using data assimilation to improve estimates of c cycling mathew williams school of geoscience,...
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Using data assimilation to improve estimates of C cycling
Mathew WilliamsSchool of GeoScience, University of Edinburgh
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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
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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
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Time update“predict”
Measurement update
“correct”
A prediction-correction system
Initial conditions
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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
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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
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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
)
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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)
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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)
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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
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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
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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
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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
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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
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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)
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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
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Acknowledgements: Bev Law, James Irvine, + OSU team
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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%