6.1  CONCEPTS OF STATISTICS AND PROBABILITY

[Frequency Analysis]   [Low-flow Frequency]   [Questions]   [Problems]   [References]   •   [Top]

Frequency analysis uses random variables and probability distributions. A random variable follows a certain probability distribution. A probability distribution is a function that expresses in mathematical terms the relative chance of occurrence of each of all possible outcomes of the random variable. In statistical notation, P (X = x₁ ) is the probability P that the random variable X takes on the outcome x₁. A shorter notation is P (x₁).

An example of random variable and probability distribution is shown in Fig. 6- 1. This is a discrete probability distribution because the possible outcomes have been arranged into groups (or classes). The random variable is discharge Q; the possible outcomes are seven discharge classes, from 0-100 m³/ s to 600-700 m³/s. In Fig. 6-1 , the probability that Q is in the class 100-200 m³/s is 0.25. The sum of probabilities of all possible outcomes is equal to 1.

A cumulative discrete distribution, corresponding to the discrete probability distribution of Fig. 6-1, is shown in Fig. 6-2. In this figure, the probability that Q is in a class less than or equal to the 100-200 class is 0.40. The maximum value of probability of the cumulative distribution is 1.

Figure 6-1  Discrete Probability Distribution.

Figure 6-2  Cumulative Discrete Probability Distribution.

Properties of Statistical Distributions

The properties of statistical distributions are described by the following measures:

1. central tendency,

2. variability, and

3. skewness.

Statistical distributions are described in terms of moments. The first moment describes central tendency, the second moment describes variability, and the third moment describes skewness. Higher order moments are possible but are seldom used in practical applications.

The first moment about the origin is the arithmetic mean. or mean. It expresses the distance from the origin to the centroid of the distribution (Fig. 6-3(a)):

1    n x̄ = __  Σ xi       n   i=1

(6-1)

in which x̄ is the mean, x_i is the random variable, and n is the number of values.

The geometric mean is the nth root of the product of n terms:

x̄ = ( x1 x2 x3 . . . xn ) (1/n)

(6-2)

The logarithm of the geometric mean is the mean of the logarithms of the individual values. The geometric mean is to the lognormal probability distribution what the arithmetic mean is to the normal probability distribution.

The median is the value of the variable that divides the probability distribution into two equal portions (or areas) (Fig. 6-3(b)). For certain skewed distributions (Le. one with third moment other than zero), the median is a better indication of central tendency than the mean. Another measure of central tendency is the mode, defined as the value of the variable that occurs most frequently (Fig. 6-3(c)).

Statistical moments can be defined about axes other than the origin. The second moment about the mean is the variance, defined as

(a) Mean x̄

(b) Median xmed

(c) Mode xmod

(d) Standard deviation s

(e) Skew coefficient C s Figure 6-3  Properties of statistical distributions: (a) Mean x̄ (b) Median xmed; (c) Mode xmod; (d) Standard deviation s; (e) Skew coefficient C s

1       n s2 = ______  Σ( xi - x̄ )2         n - 1   i=1

(6-3)

in which s² is the variance. The square root of the variance, s, is the standard deviation. The variance coefficient (or coefficient of variation) is defined as

s Cv = ___           x̄

(6-4)

The standard deviation and variance coefficient are useful in comparing relative variability among distributions. The larger the standard deviation and variance coefficient, the larger the spread of the distribution (Fig. 6-3(d)).

The third moment about the mean is the skewness, defined as follows:

1             n a = _____________  Σ( xi - x̄ )3        (n - 1)(n - 2)  i=1

(6-5)

in which a is the skewness. The skew coefficient is defined as

a Cs = ___           s3

(6-6)

For symmetrical distributions, the skewness is 0 and C_s = O. For right skewness (distributions with the long tail to the right), C_s > 0; for left skewness (long tail to the left), C_s < 0 (Fig. 6-3(e)).

Another measure of skewness is Pearson's skewness defined as the ratio of the difference between mean and mode to the standard deviation.

Example 6-1. Calculate the mean, standard deviation, and skew coefficient for the following flood series: 4580, 3490, 7260,9350, 2510,3720,4070, 5400, 6220, 4350, and 5930 m3/ s. The calculations are shown in Table 6-1. Column 1 shows the year and Col. 2 shows the annual maximum flows. The mean (Eq. 6-1) is calculated by summing up Col. 2 and dividing the sum by n = 11. This results in x̄ = 51 71 m3/ s. Column 3 shows the flow deviations from the mean, xi - x̄. Column 4 shows the square of the flow deviations, ( xi - x̄ )2. The variance (Eq. 6-3) is calculated by summing up Col. 4 and dividing the sum by (n - 1)= 10. This results in: s2 = 3,780,449 m6/ s2. The square root of the variance is the standard deviation: s = 1944 m3/ s. The variance coefficient (Eq. 6-4) is Cv = 0.376. Column 5 shows the cube of the flow deviations, ( xi - x̄ )3. The skewness (Eq. 6-5) is calculated by summing up Col. 5 and multiplying the sum by n /[(n - l)(n - 2)]= 11/90. This results in a = 6,717,359,675 m9/ s3. The skew coefficient (Eq. 6-6) is equal to the skewness divided by the cube of the standard deviation. This results in Cs = 0.914. Table 6-1 Calculation of Mean, Standard Deviation, and Skew Coefficient: Example 6-1. (1) (2) (3) (4) (5) Year Peak flow (m3/s) ( xi - x̄ ) (m3/s) ( xi - x̄ )2 (m6/s2) ( xi - x̄ )3 (m9/s3) 1 4,580 -591 349,281 -206,425,071 2 3,490 -1,681 2,825,761 -4,750,104,241 3 7,260 2,089 4,363,921 9,116,230,969 4 9,350 4,179 17,464,041 72,982,227,340 5 2,510 -2,661 7,080,921 -18,842,330,780 6 3,720 -1,451 2,105,401 -3,054,936,851 7 4,070 -1,101 1,212,201 -1,334,633,301 8 5,400 229 52,441 12,008,989 9 6,220 1,049 1,100,401 1,154,320,649 10 4,350 -821 674,041 -553,387,661 11 5,930 759 576,081 437,245,479 Sum 56,880 37,804,491 54,960,215,521

Continuous Probability Distributions

A continuous probability distribution is referred to as a probability density function (PDF). A PDF is an equation relating probability, random variable, and parameters of the distribution. Selected PDFs useful in engineering hydrology are described in this section.

Normal Distribution.

The normal distribution is a symmetrical, bell-shaped PDF also known as the Gaussian distribution, or the natural law of errors. It has two parameters: the mean, μ, and the standard deviation, σ, of the population. In practical applications, the mean x̄ and the standard deviation s derived from sample data are substituted for μ and σ. The PDF of the normal distribution is

1 f(x) = _____  e-(x-μ)2/(2σ2)          σ√2π

(6-7)

in which x is the random variable and f(x) is the continuous probability.

