GLOBAL ANALYSIS OF RUNS OF ANNUAL PRECIPITATION AND RUNOFF EQUAL TO OR BELOW THE MEDIAN: RUN MAGNITUDE AND SEVERITY

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1 INTERNATIONAL JOURNAL OF CLIMATOLOGY Int. J. Climatol. 25: (2005) Published online in Wiley InterScience ( DOI: /joc.1147 GLOBAL ANALYSIS OF RUNS OF ANNUAL PRECIPITATION AND RUNOFF EQUAL TO OR BELOW THE MEDIAN: RUN MAGNITUDE AND SEVERITY MURRAY C. PEEL, a, * THOMAS A. MCMAHON a and GEOFFREY G. S. PEGRAM b a Centre for Environmental Applied Hydrology and the Cooperative Research Centre for Catchment Hydrology, Department of Civil and Environmental Engineering, The University of Melbourne, Victoria 3010, Australia b Civil Engineering, University of KwaZulu-Natal, Durban 4041, South Africa Received 23 March 2004 Revised 15 November 2004 Accepted 15 November 2004 ABSTRACT Fluctuations of wet and dry years have long been investigated in the climatology and hydrology literature. In this, the second of two papers investigating runs of consecutive dry years, the magnitude, also known as the intensity, and severity (length magnitude) of dry runs are investigated. In the first paper the length of dry runs was investigated. Periods of consecutive dry years are associated with drought and the attendant physical and economic stresses that are placed on society. Run magnitudes of consecutive years equal to or below the median were analysed for 3863 precipitation and 1236 runoff stations from around the world. For both annual precipitation and runoff, run magnitude was found to be predominately related to interannual variability and to a lesser extent skewness. Run magnitude of annual runoff was observed to be greater than that for annual precipitation, due to annual runoff having a higher coefficient of variation than annual precipitation. Continental differences in run magnitude of annual runoff were observed and were consistent with continental differences in interannual variability reported previously. Annual run severity was also investigated and found to be independent of run length and strongly related to run magnitude. These findings differ from previously published work; this difference is primarily due to the methodology of comparing run metrics between stations (used in this paper) rather than at a station (previous research). The relationships between run magnitude, severity and interannual variability highlight the importance of adequately reproducing interannual variability within global climate models for future modelling of drought scenarios, as well as having economic implications for drought relief and management policy-making. Copyright 2005 Royal Meteorological Society. KEY WORDS: comparative hydrology; run magnitude; run severity; runs analysis; annual precipitation; annual runoff 1. INTRODUCTION Investigations into the nature of extended periods of consecutive dry years, which are associated with drought, have formed the basis of considerable research in the climatology and hydrology literature. Drought is a significant physical and economic phenomenon that can severely impact ecosystems and societies (Wilhite, 2000). Owing to its multifaceted nature, a generally accepted definition of drought remains elusive (Dracup et al., 1980a; Wilhite and Glantz, 1985; Frick et al., 1990; Bonacci, 1993; Wilhite, 2000; Heim, 2002; Keyantash and Dracup, 2002; Panu and Sharma, 2002; McMahon and Finlayson, 2003). This paper does not argue for any particular definition of drought, but simply uses runs analysis as a tool for investigating drought behaviour. Applying runs analysis, Yevjevich (1967) and Dracup et al. (1980a) defined drought severity as the sum of the negative deviations from a threshold (mean, median, etc.) for a given length of negative deviations. * Correspondence to: Murray C. Peel, Centre for Environmental Applied Hydrology, Department of Civil and Environmental Engineering, The University of Melbourne, Victoria 3010, Australia; mpeel@unimelb.edu.au Copyright 2005 Royal Meteorological Society

