Results of a study of dry and wet spells at Itabaina, a semi-arid station in northeast Brazil, are presented in this paper. Frequency distributions of dry spells of different durations at the station are compared with those derived from three statistical models. It is found that the Eggenberger-Polya and the geometric model provide better results than the logarithmic model. Frequency distribution of wet spells during the three rainiest months of the year shows that rainfall persistency reaches a maximum value on the fifth day of a rainy spell

Karuna Kumar, K.

Ramana Rao, T. V.

English

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DOI 10.1393/ncc/i2005-10195-0IL NUOVO CIMENTO Vol. 29 C, N. 6 Novembre-Dicembre 2006Dry and wet spells at Itabaina (NE Brazil)(∗)K. Karuna Kumar and T. V. Ramana RaoDepartment of Atmospheric Sciences, Federal University of Campina Grande58109-970 Campina Grande-PB, Brazil(ricevuto il 10 Ottobre 2005; revisionato l’8 Agosto 2006; approvato il 2 Settembre 2006; pub-blicato online il 29 Novembre 2006)Summary. — Results of a study of dry and wet spells at Itabaina, a semi-aridstation in northeast Brazil, are presented in this paper. Frequency distributions ofdry spells of different durations at the station are compared with those derived fromthree statistical models. It is found that the Eggenberger-Polya and the geometricmodel provide better results than the logarithmic model. Frequency distribution ofwet spells during the three rainiest months of the year shows that rainfall persistencyreaches a maximum value on the fifth day of a rainy spell.PACS 92.60.-e – Properties and dynamics of the atmosphere; meteorology.PACS 92.60.Jq – Water in the atmosphere.1. – IntroductionThe semi-arid zone of northeast Brazil is 860000 km2 in area and contains nearly 10%of the country’s population. The main climatic characteristics of this region are: annualrainfall of 400–800mm with a coefficient of variability of up to 60%, high air temperatureand high potential evapotranspiration values. In the semi-arid region the main constraintto crop production is the rainfall and its extreme variability.Information on the occurrence of dry and wet spells during the crop-growing seasonis a matter of much importance in regions with limited water resources. Results of astudy of dry and wet spells at Itabaina (35◦ 20′ W, 7◦ 20′ S), a semi-arid station innortheast Brazil, are presented in this paper. Daily rainfall data at the station duringthe period 1911-1976 is used in the study. Due to missing information the year 1916 isnot considered.Various studies have been carried out in the past to estimate the frequency distributionof sequences of dry days. The most often used model is the simple Markov chain whichassumes that the probability of any particular day being dry or wet depends only on(∗) The authors of this paper have agreed to not receive the proofs for correction.c© Societa` Italiana di Fisica 617618 K. KARUNA KUMAR and T. V. RAMANA RAOthe nature of the previous day [1-3]. Higher-order Markov chain models have been usedby Feyerherm and Bark [4]. The logarithmic series has been suggested by Williams [5]as a fit to dry- and wet-spell distribution. The Eggenberger-Polya model [6] has beenemployed by Berger and Goossens [7] to estimate dry- and wet-spell frequencies.2. – MethodologyIn this paper a dry day is defined as a day with no rainfall. A dry spell is a sequenceof dry days bracketed by wet days on both sides. Daily rainfall data at Itabaina forthe months April-June during the period 1911-1976 is used to obtain the frequencydistribution of dry spells of different durations. These frequencies are compared withthose computed using a) the logarithmic model, b) the Eggenberger-Polya model andc) the geometric model.The logarithmic series is mathematically written as [5]αx, αx22, αx33, αx44. . . ,where each term, in the case of dry-spell series, represents the number of spells of drydays lasting over 1, 2, 3, etc. days.The two constants x (which is always less than unity) and α can be computed fromthe two simultaneous equations:S = α ln (1 +N/α) ,N =αx(1− x) ,where S is the total number of dry spells and N is the total number of dry days involvedin the study period. For the estimation of α and x, the procedure given by Williams [8]is used.In the Eggenberger-Polya [6] model the frequencies (φ) of dry spells of 1, 2, 3 etc.days are given byφ1 =S(1 + d)m/d, φ2 = φ1m1 + d, φ3 = φ2m+ d2(1 + d),where d = (σ2/m)−1, (m+1) is the mean length of a dry spell and S is the total numberof dry spells.The geometric series is written as