Statistical Hydrology

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# Statistical Hydrology - PowerPoint PPT Presentation

Statistical Hydrology. Read Chapter 2 (McCuen 2004) for background review Supplementary materials: Parameter Estimation: (a) Method of Moments * Product Moments (covered in CIVL181) * L-Moments (b) Method of Likelihood (covered in CIVL181). Statistical Moments.

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Presentation Transcript
Statistical Hydrology
• Read Chapter 2 (McCuen 2004) for background review
• Supplementary materials:
• Parameter Estimation:

(a) Method of Moments

* Product Moments (covered in CIVL181)

* L-Moments

(b) Method of Likelihood (covered in CIVL181)

Statistical hydrology

Statistical Moments
• Product Moments: E[Xr]

(a) Sensitive to the presence of outliers;

(b) Accuracy deteriorates rapidly as the order of moment

increases

• Probability-Weighted Moments (PWM):

- Def: Mr,p,q = E{Xr [F(X)]p [1-F(X)]q }

- Especially attractive when the CDF, F(x), has a closed-

form expression

- Special cases

(a) Mr,0,0= E[Xr]

(b) M1,0,q = aq ; M1,p,0 = bp

Statistical hydrology

Relations between moments and parameters of selected distribution models (Tung et al. 2006)

Statistical hydrology

Statistical Moments (2)
• L-Moments:

A linear combination of order statistics

• Specifically, for the first 4 L-moments:

Statistical hydrology

Analogy Between L- and Product-Moments

Product Moments L-Moments

m (mean) l1 (mean)

s (stdev) l2 (L-std)

Cv = s/mt2 = l2/l1 (L-Cv)

Cs = m3/s3t3 = l3/l2 (L-Cs), | t3|<1

Ck = m4/s4t4 = l4/l2 (L-Ck), -0.25<t4<1

Statistical hydrology

L-moments & Distribution Parameter Relations

From “Frequency Analysis of Extreme

Events,” Chapter 8 in Handbook of Hydrology,

By Stedinger, Vogel, and Foufoula-Georgiou,

McGraw-Hill Book Company, New York,

1993

Statistical hydrology

L-Moment Ratio Diagram

Statistical hydrology

Statistical Moments (3)
• Relations between L-moments and b-moments:

Statistical hydrology

Sample Estimates of Statistical Moments

Product Moments

L- Moments

Statistical hydrology

Example-1(a)

Statistical hydrology

Example-1(b)

Statistical hydrology

Example-1(c)

Statistical hydrology

Types of Hydrologic Data Series

Statistical hydrology

Return Period (Recurrent Interval)
• The return period of an event is the time between occurrences of the events. The events can be those whose magnitude exceeds or equals to a certain magnitude of interest, i.e, XxT
• In general, the actual return period (or inter-arrival time) between the occurrences of an event could vary. The ‘return period’ commonly used in engineering is the expected (or long-term averaged) inter-arrival time between events.
• Return period depends on the time scale of the data. E.g., using annual maximum (or min.) series, the return period is year.
• Return period T = 1/Pr[XxT]
• To avoid misconception and mis-interpretation of an event, e.g., 50-year flood, it is advisable to use “flood event with 1-in-50 chance being exceeded annually”.

Statistical hydrology

Distributions Commonly Used in Hydrologic Frequency Analysis
• Normal Family – Normal, Log-normal
• Gamma Family – Pearson type III, Log-Pearson type III
• Extreme Value – Type I (for max. or min) - Gumbel

Type II (for min) – Weibull

Generalized Extreme Value

Statistical hydrology

Graphical Frequency Analysis
• Data are arranged in ascending order of magnitude,

x(n) x(n-1) ··· x(2) x(1)

• Compute the plotting position for each observed data

Weibull plotting position formula:

P[X≤x(m)]=m/(n+1)

See other formulas

• Plot x(m) vs. m/(n+1) on a suitable probability paper. (Commercially available are normal, log-normal, and Gumbel probability papers)
• Extrapolate or interpolate frequency curve graphically.

Statistical hydrology

Plotting Position Formulas

Statistical hydrology

Example-2 (Graphical Procedure)

Statistical hydrology

Normal Probability Plot

Statistical hydrology

Log-normal Probability Plot

Statistical hydrology

Gumbel Probability Plot

Statistical hydrology

Frequency Factor Method

Statistical hydrology

Issues in Frequency Analysis
• Selection of distribution and parameter estimation
• Treatment of zero flows
• Detection and treatment of outliers (high or low)
• Regional frequency analysis
• Use of historical and paleo data

Statistical hydrology

Example (Analytical Procedure)

Statistical hydrology

Uncertainty of Sample Quantiles

Statistical hydrology

(1-a)% CI for Sample Quantiles

Statistical hydrology

Example (C.I.)

Statistical hydrology

Hydrologic Risk
• For a T-year event, P(XxT)=1/T. If xT is determined from an annual maximum series, 1/T is the probability of exceedance for the hydrologic event in any one year.
• Assume independence of occurrence of events and the hydraulic structure is design for an event of T-year return period.

Failure probability over an n-year service period, pf, is

pf = 1-(1-1/T)n (using Binomial distribution)

or pf = 1-exp(-n/T) (using Poisson distribution)

• Types of problem:

(a) Given T, n, find pf

(b) Specify pf & T, find n

(c) Specify pf & n, find T

Statistical hydrology