2 550 M. C. PEEL, T. A. MCMAHON AND G. G. S. PEGRAM They also noted that drought severity is the product of drought length (the period of negative deviations) and drought magnitude (the average of the negative deviations). Keyantash and Dracup (2002), in an evaluation of drought indices, confirmed the appropriateness of using runs analysis for assessing drought severity, finding that the Total water deficit ( drought severity of Yevjevich, (1967) and Dracup et al. (1980a)) was the most useful hydrologic drought index. Peel et al. (2004a), in a companion paper, investigated the nature of drought length by analysis of run lengths of annual precipitation and runoff data. They found that run lengths of annual precipitation and annual runoff equal to or below the median are similar across all continents and Köppen climate zones, except for arid and tropical North Africa (Sahel), which has longer run lengths than other regions. In this paper the nature of drought magnitude is investigated by analysis of run magnitude of annual precipitation and runoff data. Also, the nature of drought severity is investigated by combining the run length results of Peel et al. (2004a) and the run magnitude results of this paper. Section 2 contains a brief review of run magnitude and severity studies in the hydrologic literature. The data used in the analysis are then described briefly before the results of the run magnitude analyses are presented and discussed. Utilizing the results of the run magnitude analyses and the run length analysis of Peel et al. (2004a), an investigation and discussion of run severity is then presented. The implications of these results are presented prior to conclusions being drawn. 2. BRIEF REVIEW OF RUN MAGNITUDE AND SEVERITY LITERATURE The applications of runs analysis briefly reviewed here are limited to estimating the expected frequency and average recurrence interval (return period) of wet and dry periods of certain magnitudes and severity. The hydrologic literature contains numerous papers investigating the expected frequency and average recurrence interval of wet and dry periods of certain magnitude and severity. In comparison with run length (briefly reviewed in Peel et al. (2004a)), run magnitude has received considerably less attention in the hydrologic literature. Of the three drought components (length, magnitude and severity), run severity has received the most attention in the hydrologic literature, and many of the papers previously reviewed for run length were primarily focused on run severity. Şen (1977) provides equations for the expected value and variance of run severity for the independent normal, log-normal and the dependent (lag-one Markov process) normal distributions. Sharma (1997a,b, 1998, 2000) provides an equation for the expected run severity for any given average recurrence interval, truncation level and autocorrelation for normal, log-normal and Gamma distributed annual data. The studies of Şen and Sharma developed separate distributions for run length and magnitude and combined them to calculate the severity distribution. The run magnitude equations of Şen and Sharma are dependent on the truncation level and probability distribution function of the sample and are independent of the autocorrelation and average recurrence interval. Shiau and Shen (2001) developed a method to simulate the conditional distribution of run severity for a given run length for independent and identically distributed monthly data. In summary, several points are of relevance to this study. Equations exist for estimating the average recurrence interval of run magnitude and severity for independent and certain forms of dependent annual data. With regard to dependent annual data, the most significant variables in the run magnitude equations are truncation level and the sample probability distribution function. For run severity the most significant variables are a combination of truncation level, lag-one autocorrelation and the sample probability distribution function. Clearly, the truncation level used to divide the sample into wet and dry periods plays a significant role in the results of any runs analysis. In this paper and the companion paper (Peel et al., 2004a) the median is used consistently as the truncation level, so the results within and between the two papers are comparable. As noted in Peel et al. (2004a), the choice of the sample median as the truncation level, rather than the sample mean or any fraction of the mean, may limit the observed influence of skewness on drought length, magnitude and severity in the following analysis.

3 RUNS OF PRECIPITATION AND RUNOFF ANOMALIES DATA The annual precipitation and annual runoff data analysed in this paper are the same as described in Peel et al. (2004a). A total of 3863 precipitation stations, from 162 countries, were obtained from the Global Historical Climatology Network Version 2 (Vose et al., 1992). The minimum record length is 10 years, with an average record length of 41.0 years. The continuous records are in water years and have units of millimetres per year. The spatial distribution of the precipitation stations by continent is shown in Peel et al. (2004a: figure 1). Most of the annual precipitation data were recorded between 1900 and The annual runoff database is comprehensively described in Peel et al. (2001). A total of 1236 runoff stations, from 87 countries, have at least 10 years of continuous data, an average of 33 years, and are believed to be unregulated for that period. The data are stored in water years and have units of millions of cubic meters per year. The spatial distribution of the runoff stations by continent is shown in Peel et al. (2004a: figure 2). Most of the annual runoff data were recorded between 1935 and The definition of continents follows that of Peel et al. (2002) and the continental symbols used throughout this paper are: AS, Asia; AUS, Australia; EUR, Europe; NAF, northern Africa; NAM, northern America; SAF, southern Africa; SAM, southern America; SP, South Pacific. Although some of the continents defined are not strictly continents, these definitions are used to be consistent with previous work. A description of how stations were assigned a Köppen climate zone are given in Peel et al. (2002) for precipitation and Peel et al. (2004b) for runoff. 4. METHOD The median was calculated for each consecutive record length N of annual precipitation and runoff. The magnitude of each run of consecutive years equal to or below the median was also calculated for each sample. Figure 1 shows an example time series of annual precipitation relative to the median for Alice Springs, Australia (N = 50). The run magnitude (average rainfall deficit) of each run length for Alice Springs is presented in the second column of Table I. To compare run magnitudes at a station with magnitudes observed at other stations, an objective summary statistic of run magnitude is required Metrics of run magnitude Two metrics are used in this paper to summarize run magnitude at each station in order to enable meaningful comparisons between stations (Table II provides a listing of the key terms and symbols used in this paper). Figure 1. Time series of annual precipitation relative to the median (N = 50) for Alice Springs, Australia

4 552 M. C. PEEL, T. A. MCMAHON AND G. G. S. PEGRAM Figure 2. Relative magnitude (Equation (1)) versus sample size N for run length j = 3 for (a) all annual precipitation stations and (b) annual runoff stations The first metric is the average run magnitude relative to the median for a given run length j (hereafter referred to as the relative magnitude ) defined as n j M ij /n j i=1 Relative magnitude = (1) x m where x m is the sample median, n j is the number of runs of length j and M ij is the ith run magnitude of length j where i = 1, 2,..., n j. A run occurs when consecutive values are equal to or below the median. In the Alice Springs example (Table I) there are run lengths of 1, 2, 4 and 10 years equal to or below the median. The relative magnitudes (column 3 of Table I) are calculated by averaging the run magnitudes for a given run length j and dividing by the annual sample median. The relative magnitude for a given run length is a dimensionless number that is appropriate for intercomparison between stations. The second metric is based on a measure of reservoir system vulnerability developed by Hashimoto et al. (1982) and hereafter referred to as vulnerability. Vulnerability measures the likely magnitude of a reservoir

