Application of a Fractional Instantaneous Unit Hydrograph in the TOPMODEL: A Case Study in Chengcun Basin, China

Abstract

The movement of water flow usually has history and path dependence. Fractional calculus is very suitable for describing the process with memory and hereditary properties. In this study, the order of the differential equation in the Nash confluence system was extended from integer order to fractional order. On the basis of the Laplace transform, the fractional instantaneous unit hydrograph was obtained, which was used to describe the long-term memory of the basin confluence system. Furthermore, the enhanced TOPMODEL (FTOP) model was obtained by applying the fractional instantaneous unit hydrograph as the surface runoff calculation. Taking Chengcun Basin in China as an example, the FTOP model was used to simulate the daily runoff and 22 floods from 1989 to 1996. The simulation results were compared with two original TOPMODEL models (the NTOP and ITOP models). The results show that in the daily runoff simulation, the Nash–Sutcliffe efficiency (NSE), relative error (RE), and root mean square error (RMSE) of the FTOP model were 0.82, −11.14%, and 15.25 m³/s, respectively, being slightly better than the other two TOPMODEL models. According to the hydrologic frequency curve, the measured daily runoff was divided into different flow levels. It was found that the FTOP model can improve the simulation effect of the medium-flow (frequency between 10% and 50%) and low-flow (frequency more than 50%) sections to a certain extent. In the flood simulation, the average runoff depth relative error (RDRE), peak discharge relative error (PDRE), peak occurrence time error (POTE), and NSE of 22 floods were 1.99%, 14.06%, −1.27, and 0.88, respectively, indicating that the simulation effect had been improved. Especially in NSE, the improvement was more prominent, meaning that the FTOP model can better simulate the flooding process. However, the flood peak and runoff depth simulation effect were not significantly improved. These conclusions indicate that the confluence method using the fractional instantaneous unit hydrograph as the TOPMODEL model can improve the simulation effect.

1. Introduction

The hydrological model is the primary tool to study the natural law of hydrology and solve the practical problems of hydrology. The hydrological model takes the hydrological system as the research object. It is a mathematical model established according to the formation law of rainfall and runoff in nature. Hydrological models are widely used in flood forecasting [1,2,3,4,5], water resources management [6,7], non-point source pollution [8,9], runoff analysis [10,11], and many other fields.

According to the ability to reflect the spatial variation of water flow movement, hydrological models can be divided into lumped, semi-distributed, and distributed models. On the one hand, although the lumped model can meet the needs of researchers in the early stage, with the development of hydrology, the requirement for hydrological model increases. The lumped model without clear physical meaning is difficult to meet the simulation requirements, especially in the face of a large watershed area [12]. On the other hand, with the improvement of computing power and the development of the Geographic Information System (GIS), increasingly more distributed hydrological models have been widely used, such as the Soil and Water Assessment Tool (SWAT) model [13], the Hydrologic Simulation Program-Fortran (HSPF) model [14], the Topographic Kinematic Approximation and Integration (TOPKAPI) model [15], and the Variable Infiltration Capacity (VIC) model [16]. However, the input data are complicated, being challenging to meet in some areas lacking data. As a typical semi-distributed hydrological model, the TOPMODEL was proposed in 1979 [17] by Beven and Kirkby. The TOPMODEL is a link between lumped and distributed hydrological models and has been widely concerned due to its clear physical concept, simple structure, and few optimal parameters [18].

Given the advantages of the TOPMODEL model, many researchers have applied it to many fields. For example, the TOPMODEL model is used to analyze the impact of climate change on runoff [19,20,21], as well as to calculate soil moisture content [22,23], water resources management [24,25], the impact of urbanization on hydrology [26,27], and the impact of land use on runoff [28,29]. To improve the simulation effect of the TOPMODEL model, many researchers have also studied the influencing factors of simulation results. Many studies have analyzed the influence of the resolution and the creation method of the Digital Elevation Model (DEM) on the TOPMODEL model output [30,31,32,33,34,35,36,37,38,39]. Some studies also analyzed the influence of the catchment runoff coefficient [40], topographic index calculation method [41,42,43], model parameters [44,45,46,47,48], and rainfall input conditions [49,50,51] on the results of TOPMODEL. Meanwhile, many improvements to the TOPMODEL model have been proposed. These improvements mainly include the following aspects. Firstly, it is to increase the TOPMODEL model structure, such as the excess infiltration runoff module [52,53], snowmelt runoff module [54], solute transport module [55], canopy interception [56], and groundwater evaporation module [57]. Secondly, it is to replace the assumptions of the original model, for example, introducing the Green-Ampt infiltration model to replace the original model [58], introducing the power function model to replace the original exponential function model [58,59,60,61,62], introducing the motion wave model [63]. Thirdly, it is to simplify the model [64]. From this, we know that although there is much research on the TOPMODEL model, most still use the isochrones method [65] or the instantaneous unit hydrograph [66] to calculate the confluence module, which needs to be improved.

