A study modeling of 15 days cumulative rainfall at Purajaya Region, Bandar Lampung, Indonesia

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1 A study modeling of 15 days cumulative ainfall at Puajaya Region, Banda Lampung, Indonesia Ahmad Zakaia* Abstact Aim of this eseach is to study peiodic modeling of 15 days cumulative ainfall time seies. The study was undetaken using 5 yeas ( ) data of Puajaya egion. The seies of the daily ainfall data assumed was tend fee. The peiodic component of 15 days cumulative ainfall time seies could be epesented by using 53 hamonic expessions. The stochastic component of the 15 days cumulative ainfall was using the 3 d ode autoegessive model. Validation of geneated 15 days cumulative ainfall seies was done by compaing between the geneated with the measued ainfall seies. The coelation coefficient between the geneated o synthetic ainfall seies with the measued ainfall seies with the numbe of the data N is equal to 51 days fo 5 yeas was found to be Theefoe, developed model could be used fo futue pediction of 15 days cumulative ainfall time seies. Index Tems - 15 days cumulative ainfall, fast Fouie tansfom, autoegessive model, least squaes method. I. INTRODUCTION To design wate consuming of iigation, detailed infomation about the ainfall with espect to time is equied. To povide long sequence ecod of ainfall data was vey difficult, so sometime to extend the ainfall ecod, geneating the synthetic ainfall ecod is necessay. Vaious methods have been used by Enginees and scientists to povide this infomation. Most the existing methods ae eithe deteministic o pobabilistic in natue [3] and []. While the fome methods do not conside the andom effects of vaious input paamete, the late method employ the concept of pobability to the extent that the time based chaacteistics of ainfalls ae ignoed. With the eve inceasing demand fo accuacy of analyzing ainfall data, these methods ae no longe sufficient. The ainfalls ae peiodic and stochastic in natue, because they ae affected by climatological paamete, i. e., peiodic and stochastic climate vaiations ae tansfeed to become peiodic and stochastic components of ainfalls. Hence the I. Ahmad Zakaia, Ph.D., is woking as senio lectue at Depatment of Civil Engineeing, Faculty of Engineeing, Lampung Univesity, Indonesia. He aea of inteest is in the field of physical and numeical modeling of wave popagation and signal pocessing. He is the coesponding autho. (Phone: ; Fax: ; ainfalls should be computed consideing both the peiodic and the stochastic pats of the pocess. Consideing all othe factos known o assumed that the ainfall is a function of the stochastic vaiation of the climate. Hence peiodic and stochastic analysis of ainfall time seies will povide a mathematical model that will account fo the peiodic and stochastic pats and will also eflect the vaiation of ainfalls. Duing the past yeas, some eseaches that study the peiodic and stochastic modeling have been published by [4] [5] [7] [8] [9] [10]. Aim of this eseach is to geneate the sequences of 15 days cumulative ainfall time seies fo Puajaya station using fast Fouie tansfom and least squaes methods. The model can be used to povide synthetic and easonably ainfall data fo planning the iigation o wate esouce pojects in the futue..1. Study Aea II. MATERIALS AND METHODS The study aea comes unde the humid egion of the subdistict of West Lampung, Pofince of Lampung, Indonesia... Collection of Rainfall Data Daily ainfall data of Puajaya egion was collected fom Indonesian Meteoological, Climatological and Geophysical Agency, Pofince of Lampung. Rainfall data fo a peiod of 5 yeas ( ) was used in the study. The mathematical pocedue adopted fo fomulation of a pedictive model has been discussed as follows: The pincipal aim of the analysis was to obtain a easonable model fo estimating the geneation pocess and its paametes by decomposing the oiginal data seies into its vaious components. Geneally a time seies can be decomposed into a deteministic component, which could be fomulated in manne that allowed exact pediction of its value, and a stochastic component, which is always pesent in the data and can not stictly be acounted fo as it is made by andom effects. The time seies X(t) was epesented by a decomposition model of the additive type [5][7][8], as folows, ( t) = T ( t) + P( t) + S( t) X (1) 101