By means of the transformation

x - μ z = _________           σ

(6-8)

the normal distribution can be converted into a one-parameter distribution, as follows:

1 f(z) = _____  e-x2/2          √2π

(6-9)

in which z is the standard unit, which is normally distributed with zero mean and unit standard deviation.

From Eq. 6-8,

x = μ + zσ

(6-10)

in which z, the standard unit, is the frequency factor of the normal distribution. In general, the frequency factor of a statistical distribution is referred to as K.

A cumulative density function (CDF) can be derived by integrating the probability density function. From Eq. 6-9, integration leads to

1       z F(z) = _____  ∫  e-u2/2du            √2π  -∞

(6-11)

in which F(z) denotes cumulative probability and u is a dummy variable of integration. The distribution is symmetrical with respect to the origin; therefore, only half of the distribution needs to be evaluated. Table A-S (Appendix A) shows values of F(z) versus z, in which F(z) is integrated from the origin to z.

Example 6-2. The annual maximum flows of a certain stream have been found to be nonnally distributed, with mean 90 m3/s and standard deviation 30 m3/s. Calcuiate the probability that a flow larger that 150 m3/s will occur. To enter Table A-5, it is necessary to calculate the standard unit. For a flow of 150 m3/s, the standard unit (Eq. 6-8) is: z = (150 - 90)/ 30 = 2. This means that the flow of 150 m3/s is located two standard deviations to the right of the mean (had z been negative, the flow would have been located to the left of the mean). In Table A-5, for z = 2, F(z) = 0.4772. This value is the cumulative probability measured from z = 0 to z = 2, i.e. , from the mean (90 m3/s) to the value being considered (150 m3/s). Because the normal distribution is symmetrical with respect to the origin, the cumulative probability measured from z = - ∞ to z = 0, is 0.5 . Therefore, the cumulative probability measured from z = -∞ to z = 2, is F(z) = 0.5 + 0.4772 = 0.9772. This is the probability that the flow is less than 150 m3/ s. To find the probability that the flow is larger than 150 m3/s, the complementary cumulative probability is calculated: G(z)= 1 - F(z) = 0.0228. Therefore, there is a (0.0228 X 100) = 2.28% chance that the annual maximum flow for the given stream will be larger than 150 m3/s.

Lognormal Distribution

For certain natural phenomena, values of random variables do not follow a normal distribution, but their logarithms do. In this case, a suitable PDF can be obtained by substituting y for x in the equation for the nonnal distribution, Eq. 6-7, in which y = ln x. The parameters of the lognonnal distribution are the mean and standard deviation of y: μ_y and σ_y.

Gamma Distribution.

The gamma distribution is used in many applications of engineering hydrology. The PDF of the gamma distribution is the following:

xϒ - 1 e-x/β f(x) = ____________             βϒΓ(ϒ)

(6-12)

for 0 < x < ∞, β > 0, and ϒ> 0. The parameter r is known as the shape parameter, since it most influences the peakedness of the distribution, while the parameter β is called the scale parameter, since most of its influence is on the spread of the distribution [4].

The mean of the gamma distribution is βϒ,the variance is β²ϒ, and the skewness is 2/(ϒ)^1/2. The term Γ(ϒ) = (ϒ - 1)!, where ϒ is a positive integer,is an important definite integral referred to asthe gamma function. defined as follows:

∞ Γ(ϒ) = ∫ xϒ - 1e-xdx            0

(6-13)

Pearson Distributions.

Pearson [24] has derived a series of probability functions to fit virtually any distribution. These functions have been widely used in practical statistics to define the shape of many distribution curves. The general PDF of the Pearson distributions is the following [6]:

x              ∫ [( a + x )/ ( b0 + b1x + b2x2 )]dx            -∞ f(x) = e

(6-14)

in which a, b₀ , b₁, b₂ are constants. The criterion for determining the type of distribution is k, defined as follows:

β1( β2 + 3)2 k = _______________________________          4 ( 4 β2 - 3β1)(2β2 - 3β1 - 6)

(6-15)

in which β₁ μ²₃ / μ³₂ and β₂ μ² / μ²₂, with μ₂, μ₃, and μ₄ being the second, third, and fourth moments about the mean. With μ₃ = 0 (i.e., zero skewness), β₁ = 0, k = 0, and the Pearson distribution reduces to the normal distribution.

The Pearson Type III distribution has been widely used in flood frequency analysis. In the Pearson Type III distribution, k = ∞, which implies that 2β₂ = ( 3β₃ + 6). This is a three-parameter skewed distribution with the following PDF:

(x - x0)ϒ - 1 e-(x - x0) /β f(x) = _________________________                        βϒΓ(ϒ)

(6-16)

and parameters β, ϒ, and x₀. For x₀ = 0, the Pearson Type III distribution reduces to the gamma distribution (Eq. 6-12). For ϒ = 1, the Pearson Type III distribution reduces to the exponential distribution, with the following PDF:

1 f(x) = (___) e- (x - x0)/β             β

(6-17)

The mean of the Pearson Type III distribution is x₀ + βϒ, the variance is β²ϒ, and the skewness is 2/( ϒ)^1/2.

Extreme Value Distributions.

The extreme value distributions Types I, II, and III are based on the theory of extreme values. Frechet (on Type II) in 1927 [8] and Fisher and Tippett (on Types I and III) in 1928 [8] independently studied the statistical distribution of extreme values. Extreme value theory implies that if a random variable Q,is the maximum in a sample of size n from some population of x values, then, provided n is sufficiently large, the distribution of Q is one of three asymptotic types (1, II, or III), depending on the distribution of x.

The extreme value distributions can be combined into one and expressed as a general extreme value (GEV) distribution [23]. The cumulative density function of the GEV distribution is:

F ( x ) = e-[ 1 - k ( x - u ) / α]1/k

(6-18)

in which k, u and α are parameters. The parameter k defines the type of distribution, u is a location parameter, and α is a scale parameter. For k = 0, the GEV distribution reduces to the extreme value Type I (EVl), or Gumbel, distribution. For k < 0, the GEV distribution is the extreme value Type II (EV2), or Frechet, distribution. For k > 0, the GEV distribution is the extreme value Type III (EV3), or Weibull, distribution. The GEV distribution is useful in applications where an extreme value distribution is being considered but its type is not known a priori.

Gumbel [ 13 - 16 ] has fitted the extreme value Type I distribution to long records of river flows from many countries. The cumulative density function (CDF) of the Gumbel distribution is the following double exponential function:

F ( x ) = e-e-y

(6-19)

in which y = (x - u)/α is the Gumbel (reduced) variate.