Watershed confluence is a very complex physical process. The Saint-Venant equations derived from the law of conservation of mass and energy are the theoretical basis of confluence calculation. On the basis of the Saint-Venant equations and the Nash linear reservoir, the n-order linear differential equation with constant coefficients and the famous Nash instantaneous unit hydrograph of the confluence system can be obtained by continuous calculation [67]. The Nash instantaneous unit hydrograph is a generalization model with physical concepts, rigorous mathematical derivation, and deep theory. Recently, a relatively complete confluence theory has been formed with the rapid development and gradual improvement of the Nash instantaneous unit hydrograph [68,69,70,71,72,73]. The Nash instantaneous unit hydrograph is based on integer order differential equations, which mathematically reflect the local properties of a function at a certain point. They reflect the short-range memory characteristics of the catchment in the confluence system. However, the hydrological response of many natural systems involves multiple scales. The movement process of water flow has historical dependence and path dependence, characterized by trailing characteristics, that is, the long-term memory effect. Fractional calculus is a non-local operator expressed in differential-integral form, which is very suitable for representing processes with memory and hereditary properties. Recently, it has been widely concerned, developed, and successfully applied in many fields [74,75,76,77,78,79]. Fractional calculus has become a key research field due to its unique advantages and irreplaceability.

The application of fractional calculus theory in the hydrology field is mainly reflected in the solute transport in underground aquifers [80,81,82], unsaturated soil water transport [83,84,85], solute transport in soil [86,87], and so on. Given the long memory of the confluence system, the theory has been applied to the confluence calculation process. On the basis of fractional calculus theory and linear reservoir theory, fractional instantaneous unit lines are derived and applied to the simulation of the rainfall–runoff process [88]. For basins with multiscale characteristics, such as karst and hilly areas, Guinot et al. [89] proposed to establish the response model of rainfall–runoff by using the infinite characteristic time transfer function with a long memory, whose governing equation is a fractional differential equation. This model incorporates the long memory characteristics of runoff, and its accuracy is significantly improved compared with the commonly used linear or nonlinear reservoir models. Because of the heterogeneity of the bus system, the movement of surface runoff shows abnormal behavior scale dependency, and Zhang et al. [90], on the basis of the wave equation of river flood routing, used the operator decomposition method to establish the river confluence of fractional differential equations and numerical simulation. The results showed that the model can accurately reflect nonlocality and tail confluence characteristics. Although the fractional calculus theory has been relatively mature in recent decades, its application in the field of hydrology has just begun. With the deepening of research, fractional calculus theory is expected to become an essential tool for solving complex hydrological problems.

On the basis of the basic theory of fractional differential equation, the fractional differential equation of catchment is established. The fractional instantaneous unit hydrograph containing the generalized Mittag–Leffler function is obtained by the Laplace transform solution. On the framework of the original TOPMODEL model, the fractional instantaneous unit hydrograph is applied. The aspects of this research are as follows: (1) to verify the simulation effect of fractional instantaneous unit hydrograph; (2) to compare the differences between different confluence methods of the TOPMODEL model; and (3) to provide a reference for further improving the simulation effect of the TOPMODEL model.

2. Materials and Methods

2.1. Study Area

Chengcun Basin is located in Huangshan City, Anhui Province, belonging to the Qiantang River Basin, adjacent to the southeast coast of China, located in the subtropical monsoon climate zone, with an annual average temperature of 17 °C. The weather in the Chengcun Basin in winter is sunny, cold, and dry, with the prevailing northwest wind. In summer, there is a southeast wind with high temperatures, intense light, and humid air. In spring and autumn, cyclone activity is frequent, with significant changes in cold and warm. The frontal rain is expected in spring and early summer, and typhoons are common in summer and autumn. The direction of the monsoon circulation is orthogonal to the direction of the central mountain ranges, which block the northern cold current and typhoons. The average annual precipitation in Chengcun Bain is 1600 mm, which is rainy from April to June, accounting for 50%, which is prone to flood disasters. Rainfall from July to September accounts for 20%, with frequent droughts. The river’s runoff varies significantly within and between years, so it is a typical humid area. The flood of the Chengcun Basin has the characteristics of steep rise and fall, high flood peak, and short duration [91,92,93,94]. The total area of the Chengcun Basin is 298.024 km², as shown in Figure 1, which contains ten rain gauging stations and one hydrologic station. The Theisen polygon method calculates the average rainfall over the basin.

2.2. Fractional Instantaneous Unit Hydrograph

2.2.1. Fractional Calculus Theory

Fractional order calculus is a generalization of integer order calculus, which is a theory about calculus of arbitrary order. At this stage, the theory of fractional order calculus is imperfect, and the definition of fractional order derivatives is not unified. Currently, four definitions are commonly used: Grünwald–Letnikov fractional order derivative, Riemann–Liouville fractional order derivative, Caputo fractional order derivative, and Weyl fractional order derivative. Among them, the Caputo definition is often used in engineering applications because the initial conditions to be satisfied in modeling applications and integral transformations are given in the form of integer order calculus, which is easier to obtain in practical engineering problems [95].

