INTERNATIONAL JOURNAL OF CLIMATOLOGY Int. J. Climatol. 18: 1479–1491 (1998)
DROUGHT TENDENCIES IN HUNGARY C.S. SZINELL*, A. BUSSAY and T. SZENTIMREY Hungarian Meteorological Ser6ice, 1525 -Budapest PO Box 38, Hungary Recei6ed 20 May 1997 Re6ised 22 April 1998 Accepted 23 April 1998
ABSTRACT Drought is a major climatic hazard, which is expressed as relative to a long term average. Drought is a recurrent feature in Hungary’s climate and it can cause significant damage to the country’s agriculture. Its importance is emphasized by recurrent drought events in the 1980s and 1990s. In this study, we examine long time series of the Palmer Drought Severity Index (PDSI) to characterize statistical features of drought in Hungary. Two statistical tests are presented which are appropriate tools for studying normal and extreme climatic features or events that can be characterized with thresholds. The two tests were applied to the PDSI series in order to capture changes in drought occurrence: (i) whether drought events tend to concentrate at one end of the time series, and (ii) whether droughts tend to recur in subsequent years. On a country-wide scale, results indicate that more droughts happened at the end of the time series, especially moderate and severe droughts. Moreover, moderate and severe droughts tend to occur in subsequent years. Most of the stations show the same results as the country-wide series, although some of them failed to indicate significant change. There are spatial differences, but at most of the stations a general drying tendency is apparent. © 1998 Royal Meteorological Society. KEY WORDS: statistical
test for threshold events; extreme climatological events; drought; drought tendency; Palmer Drought Severity Index; Hungary; Wilcoxon Test; Szentimrey Test; spells
1. INTRODUCTION Drought, a major climatic hazard and extreme meteorological event, originates from a deficiency of precipitation leading to a water shortage for some activity or group. The degree of shortage and its timing determine, for the most part, agricultural production and farm income and effects the local and regional economy. Drought is best expressed as relative to a long term climatic average of precipitation and evapotranspiration in the particular area. It is a recurrent feature of Hungary’s climate and can cause substantial damage to the nation’s agriculture. Dunay and Czako´ (1987) note that 36% of the overall agricultural loss originates from drought, followed by hail, floods, and frosts, in order of importance. Since 1983, every year, with the exception of 1987, 1988 and 1991, has been a drought year. This long drought series is unprecedented in the 20th century in the region and is comparable in length only to the 10-year period from 1943 to 1952 or, in severity, to the 1779–1794 drought event (Gunst, 1993). Because 8 years of the last 12 years were disastrous drought years, this series of dry years has increased scientific and political interest in climate variability and climate change and the importance of drought as an extreme meteorological event. Any change in variability or climate state could have significant impacts on the frequency and severity of future drought events (Ped, 1979). Therefore, changes in the frequency of drought occurrence can be used as an indicator of climatic change (Urba´n, 1993). Wilhite and Glantz (1987) reviewed numerous definitions of drought to determine those characteristics considered most essential for a description of the phenomenon. The methods of quantifying meteorolog* Correspondence to: Hungarian Meteorological Service, 1525-Budapest PO Box 38, Hungary; tel.: + 36 1 2124244; fax: +36 1 2125159/2125153; e-mail:
[email protected] Contract grant sponsor: US-Hungarian Joint Fund; Contract grant number: JFNo 673/96
CCC 0899–8418/98/131479 – 13$17.50 © 1998 Royal Meteorological Society
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ical droughts are based primarily on the departure of actual precipitation from the climatic mean. The Palmer Drought Severity Index (PDSI), developed by Palmer (1965), is one of the best known indices of meteorological drought and is widely applied in the United States to monitor water supply conditions (e.g. Karl, 1983; Riebsame et al., 1991). Although there are limitations and potential deficiencies when using Palmer’s index (Karl, 1983; Alley, 1984; Guttman et al., 1992), it is widely applied in the United States and elsewhere as a convenient tool for expressing moisture conditions in a standardized form, which allows direct comparison of different regions (Riebsame et al., 1991; Briffa et al., 1994; Scian and Donnari, 1997). Briffa et al. (1994) have calculated PDSI series from gridded monthly mean precipitation and temperature climatology for Europe. Their analysis provides a valuable overview of the general water status in Europe in the last century. They detected a slight and statistically insignificant water supply increase. However, the area examined has been extended, therefore considerable spatial differences occurred. The main objective of this paper is to provide a thorough analysis of drought occurrence and climatic tendencies in Hungary in the 20th century using PDSI time series. Analysis of trend, frequency of occurrence, and persistence are carried out in order to test whether the last decade reflects changes in the probability of drought and its severity. Overview data and methods are presented in Section 2 and results concerning evaluation of drought in Hungary, and at specific sites within, are presented in Section 3. The Appendices discuss the exact mathematical apparatus and derivations of the appropriate mathematical tools that were developed and applied.
