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Applications of hybrid wavelet–Artificial Intelligence models in hydrology:A review

Applications of hybrid wavelet–Artificial Intelligence models in hydrology:A review
Applications of hybrid wavelet–Artificial Intelligence models in hydrology:A review

Review Paper

Applications of hybrid wavelet–Arti?cial Intelligence models in hydrology:A

review

Vahid Nourani a ,?,Aida Hosseini Baghanam a ,Jan Adamowski b ,Ozgur Kisi c

a

Dept.of Water Resources Eng.,Faculty of Civil Eng.,Univ.of Tabriz,Iran b

Dept.of Bioresource Eng.,McGill Univ.,Quebec,Canada c

Eng.Faculty,Canik Basari University,Samsun,Turkey

a r t i c l e i n f o Article history:

Received 21August 2013

Received in revised form 16February 2014Accepted 24March 2014

Available online 1April 2014

This manuscript was handled by Andras Bardossy,Editor-in-Chief,with the

assistance of Fi-John Chang,Associate Editor Keywords:

Hydro-climatology Black box model

Arti?cial Intelligence Wavelet transform Hybrid model

s u m m a r y

Accurate and reliable water resources planning and management to ensure sustainable use of watershed resources cannot be achieved without precise and reliable models.Notwithstanding the highly stochastic nature of hydrological processes,the development of models capable of describing such complex phe-nomena is a growing area of research.Providing insight into the modeling of complex phenomena through a thorough overview of the literature,current research,and expanding research horizons can enhance the potential for accurate and well designed models.

The last couple of decades have seen remarkable progress in the ability to develop accurate hydrologic models.Among various conceptual and black box models developed over this period,hybrid wavelet and Arti?cial Intelligence (AI)-based models have been amongst the most promising in simulating hydrologic processes.The present review focuses on de?ning hybrid modeling,the advantages of such combined models,as well as the history and potential future of their application in hydrology to predict important processes of the hydrologic cycle.Over the years,the use of wavelet–AI models in hydrology has steadily increased and attracted interest given the robustness and accuracy of the approach.This is attributable to the usefulness of wavelet transforms in multi-resolution analysis,de-noising,and edge effect detection over a signal,as well as the strong capability of AI methods in optimization and prediction of processes.Several ideas for future areas of research are also presented in this paper.

ó2014Elsevier B.V.All rights reserved.

Contents 1.Introduction (359)

2.Wavelet transform ....................................................................................................362

3.

Hydro-climatologic applications of wavelet–AI models.......................................................................3653.1.Topic 1:Wavelet–AI approach for precipitation modeling...............................................................3653.2.Topic 2:Wavelet–AI models for flow forecasting ......................................................................3663.3.Topic 3:Wavelet–AI models for rainfall–runoff modeling ...............................................................3693.4.Topic 4:Wavelet–AI models for sediment modeling ...................................................................

370

https://www.sodocs.net/doc/754333614.html,/10.1016/j.jhydrol.2014.03.057

0022-1694/ó2014Elsevier B.V.All rights reserved.

Abbreviations:ACO,Ant Colony Optimization;AI,Arti?cial Intelligence;ANFIS,Adaptive Neuro-Fuzzy Inference System;ANN,Arti?cial Neural Network;AR,Auto Regressive;ARIMA,Auto Regressive Integrated Moving Average;ARIMAX,ARIMA with exogenous input;CWT,Continues Wavelet Transform;DWT,Discrete Wavelet Transform;dbn,Daubechies order n wavelet;GA,Genetic Algorithm;GEP,Gene-Expression Programming;GP,Genetic Programming;GWL,Groundwater Level;LR,Linear Regression;MA,Moving Average;MAE,Mean Absolute Error;MARS,Multivariate Adaptive Regression Spline;MLP,Multi-Layer Perceptron;MLR,Multiple Linear Regression;NF,Neuro Fuzzy;NN,Neural Network;PSO,Particle Swarm Optimization;RMSE,Root Mean Square Error;SOM,Self-Organizing Map;SPI,Standard Precipitation Index;SRC,Sediment Rating Curve;SSA,Singular Spectrum Analysis;SSC,Suspended Sediment Concentration;SSL,Suspended Sediment Load;SST,Sea Surface Temperature;SVM,Support Vector Machine;SVR,Support Vector Regression;WANFIS,Wavelet–ANFIS;WANN,Wavelet–arti?cial neural network;WBANN,Wavelet–Bootstrapping ANN;WGRNN,Wavelet–Generalized Regression NN;WMF,Wavelet Modeling Framework;WMRA,Wavelet Multi-Resolution Analysis;WNF,Wavelet–Neuro Fuzzy;WR,Wavelet–Regression;WVC,Wavelet–Volterra.

?Corresponding author.Tel.:+989144030332;fax:+984113344287.

E-mail addresses:vnourani@https://www.sodocs.net/doc/754333614.html, ,vnourani@https://www.sodocs.net/doc/754333614.html, (V.Nourani),hosseinibaghanam@https://www.sodocs.net/doc/754333614.html, (A.Hosseini Baghanam),jan.adamowski@mcgill.ca (J.Adamowski),okisi@https://www.sodocs.net/doc/754333614.html,.tr (O.Kisi).

3.5.Topic5:Wavelet–AI models for groundwater modeling (371)

3.6.Topic6:Other hydro-climatologic applications of wavelet–AI models (371)

4.Summary and conclusions (373)

5.Recommendations for future research (374)

References (375)

1.Introduction

Characterized by high complexity,dynamism and non-stationa-rity,hydrological and hydro-climatologic forecasting has always presented a challenge to hydrologists who recognize its essential role in environmental and water resources management as well as in water-related disaster mitigation.Recent years have seen a signi?cant rise in the number of scienti?c approaches applied to hydrologic modeling and forecasting,including the particularly popular‘data-based’or‘data-driven’approaches.Such modeling approaches involve mathematical equations drawn not from the physical process in the watershed but from an analysis of concur-rent input and output time series(Solomatine and Ostfeld,2008). Such models can be de?ned on the basis of connections between the system state variables(input,internal and output variables) with only a limited number of assumptions being made regarding the physical behavior of the system.Typical examples of data-dri-ven models are rating curves,the unit hydrograph method and var-ious statistical models(Linear Regression;LR,multi-linear,Auto Regressive Integrated Moving Average;ARIMA)and methods of machine learning.The conventional black box time series models such as ARIMA,ARIMA with exogenous input(ARIMAX)and Multi-ple Linear Regression(MLR)are linear models and assume sta-tionarity of the dataset.Such models are unable to handle non-stationarity and non-linearity involved in hydrological processes. As a result,many researchers have focused on developing models that are able to model non-linear and non-stationary processes.

The data-driven methods of Arti?cial Intelligence(AI)have shown promise in modeling and forecasting non-linear hydrologi-cal processes and in handling large amounts of dynamicity and noise concealed in datasets.Such properties of AI-based models are well suited to hydrological modeling problems.Numerous AI tools or techniques have been used,including versions of search optimization,mathematical optimization,as well as logic-,classi?-cation-,statistical learning-and probability-based methods(Luger, 2005).In particular,three sub-sets of AI have been widely used in the hydro-climatologic and environmental?elds:

(1)Evolutionary computation:A branch of optimization methods

that includes swarm intelligence algorithms such as Ant Col-ony Optimization(ACO;Dorigo et al.,1996)or Particle Swarm Optimization(PSO;Kennedy and Eberhart,1995) and evolutionary algorithms such as Genetic-Algorithms (GA;Goldberg,2000),Gene-Expression Programming (GEP),and Genetic-Programming(GP;Koza,1992).

(2)Fuzzy logic:Fuzzy systems(Zadeh,1965)can be used for

uncertain reasoning,which provide a logic perspective in AI techniques.

(3)Classi?ers and statistical learning methods:These models

employ statistical and machine-learning approaches.The most widely used classi?ers are Neural Networks(NNs;

Haykin,1994),kernel methods such as the Support Vector Machine(SVM;Vapnik,1995),k-nearest neighbor algo-rithms such as Self-Organizing Map(SOM;Kohonen, 1997),Gaussian mixture model,naive Bayes classi-?er,and decision tree.NNs,the predominant AI method, are used in hydrology via two approaches:(i)supervised,

including acyclic or feed-forward NNs(where the signal passes in only one direction)and recurrent NNs(which allow feedback),and(ii)unsupervised(e.g.,SOM).

Among the broader applications of AI methods,GA,GP,Fuzzy, NNs,and SVM are widely used in different?elds of hydrology. Since their emergence in hydrology,the ef?cient performance of AI techniques such as data-driven models has been reported over a wide range of hydrological processes(e.g.,precipitation, stream-?ow,rainfall–runoff,sediment load,groundwater,drought, snowmelt,evapotranspiration,water quality,etc.).The number of researchers active in this area has increased signi?cantly over the last decade,as has the number of publications.Several dozen suc-cessful applications for hydrological process modeling(e.g., stream-?ow,rainfall–runoff,sediment,groundwater,water qual-ity)using ANN,Fuzzy,GP,GA,and SVM have been reported,with some examples listed in Table1.

Despite the?exibility and usefulness of AI-based methods in modeling hydrological processes,they have some drawbacks with highly non-stationary responses,i.e.,which vary over a wide scale of frequencies,from hourly to multi-decadal.In such instances of ‘seasonality’,a lack of input/output data pre/post-processing, may not allow AI models to adequately handle non-stationary data. Here,hybrid models which combine data pre/post-processing schemes with AI techniques can play an important role.

Hybrid hydrological models may take advantage of black box (here AI-based)models and their ability to ef?ciently describe observed data in statistical terms,as well as other prior informa-tion,concealed in observed records.The hybrid models discussed here represent the joint application of AI-based methods with the wavelet transform to enhance overall model performance.

As an advance in signal processing,wavelet transforms can reli-ably obviate AI model shortcomings in dealing with non-stationary behavior of signals.A mathematical technique useful in numerical analysis and manipulation of multidimensional signal sets,wavelet analysis provides a time-scale representation of the process and of its relationships.Indeed,the main property of the wavelet trans-form is its ability to provide a time-scale localization of a process. The wavelet transform has attracted signi?cant attention since its theoretical development in1984(Grossmann and Morlet,1984).A number of recent hydrological studies have implemented wavelet analysis(e.g.,Adamowski and Sun,2010;Kim and Valdes,2003; Kisi,2009a,b,2010;Nourani et al.,2009a,b,2011;Maheswaran and Khosa,2012a;Partal and Kisi,2007;Sang,2012;Tiwari and Chatterjee,2010;Zhou et al.,2008).

The Wavelet transform is applicable in extracting nontrivial and potentially useful information,or knowledge,from the large data sets available in experimental sciences(historical records,reanaly-sis,global climate model simulations,etc.).Providing explicit infor-mation in a readable form,it can be used to solve diagnostic, classi?cation or forecasting problems.In a review of the applica-tions of the wavelet transform in hydrologic time series modeling, Sang(2013a)highlighted the multifaceted information that can be drawn from such analysis:characterization and understanding of hydrologic series’multi-temporal scales,identi?cation of seasonal-ities and trends,and data de-noising.Therefore,the ability of the wavelet transform to decompose non-stationary signals into

V.Nourani et al./Journal of Hydrology514(2014)358–377359

sub-signals at different temporal scales(levels)is helpful in better interpreting hydrological processes(Adamowski,2008a,b; Adamowski et al.,2009;Kisi,2010;Mirbagheri et al.,2010; Nason and Sachs,1999;Sang,2012).

Depending on wavelet and AI methods’individual capacities,it can be inferred that a hybrid model comprised of both would simultaneously have the advantages of both techniques.The com-bined wavelet–AI approach is a useful methodology,grounded on both wavelet transform and various AI modeling techniques.It allows for the construction of tractable joint models with such broad applications in hydrology as de-noising,optimization,reme-diation of active Arti?cial NN(ANN)functions,as well as hydrolog-ical process forecasting.In the latter case,wavelet–AI models have been explored by hydrologists,as the combination allows for a detailed elucidation of signals,making the hybrid method an effec-tive tool for predicting hydrological phenomena.In forecasting tasks,the hybrid wavelet–AI method follows a two-step procedure (Fig.1):

(i)Use of the wavelet transform to pre-process input data.This

includes providing a time–frequency representation of a sig-nal at different periods in the time domain,as well as con-siderable information about the physical structure of the data.

(ii)Extraction of features from the main signal to serve as AI inputs,and allowing the full model to process the data.hydrological simulations have led to the conclusion that to deter-mine the ideal mother wavelet for a given problem a variety of mother wavelets should be tested through a trial and error process (Maheswaran and Khosa,2012a;Nalley et al.,2012;Nourani et al., 2011;Sang,2012).Nevertheless,similarity in shape between the mother wavelet and the raw time-series is often the best guideline in choosing a reliable mother wavelet.Generally,mother wavelets with a compact support form(e.g.,Daubechies-1,Haar;and Daube-chies-4,db4)are the most effective in generating time localization characteristics for time series which have a short memory and short duration transient features.In contrast,mother wavelets with a wide support form(e.g.,Daubechies-2,db2)yield reliable forecasts for time series with long term features(Maheswaran and Khosa,2012a).

