Probabilistic Definition and Analysis of Severe Rainy Events

Probabilistic definition and analysis of severe rainy events

0, Terranova National Research Council - Research Ikstitute for Hydraulic-Geologic Protection in Southern and Insular Italy (Italy)

Abstract

Human activities are often subject to damages resulting flom exceptional rainy events. The character of exceptionality of a rainy event may be due only to one or to many factors. In particular the total rainfall amount the maximum intensity, the average intensity and the total duration affect the natural catastrophic phenomena. Based on this concept, a methodology to identi~ severe rainy events is proposed, with the aim to select and characterize those events potentially more dangerous to human activities. The single normal rainy event is simply defined by being preceded and followed by at least one not rainy day; the events are considered severe according to the overcoming of one or more threshold values of the aforesaid factors. The study refers to the method of peaks over threshold (P.O.T.), based on the theory of the rare events and on the extreme values theory. The fust theory was introduced by Poisson and aims exclusively to define the relations between the number of events and their low probability of occurrence; the second theory is usually adopted for the annual maximum analysis and deals just with the size of the events, with no account for their number. Actually the reduction law of the number of events when the threshold values are increasing is illustrated by simple correlative relationships. An application to rainfall data of the North area of Stretta di Catanzaro (Calabria, Southern Italy) is proposed, comparing the results with those of an hktorical investigation regarding landslides and flooding. The analysis allows to assess the temporal and spatial distribution of the most severe events and to evaluate their hazard for forecasting and/or real-time alert system identification. Particular sites and year periods that are more frequently subject to severe events are identified, in this way we can identi~ homogeneous regions keeping constant values of parameters. © 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Web: www.witpress.com Email [email protected] Paper from: Risk Analysis III, CA Brebbia (Editor). ISBN 1-85312-915-1

532 Risk Analysis III Introduction

Rainfalls, when are marked by exceptional characteristics, concur to prime physical processes that often are catastrophic natural phenomena (CNP). Floods and slope instability are examples of the possible effects of exceptional rainfalls. The research is set in the ambit of the risk deftition and analysis corresponding to rainy events potentially able to prime natural phenomena as supefilcial slope instability, floods and spate in basins of proper extension. Rainfalls, together with the elements that determine the ambient scenery we refer to, dictate terms and ways in which a particular kind of natural phenomenon could be primed and concur to guide the evolution. The time detail we use to describe rainfall events in this research is that of daytime; therefore the methodology used in this case is suitable for physical processes describable in this scale. In particular individual rainy events are determined by being composed by not null consecutive values sequences preceded and followed by a null value at least.

Methodology

The time series of rainy events are analysed following a methodology that refers to P.O.T. ( Peak Over Threshold ). This approach is based on extreme value theory presented in 1970 by Todorovic and developed, as regards the flood frequency analysis, by Zelenhasic [13] andRousselle[11]. At the moment the research aspects regarding the individuation of probability laws that better fit to interpret P.O.T. series are faced applying sample frequencies obtained imposing variable threshold values. The potentiality that a rainy event originates CNP can be synthesised through triggering factors as the total rainfall P,,, the Duration D,,, the maximumintensity I~Wand the medium intensity I~d; other factors as the peak position and the ratio r between Im.dand 1~= can describe the form of the rainfall event. In effect the slope or basin condition before the occurring of the examined event has a determinant importance [2]; these conditions can be determined by factors, called predisposing factors, as the Zk rainfalls totalized ink days before the rainfall peak, the Ae,.e, time lapse between the examined event and the previous one and, in the end, the H,..P, total rainfall of the event preceding the examined one, In this work at the beginning we examine some triggering factors and then the predisposing factors and the remaining ones, just for the events characterized by prefixed peculiarities. For the generic triggering factors x we want to determine the risk ~ or the exceeding probability P, that x overcomes a prefmed threshold value S:

P(xX3) = 1- Js P(X) dx . -m © 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Web: www.witpress.com Email [email protected] Paper from: Risk Analysis III, CA Brebbia (Editor). ISBN 1-85312-915-1

