1. Distributions and Flood Frequency Chapter 3 – Part 2
Dr. Philip B. Bedient
Rice University 2018

2. Normal Distribution
Using the Normal Table, approx. 68% of the data will fall in the interval of -s to s, one S.D.
~ 95% of the data falls between -2s to 2s, and approx 99.9% data lie between -3s to 3s
For a standard normal distribution, 68% of the data fall in the interval of z = -1 to z = 1.

3. Normal, Log N, LPIII
Data in bins
Normal

4. Probability Paper
Normal Probability Paper converts the Normal CDF Sigmoid curve into a straight line on a special scale.
Normal paper allows for a rapid interpretation of how well data fits the Normal or Log N distribution
Extreme Value paper is also available for skewed distributions

5. Normal Prob. Paper
Normal Prob Paper converts the Normal CDF S curve into a straight line on a prob scale

6. Normal Prob. Paper Std Dev = +1000 cfs Mean = 5200 cfs
Place mean at F = 50%
Place one Sx at 15.9 and 84.1%
Connect points with st. line
Plot data with plotting position    formula P = m/n+1
Std Dev = –1000 cfs

7. Normal Paper Fit
Mean

8. Parameters of Dist’n Distribution Normal x LogN Y =logx Gamma Exp t
Mean mx my nk
1/k
Variance
sx2
sy2
nk2
1/k2
Skewness
zero
2/n0.5 2

9. Frequency Analysis of Peak Flow Data – Siletz River Example
Year
Rank
Ordered cfs
1940 1
42,700
1925 2
31,100
1932 3
20,700
1966 4
19,300
1969 5
14,200
1982 6
1988 7
12,100
1995 8
10,300
2000 ……
…….

10. Frequency Analysis of Peak Flow Data
Take Mean and Variance (S.D.) of ranked data
Take Skewness Cs of data (3rd moment about mean)
If Cs near zero, assume normal dist’n
If Cs large, convert Y = Log x - (Mean and Var of Y)
Take Skewness of Log data - Cs(Y)
If Cs near zero, then fits Lognormal
If Cs not zero, fit data to Log Pearson III

11. Frequency Factor Table 3-4 in Chapter 3 (Normal, LN, LP3 table)
Cs (skew) vs. Recurrence Interval or % exceedance
5 year event has 20% exceedance chance = 1/5
100 yr event has 1% chance exceedance = 1/100
100 yr has K = 2.326, 10 yr has K = 1.282

12. Frequency Factor Equation
Q = Qm + K(Cs, T) (SQ) Where Qm = mean, SQ = Std Dev.
Note (Y = Log Q) Y = Ym + K(Cs, T) (SY)

13. Siletz River Example 75 data points - Excel Tools
Original Q
Y = Log Q
Mean
20,452
4.2921
Std Dev
6089
0.129
Skewness
0.7889

14. Siletz River Example - Fit Normal and Log N
Normal Distribution Q = Qm + z SQ
Q100 = (6089) = 34,620 cfs
Mean z (S.D.)
Where z = std normal variate - tables
Log N Distribution Y = Ym + k SY
Y100 = (0.129) =
k = freq factor and Q = 10Y = 39,100 cfs

15. Log Pearson Type III Log Pearson Type III Y = Ym + k SY
K is a function of Cs and Recurrence Interval
Table 3.4 lists values for pos and neg skews
For Cs = -0.15, thus K = 2.15 from Table 3.4
Y100 = (0.129) = 4.567
Q = 10Y = 36,927 cfs for LP III
Plot several points on Log Prob paper

16. Log N Paper for CDF
What is the prob that flow exceeds     some given value yr value
Plot data with plotting position    formula P = m/n+1 , m = rank, n = #
Log N dist’n plots as straight line

17. Log N Plot of Siletz R.
Mean
Straight Line Fits Data Well

18. Flow Duration Curves

19. Rainfall Analysis in Urban Basins
Hydrology and Floodplain Analysis, Chapter 6.3
Rainfall Analysis in Urban Basins

20. Data Sources Two types of rainfall data:
“raw” point data (actual hyetographs)
Processed data (frequency information)
Raw data can be sourced from physical gages or radar
Multiple sources (besides NWS, although this is primary) may be needed to accurately describe a catchment
Left: National Weather Service national precipitation accumulation
Right : IDF curve Houston

21. Intensity-Duration-Frequency Curves
Represent the probability a given rainfall intensity will occur, given a duration
Note: duration is not the length of the storm
Do not represent actual time histories of rainfall
Smoothed results of several different storms
Hypothetical results if point does not fall on contour
Should not be used to estimate rainfall volume b/c duration is arbitrarily assigned

22. IDF curve for Tallahassee, Florida region

23. Definition of Storm Event
For analysis, rainfall time series must be separated into discrete events
Minimum interevent time (MIT)
Rainfall pulses separated by less than this time are considered the same event
Note that same rainfall volumes can be returned from storms with a multitude of durations
Emphasizes that the characterization of any event is duration dependent

24. Choice of Design Storm
Need to know return period and parameter (precipitation, runoff volume, etc.)
25 year rainfall may only produce 5 year runoff
Image:

25. Synthetic Design Storm Creation
Procedure
1. Duration specified
2. Duration depth obtained from IDF curve
3. Rainfall distributed in time to construct hyetograph
Shape varies depending on storm type
Natural Resources Conservation Service publishes temporal distributions
e.g. SCS type II and type IA
Type II used for southeast, type 1A used for northwest

26. SCS type II 5yr design storm for Tallahassee Florida

27. Historical Storms
Use rainfall data from storm that created peak flow of desired return interval
Ideal method for design
No assumptions about hyetograph shape
Popular with the public
Design to prevent a readily remembered flood event
Synthetic design storms are still the most commonly used method
Photo – tropical storm Allison Houston, Texas