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New hybrid GR6J-wavelet-based genetic algorithm-artificial neural network (GR6J-WGANN) conceptual-data-driven model approaches for daily rainfall–runoff modelling

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Abstract

Rainfall–runoff modeling is significant for efficient water resources management and planning. The hydrological conceptual models can have challenges, such as dealing with nonlinearity and needing more data, whereas data-driven models are generally lacking in reflecting the physical process in the basin. Accordingly, two-hybrid model structures, namely Génie Rural à 6 paramètres Journalier (GR6J)-wavelet-based-genetic algorithm-artificial neural network1 (GR6J-WGANN1) and GR6J-wavelet-based genetic algorithm-artificial neural network2 (GR6J-WGANN2) models, were proposed in this study to develop rainfall–runoff modeling performance. The novel GR6J-WGANN1 model used the routing store outflow (QR), exponential store outflow (QRexp), and direct flow (QD) obtained from the GR6J, and the GR6J-WGANN2 model used the soil moisture index (SMI) obtained from the GR6J as input data. The wavelet transformation and Boruta algorithm were implemented to decompose the input data into components and select important wavelet components, respectively. The performance of the GR6J, standalone WGANN models, and hybrid models were tested in three sub-basins of Konya Closed Basin, Turkey, which generally has arid and changing climate conditions. The hybrid models performed better than the conceptual and data-driven models, particularly regarding the extreme flow predictions. Using soil moisture index, routing store outflow, exponential store outflow, and direct flow as the output of the GR6J in GR6J-WGANN1 and GR6J-WGANN2 improved the rainfall–runoff modeling performance remarkably. The findings of this study indicated that hybrid models, which integrate strong sides of conceptual and data-driven models, can be more useful for producing more accurate forecasting results.

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Acknowledgements

The authors are grateful to the Turkish State Meteorological Service and General Directorate of State Hydraulic Works for providing the data.

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This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

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  1. Department of Civil Engineering, Ondokuz Mayıs University, Samsun, Turkey

    Cenk Sezen & Turgay Partal

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  1. Cenk Sezen

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Appendices

Appendix

The equations and detailed description of the GR6J model were given as follows [8,9,39,43,44,105]:

The equations regarding the calculation of production store content

In this stage, the capacity of the production store is calculated. Daily precipitation (P) and evapotranspiration (E) data are used as input data. Thex1 parameter represents the maximum capacity of the production store. Accordingly,

If\(P \ge E\), then\({\text{P}}_{{\text{n}}} = {\text{P}} - {\text{E ve E}}_{{\text{n}}} = 0\);

Otherwise,\({\text{P}}_{{\text{n}}} = 0{\text{ ve E}}_{{\text{n}}} = {\text{E}} - {\text{P}}\).

IfPn is positive, a part ofPn, namelyPs, which feeds the production store, is calculated as follows:

$${\text{P}}_{{\text{s}}} = \frac{{{\text{x}}_{{1{ }}} \left( {1 - \left( {\frac{{\text{S}}}{{{\text{x}}_{1} }}} \right)^{2} } \right){\text{tanh}}\left( {\frac{{{\text{P}}_{{\text{n}}} }}{{{\text{x}}_{1} }}} \right)}}{{1 + \frac{{\text{s}}}{{{\text{x}}_{1} }}{\text{tanh}}\left( {\frac{{{\text{P}}_{{\text{n}}} }}{{{\text{x}}_{1} }}} \right)}};\;E_{s} = 0$$
(13)

Otherwise, a part of Es of En is taken from the production store:

$${\text{E}}_{{\text{s}}} = \frac{{{\text{S}} \left( {2 - \left( {\frac{{{\text{S}} }}{{{\text{x}}_{1} }}} \right)^{2} } \right){\text{tanh}}\left( {\frac{{{\text{E}}_{{\text{n}}} }}{{{\text{x}}_{1} }}} \right)}}{{1 + \left( {1 - \frac{{{\text{S}} }}{{{\text{x}}_{1} }}} \right){\text{tanh}}\left( {\frac{{{\text{E}}_{{\text{n}}} }}{{{\text{x}}_{1} }}} \right)}}$$
(14)