RiskAnalysis111 533 In the end, by applying proper threshold values, we can individuate the pluviometrical events that are more severe as regards the considered factors. The territory interested by an event can be individuated analysing the series of the stations in that area; anyway these series have to be made comparable. In fact an event that in a particular area presents characteristics of danger it could be considered absolutely normal in another area, In this order, limiting the analysis to P,v, I~,Xand l~.d factors, it’s convenient to refer the daily rainfall values and the applied threshold to the local values on Normal Rainy Day (GPN) [4], that is the ratio between the mean of the annual total rainfalls (MAR) and the number of rainy days (NGP), In the end for each area and for each factor we can construct cumulated frequency curves and it is possible to select the events overcoming a prefixed return time T=l/AR, in which A is the annual medium number of events, To events having T superior to the prefixed one we give the denomination of severe events following the parameter x (SevEv.J. Events resulting severe respect to many factors xl, Xz,X3.,, are symbolically indicated with SevEvxl, ~z,*3,,,, and have a peculiarity of damage higher than those events that are severe respect to an inferior number of factors.

Application

The shortly illustrated methodology has been applied to the daily rainfall series of the stations settled in the North area of Stretta di Catanzaro. For these 17 stations( fig. 1), situated at variable altitudes from 6 to 1250 m slm, the available temporal daily rainfalls series are constituted by record in duration from 12 to 73 years. The typical pluviometrical regimen of this area is strongly seasonal arid the limited number of rainy events occurs during the autumn and the winter basically.

Figure 1: Position of rain gauge stations. © 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Web: www.witpress.com Email [email protected] Paper from: Risk Analysis III, CA Brebbia (Editor). ISBN 1-85312-915-1

534 Risk Analysis III To give a preliminary hydrologic characterization of the area, we have transfemed in figure 2 the ratio between the yearly medium rainfalls and the altitudes ( fig. 2a) and the ratios between the monthly medium rainfalls and MAR (fig. 2b).

I I a) Regression between Elevation and MAR

I Elevaticm [m a.s,l,]

I b) Ratio between the Mean Monthly Rainfall (MMR) and MAR ... .._. _ 0.21T“”””—--’’--’” 1

Jan Feb Nar AP May hr. M Aug $’ep Ott Nm hC

Figure 2: General hydrologic characterization: a) MAR as a fi.mction of altitude; b) Mean ‘monthry rainfall (MMR) as ‘a function of MAR and of the month.

The climate, the geo-lithology, and the morphogenesis that are typical of this territory make it disposed both to superficial slope instability and to alluvial phenomena. CNR-IRPI’S research groups of Cosenza have studied the historical frequencies of these phenomena [8], moreover they have characterized the proper geotechnic aspects [1] too. These researches attest the high temporal frequency and the wide spatial diffusion of CNP; to give basically a methodological contribution to the risk analysis of rainy events that potentially can prime CNP, we will use only the historical information about CNP reported by Rizzo & Fragale [8], even though other bibliographical sources about CNP in the examined area are available, © 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Web: www.witpress.com Email [email protected] Paper from: Risk Analysis III, CA Brebbia (Editor). ISBN 1-85312-915-1

Risk Analysis III 535

The altitudes on sea level, the number of available years, the M NGP and GPN values for the examined rain gauge stations are reported on table I. Carrying out the elaboration on adimesional basis, as we said in the last paragrapk and using threshold values S1, Sz, S3that can vary horn Oto 40, from Oto 25 and ftom Oto 10 respectively for P,v, I~u and 1~,~,we individuated the distributions of accumulated frequency for every station on the basis of a yearly medium number of “events with a threshold equal to O“.

Table I. Altitudes on sea level, number of available years, MAK NGP and GPN values for the examined rain gauge stations:

series Rain Gauge Elevation length MAR N“ Station (m aos.l.) (years) (mm) NGP GPN 1 Monaco 1250 32 1596.0 130.9 12.2 2 Albi 717 66 1222.9 107.5 11.4 3 S. Elia 650 49 1168.9 116.3 10.1 4 Catanzaro 343 73 962.8 117.9 8.2 5 Catanzaro Lido 6 61 788.9 93.2 8.5 6 Carlopoli 950 53 1406.2 123.0 11.4 7 Fiorenza 1126 37 1285.6 129.8 9.9 8 Umbri 885 37 1138.9 116.5 9.8 9 Olivella 360 32 1188.2 98.9 12.0 10 Girnigliano 550 49 1258.2 104.2 12.1 11 Borgia 332 69 373.8 103.6 3.6 12 Girifalco 450 59 1421.8 126.0 11.3 13 Tiriolo 690 43 1376.1 128.6 10.7 14 Marcellinara 330 61 1176.6 115.9 10.1 15 Caraffa di Catanz. 370 64 1238.1 96.6 12.8 16 Vena di Madia 240 12 1241.4 101.7 12.2 ~ 17 Maida 300 48 1211.1 117.6 10.3

