Frequency Analysis Lecture 27 John Reimer

Background  Variability in rainfall and the resulting streamflow must be dealt with in planning and design  We cannot predict future with reliable degree of certainty  Solution is to apply methods of probability and statistics

Probability Concepts  Random variable: A variable whose value is not known until it manifests itself  We typically know something about possible values as either a range or a discrete set of possibilities  Example: For 1 die the possible outcomes are 1,2,3,4,5 or 6

Probability Concepts  Repeated throws of a die generate random numbers from a possible set of outcomes  We denote random variables with upper case  For the case of a die, the probability of rolling a 3 is P(X=3) = 1/6

Probability Concepts  Because a discrete event will either occur or not occur, the sum of both possibilities must sum to 1  If the probability of occurrence is P, the probability of not occurring is 1-P  For tossing a die twice, the probability of NOT throwing two 3’s is therefore 1 - 1/36 = 35/36

Frequency Distributions  One way of looking at random variables is to consider the distribution of possible values  A plot of the probability of a random variable being contained within specific classes(ranges) of values is termed the frequency distribution curve

Frequency Distribution Curve Gupta Fig. 8.1

Cumulative Distributions  In many cases the continuous distribution function (CDF) is plotted  The CDF is simply a plot of the probability of being either less than or greater than a particular value  It represents an accumulation of the frequency distribution Let Q be a random variable representing discharge associated with the annual flood F Q (q)=P{Q≤q}

Cumulative Distribution Function Gupta Fig. 8.2

Probability Density Function  For a continuous random variable, the frequency distribution concept breaks down  Instead, the probability density function (PDF) is used  Describes the relative likelihood for a random variable to take on a given value Let Q be a random variable representing discharge associated with the annual flood q1q1 q2q2 F Q (q) ∫ f Q (q)=P{q 1 ≤ Q≤q 2 } q1q1 q2q2 for q 1 <q 2 }

Statistical Measures  Mean or Expected Value  Variance  Standard Deviation  Coefficient of Variation q F Q (q) E[Q]=  Q =∫qf Q (q) dq ∞ -∞ V[Q]=  Q 2 =E[(Q-  Q ) 2 ] =∫ (q-  Q ) 2 f Q (q)dq ∞ -∞  Q = sqrt(  Q ) 2 C Q =  Q /|  Q |

Recurrence Interval  Instead of probabilities, we usually refer to the recurrence interval of rainfall events  Recurrence interval: The average interval of time between events as rare or rarer than the given event  For rainfall, recurrence interval is the average interval of time between events equaling or exceeding a particular value

Recurrence Interval (Return Period)  For p r = 0.01, the recurrence interval is thus (1/0.01) = 100 yrs Multi-year Chance of Exceedance RI=1/P RI = Recurrence Interval, P = Probability R=1-(1-1/RI) n  For the 100 year flood, the chance it will occur in 5 years R=1-(1-1/100) 5 = 0.05  For the 100 year flood, the chance it will occur in 300 years R=1-(1-1/100) 300 = 0.95

Analysis of Rainfall  Problem of interest is flood potential  Interested in rainfall over various durations  Design storms based on specific durations  Frequency curves are fitted for various durations

Estimating RI r (and p r )  To estimate the t-year rainfall at a climate station based on N-years of data, we construct the annual rainfall series (r d,1, r d,2, …, r d,n ) for specific durations (d) of rainfall  r d,1 is the largest rainfall in year 1 for the specified duration d  The values are then ranked from highest to lowest.  The plotting position is used to estimate the exceedance probability for each rank

Plotting Position  The general form of the plotting position for rank m from N data points is The Weibull formula is widely used, and is a special case for a = 0 and b = 1:

Estimating Rare Events  For a record with N annual events, the highest RI that can be estimated is N+1  In many cases we want estimates for extreme events with RI’s that exceed N+1  Two general approaches:  Graphical method  Analytical techniques

Graphical Method  Rank the data from highest (rank=1) to lowest (rank=N)  Estimate plotting positions using ranks  Compute recurrence intervals  Generate a plot of r (m) vs RI r(m)  Fit a line to the data; extrapolate to desired RI