Assuming that the function $f\left(t\right)$ is defined on the interval $\left[a,t\right]$ and $f\left(t\right)=0$, $\forall t>a$, the α-order Caputo derivative of the function $f\left(t\right)$ is defined as [95]

$${}_a{}^{C}D{}_t{}^{a}f\left(t\right)=\frac{1}{\Gamma \left(n-\alpha \right)}{{\displaystyle \int }}_a^t{\left(t-\tau \right)}^{n-\alpha -1}{f}^{\left(n\right)}\left(\tau \right)d\tau ,n-1<\alpha \le n$$

(1)

where ${}_a{}^{C}D{}_t^a$ is the notation for the derivative of Caputo fractional order, $D$ denotes the differential, $C$ denotes the Caputo operator, $\mathsf{\alpha }>0$ denotes the order of the differential, $a,$ $t$ denotes the upper and lower integration limits, $n$ is the smallest positive integer not less than $\alpha$, and $f^{\left(n\right)}\left(\tau \right)$ is the $n$th order derivative of the function $f\left(\tau \right)$. When $\alpha \to n$, the Caputo derivative turns into the usual $n$th derivative.

2.2.2. Derivation of Fractional Instantaneous Unit Hydrograph

According to the Nash confluence theory, the influence of a watershed on net rainfall storage can be viewed as the regulating effect of a series of linear reservoirs connected in series. For the first reservoir, the net rainfall $h$ in the basin is the inflow and $Q_1$ is the outflow. The continuity equation for the first reservoir can be expressed as [67]

$$h\left(t\right)-Q_1\left(t\right)=\frac{\mathrm{d}{W}_1}{\mathrm{d}t}$$

(2)

where $W_1$ represents the storage capacity of the first reservoir, and the storage and discharge equation of the linear reservoir is [67]

$$W_1=K_1{Q}_1\left(t\right)$$

(3)

where $K_1$ is the storage and discharge coefficient of the first reservoir.

The storage equation of the first reservoir can be obtained by combining the above Equations (2) and (3) [67]:

$$K_1\frac{\mathrm{d}{Q}_1}{\mathrm{d}t}=h\left(t\right)-Q_1\left(t\right)$$

(4)

This equation is a conventional integer order differential equation, and if Caputo fractional order derivatives express the integer order derivatives, then Equation (4) will become the following fractional order differential equation [88]:

$$K_1^{\alpha }{}_0{}^{C}D{}_1^{\alpha }{Q}_1\left(t\right)=h\left(t\right)-Q_1\left(t\right)$$

(5)

where $0<\alpha <1$. After further analysis of Equation (5), the following equation is obtained [88]:

$$Q_1\left(t\right)=\frac{1}{1+K_1^{\alpha }{}_0{}^{C}D{}_1^{\alpha }}h\left(t\right)$$

(6)

Similarly, the outflow of the first reservoir is used as the inlet of the second reservoir. After n linear reservoirs with the same storage and discharge coefficients, the basin outlet process is obtained [88]:

$$Q_n\left(t\right)=\frac{1}{{\left(1+K_{}^{\alpha }{}_0{}^{C}D{}_t^{\alpha }\right)}^n}h\left(t\right)$$

(7)

According to the definition of instantaneous unit hydrograph, when the period tends to infinity, $h\left(t\right)$ is the instantaneous unit net rainfall and $Q_n\left(t\right)$ is the instantaneous unit hydrograph $u\left(t\right)$; applying the principle of impulse induction and replacing $h\left(t\right)$ with the impulse response function $\delta \left(t\right)$, Equation (7) becomes the following equation [88]:

$$u\left(t\right)=\frac{1}{{\left(1+K_{}^{\alpha }{}_0{}^{C}D{}_t^{\alpha }\right)}^n}\delta \left(t\right)$$

(8)

Equation (8) is the Laplace transformation. The initial condition is set as $u\left(0\right)=u^{\prime }\left(0\right)=\cdots =u^{\left(n-1\right)}\left(0\right)=0$. Since $L\left[\delta \left(t\right)\right]=1$, the following equation is obtained [88]:

$$L\left[u\left(t\right)\right]={\left(1+K^{\alpha }{s}^{\alpha }\right)}^{-n}=K^{-n\alpha }{s}^{-n\alpha }{\left(1+K^{-\alpha }{s}^{-\alpha }\right)}^{-n}$$

(9)

The Laplace inversion of the above equation is further performed to obtain the following fractional instantaneous unit hydrograph [88]:

$$u\left(t\right)=\frac{t^{n\alpha -1}}{K^{n\alpha }}{E}_{\alpha ,n\alpha }^n[-[\frac{t}{K}{)}^{\alpha }]$$

(10)

where $E_{\alpha ,\beta }^{\gamma }()$ is the generalized Mittag–Leffler function, which is defined as follows [88]:

$$E_{\alpha ,\beta }^{\gamma }\left(t\right)={{\displaystyle \sum }}_{r=0}^{\infty }\frac{\Gamma \left(r+\gamma \right)t^r}{\Gamma \left(\gamma \right)\Gamma \left(\alpha r+\beta \right)r!}$$

