Jacob Bruxer February

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 2

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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

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

6 ΔSΔS uncertainty + ???

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 7 O = N MOM + P SAB1&2 + P RM + D NYSBC - R N - D WR

Niagara Falls Flow (N MOM ) ≈ 30-40% of total O 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 N MOM ≈ 6.7% Conservative estimate Page 8

Hydropower (P SAB1&2 + P RM ) ≈ 60% of total O 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% Page 9 9

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 Page 10

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 O ≈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

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. 12

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

Temporal Variability 14 ε(BOM) 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 Standard error of the mean = 0.3 cm

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 15 Spatial Variability

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

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 17

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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

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)

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

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 22

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