March 2024 Inferring changes to the global carbon cycle with WOMBAT v2.0, a hierarchical flux-inversion framework
Michael Bertolacci, Andrew Zammit-Mangion, Andrew Schuh, Beata Bukosa, Jenny A. Fisher, Yi Cao, Aleya Kaushik, Noel Cressie
Author Affiliations +
Ann. Appl. Stat. 18(1): 303-327 (March 2024). DOI: 10.1214/23-AOAS1790


The natural cycles of the surface-to-atmosphere fluxes of carbon dioxide (CO2) and other important greenhouse gases are changing in response to human influences. These changes need to be quantified to understand climate change and its impacts, but this is difficult to do because natural fluxes occur over large spatial and temporal scales and cannot be directly observed. Flux inversion is a technique that estimates the spatiotemporal distribution of a gas’ fluxes using observations of the gas’ mole fraction and a chemical transport model. To infer trends in fluxes and identify phase shifts and amplitude changes in flux seasonal cycles, we construct a flux-inversion system that uses a novel spatially-varying time-series decomposition of the fluxes. We incorporate this decomposition into the Wollongong Methodology for Bayesian Assimilation of Trace-gases (WOMBAT, Zammit-Mangion et al., Geosci. Model Dev., 15, 2022), a Bayesian hierarchical flux-inversion framework that yields posterior distributions for all unknowns in the underlying model. We also extend WOMBAT to accommodate physical constraints on the fluxes and to take direct in situ and flask measurements of trace-gas mole fractions as observations. We apply the new method, which we call WOMBAT v2.0, to a mix of satellite observations of CO2 mole fraction from the Orbiting Carbon Observatory-2 (OCO-2) satellite and direct measurements of CO2 mole fraction from a variety of sources. We estimate the changes in the natural cycles of CO2 fluxes that occurred from January 2015 to December 2020, and compare our posterior estimates to those from an alternative method based on a bottom-up understanding of the physical processes involved. We find substantial trends in the fluxes, including that tropical ecosystems trended from being a net source to a net sink of CO2 over the study period. We also find that the amplitude of the global seasonal cycle of ecosystem CO2 fluxes increased over the study period by 0.11 PgC/month (an increase of 8%) and that the seasonal cycle of ecosystem CO2 fluxes in the northern temperate and northern boreal regions shifted earlier in the year by 0.4–0.7 and 0.4–0.9 days, respectively (2.5th to 97.5th posterior percentiles), consistent with expectations for the carbon cycle under a warming climate.

Funding Statement

This project was largely supported by the Australian Research Council (ARC) Discovery Project (DP) DP190100180 and by NASA ROSES grant 20-OCOST20-0004. Zammit-Mangion was also supported by the ARC Discovery Early Career Research Award DE180100203.


We would like to thank the referees and Editors for their careful reading of and comments on the initial submission of this manuscript. This research was undertaken with the assistance of resources and services from the National Computational Infrastructure (NCI), which is supported by the Australian Government as well as with the assistance of resources funded by a 2021 University of Wollongong Major Equipment Grant. We are also grateful to the OCO-2 Flux Group and colleagues from the University of Bristol, U.K., for their feedback and suggestions. The MERRA-2 data used in this study/project have been provided by the Global Modeling and Assimilation Office (GMAO) at NASA Goddard Space Flight Center. The OCO-2 data were produced by the OCO-2 project at the Jet Propulsion Laboratory, California Institute of Technology. These data, as well as the in situ/flask data used in this study, may be obtained from the OCO-2 v10 MIP website at In addition to their primary associations, the author Kaushik is also associated with the Global Monitoring Laboratory at the National Oceanic and Atmospheric Administration, and the author Cressie is also associated with the Jet Propulsion Laboratory at the California Institute of Technology.


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Michael Bertolacci. Andrew Zammit-Mangion. Andrew Schuh. Beata Bukosa. Jenny A. Fisher. Yi Cao. Aleya Kaushik. Noel Cressie. "Inferring changes to the global carbon cycle with WOMBAT v2.0, a hierarchical flux-inversion framework." Ann. Appl. Stat. 18 (1) 303 - 327, March 2024.


Received: 1 October 2022; Revised: 1 June 2023; Published: March 2024
First available in Project Euclid: 31 January 2024

Digital Object Identifier: 10.1214/23-AOAS1790

Keywords: Bayesian , Carbon cycle , carbon dioxide , flux inversion , hierarchical

Rights: Copyright © 2024 Institute of Mathematical Statistics


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Vol.18 • No. 1 • March 2024
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