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 time and space How is NEE distributed across domain in both time and space

and how to capture at different scalesand how to capture at different scales

Deterministic Biosphere and Deterministic Biosphere and Transport ModelsTransport Models

SiB2.5 is used for biosphere model. SiB2.5 is used for biosphere model. MODIS fPAR and LAI products are used MODIS fPAR and LAI products are used to drive SiB2.5.to drive SiB2.5.

coupled to RAMS 5.0, meteorology forced coupled to RAMS 5.0, meteorology forced with Eta 40km reanalysis datawith Eta 40km reanalysis data

150 x 100 40km grid over North America 150 x 100 40km grid over North America for the for the

time period: 5/1/2004 through 8/31/2004.time period: 5/1/2004 through 8/31/2004.

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

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Flux i, j = βAssimn,i, j ,t * Sib_ assimni, j,t + βRe spg ,i, j * Sib_ respgi, j ,t

Inversion Methods AvailableInversion Methods Available

Bayesian Synthesis InversionBayesian Synthesis Inversion

• For many problems the quickest For many problems the quickest and easiest methodand easiest method

• at core of many inversion at core of many inversion methodologiesmethodologies

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

Inversion Methods AvailableInversion Methods Available

Kalman filtering techniquesKalman filtering techniques

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

• EnKF further reduces dimensional EnKF further reduces dimensional constraints by effectively working with a constraints by effectively working with a sampled spatial covariance structure. sampled spatial covariance structure. EnKF has also been shown to have some EnKF has also been shown to have some desirable properties for non-linear desirable properties for non-linear models.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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Data(likelihood) : y | X,β III ,Σ ~ N Xβ III ,Σy( )

Level III ecoregions : β III | Xβ III ,β II ,Σβ III ~ N Xβ III β II ,Σβ III( )

Level II ecoregions : β II | Xβ II ,β I ,Σβ II ~ N Xβ IIβ I ,Σβ II( )

Level I ecoregions : β I |β 0,Σβ I ~ N β 0,Σβ I( )

βI=1

βI=3 βI=4 βI=4,II=1

βI=2

βI=4,II=2

βI,=4,II=3 βI=4,II=4

βI=4,II=1,III=1 βI=4,II=1,III=2

βI=4,II=1,III=3 βI=4,II=1,III=4

LEVEL 1 ECOREGIONS

LEVEL 2 ECOREGIONS

LEVEL 3 ECOREGIONS

Calibrated COCalibrated CO2 2 Expected 2007Expected 2007

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ExampleExample

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IIII

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IIII

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IIIIII

2.0,0~,| :ecoregions I Level

04.0,~,,| :ecoregions II Level

001.0,~,,| :ecoregions III Level

1,~,,| :ihood)Data(likel

Factors Bias Pseudo ofCreation

0 β

ββ

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βββ

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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)

Work to doWork to do

Currently inverting on this structure with 2004 Currently inverting on this structure with 2004 data from 8 towersdata from 8 towers

Explore effects of stepping inversion through Explore effects of stepping inversion through time, in state space way.time, in state space way.

Subnesting down to SiBRAMS grid size in areas Subnesting down to SiBRAMS grid size in areas with high density of measurementswith high density of measurements

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

boundary conditions can be very important to regional boundary conditions can be very important to regional scale inversion scale inversion

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

Principal Components for boundary? These provide Principal Components for boundary? These provide “directions” of maximal variability (in time) in the “directions” of maximal variability (in time) in the boundary conditions.boundary conditions.

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

Revisiting Inversion DomainRevisiting Inversion Domain

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

1st PC Reconstruction of Boundary Influences 1st PC Reconstruction of Boundary Influences Sequence of Afternoon Observations for 5/1/2004 - 8/31/04Sequence of Afternoon Observations for 5/1/2004 - 8/31/04

(WLEF,ARM,Harvard,BERMS, BOREAS, Fraserdale, KWKT, Howland)(WLEF,ARM,Harvard,BERMS, BOREAS, Fraserdale, KWKT, Howland)

Boundary conditionsBoundary conditions

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

The PCs appear to load nicely, particularly zonally. The PCs appear to load nicely, particularly zonally.

Appears to be a promising dimension reductionAppears to be a promising dimension reduction

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

Work to doWork to do

Investigating real errors in boundary condition Investigating real errors in boundary condition estimatesestimates

Exploring relationship between time period of inversion Exploring relationship between time period of inversion and boundary predictorand boundary predictor

Comparing against other simple correction schemes, 2 Comparing against other simple correction schemes, 2 box vertical, etcbox vertical, etc