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

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

XY PLR F Structure 1: Structure 3: XaXa Y PaPa LaLa R F XbXb PbPb LbLb Structure 2: XYaYa P L R U F YbYb M Is it the overall structure or the component processes that matters? Rastetter 2003

Response to a ramp in F from time 10 to 100 Same model structure, different process equation Rastetter 2003

XY PLR F Structure 1: Structure 3: XaXa Y PaPa LaLa R F XbXb PbPb LbLb Structure 2: XYaYa P L R U F YbYb M 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 series n - white noise time series e - error time series F - Deterministic transfer function G - Stochastic transfer function y t = b 0 x t + b 1 x t a 1 y t-1 - a 2 y t  0 n t +  1 n t  1 r t-1 -  2 r t e t Young 1984 ARMA Transfer Function Models Testing system linkages

Input Time Series Output Time Series 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

F t = J = ff xx x t-1:t-1,u t F t =exp(J  t) exp(J  t) = I + J  t + (J  t) 2 /2! (J  t) n /n! +... Discrete model x t = f(x t-1, u t, w t ) Continuous model = f(x, u, w) dx dt Nonlinear models Linearized transition matrix

(Continuous) Extended Kalman Filter Predict P t:t-1 = F t P t-1:t-1 F t T + Q t estimate covariance Update S t = H t P t:t-1 H t T + R t innovations covariance K t = P t:t-1 H t T S t -1 Kalman gain x t:t = x t:t-1 + K t y t updated state P t:t = (I - K t H t ) P t:t-1 updated estimate covariance y t = z t - H t x t:t-1 innovations x t:t-1 = x t-1:t-1 + f(x,u,0)dt predicted state t-1 t

Augmented State Vector x * = x1x1 x2x2 x3x3 xnxn 11 22 33 m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 ) 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:

WebbHyperbolic Eight Models Tested by Cosby et al O 2 concentration in a Danish stream note 1 model structure, alternate representation of P S

Cosby et al. 1984

WebbHyperbolic Webb Hyperbolic Webb Hyperbolic both both Cosby et al mean value Maximum rate Initial slope of PI curve

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 CO 2 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:

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