Since DWT starts with a discrete set of data and considers a dya-dic set of scales,it is compatible with the discrete observation of hydrological signals.In order to study the signal,discretisation comes?rst,and as a result decomposition levels follow.Although appropriate selection of the maximum scale is also important in CWT,it plays an essential role in DWT due to the decomposition procedure and extraction of dominant sub-series which can not be depicted as easily as with the CWT.Therefore,along with mother wavelet type selection,determination of the appropriate decomposition level(scale)is another important sub-step within the?rst step when DWT is applied(Fig.1.).In early studies,the optimum decomposition level was usually determined through a trial-and-error process,but afterwards a formula which relates

Table1

Examples of some AI applications in hydrological process modeling.

Hydrologic process ANN Fuzzy GP GA SVM

Stream-?ow modeling Sudheer et al.(2008)Chang and Chen

(2001)

Ni et al.(2010)Parasuraman and Elshorbagy

(2007)

Li et al.(2010)

Rainfall–runoff modeling Hsu et al.(1995)Savic et al.(1999)Gautam and Holz(2001)Cheng et al.(2002)Elshorbagy et al.

(2010)

Sediment modeling Sarangi et al.(2005)Aytek and Kisi

(2008)

Altunkaynak(2009)Rajaee et al.(2009)Misra et al.(2009)

Groundwater modeling Bhattacharjya and Datta

(2009)

He et al.(2008)Fallah-Mehdipour et al.

(2013)

Bhattacharjya and Datta(2009)Yoon et al.(2011)

Water quality

modeling

Singh et al.(2009)Pai et al.(2009)Eslamian and Lavaei(2009)Dhar and Datta(2009)Singh et al.(2011)

Fig.1.Schematic diagram of hybrid wavelet–AI forecasting model.

360V.Nourani et al./Journal of Hydrology514(2014)358–377

any attention to seasonal effects.Since many seasonal characteris-tics may be embedded in hydrological signals,a precise insight into the process under study and attention to the periodicity of the process might be helpful in the selection of an appropriate decom-position level for dyadic DWT analysis.Decomposition level l contains l details and as an example in the case of daily modeling denotes2n-day mode where n=1,2,...,l(e.g.,21-day mode,22-day mode,23-day mode which is nearly weekly mode,24-day mode, 25-day mode which is nearly monthly mode,etc.),therefore,the seasonal and scale dependency of the process can be handled by the model.Depending on the wavelet type,the decomposition level and the type of AI method applied,several approaches can be examined according to the aim in developing the hybrid wave-let–AI model.In this context,AI methods can be seen to fall into three basic categories:optimization,logic,classi?cation and statis-tical learning;based on the utilization of AI over one of these three ?elds,different purposes for the hybrid wavelet–AI model can be inferred.Generally,the collective application of optimization methods and wavelet analysis leads to recognition of optimal inputs for AI models(Kuo et al.,2010a,b;Wang et al.,2011a).Fea-ture extraction and classi?cation of dominant inputs to be used in forecasting(Hsu and Li,2010;Nourani et al.,2013,2014)along with seasonality detection(Nourani and Parhizkar,2013; Nourani et al.,2009a,b,2011,2012)as well as noise reduction/ removal from the hydrologic time series(Campisi et al.,2012; Guo et al.,2011)are important elements contributing to better forecasting for future planning through hybrid wavelet–AI models.

Given the rapidly evolving?eld of wavelet-AI approaches in hydrology,it is important to survey what has been done with wavelet–AI models and current research trends.Several review papers(see Table2)concerning particular sub-sets of AI models used in hydrology or speci?cally on hydrological modeling have explored this topic(Abrahart et al.,2012;ASCE,2000;Dawson and Wilby,2001;Kalteh et al.,2008;Maier and Dandy,2000; Maier et al.,2010;Solomatine and Ostfeld,2008).While general reviews of wavelet applications in hydrology(Kumar and Foufoula-Georgiou,1997;Labat,2005;Schae?i et al.,2007;Sang, 2013a)have surveyed wavelet analysis methods(see Table2),no reviews have centered on the speci?c use of wavelet–AI models. Maier et al.(2010),in their review paper on methods used in devel-oping NNs for the prediction of water resource variables in river systems,suggested that

‘‘...work should continue on the development and evaluation of hybrid model architectures that attempt to draw on the strengths of alternative modeling approaches.Given the amount of work that has already been done in this area,a review of this emerging?eld of research would seem timely.’’

The lack of review papers evaluating the simultaneous applica-tion of AI models and wavelets in hydrology led to the collective preparation of the current review paper,which is an updated assessment of coupled AI and wavelet applications in various?elds of hydrology.The advances in hydrological modeling and simula-tion achieved through wavelet–AI models have largely outstripped conventional models in terms of performance,and led to an increase in associated research and resulting publication numbers since2003(Fig.2).While such publications remained low from 2003until2007,there was a10-fold increase over the next two years,which represents a turning point in wavelet–AI research. Articles up to2007played an innovator role,with the paper of Labat et al.in2004representing the pioneering work of wavelet applications to hydrology(with152citations in Scopus)and Labat’s review on the wavelet concept(with114citations)providing fur-ther incentive to research the application of wavelet–AI systems in hydrological modeling(Labat,2005).In2006,Partal and Kü?ük (with44citations)demonstrated the merits of wavelet trend anal-ysis in determining possible trends in annual total precipitation ser-ies,while the work of Cannas et al.(with38citations)further developed the hybrid wavelet–AI model.Since2007,there has been an increase in the number of papers dealing with wavelet-AI modeling of hydrological processes,as can be seen from Fig.2.

The principal objectives of the current review paper are to com-prehensively categorize wavelet–AI models and enumerate their novel applications in hydrology along with their bene?ts.In turn, this assessment will provide some ideas on future areas of research in the?eld.This review focuses on their extensive use in hydro-climatology,and further restricts itself to the main hydrologic parameters of interest,i.e.,(i)precipitation,(ii)stream-?ow,runoff, (iii)rainfall–runoff,(iv)sediment,(v)groundwater,(vi)miscella-neous:drought,snowmelt,evapotranspiration,water quality, wave height,etc.These selected parameters of review were drawn from a review of NN hydrological modeling undertaken by the ASCE Task Committee(ASCE,2000).The present sources consulted were drawn from the Scopus abstract and citation database (https://www.sodocs.net/doc/754333614.html,).Conference proceedings are not included in

Table2

Review papers concerning particular sub-sets of AI models and the wavelet transform used in hydrology.

Review subject Authors(year)Paper/book title

Reviews on hydrological applications of AI ASCE Task Committee on Application of Arti?cial

Neural Networks in Hydrology(2000)

Arti?cial neural networks in hydrology ii:hydrologic applications

Maier and Dandy(2000)Neural networks for the prediction and forecasting of water resources variables:a

review of modeling issues and applications

Govindaraju and Rao(2000)Arti?cial neural networks in hydrology

Dawson and Wilby(2001)Hydrological modeling using arti?cial neural networks

Solomatine(2005)Data-driven modeling and computational intelligence methods in hydrology Cherkassky et al.(2006)Computational intelligence in earth sciences and environmental applications Kalteh et al.(2008)Review of self-organizing map(SOM)in water resources:analysis,modeling,and

application

Solomatine and Ostfeld(2008)Data-driven modeling:some past experiences and new approaches

Maier et al.(2010)Methods used for the development of neural networks for the prediction of water

resource variables in river systems:Current status and future directions

Abrahart et al.(2012)Two decades of anarchy?Emerging themes and outstanding challenges for neural

network river forecasting

Reviews on hydrological applications of wavelets Kumar and Foufoula-Georgiou(1997)Wavelet analysis for geophysical applications

Labat(2005)Recent advances in wavelet analyses:Part1.A review of concepts

Schae?i et al.(2007)What drives high?ow events in the Swiss Alps?Recent developments in wavelet

spectral analysis and their application to hydrology

Sang(2013a)A review on the applications of wavelet transform in hydrology time series analysis V.Nourani et al./Journal of Hydrology514(2014)358–377361

this review.Details of the selected papers,including year of publi-cation,authors,AI methods used and variables predicted are given in Table3.This is followed by sections on the basic concepts of the wavelet transform(Section2),and the applications of hybrid mod-els in various?elds of hydrology(Section3).A summary and sug-gestions for future avenues of research are presented in the last sections of the paper.

2.Wavelet transform

The wavelet transform has increased in usage and popularity in recent years since its inception in the early1980s,yet it is still not as widely used as the Fourier transform.However,Fourier analysis has a signi?cant drawback:a signal’s Fourier transform into the frequency domain results in the loss of time information,such that it becomes impossible to tell when a particular event took place.In contrast,wavelet analysis allows for the use of long time intervals when more precise low-frequency information is needed,and shorter regions when high-frequency information is of interest.

In the?eld of earth sciences,Grossmann and Morlet(1984), who worked especially on geophysical seismic signals,introduced the wavelet transform.A comprehensive literature survey of wave-let use in the geosciences can be found in Foufoula-Georgiou and Kumar(1995)and most recent contributions are cited by Labat (2005).As there are many good books and articles introducing the wavelet transform,this paper will not delve into the theory behind wavelets and only present the main concepts of the trans-form;recommended literature for more information on the wave-let transform includes Mallat(1998)or Labat et al.(2000).

The time-scale wavelet transform of a continuous time signal, x(t),is de?ned as(Mallat,1998):

Tea;bT?

1

???

a

p

Zt1

à1

g?

tàb

a

xetTádte2T

where a is a dilation factor,b is the temporal translation of the func-tion g(t),which allows for the study of the signal around b,?corre-sponds to the complex conjugate and g(t)is the wavelet function or mother wavelet.

The main property of the wavelet transform,which is derived from the compact support of its basic function,is to provide a time-scale localization of processes.This is in contrast to the classical trigonometric functions of Fourier analysis.The wavelet transform searches for correlations between the signal and wavelet function.This calculation is done at different scales of a and locally around the time of b.The result is a wavelet coef?cient(T(a,b)) contour map known as a scalogram.In order to be classi?ed as a wavelet,a function must have?nite energy,and it must satisfy the following‘‘admissibility conditions’’(Mallat,1998):

Zt1

à1

getTdt?0;C g?

Zt1

à1

j^gewTj2

j w j

dw<1e3T

where^gewTis Fourier transform of g(t);i.e.,the wavelet must have no zero frequency component.

In order to obtain a reconstruction formula for the studied sig-nal,it is necessary to add‘‘regularity conditions’’to the previous conditions(Mallat,1998):

Zt1

à1

t k getTdt?0where k?1;2;...;nà1e4T

So the original signal may be reconstructed using the inverse wavelet transform as(Mallat,1998):

xetT?

1

c g

Zt1

à1

Z1

1

???

a

p g

tàb

a

Tea;bT

daádb

e5T

For practical applications,the hydrologist does not have at their disposal a continuous-time signal process but rather a discrete-time signal.A discretization of Eq.(2)based on the trapezoidal rule may be the simplest discretization of the continuous wavelet transform,producing N2coef?cients from a data set of length N. Redundant information is therefore locked up within the coef?-cients,which may or may not be a desirable property(Addison et al.,2001).

To overcome this redundancy,a logarithmically uniform spac-ing can be used for the a scale discretization with a correspond-ingly coarser resolution of the b locations,which allows for N transform coef?cients to completely describe a signal of length N. Such a discrete wavelet has the form(Mallat,1998):

g

m;n

etT?

1

??????

a m

p g tànb0a m0

a m

e6T

where a0is the speci?ed?ne dilation,where a0>1,with a0usually equal to2,b0is the location parameter,where b0>0,with b0usually equal to1,and m and n are integers that control the wavelet dilation and translation respectively.

published papers regarding wavelet–AI applications in hydro-climatology(indexed in Scopus)with respect 362V.Nourani et al./Journal of Hydrology514(2014)358–377

Table3

Details of the surveyed papers,including year of publication,authors,where hybrid wavelet–AI methods were used to predict hydrological variables.

