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Estimating Uncertainty in Ecosystem Budgets Ruth Yanai, SUNY-ESF, Syracuse Ed Rastetter, Ecosystems Center, MBL Dusty Wood, SUNY-ESF, Syracuse

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Ecosystem Budgets have No Error Hubbard Brook P Budget Yanai (1992) Biogeochemistry

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Replicate Measurements

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Disparate measurements, all with errors?

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How can we estimate the uncertainty in ecosystem budget calculations from the uncertainty in the component measurements? Try it with biomass N in Hubbard Brook Watershed 6.

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Mathematical Error Propagation When adding, the variance of the total (T) is the sum of the variances of the addends (x): For independent errors. If they’re correlated, use the sum of covariances.

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Mathematical Error Propagation When adding, the variance of the total (T) is the sum of the variances of the addends (x): Biomass N content = wood N content + bark N content + branch N content + foliar N content + twig N content + root N content

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Mathematical Error Propagation When adding, the variance of the total (T) is the sum of the variances of the addends (x): Biomass N content = wood mass · wood N concentration + bark mass · bark N concentration + branch mass · branch N concentration + foliar mass · foliar N concentration + twig mass · twig N concentration + root mass · root N concentration

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Mathematical Error Propagation When multiplying, variance of the product is the product of the means times the sum of the variance of the factors:

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Mathematical Error Propagation When multiplying, variance of the product is the product of the means times the sum of the variance of the factors: wood mass · wood N concentration But log (Mass) = a + b*log(PV) + error And PV = 1/2 r 2 * Height log(Height) = a + b*log(Diameter) + error

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Mathematical Error Propagation “The problem of confidence limits for treatment of forest samples by logarithmic regression is unsolved.” --Whittaker et al. (1974)

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Monte Carlo Simulation

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Tree Height log (Height) = a + b*log(Diameter) + error

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Monte Carlo Simulation Tissue Mass log (Mass) = a + b*log(PV) + error PV = 1/2 r 2 * Height

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Monte Carlo Simulation Tissue Concentration N concentration = constant + error

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Monte Carlo Simulation

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Calculate the nutrient contents of wood, branches, twigs, leaves and roots, using species- and element- specific parameters, sampling these parameters with known error. After many iterations, analyze the variance of the results.

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A Monte-Carlo approach could be implemented using specialized software or almost any programming language. This illustration uses a spreadsheet model.

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Height Parameters Height = 10^(a + b*log(Diameter) + log(E)) Lookup ***IMPORTANT*** Random selection of parameters values happens HERE, not separately for each tree

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Biomass Parameters Biomass = 10^(a + b*log(PV) + log(E)) Lookup PV = 1/2 r 2 * Height

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Biomass Parameters Biomass = 10^(a + b*log(PV) + log(E)) Lookup PV = 1/2 r 2 * Height

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Biomass Parameters Biomass = 10^(a + b*log(PV) + log(E)) Lookup PV = 1/2 r 2 * Height

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Concentration Parameters Concentration = constant + error Lookup

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COPY THIS ROW-->

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After enough interations, analyze your results Paste Values button

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Repeated Calculations of N in Biomass Hubbard Brook Watershed 6 How many iterations is enough?

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Repeated Calculations of N in Biomass Hubbard Brook Watershed 6 Two different sets of 250 iterations: Mean settles down over many iterations

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Uncertainty in Biomass N: 110 kg/ha Coefficient of Variation: 18% Repeated Calculations of N in Biomass Hubbard Brook Watershed 6

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Hubbard Brook W6 is surveyed in m x 25m plots. How much variation is there from one part of this watershed to another? This is a more common way to represent uncertainty in budgets. Approaches to Estimating Uncertainty: Replicate Measurements

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Replicate Samples Variation across plots: 16 Mg/ha, or 5%

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Replicate Samples Variance across plots: 30 Mg/ha, or 10% with smaller plots

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Which is More Uncertain? Total biomass CV Nitrogen content CV Multiple Plots 5%, 10%6%, 10% Uncertainty in Calculations 18% Parameter uncertainty doesn’t affect comparisons across space. But it matters when you take your number and go.

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The Value of Ecosystem Error Quantify uncertainty in our results

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Borrmann et al. (1977) Science The N budget for Hubbard Brook published in 1977 was “missing” 14.2 kg/ha/yr

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Net N fixation (14.2 kg/ha/yr) = hydrologic export + N accretion in the forest floor + N accretion in mineral soil + N accretion in living biomass - precipitation N input - weathering N input - change in soil N stores

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We can’t detect a difference of 1000 kg N/ha in the mineral soil…

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The Value of Ecosystem Error Quantify uncertainty in our results Identify ways to reduce uncertainty

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“What is the greatest source of uncertainty in my answer?” Better than the sensitivity estimates that vary everything by the same amount-- they don’t all vary by the same amount!

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Better than the uncertainty in the parameter estimates--we can tolerate a large uncertainty in an unimportant parameter. “What is the greatest source of uncertainty to my answer?”

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Other Considerations Independence of error (covariance) Distribution of errors (normal or not)

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Additional Sources of Error Bias in measurements Errors of omission Conceptual errors Measurement errors Spatial and temporal variation

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The Value of Ecosystem Error Quantify uncertainty in our results Identify ways to reduce uncertainty Advice One way or another, find a way to calculate ecosystem errors, and report them. This is not possible unless researchers also report error with parameters.

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