Movie S1Movie S1. Results of the assimilation of OCO-2 XCO2 data into a high-resolution global model for March 2015 through July 2015, highlighting the springtime reduction in atmospheric CO2.

Variational methods are widely used to solve geophysical inverse problems. Although gradient-based minimization algorithms are available for high-dimensional problems (dimension > 10 6 ), they do not provide an estimate of the errors in the optimal solution. In this study, we assess the performance of several numerical methods to approximate the analysis-error covariance matrix, assuming reasonably linear models. The evaluation is performed for a CO 2 flux estimation problem using synthetic remote-sensing observations of CO 2 columns. A low-dimensional experiment is considered in order to compare the analysis error approximations to a full-rank finite-difference inverse Hessian estimate, followed by a realistic high-dimensional application. Two stochastic approaches, a MonteCarlo simulation and a method based on random gradients of the cost function, produced analysis error variances with a relative error < 10%. The long-distance error correlations due to sampling noise are significantly less pronounced for the gradient-based randomization, which is also particularly attractive when implemented in parallel. Deterministic evaluations of the inverse Hessian using the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm are also tested. While existing BFGS preconditioning techniques yield poor approximations of the error variances (relative error > 120%), a new preconditioner that efficiently accumulates information on the diagonal of the inverse Hessian dramatically improves the results (relative error < 50%). Furthermore, performing several cycles of the BFGS algorithm using the same gradient and vector pairs enhances its performance (relative error < 30%) and is necessary to obtain convergence. Leveraging those findings, we proposed a BFGS hybrid approach which combines the new preconditioner with several BFGS cycles using information from a few (3-5) Monte-Carlo simulations. Its performance is comparable to the stochastic approximations for the low-dimensional case, while good scalability is obtained for the high-dimensional experiment. Potential applications of these new BFGS methods range from characterizing the information content of high-dimensional inverse problems to improving the convergence rate of current minimization algorithms.

Abstract. Accurate assessment of anthropogenic carbon dioxide (CO2) emissions and their redistribution among the atmosphere, ocean, and terrestrial biosphere in a changing climate is critical to better understand the global carbon cycle, support the development of climate policies, and project future climate change. Here we describe and synthesize data sets and methodology to quantify the five major components of the global carbon budget and their uncertainties. Fossil CO2 emissions (EFOS) are based on energy statistics and cement production data, while emissions from land-use change (ELUC), mainly deforestation, are based on land-use and land-use change data and bookkeeping models. Atmospheric CO2 concentration is measured directly, and its growth rate (GATM) is computed from the annual changes in concentration. The ocean CO2 sink (SOCEAN) is estimated with global ocean biogeochemistry models and observation-based data-products. The terrestrial CO2 sink (SLAND) is estimated with dynamic global vegetation models. The resulting carbon budget imbalance (BIM), the difference between the estimated total emissions and the estimated changes in the atmosphere, ocean, and terrestrial biosphere, is a measure of imperfect data and understanding of the contemporary carbon cycle. All uncertainties are reported as ±1σ. For the first time, an approach is shown to reconcile the difference in our ELUC estimate with the one from national greenhouse gases inventories, supporting the assessment of collective countries’ climate progress. For the year 2020, EFOS declined by 5.4 % relative to 2019, with fossil emissions at 9.5 ± 0.5 GtC yr−1 (9.3 ± 0.5 GtC yr−1 when the cement carbonation sink is included), ELUC was 0.9 ± 0.7 GtC yr−1, for a total anthropogenic CO2 emission of 10.2 ± 0.8 GtC yr−1 (37.4 ± 2.9 GtCO2). Also, for 2020, GATM was 5.0 ± 0.2 GtC yr−1 (2.4 ± 0.1 ppm yr−1), SOCEAN was 3.0 ± 0.4 GtC yr−1 and SLAND was 2.9 ± 1 GtC yr−1, with a BIM of −0.8 GtC yr−1. The global atmospheric CO2 concentration averaged over 2020 reached 412.45 ± 0.1 ppm. Preliminary data for 2021, suggest a rebound in EFOS relative to 2020 of +4.9 % (4.1 % to 5.7 %) globally. Overall, the mean and trend in the components of the global carbon budget are consistently estimated over the period 1959–2020, but discrepancies of up to 1 GtC yr−1 persist for the representation of annual to semi-decadal variability in CO2 fluxes. Comparison of estimates from multiple approaches and observations shows: (1) a persistent large uncertainty in the estimate of land-use changes emissions, (2) a low agreement between the different methods on the magnitude of the land CO2 flux in the northern extra- tropics, and (3) a discrepancy between the different methods on the strength of the ocean sink over the last decade. This living data update documents changes in the methods and data sets used in this new global carbon budget and the progress in understanding of the global carbon cycle compared with previous publications of this data set (Friedlingstein et al., 2020; Friedlingstein et al., 2019; Le Quéré et al., 2018b, 2018a, 2016, 2015b, 2015a, 2014, 2013). The data presented in this work are available at https://doi.org/10.18160/gcp-2021 (Friedlingstein et al., 2021).