tixamples of these distributions for P,v, Imax and Imd are diagramed on Gunbel% cartograms in figure 3. The low adaptability of the- exponential probability law is put in evidence by typical “separation phenomenon” due to the presence of outliers [7; 6; 9]; to tit these kind of curves, have been proposed [10; 3] probabilistic laws as GEV law, generalised Pareto’s law, Weibull’s law, etc.. The resulting diagrams enable us to determine the corresponding threshold values Si and then the SevE~, for i= 1,2,3, after having established a value of T acceptable for the examined CNP. © 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Web: www.witpress.com Email [email protected] Paper from: Risk Analysis III, CA Brebbia (Editor). ISBN 1-85312-915-1

536 Risk Analysis III 8 -~ --~~~~------;;-3- ~~~~~~~------~~~~~~------~-- T

-50 71 ‘=’’’’;$OoooF~~ OS ~zo

-lo

-5

01° 0 10 20 30 40 xl Figure 3: Cumulative distribution function for P,v, I~u and 1~,~on Gunbel’s probability cartogram: example of identification of the threshold values for T=l O.

Results analysis, conclusion and study prospects

The methodology we propose enable us to determine in an objective way the S1, S2, S3 threshold values on the basis of a prefmed risk level, therefore it enables us to select the past most severe events. Fixing T as 10 for example, the threshold values and the number of SevE~ are reported for each station in table II; of course to a single SevEv it is possible to associate the date in which it occurred. From the dates in which CNP occurred it is possible to have a conftont with the dates in which SevE~ occurred; in table III we reported the dates in which this correspondence has been verified, limitedly to the period 1951-1990. In fact, Rizzo & Fragale [8] historical inforrnations stop at 1990 and before 1950 are scarce and of worse quality. Of course best correspondences could be found assuming different return times for P,,, I- and l~.d or just for some of them. Interesting observations on pluviometrical sceneries that primed CNP can be obtained both, from the contemporaneity of SevEv in many places and from the calculation of the remaining factors, priming and predisposing factors, ignored before. © 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Web: www.witpress.com Email [email protected] Paper from: Risk Analysis III, CA Brebbia (Editor). ISBN 1-85312-915-1

Risk Analysis III 537 Table II. Threshold values and the number of SevEvi for T=lO,

Rain Gauge ~1 N“ N“ N“ s* S3 Station SevEvli SevEv2 SevEv3 Monaco 409.7 3 205.2 0 109.6 2 Albi 382.9 4 191.8 2 102.4 8 S. Elia 339.2 6 169.9 4 90.7 1 Catanzaro 275.4 4 137.9 7 73.7 7 Catanzaro L. 285,5 4 143.0 7 76.4 7 Carlopoli 382.9 6 191.8 2 102.4 2 Fiorenza 332.5 0 166.5 1 88.9 0 Umbri 329.1 5 164.9 2 88.0 1 Olivella 403.0 1 201.9 3 107.8 5 Gimigliano 406.4 3 203.5 2 108.7 2 Borgia 120.9 30 60.6 39 32.3 36 Girifalco 379.5 7 190.1 2 101.5 6 Tiriolo 359.4 4 180.0 3 96.1 1 Marcellinara 339.2 3 169.9 4 90.7 3 Caraffa di CZ 429.9 1 215.3 1 115.0 5 Vena di Maida 409.7 0 205.2 0 109.6 0 Maida 345.9 1 173.3 1 92.5 0

Other observations regard the period of the year with a major tiequency of SevEv that, in this specific case is shown in figure 4. In the end it ‘has been proposed an~ verified the applicability of a methodological approach to individuate and analyse, in a probabilistic sense, pluviometrical events able to prime CNP. The results of this research underline interesting prospects about the applicabili~ in the alert system and real time prevision ambit. Extending the examined area, the individuation of theoretic frequency distribution, of the structure of statistic independence between the random variables P,v, I.= and I..d and the arrangement of the relations found in a regional analysis, will contribute to improve the knowledge about rainfall events related to CNP’Sprime and evolution.

