Uncertainty in Lake Erie Residual Net Basin Supplies

1. Uncertainty in Lake Erie Residual Net Basin Supplies Jacob Bruxer February 2011

2. Presentation Overview • Water balance and the definition of Net Basin Supplies (NBS) + both component and residual methods of computing NBS • Uncertainty analysis of Lake Erie residual NBS • Sources and estimates of uncertainty in each of the various inputs (inflow, outflow, change in storage, etc.) • Combined uncertainty estimates (FOSM and Monte Carlo) • Comparison to results of previous research • Conclusions and next steps for improving residual NBS estimates for Lake Erie

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4. Introduction and Motivation • Net Basin Supplies (NBS) • The net volume of water entering (or exiting) a lake from its own basin over a specified period of time • NBS = P + R – E +/- G • Computed by Environment Canada in coordination with colleagues in the U.S. • Motivation for Study • To reduce uncertainty in NBS it is first necessary to identify and quantify sources of error • Accurate estimates of NBS are required in the Great Lakes basin for: • Operational regulation of Lake Superior and Lake Ontario • Formulation and evaluation of regulation plans • Water level forecasting • Time series analyses • Provide an indicator of climate change • Allows for comparisons of residual NBS to other methods of estimating NBS (i.e. component) and allows comparison of each of the different inputs to alternative methods for computing them 4

5. Net Basin Supplies (NBS) • Water Balance • Component Method • Residual Method

6. + ??? uncertainty ΔS

7. Flow Uncertainty: Overview • Niagara and Welland C. flow accounting is complicated • Summation of a number of different flow estimates • E.g. • Makes accounting for uncertainty difficult, but reduces overall uncertainty to some degree • Detroit River flows also complicated • Stage-fall-discharge equations, Transfer Factors, other models • Non-stationarity, channel changes, ice effects • Uncertainty in model calibration data, models themselves, and model predictor variables ON@BUF = NMOM + PSAB1&2 + PRM + DNYSBC - RN - DWR

8. Niagara Falls Flow (NMOM) • ≈ 30-40% of total ON@Buf • Stage-discharge equation based on measured water levels at Ashland Ave. gauge and ADCP flow measurements • Uncertainty (95% CL) • Gauged discharge measurements = 5% • Standard error of estimates = 4.2% • Error in the mean fitted relation = 1% • Predictor variable (i.e. water level) = 1% • Combined uncertainty in NMOM ≈ 6.7% • Conservative estimate

9. Hydropower (PSAB1&2 + PRM) • ≈ 60% of total ON@Buf • Total Hydropower Diversion = Plant Q + ΔS forebays/reservoirs • Plant flows from unit rating tables • Relate measured head and power output to flow • Developed from flows measured using Gibson and Index testing • Uncertainty ≈ 2 to 2.5 % • Also uncertainty in extrapolating to other heads, other units, predictor variables, ΔS , etc. • Overall uncertainty (95% CL) ≈ 4% 9

10. Local Runoff (RN) • Current estimates (average monthly values) based on 1962 analysis of Grand and Genesee River flows • At the time, data was not available at tributary gauges • Since 1957, anywhere from 27 to 44% of the basin was gauged • Computed local runoff from actual gauged tributary flows by maximizing gauged area without overlap and using area ratios to extrapolate to ungauged areas

11. Combined Uncertainty in Outflow • Additional inputs (i.e. NY State Barge Canal and Welland River diversions) were also evaluated but found to have a negligible impact in terms of uncertainty in Niagara River flows • Combined uncertainty ON@Buf ≈4% (95% CL) • Welland Canal flow uncertainty (determined to be approximately 8% at 95% CL) contributes only a small additional source of uncertainty to the total Lake Erie outflow and NBS due to its smaller magnitude 11

12. Detroit River Inflow • Stage-fall-discharge equations: • Uncertainty (95% CL) • Gauged discharge measurements = 5% • Standard error of estimates = 6.6% • Error in the mean fitted relation = 1% • Predictor variables (i.e. water levels) = 2% • Overall uncertainty ≈ 8.6% at 95% confidence level • Conservative estimate • Systematic effects can increase error and uncertainty significantly on a short term basis • E.g. Ice impacts and channel changes due to erosion, obstruction, etc.

