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

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

3. Replicate Measurements

4. Disparate measurements, all with errors?

5. 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.

6. 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.

7. 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

8. 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

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

10. 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

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

12. Monte Carlo Simulation

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

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

15. Monte Carlo Simulation Tissue Concentration N concentration = constant + error

16. Monte Carlo Simulation

17. 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.

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

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

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

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

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

26. Concentration Parameters Concentration = constant + error Lookup

28. COPY THIS ROW-->

29. After enough interations, analyze your results Paste Values button

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

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

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

33. Hubbard Brook W6 is surveyed in 208 25m 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

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

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

36. 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.

37. The Value of Ecosystem Error Quantify uncertainty in our results

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

39. 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

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

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

42. “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!

43. 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?”

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

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

47. 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.