Graphical Method Viessman Fig. 5-7

Graphical Method  For small data sets, extrapolation is problematic (largest RI is N+1)  Different graph papers can be used to provide a straight line (Normal, Gumbel)  Log transform for r on Normal paper results in Log-Normal distribution

Analytical Techniques  The general approach is to fit a CDF to the data using standard statistical techniques  Because we don’t know the exact distribution, we use the data to develop an estimator  Given the form of the CDF, the rainfall for a particular RI can be determined

Analytical Techniques  Commonly used distributions:  Normal, Log Normal  Extreme value type 1 (Gumbel)  Log Pearson type III  Parameter Estimation  Method of Moments

Normal distribution  Central limit theorem – if X is the sum of n independent and identically distributed random variables with finite variance, then with increasing n the distribution of X becomes normal regardless of the distribution of random variables  pdf for normal distribution  is the mean and  is the standard deviation Hydrologic variables such as annual precipitation, annual average streamflow, or annual average pollutant loadings follow normal distribution

24 Standard Normal distribution  A standard normal distribution is a normal distribution with mean (  ) = 0 and standard deviation (  ) = 1  Normal distribution is transformed to standard normal distribution by using the following formula: z is called the standard normal variable

25 Lognormal distribution  If the pdf of X is skewed, it’s not normally distributed  If the pdf of Y = log (X) is normally distributed, then X is said to be lognormally distributed. Hydraulic conductivity, distribution of raindrop sizes in storm follow lognormal distribution.

26 Extreme value (EV) distributions  Extreme values – maximum or minimum values of sets of data  Annual maximum discharge, annual minimum discharge  When the number of selected extreme values is large, the distribution converges to one of the three forms of EV distributions called Type I, II and III

27 EV type I (Gumbel) distribution  If M 1, M 2 …, M n be a set of daily rainfall or streamflow, and let X = max(Mi) be the maximum for the year. If M i are independent and identically distributed, then for large n, X has an extreme value type I or Gumbel distribution. Distribution of annual maximum streamflow follows an EV1 distribution

28 Pearson Type III  Named after the statistician Pearson, it is also called three-parameter gamma distribution. A lower bound is introduced through the third parameter (  ) It is also a skewed distribution first applied in hydrology for describing the pdf of annual maximum flows.

29 Log-Pearson Type III  If log X follows a Person Type III distribution, then X is said to have a log- Pearson Type III distribution

Method of Moments  For a given 2-parameter distribution, can usually write the distribution parameters as a function of the first two moments (e.g., mean and standard deviation)  From sample estimates of the mean and standard deviation, can determine distribution parameters

Intensity-Duration-Frequency Curves Linsley Fig. 5-7

Spatial Variability  So far we’ve only considered rainfall estimates at a particular climate station  Estimates are provided for various regions through rainfall atlases  The rainfall atlas for the Midwest is provided by the Midwestern Climate Center and the Illinois State Water Survey (Bulletin 71)

ISWS Bulletin 71  Point frequency analyses were conducted from daily data at 275 National Weather Service gages  Used an empirical graphical analysis to estimate 2, 5, 10, 25, 50 and 100-year recurrence intervals

ISWS Bulletin 71  Durations ranged from 1-hour to 10 days  For durations less than 1 day, standardized ratios were used: Duration (hours)Ratio (x-hr/24-hr)

ISWS Bulletin 71  Results presented as isohyetal maps for duration/frequency combinations  Divided the midwest into 10 homogeneous regions  Tables present regional averages for various duration/frequency combinations

Example: 2-yr, 1-hr rainfall

Example: 24-hr rainfall

TR-55: Runoff Statistical Model  Tool used to model watershed runoff  Intended to use peak Recurrence Interval  Not to be used for real rainfall  Should be thought as a statistical model  Uses SCS Runoff Equation

SCS Runoff Equation

Curve Numbers  Determined by hydrologic soil group (HSG), cover type, treatment, hydrologic condition, and antecedent runoff condition (ARC).  Another factor considered is whether impervious areas outlet directly to the drainage system (connected) or whether the flow spreads over pervious areas before entering the drainage system (unconnected).

