In Areas Characterized by Different Precipitation Regimes and Different
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JOURNAL OF TilE AMERICAN WATER RESOURCES ASSOCIATION VOL. 33, NO.2 AMERICAN WATER RESOURCES ASSOCIATION APRIL 1997 STATISTICAL CONSIDERATIONS ON THE RANDOMNESS OF ANNUAL MAXIMUM DAILY RAINFALL' Francesco Lisi and Vigilio Villi2 ABSTRACT: Annual maximum daily rainfall data from nine sta- annual maximum daily rainfall. In general, these tions throughout the southern slopes of the Eastern Italian Alps hydrologic data often appear randomly scattered with record length of 67-68 years have been analyzed with the aim around a mean value (Figure 1). For this reason they of verifying if their internal structure justifies the assumption of independence and identical distribution, or the "White noise can be thought of as a sequence generated by a "white hypothesis." The approach is to consider the hypothesis H0 of white noise" (WN) process, that is a realization of indepen- noise as the intersection of several sub-hypotheses, each concerning dent and identically distributed (i.i.d.) random vari- one of the characteristics of a white noise process. To this end the ables. nine series were subjected to various statistical tests regarding ran- In Italy annual maximum daily rainfall represents domness, independence, change-points, and predictability. The results are examined first individually and then globally. They indi- the maximum cumulated rainfall, falling between cate that in eight of the nine considered time series the "white 09.00 h of the previous day and the same hour of the noise hypothesis" was rejected. considered day, for each calendar year. The precipita- (KEY TERMS: annual maximum daily rainfall; white noise; tion is sampled by tipping-bucket raingauges with change-point; prediction.) capture funnels of 0,1 m2. In this work, nine time series of annual maximum daily rainfall published in the Hydrologic Bulletins of the Italian Hydrologic Office are analyzed. INTRODUCTION The series belong to stations located on the south- ern side of the Alps (more precisely in Eastern Italy), The basic objectives of applying statistics to hydrol- in areas characterized by different precipitation ogy are the derivation of information from observed regimes and different average annual rainfall. hydrologic phenomena of the past, and the prediction The selected time series are those of Monte Maria- of what is expected in the future (Yevjevich, 1972). Marienberg (1335 m a.s.l., 673 mm/y.), Vipiteno- In this context, the data used most in hydrologic Sterzing (948 m a.s.l., 773 mmly.), Bressanone-Brixen design problems are those related to variations in the (560 m a.s.l. 668 mm/y.), S. Martino in Badia- pattern of rainfall and those related to the maximum St.Martin in Thurn (1393 m a.s.l., 748 mm/y.) and intensity (depth/time) in a given time interval. These Dobbiaco-Toblach (1250 m a.s.l., 880 mm/y.) located in data, in the form of long records, are analyzed to the South Tyrol (which is a bilingual region). Musi define the frequency difference between rainfall mea- (633 m a.s.l., 3320 mm/y.), Venzone (230 m a.s.l., 2160 süred over different time periods and to detect the mmly.), Udine (146 m a.s.l., 1370 mmly.) and Schio rainfall occurrence pattern for one day or for other (234 m a.s.l., 1520 mm/y.) are located, respectively, in definite time intervals. Friuli-Venezia Giulia and in Veneto, at the foot of the In the specific fields of the derivation of river dis- prealpine relief. All the series run, without missing charges and of mitigation of risk connected with flood data, from 1924 to 1990 or 1991. events, one of the hydrologic variables used most is 'Paper No. 96001 of the Journal of the American Water Resources Association (formerly Water Resources Bulletin).Discussions are open until October 1, 199r7. 2Respectively, Researcher, Department of Statistics, University of Padua, via San Francesco, 33, 35121 Padua, Italy;and First Researcher, Institute for Hydrogeological and Geological Hazards Prevention, National Research Council, Corso Stati Uniti, 4-35020 Padua, Italy. JAWRA JOURNALOF ThE AMERICAN WATER RESOURCES ASSOCIATION 431 Lisi and Villi 100 80 — MonteMaria-Marienberg 0 1924 1932 1940 1948 1956 1964 1972 1980 1988 100 6080 Vipiteno-Sterzing 20 I — 0 Ca 1924 1932 1940 1948 1956 196419721980 1988 100 80 60Ca D 40 I —Bressanone-Brixen 20 0 1924 1932 1940 1948 1956 1964 1972 1980 1988 100 C 80 Ca >.. 