(11)

where $\alpha$, $\beta ,$ and $\gamma$ are parameters greater than 0. The derivation of Equation (9) to (10) utilizes the following properties of the Laplace transform of the generalized Mittag–Leffler function [88]:

$$L\left[t^{\beta -1}{E}_{\alpha ,\beta }^{\gamma }\left(\lambda t^{\alpha }\right)\right]=s^{-\beta }{\left(1-\lambda s^{-\alpha }\right)}^{-\gamma }$$

(12)

The Nash instantaneous unit hydrograph is a particular fractional order differential equation case. Accordingly, the Nash instantaneous unit hydrograph should be a specific case of the fractional instantaneous unit hydrograph. When the order $\alpha =1$ of the fractional instantaneous unit hydrograph, Equation (10) becomes the following [88]:

$$u\left(t\right)=\frac{1}{K\Gamma (n)}{(\frac{t}{K})}^{n-1}{{\displaystyle \sum }}_{r=0}^{\infty }\frac{1}{r!}{(-\frac{t}{K})}^r=\frac{1}{K\Gamma \left(n\right)}{(\frac{t}{K})}^{n-1}{e}^{\frac{1}{K}}$$

(13)

2.2.3. Confluence Calculation Based on Fractional Instantaneous Unit Hydrograph

In the actual confluence calculation, it is usually necessary to convert the instantaneous unit hydrograph into the period unit hydrograph with the help of the S-curve. The S-curve for the fractional instantaneous unit hydrograph is as follows:

$$S\left(t\right)={{\displaystyle \int }}_0^{t}u(t)\mathrm{d}t={(\frac{t}{K})}^{n\alpha }{E}_{\alpha ,n\alpha +1}^n\left[-{(\frac{t}{K})}^{\alpha }\right]$$

(14)

The calculation period is $\mathsf{\Delta }t$, and the unit hydrograph of the period when the net rainfall is 1 mm is as follows:

$$q\left(\mathsf{\Delta }t,t\right)=\frac{F}{3.6\mathsf{\Delta }t}\left[S\left(t\right)-S\left(t-\mathsf{\Delta }t\right)\right]$$

(15)

where $F$ represents the area of the watershed, and the unit is km².

According to the multiple ratios of unit hydrograph and superposition assumption, the discharge process at the drainage outlet can be obtained as follows:

$$Q_i={{\displaystyle \sum }}_{j=1}^m{r}_j{q}_{i-j+1}$$

(16)

where $Q_i$ is the outflow in the $i$th period, $i=1,2,\cdots ,l+m-1$; $r_j$ is the net rainfall in the $j$th period; $m$ is the number of periods of net rain; and $l$ is the period of several unit lines, which can also be understood as the memory length.

2.3. TOPMODEL Model

2.3.1. TOPMODEL Model Principle

TOPMODEL model is based on the principle of variable runoff area. The model determines the size and location of the runoff source area through soil water content. The whole hydrological process is mainly described by water balance and Darcy law, and the model is simplified by using three critical assumptions:

(1) Assumption 1: The watershed has a saturated layer area with a stable water supply.

(2) Assumption 2: The relationship between soil hydraulic conductivity and water scarcity is exponentially decreasing, so [60]

$$T_i=T_0\times e^{-\frac{z_i}{S_{zm}}}$$

(17)

where $T_0$ is the saturated water conductivity of the basin, and it is considered that the saturated water conductivity is evenly distributed in the whole catchment. $S_{zm}$ is the maximum water storage depth in the unsaturated zone. $T_i$ is the water conductivity at point $i$ of the watershed.

(3) Assumption 3: The hydraulic gradient on the saturated area of the basin is similar to the slope of the surface topography. This assumption is more suitable for the actual situation of saturated groundwater. On the basis of Darcy law, any underground runoff can be expressed as [60]

$$q_i=T_i\tan \left({\beta }_i\right)=R{a}_i$$

(18)

where $q_i$ is the soil discharge per unit of water passing width at point $i$ of the basin, $\tan \left(\beta \right)$ is the local slope of the watershed surface, $R$ is the discharge per unit area of the watershed, and $a_i$ is the catchment area per unit width at point $i$ of the watershed.

By combining Equations (1) and (2), the following equation can be obtained [60]:

$$z_i=-S_{zm}\ln \frac{R{a}_i}{T_0\tan {\beta }_i}$$

(19)

Therefore, the average water surface depth of the whole basin is [60]

$$\overline{z}=\frac{1}{A}{{\displaystyle \int }}_A^{}{z}_i\mathrm{d}A=\frac{S_{zm}}{A}{{\displaystyle \int }}_A^{}\left(-\ln \frac{R{a}_i}{T_0\tan {\beta }_i}\right)\mathrm{d}A$$

(20)

Since the saturated water conductivity $T_0$ is considered to be uniformly distributed in the basin in the TOPMODEL model, it can be obtained from Equations (3) and (4) [60]:

$$z_i=\overline{z}-S_{zm}(\ln \frac{a_i}{\tan {\beta }_i}-{\lambda }^{*})$$

(21)

where $A$ is the area of the whole watershed, $\overline{z}$ is the average surface depth of the watershed, $\ln \frac{a_i}{\tan {\beta }_i}$ is the topographic index value at point $i$ of the watershed, and ${\lambda }^{*}$ is the mean value of the basin topographic index.

