Fekete, L.

Szepesi, K.

English

Csapadékos szakaszok hosszának és csapadékhozámának statisztikai elemzése. A dolgozat a budapesti és szegedi napi csapadékadatokból számított tartamvalószínűségeket vizsgálja. Megállapítja, hogy az egymásra következő észlelt csapadékos szakaszok hossza igen magas szignifikancia-szinten különbözik a véletlenszerűen egymásra következő csapadékos szakaszok hosszától. Meghatározza továbbá a tanulmány a csapadékos szakaszok hossza és csapadékhozama közötti összefüggésnek a lineáris függvénykapcsolattól való eltérését. Statistische Analyse der Länge der Niederschlagsperioden und ihrer Niederschlagsmenge. Der Aufsatz untersucht die Wahrscheinlichkeiten der Dauer der Niederschlagsperioden, die aus den täglichen Niederschlagsangaben von Budapest und Szeged ausgerechnet wurden. Es wurde festgestellt, dass sich die Länge der nacheinanderfolgenden registrierten regnerischen Phasen von der Länge der zufällig nacheinanderfolgenden regnerischen Phasen in einem sehr hohem Signifikänzniveau unterscheidet. Die Arbeit bestimmt weiterhin die Abweichung des Zusammenhangs zwischen der Länge der Niederschlagsperioden und ihrer Niederschlagsmengen von dem linearen funktionellen Zusammenhang. The paper examines the probability of durations calculated from the daily data of precipitation of Budapest and Szeged. It states that the length of the observed rainy periods differs from the length of the accidental rainy periods on a high level of significance. Besides the paper defines the deviation of functionality between the length of the rainy period and its amount of precipitation from the linear connection

Fekete L.

Szepesi K.

University of Szeged

A Statistical Analysis of the Length of Rainy Periods and their Quantity of Precipitation by L. Fekete and K. Szepesi Csapadékos szakaszok hosszának és csapadékhozámának statisztikai elemzése. A dolgozat a budapesti és szegedi napi csapadékadatokból számított tartamvalószínűségeket vizsgálja. Meg-állapítja, hogy az egymásra következő észlelt csapadékos szakaszok hossza igen magas szignifikan-cia-szinten különbözik a véletlenszerűen egymásra következő csapadékos szakaszok hosszától. Meghatározza továbbá a tanulmány a csapadékos szakaszok hossza és csapadékhozama közötti összefüggésnek a lineáris függvénykapcsolattól való eltérését. Statistische Analyse der Länge der Niederschlagsperioden und ihrer Niederschlagsmenge. Der Aufsatz untersucht die Wahrscheinlichkeiten der Dauer der Niederschlagsperioden, die aus den täglichen Niederschlagsangaben von Budapest und Szeged ausgerechnet wurden. Es wurde festge-stellt, dass sich die Länge der nacheinanderfolgenden registrierten regnerischen Phasen von der Länge der zufällig nacheinanderfolgenden regnerischen Phasen in einem sehr hohem Signifikänzniveau unterscheidet. Die Arbeit bestimmt weiterhin die Abweichung des Zusammenhangs zwischen der Länge der Niederschlagsperioden und ihrer Niederschlagsmengen von dem linearen funktionellen Zusammenhang. The paper examines the probability of durations calculated from the daily data of precipitation of Budapest and Szeged. It states that the length of the observed rainy periods differs from the length of the accidental rainy periods on a high level of significance. Besides the paper defines the deviation of functionality between the length of the rainy period and its amount of precipitation from the linear connection. Data referring to the length and precipitation amount of rainy periods take a very important part among the