5 RUNS OF PRECIPITATION AND RUNOFF ANOMALIES 553 Table I. Run lengths, run magnitudes, relative magnitudes, run severities and maximum deficit per run for annual precipitation data from Alice Springs, Australia. Median annual precipitation is mm. The vulnerability (Equation (3)) is 0.31 for this station Run length Run magnitude (mm) Relative magnitude Maximum deficit (mm) Run severity (mm) Table II. Summary of key terms and symbols used Term Symbol Defined Relative magnitude Equation (1) Vulnerability η Equation (3) Run length skew g Equation (4) Relative severity S Appendix A Record length or sample size N Lag-one autocorrelation r 1 Mean annual precipitation (runoff) MAP (MAR) Coefficient of variation Cv Coefficient of skewness Cs failure if a failure occurs. A reservoir system failure is defined as not being able to meet a predetermined demand D. Following Hashimoto et al. (1982) and McMahon and Adeloye (2005), vulnerability is defined as η = f s max(sh k ) k=1 f n (2) where η is the vulnerability, f n is the number of continuous failure sequences and max(sh k ) is the maximum deficit during the kth continuous sequence. The dimensionless form of vulnerability, which we shall use in this paper, is η = η D (3) In this analysis we define D as the median so that a failure is any value equal to or below the median. The maximum deficits (max(sh k )) for each failure (run length) for Alice Springs are given in column 4 of Table I. The vulnerability η = 0.31 for Alice Springs annual precipitation when D is set to the median. The vulnerability has the useful feature of summarizing the run magnitude behaviour of a station in a single number, whereas the relative magnitude has a value for each run length.

6 554 M. C. PEEL, T. A. MCMAHON AND G. G. S. PEGRAM Figure 3. Vulnerability η P (Equation (3)) versus sample size N for all annual precipitation stations A desirable feature of the two run magnitude metrics is that they are independent of sample size N. Figure 2 shows the relationship between the relative magnitude and N for run length j = 3 for (a) all annual precipitations stations and (b) all annual runoff stations. There is no detectable relationship between relative magnitude and sample size N for either annual precipitation or runoff for run length j = 3 or any other run length (not shown). Figure 3 shows the relationship between vulnerability η and sample size N for annual precipitation stations. Values of η calculated from annual precipitation and annual runoff are referred to as η P and η R respectively. There is no perceived relationship between η P and N for annual precipitation or annual runoff (not shown, but similar to Figure 3). Figures 2 and 3 show that the two metrics of run magnitude are independent of sample size and are appropriate metrics for summarizing run magnitude behaviour at and between stations. 5. RESULTS 5.1. Run magnitude by continent The precipitation and runoff station records were stratified by continent in order to investigate any continental differences in run magnitude. The relative magnitude metric (Equation (1)) was calculated for each run length at each station. Where a continent had 10 or more stations with a value of relative magnitude for a given run length, the average of those values was taken and plotted in Figure 4(a) for precipitation and Figure 4(b) runoff. A minimum of 10 stations per continent is used throughout the following analysis in order to enable meaningful comparison of continental averages. The relative magnitude of the precipitation data remains constant across the different run lengths for all continents except NAF, where the relative magnitude increases with run length. The relative magnitude of the runoff data generally increases as run length increases for all continents except EUR and SP, where the relative magnitude remains largely constant as run length increases. A cautionary note when interpreting Figure 4 is that the sample size of each average decreases (minimum of 10) as the run length increases. Simple linear regression was applied to the continental relationships between relative magnitude and run length shown in Figure 4 and the slope coefficients were tested for significant differences from zero (Barber, 1988). Although the slope coefficients are generally positive, for annual precipitation only the EUR slope coefficient was significantly different from zero at the 5% level of significance. The NAF annual precipitation slope coefficient was not significantly different from zero at the 5% level of significance, due to the run length j = 9 value. For annual runoff, the slope coefficients of AS, AUS, NAM and SAM were significantly

7 RUNS OF PRECIPITATION AND RUNOFF ANOMALIES 555 Figure 4. Relative magnitude (Equation (1)) versus run length by continent for (a) precipitation and (b) runoff different from zero at the 5% level of significance. Weakly positive slope coefficients reflect the persistence structure of annual precipitation and runoff data. For annual precipitation, which has little or no persistence (Peel et al., 2004a), the relative magnitude remains constant as run length increases, whereas for annual runoff, which has more persistence than annual precipitation (Peel et al., 2004a), the relative magnitude increases as run length increases. Stations with strong persistence are more likely to have increasing relative magnitude as run length increases because of higher autocorrelation. The proportions of stations that have annual lagone autocorrelation r 1 significantly different from zero at each annual precipitation and runoff station by continent were presented in Peel et al. (2004a: table I) and are reproduced here in Table III. Based on a two-sided hypothesis test from Yevjevich (1972), r 1 is not significantly different from zero at the 5% level of significance for 90% of annual precipitation stations and 81% of annual runoff stations. In Figure 4(a), the only continent that has increasing relative magnitude as run length increases for annual precipitation is NAF, which in Table III is the continent with the highest proportion of stations where r 1 = 0. In Figure 4(b), most continents have increasing relative magnitude as run length increases for annual runoff, which is to be expected from Table III, as most continents have a higher proportion of runoff stations than precipitation stations with non-zero r 1.