60 ——S.Martino in Badia- 40 St.Martin in Thurn 1 E 20 Cd Ca 1924 1932 1940 1948 1956 1964 1972 1980 1988 E E100 C 80 Ca >'60 _________________ Ca 40 ____________________________I — Dobbiaco-Toblach I E 20JvL&jv Cd 0 Ca 1924 1932 1940 1948 1956 1964 1972 1980 1988 Figure1. Time Series of the Annual Maximum Daily Rainfall (dashed line represents the average). JAWRA 432 JOURNAL OF THE AMERICAN WATER RESOURCESASSOCIATION Statistical Considerations on the Randomness of Annual Maximum Daily Rainfall E400 E C Ce >' —Musi 0 I I E C Ce 1924 1932 1940 1948 1956 1964 1972 1980 1988 400 300 200 —Venzone I I 100 E 19241932 1940 1948 1956 1964 1972 1980 1988 300 —Udine 1924 1932 1940 19481956196419721980 1988 E200 E C Ce >' I —Schio I VCe E C C Ce 1924 19321940 1948 1956 1964 197219801988 Figure 1. Time Series of the Annual Maximum Daily Rainfall (dashed line represents the average) (cont'd.) The aim of the present paper is to verify if these tests: they are first individually evaluated and then time series can really be thought of as realizations of related to each other. a white noise process. The topic of characterizing the internal structure of time series has already been con- sidered in the literature, above all, in studies on cli- matic variations (W.M.O. ,1992;Cavadis, 1992; TESTING APPROACH Mitosek 1992). However, with the extension of the time series, hydrologists should be aware that the We can think of a white noise as: standard assumption of the use of most common prob- ability distribution functions might not be verified. Xg=L+Et (1) For this reason a method of verifying the "white noise hypothesis" is proposed. It is based on a series of where i is a constant (which represents the average) and e is a random error such that E(e) =0Vt,Var(t) JOURNAL OF THE AMERICAN WATER RESOURCES ASSOCIATION 433 JAWRA Lisi and Villi Figure 2. Map of the Eastern Italy, Showing the Location of the Nine Rainfall Gauging Stations. = 2Vt and e1 independent of c for every s k t. By H0:fl H0 such a definition it immediately follows that, with i=1 k (3) respect to a time series {Xt}1, the hypotheses sys- H1:uH1, i=1 tem to verify is the following: H0 is accepted only if all the sub-hypotheses are is accepted, at a fixed significance level, and rejected if f H0: {x, }generated by a WN process (2) H1: {x } is generated by another type of process at the least one of the sub-hypotheses is rejected. In this study, we look at this diagnostic problem consid- When the data generator process (DPG) is a WN, the ering the following four sub-hypotheses: resulting series must not show any trend component, periodic component, change-point, or serial correla- H0,1 : randomness (no trend or systematic oscilla- tion. Moreover, the series have to be completely tion s). unpredictable. After these propositions, the system H0,2 : no serial correlation. (Equation 2) can be replaced by a new system, built H0,3 : no change-point. considering the intersection of several sub-hypothe- H0,4 :unpredictability. ses, each referred to a specific characteristic of a WN process. However, this set of sub-hypotheses can be further extended, when other characteristics of a WN are individuated. JAWRA 434 JOURNAL OF THE AMERICAN WATER RESOURCES ASSOCIATION Statistical Considerations on the Randomness of Annual Maximum Daily Rainfall The decision of which test to use, on the basis of sequence. Let's transform this sequence into a binary the data characteristics and also of the test character- sequence according with: istics (known versus unknown distribution, known dependencies in the data, number of data.. .),ledus _Ji ifx1—xO to a nonparametric approach as most recommended in Yj.—1o ifx1—<O the specific literature. While some tests are very com- mon, others are not so well known. Amongst the lat- A run is defined as a succession of one or more identi- ter we have considered the contributions coming from cal symbols which are followed by a different symbol the literature of chaos. or no symbol at all. In the above-below runs test when A problem that arises from the hypothesis system the jth element is larger than the mean of the time (Equation 3) is the significance level of the global test. series, a run above starts; when it is smaller, a run When the sub-hypotheses are tested independently below starts. We can observe when a sequence is (i.e., on different data sets) then the global level czgis above its mean, and for how long, and when it is below its mean, and for how long. k Clues to the lack of randomness are provided by ag= Xi any tendency of the symbols to exhibit a definite pat- i= 1 tern in the sequence. Too few runs, or an excessive number of long runs, is usually indicative of some sort It easy to see that the resulting test is conservative of trend, while an excessive number of short runs can with respect to H0. In fact, to have a global signifi- indicate some regular oscillation (for example cance level of 5 percent, with ajconstant,it is neces- 001100110011.