Group A soils have low runoff potential and high infiltration rates even when thoroughly wetted. They consist chiefly of deep, well to excessively drained sand or gravel and have a high rate of water transmission (greater than 0.30 in/hr). Soil Texture: Sand, loamy sand, or sandy loam Group B soils have moderate infiltration rates when thoroughly wetted and consist chiefly of moderately deep to deep, moderately well to well drained soils with moderately fine to moderately coarse textures. These soils have a moderate rate of water transmission ( in/hr). Soil Texture: Silt loam or loam Group C soils have low infiltration rates when thoroughly wetted and consist chiefly of soils with a layer that impedes downward movement of water and soils with moderately fine to fine texture. These soils have a low rate of water transmission ( in/hr). Soil Texture: Sandy clay loam Group D soils have high runoff potential. They have very low infiltration rates when thoroughly wetted and consist chiefly of clay soils with a high swelling potential, soils with a permanent high water table, soils with a claypan or clay layer at or near the surface, and shallow soils over nearly impervious material. These soils have a very low rate of water transmission ( in/hr). Soil Texture: Clay loam, silty clay loam, sandy clay, silty clay, or clay Hydrologic Soil Groups In Madison

Time of Concentration and Travel Time  Travel Time (T t ) is the time it takes water to travel from one location to another in a watershed. It is typically a component of T c.  Time of Concentration (T c ) is the time for runoff to travel from the hydraulically most distance point of the watershed to a point of interest in the watershed.

Factors Affecting T t and T c  Surface roughness  Channel shape and flow patterns  Slope

Water Movers through a Watershed as:  Sheet flow  Shallow concentrated flow  Open channel flow, or  A combination of these. where: T t = travel time (hr) L = flow length (ft) V = average velocity (ft/s)

Sheet Flow  Shallow flow depth (< 0.1 ft) over plane surfaces  Only for flows up to 300 feet where: T t = travel time (hr) n = manning’s roughness coefficient (table 3-1) L = flow length (ft) P 2 = 2-year, 24-hour rainfall (in) s = slope of hydraulic grade line (land slope, ft/ft)

Shallow Concentrated Flow  After a maximum of 300 feet, sheet flow usually becomes shallow concentrated flow. where: T t = travel time (hr) L = flow length (ft) V = average velocity from Figure 3-1 (ft/s)

Open Channel Flow  Based upon Manning’s Equation where: V = average velocity (ft/s) r = hydraulic radius (ft) and is equal to area/wetted perimeter s = channel slope (ft/ft) n = Manning’s roughness coefficient for open channel flow Then plug V and L into this equation:

Example Segment AB: Sheet flow; dense grass; slope (s) = 0.01 ft/ft; and length (L) = 100 ft. Segment BC: Shallow concentrated flow; unpaved; s = 0.01 ft/ft; and L = 1,400 ft. Segment CD: Channel flow; Manning’s n =.05; flow area (a) = 27 ft2; wetted perimeter (pw) = 28.2 ft; s = ft/ft; and L = 7,300 ft.

Segment AB: Sheet flow; dense grass; slope (s) = 0.01 ft/ft; and length (L) = 100 ft.

Segment BC: Shallow concentrated flow; unpaved; s = 0.01 ft/ft; and L = 1,400 ft.

Segment CD: Channel flow; Manning’s n =.05; flow area (a) = 27 ft2; wetted perimeter (pw) = 28.2 ft; s = ft/ft; and L = 7,300 ft.

Graphical Peak Discharge Method  The peak discharge equations used is: where: q p = peak discharge (cfs) q u = unit peak discharge (csm/in) A m = drainage area (mi 2 ) Q = runoff (in) F p = pond and swamp adjustment factor

References Gupta, R.S., 1989, Hydrology & Hydraulic Systems: Prospect Heights, IL, Waveland Press, Inc., 739 p. Huff, F.A., and Angel, J.R., 1992, Rainfall Frequency Atlas of the Midwest: Illinois State Water Survey Bulletin 71, 148 p. Linsley, R.K., and Franzini, J.B., 1979, Water Resources Engineering: New York, NY, McGraw-Hill Book Company, 716 p. Viessman, W.J., Knapp, J.W., Lewis, G.L., and Harbaugh, T.E., 1977, Introduction to Hydrology, Second Edition: New York, N.Y., Harper & Row, Publishers Inc., 704 p.