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18.24 CHAPTER EIGHTEEN 100,000 ~ u Q) E' o .c. ~ 10,000 o .' 1000 -3 -2 -1 0 Standard normal variable z FIGURE 18.3.1 A probability plot using a normal scale of 44 annual maxima for the G lupe River near Victoria, Texas. (Reproduced with permission from Ref 20, p. 398.) .. 2 ance probability of the ith-largest event is often estimated using the Weibull p position: i qi = n + 1 corresponding to the mean of Vi' Choice of plotting position. Hazens9 originally developed probability pape imagined the probability scale divided into n equal intervals with midpoin ..... (i - O.5)/n, i = I, . . . , n; these served as his plotting positions. Gumbe}S' re this formula in part because it assigned a return period of 2n years to the 1 observation (see also HarterS8); Gumbel promoted Eq. (18.3.4). Cunnane26 argued that plotting positions qi should be assigned so that on av X(j) would equal G-' '( 1 - qi); that is, qi would capture the mean of Xli) so that E[X(i)] "'" G-'(1 - qi) Such plotting positions would be almost quantile-unbiased. The Weibull plo positions i/(n + 1) equal the average exceedance probability of the ranked obs tions X(i)' and hence are probability-unbiased plotting positions. The two crite . different because of the nonlinear relationship between X(i) and V(i)' Different plotting positions attempt to achieve almost quantile-unbiasedne different distributions; many can be written i- a qi = n + 1 - 2a which is symmetric so that qi = I - qn+l-i' Cunanne recommended a = 0.40._ obtaining nearly quantile-unbiased plotting' positions for a range of distributi'. FREQUENCY ANAL YSISOF EXTREME EVENTS 18.25 ternatives are Blom's plotting position (a = t), which gives nearly unbia~ed s for the normal distribution, and the Gringorten position (a = 0.44) whlch timized plotting positions for the largest ob.servations from a Gumbel di~tri- 9 These are summarized in Table 18.3.1, which also reports the return penod, assigned to the largest observation. Section 18.6.3 develops plotting posi- " . records that contaIn censored values. ifferences between the Hazen formula, Cunanne's recommendation, and bull formula is modest for i of 3 or more. However, differences can be able for i = 1, corresponding to the largest observation (an~. i = n fo~ the t observation). Remember that the actual exceedance probabIlity assOCIated largest observation is a random variable with mean 1/(n + 1) and a standard n of nearly 1/(n + 1); see Eqs. (18.3.2) and (18.3.3). ~~u.s all plo~ting p~si- ve crude estimates of the unknown exceedance probabilities asSOCIated With est (and smallest) events. Oct method for illustrating this uncertainty is to consider quantiles of the beta tion of the actual exceedance probability associated with the ~est obse!"a- , . The actual exceedance probability for the largest observa~on X(1) 10 a Is between 0.29/n and 1.38/(n + 2) nearly 50 percent of the 1tme;and be- :052/n and 3/(n + 2) nearly 90 percent of the time. Such bounds all?w o~e to e consistency ofthe largest (or, by symmetry, the smallest) observation wlth a stribution better than does a single plotting position. ility Paper. It is now possible to see how probability papers can be con- 4Jor many distributions. A probability plot is a graph of the ranked observa- (i) versus an approximation of their expected value G-'(l - qi)' For the nor- ( , C-.: ;.~-: W t' r\J 18.3.1 Alternative Plotting Positions and their Motivation. Formula a T. Motivation ; 0 n+1 Unbiased exceedance probabilities (\ n+1 for all distributions ; - 0.3175 0.3175 1.47n + 0.5 Median exceedance probabilities n + 0.365 for all distribution.s ; - 0.35 -0.35 1.54n Used with PWMs [Eq. (18.1.13)] n ; - 3/8 0.375 1.60n + 0.4 Unbiased normal quantiles n + 1/4 ; - 0.40 0.40 1.67n + 0.3 Approximately quantile-unbiased n+0.2 i - 0.44 0.44 1.79n + 0.2 Optimized for Gumbel distribution. n + 0.12 ;-0.5 0.50 2n A traditional choice n a is the plotting-position parameter in Eq. (18.3.6) and TI is the return period each plotting igns to the largest observaji.2n in a sample of size n. . 1>= I and n, the exact value is ql >= I - q. >= I - 0.51/..