Finding the optimal value (xa) of the CO2 flux estimates involves identification of the best fits between both observations (y) and a prior (or background) estimate (xb) of these fluxes . Using Bayes' theorem and under the hypothesis of unbiased Gaussian-distributed errors of xb and y, the best estimate of xa (likelihood maximum a posteriori) is equivalent to finding the minimum of the cost function J(x) shown in Eq. (). Notation in this study follows . 1J(x)=12(x-xb)TB-1(x-xb)+12(H(x)-y)TR-1(H(x)-y)

In Eq. (), H, represents the application of the forward model and the “observation operator”, which allows us to map the model variables (e.g. fluxes) to observations. R represents the error covariance matrix of the observations y, including the transport model error. R is defined as a diagonal matrix (details Sect. ). x represents the control vector of unknowns. x includes not only CO2 surface fluxes but also initial and boundary conditions (details Sect. ). B is the associated error covariance matrix of xb, boundary and initial concentrations, and includes off-diagonal terms. In these off-diagonal values, we only include spatial and non-temporal correlations of the prior fluxes (details of the structure of the prior error covariance matrix can be found in Sect. 2.4 in ).

We calculate the minimum of J(x) by an iterative process and not by an analytical expression. This numerical problem requires the value of the cost function gradient ∇xJ(x). 2∇xJ=B-1x-xb+HTR-1H(x)-(y)

We compute HT using the adjoint of the CMAQ model version 4.5.1;. We can see in Eq. () that HT is applied to the vector R-1H(x)-y, which is often called the “adjoint forcing”, and represents the error-weighted differences between the forward model and the observed concentrations. Applying the adjoint model to the adjoint forcing, running backward in time from ti-1 to t0, allows us to construct the gradient of the cost function, ∇xJ(x). The algorithm that our inverse system uses to optimize the J(x) is the limited-memory Broyden–Fletcher–Goldfarb–Shanno (L-BFGS-B), implemented in the SciPy Python module . Figure  shows a simplified version of how our inversion system works to find the optimal values of CO2 surface fluxes.

The error statistics of xa are embodied in the posterior error covariance matrix (A). In this study, A was computed by a series of observing system simulation experiments (OSSEs) carried out by Sect. 2.4. However, here we adjusted the prior and observation uncertainties assumed in by a factor of 1.2. We made this adjustment to satisfy the theoretical assumption in the variational optimization, which indicates the value of the cost function in its minimum has to be approximately equal to half of the number of observations (for more details, see Sect. ). In general, errors assumed in the inversion are not Gaussian and independent but rather have errors correlated in time and space (including flat biases) that render the statistical assumptions made in deriving the estimation method invalid and lead to a higher cost function than expected. A description of how the prior and observation uncertainties were assumed in our study is found in Sect.  and . Appendix  (Figs.  and ) shows the spatial distribution of the prior and posterior that we reference in this study.

In our data assimilation system, we solve for monthly average surface fluxes at 81 km grid-cell-scale resolution as the multipliers of the principal eigenvectors of the prior error covariance matrix B, computed as WTw-1/2(x-xb), where W was defined as the matrix of eigenvectors and w as a diagonal matrix of corresponding eigenvalues Sect. 2.2. In order to avoid the impact of the initial conditions (ICs) and boundary conditions (BCs) on our assimilated fluxes, we also solved them within the control vector x. We did not optimize them in the same way as the fluxes in order to not increase the control vector size, so we treat the unknowns related to the BCs and ICs as scaling factors of the emissions added to the CMAQ model. Lateral BCs were solved as eight boundary regions divided by the upper and lower boundary areas within the CMAQ domain (south, east, north and west). In Fig. , we provide a representation of these boundaries. In this figure, we can see that our study domain not only covers the Australian continent (AUS) but additionally other countries such as Indonesia (IND), Papua New Guinea (PNG) and New Zealand (NZ). The extension of this domain was created as an extra precaution to minimize the influence of the boundaries over Australia.