8 556 M. C. PEEL, T. A. MCMAHON AND G. G. S. PEGRAM Table III. Percentage of stations with lag-one autocorrelation r 1 significantly different from zero at the 5% level of significance for annual precipitation and runoff by continent Continent Precipitation Runoff N r 1 = 0(%) r 1 = 0(%) N r 1 = 0(%) r 1 = 0(%) AS AUS EUR NAF NAM SAF SAM SP World The average relative magnitude is higher for runoff (Figure 4(b)) than for precipitation (Figure 4(a)) for most continents across the different run lengths. Since the data sets used in this analysis are largely recorded over similar time periods, this result indicates that precipitation run magnitude is magnified into a larger runoff run magnitude. The difference in magnification between continents is most pronounced for AUS and SAF (Figure 4(b)), where the relative magnitude for runoff is about double that for the other continents. In Figure 4(a), a similar difference in the relative magnitude of annual precipitation is not apparent for AUS and SAF Run magnitude stratified by Köppen climate zone The continental data were further stratified into Köppen climate zones to investigate whether the continental differences in Figure 4(b) remain when the continents are compared using data from the same climate zone. Instead of using the relative magnitude metric for this analysis, the vulnerability η metric of Equation (3) is used since it provides a single value summary (inclusive of all run lengths) of the run magnitude at a station. The number of stations grouped by continent and Köppen climate zone is presented in Table IV, for both the precipitation and runoff data sets. If the number of stations in a particular continent and Köppen climate zone was less than 10, then the number was not listed in Table IV. The restriction of at least 10 stations for calculating an average is also applied to the continental Köppen climate zone analysis. The average vulnerability of annual precipitation η P and runoff η R for each continent and Köppen climate zone is presented in Table V. No consistent continental differences in η P are apparent when compared by Köppen climate zone. This confirms the conclusion from Figure 4(a), that there are no continental differences in run magnitude of annual precipitation. The relative positions of the continental lines on Figure 4(a) are due to the proportion of precipitation stations in arid (high η P : NAF, AUS) versus non-arid (low η P :EUR, NAM, SP) climate zones. Continental differences in runoff vulnerability η R are apparent when compared by Köppen climate zone in Table V. Both AUS and SAF have η R that is about 50% higher than other continents for the same climate zone. This confirms the conclusion from Figure 4(b), that there are continental differences in run magnitude of annual runoff. The relative positions of the continental lines on Figure 4(b) are not completely explained by the proportion of runoff stations in arid versus non-arid climate zones. In order to explain these continental differences in annual runoff run magnitude, the relationships between run magnitude and mean, coefficients of variation (Cv) and skewness (Cs) of annual precipitation and runoff are investigated. The mean annual precipitation (MAP) and runoff (MAR) for each continent and Köppen climate zone are presented in Table VI. A feature of this table is the anomaly for some continents where the MAP is less than MAR for a given Köppen zone (EUR: Dfc; NAM: Aw; SAM: BSk). There are three possible reasons for this. The first is sampling variability, in which the runoff stations sampled by chance have a higher mean than the

9 RUNS OF PRECIPITATION AND RUNOFF ANOMALIES 557 Table IV. Number of stations (minimum of 10) with precipitation (P) and runoff (R) data by continent and Köppen climate zone AS AUS EUR NAF NAM SAF SAM SP P R P R P R P R P R P R P R P R Af Am 17 Aw BWh BWk BSh BSk Csa Csb Cwa Cwb Cfa Cfb Dsb Dsc 13 Dwa 95 Dwb 52 Dwc 47 Dfa Dfb Dfc ET precipitation stations. The minimum of 10 stations before calculating an average reduces the probability of this occurring. The second possible reason is that the precipitation stations are biased toward valleys rather than catchment headwaters; this bias would underestimate precipitation relative to runoff (Milly and Dunne, 2002). The third possible reason is that stations may have been misallocated to a Köppen climate zone. As discussed in Peel et al. (2004b), the Köppen climate zone was calculated from raw data for the precipitation stations and, therefore, is correct. Runoff stations were allocated a Köppen climate zone by visual inspection of the precipitation Köppen zone map. The allocation of runoff stations with large catchments areas is complicated by the possibility that a catchment area may be subject to more than one Köppen climate zone. Thus, the anomaly is most likely due to a combination of valley bias in the precipitation stations and misallocation of runoff stations to a Köppen climate zone. Inspection of Table VI does not reveal any consistent continental differences in MAP or MAR that might explain the continental differences observed in Figure 4(b) or Table V. Figure 5 shows the relationships between η P and MAP and η R and MAR. The data points in Figure 5 are taken from Tables IV and V and are the continental averages for a given Köppen climate zone of mean and η. In Figure 5, MAP explains 52% of the variance in η P and MAR explains 34% of the variance in η R. The relationship between η P and MAP is similar to that for η R and MAR, with the η R MAR relationship being vertically displaced relative to the η P MAP relationship. The coefficient of variation of annual precipitation (Cv p ) and runoff (Cv r ) for each continent and Köppen climate zone are presented in Table VII. All continental Köppen climate zones have Cv r greater than or equal to Cv p. Inspection of Table VII does not reveal any consistent continental differences in Cv p ; however, continental differences in Cv r are apparent. Values of Cv r for AUS, SAF and SAM are generally higher than values from other continents. Since the majority of AUS and SAF stations (Table IV) are located in the climate zones with higher Cv r values, this may explain the continental differences in run magnitude observed