The mean ȳ_n and standard deviation σ_n of the Gumbel variate are functions of record length n. Values of ȳ_n and σ_n as a function of n are given in Table A-8 (Appendix A). When the record length approaches ∞, the mean ȳ_n approaches the value of the Euler constant (0.5772) [26] and the standard deviation σ_n approaches the value π/ √6. The skew coefficient of the Gumbel distribution is 1.14.

The extreme value Type II distribution is also known as the log Gumbel. Its cumulative density function is

F ( x ) = e-y1/k

(6-18)

for k < 0.

The extreme value Type III distribution has the same CDF as the Type II, but in this case k > 0. As k approaches 0, the EV2 and EV3 distributions converge to the EV1 distribution.

6.2  FREQUENCY ANALYSIS

[Low-flow Frequency]   [Questions]   [Problems]   [References]   •   [Top]   [Concepts of Statistics]

Flood frequency analysis refers to the application of frequency analysis to study the occurrence of floods. Historically, many probability distributions have been used for this purpose. The normal distribution was first used by Horton [19] in 1913, and shortly thereafter by Fuller [11]. Hazen [17] used the lognormal distribution to reduce skewness, whereas Foster [9] preferred to use the skewed Pearson distributions.

The logarithmic version of the Pearson Type III distribution, i.e. , the log Pearson III, has been endorsed by the U.S. Interagency Advisory Committee on Water Data for general use in the United States [28]. The Gumbel distribution (extreme value Type I, or EVl) is also widely used in the United States and throughout the world. The log Pearson III and Gumbel metho, are described in this section.

Selection of Data Series

The complete record of streamflows at a given gaging station is called the complete duration series. To perform a flood frequency analysis, it is necessary to select a flood series, i.e., a sample of flood events extracted from the complete duration series.

There are two types of flood series: (1) the partial' duration series and (2) the extreme value series. The partial duration (or peaks-over-a-threshold (POT) [23] series consists of floods whose magnitude is greater than a certain base value. When the base value is such that the number of events in the series is equal to the number of years of record, the series is called an annual exceedence series.

In the extreme value series, every year of record contributes one value to the extreme value series, either the maximum value (as in the case of flood frequency analysis) or the minimum value (as in the case of low-flow frequency analysis). The former is the annual maxima series; the latter is the annual minima series.

The annual exceedence series takes into account all extreme events above a certain base value, regardless of when they occurred. However, the annual maxima series considers only one extreme event per yearly period. The difference between the two series is likely to be more marked for short records in which the second largest annual events may strongly influence the character of the annual exceedence series. In practice, the annual exceedence series is used for frequency analyses involving short return periods, ranging from 2 to 10 y. For longer return periods the difference between annual exceedence and annual maxima series is small. The annual maxima series is used for return periods ranging from 10 to 100 y and more.

Return Period, Frequency, and Risk

The time elapsed between successive peak flows exceeding a certain flow Q is a random variable whose mean value is called the return period T (or recurrence interval) of the flow Q. The relationship between probability and return period is the following:

1 P(Q) = ___              T

(6-21)

in which P(Q) is the probability of exceedence of Q, or frequency. The terms frequency and return period are often used interchangeably, although strictly speaking, frequency is the reciprocal of return period. A frequency of 1/T, or one in T years, corresponds to a return period of T years.

The probability of nonexceedence P(Q̄) is the complementary probability of the probability of exceedence P(Q), defined as

1 P(Q̄) = 1 - P(Q) = 1 - ___                                    T

(6-22)

The probability of nonexceedence in n successive years is

1     n P(Q̄) = ( 1 - ___ )                      T

(6-23)

Therefore, the probability, or risk, that Q will occur at least once in n successive years is

1     n R = 1 - P(Q̄) = ( 1 - ___ )                                  T

(6-24)

Plotting Positions

Frequency distributions are plotted using probability papers. One of the scales on a probability paper is a probability scale; the other is either an arithmetic or logarithmic scale. Normal and extreme value probability distributions are most often used in probability papers.

An arithmetic probability paper has a normal probability scale and an arithmetic scale. This type of paper is used for plotting normal and Pearson distributions. A log probability paper has a normal probability scale and a logarithmic scale and is used for plotting lognormal and log Pearson distributions. An extreme value probability paper has an extreme value scale and an arithmetic scale and is used for plotting extreme value distributions.

Data fitting a normal distribution plot as a straight line on arithmetic probability paper. Likewise, data fitting a lognormal distribution plot as a straight line on log probability paper, and data fitting the Gumbel distribution plot as a straight line on extreme value probability paper.

For plotting purposes, the probability of an individual event can be obtained directly from the flood series. For a series of n annual maxima, the following ratio holds:

x̄          m ___ = _______ N        n + 1

(6-25)

in which x̄ = mean number of exceedences; N = number of trials; n = number of values in the series; and m = the rank of descending values, with largest equal to 1.

For example, if n = 79, the second largest value in the series (m = 2) will be exceeded twice on the average (x̄ = 2) in 80 trials (N = 80). Likewise, the largest value in the series (m = 1) will be exceeded once on the average (x̄ = 1) after 80 trials (N = 80). Since return period T is associated with x̄ = 1, Eq. 6-25 can be expressed as follows:

1                 m ___ = P = _______ T                n + 1

(6-26)

in which P = exceedence probability.

Equation 6-26 is known as the Weibull plotting position formula. This equation is commonly used in hydrologic applications, particularly for computing plotting positions for unspecified distributions [1]. A general plotting position formula is of the following form [12]:

1                   m - a ___ = P = _____________ T                n + 1 - 2a

(6-27)

in which a = parameter. Cunnane [7] performed a detailed study of the accuracy of different plotting position formulas and concluded that the Blom formula [3], with a = 0.375 in Eq. 6-27, is most appropriate for the normal distribution, whereas the Gringorten formula, with a = 0.44, should be used in connection with the Gumbel distribution. According to Cunnane, the Weibull formula, for which a = 0, is most appropriate for a uniform distribution.

In computing plotting positions, when the ranking of values is in descending order (from highest to lowest), P is the probability of exceedence, or the probability of a value being greater than or equal to the ranked value. When the ranking of values is in ascending order (from lowest to highest), P is the probability of nonexceedence, or the probability of a value being less than or equal to the ranked value. The computation of plotting positions is illustrated by the following example.