pieces of climatological information for the distri-bution of rain. It is mainly the management of water supplies and agriculture that need such information. A rainy period is defined as a sequence of rainy days. The definition allows the presence of more factors in bringing forth a rainy period since the amount of rain refers to 24 hours. So rainy periods cannot be considered homogenous from meteoro-logical point of view. In the analysis of data these factors are neglected. In this study the daily precipitation amounts of Budapest and Szeged are ana-lysed for the 40 year long period of 1931—1970. Rainy day means the fulfilment of the R 0.1 mm condition where R represents one day's precipitation amount. Tables 1—2 give the number of rainy periods of different length during the period in question. It can be stated that rainy periods of more than 5 days occur very rarely. The number of rainy periods shows a gradual decrease from May to June with a minimum in October, then a gradual increase follows. The number of rainy periods is less at Szeged than at Budapest. Table 8 gives the probability of rainy days per month (p) and the complementer probability of days without rain (q). It holds for the whole year that the probability of 25-rainy days is under 50 per cent, it is only the maximum of November when this value is 46 per cent at Budapest. The data of Table 8 are used to calculate other proba-bilities for the characterization of the length of rainy periods, those which are valid on condition of independence. These are compared to real values. In case of independence the probability (p) of rainy periods of a definite length (k) can be calculated with the r(k) = Nq2pk (1) formula, where N is the number of data in the sequence i. e. the number of days in the period in question. The frequency values r{k) are contained in Tables 3—4 comparing Tables 1—2 and 3—4, a great difference can be noticed between the number of rainy periods calculated on the basis of independence and those in reality: the number of short periods (1—3 days) is much greater on condition of independence than in reality; on the other hand the longer periods (4days or more) show an opposite deviation. So rainy periods cannot be considered as a sequence of independent cases. The simplest mathematical-statistical proof of this can be given by counting the average length of the rainy periods on condition of independence. This value can be compared to the real, lengths. On condition of independence in a sequence of N data the possible number of periods in Npq, and from this the mean length of the periods (A) is: A =\ (2) The real mean lenght of the periods — calculated from the frequency of the event in question divided by the number of the periods — is a quantity A*. In case of independence A*/A = 1, while otherwise A*/A ^ 1. The hypothesis of independence of data neglected on condition that the value of the ratio A*/A falls outside, a definite interval calculated I = — where d is a factor of the wanted level of significance. In our study, d=3 with a level of significance 0.0027. Table 5 gives the values of A*, A and A*/A with the limits of the interval of independence. The value of the ratio A* IA is greater than 1 in every month and they fall outside the interval I. So the probability of independence in the length of the periods is 0.0027, this, being a very small value the idea of independence can be thrown away. The annual course of the value A*/A is very characteristic. This is contained in Fig. 