13. Change in Storage (ΔS) • Change in the lake-wide mean water level from the beginning-of-month (BOM) to the end-of-month (EOM) • Sources of Uncertainty: • Gauge accuracy (+/- 0.3 cm) • Rounding error (+/- 0.5 cm) • Temporal variability • Spatial variability • Lake area • Uncertainty is relatively small • Glacial Isostatic Adjustment (GIA) • Negligible on a monthly basis • Thermal expansion and contraction 13

14. Temporal Variability • Evaluated: • Where: • Used daily estimates of each input • Error almost negligible (max < 1 cm); two-day mean provides adequate representation of instantaneous water level at midnight • Need only to know uncertainty of the mean • Computed hourly four-gauge mean for years 1984-1985 • Standard error of the mean = 0.3 cm ε(BOM) 14

15. Spatial Variability • Caused primarily by meteorological effects (i.e. winds, barometric pressure, seiche) • Differences in water levels measured at opposite ends of the lake can be upwards of a few metres • Gauge measurements at different locations around the lake are averaged to try to balance and reduce these errors • Spatial variability errors result from slope of the lake surface and imbalance in the weighting given to different gauges

16. Spatial Variability • Compared BOM water levels from four-gauge average to 9-gauge Thiessen weighted network average (Quinn and Derecki, 1976) for period 1980-2009 • Logistic distribution fit differences well • BOM standard error ~= 0.6 to 1.6 cm, depending on the month • Largest errors in the fall/winter

17. Thermal Expansion and Contraction (ΔSTh) • Normally considered negligible, but can be significant source of error • Measured water column temperature data is not available • Adapted method proposed by Meredith (1975) • Related dimensionless vertical temperature profiles for each month to measured surface temperatures to estimate vertical temperature dist. • Computed volume at BOM and EOM and determined difference • Conclusions based on results of both surface temp. datasets and all three sets of temp. profiles

19. Combined Uncertainty in NBS: Methods • First-Order Second Moment (FOSM) Method • Model: • Taylor Series Expansion: • Requires only mean and standard deviation of model inputs • Provides mean and standard deviation of model output only • Monte Carlo Analysis Method • Involves repeatedly simulating the output variable, , using randomly generated subsets of input variable values, , according to their respective probability distributions • Requires probability distribution of model inputs, and provides full probability distribution of model output

20. Combined Uncertainty in NBS • Determining combined estimate of uncertainty in NBS quite simple due to mathematical simplicity of the model • FOSM and Monte Carlo method results almost identical • Linear model • Variance of model inputs described consistently • Uncertainty varies by month • Absolute uncertainty is fairly similar • Relative uncertainty greatest in the summer and November (> than 100% in some cases)

21. Comparisons • Neff and Nicholas (2005) • Uncertainty in both residual and component NBS • Based primarily on authors’ best professional judgement • Similar results; main difference is uncertainty in change in storage, which was highly underestimated based on the results of this thesis • De Marchi et al (2009) • Uncertainty in GLERL component NBS • Overall uncertainty in component NBS is of a similar magnitude to residual NBS on Lake Erie • Useful for measuring the effects of improvements to each method of computing NBS in the future 21

22. Conclusions • Evaluating uncertainty in each input the most difficult part of overall NBS uncertainty analysis • FOSM and Monte Carlo methods gave nearly identical results • Uncertainty in BOM water levels as currently computed and change in storage is large • Same magnitude as Detroit River inflow and in some months greater than Niagara River flow uncertainty • Uncertainty due to change in storage due to thermal expansion and contraction is in addition to this • Uncertainty in change in storage possibly easiest to reduce • To reduce uncertainty in Erie NBS must reduce uncertainty in each of the different major inputs (i.e. inflow, outflow and change in storage) • Reduction of uncertainty in one input will not significantly reduce uncertainty in residual NBS

23. Next Steps • Compare/validate component and residual supplies • Comparisons must account for consumptive use, groundwater, and other inputs normally considered negligible, and the errors this causes • Explain differences, if possible, by systematic errors from this study and others • Incorporate new data/methods as they become available • e.g. horizontal ADCPs/index velocity ratings • Investigate ΔS computation method further • Consider use of local tributary flows or hydrologic model to compute local inflow for Niagara at Buffalo 23