Lower boundary layers were defined to cover from σ=1 to σ=0.96, which correspond (on average) to a pressure of 972.5 hPa, while the upper boundary layer was defined to cover from 972.5 up to 50 hPa. As mentioned before, our inversion system solves for these lateral boundary components, while surface fluxes are also being optimized. Boundary conditions are provided to the CMAQ model as daily averages, but we optimize them as monthly averages. BC and IC datasets were taken from the CAMS global CO2 atmospheric inversion product (version v19r1) (Frédéric Chevallier, personal communication, 2019). Uncertainties for the initial condition were set at 1 % (approximately 4 ppm), and the uncertainties in the lateral boundary conditions were assumed as the standard deviation (1σ uncertainty) of CAMS concentration data in the perimeter of the boundary.

We updated the prior CO2 fluxes described in Sect. 2.4. Biosphere carbon fluxes were derived using a modified version of the Community Atmosphere-Biosphere Land Exchange model (CABLE) , which was forced by Australian regional drivers and observations (BIOS3 setup). CABLE consists of a biophysical core: the Carnegie–Ames–Stanford approach, carbon, nitrogen, phosphorus (CASA-CNP) biogeochemical model , the POP module for woody demography and disturbance-mediated landscape heterogeneity , and a module for land use and land management (POPLUC; ). For our case, Vanessa Elizabeth Haverd (personal communication, 2020) ran the CABLE model in the BIOS3 setup (hereafter CABLE-BIOS3) at a resolution of 0.25∘. We calculated 3-hourly biosphere CO2 fluxes by combining two datasets: daily net ecosystem exchange (NEE) fluxes with 3-hourly gross primary production (GPP). Given that the BIOS3 product did not cover our whole CMAQ model domain, we also incorporated monthly biosphere fluxes from CABLE-POP global simulations as shown in Fig. . These CABLE-POP simulations were used in the carbon cycle model intercomparison project (TRENDY version 8) for the 2019 global carbon budget . Biosphere flux uncertainties in our system were assumed to be equal to the NPP simulated by CABLE, with a ceiling of 3 gC m-2 d-1 following .

Anthropogenic fluxes were created by the combination of two different inventory datasets: the Open-source Data Inventory for Anthropogenic CO2 (ODIAC) and the Emissions Database for Global Atmospheric Research (EDGAR) version 5 . The combination of these two anthropogenic inventories (each used to cover different source sectors) was necessary because the version of the ODIAC selected did not contain emissions from aviation. The EDGAR emissions combined with ODIAC were aviation climbing and descent, aviation cruise, and aviation landing and take-off datasets. Aviation emissions were also distributed across the vertical layers of the CMAQ domain. EDGAR is a gridded product with spatial resolution of 0.1∘×0.1∘ with monthly temporal resolution. ODIAC (version 2019) is also a gridded product, which has a spatial resolution of 1×1 km. Monthly ODIAC fluxes were modified by incorporating a diurnal-scale factor estimated by . The ODIAC data product selected did not include bunker emissions. Fossil fuel carbon emission uncertainties were created by multiplying the emissions dataset by a factor of 0.44. This factor was calculated by a linear regression between the mean fluxes and the spread of an ensemble of 25 realizations of posterior emissions estimated by the Fossil Fuel Data Assimilation System (FFDAS) .

Prior ocean fluxes were taken from the CAMS greenhouse gas flux inversion (version v19r1) (Frédéric Chevallier, personal communication, 2019). The prior fluxes that CAMS uses in its inversion also includes EDGAR emissions over the ocean; thus, we did not include this anthropogenic flux over the ocean to avoid double counting. We assumed that the ocean uncertainties were uniform and set up a value of 0.2 gC m-2 d-1 over ocean, as in . We also used monthly fire emissions from the Global Fire Emission Database (GFED) v4.1 , which includes small fire emissions. Fire emission uncertainties were assumed as 20 % of the biomass burning carbon emissions. All datasets mentioned above (terrestrial biosphere exchange, fossil fuel, fires and ocean fluxes) were interpolated to the spatial resolution of the CMAQ model.