2. DATA AND METHODS The Palmer Drought Severity Index is a widely accepted method of quantifying drought severity in a standardized form. Therefore, the PDSI was chosen for evaluating droughts in Hungary. The PDSI uses a hydrological accounting procedure and compares actual soil water status to the climatological mean. The calculation procedure of the Palmer Drought Severity Index is based on calculating elements of the water balance, determining their averages and characteristic local coefficients for each month using historic precipitation and temperature data. The actual index value is derived from soil moisture anomalies with the coefficients. The whole calculation procedure is well described in earlier papers (e.g. Palmer, 1965; Karl, 1983; Briffa et al., 1994). In calculating PDSI the Blaney-Criddle method for estimating evapotranspiration was used and corn (maize) was selected as the reference crop. The evaluation of drought is based on threshold values defined by Palmer (1965): drought is considered to be moderate if PDSI values are between −2.0 and − 3.0; severe, if between − 3.0 and − 4.0, and extreme if PDSIB − 4.0. Monthly PDSI values have been calculated for the period 1881–1995 at 15 stations of the Hungarian Meteorological Service. These stations have the longest record in the database (1881–1995) and provide a good spatial coverage of the country, especially for the most vulnerable agricultural lands (Figure 1). PDSI calculation requires monthly temperature means and precipitation sums. The combined effects of station movements, increasing urban heat island effects, and instrumental changes, may result in an inhomogeneous data series, especially during long periods. This problem has attracted growing scientific interest throughout the world. One possible method to overcome this problem is to slightly modify the observations using the method developed by Szentimrey (1994, 1997) for detecting and quantifying inhomogeneous data and making corrections. The method was applied to the temperature series. The original precipitation series were used, since examination revealed no large inhomogeneities in the Hungarian series. The only other parameter required in the calculation is the available soil water capacity, these values were taken from Va´rallyai et al. (1980) as representative values for the station’s vicinity. However, this parameter do not influence significantly the resulting PDSI values (Briffa et al., 1994). In the Appendices, two statistical tests are presented which have been designed to study normal and extreme climatic features or events that can be characterized with thresholds. The first test is suitable for © 1998 Royal Meteorological Society
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detecting an increase in the frequency of threshold events, while the second test can reveal the existence of spells. The tests have been applied to PDSI time series, where thresholds indicate severity of drought. The statistical tools as applied to drought index series are described in the Appendices and are briefly discussed here. The first question to address is whether drought occurrence in a specific intensity category has systematically changed or not. If there is a trend in the occurrence, drought events will concentrate at one end of the time series. This feature can be tested by Test 1, which is based on ranks of drought events. The second question concerns whether the occurrence of drought in a particular category reveals periodicity over the years; to test this, the second test has been developed. This test is based on the number of events when two subsequent elements of the time series fall on different sides of a certain threshold (called an alternation event). An example is when PDSI falls from − 2.8 to − 3.2, and the threshold is − 3. The derived ‘rank statistics’ (Test 1) and ‘alternation statistics’ (Test 2) and their statistical features make it possible to validate the existence of changes in the PDSI time series or to accept the null hypothesis of its randomness on a given significance level.
