Nigam, R.; Nigam, S.; Mittal, S. K. -- The River Runoff Forecast Based on the Modeling of Time Series

 ISSN 1068-3739, Russian Meteorology Meteorology and Hydrology, 2014, Vol. 39, No. 11, pp. 750–761.  Allerton Press, Inc., 2014. Original Russian Text  R. Nigam, S. Nigam, S.K. Mittal, 2014, published in Meteorologiya i Gidrologiya, 2014, No. 11, pp. 56–73.

The River Runoff Forecast Forecast Based on the Modeling of Time Series Series a

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R. Nigam , S. Nigam  and S. K. Mittal a

 Rajeev Gandhi Gandhi Techni Technical cal Univer Univer sity, Bhopal, M.P. India In dia  LNCTS, Bhopal, M.P., India, e-mail: [email protected] c  MANIT, Bhopal, M.P., In dia

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Received Re ceived May 21, 2014

Abstract Ab stract —Discussed  —Discussed are the methods of stochastic modeling the precipitation runoff run off time series se ries and fields. Discussed are the structural structural attrib attributes, utes, scope, boundary boundary condi conditions tions and vari various im prove  provements ments of  the univariate Autoregressive In Inte tegrated grated Moving Aver Average age (ARIMA) and the multivariate Transfer  Trans fer  Function Func tion Model (TFM). Presented Presented are the com par  paraative studies studies of exist existing ing models models of the neural neural network. network. An attempt attempt is made to inves investi tigate gate vari various geograph geographical ical loca locations tions and var variious ap pli  plica cations tions of the river  runoff run off forecast. forecast.

DOI: 10.3103/S1068373914110053

1. INTRO INTRODUC DUCTION TION Right through the history his tory of mankind, mankind, we learnt that precipitation and con se sequent quent river runoff brought myriad wealth and pros per myriad  perity. ity. However, excess and uncon un controlled trolled rainfall rain fall and runoff always resulted re sulted in tretre mendous men dous losses and untold un told suffer suf ferings ings to peo ple. Focused Focused efforts efforts are required required to reduce reduce the risk of river runrun off and utilize utilize it for the bene ben efits of soci society ety [36]. Gener Generaation of the systems of river runoff run off forecast forecasting ing and warning is one solu so lution tion which leads towards to wards the above said ob jec ob jective tive [6, 97]. An accu ac curate rate forecast forecasting ing of   peak values values of precipitation and result re sulting ing river flow plays an im por im portant tant role in hydroinformatics for the flood disas disaster ter manage management ment and for assessing of the risk of fail ure of de pendable  pendable ca paci  pacities of hydrau hydraulic lic structures structures [79, 91]. Suitable Suit able model modeling ing of tem poral  poral varia variations of precip pre cipiitation tation and asso as soci ciated ated river runoff  runoff  can be used for flood and drought stud ies, for opti optimal mal oper operaation of reser reservoir voir systems, systems, for planning planning and dede sign of water wa ter sup ply and hydrau hydraulic lic systems, systems, for invent inventing ing alter alterna native tive water water sup ply strate strategies, and for many other pur poses [83]. Pa pers [71, 98] discuss discuss the details details of the models mod els of vari various types that deal with the spatiotemporal vari ations of precip pre cipiitation tation runoff. runoff. These models mod els have vari various degrees degrees of com plexity,  plexity, ca pa bili  bilities, structure, structure, strength and limitation, and each has its own objecive, its ad vantages vantages and disad dis advan vantages. tages. Among vari various methods methods of precip precipiitation tation runoff runoff calcu calcula lation, tion, the time series se ries analysis suggested by Box and Jenkins [12, 13, 37, 40] is worth mention men tioning. ing. The ob jec  jective tive of this arti ar ticle cle is to study vari var ious stochas sto chastic tic models models of precip precipiita tation tion runoff run off based on the analy anal ysis of time series se ries and used for the es esti tima mation tion and forecast fore cast of river runoff run off in vari var ious water watersheds. sheds. The attempt at tempt is made to take precip pre cipiita tation tion in account ac count during dur ing the forecast fore cast of river runoff. runoff. Vari Various modi modifi fica cations tions of models mod els and methods methods of stochas sto chastic tic analy analysis are consid considered ered in order or der to provide pro vide more reli reliable able and accu accurate rate calcu cal cula lation tion of precip pre cipiita tation tion runoff. run off. 2. STOCHAS STOCHASTIC TIC NATURE NATURE OF RAINFALL RAINFALL RUNOFF PROCESS The descrip description tion of river runoff run off varia variations is most often of ten based on stochas sto chastic tic methods methods that use ARMA, ARIMA, transfer transfer functions, functions, neural neu ral networks networks and system sys tem identi identifi fica cation tion [11]. Pa per [83] sug [83] suggests gests to treat the stream flow process pro cess as the inte in tegra gration tion of stochas sto chastic tic and deter determin minis istic tic com po  ponents. nents. Dynamic Dynamic rela relation tionships ships are described described using us ing transfer trans fer functions functions whereas the stochas sto chastic tic com po  ponent nent ex plains the anoma anomalies of the syssys tems that do not seem to be inher in herently ently stochas stochastic. tic. For the pur poses of model modeling ing river runoff run off is assumed assumed to  be either either a totally to tally linear lin ear stochas stochastic tic process process or fully nonlin non linear ear deter determin minis istic tic chaos [33, 44]. 750