Example 6-3. Use Eq. 6-26 to calculate the plotting positions for the flood series (annual maxima) shown in Table 6-2, Col. 2. The solution is shown in T ble 6-2, Cols. 3-5. Column 3 shows the ranked values, from highest to lowest. Colum (4 hows the rank of each value, from 1 to 16 (n = 16), with the highest value ranked as 1 and the lowest value ranked as 16. Column 5 shows the probability calculated by Eq. 6-26 (expressed in percent). Because the ranking was done in descending order, Col. 5 shows the probability that a value of flood discharge will be greater than or equal to the ranked value. To illustrate, there is a 5.88% probability that a value of flood discharge will be greater than or equal to 3320 m3/s. Conversely, there is a 94.12% probability that the value of flood discharge will be greater than or equal to 690 m3/s. Column 6 shows the return period calculated by Eq. 6-26.  Table 6-2 Computation of Plotting Positions: Example 6-3. (1) (2) (3) (4) (5) (6) Year Annual Flood (m3/s) Ranked Values (m3/s) Rank Probability (percent) Return Period (y) 1972 2520 3320 1 5.88 17.00 1973 1850 3170 2 11.76 8.50 1974 750 2520 3 17.65 5.67 1975 1100 2160 4 23.53 4.25 1976 1380 1950 5 29.41 3.40 1977 1910 1910 6 35.29 2.83 1978 3170 1850 7 41.18 2.43 1979 1200 1730 8 47.06 2.13 1980 820 1480 9 52.94 1.89 1981 690 1380 10 58.82 1.70 1982 1240 1240 11 64.71 1.55 1983 1730 1200 12 70.59 1.42 1984 1950 1100 13 76.47 1.31 1985 2160 820 14 82.35 1.21 1986 3320 750 15 88.24 1.13 1987 1480 690 16 94.12 1.06

Curve Fitting

Once the data have been plotted on probability paper, the next step is to fit a curve through the plotted points. Curve fitting can be accomplished by any of the following methods: (1) graphical, (2) least square, (3) moments, and (4) maximum likelihood. The graphical method consists of fitting a function visually to the data. This method, however, has the disadvantage that the results are highly dependent on the skills of the person doing the fitting_ A more consistent procedure is to use either the least square, moments, or maximum likelihood methods.

In the least square method, the sum of the squares of the differences between observed data and fitted values is minimized. The minimization condition leads to a set of m normal equations, where m is the nuxpber of parameters to be estimated. The simultaneous solution of the normal equations leads to the parameters describing the fitting (Chapter 7).

To apply the method of moments, it is first necessary to select a distribution; then, the moments of the distribution are calculated based on the data. The method provides an exact theoretical fitting, but the accuracy is substantially affected by errors in the tail of the distribution (i.e., events of long return period). A disadvantage of the method is the uncertainty regarding the adequacy of the chosen probability distribution.

In the method of maximum likelihood, the distribution parameters are estimated in such a way that the product of probabilities (i.e., the joint probability, or likelihood) is maximized. This is obtained in a similar manner to the least square method by partially differentiating the likelihood with respect to each of the parameters and equating the result to zero.

The four fitting methods can be rated in ascending order of effectiveness: graphical, least square, moments, and maximum likelihood. The latter, however, is somewhat more difficult to apply [6,21] . In practice, the method of moments is the most commonly used curve fitting method (see, for instance, the log Pearson III and Gumbel methods described later in this section).

Frequency Factors

Any value of a random variable may be represented in the following form:

x = x̄ + Δx

(6-28)

in which x = value of random variable; x̄ = mean of the distribution, and Δx = departure from the mean, a function of return period and statistical properties of the distribution. This departure from the mean can be expressed in terms of the product of the standard deviation s and a frequency factor K such that Δx = Ks. The frequency factor is a function of return period and probability distribution to be used in the analysis. Therefore, Eq. 6-28 can be written in the following form:

x = x̄ + Ks

(6-29)

or, alternatively,

x ___ = 1 + KCv x̄

(6-28)

in which C_v = variance coefficient.

Equation 6-29 was proposed by Chow [5] as a general equation for hydrologic frequency analysis. For any probability distribution, a relationship can be determined between frequency factor and return period. This relationship can be expressed in analytical terms, in the form of tables, or by K-T curves. In using the procedure, the statistical parameters are first determined from the analysis of the flood series. For a given return period, the frequency factor is determined from the curves or tables and the flood magnitude computed by Eq. 6-29.

Log Pearson III Method

The log Pearson III method of flood frequency analysis is described in Bulletin 17B: Guidelines for Determining Flood Flow Frequency, published by the U.S. Interagency Advisory Committee on Water Data, Reston, Virginia [28].

Methodology.

To apply the method, the following steps are necessary:

1. Assemble the annual flood series x_i.

2. Calculate the logarithms of the annual flood series:

   yi = log xi

   (6-31)

3. Calculate the mean ȳ, standard deviation s_y, and skew coefficient C_sy of the logarithms y_i.

4. Calculate the logarithms of the flood discharges, log Q_j for each of several chosen probability levels P_j using the following frequency formula:

   log Qj = ȳ + Kjsy

   (6-32)

   in which K_j is the frequency factor, a function of the probability P_j and the skew coefficient C_sy Table A-6 (Appendix A) shows frequency factors K for ten selected probability levels in the range 0.5 to 95 percent (and corresponding return periods in the range 200 to 1.05 y) and skew coefficients in the range -3.0 to 3.0.

5. Calculate the flood discharges Q_j for each P_j probability level (or return period T_j) by taking the antilogarithms of the log Q_j values.

6. Plot the flood discharges Q_j against probabilities P_j on log probability paper, with discharges in the log scale and probabilities in the probability scale. The log Pearson III fit to the data is obtained by linking the points with a smooth curve. For C_sy = 0, the curve reduces to a straight line.

The procedure is illustrated by the following example.

Example 6-4. Apply the log Pearson III method to the flood series of Example 6-3. Plot the results on log probability paper along with the plotting positions calculated in Example 6-3. The discharge values, Table 6-2, Col. 2, are converted to logarithms, and the mean, standard deviation, and skew coefficient of the logarithms calculated. This results in ȳ = 3.187, sy = 0.207, and Csy = -0.116. The computations are summarized in Table 6-3. Column 1 shows selected return. periods, and Col. 2 shows the associated probabilities in percent (exceedence probability). Column 3 shows the frequency factors K for Csy = -0.116 and for each return period or exceedence probability. Values in Col. 3 are obtained from Table A-6 by linear interpolation. Column 4 shows the logarithms of the flood discharges calculated by Eq. 6-32, and Col. 5 shows the flood discharges. The flood  Table 6-3 Log Pearson III Method: Example 6-4. (1) (2) (3) (4) (5) Return period T (y) Probability P (percent) Frequency factor K (for Csy = -0.116 yi = log Q xi = Q m3/s 1.05 95 -1.677 2.840 692 1.11 90 -1.293 2.919 830 1.25 80 -0.835 3.014 1033 2 50 0.019 3.191 1552 5 20 0.847 3.362 2301 10 10 1.268 3.449 2812 25 4 1.710 3.541 3475 50 2 1.991 3.599 3972 100 1 2.240 3.651 4477 200 0.5 2.467 3.698 4989 discharges are plotted against corresponding probabilities, as shown by the solid line of Fig. 6-4, to obtain the log Pearson III fit to the data. The plotting positions calculated in Example 6-3 are also shown for comparison.

