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 Rastetter The Ecosystems Center Marine
Biological Laboratory Woods Hole, MA USA
Jack Cosby Environmental Sciences University of
Virginia Charlottesville, VA USA
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Topics
I. The ANOVA Curse II. What should be the focus
of model development and testing efforts? III.
Using transfer-function estimations to identify
important system linkages IV. Using the Extended
Kalman Filter as a test of model adequacy that
yields valuable information on how to improve
model structure
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The ANOVA Curse
Grant will cover 6 treatment plots, which we set
up in a 2-level 3-replicate design
Tells us if a treatment is important, but not how.
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If the same treatments are spread out along a
continuum,
then we learn a lot more about how the treatment
is important,
but still little about the system dynamics.
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Focus of model development and testing
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)?
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Is it the overall structure or the component
processes that matters?
Rastetter 2003
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Rastetter 2003
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Response to a ramp in F from time 10 to 100
Rastetter 2003
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Its the structure that matters!!!!!!! (i.e. how
the components are linked to one another) Not the
detailed process representation!
Structure 1
Structure 2
Structure 3
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Testing system linkages
ARMA Transfer Function Models
yt b0 xt b1 xt-1 ... - a1 yt-1 - a2 yt-2 -
... ?0 nt ?1 nt-1 ... - ?1 rt-1 -
?2 rt-2 - ... et
x - input time series y - output time
series n - white noise time series e - error
time series F - Deterministic transfer function G
- Stochastic transfer function
Young 1984
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Input Time Series
Output Time Series
No significant pattern
Deterministic function significant
Combined model significant but deterministic
function not significant
Rastetter 1986
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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

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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

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Nonlinear models
Discrete model
xt f(xt-1, ut, wt)
Linearized transition matrix
Continuous model
Linearized transition matrix
Ft exp(JDt)
exp(JDt) I JDt (JDt)2/2! ... (JDt)n/n!
...
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(Continuous) Extended Kalman Filter
Predict
xtt-1 xt-1t-1 f(x,u,0)dt predicted
state
Ptt-1 Ft Pt-1t-1 FtT Qt estimate
covariance
Update
yt zt - Ht xtt-1
innovations
St HtPtt-1 HtT Rt innovations
covariance
Kt Ptt-1 HtT St-1 Kalman
gain
xtt xtt-1 Kt yt updated
state
Ptt (I - Kt Ht) Ptt-1 updated
estimate covariance
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Augmented State Vector

• 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)

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EKF Test of Model Adequacy Cosby Hornberger 1984
The model embedded in the EKF is adequate if

• 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 between
  parameters and state variables or control
  (driver) variables

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Eight Models Tested by Cosby et al. 1984 O2
concentration in a Danish stream
note 1 model structure, alternate representation
of PS
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Cosby et al. 1984
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mean value
Webb
Hyperbolic
Webb - 1.2 Hyperbolic - 1.7
Maximum rate
Webb - 3.7 Hyperbolic - 0.32
Initial slope of PI curve
both - 0.51
both - 0.94
Cosby et al. 1984
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• 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
  thats crucial, its how the processes are linked
  to one another.

Linear model wags as light changes
All models have diel hysteresis
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EKF Test of Model Adequacy

• 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

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Are we getting the right type of data?
Time series data are extremely expensive and
therefore rare e.g., eddy flux, hydrographs,
chemographs, tree rings, 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
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Conclusions

• 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 (but perhaps restricts
  confidence in extrapolation to other systems)
• 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

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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.

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The End