THE RIVER RUNOFF RUN OFF FOREC FORECAST BASED ON THE MODELING

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A stochastical model is struc tur turally ally adapted to local lo cal condi conditions tions and hydro drolog logiical data in real time. Using the stochas stochastic tic model allows allows under un derstand standing ing the physics phys ics of the process pro cess through the data analy anal ysis and eluci elu cidat dat-ing it through the inter in ter pre  preta tation tion of data. In the anal ysis of stochastic time series, the im por im portant system sys tematic atic  patterns  pat terns and trends of a phenom phe nomeenon are recog recognized nized and sepa sep arated, and the random ran dom fraction fraction of data which is also called noise, is left out. The struc ture of a model should be sim ple (with the least pos possi si ble  ble in put rere quirements), quire ments), and the ap proach towards towards the data process pro cessing ing should ensure en sure the accu accurate rate result. result. The analy anal ysis of stochastic time series se ries allows allows ex plor  ploring ing possi pos si ble  ble real realis istic tic scenar scenarios ios of a given precip pre cipiita tation tion runoff run off system system using us ing suitable suit able simu simula lation tion techniques tech niques [51]. Exten Ex tensive sive reviews reviews of the several sev eral classes of stochas sto chastic tic models models pro posed for op oper eraational hydrol hy drology ogy can be found in [15, 55]. In this pa per pa per we shall dis discuss cuss only the Autoregressive In te tegrated grated Moving Aver Av erage age (ARIMA) models models devel developed oped using using the analy anal ysis suggested sug gested by Box and Jenkins [12–14]; these mod els are the basic basic tool in the today’s model mod eling ing of time series se ries [82]. 3. METHOD METHODOL OLOGY OGY 3.1. The ARIMA Models The earlier earlier studies studies of the precipitation runoff run off process process consisted consisted of the synthe syn thesis sis of avail available able annual annual hydrologic data on time de pend de pendent ent or inde in de pend  pendent ent stochas stochastic tic com po  ponents nents and the identi iden tifi fica cation tion of trends and cycles. cycles. A lot of studies stud ies ap plied sim ple and multi multi ple  ple regres regression sion techniques techniques before be fore 1970. Kisiel [48] first ap plied the pro proce cedures dures of autoregression and mov ing aver average age in the time series series of hydro hy drolog logiical data. Later, in about four decades, de cades, stochastic models mod els (belong (belonging ing to the ARIMA class) became be came the main choice of the rere searchers search ers of river runoff run off forecast forecasting. ing. In 1970 and 1976 Box and Jenkins pre sented an orderly, orderly, well demon dem onstra strative tive and com pre  prehen hensive sive apap proach that is the basic ba sic tool in the today’s anal ysis of time series se ries aimed at the model mod eling ing of station stationary ary and non-stationary time series se ries of runoff. run off. Since then the methods meth ods of time series se ries have been used more and more widely for the model mod eling ing of precip pre cipiita tation tion runoff. run off. In order order to charac char acter terize ize precip precipiita tation tion and runoff, run off, the model of stochastic time series se ries can be divided di vided into two groups. One is es sen sentially tially stochas sto chastic tic univariate (where only one independent variable vari able is used) and multivariate (where more than one se ries are used). Stochas Stochastic tic univariate series series are modeled modeled using using ARIMA while the im pact of more than two series series (where dependent sese ries are asso associ ciated ated with other series se ries through random ran dom errors) errors) is incor incor porated  porated using using transfer trans fer function function models mod els (TFM). The con concept cept of ana analytical ical solu solution tion of model mod eling ing stochastic time series se ries for the analy anal ysis of both univariate (ARIMA) and multivariate (TFM) time se ries was suggested sug gested by Box and Jenkins and was fur ther  promoted  pro moted in [14, 27, 60, 101, 105]. 3.2. Univariate Univariate ARIMA Modeling  The Box–Jenkins ap proach to the model mod eling ing of the univariate sto chas chastic tic time series se ries of river runoff run off inincludes four steps: iden ti tifi fica cation, tion, esti estima mation, tion, diag diagnos nostic tic veri verifi fica cation tion and the forecast. fore cast. Data series series should be station sta tionary ary in order to fit the model of stochas sto chastic tic time series; series; i.e., the mean mean and  and vari variance ance should be constant con stant in time. time. There fore, at the stage of identification data se ries are analysed analysed in order to re remove move any trend, seasea sonal, cyclic cyclic and peri pe riodic odic com po  ponents; nents; it is accom ac com plished using using the ap pro pri  priate ate methods methods of transfor trans forma mation tion (standardi (stand ardisa sation) tion) or filteration as discussed dis cussed further further [40]. The next step is to plot the autocorrelation func tion (ACF) and partial partial autocorrelation function func tion (PACF) in order or der to identify identify a tenta ten tative tive order order of the model and its  param  pa rameeters. The ARIMA model is described de scribed with the follow fol lowing ing equation equation  p ( B )(1  B ) d   p ( B s )(1  B s ) D y t     q ( B ) Q ( B s )a t  ,

(1)

where  ( B)  B) and ( B)  B) are non-seasonal polyno poly nomi mials als of the order or der p  p (auto  (auto regres regression), sion), q is the moving mov ing aver aver-age, and d  is  is differ dif ferenc encing; ing;   B s ) and ( B s ) are seasonal seasonal polyno poly nomi mials als of the order or der P   P  (auto  (auto regres regression), sion), Q  B is the moving mov ing aver average, age, D  D is  is seasonal seasonal differ dif ferenc encing, ing, and S  is  is the length of the sea season. son. Random Random errors errors at  are assumed to be inde in de pend  pendently ently or identi iden tically cally distrib dis tributed uted with the constant con stant (or zero,  = 0) mean and the concon  stant variance vari ance  . Once a tenta tentative tive model is identified, the val values ues of the model param pa rameeters are esti timated mated using using the method of least squares, and then diag di agnos nostic tic veri verifi fica cation tion of the model ade ad equacy is accom accom plished. This pro process cess is rere peated until until the satis satisfac factory tory model is finally fi nally selected. selected. Then the forecast fore casting ing model is used to com pute the se se-lected values values and forecast values. The tem poral and spa spatial tial variabil ability ity that char ch arac acter terizes a river system sys tem makes the forecast fore cast of river run off a very demand demanding ing task. In the most suit able and reli reliable able forecast forecasting ing model, residuals (differ (dif ference ence between between obobRUSSIAN METEOROLOGY AND HYDROLOGY