Regional Skew Characteristics

The skew coefficient of the flood series (i.e. the station skew) is sensitive to extreme events. The overall accuracy of the method is improved by using a weighted value of skew in lieu of the station skew. First, a value of regional skew is obtained, and the weighted skew is calculated by weighing station and regional skews in inverse proportion to their mean square errors (MSE). The formula for weighted skew is the following:

Figure 6-4  Log Pearson III fit: Example 6-4.

(MSE)svCsy + (MSE)syCsr Csw = _____________________________                   (MSE)sr + (MSE)sy

(6-33)

in which C_sw = weighted skew; C_sy = station skew; C_sr = regional skew; (MSE)_sy = mean square error of the station skew; and (MSE)_sr = mean square error of the regional skew.

To develop a value of regional skew, it is necessary to assemble data from at least 40 stations or, alternatively, all stations within a 160-km radius. The stations should have at least 25 y of record. In certain cases, the paucity of data may require a relaxation of these criteria. The procedure includes analysis by three methods: (1) skew isolines map, (2) skew prediction equation, and (3) statistics of station skews.

To develop a skew isolines map, each station skew is plotted on a map at the centroid of its catchment area, and the plotted data are examined to identify any geographic or topographic trends. If a pattern is evident, isolines (lines of equal skew) are drawn and the MSE is computed. The MSE is the mean of the square of the differences between observed skews and isoline skews. If no pattern is evident, an isoline map cannot be developed, and this method is not considered further.

In the second method, a prediction equation is used to relate station skew to catchment properties and climatological variables. The MSE is the mean of the square of the differences between observed and predicted skews.

In the third method, the mean and variance of the station skews are calculated. In some cases, the variability of runoff may be such that all the stations may not be hydrologically homogeneous. If this is the case, the values of about 20 stations can be used to calculate the mean and variance of the data.

Of the three methods, the one providing the most accurate estimate of skew coefficient is selected. First a comparison of the MSEs from the isolines map and prediction equations is made. Then the smaller MSE is compared to the variance of the data. If the smaller MSE is significantly smaller than the variance, it should be used in Eq. 6-33 as (MSE)_sr. If this is not the case, the variance should be used as (MSE)_sr, with the mean of the station skews used as regional skew (C_sr).

In the absence of regional skew studies, generalized values of regional skew for use in Eq. 6-33 can be obtained from Fig. 6-5. When regional skew is obtained from this figure, the mean square error of the regional skew is MSE_sr = 0.302. The mean square error of the station skew is approximated by the following formula:

(MSE)sy = 10A - B log(n/10)

(6-34)

in which

A = -0.33 + 0.08G,     for G < 0.9

(6-34a)

A = 0.52 + 0.30G,      for G ≥ 0.9

(6-34b)

B = 0.94 - 0.26G,       for G < 1.5

(6-34c)

B = 0.55                     for G ≥ 1.5

(6-34d)

with G = absolute value of the station skew and n = record length in years.

Figure 6-5  Generalized skew coefficient of logarithms of annual maximum streamflow [28].

Example 6-5. A station in San Diego, California, has flood records for 34 y, with station skew Csy = - 0.1. Calculate a weighted skew following Eq. 6-33 and Fig. 6-5. From Fig. 6-5, the generalized value of regional skew is Csr = -0.3. The MSE of the station skew is calculated by Eq. 6-34, with C = 0.1: (MSE)sy = 0.156. Therefore, the weighted skew is (Eq. 6-33): Csw = -0.168.

Treatment of Outliers.

Outliers are data points that depart significantly from the overall trend of the data. The treatment of these outliers (i.e., their retention, modification, or deletion) may have a significant effect on the value of the statistical parameters computed from the data, particularly for small samples. Procedures for treatment of outliers invariably require judgment involving mathematical and hydrologic considerations.

The detection and treatment of high and low outliers in the log Pearson III method is performed in the following way [28]. For station skew greater than +0.4, tests for high outliers are considered first. For station skew less than -0.4, tests for low outliers are considered first. For station skew in the range -0.4 to +0.4, tests for high and low outliers are considered simultaneously, without eliminating any outliers from the data.

The following equation is used to detect high outliers:

yH = ȳ + Knsy

(6-35)

in which y_H = high outlier threshold (in log units); and K_n = outlier frequency factor, a function of record length n. Values of y_H are given in Table A-7 (Appendix A).

Values of y_i (logarithms of the flood series) greater than y_H are considered to be high outliers. If there is sufficient evidence to indicate that a high outlier is a maximum in an extended period of time, it is treated as historical data. Otherwise, it is retained as part of the flood series.

Historical data refers to flood information outside of the flood series, which can be used to extend the record to a period much longer than that of the flood series. Historical knowledge is used to define the historical period H, which is longer than the record period n. The number z of events that are known to be the largest in the historical period are given a weight of 1. The remaining n events from the flood series are given a weight of (H - z)/n. For instance, for a record length n = 44 y, a historical period H = 77 y, and a number of peaks in the historical period z = 3, the weight applied to the three historical peaks would be 1, and the weight applied to the remaining flood series would be (77 - 3)/44 = 1.68. In other words, the record is extended to 77 y, and the 44 y of flood series (excluding outliers that have been considered part of the historical data) represent 74 y of data in the historical period of 77 y [28].

The following equation is used to detect low outliers:

yL = ȳ - Knsy

(6-35)

in which y_L = low outlier threshold (in log units) and other terms are as defined previously. If an adjustment for historical data has been previously made, the values on the right-hand side of Eq. 6-36 are those previously used in the historically weighted computation. Values of y_i smaller than y_L are considered to be low outliers and deleted from the flood series [28].

Gumbel's Extreme Value Type I Method

The extreme value Type I distribution, also known as the Gumbel method [16] , or EVl, has been widely used in the United States and throughout the world. The method is a special case of the three-parameter GEV distribution described in the British Flood Studies Report [23].

The cumulative density function F(x) of the Gumbel method is the double exponential, Eq. 6-19, repeated here:

F ( x ) = e-e-y

(6-19)

in which F ( x ) is the probability of nonexceedence. In flood frequency analysis, the probability of interest is the probability of exceedence, i.e. the complementary probability to F ( x ):

G ( x ) = 1 - F ( x )

(6-37)

The return period T is the reciprocal of the probability of exceedence. Therefore,

1 ___ = 1 - e-e-y T

(6-38)

From Eq. 6-38:

T y = -ln ln _______                  T - 1

(6-39)

Table 6-4 Types or Analyses Used in Flood Frequency Studies [28]. Description Record Length n (y) 10 - 24 25 - 49 50 or more Statistical analysis of flood frequency X X X Comparisons with similar catchments X X ___ Flood estimates from precipitation X ___ ___

In the Gumbel method, values of flood discharge are obtained from the frequency formula, Eq. 6-29:

x = x̄ + Ks

(6-29)

The frequency factor K is evaluated with the frequency formula:

y = ȳn + Kσn

(6-40)

in which y = Gumbel (reduced) variate, a function of return period (Eq.6-39); and ȳ_n and σ_n are the mean and standard deviation of the Gumbel variate, respectively. These values are a function of record length n (see Table A-8, Appendix A).