2 Whee, T (t) is the component of tend, t = 1,, 3... N. P (t) is the peiodic components, and S (t) is the stochastic components. In this eseach, the ainfall data is assumed to have no tend. So this equation can be pesented as follows, ( t ) P ( t ) + S ( t ) X () () is an equation to obtain the epesentative peiodic and stochastic models of 15 days cumulative ainfall seies..3. Spectal Method Spectal method is one of the tansfomation method which geneally used in applications. it can be pesented as Fouie tansfom [6][8][9][10] as follows,. π. i n= N /. m. n t P ( f m ) = P ( t ). e M (3) π n= N / Whee P (t) is a 15 days cumulative ainfall data seies in time domain and P (f m ) is a 15 days cumulative ainfall data seies in fequency domain. The t is a seies of time that pesent a length of the ainfall data to N, The f m is a seies of fequencies. Based on the ainfall fequencies esulted using Equation (3), amplitudes as functions of the ainfall fequencies can be geneated. The maximum amplitudes can be obtained fom the amplitudes as significant amplitudes. The ainfall fequencies of significant amplitudes used to simulated synthetic ainfalls ae assumed as significant ainfall fequencies. The significant ainfall fequencies esulted in this study is used to calculate the angula fequencies (ω ) and obtain the peiodic components of Equation (4) o (5)..4. Peiodic Components The peiodic component P(t) concens an oscillating movement which is epetitive ove a fixed inteval of time (Kottegoda 1980). The existence of P(t) was identified by the fouie tansfomation method. The oscillating shape veifies the pesence of P(t) with the seasonal peiod P, at the multiples of which peak of estimation can be made by a Fouie Analysis. The fequencies of the spectal method clealy showed the pesence of the peiodic vaiations indicating its detection. The peiodic component P(t) was expessed in Fouie seies [4] as follows, Pˆ = k = k o. = 1 = 1 ( t) = S + A sin (ω. t) + B cos (ω t) (4) Pˆ Whee ( t ) = k+ 1 = k. = 1 ( t ) = A sin (ω. t) + B cos (ω t) = 1 P is the peiodic component, Pˆ (t) is the peiodic component of the model, S o = A k + 1 is the mean of the ainfall time seies ω is the angula fequency. A and B ae the coefficients of Fouie components..5. Stochastic Components The stochastic component was constituted by vaious andom effects, which could not be estimated exactly. A stochastic model in the fom of autoegessive model was used fo the pesentation in the time seies. This model was applied to the S (t) which was teated as a andom vaiable. Mathematically, an autoegessive model of ode p can be witten as: S p ( t) = ε + b S( t k) k= 1 Equation (6) may be aanged as, k (5). (6) ( t ) = ε + b S ( t 1) + b. S ( t ) b. S ( t p) S 1. p (7) Whee, b is the paamete of the autoegessive model. ε is the constant of andom numbes. = 1,, 3, 4,..., p is the ode of stochastic components. To get the paamete of the autoegessive model and the constant of andom numbe, least squaes method can be applied..6. Least Squaes Method.6.1. Analysis of peiodic components In cuve fitting, as an appoximate solution of peiodic components P (t), to detemine Function P ˆ ( t ) of Equation (4) and (5), a pocedue widely used is least squaes method. Fom Equation (5) we can calculate sum of squaes [4] as follows, t=m { } Sum of squaes = J = P( t) Pˆ ( t) t= 1 Whee J is depends on A B, and ω. A necessay condition fo J be minimum is as follows, (8) Equation (4) could be aanged to be Equation as follows, 10