On the basis of the above three assumptions, the soil in the watershed was divided into three water-bearing zones: the vegetation root zone, unsaturated soil zone, and saturated underground zone. The hydrological process in the basin is as follows: precipitation first penetrates the root zone of vegetation, and part of the water is stored in this zone and involved in evaporation. In contrast, the rest directly penetrates the unsaturated zone of the soil. Then, the amount of water in the unsaturated soil zone penetrates the groundwater zone vertically at the rate of $q_v$. As a result, regression flow is generated if the groundwater surface rises and emerges at a low-lying confluence. Finally, in the groundwater area, the groundwater flows in saturated soil through lateral movement. Therefore, in the TOPMODEL model, saturated soil flow and saturated overland flow constitute the total runoff of the watershed.

2.3.2. Improvement of the TOPMODEL Model

On the basis of the calculation framework of the classical TOPMODEL model, the surface confluence module is replaced by the fractional instantaneous unit hydrograph model to establish an improved TOPMODEL (FTOP). The enhanced model has eight parameters, as shown in

Table 1. Definitions of parameters in the FTOP.

Parameter	Definition
$T_0$	Downslope transmissivity when the soil is just saturated to the surface (L2T−1)
$S_{zm}$	Maximum depth of the unsaturated soil zone (L)
$S_{rmax}$	Maximum capacity of the root zone (available water capacity to plants) (L)
$S_{ro}$	Initial storage in the root zone at the start of a run (L)
$td$	Time delay for recharge to the saturated zone per unit of deficit (T)
$k$	Storage and discharge coefficient of a linear reservoir
$n$	Number of linear reservoirs
$\alpha$	Order of the fractional derivative

. To illustrate the improvement of the enhanced TOPMODEL model, the TOPMODEL model (ITOP and NTOP) with the surface confluence module as the isochrones method and integer instantaneous unit hydrograph model were established, respectively, in this study.

2.3.3. Input Data of the TOPMODEL Model

Digital elevation data (DEM) are free data with a resolution of 30 × 30 m provided by Geospatial Data Cloud (http://www.gscloud.cn/, accessed on 5 October 2022). They are mainly used for filling, determining the flow direction, extracting river systems, and calculating the topographic wetness index (TWI) (as shown in Figure 2). To better achieve the purpose of this study, TOPMODEL models of two temporal scales were applied. Because of TOPMODEL models with different time scales, the meteorological data input is also varied. The daily precipitation data of 10 rain gauge stations and the daily evaporation data of Chengcun Station from 1989 to 1996 were collected in the Chengcun Basin in order to build the daily TOPMODEL model. The hourly precipitation data of 10 rain gauge stations and the hourly evaporation data of Chengcun Station of 22 floods from 1989 to 1996 were collected for building the hourly TOPMODEL model.

2.3.4. Model Efficiency Criteria

Due to the different emphases of daily scale runoff simulation and hourly scale flood simulation, the indexes of the simulation effects of these two scales are also other in this study.

For daily scale runoff simulations, the first criterion used in the study to evaluate the performance of the model was the Nash–Sutcliffe efficiency (NSE) index, which is defined by

$$NSE=1-\frac{{\sum }_t{\left(Q_t^o-Q_t^s\right)}^2}{{\sum }_t{\left(Q_t^o-\overline{Q}\right)}^2}$$

(22)

where $Q_t^o$ is the observed streamflow, $Q_t^s$ is the simulated streamflow, and $\overline{Q}$ is the mean observed streamflow. The closer the value of NSE is to 1, the better the simulation effect of the TOPMODEL model.

The second efficiency criterion used is the relative error (RE) between the observed and simulated streamflow series, which is defined as

$$RE=\frac{{\sum }_t{Q}_t^o-{\sum }_t{Q}_t^s}{{\sum }_t{Q}_t^o}\times 100\%$$

(23)

The third efficiency criterion used is the root mean square error (RMSE) between the observed and simulated streamflow series, which is defined as

$$RMSE=\sqrt{\frac{{\sum }_t{\left(Q_t^o-Q_t^s\right)}^2}{N}}$$

(24)

where $N$ is the length of the sequence; the closer the value of RE and RMSE is to 0, the better the simulation effect of the TOPMODEL model.

For hourly scale runoff simulations, the evaluation indexes used are peak discharge relative error (PDRE), peak occurrence time error (POTE), runoff depth relative error (RDRE), and NSE mentioned above. The smaller the value of these errors, the higher the simulation accuracy of the TOPMODEL model.

2.3.5. Calibration and Verification of the TOPMODEL Model

Corresponding to the simulation of the above two temporal scales, the calibration and verification of the TOPMODEL model also occur as the following two parts:

(1) For the daily scale simulation results, the daily runoff data of the Chengcun hydrological station from 1989 to 1996 were calibrated and verified. Among them, 1989–1994 was the calibration period, and 1995–1996 was the verification period.

