# Frequency Analysis Problems Problems 1. Extrapolation 2. Short

Frequency Analysis Problems 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 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 Extrapolation Thompson (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 2. Short Records 1200

Number of Gauges 1000 800 600 400 200 0 0 20 40 60 Minimum Record Length (Years)

80 100 2. Short Records Min. Record Length Number of Gauges Percent 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 (Years) 2. Short Records 2 1:100 - 95% 1:50 - 95% 1.8 1:25 - 95% 1.6 Qn/Q90

1.4 1.2 1 0.8 1:100 - 5% 1:50 - 5% 0.6 1:25 - 5% 0.4 0 20 40 Sample Length 'n' (Years) 60

80 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 1990 1980

1970 1960 2000 1950 1940 1930 1920 1910 1900 1890 1880

1870 Peak Instantaneous Discharge (cms) 3. Extreme Data 2500 Gauge 05BH004 Bow River At Calgary 1500 1000 500 0 3. Extreme Data 10000

LN3 - Without Ungauged Data Discharge (cms) LN3 - With Ungauged Data 1000 Gauge 05BH004 Bow River At Calgary 100 1.003 1.05 1.25 2 5

Return Period (Years) 10 20 50 100 200 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

4. Non-extreme Data 10000 LN3 - Without Ungauged Data Discharge (cms) LN3 - With Ungauged Data 1000 Gauge 05BH004 Bow River At Calgary 100 1.003 1.05 1.25 2

5 Return Period (Years) 10 20 50 100 200 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) 4. Non-extreme Data 100 Combined Data

Spring Data Discharge (cms) Summer Data 10 LP3 - Combined LP3 - Spring LP3 - Summer 1 0.1 1.003 1.05 1.25 2

Return Period (Years) 5 10 20 50 100 200 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 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 300 250 Flood Hydrograph Mean Annual Flood 200 150

100 50 Jun-10 Jun-09 Jun-08 Jun-07 0 Jun-06 Discharge (m3/s) Gauge Measurement 6. Data Accuracy 5

Highest Recorded Water Level Stage (m) 4 3 Highest Gauge Measurement 2 Gauge 05AA004 Pincher Ck - 1995 1 0 0 100

200 Discharge (m3/s) 300 6. Data Accuracy 350 300 Qi reported as 200 m3/s Does not fit mean daily flows Discharge (cms) 250 200 Gauge 05BK001 150

Fish Ck - 1915 100 50 0 Jun-24 Jun-25 Jun-26 Jun-27 Jun-28 Jun-29 Jun-30

Jul-01 Jul-02 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 1500 Hydrograph Mean Daily - Peak At Noon

Discharge (m3/s) Mean Daily - Peak At Midnight 1000 Gauge 05AA023 Oldman R - 1995 500 0 0 20 40 60 Time (hours) 80 100

7. Peak Instantaneous Data 4000 1975 Flood pre 1995 1:100 Est Discharge (cms) 3000 1995 Flood 2000 Oldman R Dam 1000 0 -40

-20 0 20 Time (hours) 40 60 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) 0 10 20 30 40 50 60 70 80 90 Number of Gauges 1085 564

354 212 109 62 44 29 21 3 Percent 100.0 52.0 32.6 19.5 10.0 5.7 4.1 2.7 1.9 0.3 8. Gauge Coverage

2.0 Discharge Ratio 1.5 1.0 n = 0.5 n = 0.7 n = 0.9 0.5 0.0 0 0.5 1 Drainage Area Ratio 1.5

2 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 9. No Routing Discharge (m3/s) 15 Inflow 10 Outflow

5 0 0 20 40 Time (hrs) 60 80 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 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 11. Variation in Results 1000 Data

Discharge (cms) GEV LN3 100 LP3 Gauge 05AD003 Waterton River Near Waterton 74 Years of Record 10 1.003 1.05 1.25 2 Return Period (Years)

5 10 20 50 100 200 11. Variation in Results 100 Data Discharge (cms) GEV LN3 10 LP3 Gauge 05BL027 Trap Ck Near Longview

20 Years of Record 1 1.003 1.05 1.25 2 Return Period (Years) 5 10 20 50 100 200 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

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