Frequency Analysis Problems

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Frequency Analysis Problems
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Problems
1. Extrapolation 2. Short Records 3. Extreme Data
4. Non-extreme Data 5. Stationarity of Data 6.
Data Accuracy 7. Peak Instantaneous Data 8. Gauge
Coverage 9. No Routing 10. No Correct
Distribution 11. Variation In Results 12. No
Verification Of Results 13. Mathematistry
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1. Extrapolation

• Danger in fitting to known set of data and
  extrapolating to the unknown, without
  understanding physics
• Example of US population growth chart
• Tight fit with existing data
• Application of accepted distribution
• No understanding of underlying factors
• Results totally wrong

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1. Extrapolation
US Population Extrapolation Thompson (1942)
reported in Klemes (1986)
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2. Short Records

• Ideally require record length several times
  greater than desired return period
• Alberta has over 1000 gauges with records, but
  very few are long
• Frequency analysis results can be very sensitive
  to addition of one or two data points
• Subsampling larger records indicates sensitivity

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2. Short Records
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2. Short Records
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2. Short Records
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3. Extreme Data

• The years recorded at a gauge may or may not
  have included extreme events
• Large floods known to have occurred at gauge
  sites but not recorded
• Some gauges may have missed extreme events only
  by chance e.g. 1995 flood - originally predicted
  for Red Deer basin, but ended up on the Oldman
  basin. The Red Deer and Bow River basins have
  not seen extreme floods in 50 to 70 years
• Presence of several extreme events could cause
  frequency analysis to over-predict
• Presence of no extreme events could cause
  frequency analysis to under-predict

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3. Extreme Data
Gauge 05BH004 Bow River At Calgary
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3. Extreme Data
Gauge 05BH004 Bow River At Calgary
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4. Non-extreme Data

• All data points are used by statistical methods
  to fit a distribution. Most of these points are
  for non-extreme events, that have very different
  physical responses than extreme events e.g.
• magnitude, duration, and location of storm
• snowmelt vs. rainfall
• amount of contributing drainage area
• initial moisture
• impact of routing at lower volumes of runoff
• Fitting to smaller events may cause poor fit and
  extrapolation for larger events
• Impact of change in values at left tail impact
  the extrapolation on the right - makes no
  physical sense

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4. Non-extreme Data
Gauge 05BH004 Bow River At Calgary
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4. Non-extreme Data
A - Original Fit B - 3 lowest points slightly
reduced C - 3 lowest points slightly increased
East Humber River, Ontario Klemes (1986)
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4. Non-extreme Data
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5. Stationarity Of Data

• Changes may have occurred in basin that affect
  runoff response during the flow record e.g.
• man-made structures - dams, levees, diversions
• land use changes - agriculture, forestation,
  irrigation
• In order to keep the equivalent length of
  record, hydrologic modelling would be required to
  convert the data so that it would be consistent.
• This modelling would be very difficult as it it
  would cover a wide range of events over a number
  of years

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6. Data Accuracy

• Extreme data often not gauged
• Extrapolated using rating curves
• Channel changes during large floods - geometry,
  roughness, sediment transport,
• Problems with operation of stage recording
  gauges e.g. damage, ice effects
• Problems with data reporting e.g. Fish Ck, 1915
• Hydrograph examination can ID problems

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6. Data Accuracy
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6. Data Accuracy
Highest Recorded Water Level
Highest Gauge Measurement
Gauge 05AA004 Pincher Ck - 1995
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6. Data Accuracy

• Qi reported as 200 m3/s
• Does not fit mean daily flows

Gauge 05BK001 Fish Ck - 1915
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7. Peak Instantaneous Data

• Design discharge is based on peak instantaneous
  values, but sometimes this data is not available
• Conversion of mean daily data to instantaneous
  requires consideration of the hydrograph timing
  e.g. peaks near midnight vs. peaks near noon
• Different storm durations can result in very
  different peak to mean daily ratios for the same
  basin
• Applying a multiplier to the results of a
  frequency analysis based on mean daily values can
  lead to misleading results
• Statistical methods require that all data points
  be consistent, even though many are irrelevant to
  extrapolation

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7. Peak Instantaneous Data
Gauge 05AA023 Oldman R - 1995
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7. Peak Instantaneous Data
Oldman R Dam
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8. Gauge Coverage

• Limited number of gauges in province with
  significant record lengths
• Difficult to transfer peak flow number to other
  sites without consideration of hydrographs and
  routing
• Area exponent method very sensitive to assumed
  number

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8. Gauge Coverage
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8. Gauge Coverage
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8. Gauge Coverage
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9. No Routing

• Peak instantaneous flow value is only applicable
  at the gauge site
• Need hydrograph to rout flows, not just peak
  discharges
• Major Routing Factors include
• Basin configuration
• Lakes and reservoirs
• Floodplain storage
• inter-basin transfers e.g. Highwood - Little Bow
  River

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9. No Routing
Inflow
Outflow
Discharge (m3/s)
Time (hrs)
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10. No Correct Distribution

• Application of theoretical probability
  distributions and fitting techniques originated
  with Hazen (1914) in order to make straight line
  extrapolations from data
• There is no reason why they should be applicable
  to hydrologic observations
• None of them can account for the physics of the
  site during extrapolation
• discharge limits due to floodplain storage
• addition of flow from inter-basin transfer at
  extreme events
• changes in contributing drainage area at extreme
  events

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11. Variation in Results

• Different distributions and fitting techniques
  can yield vastly different results
• Many distributions in use - LN2, LN3, LP3, GEV,
  P3
• Many fitting techniques - Moments, Maximum
  Likelihood, Least Squares Fit, PWM
• No way to distinguish between which one is the
  most appropriate for extrapolation
• Extrapolated values can be physically unrealistic

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11. Variation in Results
Gauge 05AD003 Waterton River Near Waterton 74
Years of Record
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11. Variation in Results
Gauge 05BL027 Trap Ck Near Longview 20 Years of
Record
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12. No Verification Of Results

• Due to the separation of frequency analysis from
  physical modelling, the process cannot be tested.
  
• 1100 year flood predictions cannot be actually
  tested for 100's or 1000's of years.
• There is therefore little opportunity to refine
  an analysis or to improve confidence in its
  applicability

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13. Mathematistry

• Gain artificial confidence in accuracy due to
  mathematical precision
• statistics - means, standard deviations, skews,
  kurtosis, outliers, confidence limits
• curve fitting - moments, max likelihood, least
  squares, probability weighted moments
• probability distributions - LN3, LP3, GEV,
  Wakeby
• Loose sight of physics with focus on numbers

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Conclusions

• Statistical frequency analysis has many problems
  in application to design discharge estimation for
  bridges.
• If frequency analysis is to be employed,
  extrapolation should be based on extreme events.
  This can be accomplished using graphical
  techniques if appropriate data exists.
• Alternative approaches to design discharge
  estimation should be investigated. These should
  
• be based on all relevant extreme flood
  observations for the area, minimizing
  extrapolations
• account for physical hydrologic characteristics
  for the area and the basin

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Conclusions

• Recommended articles by Klemes
• Common Sense And Other Heresies - Compilation
  of selected papers into a book, published by CWRA
• Dilettantism in Hydrology Transition or
  Destiny? (1986)
• Hydrologic And Engineering Relevance of Flood
  Frequency Analysis (1987)
• Tall Tales About Tails Of Hydrological
  Distributions - paper published in ASCE Journal
  Of Hydrologic Engineering, July 2000