Frequency Analysis Problems
Problems1. Extrapolation2. Short Records3. Extreme Data 4. Non-extreme Data5. Stationarity of Data6. Data Accuracy7. Peak Instantaneous Data8. Gauge Coverage 9. No Routing10. No Correct Distribution11. Variation In Results12. No Verification Of Results13. Mathematistry
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
1. Extrapolation
US Population ExtrapolationThompson (1942) reported in Klemes (1986)
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
0
200
400
600
800
1000
1200
0 20 40 60 80 100
Minimum Record Length (Years)
Nu
mb
er o
f G
aug
es2. Short Records
2. Short Records
Min. Record Length Number of Gauges Percent
(Years)
0 1085 100.0
10 564 52.0
20 354 32.6
30 212 19.5
40 109 10.0
50 62 5.7
60 44 4.1
70 29 2.7
80 21 1.9
90 3 0.3
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0 20 40 60 80
Sample Length 'n' (Years)
Qn/Q
90
1:100 - 95%
1:50 - 95%
1:25 - 95%
1:100 - 5%
1:50 - 5%
1:25 - 5%
2. Short Records
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
0
500
1000
1500
2000
250018
70
1880
1890
1900
1910
1920
1930
1940
1950
1960
1970
1980
1990
Pea
k In
stan
tan
eou
s D
isch
arg
e (c
ms)
3. Extreme Data
Gauge 05BH004
Bow River At Calgary
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1000
10000
Return Period (Years)
Dis
char
ge
(cm
s)
LN3 - Without Ungauged Data
LN3 - With Ungauged Data
3. Extreme Data
Gauge 05BH004
Bow River At Calgary
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
200100502010521.251.051.003100
1000
10000
Return Period (Years)
Dis
char
ge
(cm
s)
LN3 - Without Ungauged Data
LN3 - With Ungauged Data
Gauge 05BH004
Bow River At Calgary
4. Non-extreme Data
4. Non-extreme Data
A - Original FitB - 3 lowest points slightly reducedC - 3 lowest points slightly increased
East Humber River, OntarioKlemes (1986)
200100502010521.251.051.0030.1
1
10
100
Return Period (Years)
Dis
char
ge
(cm
s)
Combined Data
Spring Data
Summer Data
LP3 - Combined
LP3 - Spring
LP3 - Summer
4. Non-extreme Data
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, diversionsland 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
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
6. Data Accuracy
0
50
100
150
200
250
300Ju
n-0
6
Jun
-07
Jun
-08
Jun
-09
Jun
-10
Dis
char
ge
(m3 /s
)
Flood Hydrograph
Gauge Measurement
Mean Annual Flood
6. Data Accuracy
0
1
2
3
4
5
0 100 200 300
Discharge (m3/s)
Sta
ge
(m)
Gauge 05AA004
Pincher Ck - 1995
Highest Recorded Water Level
Highest Gauge Measurement
0
50
100
150
200
250
300
350
Jun-24 Jun-25 Jun-26 Jun-27 Jun-28 Jun-29 Jun-30 Jul-01 Jul-02
Dis
char
ge
(cm
s)6. Data Accuracy
• Qi reported as 200 m3/s• Does not fit mean daily flows
Gauge 05BK001
Fish Ck - 1915
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
7. Peak Instantaneous Data
0
500
1000
1500
0 20 40 60 80 100
Time (hours)
Dis
char
ge
(m3 /s
)
Hydrograph
Mean Daily - Peak At Noon
Mean Daily - Peak At Midnight
Gauge 05AA023
Oldman R - 1995
7. Peak Instantaneous Data
0
1000
2000
3000
4000
-40 -20 0 20 40 60
Time (hours)
Dis
char
ge
(cm
s)
1975 Flood
pre 1995 1:100 Est
1995 Flood
Oldman R Dam
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
8. Gauge Coverage
All Gauges(1085)
Gauges >30 Years (212)
8. Gauge Coverage
Minimum Record Length (Years)
Number of Gauges
Percent
0 1085 100.010 564 52.020 354 32.630 212 19.540 109 10.050 62 5.760 44 4.170 29 2.780 21 1.990 3 0.3
0.0
0.5
1.0
1.5
2.0
0 0.5 1 1.5 2
Drainage Area Ratio
Dis
char
ge
Rat
io
n = 0.5n = 0.7n = 0.9
8. Gauge Coverage
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
15
0
5
10
0 20 40 60 80
Time (hrs)
Inflow
Outflow
Dis
char
ge (
m3 /
s)9. No Routing
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 storageaddition of flow from inter-basin transfer at extreme eventschanges in contributing drainage area at extreme events
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
200100502010521.251.051.00310
100
1000
Return Period (Years)
Dis
char
ge
(cm
s)
Data
GEV
LN3
LP3
11. Variation in Results
Gauge 05AD003
Waterton River Near Waterton
74 Years of Record
200100502010521.251.051.0031
10
100
Return Period (Years)
Dis
char
ge
(cm
s)
Data
GEV
LN3
LP3
11. Variation in Results
Gauge 05BL027
Trap Ck Near Longview
20 Years of Record
12. No Verification Of Results
•Due to the separation of frequency analysis from physical modelling, the process cannot be tested.
•1:100 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
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
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
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