Constraining North American Fluxes of Carbon Dioxide and Inferring their Spatiotemporal Covariancesthrough Assimilation of Remote Sensing and Atmospheric Data in a Geostatistical Framework

A.M. Michalak1,2 (amichala@umich.edu), A.I. Hirsch3,4, J.C. Lin5, A. Andrews4

1Department of Civil and Environmental Engineering, University of Michigan, Ann Arbor, MI 48109-2125, 2Department of Atmospheric, Oceanic and Space Sciences, University of Michigan, Ann Arbor, MI 48109-2143,3Cooperative Institute for Research in Environmental Sciences, University of Colorado, Boulder, CO 80309, 4NOAA Global Monitoring Division, Boulder, CO 80305

5Department of Earth Sciences, University of Waterloo, Waterloo, ON N2L 3G1

Goal and Scope

Use of Auxiliary Data

Geostatistical Inverse Modeling

Transport Model and Influence Functions

Regional Inversions

Continuous Tall Tower Measurements

TOWER MEASUREMENTS

ENVIRONMENTALDATASETS VARIANCE

RATIOTEST

RESTRICTEDMAXIMUM

LIKELIHOOD

COVARIANCEPARAMETERS

SELECTEDENVIRONMENTAL

DATASETS

GEOSTATISTICALINVERSEMODEL

INFLUENCEFUNCTIONS

STILT MODEL

HIGHRESOLUTIONWIND FIELDS

WRF MODEL

INFLUENCE OF ENV. DATAON FLUXES

A POSTERIORIFLUXES

DATA GEOSTATISTICALTECHNIQUES

OUTPUT

TRANSPORTINFORMATION

A POSTERIORIERROR

Prescribed and Retrieved Fluxes

Reduction in Uncertainty in Stochastic Component of Flux Estimates

For geostatistical inverse modeling approaches, reduction in overall flux uncertainty cannot be evaluated in a manner analo-gous to synthesis Bayesian inversions, because there is no a priori information on the components of the spatial trend of the flux. However, the reduction in uncertainty in the stochastic component of the estimate can be used as an indication of the information provided by the atmospheric measurements.

Prescribed flux (black dashed), retrieved flux (black), a priori sto-hastic uncertainty (red), a poste-riori total uncertainty (blue) as a function of aggregated distance from tall tower sites

Ls, 12 z HsTR1z Hs 1

2 s XTQ1s X #

X 1 1T

#

X 1 1t1 tm

T

#

REFERENCES: Kitanidis, P.K., A variance-ratio test for supporting a variable mean in kriging, Mathematical Geol-

ogy, 29 (3), 335-348, 1997.Lin, J.C., and C. Gerbig, Accounting for the effect of transport errors on tracer inversions, Geo-

physical Research Letters, 32 (L01802), 2005.Michalak, A.M., L. Bruhwiler, and P.P. Tans, A geostatistical approach to surface flux estimation of

atmospheric trace gases, Journal of Geophysical Research-Atmospheres, 109 (D14109), 2004.

This sensitivity information was used to estimate monthly biospheric fluxes from the SiB2 model at a 1o x 1o resolution over North America. Results show that the geostatistical inverse modeling framework is able to constrain thou-sands of unknown fluxes using only sev-eral hundred atmospheric measure-ments by utilizing information on the co-variance of those fluxes and the ex-pected mismatch between modeled and observed CO2 observations.

• This ongoing project will use remote-sensing data and atmospheric measurements in a geostatistical inverse modeling framework to quantify North American surface fluxes of carbon dioxide at high spatiotemporal resolution.

• Flux magnitudes will be estimated without relying on prior flux estimates, and the approach will account for the uncertainties and covariances associated with measurement and transport errors and flux distributions.

• The influence of remote-sensing derived auxiliary environmental data will be evaluated as part of the inversion, which will provide process-based understanding of flux variations. This process-based understanding will be useful when compared to biospheric model parameterizations.

• Preliminary results from global scale inversions suggest that LAI, FPAR, and synoptic scale meteorological parameters explain a significant amount of biospheric flux variability at the global scale.

• Synthetic data experiments at the scale of North America show that the proposed approach is able to accurately recover monthly average NEE and the monthly-average diurnal cycle of NEE for large portions of the continent at 1o x 1o resolution.

The geostatistical approach to inverse modeling is a Bayesian approach in which the prior probability density function is based on an assumed form for the spatial and/or temporal correlation of the surface fluxes to be estimated (Michalak et al. 2004). This differs from the traditional Bayesian approaches, where the prior information is in the form of initial surface flux estimates.

In the geostatistical approach, the prior flux estimate is replaced by a model of the mean of the flux distribution, and the criterion of remaining close to a prior flux esti-mate is replaced by a criterion of preserving a spatial and/or temporal correlation in the flux distribution: Mathematically,

where z : Observation vector H : Transport matrix s : Surface flux distribution R : Model-data mismatch error covariance matrix X : Matrix that defines the model of the mean.

