TREND AND VARIABILITY ANALYSIS OF RAINFALL SERIES AND THEIR EXTREME

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1 TREND AND VARIABILITY ANALYSIS OF RAINFALL SERIES AND THEIR EXTREME EVENTS J. Abaurrea, A. C. Cebrián. Dpto. Métodos Estadísticos. Universidad de Zaragoza. Abstract: Rainfall series and their corresponding extreme event processes are analysed in order to study the evolution of their mean value and variability. Two broad statistical approaches are used, one based on inference of the entire time series and the other on modelling the extreme event process. The performance of two precipitation indices, SPI and percentiles, has been compared using long, medium and short time scales. We present the results corresponding to Huesca and Murcia rainfall records, two markedly different locations in Spain. 1. Introduction Interest in climate variations has experienced a significant increase in recent years due to the important economic and social consequences connected with extreme weather events. Most of the studies regarding climate change only seek to detect potential trends or fluctuations in the long term mean of climatic signals, but the study of variability changes and extreme event behaviour is also essential. This interest is enhanced since some recent climate models, Karl and Knight (1998), announce that climate changes in the 21st century will result in an increase of extreme events. The aim of this paper is to analyse long rainfall records and their corresponding dry and wet extreme event series in order to detect trends in their mean value and variability. The monthly rainfall records of Murcia from 1901 to 1994 and Huesca from 1895 to 1995, two observatories located in widely different regions of Spain, have been studied. A longer record from Huesca is available, but it 1

2 perc time Figure I: Percentile series of Huesca from 1895 to apparently shows an anomalous behaviour before 1895 (see Figure I); so, to avoid a likely inhomogeneity, only observations after that date have been considered. There are two broad statistical approaches to this problem; under the first one, presented in section 3, inference is based on the entire time series and, under the second, presented in section 4, on characterizing and modelling only the extreme event process. 2. Precipitation indices The use of a uniform and worldwide applicable precipitation index would greatly facilitate the assessment of changes in general precipitation patterns and the comparison of different region results. So, indexes that are independent of local climatic features should be used. Two signals with these properties have been described in this work: the standardised precipitation index series or SPI (Mc Kee et al. (1993)) and the percentile series associated to the rainfall record. A simultaneous analysis of both has been performed in order to compare their qualities. Percentile series are obtained by transforming original observations into rank proportional values varying between zero and one. To calculate the SPI, observations from a long record are fitted to some 2

3 probability distribution which is then transformed into a standard normal distribution, so that mean and standard deviation of the SPI are zero and one respectively. Thanks to this normalisation, wetter and drier climates can be represented on the same scale. For the SPI calculation we have tried out four distributions, Weibull, Gamma, Lognormal and Exponential. The gamma distribution has provided the best or one of the best fits for the studied series and it has been selected to fit all of them. Precipitation below normal over a period of time is the first sign of drought. Depending on its duration, the rainfall shortage can give rise to hydrological and agricultural drought. This makes it necessary to characterize meteorological drought at different time scales; so, we have calculated the SPI and percentiles from monthly series where each observation corresponds to the accumulated rainfall during the p previous months. Three accumulation periods, p = 3, 12, 24, have been considered. From now on, M3, M12, and M24 will denote the Murcia series and H3, H12, and H24, those of Huesca. Due to the presence of a seasonal component, when the accumulation period length is not a twelve month multiple, p=3 for example, 12 different series, one for each month, have to be considered to calculate the SPI and percentile series. Afterwards, calculated values are reorganised in a global series. 3. Analysis of the entire time series Our objective in this analysis is to detect a possible time trend in the mean value and changes in the variability of the observations around that level Analysis of the mean value To analyse the mean level, the signal has been smoothed using Lowess, a robust locally weighted regression smoother. A suitable bandwidth has been selected in each case: a 35% one, that corresponds to 35 years, for p=12 and 24, and a 30% one, corresponding to 30 years approximately, for p=3. 3

4 p=24 p=12 p=3 0 SPI time 0 Percentiles time p=24 p=12 p=3 0 SPI time 0 Percentiles time Figure II: Murcia (top) and Huesca (bottom) smoothed series. 4

5 The smoothed SPI and percentile series corresponding to the three accumulation periods are shown in Figure II, where p=24 and p=3 curves have been displaced to show them distinctly on the same plot. Murcia shows a decreasing trend until 1935 and, after a small fluctuation, the signal can be considered stable or slightly increasing since The mean Huesca profile increases slightly until 1940; after a brief fluctuation it reaches its absolute maximum a bit earlier than 1970; from that moment on there has been a persistent decreasing trend. It is interesting to point out some general features from these results, a.- The mean profiles corresponding to the three accumulation periods do not show significant differences, although the ones corresponding to shorter time scales are smoother. b.- SPI and percentile smoothed curves show a very similar shape, suggesting that both signals properly represent rainfall evolution. c.- Finally, it must be pointed out that rainfall behaviour of two not very distant locations, such as Huesca and Murcia, are markedly different, Figure III. Huesca Murcia 0 SPI time Figura III: Murcia and Huesca p=12 SPI smoothed series Variability analysis Variability analysis is based on residuals, defined as the difference between the observed and the smoothed series. The SPI is a better signal for studying variability since its values are not bounded 5

6 between 0 and 1 (like the percentiles). Boxplots of residuals using both disjoint and overlapped time periods and 12-month moving range plots have been performed. The H3 moving range behaviour is clearly not homogeneous, Figure IV; there has been a continuous increase of variability since 1975, confirmed by the analysis of the boxplots. Curiously, the variability change disappears in the H12 and H24 series. Moving range t Moving range t Figure IV: Moving range of SPI residuals from Huesca; H3 (top) and H12 (bottom). The M3 moving range shows an increase of variability from 1975 to 1985 and returns afterwards to previous level values. Boxplots suggest an increase of extreme events starting in about 1970, which since 1990 has only been observed in the wet extreme events. As for Huesca, this effect becomes unnoticeable with longer accumulation periods. 6

7 In conclusion, there is a significant growth of variability, more evident in Huesca, limited to the short term signals. This can be explained by the increase in the short term scale of both dry and wet extreme events. In longer accumulation periods the effects of wet and dry extremes cancel each other out and, consequently, the variability remains stable. In order to investigate this behaviour, each month rainfall smoothed profile has been compared in an exploratory analysis. They are rather heterogeneous and it has been observed that only some months show the global profile. In particular, the Murcia slight increase after 1960 is observed only in February, March, May and September; an opposite trend is detected in April and June. The Huesca final decreasing trend is only observed in February, March, May, June and July, while September and, in general, autumn months show an increasing or stable final profile. 4. Extreme event analysis Once the entire series has been studied, we focus on the extreme event process. First of all, an extreme event definition, allowing the extreme dry and wet events to be extracted from the entire series, has to be set up. An operative definition capable of identifying the beginning, end and magnitude of extreme events is based on Excess Over Threshold methodology. In this approach, a stochastic process, s(t), related to precipitation or some other variable describing the hydrological state of the system, is compared to a threshold, u(t), which represents a critical level for the process. A dry or wet extreme event will occur when s(t) is below or above u(t), respectively. Two standard SPI values, -1.5 and 1.5, have been proposed as extreme dry and wet thresholds; they correspond to the 7th and 93rd percentiles, respectively. To characterize each event size, we use three variables, length or duration, L, severity, S, (deficit, D, or excess, E, depending on the type of extreme) and maximum intensity, MI. Their definitions for a dry event are illustrated in Figure V. 7

8 Figure V: Defining variables of dry event magnitude Frequency analysis The extreme value theory, Davison and Smith (1990) and Embretchs et al. (1997), asserts that the occurrence of high level excesses in independent or short term dependent stationary stochastic processes behaves asymptotically as a Poisson process, PP. In a previous work, Cebrián (1999), it was verified that occurrence of dry periods defined with 10th or lower percentiles in p=12 series, behaved as a Poisson process. If occurrence rate is constant, recurrence times, i.e. time intervals between two consecutive events, must be homogeneous. Randomness tests (run tests), general trend tests (Kendall and Spearman time correlation tests and von Neuman rank test), and trend tests for Poisson processes (Laplace, Lewis-Robinson, and Military Handbook tests), as well as some graphical analyses have been performed to contrast recurrence time homogeneity. Three series, one for dry, one for wet and one for both types of extreme events together, have been analysed for each location. As in the entire series comparison, there are no noticeable differences between extreme event processes obtained from the SPI and percentile series so, only results from SPI extremes are shown. None of the Murcia series shows a significant trend in the extreme event rate occurrence, no matter what accumulation period we use. In the H3 series there is a graphical evidence of an increase in the occurrence rate of extreme events, although it is not significant according to the performed test p- values. This increase is mainly due to dry events, as can be seen in Figure VI, where a decrease of 8