In Eq. 6-29, for K = 0, x is equal to the mean annual flood x̄. Likewise, in Eq. 6-40, for K = 0, the Gumbel variate y is equal to its mean ȳ_n. The limiting value of ȳ_n, for n approaching ∞ is the Euler constant, 0.5772 [26]. In Eq. 6-38, for y = 0.5772: T = 2.33 years. Therefore, the return period of 2.33 y is taken as the return period of the mean annual flood.

From Eqs. 6-29 and 6-40,

y - ȳn x = x̄ + ________ s                σn

(6-41)

and with Eq. 6-39.

(6-42)

The following steps are necessary to apply the Gumbel method:

1. Assemble the flood series.

2. Calculate the mean x̄ and standard deviation s of the flood series.

3. Use Table A-8 to determine the mean ȳ_n and standard deviation σ_n of the Gumbel variate as a function of record length n.

4. Select several return periods T_j and associated exceedence probabilities P_j.

5. Calculate the Gumbel variates Y_j corresponding to the return periods T_j by using Eq. 6-39, and calculate the flood discharge Q_j = x_j for each Gumbel variate (and associated return period) using Eq. 6-41. Alternatively, the flood discharges can be calculated directly for each return period by using Eq. 6-42.

Values of Q are plotted against y or T (or P) on Gumbel probability paper, and a straight line is drawn through the points. Gumbel probability paper has an arithmetic scale of Gumbel variate Y in the abscissas and an arithmetic scale of flood discharge Q in the ordinates. To facilitate the reading of frequencies and probabilities, Eq. 6-38 can be used to superimpose a scale of return period T (or probability P) on the arithmetic scale of Gumbel variate y.

Example 6-6. Apply the Gumbel method to the flood series of Example 6á3. Plot the results on Gumbel paper along with the plotting positions calculated in Example 6-3. The mean and standard deviation of the flood series are: x̄ = 1704 m3/s and s = 795 m3/s. From Table A-8. for n = 16, the mean and standard deviation of the Gumbel variate are ȳ = 0.5157 and σn = 1.0316. The results are shown in Table 6-5. Columns 1 and 2 show selected return periods T and associated exceedence probabilities (in percent). Column 3 shows the values of Gumbel variate calculated by Eq. 6-39. Column 4 shows the flood discharge Q calculated by Eq. 6-41 for each variate y, return period T, and associated exceedence probability P. The flood discharges define a straight line when plotted versus return period on Gumbel paper, as shown by the solid line of Fig. 6-6. Plotting positions calculated by Example 6-3 are shown for comparison purposes.

Modifications to the Gumbel Method.

Since its inception in the 1940s, several modifications to the Gumbel method have been suggested. Gringorten [12] has shown that the Gumbel distribution does not follow the Weibull plotting rule, Eq. 6- 26 (or Eq. 6-27 with a = 0). He recommended a = 0.44, which led to the Gringorten plotting position formula:

1                     m - 0.44 _____ = P = _______________ T                    n + 0.12

(6-42)

Lettenmaier and Burges [22] have suggested that better flood estimates are obtained by using the limiting values of mean and standard deviation of the Gumbel variate (ie. , those corresponding to n = ∞) in Eq. 6-40, instead of basing these values on the record length. In this case, ȳ_n = 0.5772, and σ_n = π / √6 = 1.2825. Therefore, Eq. 6-41 reduces to

x = x̄ + (0.78y - 0.45)s

(6-44)

and Eq. 6-42 reduces to

T x = x̄ + (0.78 ln ln _________ + 0.45)s                                T - 1

(6-45)

Table 6-5 Gumbel Method: Example 6-6. (1) (2) (3) (4) Return Period T (y) Probability P (percent) Gumbel variate y Flood discharge Q (m3/s) 1.05 95 -1.113 449 1.11 90 -0.838 661 1.25 80 -0.476 940 2 50 0.367 1590 5 20 1.500 2462 10 10 2.250 3040 25 4 3.199 3772 50 2 3.902 4314 100 1 4.600 4851 200 0.5 5.296 5388

Figure 6-6  Flood-frequency analysis by Gumbel method: Example 6-6.

Lettenmaier and Burges [22] have also suggested that a biased variance estimate, using n as the divisor in Eq. 6-3, yields better estimates of extreme events that the usual unbiased estimate, that is, the divisor n - 1.

Comparison Between Flood Frequency Methods

In 1966, the Hydrology Subcommittee of the U.S. Water Resources Council began work on selecting a suitable method of flood frequency analysis tltat could be recommended for general use in the United States.

The committee tested the goodness of fit of six distributions: (1) lognormal, (2) log Pearson III, (3) Hazen, (4) gamma, (5) Gumbel (EV1) and (6) log Gumbel (EV2). The study included ten sets of records, the shortest of which was 40 y. The findings showed that the first three distributions had smaller average deviations that the last three. Since the Hazen distribution is a type of lognormal distribution and the lognormal is a special case of the log Pearson III, the Committee concluded that the latter was the most appropriate of the three, and hence recommended it for general use.

The same type of analysis was repeated for six sets of records in the United Kingdom, the shortest of which was 32 y [2]. The methods were: (1) gamma, (2) log gamma, (3) lognormal, (4) Gumbel (EV1), (5) GEV, (6) Pearson Type III, and (7) log Pearson III. At low return periods (Tfrom 2 to 5 y), the GEV and Pearson Type III showed the smallest average deviations, whereas for return periods exceeding 10 y the log Pearson III method had the smallest average deviations.

Similar comparative studies were reported in the British Flood Studies Report [23]. The study concluded that the three-parameter distributions (GEV, Pearson Type III, and log Pearson III) provided a better fit than the two-parameter distributions (Gumbel, lognonnal, gamma, log gamma). Based on mean absolute deviation criteria, the study rated the log Pearson III method better than the GEV and the latter better than the Pearson Type III. However, based on root mean square deviation, it rated the Pearson Type III better than both the log Pearson III and GEV distributions.

Although in general, the three-parameter methods seemed to fare better than the two-parameter methods, the latter should not be completely discarded. The British Flood Studies Report [23] observed that their use in connection with short record lengths often leads to results which are more sensible than those obtained by fitting three-parameter distributions. A three-parameter distribution fitted to a small sample may in some cases imply that there is an upper bound to the flood discharge equal to about twice the mean annual flood. While there may be an upper limit to flood magnitude, it is certainly higher than twice the mean annual flood.