Paper No.Author(year)Type of AI

technique

Wavelet transform

type

Variables Time scale

9Mwale and Gan(2005)ANN,GA CWT Precipitation Monthly Mwale et al.(2007)ANN,GA CWT Precipitation Monthly

Partal and Kisi(2007)ANFIS DWT Precipitation Daily

Nourani et al.(2009a)ANN DWT Precipitation Monthly

Partal and Cigizoglu(2009)ANN DWT Precipitation Daily

Kuo et al.(2010a)ANN,GA CWT Precipitation Seasonal

Kisi and Shiri(2011)GEP,NF DWT Precipitation Daily

Kisi and Cimen(2012)SVM DWT Precipitation Daily

Ramana et al.(2013)ANN DWT Precipitation Monthly

35Cannas et al.(2006)ANN DWT,CWT Runoff Monthly Kisi(2008)ANN DWT Stream-?ow Monthly

Wang et al.(2009)ANN DWT Runoff Daily,annually

Wu et al.(2009)ANN DWT Runoff Daily

Adamowski(2008a)ANN CWT Stream-?ow,meteorological data Daily

Zhou et al.(2008)ANN DWT Discharge Monthly

Kisi(2009a)ANN DWT Stream-?ow Daily

Partal(2009a)ANN DWT Stream-?ow Monthly

Mwale and Gan(2010)ANN,GA CWT Runoff Monthly

Adamowski and Sun(2010)ANN DWT Stream-?ow Daily

Kuo et al.(2010b)ANN,GA CWT Stream-?ow,rainfall,air temperature Seasonal,daily

Pramanik et al.(2010)ANN DWT Stream-?ow Daily

Tiwari and Chatterjee(2010)ANN-Bootstrap DWT River water level Hourly

Shiri and Kisi(2010)ANFIS DWT Stream-?ow Daily,monthly,yearly

Wang et al.(2011a,b)Statistical

method

DWT Stream?ow Daily

Kisi(2011a)ANN DWT Stream-?ow Monthly

Kisi and Partal(2011)NF DWT Stream-?ow Monthly

Guo et al.(2011)SVM DWT Stream-?ow Monthly

Kisi and Cimen(2011)SVM DWT Stream-?ow Monthly

Tiwari and Chatterjee(2011)ANN-Bootstrap DWT Discharge Daily

Krishna et al.(2011)ANN DWT Stream-?ow Daily

Tiwari et al.(2012)ANN,SOM DWT Discharge Daily

Kalteh(2013)SVR,ANN DWT Stream-?ow Monthly

Wei et al.(2012)ANN DWT River discharge Monthly

Ren et al.(2011)ANFIS DWT,CWT Runoff Monthly

Adamowski and Prokoph

(2013)

ANN CWT Stream-?ow Daily

Maheswaran and Khosa

(2013a)

ANN DWT,WVC Stream-?ow Daily

Maheswaran and Khosa

(2012b)

ANN DWT,WVC Stream-?ow Monthly

Krishna(2013)ANN DWT In?ow Daily

Badrzadeh et al.(2013)ANN,ANFIS DWT River?ow Daily

Danandeh Mehr et al.

(2013a)

ANN,GP DWT Stream-?ow Monthly

Danandeh Mehr et al.

(2013b)

ANN DWT Stream-?ow Monthly

Sahay and Srivastava(2013)ANN,GA DWT Flood Daily

Sang(2013b)WMF DWT Rainfall,runoff Monthly,daily

Maheswaran et al.(2013)ANN DWT,WVC Stream-?ow Daily,weekly,monthly

12Anctil and Tape(2004)ANN CWT Stream-?ow,rainfall,evapotranspiration Daily Remesan et al.(2009)ANN DWT Rainfall,runoff Daily

Nourani et al.(2009b)ANN DWT Rainfall,runoff Daily

Nourani et al.(2011)ANN,ANFIS DWT Rainfall,runoff Daily,monthly

Wang et al.(2011a,b)ANN,GA DWT Rainfall,stream-?ow Hourly

Adamowski et al.(2011)ANN-MARS DWT Morphological data,rainfall,runoff Daily

Nourani et al.(2012)ANN,GP DWT Rainfall,runoff Daily,monthly

Adamowski and Prasher

(2012)

SVR,ANN DWT Rainfall,runoff Daily

Nayak et al.(2013)ANN DWT Rainfall,discharge,evaporation Daily

Nourani et al.(2013)ANN,SOM DWT Rainfall,runoff Daily

Kamruzzaman et al.(2013)AR DWT Rainfall,stream

Nourani and Parhizkar

(2013)

ANN,SOM DWT,CWT Rainfall,runoff Daily,monthly

10Partal and Cigizoglu(2008)ANN DWT SSL Daily Mirbagheri et al.(2010)ANN,NF DWT SSL,discharge Daily

Kisi(2010)ANN DWT SSL Daily

Rajaee(2010)NF DWT SSL Daily

Rajaee et al.(2010)NF DWT SSL Daily

Rajaee et al.(2009)ANN,NF DWT SSL Daily

Rajaee et al.(2011)ANN DWT SSL Daily

Shiri and Kisi(2012)ANN,GEP,NF DWT SSL,discharge Daily

(continued on next page)

V.Nourani et al./Journal of Hydrology514(2014)358–377363

This power of two logarithmic scaling of the translation and dila-tion is known as the dyadic grid arrangement.The dyadic wavelet can be written in more compact notation as(Mallat,1998):

g

m;n

etT?2àm=2ge2àm tànTe7TDiscrete dyadic wavelets of this form are commonly chosen to be orthonormal;i.e.,(Mallat,1998):

Zt1à1g

m;n

etTg m0;n0etTdt?d m;m0d n;n0e8T

where d is the Kronecker delta.

This allows for the complete regeneration of the original signal as an expansion of a linear combination of translates and dilates of orthonormal wavelets.

For a discrete time series,x i,the dyadic wavelet transform becomes(Mallat,1998):

T m;n?2àm=2

X Nà1

i?0

ge2àm iànTx ie9T

where T m,n is the wavelet coef?cient for the discrete wavelet of scale a=2m and location b=2m n.Eq.(9)considers a?nite time series,x i,where i=0,1,2,...,Nà1;and N is an integer power of2,i.e., N=2M.This gives the ranges of m and n as,respectively, 0

In addition to this,a signal smoothed component,T,remains, which is termed the signal mean.Thus,a time series of length N is broken into N components,i.e.,with zero redundancy.The inverse discrete transform is given by(Mallat,1998):

x i?Tt

X M

m?1

X

2Màmà1

n?0

T m;n2àm=2ge2àm iànTe10Tor in a simpler format as(Mallat,1998):

x i?tTt

X M

m?1

W metTe11T

Table3(continued)

Paper No.Author(year)Type of AI

technique

Wavelet transform

type

Variables Time scale

Liu et al.(2013a,b)ANN DWT SSC Daily Nourani et al.(2014)ANN DWT Stream-?ow,SSL Daily

5Adamowski and Chan(2011)ANN DWT GWL Monthly Maheswaran and Khosa

(2013b)

ANN WVC GWL Monthly

Kisi and Shiri(2012)ANFIS DWT GWL Daily

Moosavi et al.(2013a)ANN,ANFIS DWT GWL Monthly

Moosavi et al.(2013b)ANN,ANFIS DWT GWL Monthly 32Kim and Valdes(2003)ANN DWT Drought Monthly Belayneh and Adamowski

(2013)

AN DWT SPI,drought Monthly

Belayneh et al.(2014)ANN,SVR DWT SPI,drought Monthly

Shirmohammadi et al.

(2013)

ANN,ANFIS DWT Drought Monthly

Wang and Ding(2003)ANN DWT Shallow GWL,discharge Daily,monthly

Lauzon et al.(2004)SOM CWT Soil moisture-precipitation-?ow Daily

Deng et al.(2011)SVM DWT Soil water content,precipitation,temperature,

evaporation

Daily Adamowski(2008b)ANN CWT Snowmelt river?ood Daily

Noori et al.(2009)ANN,ANFIS CWT Waste Generation Weekly

Partal(2009b)ANN DWT Evapotranspiration Daily

Shankar et al.(2011)Fuzzy DWT Land cover–

Abghari et al.(2012)ANN Active function Evapotranspiration Daily

Adamowski et al.(2012)ANN DWT Urban water demand,precipitation,temperature Daily

Kisi(2009b)ANN DWT Lake level Monthly

Campisi et al.(2012)ANN DWT Urban water demand Monthly

Tiwari and Adamowski

(2013)

Bootstrap-ANN DWT Urban water demand Daily,monthly

Ozger(2010)ANN,ANFIS CWT Wave height Hourly

Kisi(2011b)ANN DWT River-stage Daily

Deka and Prahlada(2012)ANN DWT Wave height Hourly

Shekarrizfard et al.(2012)ANN DWT Meteorological data Daily

Siwek and Osowski(2012)ANN,SVM DWT Meteorological data Daily

Najah et al.(2012)ANFIS DWT Water quality parameters Monthly

Karran et al.(2013)ANN,SVR DWT Climate regimes Daily

Yu et al.(2013)NF DWT Hydro-meteorological data Daily

Nalley et al.(2013)Trend test DWT Air temperature Monthly,seasonally,

annually Pingale et al.(2013)Trend test DWT Temperature,rainfall Monthly,seasonally,

annually Eynard et al.(2011)ANN DWT Temperature and thermal power consumption Monthly

Liu et al.(2013a)GA,PSO DWT Wind speed Sampling

Liu et al.(2013b)ANN DWT,Packet Wind speed Sampling

Liu et al.(2014)SVM DWT Wind speed Sampling

Evrendilek(2012)ANN DWT Heat?uxes,evapotranspiration Sampling

Wang et al.(2013)ANN,ARIMA DWT Water quality properties Monthly

364V.Nourani et al./Journal of Hydrology514(2014)358–377

where tTis the approximation sub-signal at level M,representing the background information of data,and W m(t)are wavelet coef?-cients which provide the detail sub-signals at levels m=1,2,...,M and can capture small features of interpretational value in the data.

Because of the simplicity of W1(t),W2(t),...,W M(t),TetT,some interesting characteristics,such as period,hidden period,depen-dence and jump can be diagnosed easily through wavelet components.

3.Hydro-climatologic applications of wavelet–AI models

This review is a complement to recent surveys such as Maier et al.(2010)and Abrahart et al.(2012)who mainly focused on either technical or historical reviews of the use of ANNs in the prediction of water resource variables in river systems and river forecasting,respectively.The current review deals with various hydro-climatologic processes.Moreover,it goes through applica-tions of not only the ANN technique but also other data-driven AI techniques(e.g.,SOM,Fuzzy logic,GA,GP,SVM,etc.)coupled with wavelet transform.Approximately105papers on the subject of wavelet–AI for several hydro-climatologic issues were investigated. Table3compares the type of utilized AI techniques,wavelet types, applied hydrological variables and time scales of the reviewed papers.

3.1.Topic1:Wavelet–AI approach for precipitation modeling

Precipitation is needed to replenish water to the earth and is important because it helps maintain the atmospheric balance. The amount and duration of precipitation events affect both water level and water quality.Precipitation can also be damaging;for example,too much rain can cause severe?ooding.Therefore,an accurate estimate of precipitation is essential in water resources management,particularly with respect to?ood mitigation.How-ever,the wide spatiotemporal variation in rainfall makes its pre-diction particularly challenging.Numerous numerical,physical and data-driven-based models have been developed to provide an accurate estimation model for precipitation.Mwale and Gan (2005)used wavelet spectra information to identify and analyze the variety in space,time and frequency of dominant oscillations in the rainfall of East Africa,along with the relationships existing between September–November rainfall in that region and Sea Sur-face Temperature(SST)of the Indian and South Atlantic Oceans. Their wavelet-based analysis discerned homogeneous zones of rainfall variability over various parts of the Oceans.In order to accurately predict rainfall with a2-month lead time,linear(i.e., canonical correlation analysis)and non-linear(i.e.,ANN–GA)sta-tistical tele-connection models were applied.A non-linear ANN–GA model was the most accurate in predicting rainfall over most of East Africa,whereas a model based on linear canonical correla-tion analysis performed poorly over the same region.Mwale et al. (2007)then expanded their models two years later using wavelet empirical orthogonal functions of space–time-Frequency regimes for examining the predictability of southern Africa summer rainfall.

Partal and Kisi(2007)proposed a wavelet–NF method to predict precipitation values.As choosing appropriate model inputs is one of the most critical steps in building an accurate forecasting model, DWT was used to present the original precipitation signal under different resolution intervals,such that daily,monthly,and annual sub-series’characteristics could be more clearly delineated than in the original signal.Subsequently,the correlation coef?cients between sub-series and the original precipitation series provided information for the selection of the NF model inputs and for the determination of the effective wavelet components to use in predicting precipitation values at a daily scale.Their wavelet–NF model provided a good?t with the observed data,especially for time series which had zero precipitation in the summer months as well as for the peaks within the testing period.This result was interesting since classical NF models have usually faced dif?culties in forecasting extreme values of observed precipitation series. Nourani et al.(2009a)linked wavelet analysis to a non-linear inter-extrapolator ANN for monthly precipitation prediction.A wavelet transform,capable of capturing signals’multi-scale fea-tures,served to decompose the precipitation time series into https://www.sodocs.net/doc/754333614.html,ing an ANN model with a non-linear kernel to reconstruct the signal better simulated the non-linear behavior of the phenomenon than did other linear models such as seasonal ARIMA.This was largely because the dominant seasonalities extracted via wavelet analysis were assigned greater weights.Fur-thermore,in investigating the effect of wavelet transform type and optimum decomposition level on model performance,they con-?rmed the model’s accuracy in forecasting short-(one month ahead)and long-term precipitation events.A similar methodology was also followed by Ramana et al.(2013)to model monthly rain-fall values using both rainfall and temperature data as inputs for a WANN model.

