Sensitivity Analysis of the Surface Runoff Coefficient of HiPIMS in Simulating Flood Processes in a Large Basin

Abstract

To simulate flood processes at the basin level, the GPU-based High-Performance Integrated Hydrodynamic Modelling System (HiPIMS) is gaining interest as computational capability increases. However, the difficulty of coping with rainfall input to HiPIMS reduces the possibility of acquiring a satisfactory simulation accuracy. The objective of this study is to test the sensitivity of the surface runoff coefficient in the HiPIMS source term in the Misai basin with an area of 797 km² in south China. To achieve this, the basin was divided into 909,824 grid cells, to each of which a Manning coefficient was assigned based on its land use type interpreted from remote sensing data. A sensitivity analysis was conducted for three typical flood processes under four types of surface runoff coefficients, assumed a priori, upon three error functions. The results demonstrate the crucial role of the surface runoff coefficient in achieving better simulation accuracy and reveal that this coefficient varies with flood scale and is unevenly distributed over the basin.

1. Introduction

Due to rapid urbanization, flood disasters have been globally increasing over the last few decades [1,2,3,4]. The Intergovernmental Panel on Climate Change (IPCC) [5] has stressed that “extreme precipitation events over most of the mid-latitude land masses and over wet tropical regions will very likely become more intense and more frequent by the end of this century, as global mean surface temperature increases”. China has a large area of land in the middle latitudes and has been experiencing unprecedented urbanization for decades. Thus, flood management has been an arduous task for the country.

As a basis of flood management, great progress has been made in developing numerical methodologies (e.g., [6,7,8,9,10,11]). Among them, the two-dimensional hydrodynamic model has become an available technological tool to simulate floods over complex topography (e.g., [12,13,14]).

The development of high-resolution data sources gives the possibility to develop hydrodynamic models at the large basin level. To achieve this, computational acceleration techniques have been greatly progressed in recent years (see, e.g., [15,16,17,18]). Among them, the GPU-based High-Performance Integrated Hydrodynamic Modelling System (HiPIMS) has been favored (see, e.g., [19,20,21,22]) and is the focus of this paper.

The HiPIMS is based on a regular uniform computational grid and fully 2D shallow water equations. In order to deal with rainfall input, an R vector was inserted into HiPIMS as one of the source terms (see in detail [22]). This added term was expressed as the product of a surface runoff coefficient and observed rainfall, instead of solutions based on the Kostiakov equation (e.g., [23]), Green–Ampt model (e.g., [24]), and Richard model (e.g., [25]).

This paper aims to test the sensitivity of the surface runoff coefficient of HiPIMS in terms of different types of floods at the large basin level. The structure of the paper is as follows: Section 2 depicts materials and methods used in the study. The results are presented and discussed in detail in Section 3. Conclusions are given in Section 4.

2. Materials and Methods

2.1. HiPIMS

The fully 2D shallow water equations (Equations (1) and (2)) are employed in HiPIMS based on the Cartesian uniform grids [22].

$$\frac{\partial \mathbf{q}}{\partial \mathit{t}}+\frac{\partial \mathbf{f}}{\partial \mathit{x}}+\frac{\partial \mathbf{g}}{\partial \mathit{y}}=\mathbf{R}+{\mathbf{S}}_{\mathbf{b}}+{\mathbf{S}}_{\mathbf{f}}$$

(1)

$$\begin{array}{c}\mathbf{q}=\left[\begin{array}{c}\mathit{h}\\ \mathit{u}\mathit{h}\\ \mathit{v}\mathit{h}\end{array}\right], \mathbf{f}=\left[\begin{array}{c}\mathit{u}\mathit{h}\\ {\mathit{u}}^2\mathit{h}+\frac{1}{2}\\ \mathit{u}\mathit{v}\mathit{h}\end{array}\mathit{g}{\mathit{h}}^2\right], \mathbf{g}=\left[\begin{array}{c}\mathit{v}\mathit{h}\\ \mathit{u}\mathit{v}\mathit{h}\\ {\mathit{v}}^2\mathit{h}+\frac{1}{2}\mathit{g}{\mathit{h}}^2\end{array}\right]\\ \mathbf{R}=\left[\begin{array}{c}\mathit{r}\\ 0\\ 0\end{array}\right],{\mathbf{S}}_{\mathbf{b}}=\left[\begin{array}{c}0\\ -\mathit{g}\mathit{h}\partial \mathit{b}/\partial \mathit{x}\\ -\mathit{g}\mathit{h}\partial \mathit{b}/\partial \mathit{y}\end{array}\right],{\mathbf{S}}_{\mathbf{f}}=\left[\begin{array}{c}0\\ -{\mathit{\tau }}_{\mathit{b}\mathit{x}}/\mathit{\rho }\\ -{\mathit{\tau }}_{\mathit{b}\mathit{y}}/\mathit{\rho }\end{array}\right]\end{array}$$