As described in Sect. 2.4, we included spatial correlations into our prior error covariance matrix B following Sect. 3.1.1. The correlation length between grid points over land was assumed to be 500 km and over ocean 1000 km. We assume that fossil fuel uncertainties were not correlated, so we only use the diagonal values of the matrix. In our eigen-decomposition of B, the eigen-spectrum (eigenvectors of the covariance matrix) retains 99 % of the explained variance (eigenvalues).

The inversion was based around the CMAQ modelling system (version 5.3) and its adjoint version 4.5.1;. The CMAQ modelling system is an Eulerian (gridded) mesoscale chemical transport model (CTM). We added CO2 into the CMAQ model as an inert chemical species, whose concentration is determined by atmospheric transport, fluxes, initial and boundary concentrations. The CMAQ model was driven by meteorological fields from the Weather Research and Forecast model (WRF) Advance Research Dynamical Core WRF-ARW (henceforth WRF) version 4.1.1 , the data of which were processed by the Meteorology-Chemistry Interface Processor (MCIP) version 4.2 . WRF configuration details are shown in Table . Our WRF model was set up at a spatial resolution of 81 km with 32 vertical layers from the surface up to 50 hPa. The numerical simulation was carried out on a single domain (i.e. non-nested). WRF initial conditions were taken from 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.

We assimilated satellite data from OCO-2 level 2 (lite file version 9) for 2015, which is distributed by the National Aeronautics and Space Administration (NASA) . OCO-2 was launched in 2014 and since then has provided nearly global coverage of column-averaged dry air mole fraction of CO2 that has been used by several carbon cycle researchers to estimate surface carbon fluxes at global and regional scales. OCO-2 provides data in three modes: nadir, glint and target mode. In nadir mode, OCO-2 instrument points straight down at the surface of the Earth (surface solar zenith angle is less than 85∘); in glint mode, OCO-2 instrument points to the bright glint spot on Earth where solar radiation is directly reflected off the Earth's surface (local solar zenith angle is less than 75∘); and target mode, the instrument is configured to scan about a particular point on the ground as it passes overhead. In this study, we used the combination of both land nadir and land glint observations (LNLG), because there are no systematic offsets between the two datasets . We also performed an inversion using the combination of land (nadir and glint) and ocean glint observations (LNLGOG). However, this inversion was treated as an independent experiment (see Appendix , Table ), and the assimilated fluxes estimated by using LNLGOG were not included in our main results. We decided to not incorporate them because ocean glint retrievals still have undetermined biases that may complicate or confound the Australia flux estimation. We discussed the impact of assimilation LNLGOG in the validation of our inversion with independent data (see Sect.  for more details). We only used OCO-2 retrievals with quality flag 0 and only soundings that were bias corrected, as described by . The spatial distributions of OCO-2 soundings (LNLG and LNLGOG) across the CMAQ domain for 2015 are shown in Appendix , Figs.  and , respectively.

Given that multiple OCO-2 soundings cross one grid cell over the CMAQ domain, we had to average them before doing any comparison with the CMAQ model simulations. This averaging process was carried out in two steps. First, we averaged all OCO-2 soundings that fall within 1 s intervals, and then these 1 s averages were averaged again within the CMAQ vertical columns (approximately 11 s average) across 81 km × 81 km grid-cells. The 1 s weighted averaging process is described in detail in Sect. 2.3. In summary, to obtain the uncertainties of these 1 s averaging processes, we considered three different forms of uncertainty calculation, similar to . First, we averaged OCO-2 uncertainties assuming that these were correlated in a 1 s span (uncertainties defined as σs). Second, given that the average of OCO-2 uncertainties is sometimes low because they neglect systematic errors, we also used the spread of the OCO-2 retrievals in the 1 s average (uncertainties defined as σr). Third, we also defined baseline uncertainties (defined as σb) for cases where the number of soundings was not enough to compute a realistic spread. The values for our baseline uncertainties were assumed to be 0.8 ppm over land and 0.5 ppm over ocean. Finally, we selected the maximum value between these three uncertainties (σs, σr and σb). We also added (in quadrature) to this term 0.5 ppm as the contribution of the model uncertainty (defined as σm).