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served and forecasted forecasted values) values) should satisfy sat isfy the require requirements ments of the white noise process, i.e., a resid re sidual ual should be inde in de pend  pendent ent and normally normally distrib distributed uted around the zero mean. The inde in de pend  pendence ence of the model forecast fore cast is tested by the Ljung–Box–Pierce sta tis tistics, tics, and the ade ad equacy of the model (goodness (good ness of fit) is evalu eval uated by the Akaike information criterion [5, 17, 84]. ARIMA models models provide provide great flexi flex i bility  bility and poten po tential tial for model modeling ing the time series se ries of precip pre cipiita tation tion runrun off. ARIMA models models were used jointly with such mod els as nonlin nonlinear ear regres regression sion (NLR), multi mul ti ple  ple linear linear reregression gres sion (MLR), exog ex ogeenous in put data (ARMAX), and vari variaations of autoregressive moving mov ing aver average age (VARMA), etc. This resulted re sulted in the devel de velop opment ment of the im proved and ef effi ficient cient model of the mixed ARIMA class includ including ing NLRMA, MLRMA, ARMAX, and VARMA, respectively [65, 83, 103]. An other class of  the ARIMA models mod els of river runoff run off includes includes derived derived linear lin ear models models such as ARMA (Auto Regres Re gressive sive Moving Aver Av erage), age), SARIMA (Seasonal (Seasonal ARIMA) and PARMA (peri (pe riodic odic ARMA), etc. [50, 61, 95]. The hy  brid class of stochas stochastic tic models models of ARIMA class that were widely used in the past for mod el eling ing hydrologic time series, series, includes includes auto regres re gressive sive models, models, the fractional fractional model of Gaussian Gaussi an noises, moving mov ing aver average age modmodels, broken-line broken-line models, mod els, short-noise models, mod els, the model of inter in termit mittent tent processes, processes, disaggregation models, mod els, the Markov models, mod els, the ARMA–Markov models, mod els, and general gen eral mixture mixture models models [82]. These im proved verver sions of the ARIMA model re quire less com puta tation tion time, ensure more accu curate rate forecasts forecasts and are useful use ful for real-time forecast forecast of river runoff. run off. However, However, the ap pli  plica ca bil  bility ity of ARIMA models mod els is limited limited to the basins bas ins in which runoff runoff has been measured measured for a long period pe riod and no signif sig nifiicant changes in water wa tershed shed condi conditions tions have occurred. oc curred. 3.3. Multivariate Modeling with Trans fer Trans fer Functions Func tions The river runoff runoff process process is the result result of several sev eral tem poral  poral (mete (meteo orolog rologiical processes) processes) and spatial spa tial (hydrau (hydrau-lic charac character teris istics) tics) factors. This phenome nomenon can be described described as the system sys tem of multivariate multivariate transfer trans fer funcfunctions in tions  in which the in put series series exert influ in fluence ence via the transfer trans fer function function TFM on the out put out put series series during during the long time period period [21, 60]. In the transfer trans fer function function model, river runoff run off (de pendant  pendant variable variable Y t ) is corre correlated with vari various tem poral  poral and spatial spatial inde inde pendent  pendent variables variables (repre (represented sented by differ dif ferent ent time series se ries X   X 1t , X 2t , X 3t , etc.); it is also corre cor related lated with the error er ror  N t  (that cannot cannot be ex plained by X  by X t ) in the follow fol lowing ing form (2)

Y t  X t  N t .

In equation (2) the noise term N  term  N t  is considered to be generated by the ARIMA process of the order ( p ( p,, d , q) which is statistically independent from the input function  X t . Equation (2) can be represented as Y t 

( B ) ( B )

 X t  b 

( B ) ( B )

a t  or Y t    

.... r B r  0  1 B.. 1   i B.. . t B t 

 x t  b 

1   t B.. ....  p B

q

1   t B.. ....  q B

q

(3)

a t .

Here, at  is white noise; in Here, noise;  in the stochas sto chastic tic process process of precip pre cipiita tation tion runoff runoff it is usually usu ally assumed assumed to have nornor 2 mal distri distri bution  bution with the zero mean and vari ance  a . The ap plica  plication tion of the TFM proce pro cedure dure usually usually includes includes several sev eral stages, e.g., tests for stationarity, identi iden ti-fica fication, tion, fitting, fitting, and diag di agnos nostic tic checking. checking. First, the the structure structure of the model has to be de ter termined mined assum assuming ing that the distri dis tri bu  bution of data is normal and es esti timat mating ing the initial val values ues of the param pa rameeters related related to ACF, PACF and CCF [56]. The cross-correlation and cross spec tral analyses show that there is a linear lin ear rela relation  between  be tween annual annual cycles cycles and the inde in de pend  pendent ent stochas stochastic tic com po  ponents nents of river runoff, run off, tem per  peraature, precip precipiita ta-tion etc. The esti estimate mate of the response response function function is used to deter de termine mine the suitable suitable values values of oper operaators in the TFM, and the fitness fit ness of the prelim pre limiinary TFM is tested using us ing the Akaike infor in forma mation tion cri crite terion rion [5] and [5] and the Schwarz infor informa mation tion crite criterion rion [86]. [86]. 4. SIG NIF  NIFIICANT STUDIES ON STO STOCHAS CHASTIC TIC PRECIP PRECIPIITA TATION TION AND RUNOFF RUN OFF Below Below discussed discussed are univariate and multivariate sto chastic chas tic models models of the ARIMA class for es tima timation tion and forecast forecast of precip precipiitation tation and river runoff. runoff. This pa per does not pro provide vide the review review of all techni tech nical cal issues issues asso associ ciated ated with the model mod eling ing of precip precipiitation tation runoff. run off. However, attempts are made to cover the most part of   problems asso as soci ciated ated with differ dif ferences ences in the ca pa bili  bilities and proce pro cedures dures of model modeling. ing. Some com para  parative studies (with ANN and other con ceptual ceptual models) models) were included included in the discus dis cussion sion to justify jus tify the ap plica  plica bility  bility of vari various modi modifica fications tions in order or der to enhance enhance the forecast fore cast effi efficiency. ciency.