3. RESULTS AND DISCUSSION
3.1. Hungarian drought e6aluation To consider drought severity on a country-wide scale, both the index values and their spatial extent are important. The following categories were applied to evaluate drought severity: drought is moderate if PDSI values B −2.0 extend over more than 50% of Hungary; severe if these PDSI values B − 3.0, and extend over 33% of the country; and extreme if PDSI values B − 4.0 cover at least 20% of the nation. These categories, while somewhat arbitrary, were selected because more severe droughts have a lower probability of affecting larger areas. In Figure 2, symbols indicate years when the drought phenomena occurred in a particular class. It can be seen that moderate and severe droughts have occurred almost continuously during the last decade. Tests 1 and 2 were applied to the series of country-level drought categories on a seasonal and annual basis; the results are presented in Table I. The first column is the number of years in which drought of a given category occurred in the particular time interval of the year, while the other two columns are the statistical results of the two tests.
Figure 1. Map of meteorological stations (1881 – 1995) © 1998 Royal Meteorological Society
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Figure 2. Droughts in Hungary according to the country-wide drought severity categories defined in the text
Statistics of Test 1 indicate whether the occurrence of drought in the given category tend to concentrate at one end of the series; positive numbers correspond to greater frequency in later years. Table I reveals that droughts generally occurred more frequently in the second half of the century. Moderate droughts occurred more frequently in the later part of the series in all seasons as well as on yearly basis. These statistics are significant at the 1% level in all five time intervals. Evaluating the tendency of severe droughts, all the five values of Test 1 indicated a significant shift of dry years towards the end of the series, although for spring and summer the significance level is 5%. Because of the limited number of extreme droughts, results in this category can only be regarded as informative. However, the results are significant for spring and winter (5%) and for the year (1%), but not for summer and autumn. Significant negative values of test 2 indicate periodic recurrence of dry years. This was found for moderate and severe droughts occurring in winters (10%) or in a year (1%).
Table I. Results of tests 1 and 2 when applied to country drought severity series Moderate
Spring Summer Autumn Winter Year
Severe
Extreme
N
Test 1
Test 2
N
Test 1
Test 2
N
Test 1
Test 2
16 21 25 22 50
3.065 2.884 2.968 3.825 5.765
−1.401 −0.104 −0.865 −1.673 −4.864
10 8 14 18 33
2.203 2.340 2.951 2.885 4.809
−0.157 0.827 0.624 −1.539 −4.142
8 3 2 8 13
2.240 0.814 1.316 2.361 3.127
−0.659 0.314 0.219 −0.649 −1.449
Test statistics of 10% significance are italic; 5%, bold; 1%, bold italic. © 1998 Royal Meteorological Society
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It is noteworthy to mention that the tests resulted in a higher significance on a yearly basis in all the three drought severity categories. An explanation for this feature is that droughts do not always happen in the same season in the following years, but it is highly probable that it would recur at some time in consecutive years. The test results for winter should be used with care and regarded only as guidelines, because snow is not adequately accounted for in the calculation procedure of PDSI.