10 558 M. C. PEEL, T. A. MCMAHON AND G. G. S. PEGRAM Table V. Average vulnerability (minimum of 10 stations per average) for precipitation η P and runoff η R data by continent and Köppen climate zone AS AUS EUR NAF NAM SAF SAM SP η P η R η P η R η P η R η P η R η P η R η P η R η P η R η P η R Af Am 0.18 Aw BWh BWk BSh BSk Csa Csb Cwa Cwb Cfa Cfb Dsb Dsc 0.19 Dwa 0.21 Dwb 0.19 Dwc Dfa Dfb Dfc ET Figure 5. Annual precipitation and runoff vulnerability (η P and η R ) versus mean annual precipitation (MAP) and mean annual runoff (MAR) respectively, based on continental Köppen climate zone averages from Tables IV and V in Figure 4(b) and Table V. Figure 6 shows the relationships between η P and Cv p and η R and Cv r. The data points in Figure 6 are taken from Tables IV and VI and are the continental averages for a given Köppen climate zone of Cv and η. In Figure 6, Cv p explains 98% of the variance in η P and Cv r explains 81% of the variance in η R. The relationship between η P and Cv p is similar to that for η R and Cv r, although 1% confidence

11 RUNS OF PRECIPITATION AND RUNOFF ANOMALIES 559 Figure 6. Annual precipitation and runoff vulnerability (η P and η R ) versus coefficient of variation of annual precipitation Cv p and runoff Cv r respectively, based on continental Köppen climate zone averages from Tables IV and VI Table VI. Average mean annual precipitation (P) and runoff (R) (units of mm, minimum of 10 stations per average) by continent and Köppen climate zone. For an explanation of why some R > P, see text AS AUS EUR NAF NAM SAF SAM SP P R P R P R P R P R P R P R P R Af Am 2702 Aw BWh BWk BSh BSk Csa Csb Cwa Cwb Cfa Cfb Dsb Dsc 339 Dwa 837 Dwb 557 Dwc 466 Dfa Dfb Dfc ET intervals around the slope coefficients (Barber, 1988) do not overlap, indicating that the slopes of the two relationships are statistically different at the 1% level of significance. Figure 6 shows very strong linear relationships between η and Cv. Data for precipitation and runoff from all continents are well described by these linear relationships. There are no continental differences in the way variability is related to vulnerability

12 560 M. C. PEEL, T. A. MCMAHON AND G. G. S. PEGRAM Table VII. Average coefficient of variation of annual precipitation (P) and runoff (R) (minimum of 10 stations per average) by continent and Köppen climate zone AS AUS EUR NAF NAM SAF SAM SP P R P R P R P R P R P R P R P R Af Am 0.21 Aw BWh BWk BSh BSk Csa Csb Cwa Cwb Cfa Cfb Dsb Dsc 0.22 Dwa 0.24 Dwb 0.21 Dwc 0.20 Dfa Dfb Dfc ET Figure 7. Coefficient of variation of annual precipitation Cv p and runoff Cv r versus mean annual precipitation (MAP) and mean annual runoff (MAR) respectively, based on continental Köppen climate zone averages from Tables V and VI for either precipitation or runoff. However, there are continental differences in the variability of annual runoff for a given Köppen climate zone (Table VII). Therefore, the vertical displacement of the η R MAR relationship relative to the η P MAP relationship in Figure 5 is due to the fact that Cv r is greater than Cv p for a given continental Köppen climate zone (Table VII). This is confirmed in Figure 7, where Cv p and Cv r are plotted against MAP and MAR respectively. Figure 7

13 RUNS OF PRECIPITATION AND RUNOFF ANOMALIES 561 Figure 8. Annual precipitation and runoff vulnerability (η P and η R ) versus coefficient of skewness of annual precipitation Cs p and runoff Cs r respectively, based on continental Köppen climate zone averages from Tables IV and VII is virtually identical to Figure 5, with MAP explaining 50% of the variance in Cv p and MAR explaining 37% of the variance in Cv r, results that are very similar to those between η P MAP and η R MAR. The coefficient of skewness of annual precipitation (Cs p ) and runoff (Cs r ) for each continent and Köppen climate zone is presented in Table VIII. Cs r is generally greater than Cs p. Inspection of Table VIII does not reveal any consistent continental differences in Cs p ; however, continental differences in Cs r are apparent, with values of Cs r for AUS, SAF and SAM generally higher than those from other continents. Figure 8 shows the relationships between η P and Cs p and η R and Cs r. The data points in Figure 8 are taken from Tables IV and VII and are the continental averages for a given Köppen climate zone of Cs and η. In Figure 8, Cs p explains 64% of the variance in η P and Cs r explains 50% of the variance in η R. The slope of the linear relationship between η P and Cs p appears different to that for η R and Cs r, but at the 5% level of significance there is no difference. Since the vulnerability is strongly related to variability, the importance of variability to skewness was assessed. In Figure 9, Cv p explained 69% of the variance in Cs p and Cv r explained 82% of the variance in Cs r. Thus, the relationships between η and Cs shown in Figure 8 are largely explained by the relationship between Cs and Cv Summary In summary, run magnitude is strongly related to interannual variability (Figure 6). The continental differences in run magnitude observed in Table V and Figure 4(b) are due to continental differences in the variability of annual runoff. McMahon et al. (1992) observed continental differences in the variability of annual runoff for temperate AUS, SAF and arid SAF, which are consistent with the continental differences in run magnitude. A combination of higher precipitation variability, the distribution of evergreen and deciduous vegetation in temperate regions, differences in the proportion of forested catchment area and differences in the mean annual daily temperature range are suggested causes of the observed continental differences in the variability of annual runoff (Peel et al., 2001, 2004b). The conclusion that run magnitude is strongly related to the coefficient of variation is consistent with the equations of Şen (1977) and Sharma (1997b, 1998). The importance of the coefficient of skewness to run magnitude, noted by Sharma (1997b, 1998), was observed to be secondary to the coefficient of variation in the present analysis. The importance of skewness to run magnitude would increase when the threshold is the mean, as used by Sharma (1997b, 1998), rather than the median, because as the sample skewness increases so the percentage of sample values below the threshold increases, resulting in increases in run length, magnitude and severity.

