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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). Statistical Moments.

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

Statistical hydrology


Statistical moments 2
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
Graphical Representation of L-Moments distribution models (Tung et al. 2006)

Statistical hydrology


Analogy between l and product moments
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
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
Generalized Logistic Distribution distribution models (Tung et al. 2006)

Statistical hydrology


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

Statistical hydrology


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

  • Relations between L-moments and b-moments:

Statistical hydrology


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

Product Moments

L- Moments

Statistical hydrology


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

Statistical hydrology


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

Statistical hydrology


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

Statistical hydrology


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

Statistical hydrology


Return period recurrent interval
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, 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
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
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
Plotting Position Formulas distribution models (Tung et al. 2006)

Statistical hydrology


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

Statistical hydrology


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

Statistical hydrology


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

Statistical hydrology


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

Statistical hydrology


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

Statistical hydrology


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

Statistical hydrology


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

Statistical hydrology


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

Statistical hydrology


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

Statistical hydrology


Issues in frequency analysis
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
Example (Analytical Procedure) distribution models (Tung et al. 2006)

Statistical hydrology


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

Statistical hydrology


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

Statistical hydrology


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

Statistical hydrology


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

Statistical hydrology


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

Statistical hydrology


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

Statistical hydrology


Hydrologic risk
Hydrologic Risk distribution models (Tung et al. 2006)

  • 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


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