Ay,Ay2, Ay3, Ay4, . . . where the terms representfrequencies of dry spells of duration 1, 2, 3 etc. days. The values of A and Y arecomputed using the procedure given in Cooke [9].The dry-spell frequencies derived from these models are compared with observedfrequencies at Itabaina. The three models are evaluated in terms of Mean Bias ErrorDRY AND WET SPELLS AT ITABAINA (NE BRAZIL) 619(MBE), Root Mean Square Error (RMSE) and the Mean Percentage Error (MPE).MBE =n∑i=1(xie − xim)n,RMSE =√√√√√n∑i=1(xie − xim)2n,MPE =n∑i=1[(xie − xim)xim· 100]n.In these expressions Xie and Xim are the i-th estimated and measured values andn is the number of observations. A positive value of MBE suggests an overestimationand a negative value an underestimation by the model. The values of MBE represent asystematic error or bias while the RMSE represents a non-systematic error. In the caseof MPE the signs of the errors are neglected and the percentage errors are added up toderive the mean value. For a model to be a good estimator MPE, MBE and the RMSEshould be small.Daily rainfall data for the months April-June during the period 1911-1976 is usedfor the study of rainfall persistency. These months represent the wettest part of theyear at this station. Rainfall during these months is attributed to easterly disturbanceswhich enter the region from the Atlantic ocean. Their frequency varies on interannualand interseasonal time scales but there are from three to five events per month. Theinfluence of any particular event on local rainfall can last from one to five or six days.A wet spell is defined as a sequence of rainy days bracketed by dry days on bothsides. Rainfall persistency is defined as the observed percentage frequency in excess ofchance that a period of M rainy days is followed by N or more consecutive days withrain beginning L days later [10].From the daily rainfall data frequencies of wet spells of different durations are ob-tained. Frequencies of occurrence of rain on the first, second and third etc. days followingwet spells of various durations are also derived.3. – ResultsThe total number of dry days is 4015 or approximately 68% of the number of daysconsidered in the study. The number of dry spells of all durations is 875. The number ofdry spells of more than eight days duration is less than 10% of this total. Dry spells whichoccur rarely are not of much importance and observed frequencies of spells of up to eightdays duration are compared with frequencies derived from the three models describedabove (table I). Parameters of the three models are included in the table. From table Iit can be seen that the geometric and Eggenberger-Polya models yield better resultsthan the logarithmic model. Frequencies obtained from the Eggenberger-Polya modelare closer to the observed frequencies than those from the geometric series. According tothe EP model the total number of dry spells is 870 and 66% of the first dry days will befollowed by at least one more dry day. Similarly 73% and 75% of the second and third620 K. KARUNA KUMAR and T. V. RAMANA RAOTable I. – Observed and theoretical frequencies of dry spells at Itabaina.Duration of dry Observed Theoretical frequenciesspells in days frequencies EP Log Geometric1 289 300 317 2752 163 156 146 1883 114 104 90 1294 73 74 62 895 59 55 46 616 43 41 35 427 29 31 28 298 21 24 22 20MBE −0.8 −5.6 5.3RMSE 6.2 15.8 12.8MPE 6.4 13.1 8.2A = 400.7574, α = 344.9109, d = 4.3054, Y = 0.68587, X = 0.92089, m = 2.7611, S = 875.dry days will be followed by at least one more dry day. The chances of a four-day dryspell lasting for one more day is 76%.In many studies on rainfall climatology a first-order Markov chain model is used. Ifthe probability of a dry day being followed by a wet day P (w/d) is denoted by α and theprobability of a wet day being followed by a dry day P (d/w) is denoted by β the meanlength of a dry spell is 1/α and the mean length of a wet spell is 1/β.In the present case α = 0.22 and β = 0.46 and the mean lengths of dry and wet spellsat the station are about 5 and 2 days, respectively. The period of study is the rainiestpart of the year at the station and is hence the central part of the crop-growing period.The mean dry spell length of 5 days during this period suggests that the rainfall regimeis not very favourable for crop growth.It may be mentioned here that the mean rainfall during the period April-June atthe station for two 30 year periods 1917-1946 and 1947-1976 is 357mm and 320mm,respectively. The corresponding change in the number of rainy days is from 38 to 33.The total number of days considered in the study is 5915 