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