The El Niño Modoki in 2010 led to historic droughts in Brazil. In order to understand its impact on carbon cycle variability, we derive the 2011-2010 annual carbon flux change ( F ↑ ) globally and specifically to Brazil using the NASA Carbon Monitoring System Flux (CMS-Flux) framework. Satellite observations of CO 2 , CO, and solar-induced fluorescence (SIF) are ingested into a 4D-variational assimilation system driven by carbon cycle models to infer spatially resolved carbon fluxes including net ecosystem production, biomass burning, and gross primary productivity (GPP). The global 2011-2010 net carbon flux change was estimated to be Plain Language SummaryWe quantify the global and Brazilian carbon response to 2010 El Niño using the NASA Carbon Monitoring System Flux (CMS-Flux) framework. Satellite observations of CO 2 , CO, and solar-induced fluorescence (SIF) are ingested into a 4D-variational assimilation system driven by carbon cycle models to infer spatially resolved carbon fluxes including net ecosystem exchange, biomass burning, and gross primary productivity (GPP). We show that CO 2 biomass burning from Brazil was both the dominant driver of net carbon exchange in Brazil and the dominant contributor to the global biomass burning from 2011-2010.

Quantifying variability in the ocean carbon sink remains problematic due to sparse observations and spatiotemporal variability in surface ocean pCO 2. To address this challenge, we have updated and improved ECCO-Darwin, a global ocean biogeochemistry model that assimilates both physical and biogeochemical observations. The model consists of an adjoint-based ocean circulation estimate from the Estimating the Circulation and Climate of the Ocean (ECCO) consortium and an ecosystem model developed by the Massachusetts Institute of Technology Darwin Project. In addition to the data-constrained ECCO physics, a Green's function approach is used to optimize the biogeochemistry by adjusting initial conditions and six biogeochemical parameters. Over seasonal to multidecadal timescales (1995-2017), ECCO-Darwin exhibits broad-scale consistency with observed surface ocean pCO 2 and air-sea CO 2 flux reconstructions in most biomes, particularly in the subtropical and equatorial regions. The largest differences between CO 2 uptake occur in subpolar seasonally stratified biomes, where ECCO-Darwin results in stronger winter uptake. Compared to the Global Carbon Project OBMs, ECCO-Darwin has a time-mean global ocean CO 2 sink (2.47 ± 0.50 Pg C year −1) and interannual variability that are more consistent with interpolation-based products. Compared to interpolation-based methods, ECCO-Darwin is less sensitive to sparse and irregularly sampled observations. Thus, ECCO-Darwin provides a basis for identifying and predicting the consequences of natural and anthropogenic perturbations to the ocean carbon cycle, as well as the climate-related sensitivity of marine ecosystems. Our study further highlights the importance of physically consistent, property-conserving reconstructions, as are provided by ECCO, for ocean biogeochemistry studies. Plain Language Summary Data-driven estimates of how much carbon dioxide the ocean is absorbing (the so-called "ocean carbon sink") have improved substantially in recent years. However, computational ocean models that include biogeochemistry continue to play a critical role as they allow us to isolate and understand the individual processes that control ocean carbon sequestration. The ideal scenario is a combination of the above two methods, where data are ingested and then used to improve a model's fit to the observed ocean, also known as, data assimilation. While the physical oceanographic community has made great progress in developing data assimilation systems, for example, the Estimating the Circulation and Climate of the Ocean (ECCO) consortium, the biogeochemical community has generally lagged behind. The ECCO-Darwin model presented in this paper represents an important technological step forward as it is the first global ocean biogeochemistry model that (1) ingests both physical and biogeochemical observations into the model in a realistic manner and (2) considers how the nature of the ocean carbon sink has changed over multiple decades. As the ECCO ocean circulation