developing and testing mechanistic models of terrestrial carbon cycling using time-series data
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Developing and Testing Mechanistic Models of
Terrestrial Carbon Cycling Using Time-Series Data
Ed RastetterThe Ecosystems CenterMarine Biological LaboratoryWoods Hole, MA USA
Jack CosbyEnvironmental SciencesUniversity of VirginiaCharlottesville, VA USA
I. What should be the focus of model development and testing efforts?
II. Using transfer-function estimations to identify important system linkages
III. Using the Extended Kalman Filter as a test of model adequacy that yields valuable information on how to improve model structure
Topics:
There has been an emphasis on the individual processes within models (e.g., photosynthesis, respiration, transpiration).
But are differences among models because of the individual processes?
Or is it because of the overall model structure (i.e., how the components are linked together)?
Focus of model development and testing
X YP L R
FStructure 1:
Structure 3:Xa
Y
Pa La RF
Xb
Pb
Lb
Structure 2: X Ya
PL
R
U
F
YbM
Is it the overall structure or the component processes that matters?
Rastetter 2003
F
F
X
XP
F
F
X
XP
PPP
FeP
F
FXP
ba
bbb
ab
aaa
ba
X
b
a
1
1
1
3
2
1
0
2
4
6
8
0 0.5 1 1.5 2 2.5
F
P
P1
P2
P3
Rastetter 2003
Response to a ramp in F from time 10 to 100
90
120
150
180
0 50 100 150 200Time
XP1S1
P1S2
P2S1
P2S2
P3S3
Same model structure, different process equation
Rastetter 2003
X YP L R
FStructure 1:
Structure 3:
Xa
Y
Pa La RF
Xb
Pb
Lb
Structure 2:
X Ya
PL
R
U
F
YbM
Its the structure that matters!!!!!!!(i.e. how the components are linked to one
another)
Not the detailed process representation!
G
F + +
n
x
r
y
e
y
x - input time series y - output time seriesn - white noise time series e - error time seriesF - Deterministic transfer functionG - Stochastic transfer function
yt = b0 xt + b1 xt-1 + ... - a1 yt-1 - a2 yt-2 - ... +
0 nt + 1 nt-1 + ... - 1 rt-1 - 2 rt-2 - ... + et
Young 1984
ARMA Transfer Function Models
Testing system linkages
Input Time Series
Ou
tpu
t Tim
e S
eri
es
I C
CO3
B I C
CO2
PO4
NO3 P R
P / R
PHY
F I L
ENC
AUT
ROT
STE
MAL
FAL
ALO
EPC
EGG
LAR
CAD
CAE
CAT
CYC
HET
PO4 O O O O O O O O O O O X O O O O O O O O O O O ONO3 O O O O O O O O O A O O O O A A O O O O O O O O
P O X X X X X X X X X X X X X X X X X X X X X X XR A X A X X X X X X X X X X X X X X X X X X X X X
P/R A X A A X X X X X XL A X A A X X A A
PHY A A A A X X A X A X A X X X A X X X X X X AFIL O O O O O O O O O O O O O O O O O O O O O O
ENC A A A A X A X A A A X X X A A A X X X X X AAUT X X X X X X X O O X X A A A X X O X O X
ROT O X O X O X O X X X X X X X X O X X O X XSTE O O O O O O O O O O O O O O O O O A X O OMAL X A X X X X X X X X X X X X X X X X X XFAL O O O O O A O O O O X X O O O O O O O OALO O O O O O A O O O O O O O O O O O O OEPC O O O X O O X O X X X O O X O O O X X O XEGG X X X X O X X O O O O X X O X X O O O XLAR O O O O O O O O O O O O O O O O O O O O OCAD O X X X X X X X X X X X X A X X X X X XCAE O O O O O O X O O O O O O O O O O O OCAT O O O O O X O O O X O O O A O O X X XCYC O O O O O O O O O O O O O A O O XHET O O O O O X O O O O O O
No significant pattern
Deterministic function significant
Combined model significant but deterministic function not significant
Rastetter 1986
Kalman Filter• The Kalman Filter is recursive filter that estimates successive states of a dynamic system from a time series of noise-corrupted measurements (Data Assimilation)
•A linear model is used to project the system state one time step into the future
•Measurements are made after the time step has elapsed and compared to the model predictions
•Based on this comparison and a recursively updated assessment of past model performance (estimate covariance matrix) and past measurement error (innovations covariance), the Kalman Filter updates, and hopefully improves, estimates of the modeled variables
Extended Kalman Filter
• The Extended Kalman Filter (EKF) is essentially the same as the Kalman filter, but with an underlying nonlinear model
•To accommodate the nonlinearity, the model must be linearized at each time step to estimate the Transition matrix
•This transition matrix is used to update the estimate covariance
Ft = J = fx xt-1:t-1,ut
Ft =exp(Jt)
exp(Jt) = I + Jt + (Jt)2/2! +...+ (Jt)n/n! +...
