Research Vignette: The TransCom3 Time-Dependent Global CO 2 Flux Inversion … and More David F. Baker NCAR 12 July 2007 David F. Baker NCAR 12 July 2007.

Research Vignette:

The TransCom3 Time-DependentGlobal CO2 Flux Inversion

… and More

David F. BakerNCAR

12 July 2007

David F. BakerNCAR

12 July 2007

OutlineOutline The TransCom3 CO2 flux inversion inter-comparison project

The fully time-dependent T3 flux inversion Method (“batch least squares”) Results

Methods for bigger problems: Kalman filter (traditional, full rank) Ensemble filters Variational data assimilation

The TransCom3 CO2 flux inversion inter-comparison project

The fully time-dependent T3 flux inversion Method (“batch least squares”) Results

Methods for bigger problems: Kalman filter (traditional, full rank) Ensemble filters Variational data assimilation

TransCom3 CO2 Flux Inversions

CO2 fluxes for 22 regions, data from 78 sites Annual-mean inversion, 1992-1996

Fixed seasonal cycle, no IAV 22 annual mean fluxes solved for Gurney, et al Nature, 2002 & GBC, 2003

Seasonal inversion, 1992-1996 Seasonal cycle solved for, no IAV 22*12 monthly fluxes solved for Gurney, et al, GBC, 2004

Inter-annual inversion, 1988-2003 Both seasonal cycle and IAV solved for 22*12*16 monthly fluxes solved for Baker, et al, GBC, 2006

TransCom3 interannual inversion setup:Solve for monthly CO2 fluxes from 22 regions,

1988-2003, using monthly concentrations from 78 sites

Fossil fuel input

Atmosphericstorage

Land + oceanuptake

0.45*FF

FF = Atmos + Ocean + Land Bio

NH SummerNH Winter

CO2 Uptake & Release

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Form of the Batch Measurement Equations

fluxesconcentrationsTransport basis functions

Batch least-squares or “Bayesian synthesis” inversion

Batch least-squares or “Bayesian synthesis” inversion

Optimal fluxes, x, found by minimizing:

wheregiving

Optimal fluxes, x, found by minimizing:

wheregiving

)()()()( 11oo

To

TJ xxPxxzHxRzHx −−+−−= −−

€

ˆ x = (HTR−1H + Po−1)−1(HTR−1z + Po

−1xo)

Pˆ x ̂ x = E(dˆ x dˆ x T ) = (HTR−1H + Po−1)−1

or....Pˆ x ̂ x −1 = Po

−1 + HTR−1H

)();( Tooo

T ddEddE xxPzzR ==

^

Computation of the interannual variability (IAV): Europe

Monthly flux

Deseasonalized flux

IAV

Between-model error

Estimation error

[PgC/yr]

Total flux IAV(land+ocean)

[PgC/yr]

OceanFlux IAV

Land flux IAV[PgC/yr]

[PgC/yr]

Flux IAV [PgC/yr],11 land regions

Flux IAV [PgC/yr],

11 ocean regions

Computational considerationsComputational considerations

Transport model runs to generate H: 22 regions x 16 years x 12 months x 36 months = 152 K tracer-months (if using real winds)

22 x 12 x 36 = 9.5 K tracer-months (using climatological winds)

Matrix inversion computations: O (N3) N = 22 regions x 16 years x 12 months = 4.4 K

Matrix storage: O (N*M) --- 66 MB M = 78 sites x 16 years x 12 months = 15 K

Transport model runs to generate H: 22 regions x 16 years x 12 months x 36 months = 152 K tracer-months (if using real winds)

22 x 12 x 36 = 9.5 K tracer-months (using climatological winds)

Matrix inversion computations: O (N3) N = 22 regions x 16 years x 12 months = 4.4 K

Matrix storage: O (N*M) --- 66 MB M = 78 sites x 16 years x 12 months = 15 K

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Kalman Filter/Smoother

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Kalman Filter/Smoother

Kalman Filter EquationsKalman Filter EquationsMeasurement update step at time k:

Dynamic propagation step from times k to k+1:

Put multiple months of flux in state vector xk, method becomes effectively a fixed-lag Kalman smoother

Measurement update step at time k:

Dynamic propagation step from times k to k+1:

Put multiple months of flux in state vector xk, method becomes effectively a fixed-lag Kalman smoother

][

]ˆ[ˆˆ

1)()(

)()()(

)()()(

−−−

−−+

−−+

+=

−=−+=

kT

kkkT

kkkwhere

kkkkk

kkkkkk

RHPHHPK

PHKPP

xHzKxx

kT

kkkk

kkk

QPP

xx

+ΦΦ=

Φ=

+−+

+−+

)()(1

)()(1 ˆˆ

Error

Time

2

1

1

11

1

1 22

222

Φ = tangentlinear model

Inversion methods for the data-rich, fine-scale

problem

Inversion methods for the data-rich, fine-scale

problem Kalman filter: some benefit, but long lifetimes for CO2 limit savings

Ensemble KF: full covariance matrix replaced by an approximation derived from an ensemble

Variational data assimilation (4-D Var): an iterative solution replaces the direct matrix inversion; the adjoint model computes gradients efficiently

