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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)

- Parameter Estimation:

Statistical hydrology

Statistical Moments

- Product Moments: E[Xr]
- Disadvantages:

(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) distribution models (Tung et al. 2006)

- L-Moments:
A linear combination of order statistics

- Specifically, for the first 4 L-moments:

Statistical hydrology

Graphical Representation of L-Moments distribution models (Tung et al. 2006)

Statistical hydrology

Analogy Between L- and Product-Moments distribution models (Tung et al. 2006)

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 distribution models (Tung et al. 2006)

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

Generalized Logistic Distribution distribution models (Tung et al. 2006)

Statistical hydrology

L-Moment Ratio Diagram distribution models (Tung et al. 2006)

Statistical hydrology

Statistical Moments (3) distribution models (Tung et al. 2006)

- Relations between L-moments and b-moments:

Statistical hydrology

Sample Estimates of Statistical Moments distribution models (Tung et al. 2006)

Product Moments

L- Moments

Statistical hydrology

Example-1(a) distribution models (Tung et al. 2006)

Statistical hydrology

Example-1(b) distribution models (Tung et al. 2006)

Statistical hydrology

Example-1(c) distribution models (Tung et al. 2006)

Statistical hydrology

Types of Hydrologic Data Series distribution models (Tung et al. 2006)

Statistical hydrology

Return Period (Recurrent Interval) distribution models (Tung et al. 2006)

- 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, XxT
- 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[XxT]
- 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 distribution models (Tung et al. 2006)

- 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 distribution models (Tung et al. 2006)

- 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 distribution models (Tung et al. 2006)

Statistical hydrology

Example-2 (Graphical Procedure) distribution models (Tung et al. 2006)

Statistical hydrology

Normal Probability Plot distribution models (Tung et al. 2006)

Statistical hydrology

Log-normal Probability Plot distribution models (Tung et al. 2006)

Statistical hydrology

Gumbel Probability Plot distribution models (Tung et al. 2006)

Statistical hydrology

Frequency Factor Method distribution models (Tung et al. 2006)

Statistical hydrology

Frequency Factor for Various Dist’ns (1) distribution models (Tung et al. 2006)

Statistical hydrology

Frequency Factor for Various Dist’ns (2) distribution models (Tung et al. 2006)

Statistical hydrology

K distribution models (Tung et al. 2006)T for Log-Pearson III Distribution

Statistical hydrology

Analytical Frequency Analysis Procedure distribution models (Tung et al. 2006)

Statistical hydrology

Issues in Frequency Analysis distribution models (Tung et al. 2006)

- 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) distribution models (Tung et al. 2006)

Statistical hydrology

Confidence of Derived Frequency Relation distribution models (Tung et al. 2006)

Statistical hydrology

Uncertainty of Sample Quantiles distribution models (Tung et al. 2006)

Statistical hydrology

Approaches to Construct Confidence Interval distribution models (Tung et al. 2006)

Statistical hydrology

Standard Error of Sample Quantiles distribution models (Tung et al. 2006)

Statistical hydrology

(1- distribution models (Tung et al. 2006)a)% CI for Sample Quantiles

Statistical hydrology

Example (C.I.) distribution models (Tung et al. 2006)

Statistical hydrology

Hydrologic Risk distribution models (Tung et al. 2006)

- For a T-year event, P(XxT)=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

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