Jacob Bruxer
February 2011
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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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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
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methods for computing them
Net Basin Supplies (NBS)
 Water Balance
S  STh  I  O  P  R  E  G  D  C
 Component Method
NBS  P  R  E  G
 Residual Method
S  STh  I  O  NBS  D  C
NBS  S  STh  I  O  D  C
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NBS RES  S  I  O
NBS Erie  S  I Det  OW C  ON @ Buf + ???
uncertainty
ΔS
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Flow Uncertainty: Overview
 Niagara and Welland C. flow accounting is complicated
 Summation of a number of different flow estimates

E.g.
ON@BUF = NMOM + PSAB1&2 + PRM + DNYSBC - RN - DWR
 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
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Niagara Falls Flow
(NMOM)
 ≈ 30-40% of total ON@Buf
 Stage-discharge equation
based on measured water
levels at Ashland Ave. gauge
3.0
and ADCP flow measurements N

0
.
6429

(
h

82
.
814
)
MOM
AA
 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
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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%
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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
RN   RGauged 
ATotal
 AGauged
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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
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Detroit River Inflow
 Stage-fall-discharge equations:

Q  C  ( w1 h1  w2 h2  y b )  (h1  h2 )

 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.
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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
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Temporal Variability
 Evaluated:
 ( BOM 2day ) 
S True, 1  S True, 1
ε(BOM)
h ( m t ,d 1st )
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 Where:
S True  I  O  P  R  E
h ( m t 1,d last)
BOM t  BOM 12am
 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
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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
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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
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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
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Combined Uncertainty in NBS: Methods
 First-Order Second Moment (FOSM) Method
y  f ( x1 , x2, ...xn )
 Taylor Series Expansion:
 Model:
E ( y )  f ( x1, x 2, ...x n )
n
n 1
n
u ( y )   ci u ( x i )  2    c i  c j  u ( x i )  u ( x j )  r ( xi , x j )
2
2
2
i 1
i 1 j i 1
 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,
y , using
randomly generated subsets of input variable values, ( x1 , x 2 ,...x n ) ,
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
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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
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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
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