Solving the cost function shown in Eq. () requires convolving the vertical levels of the CMAQ model with the retrieval profile from OCO-2. For this, we used Eq. () derived by as follows: 3xCO2m=xCO2a-∑jhjaCO2,jxa+∑jhjaCO2,jxjm, where xa is the OCO-2 a priori, h is a vector of pressure weight, 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 is the OCO-2 a priori profile, and xm is the simulated profile from the CMAQ model. In our inversion system, the OCO-2 averaging kernel is defined on 20 pressure levels and we interpolate these to the CMAQ vertical levels.

In this study, we also use auxiliary data such as the EVI, rainfall and GPP from the CABLE-BIOS3 model to understand the difference between the prior and posterior fluxes over Australia in 2015.

As described in Eq. (), the main purpose of the inversion is to optimize fluxes by minimizing the mismatch between the model simulation and observations. In order to evaluate the performance of the inversion, we compared the CO2 concentrations obtained when forcing the CMAQ model with the prior and posterior fluxes (for convenience, we will call these the prior and posterior CO2 concentrations, respectively). Figure  shows the bias and root mean square error (RMSE) between the prior and posterior CMAQ simulations against the OCO-2 observations for 2015. This figure shows that the biases and RMSE in the posterior concentration were reduced by the inversion and indicate the inversion system leads to an overall improvement of the representation of OCO-2 observations. Our findings indicate that the prior concentrations overestimate OCO-2 from March to April and from July to September. Prior biases in these months were reduced by more than 90 %. In March, for example, the monthly mean bias was reduced from 0.56 to 0.05 ppm, with a decrease in the RMSE from 1.11 to 0.84 ppm. In April, we see similar results to those for March; the prior bias was reduced from 0.40 (RMSE = 1.05) to 0.03 (RMSE = 0.88). On the other hand, in January, February, May and December, prior biases were negligible, showing a good agreement with OCO-2. In a consistent system, we know that 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. In our inversion, after iteration 27, we obtained a cost function of 4392.15, which was close to half of the total number of OCO-2 assimilated observations for 2015 (N=9556). In general, we also see a modest reduction in the prior RMSE each month during 2015, and its variability is proportional to the number of assimilated observations. Thus, a slight prior RMSE decrease corresponds to a month with a reduced number of OCO-2 data available.

In this section, we will only discuss the results of carbon fluxes that were assimilated by the combination of LNLG OCO-2 retrievals, not the carbon fluxes estimated using LNLGOG observations. We decided not to discuss the results based on LNLGOG because ocean glint observations (version 9) still have undetermined biases that might contaminate the Australian carbon flux estimate. However, we include these findings in the Appendix , Table . Adding ocean OCO-2 glint observations to our inversion system does not significantly alter the terrestrial annual mean flux estimate for Australia (-0.36 PgC yr-1) compared to estimates made by only using OCO-2 LNLG observations (-0.41 PgC yr-1). Results based on LNLGOG will be further discussed in Sect. .

Figure a represents the terrestrial prior and posterior annual mean flux for 2015 (excluding fossil fuel). As mentioned previously, our assimilated carbon fluxes using LNLG indicate that the Australian annual terrestrial flux for 2015 was a slight carbon sink of -0.41 ±  0.08 PgC yr-1 (1σ uncertainty) compared to the prior terrestrial estimate of 0.09 ± 0.17 PgC yr-1. Our prior fossil fuel estimate from ODIAC and EDGAR, which is about 0.06 ± 0.01 PgC yr-1 (mostly constant for each month in 2015) over Australia represents only 25 % of the annual posterior flux. We decided to exclude these emissions from our analysis because variations in land uptake cause most of the variation in our posterior fluxes.