6.3  LOW-FLOW FREQUENCY ANALYSIS

[Questions]   [Problems]   [References]   •   [Top]   [Concepts of Statistics]   [Frequency Analysis]

Whereas high flows lead to floods, sustained low flows can lead to droughts. A drought is defined as a lack of rainfall so great and continuing so long as to affect the plant and animal life of a region adversely and to deplete domestic and industrial water supplies, especially in those regions where rainfall is normally sufficient for such purposes [18].

In practice, a drought refers to a period of unusually low water supplies, regardless of the water demand. The regions most subject to droughts are those with the greatest variability in annual rainfall. Studies have shown that regions where the variance coefficient of annual rainfall exceeds 0.35 are more likely to have frequent droughts [6]. Low annual rainfall and high annual rainfall variability are typical of arid and semiarid regions. Therefore, these regions are more likely to be prone to droughts.

Studies of tree rings, which document long term trends of rainfall, show clear patterns of periods of wet and dry weather [27]. While there is no apparent explanation for the cycles of wet and dry weather, the dry years must be considered in planning water resource projects. Analysis of long records has shown that there is a tendency for dry years to group together. This indicates that the sequence of dry years is not random, with dry years tending to follow other dry years. It is therefore necessary to consider both the severity and duration of a drought period.

The severity of droughts can be established by measuring (1) the deficiency in rainfall and runoff, (2) the decline of soil moisture, and (3) the decrease in groundwater levels. Alternatively, low-flow-frequency analysis can be used in the assessment of the probability of occurrence of droughts of different durations.

Figure 6-7  Low-flow frequency curves [25].

Methods of low-flow frequency analysis are based on an assumption of invariance of meteorological conditions. The absence of long records, however, imposes a stringent limitation on low-flow frequency analysis. When records of sufficient length are available, analysis begins with the identification of the low-flow series. Either the annual minima or the annual exceedence series are used. In a monthly analysis, the annual minima series is formed by the lowest monthly flow volumes in each year of record. If the annual exceedence method is chosen, the lowest monthly flow volumes in the record are selected, regardless of when they occurred. In the latter method, the number of values in the series need not be equal to the number of years of record.

A flow-duration curve can be used to give an indication of the severity of low flows. Such a curve, however, does not contain information on the sequence of low flows or the duration of possible droughts. The analysis is made more meaningful by abstracting the minimum flows over a period of several consecutive days. For instance, for each year, the 7-day period with minimum flow volume is abstracted, and the minimum flow is the average flow rate for that period. A frequency analysis on the low-flow series, using the Gumbel method, for instance, results in a function describing the probability of occurrence of low flows of a certain duration. The same analysis repeated for other durations leads to a family of curves depicting low-flow frequency, as shown in Fig. 6-7 [25].

In reservoir design , the assessment of low flows is aided by a flow-mass curve. The technique involves the determination of storage volumes required for all low-flow periods. Although it is practically impossible to provide sufficient storage to meet hydrologic risks of great rarity, common practice is to provide for a stated risk (i.e., a drought probability) and to add a suitable percent of the computed storage volume as reserve storage allowance. The variance coefficient of annual flows is used in determining the risk and storage allowance levels. Extraordinary drought levels are then met by cutting draft rates.

Regulated rivers may alter natural flow conditions to provide a minimum downstream flow for specific purposes. In this case, the reservoirs serve as the mechanism to diffuse the natural flow variability into downstream flows which can be made to be nearly constant in time. Regulation is necessary for downstream low flow maintenance, usually for the purpose of meeting agricultural, municipal and industrial water demands, minimum in stream flows, navigation draft, and water pollution control requirements.

QUESTIONS

[Problems]   [References]   •   [Top]   [Concepts of Statistics]   [Frequency Analysis]   [Low-flow Frequency]

1. In statistical analysis, what are the measures of central tendency? Explain.

2. What is skewness? A distribution with a long tail on the right side has positive or negative skewness?

3. What are the parameters of the gamma distribution? How are the gamma and Pearson Type III distributions related?

4. What is the parameter that distinguishes the three extreme value distributions? What is the limiting value of the mean of the Gumbel variate?

5. What is the difference between the annual exceedence series and the annual maxima series? What is risk in the context of frequency analysis?

6. How is an extreme value probability paper constructed? What type of probability paper is used in the log Pearson Type III method?

7. What is the difference between the Weibull, Blom, and Gringorten plotting position formulas?

8. How is skewness variability accounted for in the log Pearson III method?

9. When are high outliers considered part of historical data? When is it necessary to perform a historically weighted computation?

10. Why are two-parameter distributions such as the Gumbel distribution appropiate for use in connection with short record lengths?

11. Compare floods and droughts from the standpoint of frequency analysis.

PROBLEMS

[References]   •   [Top]   [Concepts of Statistics]   [Frequency Analysis]   [Low-flow Frequency]   [Questions]

1. Develop a computer program to calculate the mean, standard deviation, and skew coefficient of a series of annual maximum flows. Test your computer program using the data of Example 6-1 in the text.

2. The annual maximum flows of a certain stream have been found to be normally distributed with mean 22,500 ft³/s and standard deviation 7500 ft³/s. Calculate the probability that a flow larger than 39,000 ft³/s will occur.

3. The 10-y and 25-y floods of a certain stream are 73 and 84 m³/s, respectively. Assuming a normal distribution, calculate the 50-y and 100-y floods.

4. The low flows of a certain stream have been shown to follow a normal distribution. The flows expected to be exceeded 95% and 90% ofthe time are 15 and 21 m³/s, respectively. What flow can be expected to be exceeded 80% of the time?

5. A temporary cofferdam for a 5-y dam construction period is designed to pass the 25-y flood. What is the risk that the cofferdam may fail before the end of the construction period? What design return period is needed to reduce the risk to less than 10%?

6. Use the Weibull formula (Eq. 6-26) to calculate the plotting positions for the following series of annual maxima, in cubic feet per second: 1305, 3250, 4735, 5210, 4210, 2120, 2830,3585,7205,1930,2520,3250,5105,4830,2020, 2530,3825, 3500, 2970,1215.

7. Use the Gringorten formula to calculate the plotting positions for the following series of annual maxima, in cubic meters per second: 160,350,275,482,530, 390,283, 195, 408, 307,625,513.

8. Modify the computer program of Problem 6-1 to calculate the mean, standard deviation, and skew coefficients of the logarithms of a series of annual maximum flows. Test your computer program using the results of Example 6-4 in the text.

9. Fit a log Pearson III curve to the data of Problem 6-6. Plot the calculated distribution on log probability paper, along with the Weibull plotting positions calculated in Problem 6-6.

10. Fit a Gumbel curve to the data of Problem 6-6. Plot the calculated distribution on Gumbel paper, along with the Weibull plotting positions calculated in Problem 6-6.

11. Develop a computer program to read a series of annual maxima, sort the data in descending order, and compute the corresponding plotting positions (percent chance and retum period) by the Weibull and Gringorten formulas.