Thus, the production store content is updated as follows:

$${\text{S}} = {\text{S}} - {\text{E}}_{{\text{s}}} + {\text{P}}_{{\text{s}}}$$
(14)

ThePerc, which percolates from the production store to the routing function:

$${\text{Perc}} = {\text{S}}\left\{ {1 - \left[ {1 + \left( {\frac{4}{9}\frac{{\text{S}}}{{{\text{x}}_{1} }}} \right)^{4} } \right]^{ - 1/4} } \right\}$$
(15)

In this regard, production store capacity is updated by:

$${\text{S}} = {\text{S}} - {\text{Perc}}$$
(16)

TheSMI time series are sequencesofSt forx1 production store maximum capacity:

$${\text{SMI}}_{{{\text{x1}}}} \left( t \right) = {\text{S}}_{t}$$
(17)

The water quantity Pr that reaches the routing part is,

$${\text{P}}_{{\text{r}}} = {\text{Perc}} + \left( {{\text{P}}_{{\text{n}}} - {\text{P}}_{{\text{s}}} } \right)$$
(18)

Unit hydrographs in the GR6J model

Pr splits into two parts: 90% are routed through the UH1 one-sided unit hydrograph, and 10% through the UH2 two-sided unit hydrograph. The cumulated ordinates of the UH1 and UH2 unit hydrographs, namely SH1 (t) and SH2 (t), are specified by the base timex4 for tϵN:

$${\text{if}\text{ t}}=0,{\text{ SH}}1{ }\left( {\text{t}} \right) = 0$$
$$\text{if}\ 0 < {\text{t}} < {\text{x}}_{{4{ }}},{\text{ SH}}1\left( {\text{t}} \right) = \left( {\frac{{\text{t}}}{{{\text{x}}_{4} }}} \right)^{5/2}$$
(19)
$${\text{if}\text{ t}} \ge {\text{x}}_{4},{\text{ SH}}1\left( {\text{t}} \right) = 1$$
$${\text{if}\text{ t}}=0,{\text{ SH}}2{ }\left( {\text{t}} \right) = 0$$
$${\text{if}}\ 0 < {\text{t}} \ <{\text{x}}_{{4{ }}},{\text{ SH}}2\left( {\text{t}} \right) = \frac{1}{2}\left( {\frac{{\text{t}}}{{{\text{x}}_{4} }}} \right)^{5/2}$$
(20)
$${\text{if}}{\text{ x}}_{{4{ }}} \le {\text{t}} < 2{\text{x}}_{{4{ }}},{\text{ SH}}2\left( {\text{t}} \right) = 1 - \frac{1}{2}\left( {2 - \frac{{\text{t}}}{{{\text{x}}_{4} }}} \right)^{5/2}$$
$${\text{if}}\ {\text{t}} \ge 2{\text{x}}_{4},{\text{ SH}}2\left( {\text{t}} \right) = 1$$

Accordingly, ordinates ofUH1(t) andUH2(t) are calculated based on the differentiating the cumulated ordinates:

$${\text{UH}}1{ }\left( {\text{t}} \right) = {\text{SH}}1\left( {\text{t}} \right) - {\text{SH}}1\left( {{\text{t}} - 1} \right)$$
(21)
$${\text{UH}}2{ }\left( {\text{t}} \right) = {\text{SH}}2\left( {\text{t}} \right) - {\text{SH}}2\left( {{\text{t}} - 1} \right)$$