1. Maxima of both stations take place in October (this is the period of the least number of rainy seasons and that of the longest ones), the minima in summer. A weaker secondary maximum takes place in spring. This can be under-stood by the climatological fact that in autumn, at the beginning of winter and in spring the persistent precipitations of warm fronts dominate. Further on, the relation between the amounts of precipitation and the length of rainy periods is examined. The general tendency is that with the growth of the length of the period, the amount of precipitation grows as well. With the graphical pre-sentation of the data for a month (Fig. 2—3) the increase seems to be not expressedly 2 6 linear, but rather exponential. So now the equation of this exponential function is defined, then the difference between this and the linear growth. The exponential function is: • f = y (4) where c± is the mean amount of precipitation per day in the month in question, t is the number of days and a is the exponent wanted. The approaching values of a are calculated with the help of string method and contained in Table 6. This shows that the amount of precipitation is not in close connection with the length of the period. The value of a is greater than 1 nearly all over the year. Smaller values occur in February and June at Budapest and in Novem-ber at Szeged. But in most cases there is only a small difference between a and 1, that the difference from linear values can be neglected. There are only two months at Budapest (August and November) and four at Szeged (March, July, August and December) when the difference is more than 1/iO. So it is only August that refers to both places. Table 6 shows that in case of 7 months, Szeged has the greatest values of fi/A 16-15 iA 1.3-1,2 -1,-1 -1,0 — — S z e q a d . Budapast-- r -F —i— M. T A —r 1— M. J. -i— J. —>— A. —r-0. - r -N. Fig. 1. The values of the quotient A* IA 1. ábra. Az A*IA értékei a meaning that equally long rainy periods result in more precipitation at Szeged. The difference is more prominent in February and July and December, while in August it is nearly the same. As to its annual course, it decreases from December to January, then it increases in February and March. Then, after a short decrease, the summer maximum follows (in August at Budapest and in July in Szeged). Then again a decrease follows untill October at Szeged and November at Budapest. The comparison of the two places from this point of view shows that the change is opposite in May, June, August and in the last three months of the year. A general phenomenon is suspected to exist, when the nature of the change is the same referring to the previous month. To be sure about this the data of more stations have to be treated. (The greater value of a is probably caused by the greater moisture content of the air and the greater amount of convective precipitation). For the sake of better comparison, the next rate was calculated: Q = 2 7 This was calculated for each month and for both places. Table 7 shows the values. Based on the above facts, a new aspect of examination comes. The value of Q increases with the length of the period in February and June for Budapest and in November Szsqed y = 3 ,9- t w 6 0 * Budapest y ^ - t 4 - 0 2 0 ' ' 2. ábra. A periódusok csapadékhozama januárban és áprilisban . for Szeged since the exponent of the denominator is greater in these months than that of the numerator. (c,= c1 • t\ c't — Cj • f in the formula) Table 7 shows that