12. Given the following statistics of annual maxima for stream X: number of years n = 35; mean = 3545 ft³/s; standard deviation = 1870 ft³/s. Compute the 100-y flood by the Gumbel method.

13. Given the following statistics of annual maxima for river Y: number of years n = 45; mean = 2700 m³/s; standard deviation 1300 m³/s; mean of the logarithms = 3.1; standard deviation of the logarithms = 0.4; skew coefficient of the logarithms = -0.35. Compute the 100-y flood using the following probability distributions: (a) normal, (b) Gumbel, and (c) log Pearson III.

14. A station near Denver, Colorado, has flood records for 48 y, with station skew C_sy = - 0.18. Calculate a weighted skew coefficient.

15. Determine if the value Q = 13,800 ft³/s is a high outlier in a 45-y flood series with the following statistics: mean of the logarithms = 3.572; standard deviation of the logarithms = 0.215.

16. Using the Lettenmaier and Burges modification to the Gumbel method, fit a Gumbel curve to the data of Example 6-6 in the text. Plot the calculated distribution on Gumbel paper, along with plotting positions calculated by the Gringorten formula.

17. Solve Example 6-4 in the text using the computer program EH600A (Log Pearson III method) included in Appendix D.

18. Solve Example 6-6 in the text using the computer program EH600B (Gumbel method) included in Appendix D.

19. Solve Problem 6-9 using the computer program EH600A (Log Pe,lrson III method) included in Appendix D.

20. Solve Problem 6-10 using the computer program EH600B (Gumbel method) included in Appendix D.

REFERENCES

•   [Top]   [Concepts of Statistics]   [Frequency Analysis]   [Low-flow Frequency]   [Questions]   [Problems]

1. Benson, M. A. (1962). "Plotting Positions and Economics of Engineering Planning," Journal of the Hydraulics Division. ASCE, Vol. 88, November, pp. 57-71.

2. Benson, M. A. (1968). "Uniform Flood Frequency Estimating Methods for Federal Agencies," Water Resources Research. Vol. 4, pp. 891-908.

3. Blom, G. (1958). Statistical Estimates and Transformed Beta Variables. New York: John Wiley.

4. Casella, G., and R. L. Berger. (1990). Statistical Inference, Wadsworth & Brooks/Cole, Pacific Grove, California.

5. Chow, V. T. (1951). "A General Formula for Hydrologk Frequency Analysis," Transactions. American Geophysical Union. Vol. 32, pp. 231-237.

6. Chow, V. T. (1964). Handbook of Applied Hydrology. New York: McGraw-Hill.

7. Cunnane, C. (1978). "Unbiased Plotting Positions-A Review," Journal of Hydrology. Vol. 37, pp. 205-222.

8. Fisher, R. A., and L. H. C. Tippett. (1928). "Limiting Forms of a Frequency Distribution of the Smallest and Largest Member of a Sample," Proceedings. Cambridge Philosophical Society. Vol. 24, pp.180-190.

9. Foster, H. A. (1924). "Theoretical Frequency Curves and their Application to Engineering Problems," Transactions. ASCE, Vol. 87, pp. 142-173.

10. Frechet, M. (1927). "Sur la loi de Probabilite de l'ecart Maximum," (On the Probability Law of Maximum Error), Annals of the Polish Mathematical Society. (Cracow), Vol. 6, pp. 93-116.

11. Fuller, W. E. (1914). "Flood Flows," Transactions. ASCE, Vol. 77, pp. 564-617.

12. Gringorten, 1.1. (1963). "A Plotting Rule for Extreme Probability Paper," Journal of Geophysical Research. Vol. 68, No.3, February, pp. 813-814.

13. Gumbel, E. J. (1941). "Probability Interpretation of the Observed Return Periods of Floods," Transactions. American Geophysical Union, Vol. 21, pp. 836-850.

14. Gumbel, E. J. (1942). "Statistical Control Curves for Flood Discharges," Transactions. American Geophysical Union, Vol. 23, pp. 489-500.

15. K Gumbel , E. J. (1943). "On the Plotting of Flood Discharges," Transactions. American Geophysical Union. Vol. 24, pp. 699-719.

16. Gumbel, E. J. (1958). Statistics of Extremes. Irvington, N.Y.: Columbia University Press.

17. Hazen, A. (1914). Discussion on "Flood Flows," by W. E. Fuller, Transactions, ASCE, Vol. 77, p. 628.

18. Havens, A. V. (1954). "Drought and Agriculture," Weatherwise. Vol. 7, pp. 51-55.

19. Horton, R. E. (1913). "Frequency of Recurrence of Hudson River Floods," U.S. Weather Bureau Bulletin Z. pp. 109-112.

20. Jenkinson, A. F. (1955). "The Frequency Distributions of the Annual Maximum (or Minimum) Values of Meteorological Elements," Quarterly Journal of the Royal Meteorological Society, Vol. 87, p. 158.

21. Kite, G. W. (1977). Frequency and Risk Analyses in Hydrology. Fort Collins, Colorado: Water Resources Publications.

22. Lettenmaier, D. P., and S. J. Burges. (1982). "Gumbel's Extreme Value Distribution: A New Look," Journal of the Hydraulics Division, ASCE, Vol. 108, No. HY4, April, pp.503-514.

23. Natural Environment Research Council. (1975). Flood Studies Report. Vol. 1 (of 5 volumes), London, England.

24. Pearson, K. (1930). "Tables for Statisticians and Biometricians," Part I, 3d. Ed. , The Biometric Laboratory, University College. London: Cambridge University Press.

25. Riggs, H. C. (1972). "Low-Flow Investigations," Techniques of Water Resources Investigations of the United States Geological Survey. Book 4, Chapter B1.

26. Spiegel, M. Mathematical Handbook of Formulas and Tables. Schaum's Outline Series in Mathematics. New York: McGraw-Hill.

27. Troxell, H. C. (1937). "Water Resources of Southern California with Special Reference to the Drought of 1944-51," U.S. Geological Survey Water Supply Paper No. 1366.

28. U.S. Interagency Advisory Committee on Water Data, Hydrology Subcommittee. (1983). "Guidelines for Determining Flood Flow Frequency," Bulletin No. 17B, issued 1981, revised 1983, Reston, Virginia.

SUGGESTED READINGS

• Gumbel, E. J. (1958). Statistics of Extremes. Irvington. N.Y.: Columbia University Press.

• Natural Environment Research Council. (1975). Flood Studies Report. in 5 volumes, London, England.

• Riggs, H. C. (1972). "Low-Flow Investigations," Techniques of Water Resources Investigations of the United States Geological Survey. Book 4, Chapter BI.

• U.S. Interagency Advisory Committee on Water Data, Hydrology Subcommittee. (1983). "Guidelines for Determining Flood Flow Frequency," Bulletin No. 17B, issued 1981, revised 1983, Reston. Virginia.