3.2. Change at stations 3.2.1. O6erall trends. General trends of PDSI series were tested by regression analysis and by the Mann-Kendall test. In general, both tests resulted in the same figures, therefore results of the MannKendall test are presented in Table II. The consistent results of the regression analysis and the Mann-Kendall test, i.e. the high, significant figures, support the assumption of a drying tendency and lower PDSI values in the later years. A decrease in PDSI values for different stations and months were in the range of − 1.3 to −2.4 PDSI/100 years. The majority of the values fulfill the criteria of a 1% significance level. Moreover, for more than one third of the stations, the test statistics showed a decrease in PDSI in all the months at the 1% level. Eleven of the 15 PDSI station series in all the months have experienced a significant decrease (1–10%). Only two stations revealed no significant change in the PDSI series in the majority of the months. Considering months, index series of May, October and November at all the stations decreased significantly (i.e. by at least 10%). Moreover, for October, a significant decrease was found at 14 of the 15 stations at a 1% level and one station at a 5% level. In July, three series did not indicate statistically significant change. Test statistics exceeding a value of 4 refer to an exceptionally strong trend, which was found at Buda (in 3 months), Nyı´regyha´za (5 months), Sopron (3 months), Szeged (3 months), Szombathely (10 months), Tu´rkeve (1 month) and Pe´cs (10 months), but in the latter case the value happened to exceed even 5. 3.2.2. Frequency analysis. To evaluate the frequency of drought occurrence, Test 1 was applied to the regional PDSI series. Results of this analysis are presented in Figure 3, where three lines belong to each station. These lines correspond to different drought severity classes according to Palmer (1965), while different symbols indicate different significance levels. Twelve symbols in each line correspond to the months. It is shown that droughts, particularly moderate droughts, tend to occur more frequently in north and east while in the southwest, almost no changes can be detected. Note that although a significant decrease was found at Baja in the south (Table II), the frequency of droughts remained almost unchanged. Test 2 was applied to evaluate the probability of low a PDSI value appearing in the same month in subsequent years in order to reveal tendencies of drought in certain months to form dry spells. The calculations showed diverse results (Figure 3). In many cases, no strong significance was found as in the previous test, or the figures were not significant. One can note spatial differences as well, namely in the western regions where considerable changes concentrate in the winter half year, e.g. at Sopron or Zalaegerszeg, in contrast to Szarvas, where no negligible figures occurred during the summer half of the year. The highest significance levels were found at Nyı´regyha´za in correspondence to its strong tendencies. In many cases, the occurrence of extreme drought events is too rare for statistically reasonable results. Considering the trend test and Tests 1 and 2, different behavior can be noticed at different stations. At Szarvas, besides the significant trend indicated by the Mann-Kendal test, our two tests also resulted in significant figures in numerous months and in all the three drought categories. These results suggest drying and more frequent drought events in the recent years, and also indicate a tendency for dry spells to form at the Szarvas area. Buda follows different characteristics, where in addition to the significant trend there is a strongly significant increase in drought frequency. This means that besides drying, droughts occur more often. On the contrary, the second test did not demonstrate a periodicity in dryness at the monthly time scale. At Buda, more frequent occurrence of dry and extremely dry years are accompanied by a striking disappearance of wet years after the 1960s (Figure 4), which is reflected in the statistical analysis as well. It is also apparent, although not statistically testable, that the two driest years occurred recently, in 1990 and 1992. © 1998 Royal Meteorological Society
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Table II. Results of the Mann-Kendall test January
February
March
April
May
June
July
August
September
October
November
December
Baja Buda Debrecen Kalocsa Kecskeme´t Keszthely Mosonmagyaro´va´r Nyı´regyha´za Pe´cs Sopron Szarvas Szeged Szombathely Turkeve Zalaegerszeg
−2.835 −3.614 −2.299 −2.347 −2.802 −0.906 −2.323 −3.971 −4.750 −2.714 −3.502 −3.672 −4.083 −3.101 −1.385
−2.705 −2.951 −1.999 −2.018 −2.163 −0.742 −1.491 −3.657 −5.219 −1.651 −3.391 −3.938 −3.710 −2.884 −1.665
−3.285 −3.546 −2.526 −2.540 −3.092 −1.438 −1.757 −4.005 −5.349 −1.815 −3.478 −4.155 −4.286 −3.174 −1.960
−3.063 −4.479 −2.033 −2.497 −3.295 −1.443 −2.922 −4.266 −5.586 −2.632 −3.967 −4.179 −4.972 −3.478 −2.603
−3.309 −4.904 −2.076 −2.618 −3.261 −2.018 −3.145 −3.894 −5.852 −2.792 −3.971 −4.165 −4.228 −2.714 −1.777
−2.71 0 −3.899 −2.159 −2.072 −2.632 −0.960 −2.758 −3.826 −4.624 −2.463 −3.541 −2.937 −3.735 −3.314 −1.593
−2.429 −3.764 −2.260 −1.448 −1.951 −1.240 −2.729 −3.967 −4.213 −3.256 −3.270 −2.681 −4.170 −2.951 −1.409
−2.284 −3.198 −2.183 −1.665 −2.966 −1.810 −1.970 −3.121 −3.913 −4.639 −3.425 −2.197 −4.450 −2.850 −1.303
−2.366 −3.793 −2.671 −2.139 −3.183 −1.994 −2.623 −3.560 −3.817 −4.576 −3.696 −2.743 −4.721 −2.555 −1.661
−3.295 −4.764 −3.077 −2.821 −3.831 −2.627 −3.179 −4.435 −4.987 −4.537 −4.363 −3.851 −4.817 −4.141 −2.110
−2.589 −3.938 −3.024 −2.139 −3.208 −1.873 −2.627 −4.150 −4.426 −3.739 −4.121 −3.657 −4.489 −3.764 −1.293
−2.115 −3.570 −2.816 −1.632 −2.719 −1.404 −2.420 −4.295 −4.421 −3.667 −3.817 −3.217 −4.411 −3.517 −1.245
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Test statistics of 10% significance are italic; 5%, bold; 1%, bold italic.
