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Bayesian Approach and Metropolis Monte Carlo Method for Ecosystem Model Estimation

This study outlines a method using Bayesian approach and Metropolis algorithm to estimate parameters and uncertainties in ecosystem models using eddy-covariance data. Results show the sampling of parameter sets, propagation of uncertainties, and implications for reducing prognostic uncertainties.

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Bayesian Approach and Metropolis Monte Carlo Method for Ecosystem Model Estimation

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

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

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

  4. 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. Model: BETHY parameters and uncertainties Assumed uncertainty of parameters: SD = 0.1, 0.25, 0.5

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

  8. Convergence of average values

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

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

  11. Gelman’s empirical decision of convergence 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: n: number of sequences W: average of variances B: variance of averages

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

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

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

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

  17. 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. Propagation of parameter uncertainties to modeled fluxes NEE

  19. Propagation of parameter uncertainties to modeled fluxes LH

  20. Propagation of parameter uncertainties to modeled fluxes GPP

  21. Propagation of parameter uncertainties to modeled fluxes rm

  22. Propagation of parameter uncertainties to modeled fluxes RH

  23. Propagation of parameter uncertainties to modeled fluxes GC

  24. 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. 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).

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