the change is opposite at the two places in October, November and February and December referring to the previous month. On the other hand there is not even 1/10 difference in the value of Q in any month of the year between Budapest and Szeged. Supposingly this is a general phenomenon, existing not only for these two places. This would mean that Q can be defined for any place from mean values with the help of c1 through the calculation of c',. This is shown by Fig. 2—3 where the distribution of monthly precipitation amounts among the periods 2 8 Szeged y-5.6 t 1 1 6 3 7 Szaqed y ^ . 7 t1'?550 —-Budopast <|»5.7-t t 0 M S Budapest ( (.5.0 t l c ' l t 3 Precipitation Precipitation Fig. 3. The amount of precipitation of the periods in July and October 3. ábra. A periódusok csapadékhozama júliusban és októberben J A N U A- R Y % 25- , - 2 0 -15-1 0 -5 0 i—i—i—i—i—i—i—i—i—i 0 2 h 6 . 8 10 F E B R U A R Y 30 25-i 20 -10 5-1 0 T I i I I i i i—i—I 0 2 k 6 8 10 % 30-, 2 5 -20-15-•10 5 0 M .A R c H 1 i i i i i i—i—i—i—i—i 0 2 ^ 6 8 10' 12 S z e q e d Budopes t Fig. 4. The division of monthly precipitation between the periods in January, February and March 4. ábra. A havi csapadék megoszlása a periódusok között januárban, februárban és márciusban 3 0 A P R . I L M A Y % J U N E % S z e g e d Budapest Fig. 5. The division of monthly precipitation between the periods in April, May and June 5. ábra. A havi csapadék megoszlása a periódusok között áprilisban, májusban és júni usban J U L Y 30-, 25-2 0 -15-1 0 -5 0 A U G U S T 1—i—I—i—i—i—m—i—i 0 '2 h 6 8 10 V. 3 5 -3 0 - 4 M 25- i ^ 20-/ / / 15-10-5-0 - i i S E P T E M B E R 0 2 6 8 10 12 Szeqed Budapest Fig. 6. The division of monthly precipitation.between the periods in July, August and September 6. ábra. A havi csapadék megoszlása a periódusok között júliusban, augusztusban és szeptemberben O C T O B E R N O V E . M B E R D E C E M B E R Szeged Budopest* Fig. 7-. The division of monthly precipitation between the periods in October, November and December 7. ábra. A havi csapadék megoszlása a periódusok között októberben, novemberben és decemberben 3 Acta Climatológica is reflected in per cents. So it can be said that the greatest part of monthly mean precipitation amounts comes from rainy periods shorter than four days in every month. Considering the amount of precipitation for every period it can be seen that the difference from the equalizing curve is less than 3 mm in each period. In case of longer periods the differences are greater. So we are able to define the precipitation amount of periods of 2—3 days by calculation from the mean value of one day's amount quite exactly, but estimations are possible to be given for longer periods as well. (It is possible only if the value of Q is nearly the same for the whole country, and so the treatment of the data of more stations is needed.) The reliability of this computation is to some extent decreased by the standard deviation of real data. Their periodicity can be examined, in order to make uncertainty decrease, for different amounts of precipitation. Fig. 4—7 shows the distribution of monthly mean precipitation amounts among periods of different length. Figures on the. left show how many per cents of the monthly precipitation amounts are given by the different periods, while those on the right show how many per cents of the monthly precipitation are given by periods not longer than a given value. It is seen that more than 30 per cent of the monthly total is given only in two