Kalman filter: some benefit, but long lifetimes for CO2 limit savings

Ensemble KF: full covariance matrix replaced by an approximation derived from an ensemble

Variational data assimilation (4-D Var): an iterative solution replaces the direct matrix inversion; the adjoint model computes gradients efficiently

Ensemble Kalman filter Replace xk, Pk from the full KF with an ensemble of xk,i, i=1,…,Nens

Add dynamic noise consistent with Qk to xk,i when propagating; add measurement noise consistent with Rk to measurements when updating, initial ensemble has a spread consistent with P0

When needed in KF equations, Pk replaced with

Replace matrix multiplications with sums of dot products

Good for non-Gaussian distributions

€

Pk ≈ (x k,i− < x k >)(x k,i− < x k >)T

i

Nens

∑

For retrospective analyses, a 2-sided smoother gives more accurate estimates than a 1-sided filter.

(Gelb, 1974)

Kalman filter vs. Kalman smoother

Estimation as minimization

Solve for x with an approximate, iterative method rather than an exact matrix inversion

Start with guess x0, compute gradient efficiently with an adjoint model, search for minimum along -, compute new and repeat

Good for non-linear problems; use conjugate gradient or BFGS approaches

Low-rank covariance matrix built up as iterations progress

As with Kalman filter, transport errors can be handled as dynamic noise

€

J = (h(x) − z)T R−1(h(x) − z) + (x − xo)T Po−1(x − xo)

€

∇k =∂J

∂x k

°

°

°

°

0

2

1

3

x2

x1

x3

x0

AdjointTransport

ForwardTransport

ForwardTransport

MeasurementSampling

MeasurementSampling

“True”Fluxes

EstimatedFluxes

ModeledConcentrations

“True”Concentrations

ModeledMeasurements

“True”Measurements

AssumedMeasurement

Errors

WeightedMeasurement

Residuals

/(Error)2

AdjointFluxes=

FluxUpdate

4-D Var Iterative Optimization Procedure

Minimum of cost function J

4-D Var Data Assimilation Method

Find optimal fluxes u and initial CO2 field xo to minimize

subject to the dynamical constraint

wherex are state variables (CO2 concentrations),h is a vector of measurement operatorsz are the observations,R is the covariance matrix for z,uo is an a priori estimate of the fluxes,

Puo is the covariance matrix for uo,

xo is an a priori estimate of the initial concentrations,

Pxo is the covariance matrix for xo,

Φ is the transition matrix (representing the transport model), and

G distributes the current fluxes into the bottom layer of the model

€

x i+1 = Φ i+1i x i + G iui ≡ di(x i,ui), i = 0, L ,I −1

€

J = (h j (x j ) − z j )T R j

−1(h j (x j ) − z j )j

∑

+ (ui − uio)T P

u io

−1(ui − uio)

i= 0

I −1

∑ + (x0 − x0o)T P

x 0o

−1(x0 − x0o)

4-D Var Data Assimilation Method

Adjoin the dynamical constraints to J using Lagrange multipliers

Setting F/xi = 0 gives an equation for i, the adjoint of xi:

The adjoints to the control variables are given by

F/ui and F/xoo as:

The gradient vector is given by F/ui and F/xoo, so

the optimal u and xo may then be found using one’s favorite descent method. I use the BFGS method in order to obtain an estimate of the leading terms in the covariance matrix.

€

F ≡ J + λ i+1T (di(x i,ui) − x i+1)

i= 0

I −1

∑

€

i = [Φ i+1i ]T λ i+1 +

∂hi

∂x i

T

R j−1(hi(x i) − z j )δ ij i = I −1,K ,1

λ I =∂hI

∂x I

T

R j−1(hI (x I ) − z j )δIj ≈ 0

€

∂F

∂ui

T

= G iT λ i+1 + P

u io

−1(ui − uio), i = 0,K ,I −1

∂F

∂x0

T

= Φ10T

λ1 + Px 0

o−1(x0 − x0

o) + R j−T (h0 − z j )δ0 j

· 10-8 [ kg CO2 m-2 s-1 ]

OSSE ResultsFor Five CO2

MeasurementNetworks

Pros and cons, 4DVar vs. ensemble Kalman filter (EnKF)

4DVar requires an adjoint model to back-propagate information -- this can be a royal painroyal pain to develop!

The EnKF can get around needing an adjoint by using a filter-lag rather than a fixed-interval Kalman smoother. However, the need to propagate multiple time steps in the state makes it less efficientless efficient than the 4DVar method

Both give a low-rank estimate of the a posteriori covariance matrix

Both can account for dynamic errors Both calculate time-evolving correlations between

the state and the measurements

Adjoint transport model If number of flux regions > number of measurement sites, then instead of running transport model forward in time forced by fluxes to fill H, run adjoint model backwards in time from measurement sites

What is an adjoint model? If every step in the model can be represented as a matrix multiplication (= ‘tangent linear model’), then the adjoint model is created by multiplying the transpose of the matrices together in reverse order

FWD**

* *

*

*

ADJfluxgrid

measurementsites