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Figure 3. (a) Spatial distribution of Test 1 results. Significance levels of monthly test statistics are indicated at each stations; (b) spatial distribution of Test 2 results. Significance levels of monthly test statistics are indicated at each stations
Baja represents a third type of station. Despite the strongly significant trend in PDSI values at the station, there is no statistical proof of other changes. These features can be explained by examining the actual PDSI values as presented in Figure 5 for Baja. It is clearly seen why both Tests 1 and 2 failed to indicate significant results (i.e. lower values of PDSI do not reveal any change). On Figure 5, dry years occur regularly with almost no notable change although it is worth mentioning that there were no extremely dry years observed. The overall decreasing trend results from a disappearance of wet years, especially after 1970. © 1998 Royal Meteorological Society
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Keszthely belongs to the fourth type, where none of the three tests resulted in statistically significant values, suggesting no change at that stations regarding drought.
3.2.3. E6aluation of test results with multiple tests. In Sections 3.2.1 and 3.2.2 at each station, the statistical tests were applied to independent monthly series simultaneously, therefore the results can be regarded as independent. Sneyers (1990) suggests that interpretation of results may lead to erroneous conclusions. In order to test whether a set of results can be considered as significant, two statistical tools are applicable: the Fisher test and the Binomial test (Sneyers, 1990). Results of these tests indicate the probability that the several independent test results are taken from a random sample. For 12 independent series the test statistics of the Fisher test is the following function of the significance levels of the test results: 12
id= ’’1’’\x 224 = −2 % log ai i=1
Under the null hypothesis, this test statistic has a x 2 distribution with 24 degrees of freedom. From Equation (1), it is apparent that this test is sensitive to the values of the probe statistics, therefore one large value may result in disapproving the whole hypothesis, which is problematical considering type I errors. The result of the binomial test depends on the number of test statistics reaching a given significance level. This may result in approving the null hypothesis despite some strongly significant test results, which is problematical considering type II errors. It is therefore beneficial to use and evaluate these tests simultaneously. However, the Fisher test can only be applied to test statistics following a continuous distribution.
