Analytical Approaches

An alternative to the graphic technique is to estimate the statistical parameters of a distribution from the sample data (refer to Sec. 3.6). Then the distribution model can be used to solve for the variate value corresponding to any desired return period or probability as

in which F-1(e) is the inverse cumulative distribution function with the model parameter vector в. Equation (3.5) can be applied when the inverse distribu­tion function forms are analytically amenable, such as for the Gumbel, gener­alized extreme value, generalized logistic, and generalized Pareto distributions (see Sec. 2.6.6).

Example 3.2 Consider that the annual maximum floods follow a lognormal distribu­tion with a mean of 490 ft3/s and a standard deviation of 80 ft3/s. Determine the flood magnitude with a 1-in-100 chance of being exceeded in any given year.

Solution From Eqs. (2.67a) and (2.67b), the parameters of a lognormal distribution, for annual maximum flood Q, can be obtained as

Obq = ^ln (vQ +[5]) = ^ln (490) + 1 Mnq = ln(MQ) — 1 Oj2n Q = ln(490) — 1(0.1622)2 = 6.1812

Since ln( Q) follows a normal distribution with a mean of mn q = 6.1812 and a stan­dard deviation of Oln q = 0.1622 as previously computed, the magnitude of the log — transformed 100-year flood can be calculated by

ln(g100)—= Ф-1 f 1 — -^ = Ф-1(0.99) = 2.34

Oln Q V 100 у

Hence ln(^100) = Mln Q + 2.34 x Olnq = 6.5607, and the corresponding 100-year flood magnitude can be calculated as дю0 = exp[ln(q10Q)] = 706.8 ft3/s.

For some distributions, such as Pearson type 3 or log-Pearson type 3, the appropriate probability paper or CDF inverse form is unavailable. In such a case, an analytical approach using the frequency factor KT is applied:

XT = l^x + Kt x ax (3.6)

in which xT is the variate corresponding to a return period of T, /гх and oX are the mean and standard deviation of the random variable, respectively, and KT is the frequency factor, which is a function of the return period T or P (X > xT) and higher moments, if required. It is clear that a plot of Eq. (3.6) (xT versus KT)
on linear graph paper will yield a straight line with slope of ax and intercept l^x at Kt = 0.

In order for Eq. (3.6) to be useful, the functional relationship between Kt and exceedance probability or return period must be determined for the distribution to be used. In fact, the frequency factor Kt = (xT — xx)/ax is identical to a standardized variate corresponding to the exceedance probability of 1/T for a particular distribution model under consideration. For example, if the normal distribution is considered, then Kt = zT = Ф—41 — T —:). The same applies to the lognormal distribution when the mean and standard deviation of log — transformed random variables are used. Hence the standard normal probability table (Table 2.2) provides values of the frequency factor for sample data from normal and log normal distributions. Once this relation is known, a nonlinear probability or return-period scale can be constructed to replace the linear Kt scale, and thus special graph paper can be constructed for any distribution so that plot of xT versus P or T will be linear.

Gumbel probability paper has been printed, although it is not readily avail­able from commercial sources. Referring to Eq. (2.85a), the relationship between Kt and T for this distribution can be derived as

V6 n
T T — 1
Kt	0.5772 + ln	(3.7)

For Pearson and log-Pearson type 3 distributions, linearization can be ac­complished according to Eq. (3.6). However, for this distribution, the frequency factor is a function of both P or T and the skewness coefficient yx. This means that a different nonlinear P or T scale is required for each skewness coeffi­cient, and therefore, it is impractical to construct a probability paper for this distribution. However, it should be pointed out that if yx = 0 in log-space, the log-Pearson type 3 reduces to the lognormal distribution, and thus commercial lognormal probability paper can be used. The relationship between frequency factor Kt, T, and yx cannot be developed in a closed form, as was done for the Gumbel distribution in Eq. (3.7). However, the relationship can be computed numerically, and the results are given in Table 3.3. For 0.99-1 < T < 100 and |yx | < 2, the frequency-factor values are well approximated by the Wilson — Hilferty transformation (Stedinger et al., 1993):

in which zT is the standard normal quantile with exceedance probability of 1/T.