cases by a period of fixed length. Even the number of periods giving more than 25 per cent is small. So most of the monthly amounts of precipitation is given by periods of 10—25 per cent. Regarding the whole year a change in favour of the short periods can be noticed in the summer months. Every diagram shows the longest period giving more than 10 per cent of the monthly precipitation, but the 5 days' periods are generally of less importance. Comparing the two stations it can be stated that in the case of Szeged with a more arid climate the greatest part of the monthly precipitation amount is given by shorter (2—3—4 days) periods. In the case of Budapest the longer periods have the greater importance. This shown by the diagrams on the right, since the curves of Szeged data are. above the Budapest ones, they move off each other until the periods of 2—3—4 days, then move closer. Less difference can be noticed in January, February, October and August. 34 Table L The Number of Periods Occured in Budapest (1931—1970) ' Length of the Period (Day) 1 • 2 3 4 5 6 7 8 9 10 January 109 ( 62 21 27 13 / 4 / I ' 109 124 63 108 65 6 28 8 9 10 February 84 53 33 17 7 6 2 2 2 84 106 99 68 35 36 14 16 18 March 99 51 23 14 11 6 2 •2 1 99 102 69 56 55 36 14 16 9 April 94 60 31 12 8 ' 5 2 1 1 1 94 120 93 48 40 30 14 8 9 10 May 103 .61 30 19 12 6 •1 3 7 • / 103 122 90 •76 60 36 1 24 . 9 10 June 110 '53 33 15 * 9 6 1 2 110 106 99 60 45 36 8 18 July 118 51 28 10 6 3 1 7 118 102 84 40 30 18 8 : 9 August 123 56 27 7 5 1 2 1 123 112 81 28 25. 6 14 8 September 87 47 18 9 5 2 1 87 ' 94 54 36 25 12 1 . October 67 46 18 15 6 5 2 2 1 67 92 54. 60 30 30 14 16 10 November 83 58 34 25 8 8 4 1 83 116 . 102 • 100 40 . 48 .28 32 io December 87 62 29 16 10 13 3 J . 2 87 124 87 64 50 78 21 . 27 20 11 1 11 12 13 14 15 Other 1 14 ; 15 (20) 1 20 U) The number of the wanted length periods The number of the days being in the wanted length and wanted number periods Table 2 The Number of Periods Occured in Szeged (¡931—1970) Length of the Period (Day) . 1 2 3 4 5 ' 6 : 7 8 9 10 11 January 93 61 23 18 9 3 3 1 1 93 122 69 72 45 18 . 2 1 8 9 February 85 56 24 15 • 8 6 3 1 2 85 112 72 60 40 36 21 9 10 March 93 56 21 18 10 3 2 1 93 112 63 72 50 18 14 11 April 79 . 58 24 20 3 8 2 . / 79 116 12 80 15 48 14 9 May 92 66 30 22 9 3 92 132 90 88 »45 18 June 113 64 24 14 3 4 . 2 1 • 113 128 72 56 15 24 14 10 July 120 42 • 18 8 2 3 1 120 84 54 48 10 18 8 August 111 36 21 5 3 ' 4 1 111 72 63 20 15 24 7 September 90 36 20 6 2 1 1 . 1 90 72 60 24 10 6 7 11 Oktober 69 43 19 11 7 / 69 . 86 57 44 35 21 8 November . . 74 54 27 22 12 5 / 7 ' 2 / 74 108 81 88 60 30 7 8 18 10 December 93 55 25 16 ' 10 • 7 5 1 2 1 93 110 • 75 64 50 42 • 21' 8 18 11 The number of the wanted length periods The number of the days being in the wanted length and wanted number periods Table L Length of the Period J • ' • F . . M . 1 186 171 193 2 76 68 67 3 • •31 27 24 4 12 11 8 5 5 4 3 6 2 2 1 7 1 1 (0,35) 8 (0,36) (0,28) (0,12) 9 (0,14) (0,11) (0,04) The Monthly Values of r(k) in Budapest A M J J 185 184 181 . 192 105 77 73 60 25 33 29 19 9 14 12 6 4 6 5 •2 1 2 2 1 (0,4) 1 • 1 (0,17) (0,17) (0,4) (0,29) (0,05) (0,07) (0,17) (0,12) ' .(0,01) • A _ S o N . D 192 177 190 169 180 60 45 55 78 79 19 10 16 36 35 6 3 5 17 15 2 1 1 8 • 7 1 (0,17) (0,39) 4 3 (0,17) (0,04) (0,11) 2 1 (0,05) (0,01) (0,04) 1 1 (0,01) . (0,002) (0,009) (0,34) (0,25) Length of the Period J • . • F' 1 183 163 2 66 65 3 24 26 4 9 10 5 ' 3 4 6 . 1 2 7 (0,5) . 