Figure b shows the seasonal cycle of the prior and posterior fluxes along with its uncertainties. As mentioned in Sect. , the prior and posterior uncertainties included in Fig. a and b were calculated from an ensemble of five different OSSEs adjusted by a factor of 1.2 in this study. We also plotted the spatial distribution of the prior and posterior fluxes (Figs.  and ), and the difference between them (Appendix , Fig. ).

Figure b shows that the posterior flux estimates generally refine the prior with the exception of March and the period July to September. In January and February, the posterior fluxes were not modified much by the inversion. In January, for example, the terrestrial posterior flux was -0.84 ± 0.18 PgC yr-1 compared to the prior -0.89 ± 0.75 PgC yr-1. The agreement follows from the small residual between prior simulated concentration and observation (Fig. ). From April to May, we also see the posterior is shifted from the prior, although not significantly considering the prior uncertainty. In April, for instance, the prior flux (0.24 ± 0.69 PgC yr-1) was slightly shifted to a posterior carbon sink (-0.28 ± 0.25 PgC yr-1). However, these two estimates do not disagree because they fall within 1σ uncertainties.

As mentioned in the previous paragraph, March, July, August and September were the exceptions to this general agreement. In March, we see a prior flux of 0.12 ± 0.74 PgC yr-1 compared to the posterior carbon sink of -0.82 ± 0.17 PgC yr-1. The difference between the posterior (Fig. b) and the prior flux (Fig. b) at grid-cell scale (see Appendix , Fig. c) suggests that most of the posterior sink comes from the north and southeast corner of Australia. July represents the month where the posterior is most shifted from the prior. In this month, we see a posterior flux of -1.75 ± 0.34 PgC yr-1 compared to the prior flux of 0.09 ± 0.62 PgC yr-1. The spatial distribution of the posterior and prior flux difference at grid-cell scale for July (Fig. g) indicates that the shift largely comes from northern and southeastern Australia. The stronger posterior sink seen in July decreased in August (-0.93 ± 0.27 PgC yr-1) and September (-0.78 ± 0.20 PgC yr-1), and changed sign in October and November. In November, the posterior flux was 1.75 ± 0.31 PgC yr-1 compared to the prior, which was 0.53 ±  0.58 PgC yr-1. The largest difference in this month is found in the north and on the southeast coast of Australia (Appendix , Fig. k). The carbon release from land in northern Australia is likely attributed to a combination of fire anomalies (Fig. S3k in the Supplement) and the lack of rainfall seen in Australia in 2015 (Fig. S2k in the Supplement). In December, we see that the posterior source seen in November changed to a posterior carbon neutral (0.003 ± 0.15 PgC yr-1). A further analysis which explains the reasons for this shift is given in the following section.

To investigate why our inversion led to a higher carbon uptake (relative to the prior flux) in some months in 2015, we studied the spatial pattern of monthly EVI anomalies and rainfall anomalies. EVI anomalies were calculated relative to 2000–2014 over Australia from the MOD13C1 version 6 data product and rainfall anomalies from AWAP data relative to 2000–2014 (Figs. S1–S2 in the Supplement). We indicated in Sect.  that the posterior sink recorded in January and February agrees with the prior estimates but disagrees with the prior flux estimates from March to September, where the most considerable difference is seen in March and from July to September. From the inversion viewpoint, the significant shift between prior and posterior fluxes occurs because the prior column average simulated by the CMAQ model overestimates the column-average retrieval by OCO-2 in these periods (see Fig. ).

As indicated in Sect. , most of the posterior carbon uptake seen in March comes from the northern part of Australia (except coastal regions) and the southeastern area (Appendix ; Fig. c). We found that the higher posterior uptake relative to the prior in the northern part of the continent was not attributed to an increase of greenness in vegetation (negative EVI anomalies). These results suggest that most of the posterior carbon uptake observed in March is likely associated with positive EVI anomalies seen in January and February affected by the positive rainfall anomalies recorded in January. High anomalous rainfall in January is not unexpected because it is the wet season in the northern region of Australia (tropical monsoonal climate).