The procedure for using the frequency-factor method is outlined as follows: 1

Return period in years Skewness coefficient yx 2 5 10 25 50 Exceedence probability 100 200 0.50 0.20 0.10 0.04 0.02 0.01 0.005 3.0 -0.396 0.420 1.180 2.278 3.152 4.051 4.970 2.9 -0.390 0.440 1.195 2.277 3.134 4.013 4.909 2.8 -0.384 0.460 1.210 2.275 3.114 3.973 4.847 2.7 -0.376 0.479 1.224 2.272 3.093 3.932 4.783 2.6 -0.368 0.499 1.238 2.267 3.071 3.889 4.718 2.5 -0.360 0.518 1.250 2.262 3.048 3.845 4.652 2.4 -0.351 0.537 1.262 2.256 3.023 3.800 4.584 2.3 -0.341 0.555 1.274 2.248 2.997 3.753 4.515 2.2 -0.330 0.574 1.284 2.240 2.970 3.705 4.444 2.1 -0.319 0.592 1.294 2.230 2.942 3.656 4.372 2.0 -0.307 0.609 1.302 2.219 2.912 3.605 4.298 1.9 -0.294 0.627 1.310 2.207 2.881 3.553 4.223 1.8 -0.282 0.643 1.318 2.193 2.848 3.499 4.147 1.7 -0.268 0.660 1.324 2.179 2.815 3.444 4.069 1.6 -0.254 0.675 1.329 2.163 2.780 3.388 3.990 1.5 -0.240 0.690 1.333 2.146 2.743 3.330 3.910 1.4 -0.225 0.705 1.337 2.128 2.706 3.271 3.828 1.3 -0.210 0.719 1.339 2.108 2.666 3.211 3.745 1.2 -0.195 0.732 1.340 2.087 2.626 3.149 3.661 1.1 -0.180 0.745 1.341 2.066 2.585 3.087 3.575 1.0 -0.164 0.758 1.340 2.043 2.542 3.022 3.489 0.9 -0.148 0.769 1.339 2.018 2.498 2.957 3.401 0.8 -0.132 0.780 1.336 1.993 2.453 2.891 3.312 0.7 -0.116 0.790 1.333 1.967 2.407 2.824 3.223 0.6 -0.099 0.800 1.328 1.939 2.359 2.755 3.132 0.5 -0.083 0.808 1.323 1.910 2.311 2.686 3.041 0.4 -0.066 0.816 1.317 1.880 2.261 2.615 2.949 0.3 -0.050 0.824 1.309 1.849 2.211 2.544 2.856 0.2 -0.033 0.830 1.301 1.818 2.159 2.472 2.763 0.1 -0.017 0.836 1.292 1.785 2.107 2.400 2.670 0.0 0 0.842 1.282 1.751 2.054 2.326 2.576 -0.1 0.017 0.846 1.270 1.716 2.000 2.252 2.482 -0.2 0.033 0.850 1.258 1.680 1.945 2.178 2.388 -0.3 0.050 0.853 1.245 1.643 1.890 2.104 2.294 -0.4 0.066 0.855 1.231 1.606 1.834 2.029 2.201 -0.5 0.083 0.856 1.216 1.567 1.777 1.955 2.108 -0.6 0.099 0.857 1.200 1.528 1.720 1.880 2.016 -0.7 0.116 0.857 1.183 1.488 1.663 1.806 1.926 -0.8 0.132 0.856 1.166 1.448 1.606 1.733 1.837 -0.9 0.148 0.854 1.147 1.407 1.549 1.660 1.749 -1.0 0.164 0.852 1.128 1.366 1.492 1.588 1.664 -1.1 0.180 0.848 1.107 1.324 1.435 1.518 1.581 -1.2 0.195 0.844 1.086 1.282 1.379 1.449 1.501 -1.3 0.210 0.838 1.064 1.240 1.324 1.383 1.424 -1.4 0.225 0.832 1.041 1.198 1.270 1.318 1.351 -1.5 0.240 0.825 1.018 1.157 1.217 1.256 1.282 -1.6 0.254 0.817 0.994 1.116 1.166 1.197 1.216

Return period in years Skewness coefficient yx 2 5 10 25 50 Exceedence probability 100 200 0.50 0.20 0.10 0.04 0.02 0.01 0.005 -1.7 0.268 0.808 0.970 1.075 1.116 1.140 1.155 -1.8 0.282 0.799 0.945 1.035 1.069 1.087 1.097 -1.9 0.294 0.788 0.920 0.996 1.023 1.037 1.044 -2.0 0.307 0.777 0.895 0.959 0.980 0.990 0.995 -2.1 0.319 0.765 0.869 0.923 0.939 0.946 0.949 -2.2 0.330 0.752 0.844 0.888 0.900 0.905 0.907 -2.3 0.341 0.739 0.819 0.855 0.864 0.867 0.869 -2.4 0.351 0.725 0.795 0.823 0.830 0.832 0.833 -2.5 0.360 0.711 0.771 0.793 0.798 0.799 0.800 -2.6 0.368 0.696 0.747 0.764 0.768 0.769 0.769 -2.7 0.376 0.681 0.724 0.738 0.740 0.740 0.741 -2.8 0.384 0.666 0.702 0.712 0.714 0.714 0.714 -2.9 0.390 0.651 0.681 0.683 0.689 0.690 0.690 -3.0 0.396 0.636 0.666 0.666 0.666 0.667 0.667 SOURCE: U. S. Water Resources Council (1981).