Figure 4. PDSI series at Baja in July between 1881 and 1995 © 1998 Royal Meteorological Society
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Figure 5. PDSI series at Buda in July between 1881 and 1995
In Sections 3.2.1. and 3.2.2., results of the Mann-Kendall test and Tests 1 and 2 were presented, when they were applied to monthly series of index series. From these, the Mann-Kendall test and Test 1 have continuous test statistics, while Test 2 has discrete test statistics. Therefore, the Fisher test can only be applied to the Mann-Kendall test and Test 1. Results of this analysis are presented in Table III. In most of the cases, the Fisher test resulted in highly significant figures. Considering the trend, the Fisher test indicates that we should reject the null hypothesis of randomness of the decreasing trends detected by Mann-Kendall test (Table II). Even at Keszthely and Zalaegerszeg, where the least significant trends appeared, the Fisher test suggests an overall decreasing tendency at the 1% significance level. Considering drought occurrences in different severity classes, there are also significant values at 14 stations. Occurrence in the moderate class increased at 14 stations at the 1% significance as well as in the severe class. Results at two-thirds of the stations satisfies the 1% significance criterion. Although the number of occurrence in the extreme drought severity class is generally lower, the Fisher test indicates that the randomness can be rejected at six stations at least at the 10% significance level. The binomial test can be applied to all the three tests. Table IV presents the significance levels associated with the possible values of B(12, p) binomial variable (P= 0.1, 0.05, 0.01). Table IV demonstrates the probability that the number of significant probe statistics (on P= 0.1, 0.05 and 0.01 significance level) can reach k under the null hypothesis of randomness. For instance, in the first row of Table II (Baja), the number of significant probe statistics on 0.01% level is eight, whereas Table IV indicates that its probability is only 4.776×10 − 14. Therefore, one must reject the null hypothesis of randomness of this series. © 1998 Royal Meteorological Society
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Table III. Results of the Fisher test when applied to the results presented in Table II (for Mann-Kendall), and in Figure 3 (for Test 1) Station
Mann-Kendall
Test 1 (PDSIB2)
Test 1 (PDSIB3)
Test 1 (PDSIB4)
Baja Buda Debrecen Kalocsa Kecskeme´t Keszthely Mosonmagyaro´va´r Nyı´regyha´za Pe´cs Sopron Szarvas Szeged Szombathely Turkeve Zalaegerszeg
124.6049 222.4358 101.9211 85.4116 138.8178 52.9117 107.8088 226.2145 306.9793 169.7887 204.8920 184.3112 269.2047 160.6471 57.9085
54.1584 151.4953 103.2848 68.1207 104.3879 47.8719 83.3491 208.5867 130.8772 113.9527 211.8414 149.9581 151.9046 121.9711 29.7465
25.6479 120.2260 63.4039 18.1458 68.3855 33.0375 31.8234 153.6251 76.7008 56.5742 171.6258 70.7471 51.8904 82.2086 11.3413
10.2285 62.6107 36.6987 22.0839 30.8646 33.6100 29.5019 78.0456 27.9924 35.1333 44.1470 29.4061 9.9304 18.8008 12.3547
Test statistics of 10% significance are italic; 5%, bold; 1%, bold italic.
With this procedure, all the results presented in Figure 3 can be examined in order to verify their randomness. From Table IV, in order to satisfy the 1% significance, it is sufficient when two test statistics in one line are significant on 1% level, or four on 5% level or five on 10% level. Therefore the drought frequency increase in the moderate class (Figure 3, upper line) is significant at 14 stations (13 on 1% level), in the severe class (middle line), ten are significant, and in the extreme class (lower line) four stations are significant (three at 1% level). Concerning Test 2 (Figure 3), in the moderate class, significant figures are present at ten stations (eight on 1% level), in the severe class (middle line) nine are significant (seven at 1% level), and in the extreme class (lower line) five stations are significant (three on 1% level).