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4.1. Univariate Modeling  Soon after after the devel develop opment ment of the Box–Jenkins ap proach ap proach research researchers ers started ap ply  plying ing other methods methods of   precip  pre cipiita tation tion runoff run off model modeling. ing. McKerchar and Delleur [63] ap plied seasonal sea sonal paramet parametric ric linear linear stochas stochastic tic models mod els to account account for the variabil vari ability ity of data on mean monthly runoff. run off. First studies stud ies in this area (forecast (forecasting ing of precip precipiita tation tion and river runoff) run off) are those by Ozis and Keloglu [75] and Graupe et al. [38]; they used the ARIMA methods methods for runoff run off simu simula lation tion in karstic basins bas ins of Turkey. Tur key. Todini [97] used the ARIMA model to gener gen erate ate a unit hydro hy drograph graph in the form of exog ex ogeenous variables vari ables (ARMAX) in the flood forecast fore casting ing system system  based on the Kalman filter. filter. River runoff runoff is highly de pend  pendant ant on the accu ac curacy racy of the esti es tima mation tion of mete me teo oro rolog logiical variables variables (mainly, of precipitation). Therefore There fore Salas et al. [82] and Shiiba et al. [90] have forecasted fore casted river runoff run off in the con ju  ju-gation ga tion with the esti es tima mation tion and forecasts fore casts of precip pre cipiita tation tion and other mete me teo oro rolog logiical param parameeters that influ in fluence ence the runoff runoff of the main catchment catch ment area. Simi Similarly, Chang et al. [23–26] and proposed the Dis crete Autoregressive Moving Aver Av erage age (DARMA) model for the short-term forecast fore cast of river runoff run off com bin  bining ing subse sub se-quent precip precipiita tation tion and delayed de layed runoff. runoff. Noakes et al. [73] devel de veloped oped the peri pe riodic odic autoregressive (PAR) model with season seasonally ally varying varying order order and estab established lished the su pe  peri rior ority ity of the PAR model in a course of the com par  paraative study of the runoff run off forecast forecast using using the ARIMA and SARIMA models models for the mean monthly runoff run off of thirteen thirteen rivers of North and South Amer ica. Chiew et al. [30] simulated daily, monthly and annual river runoff in eight unregulated catchments using six modeling approaches belonging to three groups: black box models, process models and conceptual models. They concluded that that the approach to the modeling modeling of stochastic time time series provides better  estimates of monthly and annual runoff of rivers in the catchments. In the same year Kuo and Sun [52] demonstrated that the intervention analysis of data on the Tanshui River (Taiwan) runoff caused by typhoon precipitation and similar abnormalities of the weather in the catchment area, has greatly  improved the forecast accuracy and confinement of runoff patterns as compared to the traditional ARMA model. Burlando et al. [20], Jakeman and Hornberger [42], and Langu [53] also used the ARIMA processes by means of varying the low-order parameter to model short-term precipitation and runoff; they tried to detect changes in precipitation and runoff patterns. Maidment [59] considered periodic components in river runoff and developed a periodic   autoregressive   autoregressive (PGAR) model to simulate the multiple periodic time series of  river runoff. Bartolini and Salas [10] and Claps et al. [32] also also used periodic variations to to obtain more accurate forecast using the stochastic ARMA model. Adap Ad apta tation tion of the modi mod ified ARIMA process pro cess and the com par  pariison of the forecast fore cast results results with the alter al ter-native model inspired in spired the research re searchers ers to continue continue ex per  periiments with the ARIMA method in order or der to enhance enhance it. At present, pres ent, the problems prob lems of the river flow forecast fore cast are tackled using us ing the sim ple lin linear ear (e.g., AR, ARMAX, and the Kalman filter) fil ter) and seasonal seasonal linear lin ear (e.g., SARIMA) techniques techniques as shown by Awwad et al. [8], Cheng [29] and El-Fandy et al. [35]. Lardet and Obled [54] and Takasao et al. [94] used both the sta tis tis-tical ti cal and stochas sto chastic tic methods methods to esti es timate mate precip precipiita tation tion in real time and pre dicted the runoff run off poten potential tial of a river using using the ARIMA model and the Kalman fil ter. Porporato and Ridolfi [76, 77] have per formed nonlin nonlin-ear analy analysis of river runoff run off for flood forecast fore casting ing to identify iden tify the hidden hid den deter determin minis istic tic behav behavior ior of the process pro cess and to under understand stand the cause and effect ef fect rela relation tionships ships in hydro hy drolog logiical problems. problems. They concluded concluded that the ARIMA model com bined with the nonlin nonlinear ear concept concept is the best choice for the ac cu curate rate forecast forecasting. ing. Weeks and Boughton [104] and Brockwell and Da vis [16] demon demonstrated strated that the ARIMA model pro vides better forecast fore cast effi efficiency ciency when using us ing random random data even if the in put data do not cor corre respond spond to the Gaussian Gaussi an pattern. pattern. Madsen et al. [57] and Toth et al. [99] dem on onstrated strated that the effi ef ficiency ciency of short-term runoff forecast in real time using the ARIMA model can be im proved im proved due to the adoption adoption of suitable suit able techtechniques of data assim as simiila lation tion instead in stead of the prelim prelimiinary transform transforming ing of data to make them closer to the Gaussian Gaussi an pattern. pattern. Till the 20th century, the ARIMA model and its deriv rivaatives such as SARIMA, PAR, etc. were success cess-fully used for the simu sim ula lation tion of precip pre cipiita tation tion and runoff run off in order or der to solve vari var ious hydro hy drolog logiical and water  water  resource re source problems. problems. It was due to the use of the lin ear regres regression. sion. ARIMA models mod els do not allow al low to identify iden tify and, hence, to simu simulate the intri in tricate cate charac character teris istics tics of precip precipiita tation tion and runoff run off process process suffi sufficiently ciently enough. Therefore, There fore, in the 21st century cen tury research researchers ers shifted their atten at tention tion to new forecast fore casting ing methods methods that enabled enabled anaalyz an lyzing ing and simu sim ulat lating ing natu natural phenom phenomena ena (e.g., precipitation and runoff) run off) basing basing on the nonlinear apap  proach. This idea affected affected the ARIMA processes, pro cesses, and the im prove  provements ments of the conven con ventional tional ap proach can  be well un under derstood stood by the follow fol lowing ing cate catego goriz rizing ing the studies stud ies on stochas sto chastic tic river runoff runoff based on ARIMA: ARIMA:  —studies based exclu exclusively sively on ARIMA or on its derivatives;  —real time studies;

