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

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Source: NOAA-ESRL

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5 Tyler Erickson, Michigan Tech Research Institute

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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 CO 2 observations Large to huge parameter space, depending on spatial / temporal resolution of estimation Need to pick your battles intelligently!

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Synthesis Bayesian Inversion Inversion Carbon Budget

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Synthesis Bayesian Inversion Meteorological fields Transport model Sensitivity of observations to fluxes (H) Residual covariance structure (Q, R) Prior flux estimates (s p ) CO 2 observations (y) Inversion Flux estimates and covariance ŝ, V ŝ Biospheric model Auxiliary variables ? ?

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Biospheric Models as Priors Deborah Huntzinger, U. Michigan

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Inversion Carbon Budget Geostatistical Inversion Model

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Inversion Carbon Budget Geostatistical Inversion Model

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Synthesis Bayesian Inversion Meteorological fields Transport model Sensitivity of observations to fluxes (H) Residual covariance structure (Q, R) Prior flux estimates (s p ) CO 2 observations (y) Inversion Flux estimates and covariance ŝ, V ŝ Biospheric model Auxiliary variables

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Geostatistical Inversion Meteorological fields Transport model Sensitivity of observations to fluxes (H) Residual covariance structure (Q, R) Auxiliary variables CO 2 observations (y) Model selection Inversion Covariance structure characterization Flux estimates and covariance ŝ, V ŝ Trend estimate and covariance β, V β select significant variables optimize covariance parameters

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Geostatistical Approach to Inverse Modeling Geostatistical inverse modeling objective function: H = transport information, s = unknown fluxes, y = CO 2 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

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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)

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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

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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:

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Examined Scales Flux Tower N. America Global

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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)

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May = Estimates from North American Study

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

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Eco-region scale annual inversion fluxes fall within the spread of forward models, except in Boreal Forests and Desert & Xeric Shrub Net flux (PgC/yr) - 2σ+ 2σ Canada + Alaska United States Central America total Annual Average Eco-Region Flux

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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 CO 2 observations Large to huge parameter space, depending on spatial / temporal resolution of estimation

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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 CO 2 observations Large to huge parameter space, depending on spatial / temporal resolution of estimation

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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:

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AN APOLOGY AND A REQUEST

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