3 A = B = 0 with = 1,,3,4,5,...,k (9) Whee J is sum squae of eo. It depends on the ℇ and b values, whee the coefficients can only be minimum value if it satisfies the equation as follows, Using the least squaes method, we can find equations as follow, a. mean of seies, ε = b = 0 with = 1,, 3, 4, 5,..., p (16) S o =A k+1 (10) b. amplitudes of significant hamonics, C = A + B (11) c. phases of significant hamonics, B φ = actan (1) A In the next, by using (16) stochastic paametes ε and b of the esidual ainfall data can be calculated. III. RESULTS AND DISCUSSION Fo testing the statistical chaacteistics of daily ainfall seies, 5 yeas data ( ) of daily ainfall fom station Puajaya was taken. The statistical chaacteistic of the annual mean and maximum ainfall of daily ainfall seies wee estimated. Figue 1 shows the daily ainfall time seies. Mean of 15 days cumulative ainfalls, amplitudes, and phases of significant hamonics can be substituted into an equation as follow, Pˆ( t) = S o + =k = 1 C. Cos ( ω. t φ ) (13) Equation (13) is hamonic model of the 15 days cumulative ainfall whee can be found based on the 15 days cumulative ainfall data seies of Puajaya..6.. Analysis of stochastic components Based on the esults of the simulations obtained fom peiodic ainfall models, stochastic components S (t) can be geneated. The stochastic component is the diffeence between ainfall data seies with calculated ainfall seies obtained fom peiodic model. Stochastic seies as a esidual ainfall seies, which can be pesented as follows, ( t ) X ( t ) - P( t ) S (14) Figue 1. Daily ainfall time seies fo 5 yeas fom Punajaya station. Fom Figue 1 is pesented mean annual daily ainfall values vay fom mm in the yea of 1986 to 1.5 mm in the yea of Maximum annual daily ainfall values vay fom 35 mm in the yea of 1986 to 15.9 mm in the yea of 199. Fo annual cumulative daily ainfall indicate minimum value of 55.5 mm in the yea of 1989 and maximum value of mm in the yea of 1996 with mean annual cumulative daily ainfall value of mm. Based on daily ainfall time seies, a seies of 15 days cumulative ainfall was geneated as pesented in Figue, (14) can be solved by using the same way with the way that used to get peiodic ainfall seies components. Following (8) and (9), stochastic models (7) can be aanged to be as follows, t=m { } Sum squaes of eo = J = S( t) Sˆ ( t) t= 1 (15) 103

4 Figue. Vaiation of 15 days cumulative ainfall seies fo 5 yeas fom Puajaya station. A numbe of the daily ainfall seies N is about 9131 days. Fom the daily ainfall seies, a seies of 15 days cumulative ainfall was geneated. The length of the 15 days cumulative ainfall seies is about 608 points. Fom the seies, the statistical analysis of the 15 days cumulative ainfall seies has been estimated. It was found that maximum value of 15 days cumulative ainfall annually wee vay fom 31.4 mm in the yea of 1989 up to 390 mm in the yea of 198. In ode to unning peiodic model, peiodogam of ainfall seies should be geneated befoe. A powe of must be used to enable using fast Fouie tansfomation method. Fo this case, a numbe of 51 data points was used to find the peiodogam of peiodic modeling. Result of the Fouie tansfomation is pesented in Figue 3 as follows, Figue 4. Vaiations of measued and calculated ainfall seies fo 5 yeas fom Puajaya station using peiodic model (1 ~ 56). Figue 5. Vaiations of measued and calculated ainfall seies fo 5 yeas fom Puajaya station using peiodic model (56 ~ 51). Figue 3. Vaiation peiodogam of the 15 days cumulative ainfall fo 5 yeas fom Puajaya station. Fom Figue 3 shown maximum amplitude of the 15 days cumulative ainfall is occued at mm fo peiod of days o nealy one yea. It indicates that the annual component of peiodicity is quite dominant compaed with the othes. The spectum above is pesented in the ainfall amplitudes as a function of peiods. To confim the pesence of peiodic component in the 15 days cumulative ainfall seies and to geneate dominant ainfall fequencies, the Fouie tansfom method was applied. Fo modeling and geneation the 53 dominant ainfall fequencies of the 15 days cumulative ainfall, 51 data points of ainfall data seies wee used. The geneated fequencies wee obtained by using an algoithm which poposed by [1] whee the numbe of data N to be analyzed is a powe of, i. e. N = k. Based on the esults, peiodic modeling of the 15 days cumulative ainfall seies, calculated and measued ainfalls ae pesented in Figue 4 and 5. The statistical paametes of 15 days cumulative ainfall ae pesented in Table 1 and Table. TABLE 1 10 MAXIMUM AMPLITUDES OF 53 PERIODIC COMPONENTS No Peiod (day) Amplitude (mm) TABLE STATISTICAL PARAMETERS OF PERIODIC RAINFALL DATA Statistical paametes of cumulative ainfall seies values Root Mean Squaes (RMS) Standad of Deviation (SD) Coefficient of Coelation (R)