The spatial pattern of the difference between the posterior and prior flux estimate recorded in July indicates that the majority of the posterior carbon uptake estimated by the inversion comes from the southeastern and northern region of Australia (Appendix ; Fig. g). We found that the posterior sink estimated in southeastern Australia was likely driven by a higher-than-expected greenness of vegetation (Fig. S1g in the Supplement), probably induced by anomalously positive rainfall in that period (Fig. S2g in the Supplement). We cannot conclude the same results in the northern region that we found in the south. We see in Appendix ; Fig. g that positive EVI anomalies were slight compared to the one found in southeastern Australia. In the following section, we will show that the underestimation of the GPP by the CABLE-BIOS3 model might be the likely reason for the difference between prior and posterior in this region.

An increase in carbon uptake estimated by our inversion in August comes from the northern and southern regions of Australia (with the exception of coastal areas in the southeastern corner of Australia), which mainly shows a release of carbon (Appendix ; Fig. h). The release of carbon by the land in this coastal region is likely attributed to a decrease in land productivity (Fig. S1h in the Supplement). The subtle decrease of photosynthesis activity in the coastal area is likely associated with a decrease of rainfall seen in June and July (Fig. S2f and g in the Supplement).

In September, the posterior carbon uptake primarily comes from the southeast corner of Australia (with a slight exception seen in the southeast and east coast of Australia), which shows a release of carbon into the atmosphere (Appendix ; Fig. i). The carbon uptake seen in the southeast of Australia aligns with a higher-than-usual increase in land productivity, as reflected by the positive EVI anomalies in that region (Fig. S1h in the Supplement), likely benefited by the positive rainfall anomalies seen in August in that area. In September, we also see that positive EVI anomalies were not as strong as in July and August. These findings are probably associated with the fact that rainfall in September decreased considerably for most parts of the country, where rainfall was lower than average (negative rainfall anomalies) (Fig. S2i in the Supplement).

Anomalies in EVI in Australia are closely related to fluctuations in rainfall, which is one of the most important drivers of ecosystem dynamics and productivity. This is the case in (e.g. semi-arid) regions where rainfall is the limiting factor for plant growth, which is indeed the case in much of Australia. These results are consistent with findings of previous studies e.g. that Australia's semi-arid ecosystems are water resilient and can respond to favourable rainfall conditions by capturing large amounts of carbon.

To understand which Australian ecosystem contributed most to our posterior carbon sink estimate, we divided the continent into six bioclimatic classes: tropical, savanna, warm temperate, cool temperate, Mediterranean and sparsely vegetated (Fig. ). We used the same six bioclimatic regions at a 0.05∘ spatial resolution as in . The six bioclimatic classes used in this study correspond to an aggregation of the 18 agroclimatic zones generated by . The climatic classification in was adapted from an existing global agroclimate classification , which was refined and closely aligned with natural vegetation formations and common land uses across Australia using 182 weather climate stations and the Interim Biogeographic Regionalisation for Australia (IBRA). In Fig. , we can see that Australian tropical land only covers the northern coastal part of Australia. Savanna extends across the northern tropics to the southeastern subtropical zone. Warm temperate land covers the southeast Australian coast, while cool temperate land covers the southeastern corner of Australia. The Mediterranean region is confined to the southwestern corner of Australia and the gulf region of South Australia. The sparsely vegetated ecosystem represents the biggest ecosystem over Australia, which extends from the northern subtropical zone to southern Australia.

Figure  also includes the prior and posterior annual flux aggregated into these bioclimatic regions. It is evident that savanna and sparsely vegetated ecosystems were the regions across Australia that most contribute to posterior carbon sink estimated for 2015. The annual posterior carbon flux for savanna was -0.17 ± 0.03 PgC yr−1 compared to the prior annual flux (0.09 ± 0.11 PgC yr−1), and the annual posterior carbon sink over sparsely vegetated was even higher (-0.25 ± 0.07 PgC yr−1) compared to prior annual flux (-0.01 ± 0.11 PgC yr−1). These results are not unexpected because the sparsely vegetated ecosystem represents the largest bioclimatic region in Australia, and a slight shift of carbon fluxes across this area causes a significant impact on the total annual flux for this ecoregion and for total annual flux estimated for Australia.

