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(tyler.erickson@mtu.edu)

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:

AN APOLOGY AND A REQUEST

amichala@umich.eduhttp://www.umich.edu/~amichala/