(2) For the hour-scale simulation results, the data of 22 floods in Chengcun hydrological station were used to calibrate and verify the results, and the time step was 1 h. Among them, the first 12 flood data were utilized as calibrating data, and the last ten were used as verifying data.

Considering that there are many criteria to evaluate the simulation effect of the TOPMODEL model, this study adopted the Non-Dominated Sorted Genetic Algorithm-Ⅱ (NSGA-Ⅱ). This algorithm is one of the most effective methods for solving multi-objective optimization problems [93]. Srinivas and Deb proposed the NSGA-Ⅱ algorithm based on NSGA in 2000 [96,97], which has been widely used in many fields and has good performance, with good convergence and distribution properties [98,99].

3. Results

3.1. Calculation of the Topographical Index

The topographical index is calculated using the single-flow direction algorithm based on the digital elevation model (DEM). Due to the topographical index’s similarity across grids, it is unnecessary to simulate it in every grid. The distribution function in the whole basin is generated by a statistical method with different topographical index values (Figure 2). The study region is segmented into 30 sections according to the topographical indexes. The relationship of area ratios of the topographical index is shown in Figure 3. The topographical index’s minimum, maximum, mean, and standard deviations were 0.86, 14.84, 7.85, and 4.17, respectively.

3.2. Simulation Results of the Daily FTOP Model

The NSGA-Ⅱ algorithm was used to optimize the parameters of the daily scale FTOP model, and the best parameters obtained are shown in

Table 2. Parameter values of the daily FTOP model.

Parameter	Value	Unit
$T_0$	2.06	m2/d
$S_{zm}$	0.02	m
$S_{rmax}$	0.96	m
$S_{ro}$	0.10	m
$td$	3.17	d
$k$	6.28	/
$n$	2.17	/
$\alpha$	0.35	/

.

Table 3. Statistical index comparison table of three TOPMODEL models.

	NSE			RE			RMSE
	FTOP	NTOP	ITOP	FTOP	NTOP	ITOP	FTOP	NTOP	ITOP
Calibration period (1989–1994)	0.84	0.85	0.80	−0.25%	−18.60%	−3.50%	12.60	12.25	13.78
Validation period (1995–1996)	0.81	0.72	0.84	−37.68%	−43.99%	−40.01%	21.29	25.72	19.26
Simulation period (1989–1996)	0.82	0.79	0.82	−11.14%	−25.98%	−14.11%	15.25	16.67	15.34

describes the simulation effects of the three TOPMODEL models in different periods. In the calibration period, the NSE of the FTOP model and NTOP model were 0.84 and 0.85, respectively, which were more significant than that of the ITOP model (0.80). The REs of the FTOP and ITOP model were −0.25% and −3.50%, respectively, much better than the −18.60% of the NTOP model. The performances of the FTOP model, NTOP model, and ITOP model were similar in RMSE, being 12.60 m³/s, 12.25 m³/s, and 13.78 m³/s, respectively. In the validation period, the NSEs of the FTOP model and ITOP model were 0.81 and 0.84, respectively, much better than that of the NTOP model (0.72). The REs of the FTOP and ITOP models were −37.68% and −40.01%, respectively, slightly better than the −43.99% of the NTOP model. Similarly, the FTOP, NTOP, and ITOP models showed little difference in RMSE performance. In the whole simulation period, the FTOP model was undoubtedly the best. The NSE, RE, and RMSE of the FTOP model were 0.82, −11.14%, and 15.25 m³/s, respectively, being better than those of the NTOP and ITOP models.

Figure 4 compares the simulated flow and the observed flow of the FTOP model, NTOP model, and ITOP model. It can be seen from Figure 4 that the simulated flow curve of the FTOP model fit well with the observed flow curve. Especially in low annual flow, the advantages of the FTOP model were more prominent. At the same time, we can also see that the performance of the NTOP model was the worst among the three TOPMODEL models. Figure 5 illustrates the scatter plots of observed flow and simulated flow of the FTOP model, NTOP model, and ITOP model. When the scatter shown in Figure 5 was closer to the diagonal (that is, the dotted line in Figure 5), it indicated that the simulation was preferable and vice versa. Compared with the NTOP model and ITOP model, it can be seen that the scatter points of the FTOP model were uniformly distributed on both sides of the diagonal, which indicated that the simulation effect of the FTOP model was better than that of the NTOP model and ITOP model.

The daily average flow of the Chengcun Basin from 1989 to 1996 was divided into the high flow section, medium flow section, and low flow section according to the flow frequency curve in order to illustrate the simulation effect of the FTOP model on different flow levels. The high flow section was below 10% of the flow frequency curve. The medium flow section was the segment between 10% and 50%. The low flow section was the part that was higher than 50%. It can be seen from

Table 4. Statistical index comparison table of three TOPMODEL models.

