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

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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 Subset of Fig. 1 from Friedlingstein et al. (2006) showing (a) atmospheric CO2 for the coupled carbon and climate simulations for several models and (c) Land carbon fluxes for these coupled runs. The differences in predicted land fluxes and ocean fluxes (not shown) cause a 300ppm difference in predicted CO2 concentrations in the year 2100. Friedlingstein et al. (2006) showing projections from coupled carbon and climate simulations for several models.

Source: NOAA-ESRL

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

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!

Synthesis Bayesian Inversion Carbon Budget 

Synthesis Bayesian Inversion Biospheric model Prior flux estimates (sp) Auxiliary variables CO2 observations (y) Inversion Flux estimates and covariance ŝ, Vŝ ? Transport model Sensitivity of observations to fluxes (H) Meteorological fields Residual covariance structure (Q, R) ?

Biospheric Models as Priors Deborah Huntzinger, U. Michigan 9

Geostatistical Inversion Model Carbon Budget 

Geostatistical Inversion Model Carbon Budget 

Synthesis Bayesian Inversion Biospheric model Prior flux estimates (sp) Auxiliary variables CO2 observations (y) Inversion Flux estimates and covariance ŝ, Vŝ Transport model Sensitivity of observations to fluxes (H) Meteorological fields Residual covariance structure (Q, R)

Geostatistical Inversion select significant variables Auxiliary variables Model selection CO2 observations (y) Flux estimates and covariance ŝ, Vŝ Inversion Transport model Sensitivity of observations to fluxes (H) Trend estimate and covariance β, Vβ Meteorological fields Residual covariance structure (Q, R) Covariance structure characterization optimize covariance parameters

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 14

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)

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

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:

Examined Scales Global N. America Flux Tower 18

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)

Estimates from North American Study + Evapotrans, Precip, Radiation, Soil Moisture, Temperature (NARR from NCEP Reanalysis), Fossil Fuels (combo of Vulcan & EDGAR for CA/MX), Fire Power, LAI (MODIS) = May 2004 20

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 21

Annual Average Eco-Region Flux Net flux (PgC/yr) - 2σ + 2σ Canada + Alaska -0.64 -0.79 -0.49 United States -0.33 -0.47 -0.18 Central America 0.12 -0.04 0.29 total -0.84 -1.11 -0.57 Note that many of these models are near 0 for all biomes. Number of models for each ecoregion: Trop=8, EastFor=12, NWConif=12, BorFor=13, Tundra=13, TempGSS=10, Desert=11 Eco-region scale annual inversion fluxes fall within the spread of forward models, except in Boreal Forests and Desert & Xeric Shrub 22

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

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

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: 25

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