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

2. 2

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
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.

4. Source: NOAA-ESRL

5. Tyler Erickson, Michigan Tech Research Institute 5 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
Carbon Budget 

8. 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) ?

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

10. Geostatistical Inversion Model
Carbon Budget 

11. Geostatistical Inversion Model
Carbon Budget 

12. 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)

13. 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

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 14

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 18

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+
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

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 21

22. 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

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: 25

26. AN APOLOGY AND A REQUEST