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 —studies using using hybrid models;  —com par  paraative studies stud ies using using models models based on the neural neu ral network network and arti ar tifi ficial cial intelligence. 4.1.1. Studies based exclu ex clu sively on ARIMA or on its de deriv riva atives Damle [33] ap plied the method methodol ology ogy of time series se ries data mining mining (TSDM) to the data on river runoff run off in order or der to study the em bed  bedding ding process pro cess delayed delayed in time. It was aimed at predict pre dicting ing floods using us ing the univariate ARIMA model. The TSDM com bines com bines the meth methods ods of phase space recon re construc struction tion and data mining min ing aimed at reveal re vealing ing hidden hid den patterns patterns which are ca pa ble to predict predict future future events in nonlin non linear ear nonstationary time series. se ries. Yurekli et al. [110] ana an alyzed daily data on maxi max imum runoff runoff from three gage stations sta tions on the Chekerek River  in order order to simu simulate monthly maxi max imum runoff runoff using using ARIMA stochas sto chastic tic ap proaches. DeSilva [34] used the model of time series se ries for predict predicting ing the runoff run off of the Kalu Ganga Gan ga basin basin (SriLanka). Naill and Momani [70] used the seasonal ARIMA model for predict pre dicting ing monthly precip pre cipiita tation tion totals at Amman air port air port (Jordan) (Jordan) in order or der to esti es timate future water budget to man manage age water water de mand in arid areas; ar eas; they also deter de termined the peak  values val ues of precip pre cipiita tation tion through inter in terven vention tion analy anal ysis. Rabenja et al. al . [80] forecasted forecasted both monthly precip pre cipiita ta-tion and the discharge dis charge of the Namorona River in the Vohiparara River ba sin in Mada Mad agas gascar car using using ARIMA and SARIMA models; mod els; they also concluded con cluded that the SARIMA model is more reli re liable able for runoff forecast. fore cast. Otok and Suhartono [74] forecasted fore casted precip precipiita tation tion and conse con sequent quent runoff runoff in Indo Indone nesia sia and suggested sug gested that seasonal sea sonal ARIMA (SARIMA) models mod els are better to ana an alyze data on seasonal sea sonal pre cip cip ita itation tion than ARIMA and TFM; SARIMA models mod els also provide pro vide better runoff run off forecasts. forecasts. Mauludiyanto et al. [62] mod eled tropi tropical rain atten at tenu uation in Surabaya (Indo (In done nesia) sia) adopting adopting the ARIMA model. The authors au thors of [9, 39, 58, 87] used the stochas sto chastic tic SARIMA model to forecast fore cast river flow as the conse con sequence quence of mete meteo oro rolog logiical param parameeters (e.g., temperature, humidity, and precip pre cipiita tation) tion) in India In dia (Vellore in Tamil Nadu), Czech Re pub  public, lic, Iran (the Abadeh Station), and Bangla Ban gladesh desh (Dhaka), respec re spectively. tively. They com pared the results results of the forecast fore cast with the data of concep conceptual tual hydro hydrolog logiical model de pend  pending ing on local lo cal condi conditions tions and suggested sug gested that the SARIMA model provides provides better forecast fore casting ing results results with less com plex com plex com pu  puta tation. tion. Nigam [71] devel de veloped oped the runoff  forecast fore cast model for the peren pe rennial nial Kulfo River in the tropi trop ical region region of Ethio Ethi o pia and the sea seasonal sonal Narmada River in the subtrop sub tropiical region region of India India using using the ARIMA model. It is shown that ARIMA model pro vides  better forecast forecast for the peren pe rennial nial river and it is the qual ity of data (data with least per tur bations)  bations) rather then their length ensures en sures better results. re sults. Meher and Jha [64] have de devel veloped oped the general gen eral ARIMA model for simu sim ulating lat ing and forecast forecasting ing mean precip precipiita tation tion using us ing Theissen weights weights and mean precip pre cipiita tation tion for 38 rain-gage stations sta tions located lo cated in the Mahanadi River basin ba sin in India. India. Modarres and Ouarda [66] used the model of gener gen eral al-ized autoregressive condi con ditional tional heteroscedasticity (GARCH) to ana lyse runoff runoff series series in nonlinear manner  man ner  and com pared the re results sults with the ARMA model. They also sug gested vari various paramet parametric ric crite criteria ria for  evaluating the results of run off forecast forecasting ing through the model. 4.1.2. Real time studies stud ies In the studies stud ies dealing dealing with real time forecast fore casting ing of runoff, runoff, the model auto au tomat matiically gener generates ates the online online forecast forecast of river runoff runoff with the varying vary ing dynamics namics of river influ in fluence. ence. In these studies stud ies the way of obtain ob taining ing in put data for the model and the qual ity of data are signif sig nifiicant for achieving achieving accu accurate rate results. Shiiba et al. [89] and Young [108] de [108]  designed signed the auto au tomatic matic precip precipiita tation–run tion–runoff off model that de pends on local local condi conditions tions and is ca pa ble of cal calii brating  brating in put pa param rameeters. They concluded con cluded that overparameterized com plex com plex mod models els cannot can not be identi iden tified fied with suffi suf ficient accu curacy, racy, hence, rela relatively sim ple models are more functional. Tachikawa et al. [93] devel de veloped oped the stochas sto chastic tic model of real-time predic pre diction tion of precip precipiita tation tion runoff run off based on the analy analysis of the fre frequency quency distri distri butions  butions of the predic pre diction tion error. error. Bruen and Yang [19] and Yu et al. [111] pro posed a scheme that com bined the chaos the theory ory and regres regression sion analy analysis to predict predict the daily disdis charge in rivers. 4.1.3. Studies using us ing hybrid hybrid models mod els Tseng et al. [100] pro posed pro posed a hybrid forecast forecasting ing model which com bines the seasonal seasonal model of time sese ries (SARIMA) and the models mod els of back propa prop aga gation tion in the neural neu ral network network (BP) known as SARIMABP which can be used to forecast fore cast river runoff runoff series. series. Broersen and Weerts [18] studied stud ied the possi pos si bili  bilities of the auto au tomatic matic error error correc correction tion in the time series se ries of precip precipiita tation tion runoff runoff for flood forecast fore casting. ing. They developed a com puter program program called ARMAsel using us ing the MATLAB software soft ware for the detec detection tion and correc cor rection tion of outli outli-ers in univariate stochas sto chastic tic data. In this pro program gram past error error signals signals were chosen chosen to give the best pre dic dictions tions during dur ing sudden sud den and abrupt changes in runoff which are the most im por im portant tant for timely flood warnings. warn ings. RUSSIAN METEOROLOGY AND HYDROLOGY