Discrete model
xt = f(xt-1, ut, wt)
Continuous model
= f(x, u, w)dxdt
Nonlinear models
Linearized transition matrix
Linearized transition matrix
(Continuous) Extended Kalman Filter
Predict
Pt:t-1 = Ft Pt-1:t-1 FtT + Qt estimate
covariance
Update
St = HtPt:t-1 HtT + Rt innovations
covarianceKt = Pt:t-1 Ht
T St-1 Kalman gain
xt:t = xt:t-1 + Kt yt updated state
Pt:t = (I - Kt Ht) Pt:t-1 updated estimate
covariance
yt = zt - Ht xt:t-1 innovations
xt:t-1 = xt-1:t-1 + f(x,u,0)dt predicted statet-1
t
Augmented State Vector
x*
=
x1
x2
x3
xn
1
2
3
m
• Once the Kalman Filter has been extended to incorporate a nonlinear model, it is easy to augment the state vector with some or all of the model parameters
•That is, to treat some or all of the parameters as if they were state variables
•This augmented state vector then serves a the basis for a test of model adequacy proposed by Cosby and Hornberger (1984)
EKF Test of Model Adequacy Cosby &
Hornberger 1984
1) Innovations (deviations) are zero mean, white noise (i.e., no auto-correlation)
2) Parameter estimates (in the augmented state vector) are fixed mean, white noise
3) There is no cross-correlation among parameters or between parameters and state variables or control (driver) variables
The model embedded in the EKF is adequate if:
Webb
Hyperbolic
Ss PRccKdt
dc
IPS IPS
I
IPS
IS eIP
22 I
IPS
IS eP 1
IPS tanh
I
IPS 1
Eight Models Tested by Cosby et al. 1984
O2 concentration in a Danish stream
note 1 model structure, alternate representation of PS
Cosby et al. 1984
Webb Hyperbolic
Webb - 1.2
Hyperbolic - 1.7
Webb - 3.7
Hyperbolic - 0.32
both - 0.51
both - 0.94
Cosby et al. 1984
mean value
Maximum rate
Initial slope of PI curve
0
0.25
0.5
0.75
1
1.25
1.5
0 0.25 0.5 0.75 1Radiation (ly min-1)
PS (
mg
O2 L
-1 h
r-1)
Hyperbolic
Webb
•All 8 models failed in the same way; parameter controlling initial slope of PI curve had a diel cycle.
•Its not the details of process representation that’s crucial, its how the processes are linked to one another.
Linear model “wags” as light changes
All models have diel hysteresis
•The EKF can be used as a severe test of model structure (few models are likely to pass the test)
•More importantly, it yields a great deal of information on how the model failed that can be used to improve the model structure
•e.g., the initial-slope parameter in the Cosby model should be replaced with a variable that varies on a 24-hour cycle, like a function of CO2 depletion in the water, or C-sink saturation in the plants
EKF Test of Model Adequacy
Are we getting the right type of data?
Time series data are extremely expensive and therefore rare
e.g., eddy flux, hydrographs, chemographs, others?
Their value to understanding of ecosystem dynamics is definitely worth the expense
The key to good time series data is automation to assure consistent, regular sampling
There should be a high degree of synchronicity among time series collected on the same system
•Time series are far richer in information on system dynamics and system linkages than data derived from more conventional experimental designs (e.g., ANOVA)
•Time series provide replication through time, which allows for statistical rigor without the replication constraints of more conventional experimental designs
•The focus of study should be on identifying and testing the linkages among system components (i.e., the system structure) rather than the details of how the individual processes are represented
Conclusions:
•Transfer-function estimation can be used to identify links among ecosystem components or test the importance of postulated linkages
•The Extended Kalman Filter can be used as a severe test of model adequacy that yields valuable information on how to improve the model structure
•Unfortunately, high quality time-series data in ecology are still rare
•However, new expenditures currently proposed for monitoring the biosphere (e.g., ABACUS, LTER, NEON, CLEANER, CUAHSI, OOI) may provide the support to automate time-series sampling of several important ecosystem properties.
Conclusions:
The End
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