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Parameter identifiability, constraints, and equifinality in data assimilation with ecosystem models Dr. Yiqi Luo Botany and microbiology department University of Oklahoma, USA Land surface models and FluxNET data Edinburgh, 4-6 June 2008 (Luo et al. Ecol Appl. In press)

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Observed Data Prior knowledgePosterior distribution Parameter identifiability Inverse model Constrained Edge-hitting Equifinality

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Wang et al. (2001) ------ a maximum of 3 or 4 parameters can be determined. Wang et al. (2001) ------ a maximum of 3 or 4 parameters can be determined. Braswell et al. (2005) ------ 13 out of 23 parameters were well-constrained. Braswell et al. (2005) ------ 13 out of 23 parameters were well-constrained. Xu et al. (2006) ------ 4 or 3 out of 7 parameters can be constrained, respectively under ambient and elevated CO 2. Xu et al. (2006) ------ 4 or 3 out of 7 parameters can be constrained, respectively under ambient and elevated CO 2. Identiable parameters

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Three methods to examine parameter identifiability 1.Search method 2.Model structure 3.Data variability

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Harvard Forest EMS-Tower Eddy flux data

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CO 2 flux CO 2 flux H 2 O flux H 2 O flux Wind speed Wind speed Temperature Temperature PAR PAR Relative humidity Relative humidity Hourly or half-hourly Eddy flux technology

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Leaf-level Photosynthesis Sub-model Canopy-level Photosynthesis Sub-model System-level C balance Sub-model Model

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Table 1 Parameters information

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Develop prior distribution Develop prior distribution Apply Metropolis-Hasting algorithm Apply Metropolis-Hasting algorithm a) generate candidate p from sample space b) input to model and calculate cost function c) select according to decision criterion d) repeat Construct posterior distribution Construct posterior distribution Bayesian inversion

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Conditional Bayesian inversion Bayesian inversion Bayesian inversion Bayesian inversion Bayesian inversion

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Fig. 2 Decrease of cost function with each step of conditional inversion

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Conclusions Conditional inversion can substantially increase the number of constrained parameters. Conditional inversion can substantially increase the number of constrained parameters. Cost function and information loss decrease with each step of conditional inversion. Cost function and information loss decrease with each step of conditional inversion.

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Measurement errors and parameter identifiability

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Leaves X1Woody X2Fine Roots X3 Metabolic Litter X4Structural Litter X5 Microbes X6 Slow SOM X7 Passive SOM X8 GPP TECO – biogeochemical model

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No. of parameter 8 12 8 3

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Exit rates

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Transfer coefficients

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Initial values

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Pool sizes without data

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Pool sizes with data and different SD

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Conclusion Magnitudes of measurement errors do not affect parameter identifiability but influence relative constraints of parameters

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Base model GPP Leaves X1Stems X2Roots X3 Metabolic L. X4Struct. L. X5 Microbes X6 Slow SOM X7 Passive SOM X8

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Simplified models Plant C Litter C GPP CO 2 Soil C Plant C Litter C GPP CO 2 O Soil C Miner. C 3P model 4P model

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Simplified models 6P model7P model GPP Leaves X1Stems X2Roots X3 Litter X4 Slow C X5 Miner. Soil C X6 GPP Leaves X1Stems X2Roots X3 Metabolic L. X4 Struct. L. X5 Microbes X6 Soil C X7

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3P model-parameter constraints Plant CLitter C Soil C

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4P model-parameter constraints Plant CLitter C Slow Soil C Passive Soil C

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6P model-parameter constraints Foliage Litter CSlow Soil CPassive Soil C WoodyFine roots

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7PM model-parameter constraints Foliage Metabolic L. C Structure L. CMicrobes C Woody Fine roots Soil C

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8P model-parameter constraints

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Conclusion Differences in model structure are corresponding to different sets of parameters. The number of constrained parameters varies with data availability

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