For example, for a constant mean:

For a system where the mean of the fluxes is expected to have a linear trend with an additional variable t (e.g. LAI or FPAR):

β : Unknown drift parameters to be estimated along with s Q : Prior flux covariance matrix based on selected model (a full matrix)

Key differences relative to classical Bayesian approach:

• Parameters of the model of the mean Xβ are estimated as part of the inversion

• Auxiliary data can be incorporated into Xβ and its significance can be evaluated using the Variance Ratio Test (Kitanidis 1997)

• The prior covariance matrix Q defines spatial and/or temporal correlations

• Parameters such as the variance of the model-data mismatch in R and variance and correlation length of the covariance model in Q can also be estimated from the data, using restricted maximum likelihood (RML)

• Geostatistical flux estimates are not subject to some limitations of traditional Bayesian inversions, such as potential biases created by the choice of prior fluxes and aggregation error resulting from the use of large regions with prescribed flux patterns

• The geostatistical approach is ideally suited to inversions at fine spatial scales

In past atmospheric geostatistical inverse modeling studies (Michalak et al. 2004), surface fluxes were assumed to tend toward different but constant means. Because deviations from these means were correlated, this simple model of the mean still al-lowed for complex flux patterns.

In recent work, tools were developed to model the spatial trend of the carbon flux dis-tribution using objective auxiliary variables known to be associated with the global carbon cycle, allowing the model greater flexibility, while still not requiring the poste-rior fluxes to follow any pre-prescribed flux patterns.

The auxiliary data are included in the inversion in the matrix X in the model of the mean Xβ, and the relationship between auxiliary data and flux magnitudes is inferred as part of the inversion.

A Variance Ratio Test is used to test the statistical significance of the information pro-vided by the auxiliary data, as evaluated using the atmospheric mass fraction mea-surements.

The best estimate is composed of distinct components attributable to an overall spa-tial trend, a portion of flux explained by each of the auxiliary variables, and a spatially correlated stochastic component.

ACKNOWLEDGMENTS: This research is funded by the NASA SMD ROSES North American Carbon Program, under contract number NNX06AE84G. Computing resources are provided by the NASA Project Columbia HEC, under award number SMD-06-0249. Additional support was provided by The NOAA Global Monitoring Division, the NOAA Program in Climate and Global Change, and an Elizabeth C. Crosby Research Award. Special thanks to Sharon Gourdji and Kim Mueller who worked on the global inversions, and everyone who participates in the NOAA-CMDL Cooperative Air Sampling Network.

• The Stochastic Time-Inverted Lagrangian Transport (STILT) model will be used to track the influence of surface fluxes on available atmospheric measurements.

• STILT is a Lagrangian model that represents air parcels as particles that are transported backward in time with deterministic and stochastic velocities derived from meteorological fields.

• Lagrangian transport models have the ability to resolve the high variability in concentration measurements over the continent caused by strong sources / sinks in the “near-field” of the observation locations.

• An example of the information provided by STILT is presented in the adjoining figure. The footprint represents the sensitivity of the concentrations at the WLEF tall tower to upstream fluxes in the source regions.

• High resolution meteorological fields will be generated using the Weather Research and Forecasting (WRF) model run on Project Columbia, a NASA HEC facility.

• In order to property resolve mesoscale circulation, cloud venting and other detailed atmospheric phenomena which affect CO2 transport, the WRF model will be nested down to high resolution (~2 km) over the tall tower target regions.

• Errors in the atmospheric transport will be evaluated using the approach outlined in Lin and Gerbig (2005). Additional tools will be developed to quantify errors in vertical mixing.

• Continuous data from the NOAA Tall Tower network are the primary source of atmospheric CO2 observations for this work. The main towers that will be used are the WLEF tower in Park Falls, WI, the WKT tower in Moody, TX, and the AMT tower in Argyle, ME.

• Data from other measurement sites such as eddy covariance towers that are equipped to provide calibrated CO2 mixing ratio measurements will be included as appropriate.

• The adjoining figure presents a map of the distance to the nearest tall tower for North America, including the three towers listed above and three proposed sites for the expansion of the tall tower network (South Carolina, Iowa, and North Dakota). Coverage is best in the upper Midwestern U.S. for this configuration, corresponding to the focus area of the planned NACP Midwest intensive experiment.

A pseudodata study was performed to verify the feasibility of using continuous tower data in a geostatistical inverse modeling framework to infer surface fluxes at various temporal and spatial scales for North America.

Estimated Stochastic Component

+ =

Best Estimate

−2 −1.5 −1 −0.5 0 0.5 1 µmol/(m2s)

Land & Ocean Constants

Percent Landcover by Shrublands

Leaf Area Index (LAI)

Fraction of Photosynthetically Active Radiation (FPAR)

Land Temperature

Population Density Weighted Gross Domestic Product (GDP)

Estimated Deterministic Trend Components

Glo

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Atmospheric observations at the tall tower sites are sensitive to fluxes over the major-ity of the continental United States, with somewhat weaker sensitivity to fluxes in the Western states. The adjoining figure pres-ents the integrated footprint for September 2004, based on 720 individual footprints, corresponding to 4 measurements per day at 6 tall towers, created with particle trajec-tory information from STILT.

Integrated Footprint, Sept. ‘04