14 562 M. C. PEEL, T. A. MCMAHON AND G. G. S. PEGRAM Table VIII. Average coefficient of skewness of annual precipitation (P) and runoff (R) (minimum of 10 stations per average) by continent and Köppen climate zone AS AUS EUR NAF NAM SAF SAM SP P R P R P R P R P R P R P R P R Af Am 0.14 Aw BWh BWk BSh BSk Csa Csb Cwa Cwb Cfa Cfb Dsb Dsc 0.32 Dwa 0.54 Dwb 0.36 Dwc 0.29 Dfa Dfb Dfc ET Figure 9. Coefficient of skewness of annual precipitation Cs p and annual runoff Cs r versus coefficient of variation of annual precipitation Cv p and annual runoff Cv r respectively, based on continental Köppen climate zone averages from Tables VI and VII The conclusion that run magnitude is strongly related to annual variability and to a lesser extent skewness highlights the importance of using a percentile-based threshold for drought analysis. For example, Pandey and Ramasastri (2002) found that drought was more frequent in arid areas compared with humid areas. They defined a drought year as any year with <75% of MAP. Figures 7 and 9 show a negative relationship between

15 RUNS OF PRECIPITATION AND RUNOFF ANOMALIES 563 Cv p and MAP and a positive relationship between Cs p and Cv p. Therefore, as MAP increases, Cv p and Cs p decrease and the frequency of years with less than 75% of MAP would decrease. Thus, as MAP increases, drought, as defined by Pandey and Ramasastri (2002), would become less frequent, which they observed Run severity Run severity is the product of run length and run magnitude (Yevjevich, 1967; Dracup et al., 1980a). The run length analysis of Peel et al. (2004a) and the present analysis of run magnitude can be combined to draw some conclusions about interannual run severity. Peel et al. (2004a) found that annual precipitation and annual runoff run lengths were similar for the world, except for tropical and arid NAF, which displayed a bias toward longer run lengths. In the present paper, values of annual precipitation run magnitude (measured by vulnerability η) are observed to be largely similar, whereas those of annual runoff are observed to be different around the world. Run magnitude (vulnerability η) is found to be linearly related to the interannual variability (coefficient of variation Cv) for both annual precipitation and runoff. The relationship between run magnitude for the annual precipitation stations (measured by vulnerability η P ) and run length (measured by run length skew g from Peel et al. (2004a)) is shown in Figure 10. The run length skew g of Peel et al. (2004a) is defined as g = k n j j 3 j=1 N 1 (4) where, in the notation of Equation (1), n j is the frequency of run length j, k is the longest observed run length and N is the length of record. For example, the Alice Springs data shown in Figure 1 and Table I has g = 22.3 for runs equal to or below the median. In Figure 10, a power regression relationship is statistically significant at the 5% level of significance, although it only explains 0.2% of the variance. Thus, run magnitude η is effectively independent of run length (as measured by skew g) for annual precipitation. A similar analysis using annual runoff data (not shown, but similar to Figure 10) revealed the relationship between run magnitude η and run length g is also effectively independent (0.4% of variance explained) for annual runoff. This result Figure 10. Annual precipitation vulnerability η P versus run length skew g from Peel et al. (2004a). Also plotted are the average η and g for each Köppen climate zone, grouped by the first letter of the Köppen classification

