, where W is a matrix of eigenvectors and w is a diagonal matrix of corresponding eigenvalues. Figure 4a 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 x^b. Given that our control vector x depends on the size of the multipliers of the principal eigenvectors of B, our vector x^b was reconstructed (as is given in Eq. 11). 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 x_p.

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. (11).

2.5 CMAQ model configuration

We used the CMAQ modelling system and its adjoint (version 4.5.1; Hakami et al., 2007) 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 (Liu et al., 2014). 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 (Folberth et al., 2005; Suntharalingam et al., 2005). 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 (Chevallier et al., 2010a). 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. 2.4: carbon exchange between biosphere and atmosphere, carbon exchange between ocean and atmosphere, fossil-fuel emissions, and biomass-burning emissions.

Table 2Physics parameterizations used in the Weather Research and Forecasting (WRF) model setup.

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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 (Skamarock et al., 2008). Details on the physics schemes used in our WRF configuration are shown in Table 2. 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 (Dee et al., 2011), 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 Deng and Stauffer (2006). Nudging has been widely used in mesoscale modelling as an effective and efficient method to reduce model errors (Stauffer and Seaman, 1990). 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 (Otte and Pleim, 2010). 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.

2.6 Observation operator: CMAQ CO₂ simulations and OCO-2 measurements

As is seen in Eq. (1), we need to compare the CMAQ simulated CO₂ concentration with OCO-2 satellite retrievals. As outlined in Sect. 2.3, 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. (12) (Rodgers and Connor, 2003; Connor et al., 2008) to convolve the simulated CO₂ concentration with the relevant averaging kernels, as follows:

where x^a is the OCO-2 a priori, h is a vector of pressure weights, h_j is the mass of dry air in layer j divided by the mass of dry air in the total column, ${\mathit{a}}_{{\mathrm{CO}}_{\mathrm{2}}}$ is the averaging kernel of OCO-2, x_a,j is the OCO-2 a priori profile, and x^m is the simulated profile from the CMAQ model. In our py4dvar system, the first and second terms in Eq. (12) represent an “offset term”. The OCO-2 averaging kernel is defined on 20 pressure levels and we interpolate these to the CMAQ vertical levels.

3.1 Convergence diagnostic

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 (Tarantola, 1987, p. 211). Table 3 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 %.

Table 3Convergence diagnostics of the inversion system using an ensemble of five independent OSSEs for March, June, September and December 2015. Date format is YYYY-MM.

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3.2 Degrees of freedom for signal

The number of degrees of freedom for signal (DFS) in our OSSEs is another useful diagnostic of the inversion (Rodgers, 2000, 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 Chevallier et al. (2007, Sect. 3.4.):

where x_a represents our posterior estimates. Table 3 shows that on average the DFS in the prior for our four months is about 30. This value is consistent with Fig. 4a 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.

3.3 Spatial distribution of uncertainty reduction

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 (Chevallier et al., 2007, Sect. 3.5.), as follows:

where σ_a and σ_b are the posterior and prior standard deviations, respectively. Figure 5 displays the monthly uncertainty reduction in CO₂ fluxes for (a) March, (b) June, (c) September and (d) December 2015. We have masked areas with ${\mathit{\sigma }}_{\mathrm{b}}<{\mathrm{10}}^{-\mathrm{7}}$ 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. 3.4 we will discuss the uncertainty reduction aggregated by land cover class.

Figure 2Spatial distribution of OCO-2 soundings (land nadir and glint data) over the CMAQ domain for March, June, September and December 2015.

Figure 3Monthly mean of CO₂ prior uncertainties accounting for the major terms in the CO₂ budget (anthropogenic fluxes, fires, land and ocean exchange) (in units of g C m⁻² d⁻¹).

Figure 4The cumulative percentage variance explained (a) and the eigenvalues (b) in the prior error covariance matrix.

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Figure 5The percentage error reduction of the monthly-mean CO₂ surface fluxes for March, June, September and December 2015 over the CMAQ model domain. The percentage of error reduction is defined as $(\mathrm{1}-{\mathit{\sigma }}_{\mathrm{a}}/{\mathit{\sigma }}_{\mathrm{b}})$, with σ_a and σ_b representing, respectively, the posterior and prior uncertainties of the CO₂ fluxes emissions.

In March, the largest uncertainty reductions (Fig. 5a) 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. 3). In June 2015 (Fig. 5b), 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. 5c) 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. 5d) 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. 2), due to increased cloud coverage during the wet season in northern Australia. This is discussed further in the next section.

