Using data assimilation to improve estimates of C cycling

Mathew WilliamsSchool of GeoScience, University of Edinburgh

DATA

MODELS

DATA+Direct observation, good error estimates

-Gaps, incomplete coverage

MODELS+Knowledge of system evolution

-Poor error estimates

Terrestrial Carbon Dynamics

MODEL-DATA FUSION

Soil chamber

Eddy fluxes

Litterfall

AutotrophicRespiration

Photosynthesis

Soil biotaDecomposition

CO2 ATMOSPHERE

Heterotrophicrespiration

Litter

Soil organicmatter

Leaves

Roots

Stems

Translocation

Carbon flow

Litter traps

Leaf chamber

Time update“predict”

Measurement update

“correct”

A prediction-correction system

Initial conditions

Ensemble Kalman Filter: Prediction

kj

kj

kj dqM )(1

ψ is the state vectorj counts from 1 to N, where N denotes ensemble numberk denotes time step, M is the model operator or transition matrixdq is the stochastic forcing representing model errors from a distribution with mean zero and covariance Q

error statistics can be represented approximately using an appropriate ensemble of model states

Generate an ensemble of observations from a distributionmean = measured value, covariance = estimated measurement error.

dj = d + j d = observations

= drawn from a distribution of zero mean and

covariance equal to the estimated measurement error

Ensemble Kalman Filter: Update

H is the observation operator, a matrix that relates the model state vector to the data, so that the true model state is related to the true observations by

dt = H ψ t

Ke is the Kalman filter gain matrix, that determines the weighting applied to the correction

)( fjje

fj

aj HdK

f = forecast state vector a = analysed estimate generated by the correction of the forecast

Ponderosa Pine, Oregon, 2000-2

-4

-2

0

2

4

0 365 730 10950

2

4

6

0123456

Net Ecosystem Exchange

NE

E (g

C m

-2 d

-1)

Time (days, day 1 = 1 Jan 2000)

Gross Primary Production

GP

P(g

C m

-2 d

-1)

Total Respiration

Rto

t (g C

m-2

d-1

)

0

50

100

150

200

0

100

200

300

400

0

2000

Foliage

Cf (

gC m

-2)

Fine rootC

r (gC

m-2)

Wood

Cw (

gC m

-2)

0 365 730 10950

4000

8000

12000SOM and coarse litter

CS

OM

CW

D (

gC m

-2 d

-1)

Time (days, day 1 = 1 Jan 2000)

GPP Croot

Cwood

Cfoliage

Clitter

CSOM/CWD

Ra

Af

Ar

Aw

Lf

Lr

Lw

Rh

D

Temperature controlled

6 model pools10 model fluxes9 rate constants10 data time series

Rtotal & Net Ecosystem Exchange of CO2

C = carbon poolsA = allocationL = litter fallR = respiration (auto- & heterotrophic)

Setting up the analysis

The state vector contains the 6 pools and 10 fluxes

The analysis updates the state vector, while parameters are unchanging during the simulation

Test adequacy of the analysis by checking whether NEP estimates are unbiased

Setting up the analysis II

Initial conditions and model parameters– Set bounds and run multiple analyses

Data uncertainties– Based on instrumental characteristics, and

comparison of replicated samples. Model uncertainies

– Harder to ascertain, sensitivity analyses required

Multiple flux constraints

Ra = 0.47 GPP

-4

-2

0

2

0 365 730 10950

2

4

6

0

2

4

Net Ecosystem Exchange

NE

E (

g C

m-2 d

-1)

Time (days, day 1 = 1 Jan 2000)

Gross Primary Production

GP

P(g

C m

-2 d

-1)

Total RespirationR

tot (

g C

m-2 d

-1)

Williams et al. 2005

0

50

100

150

200

0

100

200

300

400

0 365 730 1095600

800

1000

1200

Foliage

Cf (

gC m

-2)

Fine rootC

r (gC

m-2)

Time (days, day 1 = 1 Jan 2000)

Wood

Cw (

gC m

-2)

Af = 0.31

Aw=0.25

Ar=0.43

Turnover

Leaf = 1 yr

Roots = 1.1 yr

Wood = 1323 yr

Litter = 0.1 yr

SOM/CWD =1033 yr

Williams et al. 2005

Data brings confidence

0 365 730 1095-4

-3

-2

-1

0

1

2

0 365 730 1095-4

-2

0

2

Time (days, 1= 1 Jan 2000)

b) GPP data + model: -413±107 gC m -2

0 365 730 1095-4

-3

-2

-1

0

1

2

c) GPP & respiration data + model: -472 ±56 gC m -2NE

E (

g C

m-2 d

-1)

0 365 730 1095-4

-2

0

2

a) Model only: -251 ±197 g c m -2

d) All data: -419 ±29 g C m -2

Parameter uncertainty

Vary nominal parameters and initial conditions ±20%

Generate 400 sets of parameters and IC’s, and then generate analyses

Accept all with unbiased estimates of NEP (N=189)

The mean of the NEE analyses over three years for unbiased models (-421±17 gC m-2) was little different to the nominal analysis (419±29 g C m-2)

Discussion

Analysis produces unbiased estimates of NEP Autocorrelations in the residuals indicate the

errors are not white Litterfall models over simplified Relative short time series and aggrading

system Next steps: assimilating EO products, and long

time series inventories

Acknowledgements: Bev Law, James Irvine, + OSU team

Heterotrophic and autotrophic respiration

0 365 730 10950

1

2

3

4

Ra (

g C

m-2 d

-1)

Time (days, day 1 = 1 Jan 2000)

0

1

2

3

Rh (

g C

m-2 d

-1)

Fraction of total respiration

Ra = 42%

Rh = 58%