5 Coefficient of Vaiance Coefficient of Skewness (Cs) Coefficient of Cutosis (Cc).070 Based on the esults of peiodic modeling, the esidual of cumulative ainfall was geneated by using Equation (14) is pesented in Figue 6 and Figue 7. Autoegessive paametes pesented in Table 3 esults the best fit fo the stochastic model of the esidual ainfall. Based on the esults, compaison between measued and calculated esidual 15 days cumulative ainfall ae pesented in Figue 8 and Figue 9. These esults show that the calculated esults have good ageement with measued esults. Figue 8. Vaiations of measued and calculated esidual 15 days cumulative ainfall fo Puajaya station (1 ~ 56). Figue 6. Residual vaiation of measued and calculated 15 days cumulative ainfall fo Puajaya station (1 ~ 56). Figue 9. Vaiations of measued and calculated esidual 15 days cumulative ainfall fo Puajaya station (56 ~ 51). Figue 7. Residual vaiation of measued and calculated 15 days cumulative ainfall fo Puajaya station (56 ~ 51). TABLE 3 AUTOREGRESSIVE PARAMETERS FOR 3 TH ORDER ACCURACIES A compaison between the measued 15 days cumulative ainfall and the calculated 15 days cumulative ainfall of the peiodic and stochastic modeling as shown in Figue 10 and Figue 11 indicate that, the calculated 15 days cumulative ainfall of the peiodic and stochastic models gives highly accuate esults. autoegessive paametes value ε 0 b b b

6 Figue 10. Vaiations of measued and calculated 15 days cumulative ainfall seies fo Puajaya station using peiodic and stochastic model (0 ~ 56). Figue 1. Vaiations of eo, coelation coefficients of stochastic (S), peiodic and stochastic (P+S) models fo diffeent stochastic odes. Figue 11. Vaiations of measued and calculated 15 days cumulative ainfall seies fo Puajaya station using peiodic and stochastic model (56 ~ 51). Fo modeling of the peiodic ainfall povides the coelation coefficient R is Fo modeling of the stochastic ainfall is using 3 d odes autoegessive model gives the coelation coefficient R is Fo modeling of stochastic and peiodic 15 days cumulative ainfall giving the coelation coefficient between the data and the model inceases to be The coefficient coelation R is almost close to 1. This shows that the model of peiodic and stochastic 15 days cumulative ainfall is almost close to the patten of ainfall 15 days cumulative ainfall data. It indicates that the peiodic and stochastic models can give moe accuate and significant esult. Vaiation of the coelation coefficient of the stochastic model R(S), the coelation coefficient of the peiodic and stochastic model R(P+S) and the eo of the 15 days cumulative ainfall vesus the odes of the autoegessive model can be seen in the Fig. 1. Based on, the esults pesented in Fig. 1 shows that using the 3 d ode autoegessive model can give bette accuacy esults than the nd ode autoegessive model. Fo the accuacy of the 4 th ode up to the accuacy of the 10 th ode did not povide moe significant esults, if it is compaed with the accuacy of the 3 d ode autoegessive model. So in this eseach, the stochastic component is modeled using the 3 d ode autoegessive model. The coelation coefficient R and the eo (%) fo modeling of the synthetic peiodic ainfall give the coelation coefficient is equal to and the eo is equal to 8%. Fo modeling the peiodic and stochastic ainfalls povides the coelation coefficient is equal to and the eo is equal to 0.79 %. The 15 days cumulative ainfall modeling in this eseach can be compaed to the synthetic ainfall modeling such as have been done by [5] and [7], whee in the modeling they only use a few peiodic and stochastic paametes. To model the synthetic ainfall, in his wok, [5] using up to six hamonic components and with stochastic components using 3 d ode autoegessive model. Fo [7], in the eseach, they use only thee hamonic components with stochastic component fo 1 st ode autoegessive model. In this eseach, moe complex solution is conducted than pevious eseaches. Even though by using 53 peiodic components, the hamonic modeling of 15 days cumulative