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Mohamad and Mojtaba [67] gener gen erated ated a hy brid system system using using the analy anal ysis of neuro wavelet wave let time series se ries and multilayer ANN to forecast fore cast the runoff run off of the Halil River in south ern Iran and concluded con cluded that the conjuction model signif sig nifiicantly enlarges enlarges the possi pos si bility  bility to forecast fore cast maxi maximum runoff. runoff. Yoon et al. [107] forecasted fore casted monthly hydro hy drolog logiical data (precip (precipiita tation, tion, evapo evapora ration, tion, and runoff) run off) in the Andong dam basin ba sin using using the SARIMA model, conducted long-term simu sim ula lations tions of runoff run off through the fore forecasted casted results results of the TANK  model, and devel developed oped another another joint ARIMA+TANK model. Basing upon their con con clu clusions, sions, they suggested sug gested the models models of mid- and long-term runoff run off forecasts forecasts for ap pli  plica cation tion to water wa ter resources. resources. 4.1.4. Com par  para ative studies studies using using models mod els based on the neural neu ral network  net work  and arti ar ti ficial  ficial intel in telli li gence  gence Abrahart and See [1, 2] as well as Zaldívar et al. [112] com pared com pared forecast fore casting ing effi efficiency ciency of nonlin linear  ear  (ANN) and linear lin ear (ARIMA) models models for real-time esti es tima mation tion of runoff runoff in the course of the stochas sto chastic tic study of dynamic rout routing ing of floods; they suggested sug gested that inac in accu cura racies cies in the forecast fore casting ing results results are due to the unun availabil avail ability of rep repre resen senta tative tive histor his toriical data. A year later Cigizoglu [31], Kim and Valdes [47], and Sana et al. [85] used ARMA and ANN mod els to forecast forecast river runoff. runoff. They sug gested the fol follow lowing ing measures measures to increase in crease the forecast forecasting ing accu accuracy: racy: the use of synthet syn thetiically gener generated ated data, the exten ex tension sion of in put and out put  put data sets in the train ing stage, the use of delayed de layed response response variable, variable, the use of transformed trans formed data, and the inin troduc tro duction tion of peri periodic odic com po  ponents nents to the in put layer. Castellano et al. [22] and Kihoro et al. [46] com pared com pared the ANN and ARIMA models mod els using using river runoff  run off  data at vari various time scales (daily, monthly, quar terly, and annual) an nual) and the model algo al gorithm; rithm; they confirmed con firmed the su prem  premacy acy of the ANN model over the ARMA model when ap plied ap plied to run runoff off forecast. forecast. Kisi [49] and Mohammadi et al. [68] also com pared ARIMA and ANN ANN methods methods for in order or der to esti estimate mate the effi efficiency ciency of  the runoff runoff forecast. forecast. Somvanshi et al. al . [92] studied stud ied behav behavioral ioral patterns patterns of precip precipiita tation tion basing bas ing on past obser ob ser-vations va tions and using us ing ARIMA and ANN. In or der to increase in crease the predic prediction tion effi efficiency ciency they used data on mean annual an nual precip precipiita tation tion in the Hyderabad region re gion (India) (India) obtained obtained in 104 years, from 1901 to 2003. The mod els were trained basing bas ing on the data on annual an nual precip precipiita tation tion for 93 years. Karim et al. [45] forecasted forecasted daily river  runoff run off of the Black Water Water River and the Gila River in the USA and enumer enu merated ated the advan ad vantages tages of the ANN  process  pro cess over ARIMA models, mod els, namely, less com pu  puta tational tional diffi dif ficul culties ties in the esti es tima mation tion of the process pro cess paparameeters, smaller effect of uncertainty and outli ram liers ers in the measure mea surements etc. Reddy et al. [81] mod modeled eled the  precip  pre cipiita tation–run tion–runoff off process process by the linear lin ear autoregressive (ARX) and ANN model. They found out that the quality qual ity of runoff runoff simu simula lation tion by the ANN model can be im proved im proved by in intro troduc ducing ing the resid residu uals of ARX modmodels as in puts in the ANN model. Wu et al. [106] have studied stud ied the quality qual ity of five data-driven models mod els of ARMA, K-Nearest-Neighbors (KNN), Crisp Distrib Dis tributed uted ANN (CDANN), ANN and Crisp Dis trib tributed uted Sup port Vec Vectors tors Regres Regression sion (CDSVR) with data-preprocessing tech niques, namely, the analy anal ysis of the sin singu gular lar spectrum spectrum and mov moving ing aver av erage age using using two real series se ries of monthly data on river run off. Volkan and Onkur [102] predicted pre dicted daily mean runoff run off of the Anamur River using us ing data obtained ob tained from 1989 to 2003 by ap ply ap plying ing the methods meth ods of multivariate linear lin ear regres regression sion (MLR), autoregressive inte in tegrated grated moving moving aver average age (ARIMA) and the methods methods of the radial radial  basis  ba sis function function neural neu ral network network (RBFNN). They sug suggested gested that the RBFNN model pro vided more reli reliable able results re sults than other methods. meth ods. Abudu et al. [3] com pared ARIMA, SARIMA, and Jor Jordan–Elman dan–Elman ANN models mod els for forecast forecasting ing the monthly runoff run off of the Kizil River in Xinjiang, China. They failed to re veal signif signifiicant im prove  provement ment rela relative to time series se ries models models and suggested sug gested that ARIMA and SARIMA mod els can be used in the forecast fore cast of monthly runoff run off with a sim ple and ex plicit model struc ture; they produced pro duced perfor performance mance simiilar to those of Jordan–Elman sim Jor dan–Elman ANN models. mod els. Chattopadhyay and Chattopadhyay [28] fore casted the InIn dian Summer Summer Monsoon Monsoon Rainfall Rain fall (ISMR) using using the univariate ARIMA model and com pared com pared these re results sults with the autoregressive neural neu ral network network (ARNN) model for the same data. As a result re sult of thorough thor ough statis sta tisti tical cal analy anal ysis the su prem  premacy acy of ARNN over ARIMA model model has been been estab established. lished. Sharma et al. [88] first used ANN to gener gen erate ate spatially spatially distrib dis tributed uted data on pre precip cipiita tation tion for vari various peri periods ods and loca lo cations tions at the absence ab sence of the Next-Generation Weather Ra dar (NEXRAD) data, then used the multilinear re gres gression sion (MLR) and ANN to forecast fore cast runoff runoff of the Saugahatchee River runoff run off in southeast south east Ala Ala bama. The intuitive consideration of forecasting results using the model based on ARIMA analysis, revealed that this model provides good results for data on mean runoff and consistent random data. However, the loss of physical characteristics increases as included are various input and output techniques and the nonlinear nature of data that are applied to develop more accurate and parsimonious model [98]. Also, the method needs to be improved to be able to accommodate variations of data occurring due to natural or anthropological disturbances disturbances and abnormalities. It was noted by the Hsu et al. [41] that the ARIMA models RUSSIAN METEOROLOGY AND HYDROLOGY