4. CONCLUSIONS Two statistical tests have been presented that are appropriate tools for studying normal and extreme climatic features or events that can be characterized with thresholds, and are based on mathematical proofs. The first test is suitable for detecting increases in the frequency of threshold events, while the Table IV. The 1−F(k) (k=0, . . . 12) probabilities, where F(k) is the distribution function of a B(12, P) (P= 0.1, 0.05, 0.01) binomial variable k
10% (P =0.1)
5% (P=0.05)
1% (P= 0.01)
1
0.7176 0.3410 0.1109 0.02564 0.004329 5.412E-04 5.018E-05 3.414E-06 1.658E-07 5.455E-09 1.090E-10 1.000E-12
0.4596 0.1184 0.01957 0.002236 1.839E-04 1.111E-05 4.949E-07 1.612E-08 3.743E-10 5.873E-12 5.591E-14 2.441E-16
0.1136 0.006175 2.056E-04 4.642E-06 7.470E-08 8.775E-10 7.580E-12 4.776E-14 2.141E-16 6.481E-19 1.189E-21 1.000E-24
2 3 4 5 6 7 8 9 10 11 12
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second test can reveal existence of spells. The tests have been applied to PDSI time series, where thresholds indicate severity of drought. Results revealed a significant change in these climatic characteristics. The second test provided disparate results. On the basis of the two tests, a significant increase in drought frequency can be observed, particularly for moderate and severe droughts, but seasonal droughts are not necessarily repeated in the same season, although they are likely to happen in the consecutive years. The examination of the PDSI values in Hungary has revealed a considerable decreasing trend during the last century, which is in the order of − 2 PDSI/100 years, but is region dependent. The above result suggests that a general drying tendency exists. This idea is supported by the lack of wet years in recent decades and appearance of extremely dry years at the majority of the stations. ACKNOWLEDGEMENTS
The authors express their sincere thanks to Dr D.A. Wilhite, University of Nebraska (US) for his helpful comments on this manuscript and the anonymous reviewer for his constructive comments. Part of this work has been financed by the US-Hungarian Joint Fund (JFNo 673/96).
APPENDIX A. STATISTICAL TESTS FOR THRESHOLD EVENTS Let x(t) (t= 1, . . , n) be the examined time series, the null hypothesis (H0) be that the series elements are independent identically distributed random variables. Let c be a threshold value, and the threshold event is defined as: x B c, for which the following relationship is assumed under the null hypothesis (H0): 0B p= P(x(t) B c) B 1
!
t = 1, . . . , n.
The indicating variable of the threshold events can be defined as: o(t)=
1 0
x(t) B c x(t) ] c
t = 1, . . . , n.
The frequency of threshold events can be easily expressed by the indicating variables: n
m = % o(t). t=1
Under the null hypothesis (H0), the m variable follows Bernoulli distribution with parameters n and p: mB(n; p)
APPENDIX B. Test 1 (BASED ON WILCOXON TEST) Let us assume the ranks of threshold events are: ri (i= 1, . . . , m), that is o(ri )= 1 (i= 1, . . . , m). Let r be the rank statistics defined as: m
n
i=1
t=1
r = % ri = % to(t). Under the null hypothesis (H0): The conditional expectation and variance of r, given, m, are: E(r m)
m(n+1) , 2
V(r m)
m(n −m)(n − 1) . 12
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The test statistic, which is the ‘conditional standardization’ of rank statistic, is: r − E(r m)
V(r m)
.
Its limiting distribution is standard normal, according to the properties of the Wilcoxon test.
APPENDIX C. TEST 2 (DEVELOPED BY T. SZENTIMREY) The ‘alternation’ of threshold event can be written as: (x(t)] c and x(t +1) B c) or (x(t) B c and x(t+1)] c), and the number of alternation event can be expressed by the indicating variables n−1
n= % (o(t+1) − o(t))2.
(C1)
t=1
Under the null hypothesis (H0), the alternation statistics (n) can be characterized by the following properties. Assuming m5 n −m, the possible conditional values of n, given m, are: 1, 2, . . , 2m, when, mB n −m and 1, 2, . . , 2m− 1, when m =n − m and the conditional discrete distribution of n, given, m, is:
P(n = 2i m)=
P(n=2i+ 1 m) =
P(n = 2m m)=
m −1 i
2 n m
m −1 i− 1
1 n m
1 n m
n − m− 1 i
i= 0, . . . , m− 1,
n −m − 1 m−1 + i i
n −m − 1 i
n− m−1 i− 1
i= 1, . . . , m − 1,
if mBn− m.
