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Geostatistical Inverse Modeling for
Characterizing the Global Carbon Cycle
Anna M. Michalak
Department of Civil and Environmental EngineeringDepartment of Atmospheric, Oceanic and Space SciencesThe University of Michigan
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
Source: NOAA-ESRL
5Tyler Erickson, Michigan Tech Research Institute([email protected])
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 topick your battles
intelligently!
Synthesis Bayesian Inversion
InversionCarbon Budget
Synthesis Bayesian Inversion
Meteorological fields
Transportmodel
Sensitivity of observations to
fluxes (H)
Residual covariance
structure (Q, R)
Prior flux
estimates (sp)
CO2
observations (y)
Inversion
Flux estimates and covariance
ŝ, Vŝ
Biosphericmodel
Auxiliaryvariables
?
?
Biospheric Models as Priors
Deborah Huntzinger, U. Michigan
InversionCarbon Budget
Geostatistical Inversion Model
InversionCarbon Budget
Geostatistical Inversion Model
Synthesis Bayesian Inversion
Meteorological fields
Transportmodel
Sensitivity of observations to
fluxes (H)
Residualcovariance
structure (Q, R)
Prior flux estimates (sp)
CO2
observations (y)
InversionFlux estimates and covariance
ŝ, Vŝ
Biosphericmodel
Auxiliaryvariables
Geostatistical Inversion
Meteorological fields
Transportmodel
Sensitivity of observations to
fluxes (H)
Residual covariance
structure (Q, R)
Auxiliaryvariables
CO2
observations (y)
Model selection
Inversion
Covariance structure
characterization
Flux estimates and covariance
ŝ, Vŝ
Trend estimate and covariance
β, Vβ
select significant variables
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
1 1,
1 1( ) ( ) ( ) ( )2 2
T TL s β y Hs R y Hs s Xβ Q s Xβ
Deterministiccomponent
Stochasticcomponent
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)
yHXHXHXHXy 11111 TTTTTWSS
RHQH T
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
WSSL TT HXHX 1lnln
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:
T
T
TT X
HQ
M0HX
HX
XMQHQVys s TTˆˆ
Examined Scales
Flux TowerN. AmericaGlobal
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)
May 2004
+
=
Estimates from North American Study
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
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 -0.64 -0.79 -0.49United States -0.33 -0.47 -0.18Central America 0.12 -0.04 0.29
total -0.84 -1.11 -0.57
Annual Average Eco-Region Flux
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:
