Statistical Modeling of Extreme Rainfall in Southwest Western Australia

Transcription

1 852 J O U R N A L O F C L I M A T E VOLUME 18 Statistical Modeling of Extreme Rainfall in Southwest Western Australia Y. LI CSIRO Mathematical and Information Sciences, Wembley, Western Australia, Australia W. CAI CSIRO Atmospheric Research, Aspendale, Victoria, Australia E. P. CAMPBELL CSIRO Mathematical and Information Sciences, Wembley, Western Australia, Australia (Manuscript received 7 July 2003, in final form 10 August 2004) ABSTRACT Rainfall over southwest Western Australia (SWWA; 32 S southward and 118 E westward) has been decreasing over the past decades, putting further constraints on water resources in an already dry area. In this study, daily rainfall over five geographically dispersed and homogenized weather stations within SWWA are analyzed. A peak over threshold method from the extreme value theory is used to model daily rainfall above a given threshold. The Mann Whitney Pittitt (change point) test was applied to detect changes in annual, winter (May October), and summer (November April) maximum daily rainfall. Change points for winter extreme daily rainfall were found around 1965, based on different individual stations, with the extreme daily rainfall reduced since then. To demonstrate the degree of change in the winter extreme daily rainfall, at 1965 the data were stratified, and generalized Pareto distributions were fitted to the tails of the distributions for daily rainfall in the prechange period of (including 1965) and the postchange period of The fitted tail distributions also allow the estimation of probabilities and return periods of the daily rainfall extreme. Results show that return periods for the winter extreme daily rainfall have increased after 1965, implying that winter daily rainfall extremes in SWWA are lower after 1965 than they were before. There has been vigorous debate as to what forces the drying trend, that is, whether it is part of multidecadal variability or whether it is driven by secular forcings, such as increasing atmospheric CO 2 concentration. In this paper, statistical modeling is also used to identify possible associated changes in atmospheric circulation. It is found that there is a change point near 1965 in a dominant atmospheric circulation mode of the Antarctic Oscillation (AAO). The result offers qualified support for the argument that the AAO may contribute to the drying trend. 1. Introduction Over the past several decades, rainfall over southwest Western Australia (SWWA; 32 S southward and 118 E eastward of the continent) has been decreasing significantly, and average winter rainfall (May October, a rainfall intensive season) for the region over the last 25 yr has been only about 85% 90% of the preceding 50-yr average, which causes a 50% decrease in dam inflow as a result. This has attracted significant interest and attention, particularly in terms of possible causes and policy options in response to the decrease. A Western Australian government research project Corresponding author address: Dr. Yun Li, CSIRO Mathematical and Information Sciences, Private Bag 5, Wembley, WA 6913, Australia. yun.li@csiro.au called the Indian Ocean Climate Initiative (IOCI) was part of this effort. The project examined the effects of modes of variability on interseasonal to interdecadal time scales in the Indian and Southern Oceans upon SWWA climate variability (Indian Ocean Climate Initiative 2002). While not conclusive in terms of possible drivers of the rainfall decrease, an important finding is that since the middle of the twentieth century, winter half-year (May October) rainfall decreases at specific stations, such as Manjimup, and the decrease occurred through reductions in the number and intensity of extreme events (Nicholls et al. 1999; Smith et al. 2000). This behavior of extreme daily rainfall is different from other parts of Australia that show significant increases for the higher percentiles, for example, the 90th and 95th percentiles (Suppiah and Hennessy 1996, 1998). Groisman et al. (1999) showed that there is a 20% increase in the probability of summer daily precipitation 2005 American Meteorological Society

2 15 MARCH 2005 L I E T A L. 853 over a 50.8-mm threshold in the midlatitudes, Tropics, and subtropics in Australia. We note here the stations used by Groisman et al. (1999) did not provide good coverage of SWWA. This may be, in part, why their results are not consistent with those of Nicholls et al. (1999) and Smith et al. (2000). Because the decrease in the number and intensity of extreme events has contributed to the decline in rainfall over SWWA (Nicholls et al. 1999; Smith et al. 2000), in order to explain the drying trend, we need to explain why the number of rain events has declined, and why the rainfall amounts in extreme events have become less. Haylock and Nicholls (2000) studied trends of extreme rainfall in terms of changes of three indices: the number of events above an extreme threshold (extreme frequency), the average intensity of rainfall of extreme events (extreme intensity), and the proportion of total rainfall from extreme events. Both extreme frequency and extreme intensity are shown to have decreased in the SWWA region. The three indices are, however, somewhat arbitrary and not well defined. For example, for the extreme intensity, Haylock and Nicholls (2000) introduced three different empirical ways to calculate it depending on the following various usages of the index: climate change detection, public use (such as explaining to a farmer the change proportion of annual rainfall from the highest 5% of events), and reflection of changes in the shape of a rainfall frequency distribution. While these variations in the method may be appropriate, there is a need to develop a theoretical distribution that describes extreme rainfall and allows changes to be discovered in a more objective way. This is essential for exploring the dynamics of the drying trend, which is a central issue of the present study. The Antarctic Oscillation (AAO) is a major mode of the Southern Hemisphere atmospheric circulation (Thompson and Wallace 2000). Over the past few decades there has been an upward trend with increasing midlatitude mean sea level pressure (MSLP) (Thompson et al. 2000). From the outputs of the Commonwealth Scientific and Industrial Research Organization (CSIRO)-coupled climate model, Cai and Watterson (2002) showed that when the AAO is positive, that is, with increased MSLP in the midlatitudes, rainfall over SWWA tends to decrease as a result of decreased cloudiness and decreased westerly winds a relationship that also exists in observations. Cai et al. (2003) then explore the response of the AAO to increasing and stabilized atmospheric CO 2. They found that in response to increasing CO 2, the AAO displays an upward trend with increasing MSLP in the midlatitudes and an associated drying trend in the SWWA region. Their results on the trend in the observed AAO suggest that the AAO may be a factor contributing to the reduced rainfall over SWWA. We explore this possibility. We use a peak over-threshold method from the extreme value theory to model extreme rainfall from daily rainfall series over five geographically dispersed and homogenized weather stations within SWWA. Here, we call a daily rainfall extreme if it is over the chosen threshold. A threshold value of 30 mm (or 20 mm) is used, which is suggested by a statistical analysis (see section 3b). Our approach is based on a generalized Pareto distribution (GPD), which has an interpretation as a limiting distribution of observations that lie above a given threshold. This idea was first suggested by Pickands (1971) and has been developed by, among others, DuMouchel (1983), Davison (1984), Smith (1984, 1987), Hosking and Wallis (1987), and Joe (1987). In particular, Davison and Smith (1990) use the GDP as a modeling distribution for exceedances over high thresholds, which is similar to our use here. Our modeling approach is aimed at identifying change points in the rainfall distribution. This allows us to investigate possible causes, which must, in turn, have changes in their behavior at matching or nearly simultaneous times. This paper is structured as follows. Section 2 gives a short introduction to the data that we used in this study. Section 3 gives a general introduction to modeling extreme rainfall using a GPD, where we also explore the detection of a change point year of occurrence of extreme daily rainfall using the Mann Whitney Pittitt test (Pittitt 1979; Kiely 1999). A clear change point for winter extreme daily rainfall was found in 1965, with extreme rainfall that has reduced since then. The GPD was fitted to the tail distributions of winter daily rainfall in the prechange period of and postchange period of , respectively. These two-tail distributions are significantly different at the 1% level, based on a likelihood ratio test. Return periods that are based on these two-tail distributions are also obtained. Trends in the AAO index are studied in relation to winter extreme daily rainfall changes in section 4. A summary is given in section Data used Daily rainfall datasets for five geographically dispersed and homogenized weather stations within SWWA were used in our analyses (Fig. 1). These stations (with their respective station numbers) are Manjimup (9619), Jarrahdale (9023), Boyanup Post Office (9503), Pardelup (9591), and the Oaks (10636). Daily data from these stations were selected because they are of a high quality and are most complete (Lavery et al. 1992; Nicholls et al. 1999). These chosen stations all contain long historical records (starting as early as 1889). To avoid problems introduced by missing readings, all of the missing values in this study were substituted by associated patched point dataset (PPD) values maintained by the Queensland Department of Natural Resources and Mines. PPD is a daily meteorological dataset that presents daily weather records for 4650 recording stations around Australia and is maintained by Australian Bureau of Meteorology. It is a daily semisynthetic dataset with no missing data values.

3 854 J O U R N A L O F C L I M A T E VOLUME 18 FIG. 1. Map of the five stations with daily rainfall time series used in this study. 3. Modeling extreme rainfall We aim to model the behavior of the daily rainfall series at stations shown in Fig. 1 at extreme levels, that is, above the chosen threshold. Because we have no specified model for the underlying distribution of daily rainfall, and it is the tail behavior that is of interest here, we appeal to an asymptotic argument that characterizes such behavior (Coles and Tawn 1996). a. Fitting a GPD to rainfall series In this section we give an outline of the above argument and illustrate how to model extreme daily rainfall behavior via a GPD. Although we examine the modeling results at each of the five stations over SWWA, we focus on modeling extreme daily rainfall at Manjimup (station 9619) (Fig. 2) for ease of presentation. Let X 1,...,X n be a series of identically distributed random variables with an unknown underlying distribution F(x). Here, random variable X i denotes a daily rainfall observation. Our interest lies in estimating the behavior of rainfall over a given high threshold u. This can be approached by estimating the excess distribution defined by F u y P X i u y X i u F y u F u, 1 F u where 0 y r F u, and r F inf{x:f(x) 1} is the right endpoint of F(x). The excess distribution represents the probability that a heavy rainfall event that has a value of X i exceeds the threshold u by at most an amount of y, given the information that it exceeds the 1 threshold u. The following limit theorem is a key result about the asymptotic form of F u (y), and was first given by Balkema and de Haan (1974) and Pickands (1975). For a large class of underlying distributions we can find a function (u) such that where lim u r F sup F u y G, u y 0, 0 y r F u G, x 1 1 x exp x 0 is the generalized Pareto distribution with parameters 0, and x 0 when 0, as well as 0 x / when 0; is the shape parameter of the distribution FIG. 2. Times series of daily rainfall (mm) from 1 Jan 1889 to 30 Dec 2001 at a typical high-quality weather station in Manjimup of SWWA. 2 3

4 15 MARCH 2005 L I E T A L. 855 and is the scaling parameter. That is, for a large class of the underlying distributions F(x), as the threshold u progressively becomes large, the excess distribution F u (y) converges to a GPD. This limit theorem is, of course, not mathematically complete because it fails to convey what exactly is meant by a large class of distributions. But, it is known (e.g., McNeil 2000) that the class contains all of the common continuous distributions of statistics and climate science (normal, lognormal, 2, t, F, gamma, exponential, uniform, beta, etc.). Following McNeil (2000), our model for a rainfall value X i, having a distribution F(x), assumes that for a certain u the excess distribution over the threshold may be taken as an exact GPD for some and, that is, F u x G, x. To make assumption (4) more realistic, it is important to choose an appropriate threshold u. The choice of the threshold u usually proceeds from a compromise between choosing a sufficiently high threshold so that Eq. (4) is essentially satisfied and choosing a sufficiently low threshold so that we have enough excess data to estimate the parameters and. Having chosen a threshold u, we next estimate and in (4). It is known that if the sample size is large, the maximum likelihood estimation (MLE) is to be preferred because of its efficiency (Hosking and Wallis 1987; Smith 1985). This is the case in our application, so we choose to use MLE. Suppose that N u excesses, out of a total of n data points, exceed the threshold u; the GPD is fitted to the N u excesses by the MLE method to obtain estimates ˆ and ˆ, which are chosen by maximizing the loglikelihood function N u L, N u log 1 1 i 1 log 1 Y i, 5 provided that (1 Y i/ ) 0 for i 1,...,N u ; otherwise, L(, ), where Y 1,..., Y Ni denote the points over the threshold u in {X 1,...,X n }. See Davison and Smith (1990), McNeil (2000), and Coles (2001). We have seen from the above that for a given threshold, we can use (5) to obtain the maximum likelihood estimates of parameters for and. Now we introduce a criterion developed by Coles (2001, p. 83) for choosing the threshold. It is based on the feature that the GPD shape parameter and the reparameterizing scale parameter u u are constant with u u 0,if u 0 is a valid threshold for excesses (above u 0 ) that follow the GPD. This argument suggests that plotting the maximum likelihood estimates ˆ and ˆ u together with confidence intervals for each of these quantities is useful for selecting the threshold, and the threshold should be chosen as the lowest value of u for which the estimates ˆ and ˆ u remain near constant. The confidence intervals of ˆ and ˆ u were derived by Coles (2001, p. 83). Figure 3 gives the parameter estimates against the threshold for Manjimup daily rainfall data. 4 It is evident that both estimates ˆ and ˆ u are nearly constant for 30 u 35. Moreover, using a goodnessof-fit test (Choulakian and Stephens 2001), we can accept the following hypothesis: Manjimup daily rainfall over 30 mm follows a GPD at a highly significant level (p-value 0.05). Hence, we choose the threshold u 30 mm. Note that we have used the assumption that individual rainfall excesses are independent, leading to the log-likelihood (5). This is because the temporal dependence in the daily rainfall series at extreme levels is typically weak (Coles 1994), and the theoretical analysis by Katz (1998) also indicates that temporal correlation is not crucial for estimates of the probability of extreme rainfall events. In addition, we have checked the daily rainfall over the threshold of 30 mm and find there are no two threshold excesses within 3 days, which implies that these are approximately independent excesses. Having chosen the threshold u 30 mm we have, for our rainfall data in , n observations and N u 351 excesses. On the basis of these rainfall data, estimates of the shape ˆ and scale ˆ are obtained as and 8.313, with standard errors of and 0.65, respectively. Figure 4 gives the fitted excess distribution F u (x u) by the GPD G, (x u) with the above estimates. In this figure the estimated GPD model for the excess distribution is shown as a smooth, solid curve. The empirical distribution of the 351 excesses is shown as points. It is evident that the GPD model fits these excesses very well. Because the extreme rainfall ( 30 mm) behavior is governed by the tail of the unknown distribution F(x), we need to construct the tail estimate Fˆ (x) based on the excess distribution F u (x u). By setting x u y and combining (1) and (4) we have F x 1 F u G, x u F u 6 for x u. Now, substituting F(u) with its empirical estimator (n-n u )/n, and and with their MLE ˆ and ˆ, respectively, we get the tail estimate Fˆ x 1 N u n 1 ˆ x u ˆ 1 ˆ for x u. This function can be viewed as an asymptotic and partially nonparametric estimate of the upper tail of an unknown distribution based on the extreme value theory. Equation (7) can always be used to construct the tail of F(x) when we believe that data come from a common distribution, although its statistical properties are best understood when data are also assumed to be independent or only weakly dependent (McNeil 2000). Similarly, we can give an estimate of the mean E(X) based on the tail of F(x), where X denotes an arbitrary daily rainfall. Depending on X u, the rainfall excess X u follows a GPD with shape and scale, where E X u X u 1, 8 7

5 856 J O U R N A L O F C L I M A T E VOLUME 18 FIG. 3. (top) Shape and (bottom) modified scale parameter estimates ˆ and ˆ u against the threshold for Manjimup daily rainfall data. It is evident that both estimates ˆ and ˆ u are near constant for 30 u 35. Hence, we choose the threshold of u 30 mm. provided that 1. It can be easily seen that E X u E X u X u 1 F u E X u X u F u. Note that E(X u X u) is simply the mean of the belowness of u, for which the sample mean B u of the threshold belowness of u provides a good empirical estimate because of the large number of observations below the threshold u. Accordingly, the estimate of the mean E(X) is Ê X u ˆ 1 ˆ Nu n B u 1 N u n Equation (10) gives an estimate of the mean E(X) based on the estimated tail of F(x). It provides a way to explore the role of the changes at extreme tails of the distribution for the decrease in mean precipitation (see section 3c). For our rainfall series in , F(u) is estimated by ( )/ Combining this with the earlier parameter estimates, the tail estimate Fˆ (x) for x 30 mm is shown in Fig. 5. In this figure, the y axis is the tail probability 1 F(x) (log scale) and the x axis denotes the rainfall values (log scale). The top-left corner of the graph shows that the threshold u 30 has a tail probability of [empirical estimator 1 ( )/ ]. The points in the graph represent the empirical distribution of the 351 extreme rainfall events over 30 mm, and the solid curve shows how tail estimation in Eq. (7) allows extrapolation into the area where the data become sparse, that is, we can read off the probabilities of extreme rainfall from the solid curve in the graph. This is very important for risk management, such as for flood events. b. Change points in the extreme rainfall distribution As mentioned in section 1, an important finding in the IOCI project is that winter rainfall over SWWA has decreased substantially since the midtwentieth century, and this decline has manifested itself through reductions in the number of rain days and the rainfall amounts in extreme events over the past 25 yr. In earlier work by Wright (1974), Nicholls and Lavery (1992), and Hennessy et al. (1998), it has been noted that since the late 1960s or early 1970s SWWA has experienced significantly lower winter rainfall totals compared to the previous decades. Changes in rainfall totals are strongly correlated with changes in the frequency and intensity of extreme rainfall events (Haylock and Nicholls 2000). The aim here is to identify the change-

6 15 MARCH 2005 L I E T A L. 857 FIG. 4. Modeling the excess distribution of rainfall series at Manjimup in by the GPD with parameter ˆ and ˆ In this figure the estimated GPD model for the excess distribution is shown as a smoothed solid curve. The empirical distribution of the 351 excesses is shown as points. It is evident that the GPD model fits these excesses well. point year for the extreme daily rainfall, and to model the character and extent of such changes. Decreases in extremes (frequency and/or magnitude) are likely to lead to increased incidences of drought, as indeed has happened over SWWA in this period. As discussed earlier, identification of the starting year of the drying trend will help to pinpoint the dynamics of that effect. To proceed, we form annual, winter (May October) FIG. 5. Modeling the tail of the underling distribution for rainfall series at Manjimup for the period of Here, the y axis is the tail probabilities 1 F(x) (on log scale) and the x axis denotes the rainfall values (on log scale). The points in the graph represent empirical distribution of the 351 extreme rainfall events over 30 mm and the solid curve shows how the tail estimation formula (7) allows extrapolation into the area where the data become sparse.

7 858 J O U R N A L O F C L I M A T E VOLUME 18 and summer (November April) maximum daily rainfall series from the original daily data at each of five stations. Here the seasonal climate partitions for winter and summer are based on the fact that the climate of SWWA is a winter-rainfall-dominated regime that is characterized by a cool season (May October) and hot, dry season (November April) (Wright 1997; Indian Ocean Climate Initiative (2002). The nonparametric Mann Whitney Pittitt test (Pittitt 1979; Kiely 1999) was used to identify the change-point year in each of these maximum daily rainfall series (15 time series in all). The simpler Mann Whitney test (Salas 1993) was used to identify changes in annual flood frequency, corresponding to an observed shift in SST and mean circulation (Franks 2002; Franks and Kuczera 2002). We choose to use the Mann Whitney Pittitt test for our analysis because it not only identifies the change point but also gives its significance probability. This method had been used by Kiely (1999) to identify the changepoint year in precipitation in Ireland. Table 1 shows the change-point years and their significance probability for the five rainfall stations for the annual, winter, and summer maximum daily rainfall series. From this table the most common change-point year of winter maximum rainfall is around 1965 at most stations. The signal is not so clear for annual maxima, because the p value of the test statistics (the probability of change point) is not significant for three out of the five stations. Similarly, in summer, the change of maximum daily rainfall is not clear because only one of the five stations shows a change point in the maximum precipitation. So the change-point year of 1965 is adopted in the remaining analysis for winter extreme rainfall at all stations. c. Quantifying the changes in winter extreme rainfall To quantify the changes on winter extreme rainfall since the change-point year of 1965, we estimate the two tails of the rainfall distributions in two different periods: and The two-tail distributions are significantly different at the 1% level, based on a likelihood ratio test (Chakravarti et al. 1967) with p-value 0. The return period T R based on these twotail distributions {T R 1/[ Fˆ (x)]} is shown in Fig. 6. The y axis actually indicates extreme daily rainfall over 30 mm and the x axis is the return period (years). Table 2 shows a comparison of the estimated return periods (years) and their 95% confidence interval (CI) for extreme daily rainfall over 30 mm between the prechange period of and postchange period of , based on Fig. 6. It is evident that the return period of the winter extreme daily rainfall in the postchange period ( ) has greatly increased for the same extreme rainfall levels, which corresponds to the winter extreme daily rainfall decrease since To check the homogeneity of the winter extreme rainfall decrease over SWWA, we repeat the above analysis at four geophysically dispersed stations (Fig. 7) and the average of winter rainfall series of five stations (Fig. 8) over SWWA. According to the methods choosing the thresholds in section 3c, we choose a winter rainfall threshold of 20 mm for stations 9591 and 10636, and a threshold of 30 mm for stations 9503 and 9023, and the average over the five stations. It follows from Fig. 7 that the return period has clearly increased for winter extreme rainfall over 30 mm (or 20 mm) to about 70 mm in the postchange period of Winter extreme rainfall over 70 mm at stations 9023 and seems not to change much compared with the other stations, and, contrary to other three stations, the general estimated return period for winter extremes over 70 mm has decreased since The spatial variations may be due to occasional interactions between cold fronts and northwest cloudlands, as midlevel northwesterly winds advect moist tropical air to the SWWA region, leading to stronger extreme rainfall events over northwest regions of SWWA. For example, on 29 July 1987, an extreme rainfall event was caused through this process, resulting in a heavy winter rainfall at station 9023 ( mm) and station (74 mm), but with moderate rainfall at the other stations [station 9503 (16.80 mm), station 9691 (17.8 mm), and station 9591 (14 mm)] (Bureau of Meteorology 1987). Generally, the return period for winter extreme rainfall of the averaged rainfall series of five stations over SWWA has increased (Fig. 8), which implies that extreme rainfall over SWWA has decreased since the change-point year of To explore the dynamics of the winter rainfall drying trend, we now look at the role of the changes at extreme tails of the distribution for the decrease in mean winter precipitation. We again focus on the analysis for Manjimup (station 9619) for ease of presentation. It follows from (8) that the estimated mean Ê(X X u) ˆ /(1 ˆ) of rainfall over threshold u 30 mm, TABLE 1. Results of the Mann Whitney Pittitt test [change-point year, decrease (D), increase (I), probability of the change point] for the annual, winter (May Oct,) and summer (Nov Apr) daily maximum rainfall from at the five studied stations. Here D means the variable decreases after the change-point year and I means the variable increases after the change-point year. Time Station 9619 Station 9023 Station 9503 Station 9591 Station Annual 1965-D D D D D-0.61 Winter 1965-D D D D D-0.98 Summer 1976-D I-0.80 No change 1965-D I-0.95

8 15 MARCH 2005 L I E T A L. 859 FIG. 6. Return periods of winter extreme daily rainfall at Manjimup station. Here o points represent the empirical return period of winter extreme rainfalls observed in and points are from The solid and broken curves represent the return period based tail estimates by Eq. (7) with the 95% confidence interval given as a dashed line ( ) and dot-dashed line ( ). It is evident that the return period of the winter extreme daily rainfall in the postchange period ( ) has greatly increased for the same extreme rainfall levels, which indicates that winter extreme daily rainfall has decreased since based on two tails of estimated rainfall distributions, is 9.40 mm in the prechange period of and 6.59 mm in the postchange period, decreasing by about 30%. Note that the mean winter rainfall of less than 30 mm decreases about 16% from 3.71 to 3.12 mm. According to (10), the decreasing mean extreme rainfall over 30 mm causes a further decrease of 25.8% in mean winter precipitation since the change-point year of 1965 TABLE 2. A comparison of estimated return periods (in years) and their 95% CI for winter extreme daily rainfall over 30 (mm) prechange period of and postchange of at Manjimup station. It is evident that the return period of the winter extreme daily rainfall in the postchange period ( ) has greatly increased, indicating that the winter extreme daily rainfall has decreased since Extreme daily rainfall (mm) Return period (yr) and its 95% CI ( ) Return period (yr) and its 95% CI ( ) (0.129, 0.131) (0.389, 0.391) (0.21, 0.25) 0.83 (0.71, 1.09) (0.34, 0.50) 1.77 (1.23, 3.67) (0.51, 0.95) 3.78 (2.05, 16.2) (0.74, 1.79) 8.03 (3.28, 108) (1.05, 3.33) 17.5 (5.17, 1700) (1.45, 6.07) 38.5 (7.96, ) (2.60, 19.30) 193 (17.7, not applicable) (4.19, 53.60) 749 (32.6, not applicable) (6.90, 167) Not applicable (9.20, 331) Not applicable from 4.48 to 3.32 mm. Thus, the changes at the tails of winter rainfall distribution have significantly contributed to the dry trend in mean winter rainfall in SWWA. 4. Possible connection to the AAO The above analysis shows that winter extreme daily rainfall in SWWA experienced a downward change in Comparisons of results for the prechange period ( ) and postchange period ( ) show that the return period of winter extreme rainfall over 30 mm (or 20 mm) has increased since The drivers for the winter rainfall decline in SWWA are still a contentious issue. One view is that the decrease simply reflects natural climate variability. Another view suggests that greenhouse warming has contributed to at least part of the rainfall decrease. The greenhouse warming is thought to exert its influence through changes in properties of the dominant mode of variability of the atmospheric circulation (Cai et al. 2003). In the Southern Hemisphere (SH), the dominant mode of variability is the AAO (Mo and White 1985; Karoly 1990; Gong and Wang 1999; Thompson and Wallace 2000). The mode describes a large-scale alternation of midlatitude MSLP and high-latitude surface pressure when the MSLP is anomalously high, simultaneously in the midlatitudes the MSLP is anomalously

9 860 J O U R N A L O F C L I M A T E VOLUME 18 FIG. 7. A comparison of the return period of winter extreme rainfall over SWWA between the prechange period of and postchange period of at four stations: 9503, 9023, 9591, and Here black o points represent the empirical return period of winter extreme rainfalls observed in and red points are from 1966 to The black and red solid curves represent return period based tail estimates by Eq. (7) with the 95% confidence interval given as black dash lines ( ) and red dot-dashed lines ( ). low. Many coupled general circulation models (GCMs) successfully simulate this dominant mode (e.g., Schneider and Kinter 1994; Connolley 1997; Cai and Watterson 2002). In their study reporting the CSIRO Mark 2 coupled GCM simulation of the SH in the present-day climate (with a constant level of CO 2 ), Cai and Watterson (2002) identified the relationship between the AAO and rainfall over SWWA when the AAO is positive, that is, with increased MSLP in the midlatitudes, rainfall over SWWA tends to decrease as a result of decreased cloudiness and decreased westerly winds, which bring less synoptic rainfall events to the region. This model relationship appears to be supported by results from the National Centers for Environmental Prediction National Center for Atmospheric Research (NCEP NCAR) reanalysis. Over the past several decades, the observed AAO displays an upward trend with an increasing MSLP over the midlatitudes (Thompson et al. 2000) for the rainfall decrease. This trend is consistent with the drying trend in SWWA rainfall in the sense that the increasing MSLP provides an appropriate circulation condition (Cai et al. 2003) for the rainfall decreases. The issue we wish to address here is whether there is a turning point near 1965 in the formation of the AAO upward trend. If so, it may provide support for the argument that the AAO plays a role in the drying trend. This may, in turn, provide a dynamical basis for the change-point year (1965) that is identified by the present study. A monthly AAO index from 1948 to 2002 and a smoothed version are shown in Fig. 9. When the index is positive, MSLP anomalies are positive over the midlatitudes and negative in the high latitudes. The index is the time series of the first mode of an empirical orthogonal function (EOF) analysis of MSLP monthly anomalies (from the mean) from the NCEP reanalysis (Cai et al. 2003). The upward trend over the past decades that is reported by previous studies is clearly seen, with an increasing MSLP over the midlatitudes. The smoothed version is obtained by using a simple approach that consists of a sequence of applications of a locally weighted regression that is smoother with a variable span (Friedman 1984). This procedure has been implemented as an internal function supsmu in S-PLUS (Venables and Ripley 2002). By applying a Mann Whitney Pittitt (change point) test (Pittitt 1979; Kiely 1999) to the original monthly AAO series, change points are then identified. There are several change points such as the one in 1965 with a probability of 1, and another in 1992 with a

10 15 MARCH 2005 L I E T A L. 861 FIG. 8. A comparison of the return period of winter extreme rainfall over SWWA between the prechange period of and postchange period of based on the average of the rainfall series of five stations: 9619, 9503, 9023, 9591, and probability of Each of these change points seems to reflect a significant turning point. For example, the one in 1965 apparently signifies the onset of the upward trend, while that in 1992 indicates an acceleration of the upward trend after a short pause. The most important result here is that the onset year (about 1965) of the upward trend is consistent with the change-point year of 1965 of winter rainfall in SWWA that is identified by FIG. 9. Monthly AAO index from 1948 to 2002 and its smooth trend.

11 862 J O U R N A L O F C L I M A T E VOLUME 18 the present study. The consistency, in turn, supports the argument that the AAO upward trend may play a role in winter rainfall decrease over SWWA. To see this more clearly, we fitted independent GPDs to two stratified winter rainfall series at Manjimup based on AAO that is bigger than 162 (high-index mode) and lower than 162 (low-index mode), where the high AAO threshold 162 is chosen as the mean value of AAO plus its standard deviation. These two fitted GPDs are significantly different at the level 0.01 on the basis of a likelihood-ratio test with a p value of 0. The return periods based on the two fitted GPDs and their 95% confidence intervals are shown in Fig. 10. It is clear from Fig. 10 that the estimated return period when the AAO is in a high state is smaller than when the AAO is in a low state. This suggests that the AAO upward trend may play a role in the winter rainfall decrease in SWWA. Of course, we do not exclude the possibility that other modes may simultaneously play a role. This will be the subject of further research. 5. Summary In this paper we analyzed daily extreme rainfall at five geophysically dispersed high-quality stations in southwest Western Australia (SWWA). The method of modeling extreme daily rainfall series was introduced, and the Mann Whitney Pittitt test was used to detect change points of daily extreme rainfall. It was found that 1965 is the most common change-point year of winter maximum rainfall at most stations. A likelihoodratio test was employed to show that there is a significant difference at the 1% significance level between the two excess distributions for winter rainfall series pre- ( ) and postchange ( ). Estimated return periods, based on the estimated tail distribution ( ), are compared with those for winter rainfall series postchange ( ). It is shown that return periods for winter extreme daily rainfall have increased in SWWA after 1965, which implies that daily winter rainfall extremes in SWWA are lower after 1965 than they were before. We show that 1965 was the onset year of the Antarctic Oscillation upward trend with increasing midlatitude surface pressure, decreasing cloudiness and westerlies that were consistent with decreased rainfall, and extreme rainfall, in particular, over SWWA. This supports the argument that the winter (extreme daily) rainfall decline over SWWA is, at least in part, associated with the upward trend of the Antarctic Oscillation. Finally, the results presented here show that there are spatial variations for the winter extreme rainfall decrease in SWWA. Further research will focus on how to estimate a smooth spatial trend surface for winter extreme rainfall in SWWA, and seek to model the influence of the AAO on the winter extreme FIG. 10. A comparison of the return period of stratified winter extreme rainfall based on AAO that is bigger than 162 (high-index states) and lower than 162 (low-index states). The high AAO threshold 162 is chosen as the value of mean AAO plus its standard deviation, and the AAO threshold 162 is the value of mean AAO minus its standard deviation. It is clear that the estimated returned period when the AAO is in a high state is bigger than that when the AAO is in a low state. This suggests that the AAO upward trend may play a role in the winter rainfall decrease over SWWA.

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