9 recurrence time mean since the 40th observation, corresponding to the change of slope in the cumulative event number plot in about 1980, is observed. This effect is a consequence both of the drop in the mean level and the growth in signal variability; it disappears in H12 and H24 for both types of extremes since, as we saw in section 3, there is no increase of variability in these signals. Recurrence times Event index Cum. occurrence number Time Recurrence times Cum. occurrence number Event index Time Figure VI: Recurrence times and cumulative event number versus time; H3 (top), H12 (bottom) Analysis of the extreme event magnitudes Analysis of the extreme event properties is based on the already mentioned variables, duration, severity and maximum intensity. In order to check their time homogeneity, we have used the aforementioned tests, with the exception of the Poisson specific ones, and bubble plots. In these plots, the series of the extreme event observations is represented in chronological order, using as x-axis an index i from 1 to n. Each observation is drawn using points proportional in size to the value of a variable whose time evolution we want to study; for example, in this work, variables, L, S or MI. The representation of the occurrence month in the y-axis also allows us to analyse seasonal homogeneity. Some of the performed test p-values are shown in Table 1. 9

10 Table 1: Trend and randomness test p-values for MI. Murcia Huesca Wet, p=3 Wet, p=12 Dry, p=3 Dry, p=12 Wet, p=3 Wet, p=12 Dry, p=3 Dry, p=12 Rmean Spearman Von Neuman Cox-Stuart In M3 there is an increase in maximum intensity and severity series starting about For dry events it disappears by 1985, while for wet ones it persists. This increase can not only be explained by the slightly positive trend in the rainfall time series since similar mean levels have been observed previously in M3 without so high wet extreme events having been recorded. Neither M12 nor M24 series shows a positive trend, Figure VII. In H3 a clear increase in severity and maximum intensity of events starting in about 1970 has been observed. It is noticeable both in dry and wet extreme events, in spite of the mean level decreasing. trend. So, we can conclude that this increase is mainly originated by the growth of the signal variability. month i month i Figure VII: Excess (E) bubble plot of wet events; M3 (top), M12 (bottom). 10

11 month i month i Figure VIII: Maximum Intensity (MI) bubble plot of dry and wet events; H3 (top), H12 (bottom). we noticed in section 3. Again, this effect disappears using longer accumulation periods; Figure VIII shows an example of this effect 5. Conclusions Overall, there is no evidence that extreme weather events and climate variability have increased in a global sense, but regional studies suggest changes in some extreme and variability indicators. These changes are too small to confidently reject the stationarity hypothesis, as they could be a reflection of natural variability climate; however they may be large enough to be of some practical importance. Thus, more detailed analyses are necessary to identify which regions and which type of events are changing 11

12 Although two locations are not representative enough to assert general conclusions about the 20 th century changes in Spain, results from this work suggest three main facts: - Spatial heterogeneity of rainfall mean level pattern evolution. Two not very distant locations, such as Huesca and Murcia, show noticeably different mean profiles, with an opposite trend in the last period. This agrees with results obtained for other parts of the world, as the ones presented in the special issue on Weather and climate extremes: changes, variations and a perspective from the insurance industry of Climatic Change. (Plummer et al. (1999), Gruza et al.(1999)). - Increase of short-term signal variability and extreme events since 1970 in both series, stronger in Huesca. - Stability of the variability and extreme event processes in medium and long-term scales. As for precipitation indices comparison, it can be concluded that the SPI and percentile series work well in the study of mean evolution and extreme series, giving rise to similar results. The SPI is better for studying variability but it is harder to obtain. Thus, in general, the SPI is preferable. References Cebrián, A.C., (1999). Análisis, modelización y predicción de episodios de sequía. Non-published Ph.D. dissertation Davison, A.C. and Smith, R. L., (1990). Models for exceedances over high thresholds. J. R. Statist. Soc. B, 52, 3, Embrechts, P. et al., (1997). Modelling extremal events, Springer. Gruza, G. et al., (1999). Indicators of climate change for the Russian federation.. Climatic Change, 42, Karl, T. R. and Knight, R.W., (1998) Secular trends of precipitation amount, frequency and intensity in the United States. Bull. Amer. Meteor. Soc., 79, McKee, T.B., Doesken, N.J. and Kleist, J., (1993). The relationship of drought frequency and duration to time scales. Preprints, 8th Conference on Applied Climatology, Anaheim, CA, Plummer, N. et al., (1999). Changes in climate extremes over the Australian region and New zealand during the twentieth century. Climatic Change, 42,