Consequently, the output of theUH1(t), i.e.,Q9, and the output of theUH2(t), i.e.,Q1, are calculated as follows:

$${\text{Q}}9\left( {\text{t}} \right) = 0,9.\mathop \sum \limits_{{{\text{k}} = 1}}^{{\text{l}}} {\text{UH}}1\left( {\text{k}} \right).{\text{P}}_{{\text{r}}} \left( {{\text{t}} - {\text{k}} + 1} \right)$$
(22)
$${\text{Q}}1\left( {\text{t}} \right) = 0,1.\mathop \sum \limits_{{{\text{k}} = 1}}^{{\text{m}}} {\text{UH}}2\left( {\text{k}} \right).{\text{P}}_{{\text{r}}} \left( {{\text{t}} - {\text{k}} + 1} \right)$$

where l = int (x4) + 1 andm = int (2.x4) + 1.

The equations for the routing stores

At this stage, the model has two branches where the first branch is composed of the stores fed by Q9 from theUH1(t), and the second one is the direct branch fed by Q1 from theUH2(t). In the routing stores’ branch, Q9 is divided into 60% for the routing store and the remaining part for the exponential store. Furthermore, a possible exchange,F, is calculated based on the routing store water content,R, its maximum capacity,x3, and the exchange parametersx2 and x5:

$$F = x_{2} \left( {\frac{R}{{x_{3} }} - x_{5} } \right)$$
(23)

F can be positive, zero or negative. Because the value ofR cannot be under zero, actualF (AF1) is limited by the content of the latter as follows:

$${\text{AF}_{1}} = \left\{ {\begin{array}{*{20}c} {F if R + 0.6Q9 + F \ge 0} \\ { - R - 0.6Q9 \;\;{\text{otherwise}}} \\ \end{array} } \right.$$
(24)

Then, the routing store water content is updated as:

$$R = R + 0.6Q9 + AF_{1}$$
(25)

The output of the routing store, i.e.,QR, is calculated as follows:

$${\text{QR}} = {\text{R}}\left\{ {1 - \left[ {1 + \left( {\frac{{\text{R}}}{{{\text{x}}_{3} }}} \right)^{4} } \right]^{ - 1/4} } \right\}$$
(26)

The final water content of the routing store is\(R = R -QR\).

The exponential store is a bottomless reservoir, and water content can be negative in this reservoir. In this regard, the exponential store can be calculated as:

$${\text{Exp}} = {\text{Exp}} + 0.4Q9 + F$$
(27)

The output of the exponential store,QRexp, is computed as follows:

$$Q{\text{Re}} xp = x_{6} \log \left( {1 + \exp \left( {\frac{{{\text{Exp}}}}{{x_{6} }}} \right)} \right)$$
(28)

wherex6 (mm) stands for the exponential store depletion coefficient. Thus, the exponential store capacity takes its final form as\({\text{Exp}} = {\text{Exp}} - {\text{QRexp}}\).

Finally, the second branch fed byQ1 can be exposed to exchangeAF2:

$${\text{AF}}_{2} = \left\{ {\begin{array}{*{20}c} {F \;\; if\;\; Q1 + F \ge 0} \\ { - Q1\;\;\; {\text{otherwise}}} \\ \end{array} } \right.$$
(29)

In this respect, the output of the second branch,QD, is equal to\({\text{QD}} = Q_{1} - {\text{AF}}_{2}\).

In this regard, the simulated streamflow is obtained by summation ofQR,QRexp, andQD:

$$Q{\text{sim}} = QR + QR{\text{exp}} + QD$$
(30)

For further details and explanations concerning the model structure, one can refer to Perrin et al. [9], Pushpalatha et al. [8], Anctil et al. [39], Pelletier and Andréassian [105], Coron et al. [43], Coron et al. [44].

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Sezen, C., Partal, T. New hybrid GR6J-wavelet-based genetic algorithm-artificial neural network (GR6J-WGANN) conceptual-data-driven model approaches for daily rainfall–runoff modelling.Neural Comput & Applic34, 17231–17255 (2022). https://doi.org/10.1007/s00521-022-07372-5

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