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Uncertainty Estimation of a Transpiration Model Using Data from ChEAS

Uncertainty Estimation of a Transpiration Model Using Data from ChEAS. Sudeep Samanta, D. Scott Mackay, and Brent Ewers Department of Forest Ecology and Management University of Wisconsin - Madison. Uncertainty in Model Selection/Calibration.

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Uncertainty Estimation of a Transpiration Model Using Data from ChEAS

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  1. Uncertainty Estimation of a Transpiration Model Using Data from ChEAS Sudeep Samanta, D. Scott Mackay, and Brent Ewers Department of Forest Ecology and ManagementUniversity of Wisconsin - Madison

  2. Uncertainty in Model Selection/Calibration • Select model structure consistent with current knowledge • Many alternatives • Estimate appropriate values for parameters • Data availability • Methods of comparing model output to data

  3. Problem Statement • Deterministic Simulation models: An Example R = P - ET - ΔS ET = M(Rn, D, ga, gc) gc = gsL gs = gsmaxf1(D)f2(Q)... • Easier than stochastic models to build and interpret • No estimate of uncertainty directly available from a model • Difficult to formulate in stochastic terms to obtain a probabilistic estimate of uncertainty

  4. Bayesian Analysis • Bayesian analysis of deterministic models: i = 1, 2, …., n, • Posterior estimates of parameter distribution • Uncertainty in Predictions • Changes in model component may change error model • The errors may be auto-correlated

  5. Research Questions • Can inferences be made without probabilistic assumptions using an alternative representation of uncertainty? • Fuzzy set theory • Objective function as membership grade • Can be used without reformulating model • Flexible in terms of selection criteria • How does this representation compare with probabilistic representation of uncertainty? • Does the ability to identify parameters and their relationships change with model complexity?

  6. 1 Crisp Set Fuzzy Set 0.75 Membership Grade 0.5 0.25 0 0 2 4 6 8 10 Real numbers Crisp and Fuzzy Sets • Crisp sets - precise boundary vs. Fuzzy sets - imprecise boundary • Degree of compatibility with a concept - membership grade

  7. 1 0.75 Subnormal Fuzzy Set α-cut #1 Membership Grade 0.5 α-cut #2 0.25 0 0 2 4 6 8 10 Real numbers Subnormal Fuzzy Sets • Highest membership grade less than 1 • Crisp sets can be formed by placing an α-cut • higher the α-cut, lower the number of members in the crisp set

  8. Uncertainty in Fuzzy Sets log2|S| = U(r) # of members |S1| α-cut #1 |S2| α-cut #2 α • Crisp sets obtained through principle of uncertainty invariance [Klir and Wierman, 1998]

  9. Limitations Compared to Bayesian Analysis of Uncertainty • Inferences may not be valid outside the sampled model parameter combinations • Uncertainty is represented by a set and no likelihood distribution is available • Theories and application techniques are not as well developed

  10. Transpiration Model • Penman-Monteith equation (Monteith, 1965) • Stomatal conductance model (Jarvis, 1976) gS = gSmax f1(D) f2(Q0)…. • Data from ChEAS site, WI

  11. Comparison of Techniques δ • Model details:Canopy modeled as a big leaf logarithmic wind speed profile • . gs = gsmax*(1-δD) • . gs = gsmax*(1-δD)*min[Qrl/Qmin, 1] • Analysis:Bayesian and proposed framework gsmax

  12. Comparison of Techniques gs = gsmax*(1-δD)*min[Qrl/Qmin, 1] • Model details: • Canopy divided in sunlit and shaded leaf areas, • . logarithmic wind speed profile. • . stability corrections with factors for roughness lengths fixed. • Analysis: Bayesian and proposed framework

  13. Parameter Estimates with Increasing Model complexity • Model details:Canopy layers with sunlit and shaded leaf areas Wind speed profile modeled in canopy. gs = gsmax*(1-δD)*min[Qrl/Qmin, 1] • . parameters for ga assumed known • . parameters for ga calibrated • Analysis: proposed framework

  14. boundary layer Parameter Estimates with Increasing Model complexity • Model details:Canopy layers with sunlit and shaded leaf areas Wind speed profile modeled in canopy. • Boundary layers at each canopy layer.gs = gsmax*(1-δD)*min[Qrl/Qmin, 1] • . parameters for ga assumed known • . parameters for ga calibrated • Analysis: proposed framework

  15. Anticipated Results Comparison of Techniques • Relations between uncertainty estimates obtained by the two techniques would not change with model complexity Parameter Estimates and Increased Model Complexity • Similar but tighter parameter estimates obtained when model complexity is increased without increasing number of parameters • The estimates will become more indeterminate with increased number of calibrated parameters

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