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do not describe the nonlinear dynamics of the transformation of precipitation into runoff; hence, they could not always rely on the good performance of the model. The behavioural response functions of other  hydrological and site specific factors that make precipitation–runoff processes a complex system, can be expressed by advanced autoregressive models using spectral correlation, cross correlation and the analysis of the transfer function noise. There is a stochastic model developed in this way that includes intricacies affecting precipitation and variables that well express the process of dynamic river runoff. It is called the multivariate ARIMA model or transfer function model [34, 109]. These models are good enough to incorporate the variables of exogenous time series and to include the impact of concerned parameters in order to express the integrated performance of river runoff attributes without the loss of the assumption on linear data. The applications of TFM models in the river runoff forecasting are described below. 4.2. Multivariate Modeling  Thompstone et al. [96] com pared com pared ARIMA, Pe Peri riodic odic AutoRegressive (PAR), and Trans fer Function Function (TF) models utiliz models utilizing ing precip precipiita tation tion and snowmelt in puts, in puts, with with a con concep ceptual tual model. They found that the TF model  performs  per forms better than other mod els when forecast forecasting ing quarter-monthly quarter-monthly runoff. run off. Yu et al. [109] have ap plied the transfer transfer function function model to the runoff run off process process of reser reservoir voir runoff. runoff. Awadallah and Rousselle [7] used sea-surface tem perature signals signals of the El Ni ~ no–Southern Oscil Os cilla lation tion (ENSO) as exog ex ogeenous in put variables variables to develop develop a TFN model for the fore cast casting ing of the summer sum mer runoff runoff of the Nile River. They sug suggested gested that the ENSO in put ex plained ex plained 63% of the vari variabil ability ity of summer summer runoff runoff of the Nile River. Porporato and Ridolfi [78] devel developed oped multivariate nonlin non linear ear predic prediction tion of river runoff run off based on their studies stud ies on the nonlin non linear ear river runoff run off analy analysis. Mondal and Wasimi [69] pro posed pro posed a pe peri riodic odic TFN model and ap plied it to the monthly forecast fore casting ing of the Ganga Ganga River runoff runoff using using monthly data on precip pre cipiita tation tion in northern north ern India. India. The results results demon dem onstrated strated that the method meth odol ology ogy has the poten po tential tial ca pa bil  bility ity of captur cap turing ing the season sea sonally ally varying varying dydynamic rela relation tionship ship between between monthly precip pre cipiita tation tion and runoff run off processes. processes. Todini [98] discussed dis cussed the rela rel ative merits mer its of physical and data driven models mod els of precip precipiita tation tion runoff. runoff. He recog recognized nized that Box and Jenkins ap  proach shows the link between between the transfer trans fer function function model (TFM) and the autoregressive model of ex og ogeenous variables vari ables (ARX) al although though it intro in troduces duces a loss of ‘physicali ‘physi cality’ ty’ in the model. Agrawal et al. [4] have modeled mod eled and forecasted fore casted runoff runoff and sedi sediment from the Vamsadhara River basin ba sin (situ (sit uated between between the Mahanadi and Godavari River basins bas ins in South India) In dia) during during the monsoon mon soon period period for daily and weekly time  peri  pe riods ods using using the model of back propa prop aga gation tion in arti artifi ficial cial neural neural network network (BPANN). The models mod els with the single sin gle in put and lin lin ear transfer transfer function function (SI-LTF) for runoff run off and sedi sed iment yield forecast fore casting ing were more effi ef fi-cacious ca cious than the multi in put in put linear linear transfer transfer function function (MI-LTF) and the ANN model. Nigam [71] de vel veloped oped the TFM model for predict pre dicting ing river runoff run off of the tropi trop ical and subtrop subtropiical Kulfo River in Ethio Ethi o pia and the  Narmada River in In India dia in conjuction with data on pre cip cipiita tation tion in the river catchment catch ment area. TFM perperformed equally well for both cases. 5. DEVEL DEVELOP OPMENT MENT PROSPECTIVES The sys system tem of deci decision sion sup port in real time may be be devel developed oped for emergency emergency flood warning, warn ing, planning planning hydrau hy draulic lic structures structures and esti es tima mation tion of losses. The stochas stochastic tic model can be used in con junc  junction tion with Numer Nu meriical Weather Predic Prediction tion (NWP) models, mod els, arartifi ti ficial cial intel intelli ligence gence models models etc. to predict pre dict both the time and magni mag nitude tude of floods in the best way. This will help to alert in real time and to plan loss mit iga gation. tion. The use of multi mul ti ple  ple param parameeters related related to the depth of a river, the subsurface water wa ter level, snowmelting, surface sur face (basin) (basin) charac character teris istics tics etc. may be tried for higher predic pre diction tion accu accuracy. racy. The precip precipiita tation tion runoff run off preprediction dic tion may be ex per  periimented with differ dif ferent ent time delays delays in order order to observe observe the effect effect of elapsed time on the  predic  pre diction tion accu accuracy. racy. A multivariate inte in tegrated grated model of time series se ries may be exper periimented for model eling ing the  phenom  phe nomeenon of transla trans lation tion and rota ro tation tion of precip pre cipiita tation tion fields. Stochas Sto chastic tic forecast forecast model modeling ing requires requires that design de sign data be trained de pend  pending ing on local lo cal condi conditions tions (not acac cording cord ing to the general general blue print). Therefore, Therefore, there is a need to im prove the stochas sto chastic tic ARIMA process process for  assim as simiilat lating ing all-purpose data on precip pre cipiita tation tion runoff. run off. The prediction results from Time Series Data Mining can be used as a tool for making decisions for the authorities who are responsible for planning and control. Ap pli  plica cation tion and incor in cor po  pora ration tion of the Fourier Fou rier analy analysis, spectra spectra analy analysis, regres regression sion analy anal ysis etc. can be enhanced en hanced due to the effec ef fective tive use of ARIMA models mod els in order order to ana analyze large data sets of precip pre cipiita tation tion runrunoff events using us ing readymade software, soft ware, e.g., MATLAB, SPSS, SAS, etc. RUSSIAN METEOROLOGY AND HYDROLOGY

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6. LIMITATIONS OF STOCHASTIC MODELS OF PRECIPITATION RUNOFF River runoff event is a complex process which is mostly governed by nonlinear patterns. Since the ARIMA model of stochastic process is linear, it cannot  represent the nonlinear dynamics inherent in the transformation of precipitation to runoff. Therefore, the quality of the ARIMA analysis results is not always satisfactory. ARIMA models are highly senstitive to the pattern of data, breach of stationarity, variance, outliers, missing observations etc. that all affect the forecast performance. Also, the ARIMA processes are skilled to capture past trends but for some intervals only. Therefore, i n the stochastic ARIMA analysis the structure of the model is designed to encapsule what occured in the past supposing that no change will occur in the future. However, this is not true for the hydrological domain. As the ARIMA processes do not target  future outcomes unlike the ANN method, all these these reasons limit the the efficiency of the forecast by the ARIMA models to a great extent. Some boundaries in the ARIMA process arise due to the characterizing all data on time series observations, the requirments to the stationarity of time series, to the normality and independence of residuals. Also the ARIMA process lacks the physical interpretation of the precipitation runoff phenomenon. In order to include the dynamics of the precipitation runoff process, the univariate ARIMA process is modified to the Transfer Function Model in which the correlation among meteorological variables and runoff could be estabilished using cross correlation. The transfer of the attributes of dependant series is also  possible but this makes the TFM process lengthy and computationally computationally hard. It could be deduced from the above discussion that over the years the ARIMA process was not improved enough to remove its shortcomings. Researchers introduced many modifications for higher efficiency of  the forecast but their endeavor included alteration of input data rather than strengthening the process to adapt variations in data. It is also experienced that the ARIMA processes do not answer the present days requirements of precipitation runoff forecast in real time because they are mostly site dependant and require labour to design a parsimonious model. These methods are suitable for short-term prediction for next few  periods. The prediction prediction accuracy decreases as the prediction period increases. increases. Also, these models models are prone to giving unrealistic predictions during extreme events. Thus all these limitations lessen the applicability of  the stochastic ARIMA process for the river runoff model. It is also observed that the development of ANN methods lead to the reduction of the use of stochastic ARIMA process in the modeling of precipitation and runoff process. Stochastic modeling of precipitation and runoff phenomenon for river water management has been carried out in the last century and is still in use. With time the process of stochastic modeling has undergone many modifications to adapt the variety of data, elucidate the physics of the process, facilitate the analysis and to enhance the forecasting accuracy. Still the efficiency of the stochastic model always remained dependent upon the skill of a modeler who should know how to model a phenomenon (and this is more important that to develop an easy approach to it), what to model and how to guarantee an accurate result. The present review is intended to provide the better understanding of differences between available models, their applications, model inputs, and resulting outputs. This knowledge will be beneficial for those who use models for regulating river runoff and may be helpful in the development of guidelines for runoff modeling. 7. CONCLU CONCLUSIONS SIONS The authors recommend to continue the use of stochastic ARIMA model and its derivatives. For the forecasting of runoff in small rivers the authors recommend the parsimonious ARIMA model. For large size rivers seasonal ARIMA models or models with suitable transformation of data should be applied. In this case the forecast efficiency is important for making decisions. When the physical interpretation of the runoff is desirous, it is recommended to apply the multivariate ARIMA model and the ARIMA model coupled with the conceptual model. Stochastic modeling aimed at high-quality irrigational applicatons requires combining the surface runoff model with groundwater and soil moisture flux either at the process level or at the data application stage. The analysis of precipitation runoff phenomenon and subsequent river runoff forecasting is of great significance for the management and planning of water resources. Medium- and long-term forecasting at weekly, monthly, seasonal, or even annual time scales is enviable for the forecasting purposes such as  proper operations with reservoirs and irrigation management. The institutional and legal aspects of  management and planning of water resources also have their specific requirements for researches. Information can be modeled by assimilation of more and more relevant characteristic attributes in order to get more precise description of the precipitation–runoff process. The scope of the analysis should be widened to synthesize the model outcomes with the physical consistency of a phenomenon. RUSSIAN METEOROLOGY AND HYDROLOGY

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