Partal and Cigizoglu(2009)predicted daily precipitation via a wavelet–NN method,which provided a good?t with observed data.Kisi and Shiri(2011)used the advantage of several AI-based methods in order to model daily precipitation.They compared the abilities of single GEP,NF models,with the linked form of GEP and NF with wavelet analysis.The results of daily precipitation forecasts via single GEP and NF were weak,and the use of wavelet coef?cients did not satisfactorily improve the forecasting results, although the accuracies increased to a great extent.Finally,they obtained good precipitation forecasting results by merging the best single and hybrid models’inputs and introducing them as the model inputs.Among the outcomes of later models,the hybrid wavelet–GEP model had superior performance in forecasting daily precipitation than the wavelet–NF model which was unable to learn the non-linear nature of the process.

Kuo et al.(2010a)investigated the seasonal predictability of rainfall via the wavelet–AI approach.Wavelet analysis was employed on seasonal rainfall and Paci?c Ocean SST data,and the results revealed strong2–4-year cycles in rainfall data as well as high wavelet coherence between the selected SST and seasonal rainfall.They went on to use an ANN-GA model to predict seasonal precipitation with a one-season lead time,using the GA to calibrate ANN parameters.As a result,model parameters and coef?cients for the different layers were optimized by minimizing an objective function that,in turn,maximized the correlation between simu-lated and observed seasonal rainfall values.Their study demon-strated a strong relationship between seasonal Paci?c SST anomalies and seasonal rainfall at the study site,and this link was effectively captured by an ANN–GA model.

Kisi and Cimen(2012)used a joint wavelet–SVM model for the prediction of daily precipitation and found that the hybrid method could increase the forecasting accuracy of one-day-ahead precipi-tation better than single SVM and ANN models.

An assessment of the various studies on precipitation modeling revealed two issues regarding AI and wavelet transforms, respectively:

(i)Since an averaged value of the pointy measured rainfalls of

the rain gauges over a watershed is usually assigned to the whole of the watershed,the data and subsequently the model used for forecasting and simulation of the rainfall process usually contain uncertainties.In such uncertain sit-uations,Fuzzy-based models may be employed in the esti-mation of uncertainties in real world problems.

V.Nourani et al./Journal of Hydrology514(2014)358–377365

(ii)It can be deduced that although the data pre-processing pro-cess by the wavelet transform can improve the precipitation modeling performance at different time scales,this improve-ment is greater for large scales such as monthly or seasonal data compared to hourly,daily or weekly.Such an outcome is reasonable because the seasonality pattern in large time scales are more highlighted compared to the small time scales.In other words,the Auto Regressive(AR)or Markovian property of precipitation is more signi?cant in small time scales such as the daily scale in which the process does not present a strong Markov chain,whereas the season-ality feature is the dominant factor in large time scales such as monthly or seasonal.In such cases precipitation forecast-ing at a daily scale might not result in useful outcomes.On the other hand,precipitation time series analysis via wavelets is not only considered a temporal pre-processing technique,but it also reveals effective information about the precipitation background of a speci?c area.Accordingly, the scalogram of CWT speci?es the dominant daily,monthly, seasonal and yearly periods,as well as the failure or increase in the precipitation.In this way,droughts and?oods can be distinguished clearly.Thus,in using the hybrid wavelet–AI models for rainfall prediction,more re?ned time steps can be recognized and used to drive certain hydrologic models in order to predict droughts and?oods according to the declining and rising trends of rainfall values.Such informa-tion should bene?t the water resources management of any watershed.

3.2.Topic2:Wavelet–AI models for?ow forecasting

Simulations and predictions of stream-?ow is one of the most active research areas in surface water hydrology.Given its poten-tial consequences(e.g.,?ooding,erosion),stream-?ow is the gener-ated component of the rainfall–runoff process and needs precise prediction;therefore,short-and long-term forecasting models are extremely important for the sustainable management of water resources.Given the in?uence of such varied phenomena as pre-cipitation,evaporation,and temperature in stream-?ow genera-tion,the relevant observed time series tend to be non-linear, temporally variable and indeterminate.The underlying mecha-nisms of stream-?ow generation are likely to be quite different during low,medium,and high?ow periods,especially when extreme events occur.It is therefore very dif?cult to forecast stream-?ow(Guo et al.,2011).In several studies the ef?ciency and accuracy of stream-?ow models using a wavelet–AI approach has been compared to those of single AI or conventional regres-sion-based models.Typically,such models?rst decompose a time series into multiple levels of detail,and then implement a multi-resolution analysis which can effectively diagnose the signal’s main frequency components,as well as abstract local information from the time series.Subsequently,the appropriate sub-series are utilized in the AI model.

Cannas et al.(2006)investigated the effects of wavelet-based data pre-processing on NNs’ability to predict the hydrologic behavior of runoff.Employing DWT and CWT to account for non-stationarity and seasonal irregularity of runoff time series,they showed that networks trained with pre-processed data performed better in predicting monthly runoff than did networks trained with non-decomposed,noisy raw signals.

Adamowski(2008a)developed short-term river?ood forecast-ing models based on wavelet and cross-wavelet components and evaluated their accuracy,compared with ANN models and simple perseverance models,in forecasting daily stream-?ows with lead times of1,3,and7days.The wavelet based models showed great accuracy as a stand-alone forecasting method for1-and 3-day lead times river?ood forecasting,provided no signi?cant trends in the amplitude occurred for the same Julian day year-to-year,and a relatively stable phase shift existed between the?ow and meteorological time series.However,such river?ood forecast-ing models,based on wavelet and cross-wavelet constituent com-ponents,were not accurate for longer lead time forecasting(e.g., 7days).

In order to forecast monthly stream-?ows,Kisi(2008)used a neuro-wavelet https://www.sodocs.net/doc/754333614.html,paring these results with those of a Multi-Layer Perceptron(MLP),a MLR and AR models,he found the neuro-wavelet model outperformed the MLP,MLR and AR models.

In a further study,Wang et al.(2009)applied the multi-resolu-tion characteristic of wavelet analysis and the non-linear capability of ANN to predict in?ow of Three Gorges Dam in Yangtze River, https://www.sodocs.net/doc/754333614.html,ing both a wavelet network model and a type of threshold AR model to predict short-and long-term runoffs,they found the wavelet network model to be more accurate,leading them to sug-gest that future research should focus on functional and feasible wavelet network models.

In another study,Wu et al.(2009)explored the ef?ciency of var-ious methods in improving the ANN performance in daily?ow pre-diction.The objective of their research was to determine whether data pre-processing techniques such as Moving Average(MA),Sin-gular Spectrum Analysis(SSA),and Wavelet Multi-Resolution Analysis(WMRA),coupled with ANN,might improve the estima-tion accuracy of daily?ows.These data pre-processing techniques were used to improve and highlight the mapping relationship between inputs and output of the ANN model by smoothing raw ?ow data.The hybrid models showed noticeable improved perfor-mance over the ANN model and considering the performance and complexity of the linkage of ANN to the data pre-processing meth-ods,MA,SSA and WMRA yielded better ef?ciency,respectively.

Zhou et al.(2008)used a wavelet predictor–corrector model to decompose a time series into an approximation series and several stationary detailed sub-series.Each sub-series was then predicted individually using an ARMA model,and a correction procedure was implemented for the sum of the prediction results.Finally,simulat-ing monthly discharge with ARMA,seasonal ARIMA,and an ANN model,they found the wavelet predictor–corrector model to have the greatest prediction accuracy.In addition,the decomposition scale showed no obvious effect on the prediction for the monthly discharge time series.

Kisi(2009a),comparing the ability of a joint wavelet–ANN model to an ANN alone in predicting1-day-ahead intermittent stream-?ow,tested the models by applying different input combi-nations of decomposed time series.He ultimately showed that the wavelet–ANN provided signi?cantly better forecasting accuracy than the ANN alone,particularly for high?ow estimates.

Partal(2009a)evaluated the ef?ciency of several ANNs(i.e.,feed forward back propagation,generalized regression NN,radial based function-based NN)combined with a wavelet transform to predict river?ow in future months.Periodic components obtained via wavelet decomposition were fed to the NNs to improve river?ow forecasting.The combination of hybrid wavelet and the feed for-ward back propagation model outperformed all other models examined in the study.

Adamowski and Sun(2010)coupled DWT and ANN for?ow forecasting in non-perennial rivers in semi-arid watersheds.The decomposition process of original?ow time series into sub-series was iterated,with successive approximation signals being decom-posed in turn,so that the original?ow time series were broken down into many lower resolution components.The sub-series used in the ANN model led to ef?cient forecasting outcomes.WANN models were found to provide more accurate?ow forecasts than the regular ANN models,since wavelet transforms provided useful

366V.Nourani et al./Journal of Hydrology514(2014)358–377

decompositions of the original time series,and the wavelet-trans-formed data improved the ability of the ANN forecasting model by capturing useful information on various resolution levels.

Using a wavelet-based ANN–GA model,Kuo et al.(2010b)pre-dicted stream-?ow with a one season(3-month)lead time-based on SST.Wavelet analysis was?rst applied to select sectors of SST that were related to the rainfall data of the study sites at a seasonal time scale,and then the selected SST was used as predictors in the ANN–GA model to predict seasonal rainfall at a one-season lead time.The GA portion of the model served to calibrate the parame-ters of the ANN with a feed forward structure and three layers.This resulted in an ef?cient stream-?ow prediction methodology.

Pramanik et al.(2010)concluded that advance time step stream-?ow forecasting was of critical importance in controlling ?ood damage,while applying a hybrid wavelet–AI model to stream-?ow forecasting.They proposed models which used DWT functions to pre-process the?ow time series into wavelet coef?-cients of different frequency bands,leading to the creations of WANN models with1-,2-and3-day lead times to forecast?ow. The hybrid models were trained using the Levenberg–Marquardt algorithm and results were compared with simple ANN models. Con?rming previous studies’results,the WANN models provided better prediction of peaks in stream-?ow than individual ANN models.

In order to develop an accurate and reliable ANN model for hourly?ood forecasting,the potential of wavelet and bootstrap-ping techniques linked to ANN(WBANN)was explored by Tiwari and Chatterjee(2010).To capture useful information,the time ser-ies was decomposed into different components and then appropri-ate sub-series were added up to develop new time series.Finally a bootstrap-based ANN model was constructed.Overall,the WBANN model was found to be accurate and reliable in simulating peak water levels,and outperformed the ANN,WANN and BANN mod-els,indicating that while wavelet decomposition improved the performance of ANN models,the bootstrap re-sampling technique produced more consistent and stable solutions.

To study short-and long-term stream-?ow forecasting,Shiri and Kisi(2010)used recorded stream-?ow values to compare the performance of a combined wavelet–NF model,which took into account the periodicity of the data,to an unenhanced NF model. The comparison of results showed that adding the periodicity com-ponent into the input layer generally increased modeling accuracy; such the wavelet–NF model can be considered as an appropriate model to simulate daily,monthly and especially yearly stream-?ows.

Synthetic generation of daily stream?ow sequences via the wavelet transform was explored by Wang et al.(2011b).The method?rstly decomposes the daily stream?ow sequences with different frequency components into the series of wavelet coef?-cients W1(t),W2(t),...,W P(t)and scale coef?cients(the residual) CP(t)at a speci?c resolution of P.Secondly,the series of W1(t), W2(t),...,W P(t)and CP(t)are divided into a number of sub-series based on a yearly period.Thirdly,random sampling is performed from sub-series of W1(t),W2(t),...,W P(t)and CP(t),respectively. Finally,based on these sampled sub-series,a large number of syn-thetic daily stream?ow sequences are obtained using the wavelet reconstruction algorithm.Regarding the advantages of the devel-oped method,Wang et al.(2011b)indicated that:(1)the approach is nonparametric;(2)it is able to avoid assumptions of probability distribution types(Normal or Pearson Type III)and of dependence structure(linear or non-linear);(3)it is not sensitive to the original data length and suitable for any hydrological sequences;and(4) the generated sequences by the method could capture the depen-dence structure and statistical properties presented in the data.

