Andrew Schuh, Scott Denning, Marek Ulliasz Kathy Corbin, Nick Parazoo

A Case Study in Regional Inverse Modeling

The Question:The Question:How is NEE distributed across domain in both How is NEE distributed across domain in both

time and spacetime and space

Deterministic Biosphere and Deterministic Biosphere and Transport ModelsTransport Models

SiB2.5 is used to predict carbon assimilation and SiB2.5 is used to predict carbon assimilation and manage energy fluxes at the surface. MODIS manage energy fluxes at the surface. MODIS fPAR and LAI products are used to drive SiB2.5.fPAR and LAI products are used to drive SiB2.5.

SiB2.5 is coupled to RAMS 5.0 which is used to SiB2.5 is coupled to RAMS 5.0 which is used to transport carbon dioxide. Meteorology is forced transport carbon dioxide. Meteorology is forced with Eta 40km reanalysis datawith Eta 40km reanalysis data

Entire coupled model is run on 150 x 100 40km Entire coupled model is run on 150 x 100 40km grid over North America for the time period May grid over North America for the time period May 1,2004 through August 31, 2004.1,2004 through August 31, 2004.

How are observations “connected” to fluxesHow are observations “connected” to fluxes

Inversion Methods AvailableInversion Methods Available

Bayesian Synthesis InversionBayesian Synthesis Inversion

• For many problems the quickest and easiest For many problems the quickest and easiest methodmethod

• This basic bayesian posterior computation is at This basic bayesian posterior computation is at core of many inversion methodologiescore of many inversion methodologies

• However, computational concerns arise if the However, computational concerns arise if the dimensions of the problem get too largedimensions of the problem get too large

Inversion Methods AvailableInversion Methods Available

Kalman filtering techniquesKalman filtering techniques

• Reduces the effect of the time dimension of Reduces the effect of the time dimension of inversion problem by putting in state space inversion problem by putting in state space framework and updating model in time.framework and updating model in time.

• EnKF further reduces dimensional constraints EnKF further reduces dimensional constraints by effectively working with a sampled spatial by effectively working with a sampled spatial covariance structure. EnKF has also been covariance structure. EnKF has also been shown to have some desirable properties for shown to have some desirable properties for non-linear models.non-linear models.

Inversion Methods AvailableInversion Methods Available

What about dealing with the spatial What about dealing with the spatial structure of the problem in a hierarchical structure of the problem in a hierarchical way?way?

Inversion can take advantage of implicit Inversion can take advantage of implicit spatial structure inherent in many spatial spatial structure inherent in many spatial characterizations, like ecoregionscharacterizations, like ecoregions

Covariance properties are propagated Covariance properties are propagated through a hierarchical covariance through a hierarchical covariance structure, independent within levels, thus structure, independent within levels, thus reducing dimensionality of the covariance reducing dimensionality of the covariance

A possible hierarchy?A possible hierarchy?

A possible hierarchy?A possible hierarchy?

A possible hierarchy?A possible hierarchy?

Hierarchical Model (Model Domain)Hierarchical Model (Model Domain)

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A backward in time lagrangian particle model (LPDM) was A backward in time lagrangian particle model (LPDM) was used in conjunction with a 4 month SiB2.5Rams used in conjunction with a 4 month SiB2.5Rams simulation to produce “influence functions” for simulation to produce “influence functions” for assimilation and respiration for 34 towers.assimilation and respiration for 34 towers.

Four afternoon observations each day for May 10, 2004 - Four afternoon observations each day for May 10, 2004 - August 31, 2004 were used at each of the 34 towers.August 31, 2004 were used at each of the 34 towers.

Results for ExampleResults for Example(via MCMC Gibbs Sampler)(via MCMC Gibbs Sampler)

Results for ExampleResults for Example(via MCMC Gibbs Sampler)(via MCMC Gibbs Sampler)

What about boundary conditions?What about boundary conditions?

Initial SiBRAMS run had constant carbon dioxide Initial SiBRAMS run had constant carbon dioxide for boundary conditions.for boundary conditions.

What effect might this have on the simulation?What effect might this have on the simulation?

How might corrections be made to these How might corrections be made to these boundary inflow terms?boundary inflow terms?

Boundary conditionsBoundary conditions

Boundary conditionsBoundary conditions

Initial results would seem to imply that boundary conditions can be very Initial results would seem to imply that boundary conditions can be very important to regional scale inversion using carbon dioxide concentrationsimportant to regional scale inversion using carbon dioxide concentrations

The boundaries also represent a large spatial area, possibly contributing The boundaries also represent a large spatial area, possibly contributing many unknowns to an often already under constrained problemmany unknowns to an often already under constrained problem

In order to investigate this component, we begin by investigating the In order to investigate this component, we begin by investigating the modes of variability in simulated boundary conditions.modes of variability in simulated boundary conditions.

Principal Components are generated, using PCTM (N. Parazoo), for May 1, Principal Components are generated, using PCTM (N. Parazoo), for May 1, 2003 – August 31, 2003 and May 1,2004 – August 31, 2004. These 2003 – August 31, 2003 and May 1,2004 – August 31, 2004. These provide “directions” of maximal variability (in time) in the boundary provide “directions” of maximal variability (in time) in the boundary conditions.conditions.

Principal Component Comparison 2003/2004Principal Component Comparison 2003/2004

Boundary conditionsBoundary conditions

The first principal component generally represents about 75% - 85% of the The first principal component generally represents about 75% - 85% of the total variation over time with the second representing another 3% - 6%.total variation over time with the second representing another 3% - 6%.

The PCs appear to load nicely, particularly zonally. The first principal The PCs appear to load nicely, particularly zonally. The first principal component is capturing the changing zonal gradient of carbon dioxide while component is capturing the changing zonal gradient of carbon dioxide while the second appears to capture gradients produced by synoptic activity along the second appears to capture gradients produced by synoptic activity along the storm track in N.A. the storm track in N.A.

This appears to be a promising dimension reduction of the boundary influence This appears to be a promising dimension reduction of the boundary influence and possibly robust interannually.and possibly robust interannually.

An obvious assumption here is that PCTM captures the major modes of An obvious assumption here is that PCTM captures the major modes of variability. Deficiencies in the transport mechanisms of PCTM can not be variability. Deficiencies in the transport mechanisms of PCTM can not be expected to be captured via these PCs.expected to be captured via these PCs.

Concluding Remarks and future directionsConcluding Remarks and future directions

Hierarchical inverse modeling offers many advantages over traditional methods Hierarchical inverse modeling offers many advantages over traditional methods including an implicit spatial correlation structure, multi-scale estimates of including an implicit spatial correlation structure, multi-scale estimates of variance and computationally efficient covariance characterizations. variance and computationally efficient covariance characterizations.

Principal components appear to be a promising method of parameterizing Principal components appear to be a promising method of parameterizing uncertainty in the boundary inflow termsuncertainty in the boundary inflow terms

Further directions:Further directions:

- Nesting down to model grid resolution within regions of interest- Nesting down to model grid resolution within regions of interest

- Investigating real errors in boundary condition estimates- Investigating real errors in boundary condition estimates

- Applying real tower data- Applying real tower data