	NSE			RE			RMSE
	FTOP	NTOP	ITOP	FTOP	NTOP	ITOP	FTOP	NTOP	ITOP
High flow section	0.86	0.85	0.89	2.91%	7.26%	7.46%	45.33	46.80	39.64
Medium flow section	0.72	0.42	0.20	−20.34%	−49.58%	−30.23%	7.61	10.97	12.91
Low flow section	−0.68	−3.60	−3.82	−91.96%	−207.41%	−127.56%	2.83	4.68	4.79

that in the high flow section, the simulation effects of the three TOPMODEL models were comparable without showing unique advantages. However, in the medium and low flow sections, the simulation effect of the FTOP model was better than the other two kinds of TOPMODEL models. In the medium flow section, the NSE, RE, and RMSE of the FTOP model were only 0.72, −20.34%, and 7.61 m³/s, respectively, being significantly better than the NTOP model and ITOP model. In the low-flow section, although the simulation effect of the FTOP model was worse than that of the high-flow and medium-flow sections, the NSE was even more harmful. However, compared with the NTOP model and ITOP model, the simulation effect was still better.

Figure 6 depicts the comparison of flow accumulation curves of three TOPMDEL models simulating different levels of flow. When the curve shown in Figure 6 was closer to the diagonal (that is, the dotted line in Figure 6), it indicated that the simulation was preferable, and vice versa. As seen from Figure 6, no matter the TOPMODEL model in question, the distance between the curve and the diagonal became larger from left to right. This shows that the simulation effect of the TOPMODEL model was better in the high-flow section, and the simulation effect worsened with the decrease in flow. In the high-flow part, it can be seen from Figure 6a,d,g that the distance between the simulated data and the diagonal line was not much different, indicating that the simulation effect of the FTOP model was similar to that of NTOP model and ITOP model. In the middle-flow section, the distance between the black line and the diagonal was smaller than that between the blue line and the red line, indicating that the simulation of the FTOP model was slightly better than that of the NTOP model and the ITOP model. The FTOP model performed similarly in the low-flow section, and the improvement was pronounced.

3.3. Simulation Results of the Hourly FTOP Model

The NSGA-Ⅱ algorithm was used to optimize the parameters of the hourly scale FTOP model, and the best parameters obtained are shown in

Table 5. Parameter values of the hourly FTOP model.

Parameter	Value	Unit
$T_0$	10.00	m2/d
$S_{zm}$	0.02	m
$S_{rmax}$	1.00	m
$S_{ro}$	0.00	m
$td$	9.02	d
$k$	3.60	/
$n$	7.34	/
$\alpha$	0.76	/

.

Table 6. Summary table of statistics of different flood events.

Number	RDRE			PDRE			POTE			NSE
	FTOP	NTOP	ITOP	FTOP	NTOP	ITOP	FTOP	NTOP	ITOP	FTOP	NTOP	ITOP
19890411	23.72%	25.17%	2.69%	35.37%	40.09%	27.09%	−2.00	−3.00	−1.00	0.84	0.79	0.86
19890427	3.58%	−0.40%	−16.76%	11.43%	9.34%	6.25%	−2.00	−2.00	−1.00	0.95	0.96	0.87
19890616	0.28%	1.10%	18.92%	−5.61%	−3.23%	34.63%	1.00	1.00	0.00	0.95	0.95	0.85
19900409	−1.07%	−4.17%	−7.27%	28.41%	25.13%	13.77%	−3.00	−3.00	−5.00	0.88	0.88	0.55
19900606	−11.89%	−10.52%	−7.69%	3.86%	4.67%	−8.68%	−7.00	−7.00	−5.00	0.92	0.92	0.82
19900614	−9.28%	−4.57%	14.61%	6.93%	11.55%	28.70%	−1.00	−1.00	−6.00	0.96	0.96	0.85
19900626	−18.99%	−17.96%	−12.87%	4.62%	6.63%	23.36%	−4.00	−4.00	−2.00	0.90	0.90	0.78
19910415	25.93%	20.24%	−7.78%	42.10%	40.81%	45.47%	−2.00	−2.00	−4.00	0.76	0.79	0.63
19910518	−5.99%	−4.54%	9.23%	1.95%	6.05%	53.80%	1.00	1.00	−3.00	0.88	0.92	0.76
19920514	2.14%	3.22%	−18.12%	46.25%	48.94%	29.44%	−4.00	−5.00	−6.00	0.82	0.79	0.83
19920614	−52.38%	−49.72%	−52.42%	−12.08%	−7.37%	−4.29%	0.00	0.00	−2.00	0.76	0.79	0.68
19920701	12.09%	13.89%	17.45%	8.06%	10.31%	46.89%	0.00	0.00	−3.00	0.98	0.97	0.77
19930531	7.19%	9.76%	12.86%	37.18%	40.56%	38.71%	−3.00	−3.00	−5.00	0.83	0.80	0.78
19930618	15.90%	13.80%	14.01%	39.61%	39.14%	54.21%	−1.00	−1.00	−3.00	0.82	0.83	0.67
19930629	5.96%	6.66%	16.12%	2.76%	4.39%	44.01%	1.00	1.00	1.00	0.96	0.96	0.80
19940608	−6.88%	−3.49%	19.92%	−8.34%	−5.59%	20.70%	0.00	3.00	−1.00	0.89	0.89	0.79
19950413	0.84%	8.37%	2.73%	−17.84%	3.19%	7.08%	−1.00	−2.00	−2.00	0.96	0.94	0.89
19950519	8.27%	9.77%	12.38%	27.52%	27.54%	50.77%	0.00	0.00	−4.00	0.92	0.90	0.81
19950619	12.16%	11.20%	−4.12%	41.22%	40.48%	58.15%	0.00	0.00	1.00	0.88	0.89	0.71
19950701	13.73%	20.33%	24.21%	17.00%	20.06%	29.95%	−3.00	−3.00	−1.00	0.85	0.75	0.78
19960417	11.78%	19.51%	30.24%	−11.60%	0.93%	19.29%	1.00	0.00	2.00	0.76	0.69	0.77
19960618	6.72%	11.85%	28.16%	10.47%	11.29%	51.90%	1.00	1.00	2.00	0.93	0.93	0.68
AVERAGE	1.99%	3.61%	4.39%	14.06%	17.04%	30.51%	−1.27	−1.32	−2.18	0.88	0.87	0.77