1 8 (0,12) (0,2) 9 (0,05) (0,1) Table 4 The Monthly Values of r(k) in Szeged M A M J J 183 183 180 183 177 64 66 70 66 46 22 • 24 27 24 12 . 8. 9 11 9 3 3 3 4 3 • 1 1 1 2 1 (0,2) (0,4) (0,5) 1. (0,5) (0,05) (0,1) (0,12) (0,2) (0,12) (0,01) (0,05) (0,05) (0,1) (0,05) (0,003) A S o N D 175 '164 178 ' 173 173 ' '44 • 38 48 69 69 10 9 13 . 28 28 3 2 4 11 11 1 (0,5) 1 4 4 (0,17) (0,11) (0,26) 2 • 2 (0,04) (0,02) (0,07) 1 1 (0,01) (0,006) (0,02) (0,28) (0,28) (0,003) (0,001) (0,006)' (0,13) (0,13) u> .00 Table 5 The Values of A; A*; A* ¡A and the Limits of the Interval of Independence ( I ) Bud est J F M A M J J A S • O N D A 1,694 1,666 • 1,538 1,588 • 1,724 1,666 1,450 1,450 . 1,333 1,408 1,852 1,785 A* 2,209 2,311 2,172 2,168 2,326 2,161 1,876 1,788 1,864 .2 ,302 2,563 2,518 A*/A 1,303 1,387 1,411 1,366 1,349 1,293 1,294 1,234 1,398 1,635 1,384 1,410 I 0,9354 0,9337 0,9563 0,9391 0,9353 0,9354 0,9473 0,9473 . 0,9537 0,9497 0,9255 0,9313 —1,074 - 1 , 0 7 6 - 1 , 0 6 5 - 1 , 0 6 9 - 1 , 0 7 6 - 1 , 0 7 4 - 1 , 0 5 9 - 1 , 0 5 9 - 1 , 0 5 2 - 1 , 0 5 6 - 1 , 0 8 4 - 1 , 0 8 2 Szeged J F M A M J JL A U S O N D A 1,563 1,666 1,538 1,563 1,640 1,538 ' 1,352 1,333 1,299 1,370 1,666 1,666 A* 2,156 2.275 2,122 2,221 2,139 ' 1,920 1,681 1,724 1,783 2,162 2,432 2,310 A* ¡A 1,400 1,366 1,400 1,421 1,305 1,248 1,244 1,293 1,374 1,579 1,459 1,387 I 0,9290 0,9322 0,9456 0.9391 0,9361 0,9822 0,9541 0,9532 0,9550 0,9506 0,9342 0,9396 - 1 , 0 8 2 - 1 , 0 7 8 - 1 , 0 6 7 - 1 , 0 6 9 - 1 , 0 7 4 - 1 , 0 6 9 - 1 , 0 5 3 - 1 , 0 5 2 - 1 , 0 5 0 - 1 , 0 5 4 —1,076 - 1 , 0 6 9 Table 6 The Values of c, and a Budapest J F ' M A M J . J •/A S O N D Cj a (mm) 3,2 3,8 • 3,6 4,3 5,4 6,3 5,7 • 5,4 4,9 5,0 5,2 3,8 a 1,0225 0,9995 1,0427 1,0204 1,0922 0,9834 0,0095 1,1672 1,0599 1,0443 1,1391 1,0405 Szeged J F M A M J j A S O N D Ci (mm) 3,0 3,1 3,2 3,9 ' 4,9 5,8 5,6 6,1 5,6 4,7 4,5 3,4 . a 1,0037 1,0361 1,1107 1,0604 1,0401 1,0122 1,1637 1,1553 1,0250 1,0550 0,9730 1,1074 Budapest Length of the Period J F M A 1 1 1 1 1 2 0,9867 1,0005 0,9727 0,9867 3 • 0,9785 1,0006 0,9569 0,9785 4 0,9725 1,0007 0,9459 0,9725 5 0,9682 1,0008 0,9376 0,9683 Szeged Length of (Day) J F M . A 1 1 1 1 1 2 0,9995 0,9727 0,9267 0,9592 3 0,9990 0,9569 0,8863 0,9363 4 0,9985 0,9458 0,8585 0,9202 5 , 0,9984 0,9376 0,8377 0,9081 VO Table 7 The Values of Q M J 1 i 1 1,0142 0,9061 1,0227 0,8826 1,0277 0,8651 1,0326 J A 1 1 0,9995 0,8889 0,9990 0,8297 0,9985 0,7899 0,9984 : 0,7607 S O 1 1 0,9592 0,9728 0,9363 . 0,9569 0,9202 0,9459 0,9081 0,9376 N D 1 1 0,9078 0,9727 0,8619 0,9569 0,8302 0,9459 0,7982 0,9376 M J 1 1 0,9728 0,9936 0,9569 0,9891 0,9459' 0,9864 0,9376 0,9841 J A .1 1 0,8949 0,8948 0,8389 0,8389 0,8011 0,8011 0,7730 0,7730 S O 1 1 0,9828 0,9624 0,9734 0,9416 0,9655 0,9266 0,9604 0,9151 N D 1 1 1,0188 0,9268 1,0302 • 0,8863 1,0384 0,8585 1,0443 0,8376 Table VII, continued The Values of p and q in the Year (%) J F M A M J J A S O N D Budapest P 41 40 • 35 37 • 42 40 31 31 25 29 46 44 q 59 60 65 . 63 58 60 69 69 75 71 54 54 Szeged P 36 40 35 36 39 36 26 25 . 23 27 40 40 q 64 60 65 64 61 64 74 75- 77 73 60 60 Reference [1] Péczely, G.: Tartamvalószínűségek vizsgálata. (The Examination of the Probábility of Durations) Időjárás. 61, № 4, pp. 241—244. 1957. % 4 0 

A statistical analysis of the length of rainy periods and their quantity of precipitation

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A statistical analysis of the length of rainy periods and their quantity of precipitation

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