The ability of a combined model,Wavelet–Generalized Regres-sion NN(WGRNN),was investigated by Kisi(2011a)for prediction of one-month-ahead stream-?ow.The WGRNN,by combining DWT and GRNN,performed better than the GRNN and feed for-ward NN models for forecasting monthly stream-?ow.Since sev-eral features of the original signal,such as its daily,monthly and annual periods,could be discerned more clearly than in the origi-nal signal,therefore,estimates were more accurate than those obtained directly by the original signals.Through a similar study, Kisi and Partal(2011)developed a forecasting model for monthly stream-?ow via NF coupled to https://www.sodocs.net/doc/754333614.html,parison results indicated that the wavelet–NF model was superior to the classical NF method,especially in detecting the peak values of stream-?ow.

Guo et al.(2011)used an SVM model improved by the addition of an adaptive insensitive factor to improve the performance of the SVM in predicting monthly stream-?ow.Considering the presence of noise in the runoff time series and its potential negative in?u-ence on model performance,a wavelet de-noising method was applied to reduce or eliminate the noise.Furthermore,given the PSO algorithm’s strong searching ability,an improved PSO was applied to optimize the parameters of the forecasting model.The improved SVM model combined with wavelet analysis was able to process a complex hydrological data series(e.g.,monthly stream-?ow)better than ANN and conventional SVM models.

In a similar study,Kisi and Cimen(2011)investigated the accu-racy of a combined wavelet and SVM model in forecasting monthly stream-?ow.They implemented their study in5steps:(i)wavelet de-noising,(ii)determination of best delay time and embedding dimension,(iii)phase-space reconstruction,(iv)model?tting,(v) stream-?ow forecasting with different models.With an ANN model serving as the basis of comparison for conventional and improved SVM models,they found that coupling with a DWT sig-ni?cantly increased the accuracy of the Support Vector Regression (SVR)model in forecasting monthly stream-?ow.

With the goal of forecasting daily discharge,Tiwari and Chatterjee(2011)explored a WBANN model,comparing its perfor-mance to that of a traditional ANN,WANN and bootstrap-based ANN.The WBANN and WANN models produced signi?cantly better results than the traditional ANN and bootstrap-based ANN models, particularly in terms of peak discharge forecasting.Similarly, Krishna et al.(2011)employed a hybrid WANN model to forecast daily river?ow,achieving good forecasting accuracy,especially with respect to peak points.

Ren et al.(2011)used the advantage of localized characteristics of wavelet transform and approximation function of an Adaptive Neuro-Fuzzy Inference System(ANFIS)in order to establish a com-bined wavelet–ANFIS(WANFIS)model for monthly runoff predic-tion.Issues arising from the large amplitude of intra-and inter-annual variation in monthly runoff were avoided through the use of a wavelet analysis-based resolving and reconstruction technique allowing the decomposition of signals with different frequencies. Based on a comparison of measured and simulated values this modeling approach produced acceptable predictions.

Tiwari et al.(2012)investigated AI-based(i.e.,NN and SOM)and wavelet-based daily river discharge forecasting models.SOM was used to homogeneously classify the data sets,while the NN models served for prediction.The NN models were supplemented by a wavelet approach,which served to enhance forecasting performance with respect to long datasets.The SOM’s effectiveness in clustering data into different groups and the superiority in forecasting river ?ow of WBANN models over simple NN models were noted.

In a recent study,Kalteh(2013)forecasted monthly river?ow by using the capabilities of AI-based(i.e.,SVR and ANN)models coupled with the wavelet transform.Coupled with a wavelet transform process,ANN and SVR models provided more accurate forecasts than non-coupled ANN or SVR models.The performance of the hybrid wavelet–SVR model exhibited greater reliability than the WANN model.

V.Nourani et al./Journal of Hydrology514(2014)358–377367

In order to accurately simulate and predict the dynamic behav-ior of river discharge over a wide range of time intervals,Wei et al. (2012)proposed a hybrid WANN method capable of reliably cap-turing the high-frequency characteristics of river discharge on a monthly time scale.As a basis for comparison,the WANN model was used to predict river discharge48months in advance.The WANN model decomposed by db5mother wavelet at level4 resulted in the most accurate river discharge predictions.

To forecast daily in?ow with lead times of1–5days, Maheswaran and Khosa(2013a)established a multi-scale non-lin-ear model based on coupling a DWT and a second-order Volterra (WVC)model,and compared its performance to that of conven-tional ANN and WANN models,as well as other baseline models. The WVC performed well in short-term?ow forecasting,especially when compared with the WANN model.This may be attributed to the ability of the former approach to provide a better scale-speci?c description of the original time series.It is noted that Maheswaran and Khosa(2012b)extended the1-month ahead stream?ow fore-casting method using the wavelet based multi-scale non-linear model linked to the second order non-linear Volterra kernel esti-mated by Kalman?lter formulation.The proposed model was com-pared with wavelet based linear regression models and other non-linear approaches such as WANN based models,and was found to provide the best performance.

Adamowski and Prokoph(2013)used the multi-scale resolution features of CWT analysis and cross wavelet analysis to determine the amplitude and timing of stream-?ow discontinuities for spe-ci?c wavebands.The cross wavelet-based method was able to detect the strength and timing of abrupt shifts to new stream-?ow levels,gaps in data records longer than the waveband of interest,as well as a sinusoidal discontinuity curve following an underlying modeled annual signal at±0.5year uncertainty.Parameter testing of the time–frequency resolution demonstrated that high temporal resolution using narrow analysis windows was favorable to high-frequency resolution for detection of waveband-related disconti-nuities.Discontinuity analysis on observed daily stream-?ow records showed that there was at least one discontinuity-year related to the annual spring?ood in each record studied,and that neighboring stream-?ows had similar discontinuity patterns.

Badrzadeh et al.(2013)investigated WANN and WANFIS models provided originally by Nourani et al.(2011),for river?ow forecast-ing.Outcomes indicated that the hybrid WANN model produced bet-ter results,especially for the peak values and longer lead-times.

Krishna(2013)explored the capability of two pre-processing techniques of wavelets and MA in combination with ANN and MLR models for prediction of daily in?ow values.The study dem-onstrated the superiority of the wavelet pre-processing technique and owing to the model performance,the wavelet–MLR was con-sidered better than the WANN model.

Danandeh Mehr et al.(2013a)applied WANN and linear GP techniques to forecast monthly stream?ow values.In contrast to the results of the majority of previous research studies,in this study,WANN model performed poorly in comparison to linear GP.An explicit linear GP model constructed by only basic arithme-tic functions including one month-lagged records of both target and upstream stations resulted in the best prediction model for the study catchment.In another similar study,Danandeh Mehr et al.(2013b)explored the prediction of monthly stream?ow at successive stations using the WANN model.

The comprehensive study of Sang(2013b)led to an improved Wavelet Modeling Framework in conjunction with AI-based black box models for precipitation and discharge time series forecasting. He developed a method for DWT decomposition of time series termed the Wavelet Modeling Framework(WMF).In this light, Sang?rstly separated different deterministic components and removed noise involved in the original time series using DWT to obtain deterministic forecasting results;then,he forecasted the former and quantitatively described noise’s random characteristics to estimate uncertainty and then summed them up to attain the ?nal forecasting result.He applied the WMF to four hydrologic cases and found that wavelet-based AI models perform more effec-tively than single AI models.

Sahay and Srivastava(2013)developed a wavelet–GA–NN model for forecasting1-day-ahead monsoon river?ows which are dif?cult to model due to the irregularly spaced spiky large events and sustained?ows of varying duration.In this regard,GA was used for optimizing the initial parameters of an ANN training scheme.The results indicated that wavelet–GA–NN model could outperform the AR and GA-optimized ANN models,which used ori-ginal stream?ow time series as inputs.

Although the majority of wavelet–AI-based models in stream-?ow forecasting used a particular set of wavelet decomposition sub-series as the‘optimal’wavelet transform to be used for fore-casting purposes,relying on a speci?c wavelet sub-series often leads to predictions that capture some phenomena at the expenses of others.However,different sub-series play different roles in cap-turing the different characteristics of a particular hydrological pro-cess.Therefore,ensemble approaches based on the use of multiple different wavelets,in conjunction with a multi-model setup,could potentially improve the modeling performance and also allow for uncertainty estimation.This was a novel idea in the wavelet–AI ?eld developed by Maheswaran et al.(2013)which involved pro-posing a new multi-wavelet based ensemble method for the wave-let Volterra coupled model.The ensemble-based multi-wavelet Volterra approach was applied for forecasting stream?ow at differ-ent scales(daily,weekly and monthly)and the outcomes revealed the superiority of the new approach in comparison to non-ensem-ble wavelet Volterra models.

An assessment of the various papers that have been reviewed in this sub-section reveals the following:

(i)Single AI-based models with short-term memory can usually

handle the AR property of the process;thus,in modeling, each value of a series can only be related to the prior values, and subsequently,the peak and maximum values of?ow which are important in water resources management and particularly in?ood mitigation are https://www.sodocs.net/doc/754333614.html,bin-ing wavelet and AI methods can help handle long term sea-sonality and reveal proper outcomes for peak?ows.It is noted that classic models such as seasonal ARIMA can also handle the long term seasonality,but the advantage of wavelet–AI models is the simultaneous consideration of sev-eral short-and long-term seasonalities in the modeling pro-cess,which may lead to better estimation of peak points.

(ii)Hydrologic time series in general,and?ow time series in particular,consist of measurement and/or dynamical noise.

In this regard,the wavelet transform is capable of de-noising the time series to improve the AI-based modeling perfor-mance,in addition to extracting dynamic and multi-scale features of the non-stationary time series.

(iii)According to Table3,among the reviewed papers,the DWT has been applied more than CWT for?ow forecasting.This can be related to the nature of?ow which is less stochastic, so,the Markovian property of?ow time series is more per-ceptible in comparison to rainfall.In this way,application of DWT at speci?c levels which refer to daily,weekly, monthly,and yearly seasonalities appears to be more useful than application of CWT which exhibits much more redun-dant seasonalities.

(iv)One of the important concerns in?ow forecasting is the selection of a proper lead time.At a daily time scale it is considered that longer lead times(e.g.,7days)for?ow

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forecasting using wavelet-Al approaches could not lead to accurate forecasting results,while1-,2-or3-day ahead fore-casting was usually more effective.

3.3.Topic3:Wavelet–AI models for rainfall–runoff modeling

For any watershed,accurate rainfall–runoff modeling is a key issue in water resource planning,as it provides vital information for?ood mitigation,the design of hydraulic structures,and overall watershed management.The highly complex,dynamic and non-linear nature of the process on both spatial and temporal scales has led to the scienti?c investigation of hybrid models.

In a preliminary study,Anctil and Tape(2004)explored the use of an ANN rainfall–runoff model combined with wavelet decompo-sition in an effort to forecast next-day stream-?ow,based on stream-?ow,rainfall,and potential evapotranspiration time series. Three wavelet decomposed components(i.e.,short,intermediate, and long wavelet periods)were used to depict the rainfall–runoff process.An ANN was then trained for each wavelet sub-series. Short wavelet periods were found to be ultimately responsible for most of the WANN hybrid forecasting error.The slight advan-tage in performance of the WANN over non wavelet-assisted mod-els might be attributed to a better usage of the evapotranspiration time series.

Later,Remesan et al.(2009)described a new hybrid model based on the Gamma Test,ANN and DWT,evaluated for daily rain-fall–runoff modeling.They identi?ed input combinations com-posed of antecedent rainfall and runoff values using Gamma Test analysis.The proposed hybrid model outperformed other popular AI models(i.e.,local LR,NNAR with exogenous input and ANFIS models),as well as basic benchmark models(i.e.,a naive model in which the predicted runoff value is equal to the latest measured value)and a trend model(in which the predicted runoff value is based on a linear extrapolation of the two previous runoff values). They observed signi?cant modeling improvement by purposely decomposing input signals into different frequency bands to be modeled separately,although it has been known for decades that hydrological catchments can act as low-pass?lters in converting high frequency rainfall signals into low frequency river?ows. The wider implication of their study in the?eld of hydrological modeling was that its general framework could be applied to other model combinations in which the model engine could consist of other AI techniques,such as SVM,NF systems,or even a conceptual model.

Mwale and Gan(2010)integrated wavelet empirical orthogonal function analysis,GA driven ANN,statistical disaggregation and hydrologic modeling into a hydrologic framework to a model weekly rainfall–runoff process.They found that the statistical properties of the hydro-climatic process in their case study are approximately stationary,and so statistically generated rainfall values may be used to predict the basin runoff with considerable skill.

In developing a WANN model to simulate?ooding on an arid ?ood plain,Wang et al.(2011a)implemented a GA in order to gain the ability to achieve a global optimum and avoid a local optimum. This hybrid GA-WANN model showed a strong capacity for rain-fall–runoff mapping and computational ef?ciency as well as being suitable for?ood simulation in arid areas.

