Elsevier

Journal of Hydrology

Volume 514, 6 June 2014, Pages 358-377
Journal of Hydrology

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

https://doi.org/10.1016/j.jhydrol.2014.03.057Get rights and content

Highlights

  • The paper reviews applications of hybrid wavelet–AI models in hydro-climatology.

  • Efficiency of hybrid models regarding processes and model type were investigated.

  • Survey shows wavelet pre-processor capability to enhance AI models performance.

  • Organized information about wavelet–AI models can show the future research pass.

Summary

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 phenomena 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 Artificial Intelligence (AI)-based models have been amongst the most promising in simulating hydrologic processes. The present review focuses on defining 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.

Introduction

Characterized by high complexity, dynamism and non-stationarity, 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 significant rise in the number of scientific 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 concurrent input and output time series (Solomatine and Ostfeld, 2008). Such models can be defined 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-driven models are rating curves, the unit hydrograph method and various 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 Multiple Linear Regression (MLR) are linear models and assume stationarity 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 Artificial Intelligence (AI) have shown promise in modeling and forecasting non-linear hydrological 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-, classification-, 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 fields:

  • (1)

    Evolutionary computation: A branch of optimization methods that includes swarm intelligence algorithms such as Ant Colony 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)

    Classifiers and statistical learning methods: These models employ statistical and machine-learning approaches. The most widely used classifiers are Neural Networks (NNs; Haykin, 1994), kernel methods such as the Support Vector Machine (SVM; Vapnik, 1995), k-nearest neighbor algorithms such as Self-Organizing Map (SOM; Kohonen, 1997), Gaussian mixture model, naive Bayes classifier, 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 fields of hydrology. Since their emergence in hydrology, the efficient performance of AI techniques such as data-driven models has been reported over a wide range of hydrological processes (e.g., precipitation, stream-flow, rainfall–runoff, sediment load, groundwater, drought, snowmelt, evapotranspiration, water quality, etc.). The number of researchers active in this area has increased significantly over the last decade, as has the number of publications. Several dozen successful applications for hydrological process modeling (e.g., stream-flow, rainfall–runoff, sediment, groundwater, water quality) using ANN, Fuzzy, GP, GA, and SVM have been reported, with some examples listed in Table 1.

Despite the flexibility 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 efficiently describe observed data in statistical terms, as well as other prior information, 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 reliably 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 transform is its ability to provide a time-scale localization of a process. The wavelet transform has attracted significant attention since its theoretical development in 1984 (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, Kisi, 2009b, Kisi, 2010, Nourani et al., 2009a, Nourani et al., 2009b, Nourani et al., 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, reanalysis, global climate model simulations, etc.). Providing explicit information in a readable form, it can be used to solve diagnostic, classification or forecasting problems. In a review of the applications 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, identification of seasonalities and trends, and data de-noising. Therefore, the ability of the wavelet transform to decompose non-stationary signals into sub-signals at different temporal scales (levels) is helpful in better interpreting hydrological processes (Adamowski, 2008a, Adamowski, 2008b, 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 combined 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, remediation of active Artificial NN (ANN) functions, as well as hydrological 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 effective 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 signal at different periods in the time domain, as well as considerable 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.

The selection of an efficient mother wavelet and decomposition level are two important issues in the first step. Appropriate selection of the mother wavelet constitutes the most important decision associated with the first step; both in the case of discrete and continuous wavelet transforms (DWT and CWT, respectively). CWT and DWT construct a time–frequency representation in the form of a continuous or discrete signal, respectively. Detailed analyses regarding the performance of different mother wavelets in hydrological simulations have led to the conclusion that to determine 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 Daubechies-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 dyadic set of scales, it is compatible with the discrete observation of hydrological signals. In order to study the signal, discretisation comes first, 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 first 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 the minimum level of decomposition, L, to the number of data points within the time series Ns, was introduced in the literature (Aussem et al., 1998, Nourani et al., 2009b, Wang and Ding, 2003):L=int[logNs]

Later, Nourani et al. (2011) criticized the outcome of this formula, stating that, having been derived for fully autoregressive (AR) signals, it only considers time series length, without paying any attention to seasonal effects. Since many seasonal characteristics 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 decomposition level for dyadic DWT analysis. Decomposition level l contains l details and as an example in the case of daily modeling denotes 2n-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 wavelet–AI model. In this context, AI methods can be seen to fall into three basic categories: optimization, logic, classification and statistical learning; based on the utilization of AI over one of these three fields, 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, Kuo et al., 2010b, Wang et al., 2011a). Feature extraction and classification of dominant inputs to be used in forecasting (Hsu and Li, 2010, Nourani et al., 2013, Nourani et al., 2014) along with seasonality detection (Nourani and Parhizkar, 2013, Nourani et al., 2009a, Nourani et al., 2009b, Nourani et al., 2011, Nourani et al., 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 field 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 Table 2) concerning particular sub-sets of AI models used in hydrology or specifically 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, Schaefli et al., 2007, Sang, 2013a) have surveyed wavelet analysis methods (see Table 2), no reviews have centered on the specific use of wavelet–AI models. Maier et al. (2010), in their review paper on methods used in developing 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 field of research would seem timely.”

The lack of review papers evaluating the simultaneous application 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 fields of hydrology. The advances in hydrological modeling and simulation 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 since 2003 (Fig. 2). While such publications remained low from 2003 until 2007, there was a 10-fold increase over the next two years, which represents a turning point in wavelet–AI research. Articles up to 2007 played an innovator role, with the paper of Labat et al. in 2004 representing the pioneering work of wavelet applications to hydrology (with 152 citations in Scopus) and Labat’s review on the wavelet concept (with 114 citations) providing further incentive to research the application of wavelet–AI systems in hydrological modeling (Labat, 2005). In 2006, Partal and Küçük (with 44 citations) demonstrated the merits of wavelet trend analysis in determining possible trends in annual total precipitation series, while the work of Cannas et al. (with 38 citations) further developed the hybrid wavelet–AI model. Since 2007, 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 comprehensively categorize wavelet–AI models and enumerate their novel applications in hydrology along with their benefits. In turn, this assessment will provide some ideas on future areas of research in the field. 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-flow, runoff, (iii) rainfall–runoff, (iv) sediment, (v) groundwater, (vi) miscellaneous: 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 (www.scopus.com). Conference proceedings are not included in this review. Details of the selected papers, including year of publication, authors, AI methods used and variables predicted are given in Table 3. This is followed by sections on the basic concepts of the wavelet transform (Section 2), and the applications of hybrid models in various fields of hydrology (Section 3). A summary and suggestions for future avenues of research are presented in the last sections of the paper.

Section snippets

Wavelet transform

The wavelet transform has increased in usage and popularity in recent years since its inception in the early 1980s, yet it is still not as widely used as the Fourier transform. However, Fourier analysis has a significant 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

Hydro-climatologic applications of wavelet–AI models

This review is a complement to recent surveys such as Maier et al., 2010, 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 applications of not only the ANN technique but also other data-driven AI techniques (e.g., SOM, Fuzzy logic, GA, GP, SVM, etc.)

Summary and conclusions

Since the emergence of AI techniques in hydro-climatology, research activity in the field 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-flow, precipitation, sediment, groundwater and others. Among the processes involved in the hydrologic cycle, extensive research has been conducted on stream-flow modeling, with

Recommendations for future research

Based on the review of almost 105 papers regarding applications 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

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