Uncertainty Estimation of a Transpiration Model

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 - ?S
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
Crisp and Fuzzy Sets

• Crisp sets - precise boundary vs. Fuzzy sets -
  imprecise boundary
• Degree of compatibility with a concept -
  membership grade

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Subnormal Fuzzy Sets

• Highest membership grade less than 1
• Crisp sets can be formed by placing an a-cut
• higher the a-cut, lower the number of members in
  the crisp set

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Uncertainty in Fuzzy Sets
log2S U(r)
of members
S1
a-cut 1
S2
a-cut 2
a

• Crisp sets obtained through principle of
  uncertainty invariance

Klir and Wierman, 1998
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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

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Transpiration Model

• Penman-Monteith equation (Monteith, 1965)

• Stomatal conductance model (Jarvis, 1976)

gS gSmax f1(D) f2(Q0).

• Data from ChEAS site, WI

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Comparison of Techniques
d

• Model details Canopy modeled as a big leaf
  logarithmic wind speed profile
• . gs gsmax(1-dD)
• . gs gsmax(1-dD)minQrl/Qmin, 1
• Analysis Bayesian and proposed framework

gsmax
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Comparison of Techniques
gs gsmax(1-dD)minQrl/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

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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-dD)minQrl/Qmin, 1
• . parameters for ga assumed known
• . parameters for ga calibrated
• Analysis proposed framework

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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-dD)minQrl/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