2. For the desired return period, determine the associated value of KT for the distribution.

3. Compute the desired quantile value using Eq. (3.6) with x replacing цx and sx replacing ax, that is,

XT = x + KT x sx (3.9)

It should be recognized that the basic difference between the graphic and analytical approaches is that each represents a different method of estimating the statistical parameters of the distribution being used. By the analytical ap­proach, a best-fit line is constructed that then sets the statistical parameters. In the mathematical approach, the statistical parameters are first computed from the sample, and effectively, the line thus determined is used. The line deter­mined using the mathematical approach is in general a poorer fit to the observed data than that obtained using the graphic approach, especially if curve-fitting procedures are applied. However, the U. S. Water Resources Council (1967) rec­ommended use of the mathematical approach because

1. Graphic least-squares methods are avoided to reduce the incorporation of the random characteristics of the particular data set (especially in the light of the difficulty in selecting the proper plotting-position formula).

2. The generally larger variance of the mathematical approach is believed to help compensate for the typically small data sets.

Example 3.3 Using the frequency-factor method, estimate the flows with return pe­riods of 2, 10, 25, 50, and 100 years for the Boneyard Creek using the Gumbel and log-Pearson type 3 distributions. Use the historical data in Example 3.1 as a basis for the calculations.

Solution Based on the samples, the method requires determination of the frequency factor Kt in

xt = x + Kt x sx

For the Gumbel distribution, values of Kt can be calculated by Eq. (3.7). For the log-Pearson type 3 distribution, Table 3.3 or Eq. (3.8) can be used, which requires computation of the skewness coefficient. The calculations of relevant sample moments are shown in the following table:

Year		Original scale	Log — Transformed scale
gi (ft3/s)	gi2	gi3	Уі = ln( gi)	Уі2	Уі3
1961	390	1.52e + 05	5.93e + 07	5.97	35.59	212.36
1962	374	1.40e + 05	5.23e + 07	5.92	35.10	207.92
1963	342	1.17e + 05	4.00e + 07	5.83	34.05	198.65
1964	507	2.57e + 05	1.30e + 08	6.23	38.79	241.63
1965	596	3.55e + 05	2.12e + 08	6.39	40.84	260.95
1966	416	1.73e + 05	7.20e + 07	6.03	36.37	219.33
1967	533	2.84e + 05	1.51e + 08	6.28	39.42	247.50
1968	505	2.55e + 05	1.29e + 08	6.22	38.75	241.17
1969	549	3.01e + 05	1.65e + 08	6.31	39.79	251.01
1970	414	1.71e + 05	7.10e + 07	6.03	36.31	218.81
1971	524	2.75e + 05	1.44e + 08	6.26	39.21	245.49
1972	505	2.55e + 05	1.29e + 08	6.22	38.75	241.17
1973	447	2.00e + 05	8.93e + 07	6.10	37.24	227.27
1974	543	2.95e + 05	1.60e + 08	6.30	39.65	249.70
1975	591	3.49e + 05	2.06e + 08	6.38	40.73	259.92
Sum =	7236	3.58e + 06	1.81e + 09	92.48	570.58	3522.88

For the Gumbel distribution,

For the log-Pearson type 3 distribution,

Eln(gf) 92.48

6.165

У = n 15 sy = 1—1 1 1 1 gy = n m3 (n — 1)(n — 2) syj

= (0.417/14)1/2 = 0.173

15(-0.00336) (14)(13)(0.173)3

-0.540

in which m3 = Уз — Уі + 2ny3. The determination of the values of frequency factor corresponding to different return periods is shown in the following table: Frequency factor by distribution Return period Exceedance Nonexceedance Gumbel LP3 (years) probability probability Eq. (3.7) Eq. (3.8) 2 0.50 0.50 —0.1643 0.0892 10 0.10 0.90 1.3046 1.2093 25 0.04 0.96 2.0438 1.5526 50 0.02 0.98 2.5923 1.7570 100 0.01 0.99 3.1367 1.9292 Based on the preceding frequency-factor values, the flood magnitude of the various return periods can be determined as Frequency curves by distribution (ft3/s) Return period Gumbel LP3 (years) qT = 482.4 + 79.8Kt, EV1 qT = exp(6.165 + 0.173Kt, LP3) 2 469.3 483.3 10 586.5 586.4 25 645.4 622.2 50 689.2 644.5 100 732.6 663.9 One could compare these results for the Gumbel distribution with those obtained from the graphic approach of Example 3.1.