Figure  shows the monthly time series of the prior and posterior terrestrial flux aggregated into these bioclimatic regions. Over the savanna ecosystem (Fig. b), our inversion indicates that from January to June, this ecosystem acted as carbon sink. In February, in this ecosystem, we see that the prior sink (-0.48 ±  0.40 PgC yr−1) strengthens to a posterior of -1.07 ± 0.10 PgC yr−1. The stronger carbon sink (relative to the prior) from January to March coincides with an increase of greenness in vegetation (positive EVI anomalies) in this ecosystem (see Fig. S1a and b in the Supplement), benefited by anomalous rainy conditions in January (see Fig. S2a in the Supplement). Thus, it seems that the anomalous increase of rainfall in northern Australia in January benefits the increase in vegetation growth and carbon uptake recorded in February. However, it is difficult to draw conclusions about the posterior carbon uptake seen in months subsequent to March because of unfavourable raining conditions and negative EVI anomalies in these periods.

Another noticeable difference between prior and posterior flux estimates over the savanna is seen in July and August. In July, we cannot conclude if the prior was a sink or source of carbon (0.19 ± 0.28 PgC yr−1). However, our inversion indicates that savanna was acting as a carbon sink of -0.35 ± 0.11 PgC yr−1. In August, the prior source (0.25 ±  0.20 PgC yr−1) becomes a posterior carbon sink of -0.22 ±  0.08 PgC yr−1. To understand the difference between the prior and posterior estimate in this period, we calculated the GPP estimated by the CABLE-BIOS3 model and the GPP generated by MODIS (see Appendix ; Fig. b). The temporal correlation between CABLE-BIOS3 and MODIS GPP was moderate (R=0.69). According to MODIS estimates, the CABLE-BIOS3 GPP is overestimated from January to March and underestimated from May to October. The underestimation of the GPP flux by the CABLE-BIOS3 model might explain why we find a stronger posterior sink estimated by our inversion in this category.

Over the warm temperate region, from February to April, our posterior estimate suggests a carbon source (Fig. c). For this period, we cannot determine if the prior flux estimate was a carbon sink or source due to its uncertainty range. In February, the prior flux (-0.05 ±  0.08 PgC yr−1) becomes a posterior carbon source of 0.17 ± 0.06 PgC yr−1. In March, the prior estimate was nearly neutral (0.04 ± 0.05 PgC yr−1) compared to the posterior carbon source estimate (0.17 ± 0.05 PgC yr−1). The reduced carbon uptake estimated by the inversion in this period does not agree with the positive EVI anomalies seen in this region; however, it is likely that this extra carbon release to the atmosphere is related to an increase of leaf respiration in response to high temperatures recorded in 2015 for the majority of Australia . Another possible reason for the relatively small shift from the prior in this period was most likely because the CABLE-BIOS3 GPP overestimates MODIS GPP (Appendix , Fig. c). For the warm temperate category, the correlation of CABLE-BIOS3 and MODIS GPP flux is high (R=0.86).

We also see a subtle disagreement between prior and posterior estimates over the cool temperate ecosystem in April and May (Fig. d). In this period, our posterior estimate indicates that this category was a stronger carbon source than the prior flux estimate. In April, the inversion strengthened the prior source (0.12 ± 0.1 PgC yr−1) to a posterior of 0.47 ± 0.05 PgC yr−1. In May, we cannot define if the prior was a sink or a source (0.06 ± 0.09 PgC yr−1); however, our assimilated fluxes indicate this category was acting as a posterior carbon source (0.36 ±  0.04 PgC yr−1). The most likely reason for a larger carbon release in this period is related to negative EVI anomalies seen across this ecosystem. While it is true that April and May see positive EVI anomalies (Fig. S1d and e in the Supplement), in April, we notice predominantly negative EVI anomalies in the southern corner of the Australian (mainland) and Tasmania. The analysis of GPP between the CABLE-BIOS3 model and MODIS also shows some discrepancies (see Appendix ; Fig. d). For this category, in general, the CABLE-BIOS3 GPP is overestimating the productivity of the land for the whole year. For example, the absolute difference between both GPP datasets in April and June is about 0.2 PgC yr−1. For the cool temperate category, the correlation between CABLE-BIOS3 and MODIS GPP is moderate (R=0.73).