Nourani et al.(2009b)coupled wavelets and ANN to model the rainfall–runoff process.Given the extraction via wavelets of the time series’multi-scale characteristics,the model was capable of predicting both short and long term runoffs.In a further study, Nourani et al.(2011)investigated the rainfall–runoff process using two hybrid wavelet–AI models(i.e.,WANN and WANFIS)and found that considering seasonality effects extracted through wavelet decomposition,the hybrid WANFIS model outperformed individual AI-based models.They attributed this to the strength of wavelet analysis in extracting dominant frequencies,and fuzzy analysis in handling the uncertainties involved in the relevant phenomena.

Given the complexity of rainfall–runoff relationships in moun-tainous watersheds and the lack of hydrological data in such watersheds,process-based models have a limited applicability for runoff forecasting.In light of this,Adamowski et al.(2011)pro-posed a methodology where extensive data sets were not required for runoff forecasting in mountainous watersheds;Multivariate Adaptive Regression Spline(MARS),WANN,and regular ANN mod-els were developed and compared for runoff forecasting applica-tions in a mountainous watershed with limited data.The best WANN and MARS models were found to be comparable in terms of forecasting accuracy,both providing very accurate runoff fore-casts compared to the best ANN model,particularly in the case of short-term runoff.Adamowski and Prasher(2012)employed SVR and WANN for daily runoff forecasting in a mountainous region supported by antecedent precipitation index,rainfall,and day of the year data.Both methods provided accurate results,with the best WANN model slightly outperforming the best SVR model in terms of accuracy,leading them to suggest that to further assess the suitability in forecasting runoff these methods should be tested in other mountainous watersheds where only limited data are available.

Nourani et al.(2012)investigated the linkage of wavelet analysis to GP in constructing a hybrid model to detect seasonality patterns in rainfall–runoff.The hybrid model was useful in forecasting run-off.Nourani et al.(2013)went on to con?rm the superiority of a SOM–ANN model coupled with wavelet transform in rainfall–run-off modeling using satellite data.A two-level SOM clustering tech-nique served to identify spatially homogeneous clusters of satellite precipitation data,and the most operative and effective data were selected for the ANN to model the rainfall–runoff process on daily and multi-step scales.Besides removing noise,the wavelet trans-form served to extract dynamic and multi-scale features from the non-stationary runoff time series.Spatiotemporal pre-processing of ANN model inputs led to a promising improvement in the perfor-mance of rainfall–runoff forecasting compared to ANN and simple WANN models.The forecasting outcomes indicated that the ANN forecasting model coupled with the SOM clustering method decreased the dimensionality of the input variables and conse-quently the complexity of the ANN model.On the other hand,by removing noise and revealing the dominant periods,wavelet trans-formation of runoff data increased the forecasting performance of the model,particularly with respect to peak runoff https://www.sodocs.net/doc/754333614.html,ing a wavelet transform to capture multi-scale features of the rainfall–runoff process,a SOM to classify the extracted features and select the dominant ones and an ANN to predict runoff discharge, Nourani and Parhizkar(2013)applied the resultant wavelet–SOM–ANN model for modeling the rainfall–runoff process.The two-stage procedure(i.e.,data pre-processing and model building stages)was implemented in the rainfall–runoff forecasting model. Since one of the essential steps in any ANN-based model is determi-nation of dominant input variables,independent rainfall and runoff sub-series obtained via wavelet analysis were evaluated and classi-?ed with SOM,a strong non-linear classi?er.The newly developed model led to better predictions,especially for peak points.

Nayak et al.(2013)demonstrated the potential use of WANN for daily river?ow forecasting by developing a rainfall–runoff model and compared the WANN with the single ANN and the NAM(i.e., NAM describes the behavior of each individual component in the hydrological cycle,at catchment level,using a group of intercon-nected conceptual elements)models.The WANN model performed better compared to the ANN and NAM model which includes physical elements such as moisture content in estimating the hydrograph characteristics such as the?ow duration curve.

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Kamruzzaman et al.(2013)considered a novel aspect,which exploits the relationship between stream?ow on day t and a DWT of the rainfall from day t back as far as day t–k.Then,a multi-scale transform is also included in the modeling framework as a moving DWT.Although the authors indicated that their aim was to?nd relatively few wavelet coef?cients based on rainfall back as far as day t–k that could be used as linear predictors for stream?ow on day t,the application of the moving method to the decomposed wavelet time series appears not to be a good approach,since each value of the wavelet coef?cient time series denotes a speci?c period of the process.

The review of papers in this sub-section showed that:

(i)The majority of researchers applied a daily time scale in

order to model the rainfall–runoff process.Rainfall–runoff models are used extensively in?ood studies and forecasting, and that is why it is important to investigate the rainfall–runoff process at short-term scales such as daily.Since the Markovian property of runoff is more perceptible in compar-ison to rainfall,the combination of runoff antecedents and rainfall data can appropriately produce rainfall–runoff patterns.

(ii)Moreover,comparison of both daily and monthly time scales for rainfall–runoff modeling in a watershed revealed that the determination coef?cients for peak values were more pre-cise at a monthly time scale compared to a daily time scale.

Since at the monthly scale modeling the AR characteristic of the time series is decreased by averaging the time series data over a month,the seasonal pattern is highlighted as the main characteristic of the time series which can be cap-tured by wavelet analysis in terms of sub-signals.

(iii)Rainfall–runoff model performances for various watersheds with similar climate and markedly distinct topography con-ditions are different.The properties of a?at sub-basin can be handled with simple AI models such as ANN;however,a ‘wild’watershed(i.e.,a more steep and large watershed with elevation variety)can be more accurately modeled via ANFIS.In addition to uncertainties relevant to pointy rainfall measurement and spatiotemporal variation over the study area,wild watersheds involve more uncertain and ambigu-ous hydrological characteristics.Therefore,models based on the fuzzy theory concept might lead to more reliable results than other AI models.The application of wavelets which provide dominant sub-series as inputs to the model to have insight into the physics of the process,can effec-tively decrease the undesirable effects of topographic variety of the study area.From the point of uncertainty view,the WANFIS model seems to perform more effectively than other wavelet–AI models in modeling‘wild’watersheds. (iv)One of the important issues in rainfall–runoff models is the accurate modeling of peak values in order to designs an appropriate?ood alert and management system.The wave-let-based seasonal models are more ef?cient than only AR models(i.e.,ANN and ANFIS)in monitoring peak values.It is evident that extreme or peak values in the rainfall and runoff time series,which occur in a periodic pattern,can be detected by the seasonal models accurately.When com-paring?at and‘wild’watersheds,the wild watershed shows quicker responses for a precipitation event towards a watershed with a mild slope and fairly small area,so,more instantaneous jumps may appear in the wild watershed’s time series.WANFIS can model such extreme values more accurately due to being compatible with the uncertainty involved.By employing fuzzy and wavelet concepts linked to the ANN framework,the uncertainty and seasonality of the phenomena can respectively be better handled.3.4.Topic4:Wavelet–AI models for sediment modeling

In terms of assessing sediment impacts on design and manage-ment of water resources projects,the estimation and simulation of Suspended Sediment Load(SSL)at a watershed outlet is vital to water and environmental engineers.Unlike many chemical pollu-tants,sediment is a vital natural component of water bodies;how-ever,particularly in excessive amounts,they can be of concern, either as a contaminant affecting water quality,or by interfering with the ef?cient performance of hydraulic structures such as dams.

Partal and Cigizoglu(2008)estimated the SSL in rivers via a hybrid WANN method.The dominant wavelet components obtained via DWT were summed up and served as an input for the ANN model.The WANN model provided a good?t to observed data,particularly in the case of peak values and cumulative sedi-ment loads.Rajaee(2010)compared NF,wavelet–NF(WNF), MLR,and Sediment Rating Curve(SRC)models in forecasting SSL. The observed time series of river?ow discharge and SSL were decomposed into sub-series via DWT,and the effective sub-series were added together and used as inputs to the NF model for daily SSL prediction.The results illustrated the ef?ciency of WNF model,while NF,MLR,and SRC models provided unacceptable pre-dictions.In a similar study,Rajaee et al.(2010)explored the ef?-ciency of WNF model for SSL forecasting in a larger study area with a lower discharge and SSL amount and achieved promising results in a‘wild’watershed.The observed time series of river dis-charge and SSL were decomposed by the db4wavelet and the use-ful wavelet components were summed and used in the NF model. Results showed that the WNF model performance was better in prediction compared to the NF and SRC models,particularly in extreme value prediction.Moreover,the model could be employed to simulate the hysteresis phenomenon,while the SRC method was not able to handle the involved hysteresis.

Using a coupled WANN,NF model and a conventional SRC, Mirbagheri et al.(2010)forecasted SSL.Their WNF model satisfac-torily predicted sediment loads underestimated by ANN,NF and SRC models alone.Besides being good at predicting load,the WNF model was successful in reproducing the hysteresis phenome-non.Hysteresis is a secondary relationship between sediment and river discharge values which can be detected in a scatter plot of dis-charge vs.sediment,where above a certain threshold,increasing the discharge diminishes sediment loads.The WNF model was capable of simulating this hysteresis and while its simulation yielded loads rather unlike those measured,the SRC method was unable to model this behavior,and the ANN and NF models were only somewhat able to regenerating the hysteresis effect.Overall,the WNF model, which used decomposed data to extract important characteristics embedded in the Suspended Sediment Concentration(SSC)signal, outperformed other models that employed raw data.

Applying a WANN technique for modeling the daily suspended sediment-discharge relationship,Kisi(2010)showed that the hybrid model could increase estimation accuracy.Considering WANN,MLR,and SRC models for daily SSL modeling,Rajaee et al. (2011)showed that the WANN model outperformed the other models,generated reasonable predictions for extreme sediment loads,acceptably simulated the hysteresis phenomenon,and satis-factorily estimated the cumulative SSL.

Employing GEP,NF,and ANN techniques to estimate SSL using daily river discharge and sediment load records,Shiri and Kisi (2012)showed that the GEP model outperformed the NF and ANN https://www.sodocs.net/doc/754333614.html,bining these models with DWT analysis improved all model performances while the wavelet–GEP model outperformed the wavelet–NF and WANN models.

Due to the complexity of the relationship between SSC and river discharge,Liu et al.(2013c)constructed a WANN model to predict

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next day SSC.Observed river discharge and SSC time series were decomposed into seven sub-series via the DWT using the db4 mother wavelet.Effective sub-series were selected by cross-corre-lation analysis and summed to reconstruct noise-free time series to serve as ANN inputs for SSC prediction.The WANN model was bet-ter able to predict the highly non-linear and non-stationary SSC time series than ANN or SRC models.Noise removal using the WANN approach dramatically improved the?t of the predicted SSC time series to the observations.Additionally,error autocorrela-tion and the correlation between input and error time series in the WANN model showed it to be more robust than either the SRC or ANN models.

Nourani et al.(2014)developed an ANN-based stream-?ow-sediment model by focusing on a wavelet-based global soft thres-holding method to de-noise hydrological time series at the daily scale.Since the appropriate selection of the decomposition level and mother wavelet type are important in thresholding results, sensitivity analysis was performed over different levels and several Daubechies type mother wavelets(Haar,db2,db3,db4and db5)to choose the proper variables.De-noised time series were applied to the ANN model to forecast?ow discharge and sediment values.The results indicated that,the wavelet-based de-noising approach,as a pre-processing method,could improve the ANN-based stream-?ow-sediment forecasting models;in addition,the wavelet de-noising was signi?cantly dependent on the chosen mother wavelet whereas forecasting results varied with the alteration of mother wavelets.

According to the reviewed papers regarding sediment model-ing,one of the important issues in sediment modeling is the hys-teresis phenomenon in which the SSL depends not only on the water discharge amount and?ow capability,but also on the load availability,which is complexly related to the season or month of occurrence.Thus,the application of a solely AR model such as var-ious AI methods(e.g.,ANN,ANFIS)that relates discharge and SSL to their antecedents is not suf?cient in the presence of factors such as hysteresis.In this regard,the application of the wavelet transform which considers the seasonality of the process in order to handle hysteresis is advantageous.

3.5.Topic5:Wavelet–AI models for groundwater modeling

In many watersheds,groundwater is often one of the major sources of water supply for domestic,agricultural and industrial users.In many such regions,groundwater has been withdrawn at rates far in excess of recharge,leading to harmful environmental side effects such as major water-level declines,drying up of wells, reduction of water in streams and lakes,water-quality degrada-tion,increased pumping costs,land subsidence,and decreased well yields(Adamowski and Chan,2011).To effectively manage groundwater,the ability to predict Groundwater Level(GWL)?uc-tuations and quantify environmental threats(e.g.,contamination, salinization)and their potential to expand are key hydrological issues.

Adamowski and Chan(2011)developed a one month-ahead GWL forecasting model using a coupled DWT–ANN method.The DWT decomposed each original data series into information bear-ing component series,which then served as inputs to the ANN-based forecasting portion of the model.The DWT allowed most of the‘noisy’data to be removed and facilitated the extraction of quasi-periodic and periodic signals from the original time series.