describes the statistical indicators between the simulated and measured flows of the 22 floods. Although the performance of the FTOP model was worse than that of the NTOP model and the ITOP model in a specific index of a particular flood, the average of the statistical indicators of 22 floods showed that the performance of the FTOP model was better than the other two models. Regarding the average value of RDRE, the FTOP model reduced its value from 3.61% and 4.39% to 1.99%. The FTOP model did the same with the PDRE average, reducing its value from 17.04% and 30.51% to 14.06%.

Regarding the average value of POTE, the FTOP model and NTOP model were −1.27 and −1.32, respectively, being similar but significantly better than the −2.18 of the ITOP model. Similarly, in terms of the mean value of NSE, the FTOP model and NTOP model were 0.88 and 0.87, respectively, being similar but significantly better than the ITOP model (0.77). To sum up, the FTOP model performed slightly better than the NTOP model and considerably better than the ITOP model.

Figure 7 compares the simulated flood data of FTOP, NTOP, and ITOP models with the observed flood data. It can be seen intuitively from Figure 7 that the simulated peak flood discharge by the FTOP model of (a), (h), (m), (n), (s), and (t) graphs were significantly smaller than the observed peak flood discharge. Compared with the NTOP and ITOP models, the FTOP model clearly simulated the flooding process in the remaining flood events. Figure 8 depicts the distribution of the four statistical values of the FTOP model, the NTOP model, and the ITOP model. The closer the box of a specific model in the figure to the dotted line, the better the simulation effect of the model. In Figure 8a–c, the performance of the FTOP model was less different from that of the NTOP model, but it was significantly improved compared to the ITOP model. As shown in Figure 8d, the performance of the FTOP model was considerably enhanced compared to the NTOP model and the ITOP model.

4. Discussion and Conclusions

On the daily-scale runoff simulation surface, the simulation effect of the FTOP model was not very prominent throughout the simulation period. However, when the runoff was divided into different levels, it can be seen that the advantages of the FTOP model gradually showed. Especially in the low-flow section, the benefits were more pronounced. Comparing the three TOPMODEL models, when other parts were consistent, the differences in the confluence method led to the above differences. Therefore, we can find that the fractional instantaneous unit hydrograph system had apparent advantages in simulating low flow.

The FTOP model had the most obvious advantage in the hour-scale flood simulation in NSE. NSE focused on the entire flood process and can intuitively reflect the fit between the simulated and measured flow. Although the FTOP model has fewer advantages in simulating the peak flow, peak time, and flood water volume, the FTOP model can simulate the entire flood process well. It also proves that the fractional instantaneous unit hydrograph confluence system can better simulate the whole process of a flood.

In this study, a fractional instantaneous unit hydrograph confluence system was introduced on the original framework of the TOPMODEL model. The daily runoff and 22 floods in the Chengcun Basin from 1989 to 1996 were simulated using the improved TOPMODEL (FTOP) model. The following are the main conclusions from comparing the simulation results with the original two TOPMODEL models: the NTOP model (the integer instantaneous unit hydrograph method as the confluence method) and the ITOP model (isochrones method as the confluence method).

(1) In the daily-scale runoff simulation, the NSE, RE, and RMSE of the FTOP model were 0.82, −11.14%, and 15.25 m³/s, respectively, throughout the simulation period, indicating that the simulation results were acceptable. The flow during the simulation period was divided into the high flow section, the medium flow section, and the low flow section. The FTOP model had no apparent advantage in the simulation effect of the high flow part. However, in the middle and low flow sections, the simulation effect of the FTOP model was better than that of the NTOP model and the ITOP model.

(2) In terms of hourly scale flood simulation, the average RDRE, PDRE, POTE, and NSE of the FTOP model in the 22 floods were 1.99%, 14.06%, −1.27, and 0.88, respectively. Compared with the NTOP and ITOP models, the simulation effect was improved to a certain extent. Especially in terms of NSE, the improvement was more prominent, which shows that the FTOP model can better simulate the flow process of the entire flood.