Thanks

We thank Prof. Ennio Ferrari from University of Calabria and Ing., Giovanni Gull/i for the valuable advices they gave during the research to draw up this work. © 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Web: www.witpress.com Email [email protected] Paper from: Risk Analysis III, CA Brebbia (Editor). ISBN 1-85312-915-1

538 Risk Analysis III Table III. Dates (year and number of the day) of SevEvi compared to CNP Occmence: A = l~dslide; # = flood; m = not available data.

I Rain gauze Stations I ------.-. 11(3<11 I I I I lo-ml } *.->. .

I I I I .:, M

..-. ,, # ,-:.,., I 1963 A 367 1966 280 1969 231 1970 276 276 ~ 1

1327I

293 I I P#=#=P © 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Web: www.witpress.com Email [email protected] Paper from: Risk Analysis III, CA Brebbia (Editor). ISBN 1-85312-915-1

Risk Analysis III 539

Aug Ott Dec Feb Apr Jun

Figure 3: Distribution in time of SevEy.

Reference cited

[1] Antronico L. & Gulhi G, (2000) Slopes affected by soil slips: validation of an evolutive model. Landlides in research, theory and practice, Proc. Of the 8* ISL, 26-30 June 2000, Cardiff, UK, 77-84, [2] Clarizia M., Gull% G. & Sorbino, G. (1988) Sui meccanismi di innesco dei “soil slips”. Proc. Int. Conf. Prevention of hydrological hazard the role of scientific research. Alba (CN), 5*-7* Nov. 1996, Torino, Italy, 585-587. [3] Cruise, J.F. & Arora, K, (1990) A hydroclimatic application strategYfor the Poisson partial duration model. Water Resour. Bull. 26(3),431-442. [4] Gulls, G., Aceto L., Antronico L., Ferrari E,, Sorriso-Valvo M., Tanzi C. & Terranova O. (2001) Linee guida per interventi di stabilizzazione di pendii in aree urbane da riqualljlcare. Programma Operativo Plurifondo 1994/99. Regione Calabria, [5] Madsen, H., Rosbjerg, D. & Harremoi%, P. (1995) Application of Bayesian approach in regional analisys of extreme rainfall. Stoch. Hydrol. and Hydraul., 9(l), 77-88, [6] Matalas, n.c,, Slack, J.R. & Wallis, J.R. (1975) Regional skew in search of a parent. Wat. Resour. Res., 11(6), 815-826. [7] Potter, W.D. (1958) Upper and lower frequency curves for peak rates of runojl EOS. Trans. AGU, 39, 100-105. [8] Rizzo V. & Fragale F. (1999) II rischio geologico nella programmazione territorial. In Teti M. A. I sistemi informative geografigi per la piardjlcazione territorial. Una sperimentazione nell’Istmo di Catanzaro. Rubbettino Ed., Catanzaro (Italy), pp. 75-122. [9] Rossi, F., Fiorentino, M. & Versace, P. (1984) Two component estreme value distribution for flood frequency analisys. Water Resour. Res., 20(7), 847-856. © 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Web: www.witpress.com Email [email protected] Paper from: Risk Analysis III, CA Brebbia (Editor). ISBN 1-85312-915-1

540 Risk Analysis III

[10] Rosbjerg, D. (1987) Partial duration series with log-normal distributed peak values, In: Hydrologic fkequency modeling, Edited by V,P, Sing, 117- 129, [11] Rousselle, J. (1972) On some problems ofjZood analysis. Ph. D. Thesis, 226 pp., Colorado State Univ., Fort Collins, 1972. [12] Todorovic, P. (1970) On some problems involving random number oj random variables. Ann. Math. Statist., 41, 1059-1063. [13] Zelenhasic, E. (1970) Theoretical probabilip distributions for floodpeaks, Colorado State Univ., Hydrol. Paper n“ 42, 35p.