16 564 M. C. PEEL, T. A. MCMAHON AND G. G. S. PEGRAM confirms the findings of Dracup et al. (1980b: table 3) based on seven runoff stations from California, USA, namely that, for annual runoff, drought length is independent of drought magnitude. Figure 10 also shows the average η P and g for each Köppen climate zone for annual precipitation data. The averages are grouped by the primary letter of each Köppen climate zone, e.g. the three A climates (Af, Am and Aw) are represented by filled diamonds. Most of the averages cluster together, with some of the B zone climates (BWh, BWk and BSh) having higher η P and some D zone climates (Dsb and Dsc) having lower η P. In summary, Figure 10 shows that run magnitude η is effectively independent of run length g; and since there are few continental differences in run length (Peel et al., 2004a), then any continental differences in run severity will most likely be due to continental differences in run magnitude, which are the result of continental differences in interannual variability (Figure 6) Relative severity In order to investigate run severity further, a metric of run severity was calculated using the extended deficit analysis developed by Pegram and reported in McMahon and Adeloye (2005). Extended deficit analysis is used to determine the average recurrence interval of reservoir deficits. The metric of run severity used is the 1 in 100 year deficit relative to the mean, which is hereafter referred to as the relative severity S. The methodology for conducting extended deficit analysis is described in Appendix A. Extended deficit analysis is a form of runs analysis used in reservoir storage yield analysis, where surpluses and deficits are accumulated below a spill level and a run is defined as the period between spill events. Although the extended deficit analysis is different to the other runs analyses used in this paper and in Peel et al. (2004a), extended deficit analysis has useful statistical properties (described in Appendix A) that are utilized in calculating the run severity metric for comparison between stations. Relative severity S values were available for 2886 precipitation and 902 runoff stations using the methodology outlined in Appendix A. The relationships between run severity (measured by relative severity S) and run length (measured by run length skew g) for annual precipitation and annual runoff (not shown, but similar to Figure 10) were statistically significant at the 5% level of significance, although they explained only 0.6% and 1.6% of the variance respectively. Run severity is effectively independent of run length for annual precipitation and annual runoff. This result differs from the findings of Dracup et al. (1980b: table 3), Chang and Stenson (1990) and Bonacci (1993) due to the methodologies employed. Dracup et al. (1980b), Chang and Stenson (1990) and Bonacci (1993) investigated the correlation between paired run length (duration) and run sum (severity) values at a station (like Table I). They observed that longer runs were more severe than shorter runs and a strong relationship between the two variables, run length and severity, existed. A similarly strong relationship between run length and run severity would be found in this paper if the same methodology used by the previous authors were applied, e.g. the correlation between run length and run severity for Alice Springs annual precipitation in Table I is However, in this paper the relationship tested is between summary statistics of run length g and run severity S measured at each station. Therefore, the interpretation of the present result is that run severity at a station is independent of whether a station displays a bias toward longer or shorter run lengths. Using the 2886 precipitation and 902 runoff relative severity S values, the relationships between run severity (measured by relative severity S) and run magnitude (vulnerability η) for annual precipitation (Figure 11(a)) and annual runoff (Figure 11(b)) are statistically significant at the 5% level of significance and explain 49% and 60% of the variance respectively. Figure 11(a) and (b) also shows the average S and η for each Köppen climate zone for annual precipitation and annual runoff data. Like Figure 10, the averages are grouped by the primary letter of each Köppen climate zone. The Köppen climate zone averages remove much of the sampling variability from Figure 11(a) and (b), and power regression relationships through the averages are statistically significant at the 5% level of significance, explaining 95% and 79% of the variance respectively. In Section 5.2, run magnitude was found to be significantly related to interannual variability (measured by Cv). The relationships between run severity (measured by relative severity S) and interannual variability (measured by Cv) for annual precipitation and annual runoff (not shown, but similar to Figure 11(a) and (b)) are statistically significant at the 5% level of significance and explain 57% and 69% of the variance

17 RUNS OF PRECIPITATION AND RUNOFF ANOMALIES 565 Figure 11. Relative severity S versus vulnerability for (a) annual precipitation η P and (b) annual runoff η R. Also plotted are the average η and relative severity for each Köppen climate zone, grouped by the first letter of the Köppen classification respectively. Like in Figure 11(a) and (b), regression relationships through the Köppen climate zone averages of S and annual Cv are statistically significant at the 5% level of significance, explaining 96% and 91% of the variance for precipitation and runoff respectively. Clearly, interannual variability is the predominant influence on run severity for annual precipitation and annual runoff. The importance of the coefficient of skewness Cs and lag-one autocorrelation r 1 to run severity (measured by S) was assessed after the influence of interannual variability Cv was removed by calculating the residuals of the relationship between S and Cv for annual precipitation and annual runoff respectively (not shown). The relationships between residual S and Cs p and residual S and precipitation r 1 are both statistically significant at the 5% level of significance and explain 16% and 1% of the variance respectively. Likewise, for annual runoff the relationships between residual S and Cs r and residual S and runoff r 1 are both statistically significant at the 5% level of significance and explain 3% and 2% of the variance respectively. No significant relationship was found between Köppen climate zone, grouped by primary letter, average values of residual S and Cs p, Cs r, precipitation r 1, runoff r 1.

18 566 M. C. PEEL, T. A. MCMAHON AND G. G. S. PEGRAM 6. IMPLICATIONS Some research and policy implications are apparent from the conclusion that run severity is predominantly due to interannual variability. First, the importance of accurate replication of the current and predicted interannual variability in global climate models (GCMs) is highlighted. This implication accords with the work of Katz and Brown (1992), who noted that variability is more important than averages in predicting the future impact on extreme events of climate change. Accurate replication of interannual variability in GCMs will improve the modelling and prediction of interannual drought severity. Second, the relationship between drought severity and interannual variability provides a potentially useful insight into the drought relief/management issue reviewedby Heathcote (2000) for Australia. Severe drought is not exceptional in regions of high interannual variability; it is expected. Thus, as agricultural activities expand from temperate (moderate interannual variability) into semi-arid and arid climates (high interannual variability) the drought severity experienced increases, confirming the conclusion of Glantz (2000) that drought follows the plough. 7. CONCLUSIONS Run magnitude and severity of annual precipitation and runoff data have been investigated using a runs-based methodology. This work follows on from the investigation into run length in the companion paper of Peel et al. (2004a). Annual run magnitude (measured by vulnerability η) was found to be predominately related to interannual variability (coefficient of variation Cv) and to a lesser extent to skewness (coefficient of skewness Cs). Annual runoff run magnitude was observed to be greater than annual precipitation run magnitude, due to the higher interannual variability of runoff than precipitation. Continental differences in run magnitude of annual runoff were observed and were consistent with the continental differences in interannual variability reported previously by McMahon et al. (1992) and Peel et al. (2001, 2004b). Run severity (measured by relative severity S) was found to be independent of run length (measured by run length skew g) and strongly related to run magnitude η. These findings differ from previously published work due to the metrics of run length, magnitude and severity used in this analysis. Relationships were tested between summary statistics of run length, magnitude and severity at a station, rather than between the raw series of lengths, magnitudes and severities observed at a station used in previously published work. The relationships between annual run severity and interannual variability highlight the importance of adequately reproducing variability within GCMs for future climate-change drought scenarios, as well as having economic implications for drought relief and management policy-making. ACKNOWLEDGEMENTS We would like to thank the Cooperative Research Centre for Catchment Hydrology and the Australian Research Council grant DP for financially supporting this research. The Global Runoff Data Centre (GRDC) in Koblenz, Germany, provided 1085 stations of runoff data. Runoff data from Taiwan and New Zealand were provided by Dr Tom Piechota of the University of Nevada, Las Vegas. Runoff data for Chile were provided by Professor Ernesto Brown of the Universidad de Chile, Santiago. The precipitation data used in this paper were obtained from the Global Historical Climatology Network version 2 (Vose et al., 1992). Conversations in the initial stages of this research with Associate Professor Brian Finlayson were very helpful. APPENDIX A Extended deficit analysis is a simple technique, proposed by Pegram and reported in McMahon and Adeloye (2005), for determining the average recurrence interval (return period) of reservoir deficits. It is based on the

