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

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.

Normal, Log N, LPIII Data in bins Normal

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

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

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

Normal Paper Fit Mean

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

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 …… …….

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

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

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)

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 - 0.1565

Siletz River Example - Fit Normal and Log N Normal Distribution Q = Qm + z SQ Q100 = 20452 + 2.326(6089) = 34,620 cfs Mean + z (S.D.) Where z = std normal variate - tables Log N Distribution Y = Ym + k SY Y100 = 4.29209 + 2.326(0.129) = 4.5923 k = freq factor and Q = 10Y = 39,100 cfs

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 = 4.29209 + 2.15(0.129) = 4.567 Q = 10Y = 36,927 cfs for LP III Plot several points on Log Prob paper

Log N Paper for CDF What is the prob that flow exceeds     some given value - 100 yr value Plot data with plotting position    formula P = m/n+1 , m = rank, n = # Log N dist’n plots as straight line

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

Flow Duration Curves

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

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 http://water.weather.gov/precip/ Right : IDF curve Houston http://www.idfcurve.org/

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

IDF curve for Tallahassee, Florida region

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

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: http://en.wikipedia.org/wiki/File:FoggDam-NT.jpg

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

SCS type II 5yr design storm for Tallahassee Florida

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 http://www.cs.rice.edu/~dwallach/photo/allison2001/Hwy59.jpg