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

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.

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. The x₁ 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, ${\text{P}}_{{\text{n}}} = 0{\text{ ve E}}_{{\text{n}}} = {\text{E}} - {\text{P}}$.

If P_n is positive, a part of P_n, namely P_s, 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 E_s of E_n 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)

The Perc, 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)

The SMI time series are sequences of S_t for x₁ production store maximum capacity:

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

(17)

The water quantity P_r 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

P_r 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 time x₄ 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 of UH₁(t) and UH₂(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 the UH₁(t), i.e., Q9, and the output of the UH₂(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 and m = 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 the UH₁(t), and the second one is the direct branch fed by Q1 from the UH₂(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 parameters x₂ and x₅:

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

(23)

F can be positive, zero or negative. Because the value of R cannot be under zero, actual F (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)

where x₆ (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 by Q1 can be exposed to exchange AF₂:

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

(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 of QR, QRexp, and QD:

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

(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].