Uncertainty in Lake Erie Residual Net Basin Supplies

Jacob Bruxer

February 2011

1

Presentation OverviewWater 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

3

Introduction and MotivationNet 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

CDGERPOISS Th

GERPNBS

CDOISSNBS Th CDNBSOISS Th

6

OISNBSRES

BufNWCDetErie OOISNBS @

ΔS

uncertainty

+ ???

Flow Uncertainty: OverviewNiagara and Welland C. flow accounting is

complicatedSummation of a number of different flow estimates

E.g. Makes accounting for uncertainty difficult, but

reduces overall uncertainty to some degreeDetroit River flows also complicated

Stage-fall-discharge equations, Transfer Factors, other models

Non-stationarity, channel changes, ice effectsUncertainty in model calibration data, models

themselves, and model predictor variables7

ON@BUF = NMOM + PSAB1&2 + PRM + DNYSBC - RN - DWR

Niagara Falls Flow (NMOM)

≈ 30-40% of total ON@Buf

Stage-discharge equation based on measured water levels at Ashland Ave. gaugeand 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

Page 8

0.3)814.82(6429.0 AAMOM hN

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%

Page 99

Current estimates (average monthly values) based on 1962 analysis of Grand and Genesee River flows

At the time, data was not available at tributary gaugesSince 1957, anywhere from 27 to 44% of the basin was gaugedComputed local

runoff from actualgauged tributary flows by maximizing gauged area without overlap and using area ratios to extrapolate to ungauged areas

Page 10 Gauged

TotalGaugedN A

ARR

Local Runoff (RN)

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

Detroit River InflowStage-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 levelConservative estimate

Systematic effects can increase error and uncertainty significantly on a short term basisE.g. Ice impacts and channel changes due to erosion,

obstruction, etc.12

)()( 212211 hhyhwhwCQ b

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 variabilitySpatial variabilityLake area

Uncertainty is relatively smallGlacial Isostatic Adjustment (GIA)

Negligible on a monthly basisThermal expansion and contraction

13

Temporal Variability

14

amt BOMBOM 12),1( lastdtmh

)1,( stdtmh 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 meanComputed hourly four-gauge mean for years 1984-1985Standard error of the mean = 0.3 cm

4)( 1,1,

2

TrueTrue

day

SSBOM

ERPOISTrue

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 reducethese errors

Spatial variability errors result fromslope of the lake surface and imbalance in the weighting given to different gauges

15

Spatial Variability

Spatial VariabilityCompared 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 monthLargest 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

Thermal Expansion and Contraction (ΔSTh)

18

Combined Uncertainty in NBS: MethodsFirst-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

)...()( ,2,1 nxxxfyE ),()()(2)()(1

1 1

2

1

22jiji

n

i

n

ijjii

n

ii xxrxuxuccxucyu

)...,( ,21 nxxxfy

),...,( 21 nxxx

y

Combined Uncertainty in NBSDetermining 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)

ComparisonsNeff and Nicholas (2005)

Uncertainty in both residual and component NBSBased primarily on authors’ best professional judgementSimilar 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 NBSOverall uncertainty in component NBS is of a similar

magnitude to residual NBS on Lake ErieUseful for measuring the effects of improvements to each

method of computing NBS in the future 21

ConclusionsEvaluating uncertainty in each input the most

difficult part of overall NBS uncertainty analysisFOSM and Monte Carlo methods gave nearly

identical resultsUncertainty in BOM water levels as currently

computed and change in storage is largeSame magnitude as Detroit River inflow and in some

months greater than Niagara River flow uncertaintyUncertainty due to change in storage due to thermal

expansion and contraction is in addition to thisUncertainty in change in storage possibly easiest to

reduceTo 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 NBS22

Next StepsCompare/validate component and residual

suppliesComparisons 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 availablee.g. horizontal ADCPs/index velocity ratings

Investigate ΔS computation method furtherConsider use of local tributary flows or

hydrologic model to compute local inflow for Niagara at Buffalo

23