Maheswaran and Khosa(2013b)compared the GWL forecasting abilities of three hybrid wavelet models;WVC,WANN,and wave-let–LR as well as ANN and dynamic AR https://www.sodocs.net/doc/754333614.html,pared to the wavelet–LR and WANN models,the WVC model performed better in forecasting GWL characterized by non-linearity and non-stationarity.With an increase in lead time,the wavelet based models performed progressively better than the regular models.Overall,accurate long term GWL forecasting was best provided using the WVC model.

Investigating the ability of a joint wavelet and NF model to per-form one-,two-and three-day-ahead groundwater depth forecast-ing,Kisi and Shiri(2012)found that the joint model outperformed the NF model,particularly for two-and three-day-ahead forecasts.

Moosavi et al.(2013a)compared several data-driven models (i.e.,ANN,ANFIS,WANN and WANFIS models)for forecasting GWL at a monthly scale.The comparison of results demonstrated that the WANFIS model outperformed the other models since it could handle both uncertainty and seasonality involved in the process.

The low number of papers on groundwater modeling via wave-let–AI demonstrates the need to consider groundwater and rele-vant issues.Meanwhile,it can be inferred that the monthly time scale or any long-term scale is the appropriate scale for modeling groundwater,since such scales coincide with the nature of the process,notwithstanding the study of Kisi and Shiri(2012)who performed daily groundwater modeling.

Through a comparative study,Moosavi et al.(2013b)investi-gated the optimum structures of WANN and WANFIS models for GWL forecasting.Their research revealed that transfer functions of ANN and membership function types of ANFIS besides the mother wavelet type are the most important factors in the perfor-mance of WANN and WANFIS models,https://www.sodocs.net/doc/754333614.html,parison of optimal WANN and WANFIS demonstrated the better performance of WANFIS.

Since groundwater is recharged by?ow or any precipitation that seeps into the ground,the periodic characteristic of ground-water is relevant to rainfall and runoff processes as well as climatic parameters.Therefore,it is suggested that wavelet-AI methods be explored in order to determine the lags,correlation and interaction between climatic parameters and GWL as well as groundwater quality factors.On the other hand,because groundwater is suscep-tible to pollutants which may follow a periodic pattern to soak into the underground,the wavelet–AI approach can simulate and extract effective features and patterns among GWL,climatic parameters and contaminants.

3.6.Topic6:Other hydro-climatologic applications of wavelet–AI models

Besides the detailed investigations of wavelet–AI model appli-cations to forecast various hydrological processes,they have been also successfully applied to model other hydro-climatologic pro-cesses(i.e.,shallow watertable depths,drought,snowmelt,evapo-transpiration,etc.)

Wang and Ding(2003)proposed hybrid WANN models to pre-dict monthly mean water table depths and daily discharge.In terms of prediction accuracy,the hybrid model outperformed ARMA and threshold AR models,trained with the GA optimization technique.The results implied that when the forecasting horizon was extended the?tting and testing precision of the hybrid model outstripped that of the other models.

Kim and Valdes(2003)linked dyadic wavelet transforms and NNs to generate a WANN model capable of forecasting the Palmer drought severity index at various lead times.In order to reduce the inconsistency of the sub-signal,a wavelet transform based on the dyadic algorithm was used.They concluded that the hybrid approach enhanced the ability of NNs to forecast the indexed regional drought.Moreover,based on several accuracy statistics, the forecasting skill of the hybrid model for lead times up to 6months was much better(4–60%)compared to the other statisti-cal prediction methods.Following the previous study,Belayneh and Adamowski(2013)and Belayneh et al.(2014)investigated

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the ability of data driven models such as ARIMA,ANN,and SVR and WTs to forecast long-term(6and12months lead times)drought. Belayneh et al.(2014)proposed the wavelet–SVR model in addi-tion to WANN model to forecast long term drought.They applied Standard Precipitation Index(SPI)12and24as indicators of long-term drought conditions.The forecast results for WANN and wavelet–SVR models were improved compared to models without any wavelet based pre-processing,and the WANN model was bet-ter than the WSVR model.

This study indicated that the approximation time series compo-nent in wavelet decomposition is the most effective component in forecasting SPI time series,which adequately de-noises the data and avoids any discontinuities within the SPI time series.

Another study which modeled drought was conducted by Shirmohammadi et al.(2013).The research was carried out to eval-uate the ability of WANN and ANFIS techniques for meteorological drought forecasting at one,two,and three time steps(6months) ahead.WANFIS was found to be more accurate than the WANN model.

Given the importance and advantage of considering soil mois-ture information in a variety of hydrologic models,Lauzon et al. (2004)proposed the analysis of soil moisture conditions based on wavelet analysis and SOM through Kohonen NNs.Through these techniques,the in?uence of soil moisture on the hydrologic regime could be assessed and relevant information could be extracted for the development of a stream-?ow model.Based on results inferred from wavelet analysis,soil moisture supported the annual cycle in observed?ows.The links between precipitation events,the short-term behavior of soil moisture and the in?ow regime could be clearly seen through wavelet analysis.A compre-hensive description of the soil moisture pro?le,its evolution over time,and its relation to precipitation,temperature and?ow obser-vations were performed via wavelet analysis and SOM.

Since seasonal drought usually originates from low availability of soil moisture,Deng et al.(2011)predicted the dynamic changes of soil water in the?eld via daily soil water content simulated by least squares SVM with meteorological factors.Wavelet-based de-noising was applied to pre-process the original chaotic soil water signal and the results of the prediction showed improve-ment of the model in comparison to ANN and ANFIS models.

Adamowski(2008b)proposed a method of stand-alone short-term spring snowmelt river?ood forecasting based on wavelet and cross-wavelet analysis.The accuracy in forecasting daily stream-?ows with lead-times of1,2,and6days of the new wave-let forecasting method was compared to that of MLR analysis, ARIMA analysis,and ANN.The wavelet-based forecasting method was shown to accurately forecast river?ooding for1and2-day lead-times,but was not particularly accurate for longer lead-time forecasts(e.g.,6days).

Accurate prediction of solid waste generation is an important issue in the planning and design of municipal water puri?cation systems.Noori et al.(2009)applied hybrid WANFIS and WANN models to predict the weekly waste generation from a municipal solid waste management system.Input data pre-processing via wavelet analysis clearly improved prediction accuracy.Of the two models tested,the WANFIS model,by reason of its effective handling of uncertainties involved in the process,exhibited a bet-ter performance than the WANN model.

Evapotranspiration is a complex process affected by a variety of climatologic factors.Hybrid wavelet–NN models provide an alter-native way of exploring the underlying mechanisms of evapotrans-piration.In consideration of this,Partal(2009b)tested the ability of a WANN model in estimating reference evapotranspiration.Apply-ing wavelet analysis to raw data of climatic data(i.e.,air tempera-ture,solar radiation,wind speed,relative humidity)as a pre-processing approach allowed the ANN model to equal or outperform a MLR and the empirical Hargreaves method in daily evapotranspiration forecasting.This con?rmed that the hybrid WANN method could be successfully applied to model reference evapotranspiration based on climatic data.

In another study,the use of wavelet analysis in conjunction with AI was employed to predict daily evaporation(Abghari et al.,2012).Mexican Hat and poly WOG1mother wavelet activa-tion functions were used in an ANN instead of the commonly used Sigmoid function,and differences in terms of daily pan evaporation predictions were noted.In terms of the accuracy of daily pan evap-oration forecasts,the WANN model outperformed any single ANN model.

Applying a hybrid WANN model to1-and6-month-ahead fore-casting of mean monthly lake levels,Kisi(2009b)found that WANN signi?cantly increased the short-and long-term forecast accuracy over wavelet-free models.

Campisi et al.(2012)explored the problem of forecasting urban water demand by means of a back-propagation ANN coupled with a wavelet de-noising technique.The forecasting horizon varied from1to6months and the impact of?ve different wavelet?l-ter-banks on ANN outcomes was explored.ANNs coupled with Haar and db2and db3?lter-banks outperformed non-coupled ANN,MLR and ANN models coupled with db4and db5?lters.Over-all,they found that the de-noising impact gained via wavelet-attributable reduction in training set variance could improve fore-casting accuracy;however an oversimpli?cation of the input-matrix could lead to a deterioration in the forecasting accuracy and induce network instability.

Short term(1,3,and5days;1and2weeks;and1and 2months)urban water demand forecasting was also explored by Tiwari and Adamowski(2013)via a WBANN model.The results demonstrated that the hybrid WBANN and WANN models produce signi?cantly more accurate forecasting results than the traditional NN,BNN,ARIMA,and ARIMAX models.It was also found that the WBANN model reduces the uncertainty associated with the fore-casts,and the performance of WBANN forecasted con?dence bands were found to be more accurate and reliable than BNN forecasted con?dence bands.

For coastal and ocean engineering applications,Ozger(2010) employed a combination of wavelet and fuzzy logic approaches to forecast wave heights and periods with lead times up to48h.

A wavelet technique was used to separate time series into spectral bands,which were subsequently estimated individually through a fuzzy logic approach.The hybrid wavelet-fuzzy logic model outperformed the single fuzzy logic,ANN and ARMA models.The superiority of the wavelet-fuzzy logic model in terms of model per-formance was particularly notable for longer lead times(e.g.,48h).

Kisi(2011b)compared the performance of a Wavelet Regres-sion(WR)technique with ANN models for daily river-stage fore-casting.In order to create the forecasting models,two different WR models were developed using the stage sub-time series.The sum of effective decomposed details and the approximation com-ponents were used as inputs to the WR1model,while in the WR2model,the effective details and the approximation compo-nents were used as separate inputs.Under these circumstances, the WR models outperformed the single ANN models,and the WR2model outperformed the WR1model.

Adamowski et al.(2012)proposed an urban water demand fore-casting method based on coupling a DWT and ANN for a lead time of one day over the summer months(May–August).The key variables used to develop and validate the models were daily total precipita-tion,daily maximum temperature,and daily water demand data. The WANN model was found to provide more accurate urban water demand forecasts than the MLR,MNLR,ARIMA or ANN models.

Deka and Prahlada(2012)employed a WANN model to forecast the occurrence of waves of signi?cant height reaching the west

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coast of India for lead times up to48h.A WANN drawing on a multi-resolution time series for input data to its ANN component provided more accurate forecasts than a single ANN.

Land cover assessment as a hydro-climatologic related?eld has been considered in various hydrological studies.In this light,a wavelet feature based supervised scheme for fuzzy classi?cation of land cover multispectral remote sensing images was proposed by Shankar et al.(2011).In a distinct application of wavelet trans-forms,the obtained wavelet features from land cover images pro-vided important information about the spatial and spectral characteristics of image pixels and hence could be used in fuzzy based land cover classi?cation.The classi?cation results compared to original spectral feature based methods demonstrated the high ef?ciency of wavelet-based fuzzy classi?cation of land cover.

A wavelet-based NN model was developed by Shekarrizfard et al.(2012)in order to model the relationship between PM10(a major air pollutant)levels and meteorological data including sev-eral parameters,such as wind speed,wind direction,moisture, and temperature.Due to the reduction in the noise inherent in most meteorological data by means of the wavelet transform,the wavelet-enhanced ANN generated accurate predictions of PM10 levels,compared to a wavelet-free ANN method.They concluded that the proposed WANN model was generally an effective method for PM10level prediction.In a similar parameter prediction,Siwek and Osowski(2012)used several AI methods(i.e.,MLP,radial basis function,Elman network and SVM)as well as a linear AR model in conjunction with DWT to forecast the daily average con-centration of PM10.Application of wavelets and an ensemble of many individual prediction results led to an accurate method of prediction.

Karran et al.(2013)conducted an exploration of how well wavelet–AI models perform in different climate regimes with dif-fering hydrological characteristics and studied the performance of such models for lead times of less than one month.Their study compared the use of ANNs,SVR,WANN,and wavelet–SVR in Med-iterranean,Oceanic,and Hemiboreal watersheds.The results indi-cated that SVR based models overall performed well,but no one model outperformed the others in more than one watershed,sug-gesting that some models may be more suitable for certain types of data.Overall,model performance varied greatly between climate regimes and they suggested that higher persistence and slower hydrological processes(i.e.,snowmelt,glacial runoff,and subsur-face?ow)support reliable forecasting in daily and multi-day lead times.

Water quality modeling via hybrid wavelet–AI models has not been explored in much detail in the literature.In Najah et al. (2012),monthly water quality parameters of a river were predicted utilizing an ANFIS model.Since the observational water quality data might be polluted by noise owing to systematic and random errors,the wavelet de-noising technique in conjunction with the ANFIS model was applied.

Overall,based on the reviewed studies,the application of wave-lets as a pre-processing technique,usually improves modeling per-formance after decomposition of the main signal to seasonal sub-series at different scales via one of three scenarios:

(i)Use of all decomposed time series as inputs of the AI model.