Another disagreement between the prior and posterior terrestrial flux estimate is seen over the Mediterranean ecoregion in August (Fig. e). Our posterior estimate is a flux of -0.35 ± 0.08 PgC yr−1 compared to the prior of -0.12 ± 0.12 PgC yr−1. An increase in vegetation productivity may also be the reason for the increase in carbon uptake by this category (positive EVI anomalies; Fig. S1h in the Supplement). This larger carbon uptake was likely a consequence of an increase of rainfall in this category (greater than 60 % on average (relative to the mean for 2000–2014) for some areas of this ecosystem; Fig. S3h in the Supplement). We also found that CABLE-BIOS3 underestimates MODIS GPP by 0.2 PgC yr−1.

We found a noteworthy discrepancy between the prior and posterior flux estimate over sparsely vegetated ecosystem from March to September (Fig. f). In this period, in general, the absolute difference between the prior and posterior mean was around 0.4 PgC yr−1. The largest difference found in July and September was about 0.6 PgC yr−1. This highly unexpected and counterintuitive difference is not because we see a significant increase of positive EVI anomalies across this ecoregion (Fig. S1 in the Supplement). On the contrary, it is because a “small shift” in the carbon fluxes over this large ecosystem causes an important impact on the total carbon net flux calculated for the whole country. We clearly demonstrate this fact in Appendix , Fig. . This figure in the Appendix shows the fluxes divided by area. In the western region of this category, we see evident positive EVI anomalies which start from April and last all the way through to September, which again line up with positive rainfall anomalies in that period.

Analysis of the GPP also shows the stronger posterior sink estimated by our inversion might be associated with a underestimation of the GPP by CABLE-BIOS3 in this category. The absolute difference between the CABLE-BIOS3 GPP and MODIS GPP was almost the same between May and September, with a range of 0.8–1.1 PgC yr−1. This underestimation in GPP also suggests an underestimation of the land productivity. In this same category the posterior sink estimated in September disappears in October. For this period, our posterior source estimate (0.14 ± 0.08 PgC yr−1) did not change much from the prior (0.18 ± 0.13 PgC yr−1). In November and December, our posterior source was strengthened by the inversion. In November, we estimated a carbon source of 0.21 ± 0.08 PgC yr−1 in comparison with the prior, which was 0.12 ± 0.13 PgC yr−1. The extra carbon release estimated by the inversion in November might likely be associated with the combination of fires (Fig. S4k in the Supplement) located in the west and central northwestern region of Australia (Fig. S4k in the Supplement) and due to high temperatures recorded across Australia in summer . These conditions certainly intensified the wildfires seen in that period.

In summary, our results showed that OCO-2 produced a shift in the carbon flux (relative to the prior) over the savanna and sparsely vegetated region. We found strong negative correlations (R>0.8) at grid-cell scale between the EVI anomalies and the posterior and prior difference in northern Australia (savanna ecosystem) and in the western region of the sparsely vegetated ecosystem, which align with the spatial pattern of rainfall in that area. These results suggest that our OCO-2 inversion might likely be better at capturing the anomalies in comparison with the biosphere land model.

In this section, we evaluate the accuracy of our posterior fluxes by comparing the residual between the prior and posterior concentrations simulated by CMAQ against TCCON and in situ observations. In this comparison, we simulate the posterior concentration with fluxes that were not only assimilated by nadir and glint satellite observations (LNLG) but also by the combination of both land nadir and land and ocean glint observations (LNLGOG). We decided to examine whether biases in our posterior concentration could improve when incorporating glint ocean observations into the inversion.