P. Waylen, Department of Geography, University of Florida, 3141. Turlington Hall, Gainesville, FL 32611. (Received August 2, 1988; revised January 25, 1989;.
WATER
RESOURCES
RESEARCH,
VOL. 25. NO. 6, PAGES 1403-1411, JUNE
1989
Extreme Precipitation Events Generated by Periodic Processes Rocco
BALLERINI
Department of Statistics, University of Florida, Gainesville PETER WAYLEN
Department of Geography, University of Florida, Gainesville A procedure for the estimation of large quantile daily precipitation totals is developed. The technique draws on the statistical properties of extremes generated by processes with periodic variances. Forty-one years of records at Homestead, Florida, are used to calibrate the model and are incorporated into the simulation procedure that comparesthe proposed technique to standard methods of analyzing annual series. Results indicate that the procedure yields reasonable estimates of large quantile precipitation and performs as reliably, or more so, than the standard techniques. The physical and theoretical bases of the procedure are maintained throughout the study.
1.
INTRODUCTION
Ballerini and McCormick
Knowledge of the probability of very large, rare, daily precipitation totals is of considerableinterest to many applied fields of hydrology, particularly in the estimation of flood frequency characteristics and the design of hydraulic structures. The daily precipitation total is one of the most widely collecteditems of hydrometeorologicalinformationavailableto hydrologists. Frequently, estimates of the annual maximum precipitationamount sc, of specifiedreturn period (in years) T, are required. Traditionally, practitionershave selecteda probability distribution H of the historic annual maxima and eval-
uated it at the (1 - (1/T))th percentilesuchthat H(sCr)= (1 (l/T)) [Hershfield, 1961]. This method is often advocated by government agencies [Watt and Nozdryn-Plotnicki, 1983]. Most commonly either the extreme value type I (EV 1) distributionis employed [Demar•e, 1985]or the generalizedextreme value (GEV) distribution[Jenkinson,1954]. Sneyers[1984]also consideredthe possibility of mixed processesgeneratingthe annual maxima.
The theoretical
basis for the selection of an extreme
value
distributionto fit annual maxima is often, mistakenly,given as extreme value theory [see U.S. National Research Council, 1988,p. 19]. Most hydrologicvariables, includingdaily rainfall totals, are highly seasonalwithin a year and do not meet the assumption of independent and identically distributed (IID) random variables of extreme value theory [Todorovic, 1978]. This paper derivesthe quantity of daily precipitationof a fixed, large return period by reexamining the precepts of extreme value theory along the lines of those used by Weissman [1978] and Boos [ 1984]. In this case, only the tails of the distribution of the annual maxima are consideredby means of the examination of the distribution of the largest daily precipitation record in a period. A similarapproachis currently advocatedin the field of flood researchby Smith [ 1987a].This placesgreater emphasis on the need for the correct estimation of the distribution of the underlyingprecipitationgeneratingprocessbut avoids the necessityof specifyinga distributionof the annual maxima, which previous studieshave required.
[1989] have illustrated the as-
ymptotic behavior of the distribution, P(M n _< 0 of the maximum (or record value) of n observations, Mn from a realization of a process with a periodic variance, period p. The probability of the largest observation being less than the quantile set can be regarded as being equivalent, in an investigation of the annual series of length m years (m n/p), to (1 - (l/T)) m [Boos, 1984]. This equivalence permits estimates of the quantile to be made from the asymptotic behavior
of the record
value.
The largest daily precipitation totals in each of 490 months (40 years 10 months) of records at Homestead, Florida, are used to illustrate the technique. The comparative performances of the proposed method and standard techniques of annual maxima frequency analysis using the lognormal, GEV and EV1 (Gumbel) distributions are measured by the calculation of mean, variance, and root mean square error (RMSE) of estimates of the T year daily precipitation total, T -- 100, 200 ..... 900, 1000, 2000 ..... 9000, 10,000. The exact physical interpretation of the extremely high return period precipitation is open to question [Klemes, 1986], although the largest daily rainfall totals in both the study area and other parts of the world are generally associated with hurricanes and tropical storms [Gray, 1973; Jordan, 1985]. The primary reason for their estimation in this paper is to gauge the performance of the proposed technique in comparison to the established methodologies under a variety of conditions, similar to the work in paleoflood prediction [Hosking and Wallis, 1986]. 2.
THEORY
Let {X,7}denotea discretetime, strictly stationarystochastic process with marginal distribution function given by F.
We assumethat {X,•} satisfiesthe very weak conditionsof asymptotic independence described by Ballerini and McCormick [1989]. Since these conditions are usually met in most practical applications, they will not be presented here. Now supposeh(.) is a periodic function on the integers with some
fixed integer periodp _> 1, and define the process{Y,} by Y• = h(n)X,
Copyright 1989 by the American Geophysical Union.
n -> l
(1)
Note that if the secondmoment of F is finite, then { Y,} has
Paper number 89WR00312.
a periodicvariance function givenby Var (Y) = h2(j)o -2
0043-1397/89/89WR-00312505.00 1403
1404
BALLERINI
AND WAYLEN:
GENERATION
OF PRECIPITATION
EVENTS
2. Supposethere exist sequences{am}and {bm}suchthat
am> 0 andFro(X/am + bm)-->e-x-• asm --->oofor somea > 0. Then
lim P M,-< -- + bm = (e- X •)o am
=0
O< x < oo
x-
