My WebLink
|
Help
|
About
|
Sign Out
Home
Browse
Search
FLOOD00764
CWCB
>
Floodplain Documents
>
Backfile
>
1-1000
>
FLOOD00764
Metadata
Thumbnails
Annotations
Entry Properties
Last modified
11/23/2009 10:51:16 AM
Creation date
10/4/2006 9:27:30 PM
Metadata
Fields
Template:
Floodplain Documents
County
Statewide
Community
Nationwide
Title
Use of Extreme Value Theory in Estimating Flood Peaks from Mixed Populations
Date
2/1/1980
Prepared By
Utah State University Water Research Laboratory
Floodplain - Doc Type
Educational/Technical/Reference Information
There are no annotations on this page.
Document management portal powered by Laserfiche WebLink 9 © 1998-2015
Laserfiche.
All rights reserved.
/
43
PDF
Print
Pages to print
Enter page numbers and/or page ranges separated by commas. For example, 1,3,5-12.
After downloading, print the document using a PDF reader (e.g. Adobe Reader).
Show annotations
View images
View plain text
<br />It 1S interesting to note that in the <br />work of Bobee and Robitaille (1977), both the <br />Pearson type III and log Pearson type III <br />dlstrlbut10ns introduce an apparent inconsis- <br />l ency. In some cases an upper bound for <br />annual floods is appropriate and in others a <br />lower bound is used. The Pearson and log <br />Pearson distributions are not even consistent <br />for a given data set. In some cases the <br />Pearson distribution calls for an upper bound <br />while the log Pearson calls for a lower <br />bound. It seems that if an upper bound is <br />valid due to meteorological and geographical <br />lim.itations, it would be valid for all <br />systems. The switch in boundedness is due to <br />the inability of the Pearson and log Pearson <br />type I I I d istr ibut ions to accommodate both <br />pas it ive and negatIve skewness for a given <br />bound (upper or lower). <br /> <br />EstimatIon of Parameters <br /> <br />Although the concept of lImiting flood <br />IS reasonable, its magnitude is difficult to <br />estimate from geographical conSIderations. <br />It was found, however, that the flow esti- <br />mated for a gIven frequency is very Insen- <br />s1tive to the value chosen for b as long as <br />It is relatIvely large. Therefore, ordinary <br />maximum likelihood estimates of all of the <br />parameters were used. <br /> <br />The distrIbution F3(x) IS a transformed <br />Welbull, l.e., if the F3(x) is transformed <br />by y = -x the distribution of y IS Weibull <br />with the same parameters as F3(x) (b is <br />negatIve). Therefore a program available <br />for maximum likelihood (ML) estimation of <br />Weibull parameters (Harter and Moore 1965) <br />was used (Appendix G). This program and <br />other procedures described later in the <br />report reqUIres that the data be ordered. A <br />FORTRAN program for this purpose is found in <br />Appendix A. <br /> <br />Some difficulties were experienced in <br />applying ML methods. In general, the computer <br />program was expensive to run and, in addi- <br />t ion, required several passes to find ac- <br />ceptable scale factors and init ial values. <br />The resulting estimates were highly dependent <br />on these values even when the convergence <br />criterIon for the computation was met. In <br />some cases, a better fit was obtained using a <br />less stringent convergence measure. These <br />problems motIvated additional research not <br />directly connected with this project. <br /> <br />This research resulted in a computation- <br />ally more efficient method of estimation <br />developed for all extreme value distributions <br />(Kwan 1979). This method of estimation does <br />not depend upon sensitive convergence cri- <br />teria. These results were obtained too <br />late to be incorporated into the comparisons <br />made in this report. It is felt that improve- <br />menl in the goodness-of-fit statistics for <br />some of the streams reported in the next <br />sectIon could be obtained USIng the new <br />method of estimation. <br /> <br />Goodness-of-fit Comparisons <br /> <br />The result of fitting F3(x) to the <br />same data used by Bobee and Robitaille (1977) <br />(Table 2) to evaluate the Pearson and log <br />Pearson type III distributions is gIven in <br />this section. Maximum likelihood estimation <br />(with its accompanying difficulties) was <br />used. The same goodness-of-fit statistics <br />used by Bobee and Robitaille (1977) are used <br />herein. These statistics are derived from <br />three formulas for expected probabilities of <br />order statistics referred to as the Hazen, <br />Chegodayev, and Weibull formulas. A detailed <br />description of the goodness-of-fit computa- <br />tions is given in Bobee and Robitaille <br />(1977). Briefly the measures are based upon <br />the relative deviations, <br /> <br />q(T) ~ <br /> <br />Q(T) - D(T) * IDD <br />D(T) <br /> <br />where D(T) represents the empirical (data <br />value) for recurrence interval T, and Q(T) <br />represents the value estimated from the <br />fitted distribution. The recurrence intervals <br />T = 2, 5, 10, 20, 50, and 100 were used. The <br />average absolute deviation (i.e., ilq(T)I/L) <br />IS given in Table 3, and the average of the <br />quadratic deviations (i.e., iq(T)2/L) is <br /> <br />gIven In Table 4. FORTRAN programs for <br />these computatIons are found in Appendices D, <br />E, and F. The goodness-of-fit values for <br />the log Pearson type III distribution and <br />for the dIstribution and method of fitting. <br />;udged best by Bobee and RobitaIlle (1977) <br />(Pearson type I II) are also tabulated 10 <br />Tables 3 and 4 for comparatIve purposes. <br /> <br />It IS Imposs ible to <br />information on Tables 3 and 4 <br />plots of these data sets. <br />shown in Figures 5-18. <br /> <br />It can be seen that Figures 5, 10, and <br />17 (for stat ions bB24, jF50a, and BF19 <br />respectively) have linear plots indicating an <br />F3(x) distribution. The goodness-of-fit <br />statistics tabulated in Tables 3 and 4 <br />bear out this choice as the fit for F3(x) <br />is best for the data at these three stations. <br />The "Sit shape of the plots in Figures 7, 8, <br />11, 12, 13 and 18 indicate that neither <br /> <br />F3(x), Pearson type III nor log Pearson <br />type III distributions are appropriate. <br />These plots underscore their importance in <br />fitting data. Whenever several distributions <br />are fit to given data, one will always have <br />a "best" fit. However, none of those tried <br />may be appropriate. The plots identify <br />these cases. <br /> <br />interpret the <br />without viewing <br />The plots are <br /> <br />One physical explanation for a situatIon <br />in which the data do not plot as a straight <br />line is that they may not come from a single <br />homogeneous source. The effect of non- <br />homogeneous sources is investigated in the <br />remaining sections of this report. The very <br />good fits in association with the plots <br />clearly establish extreme value theory as a <br />viable tool for describing annual flood <br />events. <br /> <br />8 <br />
The URL can be used to link to this page
Your browser does not support the video tag.