19 RUNS OF PRECIPITATION AND RUNOFF ANOMALIES 567 premise that changes in stored water (where V s is the deficit from the full condition) over time are equal to the difference between inflow volumes Q i and outflow volumes Q o of a reservoir, such that for any year V s (t + 1) = min[0,v s (t) + Q i (t) Q o (t)] (A.1) for a constant Q o. For this notation a full reservoir occurs when V s = 0 and any excess water (any positive value of V s ) is considered to have spilled. In extended deficit analysis the objective is to find the sequence of maximum deficits (largest negative values of V s ) between spill events. The reservoir is assumed to begin full. The sequence of maximum deficits forms a renewal process (Feller, 1968), and the deficits are considered to be independent due to their separation by spill events. Following Troutman (1976), Pegram argues that the larger deficits in the sequence of maximum deficits follow the Gumbel (extreme value type 1) distribution when Q o is less than mean annual Q i. In this paper, extended deficit analysis was applied to both annual precipitation and runoff and Q o was set to be 0.8 of MAP or MAR. Once obtained, the sequence of maximum deficits is ranked from largest to smallest; the average recurrence interval T is calculated for each deficit using Gringorten s plotting position (Gringorten, 1963) T = N m 0.44 (A.2) where N is the sample size and m the rank of the deficit. For each T the Gumbel reduced variate G r is calculated as [ ( G r = ln ln 1 1 )] (A.3) T for deficits with T greater than or equal to 10 years. The expected 1 in 100 year deficit (T = 100) is then estimated from a linear regression between the maximum deficits and G r, when at least two data points are available. REFERENCES Barber GM Elementary Statistics for Geographers. The Guilford Press: New York. Bonacci O Hydrological identification of drought. Hydrological Processes 7: Chang TJ, Stenson JR Is it realistic to define a 100-year drought for water management? Water Resources Bulletin 26(5): Dracup JA, Lee KS, Paulson Jr EG. 1980a. On the definition of droughts. Water Resources Research 16(2): Dracup JA, Lee KS, Paulson Jr EG. 1980b. On the statistical characteristics of drought events. Water Resources Research 16(2): Feller W An Introduction to Probability Theory and its Applications, volume 1, 3rd edition, Wiley: New York. Frick DM, Bode D, Salas JD Effect of drought on urban water supplies. I: drought analysis. Journal of Hydraulic Engineering 116(6): Glantz MH Drought follows the plough a cautionary note. In Drought: A Global Assessment, vol. 2, Wilhite DA (ed.). Routledge: London; Gringorten II A plotting rule for extreme probability paper. Journal of Geophysical Research 68(3): Hashimoto T, Stedinger JR, Loucks DP Reliability, resiliency and vulnerability criteria for water resource system performance evaluation. Water Resources Research 18(1): Heathcote RL She ll be right mate. Coping with drought strategies old and new in Australia. In Drought: A Global Assessment, vol. 2, Wilhite DA (ed.). Routledge: London; Heim Jr RR A review of twentieth century drought indices used in the United States. Bulletin of the American Meteorological Society 83(8): Katz RW, Brown BG Extreme events in a changing climate: variability is more important than averages. Climatic Change 21: Keyantash J, Dracup JA The quantification of drought: an evaluation of drought indices. Bulletin of the American Meteorological Society 83(8): McMahon TA, Adeloye AJ Water Resources Yield. Water Resources Publications, LLC: Colorado. McMahon TA, Finlayson BL Drought and anti-droughts: the low flow hydrology of Australian rivers. Freshwater Biology 48: McMahon TA, Finlayson BL, Haines AT, Srikanthan R Global Hydrology Continental Comparisons of Annual Flows and Peak Discharges. Catena Verlag: Cremlingen-Destedt. Milly PCD, Dunne KA Macroscale water fluxes 1, quantifying errors in the estimation of basin mean precipitation. Water Resources Research 38(10): DOI: /2001WR