Geostatistical Inverse Modeling for Characterizing the Global Carbon Cycle

1. Geostatistical Inverse Modeling for Characterizing the Global Carbon Cycle Anna M. Michalak Department of Civil and Environmental Engineering Department of Atmospheric, Oceanic and Space Sciences The University of Michigan

3. The Future of Natural Carbon Sinks Land Uncertainty associated with the future of natural carbon sinks is one of two major sources of uncertainty in future climate projections 300 ppm Oceans Friedlingstein et al. (2006) showing projections from coupled carbon and climate simulations for several models.

4. Source: NOAA-ESRL

5. Tyler Erickson, Michigan Tech Research Institute (tyler.erickson@mtu.edu) 5

6. Carbon Flux Inference Characteristics • Inverse problem • Ill-posed • Underdetermined • Space-time variability • Multiscale • Nonstationary • Available ancillary data (with uncertainties) • Deterministic process models have (non-Gaussian) errors (biospheric and atmospheric models) • Large datasets (but still data poor), soon to be huge datasets with the advent of space-based CO2 observations • Large to huge parameter space, depending on spatial / temporal resolution of estimation Need to pick your battles intelligently!

7. Synthesis Bayesian Inversion Inversion Carbon Budget 

8. Synthesis Bayesian Inversion Prior flux estimates (sp) Biosphericmodel CO2observations (y) Auxiliaryvariables Inversion Flux estimates and covarianceŝ, Vŝ ? Transportmodel Sensitivity of observations to fluxes (H) Meteorological fields Residual covariancestructure (Q, R) ?

9. Biospheric Models as Priors Deborah Huntzinger, U. Michigan

10. Geostatistical Inversion Model Inversion Carbon Budget 

11. Geostatistical Inversion Model Inversion Carbon Budget 

12. Synthesis Bayesian Inversion Prior flux estimates (sp) Biosphericmodel CO2observations (y) Auxiliaryvariables Inversion Flux estimates and covarianceŝ, Vŝ Transportmodel Sensitivity of observations to fluxes (H) Meteorological fields Residualcovariancestructure (Q, R)

13. Geostatistical Inversion select significant variables Auxiliaryvariables Model selection CO2observations (y) Flux estimates and covariance ŝ, Vŝ Inversion Transportmodel Sensitivity of observations to fluxes (H) Trend estimate and covariance β, Vβ Meteorological fields Residual covariancestructure (Q, R) Covariance structure characterization optimize covariance parameters

14. Geostatistical Approach to Inverse Modeling Geostatistical inverse modeling objective function: H = transport information, s = unknown fluxes, y = CO2 measurements X and  = model of the trend R = model data mismatch covariance Q = spatio-temporal covariance matrix for the flux deviations from the trend Deterministic component Stochastic component

15. Model Selection • Dozen of types of ancillary data, many of which are from remote sensing platforms, are available • Need objective approach for selecting variables, and potentially their functional form to be included in X • Modified expression for weighted sum of squares: • Now we can apply statistical model selection tools: • Hypothesis based, e.g. F-test • Criterion based, e.g. modified BIC (with branch-and-bound algorithm for computational feasibility) • Modified BIC (using branch-and-bound algorithm for computational efficiency)

16. Covariance Optimization • Need to characterize covariance structure of unobserved parameters (i.e. carbon fluxes) Q using information on secondary variables (i.e. carbon concentrations) and selected ancillary variables • Also need to characterize the model-data mismatch (sum of multiple types of errors) R • Restricted Maximum Likelihood, again marginalizing w.r.t. : • In some cases, atmospheric monitoring network is insufficient to capture sill and range parameters of Q

17. Other Implementation Choices • No prior information on drift coefficients , which are estimated concurrently with overall spatial process s • No prior information on Q and R parameters, which are estimated in an initial step, but then assumed known • This setup, combined with Gaussian assumptions on residuals, yields a linear system of equations analogous to universal cokriging:

18. Examined Scales Global N. America Flux Tower

19. Timeline of Development • First presentation of approach: • Michalak, Bruhwiler, Tans (JGR-A 2004) • Application to estimation of global carbon budget, with and without the use of ancillary spatiotemporal data, model selection using modified F-test: • Mueller, Gourdji, Michalak (JGR-A, 2008) • Gourdji, Mueller, Schaefer, Michalak (JGR-A 2008) • Approach development for North American carbon budget, with the addition of temporal correlation: • Gourdji, Hirsch, Mueller, Andrews, Michalak (ACP, in review) • Application to estimation of NA carbon budget, model selection using modified BIC: • Gourdji, Michalak, et al. (in prep) • Related applications for carbon flux analysis and modeling: • Yadav, Mueller, Michalak (GCB, in review) • Huntzinger, Michalak, Gourdji, Mueller (JGR-B, in review) • Mueller, Yadav, Curtis, Vogel, Michalak (GBC, in review)

20. Estimates from North American Study + = May 2004

21. Grid Scale Seasonal Cycle • Inversion results compared to 15 forward models • Significant differences between inversion & forward models during the growing season, also near measurement towers

22. Annual Average Eco-Region Flux Eco-region scale annual inversion fluxes fall within the spread of forward models, except in Boreal Forests and Desert & Xeric Shrub

23. Carbon Flux Inference Contributions • Inverse problem • Ill-posed • Underdetermined • Space-time variability • Multiscale • Nonstationary • Available ancillary data (with uncertainties) • Deterministic process models have (non-Gaussian) errors (biospheric and atmospheric models) • Large datasets (but still data poor), soon to be huge datasets with the advent of space-based CO2 observations • Large to huge parameter space, depending on spatial / temporal resolution of estimation

24. Carbon Flux Inference Opportunities • Inverse problem • Ill-posed • Underdetermined • Space-time variability • Multiscale • Nonstationary • Available ancillary data (with uncertainties) • Deterministic process models have (non-Gaussian) errors (biospheric and atmospheric models) • Large datasets (but still data poor), soon to be huge datasets with the advent of space-based CO2 observations • Large to huge parameter space, depending on spatial / temporal resolution of estimation

25. Acknowledgements Collaborators on carbon flux modeling work: Research group: Abhishek Chatterjee, Sharon Gourdji, Charles Humphriss, Deborah Huntzinger, Miranda Malkin, Kim Mueller, Yoichi Shiga, Landon Smith, Vineet Yadav NOAA-ESRL: Pieter Tans, Adam Hirsch, Lori Bruhwiler, Arlyn Andrews, Gabrielle Petron, Mike Trudeau Peter Curtis (Ohio State U.), Ian Enting (U. Melbourne), Tyler Erickson (MTRI), Kevin Gurney (Purdue U.), Randy Kawa (NASA Goddard), John C. Lin (U. Waterloo), Kevin Schaefer (NSIDC), Chris Vogel (UMBS), NACP Regional Interim Synthesis Participants Funding sources:

26. AN APOLOGY AND A REQUEST amichala@umich.edu http://www.umich.edu/~amichala/