The conditional expectation and variance of n, given m, are:
n 1 E(n m)=2 m − m 2 =2 % (o(t) −o¯ )2 n t=1
V(n m)=
E(n m)(E(n m)− 1) . n− 1
(C2)
The limiting distribution of n. According to the von Neumann ratio, the statistics n−1
Á % (o(t + 1) − o(t))2 Â Ã Ã1 t=1 −1Ã ,
nà n 2 à Ã2 % (o(t) −o ¯ ) Ä Å t=1
(C3)
follows standard normal limiting distribution. Taking Equations (C1) and (C2) into account, the statistics defined by Equation (C3) can be expressed as: © 1998 Royal Meteorological Society
Int. J. Climatol. 18: 1479 – 1491 (1998)
n
n −1 E(n m)
1491
DROUGHT TENDENCIES IN HUNGARY
(C4)
Consequently, the test statistics given by Equation (C4) has standard normal limiting distribution. REFERENCES Alley, W.M. 1984. ‘The Palmer Drought Severity Index: limitations and assumptions’, J. Clim. Appl. Meteorol., 23, 1100 – 1109. Briffa, K.R., Jones, P.D. and Hulme, M. 1994. ‘Summer moisture variability across Europe, 1892 – 1991: an analysis based on the Palmer Drought Severity Index’, Int. J. Climatol., 14, 475 – 506. Dunay, S. and Czako´, F. 1987. ‘Use of meteorological information in agricultural production’, in Reports on Scientific Research of the Hungarian Meteorological Ser6ice, Budapest, pp. 193 – 209 (in Hungarian). Gunst, P. 1993. ‘The droughts and the Hungarian state’, in Bara´th, C. et al., 1993, Drought 1983, University for Horticultural Science, Budapest, pp. 131–159 (in Hungarian). Guttman, N.B., Wallis, J.R. and Hosking, J.R.M. 1992. ‘Spatial comparability of the Palmer Drought Severity Index’, Water Res. Bull., 28, 1111 – 1119. Karl, T.R. 1983. ‘Some spatial characteristics of drought duration in the United States’, J. Clim. Appl. Meteorol., 22, 1356 – 1366. Palmer, W.C. 1965. Meteorological Drought, US Weather Bureau, Research Paper No. 45, Washington DC, 58 pp. Ped, D.A. 1979. Temporal Fluctuations of Atmospheric Aridity and Excess Moisture in May – August in the European USSR, Gidrometizdat, Leningrad, Trudy, No. 213, pp. 82–103. Riebsame, W.E., Changnon, S.A. and Karl, T.R. 1991. Drought and Natural Resources Management in the US: Impacts and Implications of the 1987 – 89 Drought, Westview Press, Boulder, CO. Scian, B. and Donnari, M. 1997. ‘Retrospective analysis of the Palmer Drought Severity Index in the semi-arid pampas region, Argentina’, Int. J. Climatol., 17, 313–322. Sneyers, R. 1990. On the Statistical Analysis of Series of Obser6ations, WMO Technical Note No. 143. Szentimrey, T. 1994. Estimation of Inhomogenities in Temperature Data Series of Hungary, Clim. and Agromet. Papers, No. 2. Szentimrey, T. 1997, ‘Statistical procedure for joint homogenization of climatic series’, in Proceedings of the First Seminar for Homogenization of Surface Climatological Data (6–12 October 1997, Budapest, Hungary), Hungarian Meteorological Service, Budapest, pp. 47 – 62. Urba´n, L. 1993. ‘The concept of drought and its significance’, in Reports on Scientific Research of the Hungarian Meteorological Ser6ice, Budapest, pp. 113–135 (in Hungarian). Va´rallyai, Gy, Szu¨cs, L., Rajkai, K., Zilahy, P. and Mura´nyi, A. 1980, ‘Category system and map of hydrological properties of Hungarian soils’, Agrokemia es Talajtan, 29, 77–112 (in Hungarian). Wilhite, D.A. and Glantz, M.H. 1987, ‘Understanding the drought phenomenon: the role of definitions’, in Wilhite D.A. and Easterling, W.E. (eds), Planning for Drought, Westview Press, Boulder, CO, pp. 11 – 27.
© 1998 Royal Meteorological Society
Int. J. Climatol. 18: 1479 – 1491 (1998)