(ii)Use of only the dominant sub-series as inputs of the AI model.

(iii)Use of the original signal as the input of the AI model,recon-structed by using only selected dominant sub-series.

Comparison of the reported results in the reviewed papers dem-onstrates that the application of the second scenario due to the sim-plicity of the structure and reduction of redundant and non-relevant data as well as accurate performance,may lead to more accurate results in hydro-climatologic applications of wavelet–AI methods.

4.Summary and conclusions

Since the emergence of AI techniques in hydro-climatology, research activity in the?eld of modeling,analyzing,forecasting and prediction of water quantity and quality variables has increased dramatically.Wavelet–AI applications have increased in modeling various hydrological processes such as rainfall–runoff, stream-?ow,precipitation,sediment,groundwater and others. Among the processes involved in the hydrologic cycle,extensive research has been conducted on stream-?ow modeling,with fewer papers focused on other processes of the hydrologic cycle,and even fewer have focused on water quality and water resources manage-ment issues.

Given,on the one hand,the capacity and robustness of AI mod-els in coping with the non-linear and dynamic nature of hydrologic processes,and on the other hand the ability of wavelet analysis to extract the prominent periodicities and seasonalities from a time series,a greater understanding and ability to predict various hydrological processes can be achieved.The results of many of the studies explored in this review paper have revealed the relative ef?ciency of wavelet–AI models compared with other methods in accurately forecasting hydrological variables.These improvements in hydrological forecasting can lead to a better interpretation of phenomena and inform the development of appropriate water and environmental resource planning and management policies.

In the current review paper several papers were investigated and compared that used wavelet–AI based models with great oper-ation or forecasting ability in order to model several hydro-climate processes.One of the more important issues was exploring which AI method can best?t the speci?c hydro-climate process.A lack of attention regarding this issue has led to the use of various AI meth-ods without any consideration as to the appropriateness of the model.As an example(see Table3),among34papers that modeled ?ow,25of them solely used ANN,as the appropriate AI method. The other9papers used GP,SVM,SVR and ANFIS.Based on the review conducted in this study,it appears that for applications with high levels of uncertainty,the ANFIS approach can provide better results.Rainfall–runoff modeling in a very large watershed with sparse sampling gauges,or sediment amount prediction in various watersheds with different beds,are examples of modeling in situations of high uncertainty where the ANFIS approach might be useful to explore.

It was also found that,wavelet-based seasonal models are more ef?cient than AR models(i.e.,ANN and ANFIS)in monitoring peak values.It is evident that extreme or peak values in the rainfall,run-off or sediment time series,which occur in periodic patterns,can be detected by the seasonal models more accurately.For short term real time forecasting or for modeling at a?ne time resolution (e.g.,hourly,daily),an AR model or WANN model with low decom-position levels which uses current and only a few previous state values of the process as inputs is likely to be the most useful model.But for long term,seasonal forecasting or modeling in monthly or seasonal time scales,a hybrid wavelet–AI model which decomposes the time series at high levels can detect the long term memory of the process.Furthermore,the study parameter has a signi?cant role in the selection of a reliable modeling tool.For example,for modeling a highly stochastic process(e.g.,event-based precipitation)probabilistic-based pre-processing(e.g.,boot-strapping-based simulation)could also be helpful.

Maier and Dandy(2000)identi?ed the adoption of appropriate input determination approaches,as one of the main concerns in hydro-climatologic models.Moreover,Maier et al.(2010),in a recent

V.Nourani et al./Journal of Hydrology514(2014)358–377373

review on ANN methods indicated two de?ciencies of ANN-based hydrologic modeling;?rstly,evaluating the relationship between input variables with the model output,secondly,investigating input independence and avoiding redundant inputs even if they help the performance of an ANN model,since they might increase model complexity and parameter uncertainty.If these two issues are addressed in other AI-based models in addition to ANN,increased attention should be devoted to reduce the uncertainty surrounding model outputs and to enable AI-based models to extract more con?dent knowledge from the data.In this way,wavelet-based pre-processing as well as various AI-based optimization techniques are capable of addressing the aforementioned two issues by:

(i)Wavelet transform breaks down the signal into periods

involved in the process,then via an AI-based optimization technique(e.g.,GA)the potential sub-signals having a signif-icant relationship with the model output can be determined.

Thus,the use of a wavelet–AI approach not only tackles non-linear model input selection,but also provides pre-process-ing on each input signal via wavelet analysis.

(ii)Application of the wavelet transform leads to identi?cation of various periods as sub-signals,subsequently,the selection of dominant sub-signals(as inputs to the AI models)having an insight into model scale or entity,prevents the interfer-ence of redundant information and reduces the uncertainty surrounding AI-based model output.

The future of predicting hydrological processes through wave-let–AI approaches can be anticipated if one looks at how the?eld evolved over the past decade.Firstly,in almost all hydrologic sig-nals,the underlying complex non-linear seasonalities and relation-ships have been extracted through the implementation of the wavelet transform.Secondly,according to the task at hand(e.g., forecasting,optimization,or classi?cation),one or another AI-based technique can be applied in order to ful?ll the purpose of modeling.In general,it can be concluded that recently imple-mented wavelet-based models have principally focused on improving the accuracy of hydrologic process modeling.Through the application of a hybrid wavelet–AI model to improve modeling performance,the classic concerns of data-driven modeling(i.e., well established data division to have the requisite training and validation sub-sets,optimal network structure of the AI technique, etc.)should be regarded to acquire a precise hybrid model.

According to Table3,the dominant?eld of application of hybrid wavelet–AI models in hydrological studies is forecasting and pre-diction.Stream-?ow forecasting via wavelet–AI models has been the focus of several studies,whereas their use in predicting water quality parameters has attracted less attention.There is therefore a need to broaden the range of application of wavelet–AI models to focus on other predictive variables,especially those concerned with water quality.However,one factor limiting the application of wavelet–AI models in water quality modeling could be the lack of good quality,long-term data to detect the long-term seasonality signature of the process.

Among the reviewed papers,only about20%of studies used the CWT for decomposing hydrological time series,and the majority of studies utilized the DWT.This is because real world observed hydro-logic time series are measured and gathered in discrete form rather in a continuous format.So,the dyadic DWT is more suitable for decom-position of time series into trend and detail sub-signals comprising high frequencies and fast events,and also to reconstruct the original time series from sub-signals.Each resolution level in DWT represents a dyadic period based on the scale of data.Considering a set of daily data,DWT decomposition leads to2n-day mode resolutions (e.g.,21-day mode,22-day mode,23-day mode which is nearly weekly mode,24-day mode,25-day mode which is nearly monthly mode,and ...,26-day mode which is nearly yearly mode,etc.)which approxi-mately denote the periodicity of a hydrologic process.Although23 day mode and25-day mode are fairly accurate for the weekly and monthly periods,28-day mode represents the yearly periodicity with a30%error.Therefore,it is unlikely that DWT can represent the yearly periodicity as well as CWT which is able to depict exact periods.On the other hand,although CWT provides a time–frequency representation of a signal at many different and exact periods in the time domain, redundant information is locked up within the coef?cients,which may or may not be a desirable property.Thus,it is inferred that accord-ing to the considered hydrologic time series and its scale,the DWT or CWT should be selected and applied.

In the majority of reviewed papers,the Nash-Sutcliff evaluation criterion(Nash and Sutcliffe,1970)was applied in addition to other ef?ciency criteria(e.g.,Mean Absolute Error;MAE,Root Mean Square Error;RMSE)to evaluate the model performance.According to Legates and McCabe(1999)a good evaluation of model performance should include at least one‘goodness-of-?t’or relative error mea-sure(e.g.,Nash-Sutcliff criterion)and at least one absolute error measure(e.g.,RMSE or MAE),thus,a hydrological model can be suf-?ciently evaluated by Nash-Sutcliff and RMSE but due to the impor-tance of peak and extreme values in hydrologic processes other criteria(e.g.,Ratio of Absolute Error of Peak value)may also be applied.For the design of a disaster alert system(e.g.,?ood alert sys-tem),in addition to the measure of peak value error(error between observed and computed peak values),the occurrence time of such extreme conditions should also be regarded.In this regard, Dawson et al.(2007)developed a scope to evaluate metrics for the standardized assessment of hydrological forecasts.

In spite of the black box nature of AI methods,the use of wave-let analysis with AI methods makes it possible to provide some insight into the physics of the process in both time and space. For example,due to urbanization and land use/cover changes,a watershed’s lag time and response to the inputs(e.g.,rainfall) may be changed and in turn,the calibrated parameters of the employed wavelet–AI model(e.g.,decomposition level,mother wavelet,input lags,etc.)can change.To monitor such changes, the time series of the studied process should be split into speci?c sub-series and by comparing the calibrated parameters of the model by sub-series,the trend in land cover/use can be detected.

A similar methodology can be used to detect spatial changes of land uses by dividing the watershed into sub-basins.

5.Recommendations for future research

Based on the review of almost105papers regarding applica-tions of wavelet–AI methods in hydro-climatology,the following recommendations for future research are provided:

1.Given the discrete nature of hydrologic time series,applications

of the wavelet transform in hydrology mainly concentrate on the use of DWT.A broader use of CWT is suggested in order to exploit its properties over all time scales,such as with dyadic scales in DWT.

2.Considering the importance of the wavelet transform for tem-

poral pre-processing,it is a useful tool in extracting the under-lying features and de-noising time series.While such applications have been explored in hydrological modeling in recent years,it could be useful to explore the use of the wavelet transform for pre-processing of spatial data(e.g.,digital eleva-tion model)employed in hydrological models.

374V.Nourani et al./Journal of Hydrology514(2014)358–377

3.The main concerns regarding wavelet-based models are the

appropriate selection of the mother wavelet and decomposition level according to the hydrological process and the scale of the process.While Nourani et al.(2011,2013)stated that similarity in shape between the mother wavelet and that of the time ser-ies under study is the best guideline in choosing the proper mother wavelet,it would be useful to investigate similarities from another point of view than form(e.g.,energy).Although the selection of mother wavelet and decomposition level have been studied(Nourani et al.,2011;Sang,2012),a more thor-ough investigation could lead to the selection of a speci?c mother wavelet according to the nature of the hydrologic pro-cess investigated,and the use of an algorithm based on histor-ical data length to select the decomposition level.

4.Due to the low number of papers in the?eld of groundwater

and water quality modeling via wavelet–AI models,it is sug-gested that additional research be conducted on this topic.

5.In addition to the suggestion of Abrahart et al.(2012)to create

benchmark data sets,it would be useful to develop an archive of appropriate wavelet–AI models for speci?c hydro-climatologic processes,with transparency in the application of different types of wavelet transforms(i.e.,DWT,CWT),ef?cient mother wavelet type and?nally appropriate AI techniques for each hydro-climatologic process at a desired time scale.

6.In addition to the ability of wavelet–AI models for black box

modeling of hydrological processes,they can also be linked to physically-based models(e.g.,TOPMODEL;Beven and Kirkby,1979)to develop integrated modular models.For this purpose,the geomorphologic characteristics of the study area at a sub-grid scale can be extracted and represented via geo-graphic information system tools,pre-processed by wavelets, and then used in the model to estimate the spatiotemporal variability of parameters(e.g.,soil moisture,GWL,recharge, transmissivity).In a similar way,other AI methods such as SOM-based spatial clustering of grids into homogeneous zones can also be used.

7.It would also be useful to prepare other review papers to survey

hydro-climatologic applications of different hybrid models con-structed via the conjunction of AI models with other commonly used data pre/post-processing techniques.

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2. 在端口视图下配置源端口 表1-4 在端口视图下配置源端口 一个镜像组内可以配置多个源端口。 5 1.2.4 配置源CPU 表1-5 配置源CPU 一个镜像组内可以配置多个源CPU。 6 1.2.5 配置目的端口 可以在系统视图下为指定镜像组配置目的端口,也可以在端口视图下将当前端口配置为指定镜像组的目的端口,二者的配置效果相同。

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diffserv domain default # drop-profile default # aaa authentication-scheme default authorization-scheme default accounting-scheme default domain default domain default_admin local-user admin password simple admin local-user admin service-type http # interface Vlanif1 # interface MEth0/0/1 # interface GigabitEthernet0/0/1 port link-type trunk port trunk allow-pass vlan 10 20 # interface GigabitEthernet0/0/2 port link-type trunk port trunk allow-pass vlan 10 20 # interface GigabitEthernet0/0/3 port link-type access port default vlan 10 stp disable # interface GigabitEthernet0/0/4 port link-type access port default vlan 20 stp disable # interface GigabitEthernet0/0/5 # interface GigabitEthernet0/0/6 # interface GigabitEthernet0/0/7 # interface GigabitEthernet0/0/8 # interface GigabitEthernet0/0/9

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