ainfall in this eseach is done easily. Because, by applying the fast Fouie tansfoms (FFT), the dominant ainfall fequencies of the 15 days cumulative ainfall can be geneated quickly. Behavio of the stochastic 15 days cumulative ainfall can be seen such as pesented in Fig. 8 and Fig. 9. The stochastic components seies is the diffeence between the 15 days cumulative ainfall data with the peiodic model seies. Fom the figues they pesent that the stochastic component fluctuates in value fom mm up to 68.6 mm. The coelation coefficient of stochastic models with the accuacy of the 3 d ode is equal to , while the 1 st ode autoegessive model of the stochastic model is equal to The esult is bette when compaed with the esults pesented by [7] which uses stochastic model fo the accuacy 106

7 of the 1 st ode and give the coefficient coelation fo stochastic model of By using the 53 peiodic components and 3 d ode autoegessive model yield the simulation model of 15 days cumulative ainfall accuately, with a coelation coefficient is equal to The coelation coefficient pesented in Fig. 1 is poof that peiodic and stochastic models (P + S) of 15 days cumulative ainfall has a vey good coelation and accuate esults when compaed with only using peiodic model (P) that geneates coelation coefficient of This esult also looks much bette when compaed to the eseach done by [7], whee the model only using the 3 peiodic components with the 1 st ode accuacy of stochastic component with the coelation coefficient is The esults also bette even though compaed with the eseach fo the daily ainfall seies of 5 yeas ainfall data which have been done by [9], whee by using the 53 hamonic components, aveage of the coelation coefficient of peiodic model is about In [10] by using the 53 hamonic components and the nd ode autoegessive model, aveage of the coelation coefficient of stochastic model is about and fo the coelation coefficient of peiodic and stochastic models is about [5] Rizalihadi, M. (00). The geneation of synthetic sequences of monthly ainfall using autoegessive model. Junal Teknik Sipil Univesitas Syah Kuala, 1(), [6] A. Zakaia, A. (003). Numeical modelling of wave popagation using highe ode finite-diffeence fomulas, Thesis (Ph.D.), Cutin Univesity of Technology, Peth, W.A., Austalia. [7] Bhaka, S. R., Singh, Raj Vi, Chhajed, Neeaj, and Bansal, Anil Kuma. (006). Stochstic modeling of monthly ainfall at kota egion. ARPN Jounal of Engineeing and Applied Sciences, 1(3), [8] Zakaia, A. (008). The geneation of synthetic sequences of monthly cumulative ainfall using FFT and least squaes method. Posiding Semina Hasil Penelitian & Pengabdian kepada Masyaakat, Univesitas Lampung, 1, [9] Zakaia, A. (010). A study peiodic modeling of daily ainfall at Puajaya egion. Semina Nasional Sain & Teknologi III, Octobe 010, Lampung Univesity, 3, [10] Zakaia, A. (010). Studi pemodelan stokastik cuah hujan haian dai data cuah hujan stasiun Puajaya. Semina Nasional Sain Mipa dan Aplikasinya, 8-9 Decembe 010, Lampung Univesity,, IV. CONCLUSION The spectum of the 15 days cumulative ainfall time seies geneated by using the FFT method is used to simulate the synthetic 15 days cumulative ainfall. By using the least squaes method, the 15 days cumulative ainfall time seies can be poduced synthetic ainfall quickly. By using 53 peiodic components and 3 d ode stochastic components, the 15 days cumulative ainfall model fom Puajaya station can be poduced accuately with the coelation coefficient of ACKNOWLEDGMENT This pape is pat of the eseach wok caied out unde the poject funded by the DIPA Unila, Lampung Univesity, Indonesia. REFERENCES [1] Cooley, James W. Tukey, John W. (1965). An Algoithm fo the machine calculation of Complex Fouie Seies. Mathematics of Computation, [] Yevjevich, V. (197). Stuctual analysis of hydologic time seies, Coloado State Univesity, Fot Collins. [3] Kottegoda, N. T. (1980). Stochastic Wate Resouces Technology. London: The Macmillan Pess Ltd. [4] Zakaia, A. (1998). Peliminay study of tidal pediction using Least Squaes Method. Thesis (Maste). Bandung Institute of Technology, Bandung, Indonesia. 107

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