1. CAMELS CCDAS A Bayesian approach and Metropolis Monte Carlo method to estimate parameters and uncertainties in ecosystem models from eddy-covariance data CAMELS Meeting, Wageningen, 11.- 12. November 2003 Jens Kattge Wolfgang Knorr

2. CAMELS CCDAS Outlines Method o Bayesian approach o Metropolis algorithm Model: BETHY Eddy covariance data: Loobos site Results o Sampling parameter sets o Selecting parameter sets representing the a posteriori PDF o Using the selected parameter sets – Calculate first moments of the PDF in parameter space – Propagate parameter uncertainties into modeled fluxes Conclusions and Perspectives

3. CAMELS CCDAS Bayesian approach to estimate a posteriori PDF a posteriori probability density function (PDF) a priori probability density function : Likelihood function

4. CAMELS CCDAS Metropolis algorithm to sample a posteriori PDF Metropolis algorithm: Markov Chain Monte Carlo (MCMC) methods: Metropolis, Metropolis- Hastings, Gibbs Sampler …. Guided random walks: after walking from the starting point towards the maximum of the PDF (burn-in time), the walker samples the target distribution: probability in PDF >>> frequency in sampling Metropolis decision: if accept step if accept step with probability

5. CAMELS CCDAS Model: BETHY parameters and uncertainties Paramete r Description vcmaxmaximum carboxylation rate jmvmrelationship of jmax and vcmax aqquantum efficiency kcMichaelis-Menten constant for CO2 at 25 °C koMichaelis-Menten constant for O2 at 25 °C ecactivation energy for kc eoactivation energy for ko evactivation energy for vcmax gamproportionality of CO2 compensation point and canopy temperature frdrelationship of. dark resp and vm eractivation energy for dark respiration frlratio of leaf to total maintenance respiration fcinon water limited relationship of ci and ca cwmaximum water supply rate of root system avalbedo of vegetation surface for vegetation cover asoilsolar radiation absorbed by soil under vegetation epsky emissivity factor ωsingle scattering albedo of leaves fcfraction of vegetation cover fgavegetation factor of atmospheric conductance rsoilsoil respiration at 10 °C and field capacity q10Q10 of soil respiration kwsoil water factor of soil respiration swcsoil water content at pF 4.2 (permanent wilting point) Assumed uncertainty of parameters: SD = 0.1, 0.25, 0.5

6. CAMELS CCDAS Eddy Covariance Data: Loobos site Halfhourly data of Eddy covariance measurements from seven days during 1997 and 1998 from the Loobos site in the Netherlands PFT: coniferous forest Diagnostics: NEE and LH NEE LH

7. CAMELS CCDAS Random walk in parameter space After transition from the starting point to the region of highest probability, the walker samples the target distribution.

8. CAMELS CCDAS Convergence of average values

9. CAMELS CCDAS Does the algorithm find the global optimum? It depends on the starting point.

10. CAMELS CCDAS Does the algorithm find the global optimum? Sequences from different points lead to different “optima”.

11. CAMELS CCDAS Gelman’s empirical decision of convergence n : number of sequences W : average of variances B : variance of averages Empirical reduction factor R (Gelman et al., 1992) : Sequences have converged to a common target distribution, if the average of variances within sequences dominates the variance of averages between sequences:

12. CAMELS CCDAS Using the sampling: parameter means and SD a priori SD: 0.1 0.25 0.5

13. CAMELS CCDAS Using the sampling: relative reduction of error a priori SD: 0.1 0.25 0.5

14. CAMELS CCDAS Using the sampling: parameter correlations corr(vm,jmvm)= -0.64 >> ac = min(f(vm,jmvm)) corr(cw,swc) = 0.68 >> root water supply = f(cw,1/swc)

15. CAMELS CCDAS Does the a posteriori PDF include non Gaussian components? Projection of the multi-dimensional PDF onto the dimension of single parameters. vm cw

16. CAMELS CCDAS Modeled fluxes Fluxes with mean parameters are somehow closer to the obsevations.

17. CAMELS CCDAS Propagation of parameter uncertainties to modeled fluxes: “a priori” 25% uncertainty in parameters lead to huge uncertainty in fluxes median of modeled fluxes ≠ flux by mean set of parameters

18. CAMELS CCDAS Propagation of parameter uncertainties to modeled fluxes NEE

19. CAMELS CCDAS Propagation of parameter uncertainties to modeled fluxes LH

20. CAMELS CCDAS Propagation of parameter uncertainties to modeled fluxes GPP

21. CAMELS CCDAS Propagation of parameter uncertainties to modeled fluxes rm

22. CAMELS CCDAS Propagation of parameter uncertainties to modeled fluxes RH

23. CAMELS CCDAS Propagation of parameter uncertainties to modeled fluxes GC

24. CAMELS CCDAS Conclusions The Method seems capable of sampling points in parameter space representing the region of the global maximum of the a posteriori PDF. The sampling can be used to derive means, errors and covariances (1st&2nd moments of PDF). The PDF has non-Gaussian components. Some Parameters are constrained by a posteriori PDF. Using the complete a posteriori PDF could strongly reduce uncertainties in prognostics. Results depend on a priori parameter values and uncertainties and on number and uncertainty of measurements (Bayesian approach).

25. CAMELS CCDAS Status quo and Perspectives for MC simulations Reduce uncertainties in a priori parameter values of global simulations = CAMELS approach to spatial extrapolation of flux measurements Status quo: Routines for data preparation finished, more than 20 sites available (Isabel). Routines for parameter “optimisation” finished and tested. Next steps: Reduce uncertainties in a priori parameter values (12.2003). Get good information of measurement errors (12.2003). Get a measure of the sampling error involved in parameter generalisation: within sites (03.2004), between sites of the same Plant Functional Type (05.2004). Perspectives: Estimate optimised parameters, uncertainties and covariances for all PFT’s, based on more than one measurement site per PFT as a priori parameter sets in WP4 (11.2004). Comparison between TEM and inventory based approaches (11.2004).