and the number of wet daysis 1900. The probability on chance of rain on any day (P ) is 0.32 and on this basis theprobability of a wet spell of one, two or three days is 0.32, 0.10, 0.03 and so on.Frequencies of occurrence of wet spells of different durations expected on chance arecomputed using the expression [11]Fr = P rQ(2 +Q(m− r − 1))n,where Fr is the frequency of a wet spell of duration r days, P is the probability of agiven day being wet, Q = 1− P , m is the mean number of days in a season and n is thenumber of years considered. The frequencies obtained agree with those computed fromthe equation NP rQ2 [7], where N is the total number of days considered in the study.DRY AND WET SPELLS AT ITABAINA (NE BRAZIL) 621Table II. – Wet spells of different durations at Itabaina.1 2 3 4 5 6 7 8 9 10 11 12Observed frequencies 477 197 76 43 20 23 16 10 3 2 2 5Frequencies expected on 884 280 89 28 9 3 1 0 0 0 0 0chanceCumulative totals of 878 401 204 128 85 65 42 26 16 13 11 9observed frequenciesThe frequencies obtained are given in table II together with the observed frequenciesof wet spells. Also included in the table are cumulative totals of observed wet spellsfrom the longest to the shortest. Spells of duration 14, 16, 19 and 21days—each of whichoccurred once—are not included in the table.The occurrence of wet spells lasting one, two and three days is less than that expectedon chance and wet spells lasting longer than three days are more frequent than expectedon chance basis.From table II it is seen that out of 878 spells of at least one day duration 401 or 46%are of at least two days duration, 204 cases or 23% are of at least three days duration,etc. Of the 401 cases of wet spells of at least two days length 51% are of at least threedays length, and 32% are of at least four days length. Such information is presented intable III together with the probabilities of a wet spell continuing for 1, 2, 3, etc. days ona chance basis.The highest value of probability of a wet spell continuing for at least one more dayoccurs after a wet spell of five days with a value of 76% as compared with 32% expectedon chance. Probability of a wet spell continuing for at least two more days is the highestafter a sequence of four rainy days. . . According to the definition adopted in this study,Table III. – Percentage frequencies of occurrence of rain on N days following a wet spell of Mdays. Probabilities of N consecutive rainy days expected on chance are given at the top.32% 10% 3% 1% < 1% < 1% < 1%M 61 50 42 358 62 38 31 26 217 65 40 25 20 17 146 76 49 31 19 15 13 105 66 51 33 20 13 10 84 63 42 32 21 13 8 63 51 32 21 16 10 62 46 23 15 10 7 51N 1 2 3 4 5 6 7622 K. KARUNA KUMAR and T. V. RAMANA RAOFig. 1. – Persistency of rainfall at Itabaiana with L = 1, N = 1.rainfall persistency is given by the difference between the observed probabilities of raincontinuing for one, two, three, etc., days and the corresponding probabilities based onchance. Persistency values for L = 1 and N = 1 are shown in fig. 1. It can be seenthat persistency reaches a maximum value of 44% on the fifth day of a wet spell. On theother hand, it has a low value of 14% on the first day of a wet spell.4. – ConclusionsFrequencies of dry spells of different durations at Itabaina (PB) are compared withthose derived from three statistical models. It is found that the Eggenberger-Polya modeland the geometric models provide good estimates of dry-spell frequencies at the station.The incidence of wet spells lasting one, two and three days is less than expected onchance and spells lasting longer than three days are more frequent than expected onchance basis. The rainfall persistency reaches a maximum value on the fifth day of a wetspell.REFERENCES[1] Gabriel K. R. and Neumann, J. Q. J. R. Meteorol. Soc., London, 88 (1962) 90.[2] Caskey J. E., Mon. Weather Rev., 91 (1963) 298.[3] Weiss L. L., Mon. Weather Rev., 92 (1964) 169.[4] Feyerherm A. M. and Bark L. D., J. Appl. Meteorol., 6 (1967) 770.[5] Williams C. B., Q. J. R. Meteorol. Soc., 78 (1952) 91.[6] Eggenberger F. and Polya G., Z. Angew. Math. Mech. (Berlin), 3 (1923) 279.[7] Berger A and Goossens C. H. R., J. Climatol, 3 (1983) 21.[8] Williams C. B., J. Ecol., 34 (1947) 253.[9] Cooke D. S., Q. J. R. Meteorol. Soc., London, 79 (1953) 536.[10] Jorgensen D. L., Mon. Weather Rev., 77 (1949) 303.[11] Cochran W. G., Q. J. R. Meteorol. Soc. London, 64 (1938) 631.

Dry and wet spells at Itabaina (NE Brazil)

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Dry and wet spells at Itabaina (NE Brazil)

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