Geostatistical Inverse Modeling for Characterizing the Global Carbon Cycle

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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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(No Transcript)
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The Future of Natural Carbon Sinks
Uncertainty associated with the future of natural
carbon sinks is one of two major sources of
uncertainty in future climate projections
Land
300 ppm
Oceans
Friedlingstein et al. (2006) showing projections
from coupled carbon and climate simulations for
several models.
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Source NOAA-ESRL
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Tyler Erickson, Michigan Tech Research
Institute (tyler.erickson_at_mtu.edu)
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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 CO2
  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
Prior flux estimates (sp)
Biosphericmodel
CO2observations (y)
Auxiliaryvariables
Inversion
Flux estimates and covariances, Vs
?
Transportmodel
Sensitivity of observations to fluxes (H)
Meteorological fields
Residual covariancestructure (Q, R)
?
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Biospheric Models as Priors
Deborah Huntzinger, U. Michigan
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Geostatistical Inversion Model
Inversion
Carbon Budget
?
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Geostatistical Inversion Model
Inversion
Carbon Budget
?
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Synthesis Bayesian Inversion
Prior flux estimates (sp)
Biosphericmodel
CO2observations (y)
Auxiliaryvariables
Inversion
Flux estimates and covariances, Vs
Transportmodel
Sensitivity of observations to fluxes (H)
Meteorological fields
Residualcovariancestructure (Q, R)
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Geostatistical Inversion
select significant variables
Auxiliaryvariables
Model selection
CO2observations (y)
Flux estimates and covariance s, Vs
Inversion
Transportmodel
Sensitivity of observations to fluxes (H)
Trend estimate and covariance ß, Vß
Meteorological fields
Residual covariancestructure (Q, R)
Covariance structure characterization
optimize covariance parameters
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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
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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
Global
N. America
Flux Tower
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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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Estimates from North American Study


May 2004
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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

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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

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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 CO2
  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 CO2
  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
• amichala_at_umich.edu
• http//www.umich.edu/amichala/