3.4 Uncertainty reduction over Australia by MODIS land cover classification

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. 6. After aggregating fluxes for each realization we calculated standard deviations and uncertainty reductions following Eq. (14).

Figure 6Aggregation of land cover classes over CMAQ domain using MODIS Land Cover Type Product (MCD12C1) Version 6 data product. Colour bars represent each category: (0) ocean, (1) grasses and cereal, (2) shrubs, (3) evergreen needleleaf forest, (4) savannah, (5) evergreen broadleaf forest, and (6) unvegetated land.

Figure 7Prior and posterior uncertainties (in Pg C yr⁻¹) aggregated over five different classes over the Australian domain using MODIS Land Cover Type Product (MCD12C1). Green and orange bars represent the prior and posterior uncertainties of five realizations, respectively, while the purple bar represents prior uncertainties of 100 realizations. Circles show the percentage of uncertainty reduction by each category.

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The bar chart in Fig. 7 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. 3a).

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. 2) 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. 2c). 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. 3c). 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 %.

3.5 Uncertainty reduction in the total Australian CO₂ flux

Table 4 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).

Table 4Prior and posterior uncertainties (in Pg C yr⁻¹) for an ensemble of five realizations aggregated over the Australian continent. Date format is YYYY-MM.

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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).

4.1 Degrees of freedom for signal

Table 6 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.

Table 6Number of degrees of freedom for signal (DFS) in the prior flux uncertainty and the number the principal eigenvector in the prior error covariance matrix for sensitivity experiments S1, S2 and S3.

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4.2 Spatial distribution of uncertainty reduction over Australia

Figure 8 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. 5a (control case). Experiment S1 shown in Fig. 8a 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.

Figure 8Maps of the percentage of error reduction for the three sensitivity cases. (a) Using only nadir OCO-2 sounding and correlation lengths 50 and 100 km. (b) Using “nadir” and “glint” OCO-2 sounding and correlation lengths of 500 and 1000 km. (c) Uniform uncertainties over land and ocean, and correlation lengths of 5 and 10 km.

Experiment S2 (Fig. 8b) 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 6, where the DFS decrease from 38.66 (control experiment) to 35.32 (S2).

Experiment S3 (Fig. 8c) 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.

4.3 Uncertainty reduction over Australia by MODIS land cover classification

Figure 9 shows the uncertainty reduction for the sensitivity cases S1, S2, and S3 aggregated by ecotype. There is good consistency between the geographical distribution (Fig. 8) 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. 7a). For example, the sensitivity case S1 in Fig. 9a 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.

Figure 9Sensitivity experiments for the prior and posterior uncertainties (in Pg C yr⁻¹) aggregated over six different classes over the Australia domain using MODIS Land Cover Type Product (MCD12C1).

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Similarly, case S2 (Fig. 9b) displays significantly weaker uncertainty reductions for some of the six land-use classifications compared to the control experiments (Fig. 7a). For instance, the fractional uncertainty reductions over GC and SH reach values of about 51 % and 57 %, respectively. In the control experiment in (Fig. 7a) 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. 9c) show similar results to those found in sensitivity case S1: the smaller the correlation length, the less efficient the inversion.

4.4 Uncertainty reduction in the total Australia CO₂ flux uncertainty

Uncertainty reduction of the total Australian CO₂ flux for sensitivity experiments S1, S2 and S3 are shown in Table 7. 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 6 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.

Table 7Prior and posterior uncertainties (in Pg C yr⁻¹) for an ensemble of five realizations.

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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 %.

4.5 Impact of OCO-2 biases on the posterior fluxes

We mentioned in Sect. 4 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 10 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.

Figure 10Posterior bias of monthly CO₂ flux induced by OCO-2 bias categorized by MODIS ecotype.

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4.6 Unbiased prior CO₂ flux

Results of experiment S5 are illustrated in Fig. 11. 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 Pg C. The distribution of the fluxes across the different land-use classes (centred around zero; Fig. 11) 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.

Figure 11Prior (blue) and posterior (red) monthly-mean CO₂ flux of a ensemble of five realizations and monthly-mean prior bias (black) added to the true prior fluxes (dashed grey line). Note: results are shown for adding the same biases to our five realizations.

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4.7 Impact of boundary condition biases on the posterior fluxes

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. 12.

Figure 12Posterior bias of monthly CO₂ concentration induced by changes in the lateral boundary conditions categorized by MODIS ecotype.

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4.8 Solving for the boundary condition in the inversion

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 12 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.