(2)

In Equation (1), q is the vector representing the conserved flow variables; f and g denote the flux vectors in the x- and y-direction, respectively; R, S_b, and S_f are the source terms standing for rainfall, bed slope, and friction, respectively. In Equation (2), h is the water depth (h = η − z_b, where η and z_b are the water surface elevation and bed elevation above datum, respectively); u and v are the depth-averaged velocity components in the x- and y-direction; g is the acceleration due to gravity; ρ is the water density; r is the rainfall volume (in depth) generating surface runoff during the considered time interval and is obtained by multiplying the surface runoff coefficient (SRC), ranging between [0, 1], and observed rainfall (e.g., [22]); $\partial b/\partial x$ and $\partial b/\partial y$ are the bed slopes in the Cartesian directions; τ_bx and τ_by denote the bed friction stresses, which can be calculated by the following formulae.

$${\tau }_{bx}=\rho C_{f}u\sqrt{u^2+v^2} and {\tau }_{by}=\rho C_{f}v\sqrt{u^2+v^2},$$

(3)

In Equation (3), the bed roughness coefficient C_f can be calculated by C_f = gn²/h^1/3, and n is the Manning coefficient.

In HiPIMS, the above governing equations are solved by a first-order finite volume Godunov-type numerical scheme. The following time-marching formula is used to discretize Equation (1) and update the flow variables:

$$\begin{array}{c}{\mathrm{q}}_i^{m+1}={\mathrm{q}}_i^m-\frac{\Delta t}{{\mathsf{\Omega }}_i}{F}_k({\mathrm{q}}^m)l_k+\Delta t({\mathrm{R}}_i^m+{\mathrm{S}}_{\mathrm{b}i}^m+{\mathrm{S}}_{\mathrm{f}i}^{m+1}),\\ F_k(\mathrm{q})={\mathrm{f}}_k(\mathrm{q})n_x+{\mathrm{g}}_k(\mathrm{q})n_y,\end{array}$$

(4)

where the superscript m represents the present time level; Δt is the time step; Ω_i is the area of cell i; k is the index of the cell edges (k = [1, 4]); l_k stands for the length of the corresponding cell edge; F_k(q) presents the fluxes normal to the cell edges; and n = (n_x, n_y) defines the unit vector of the outward normal direction. The flux terms and bed slope term are evaluated by an explicit scheme. The friction term is solved by an implicit scheme (see [26]) to maintain the numerical stability when dealing with the surface runoff with small water depth. The surface reconstruction method proposed by [27] is adopted to define the local Riemann problems at the cell interfaces. The Harten-Lax-van Leer-Contact approximate Riemann solver is employed to evaluate the interface fluxes (see [28]). The time step is controlled by the Courant–Friedrichs–Lewy (CFL) criterion. The OpenCL programming framework is implemented to develop a CPU-/GPU-integrating model for high-performance heterogeneous computing. Multiple GPUs can be directly achieved by adopting a domain decomposition technique for medium-/large-scale application (e.g., [20,22]).

2.2. Error Functions

In this study, the simulated discharge is assessed by evaluating the Absolute Relative Error of flood-peak Discharge (ARED) (Equation (5)), the Difference of Peak Arrival Time (DPAT) (Equation (6)), and the Nash–Sutcliffe Efficiency coefficient (NSE) (Equation (7)):

$$\mathrm{ARED}=\frac{|\mathrm{MAX}({\mathrm{Q}}_{\mathrm{s}})-\mathrm{MAX}({\mathrm{Q}}_{\mathrm{m}})|}{\mathrm{MAX}({\mathrm{Q}}_{\mathrm{m}})}$$

(5)

where MAX(Q_s) is the maximum simulated discharge, MAX(Q_m) is the maximum measured discharge, and ARED ∈ (−∞, +∞);

$$\mathrm{DPAT}={\mathrm{T}}_{\mathrm{MAX}(\mathrm{Qs})}{-\mathrm{T}}_{\mathrm{MAX}(\mathrm{Qm})}$$

(6)

where T_MAX(Qs) is the arrival time of maximum simulated discharge and T_MAX(Qm) is the arrival time of maximum measured discharge;

$$\mathrm{NSE}=1-\frac{{\displaystyle \sum }{({\mathrm{Q}}_{\mathrm{m}}-{\mathrm{Q}}_{\mathrm{s}})}^2}{{\displaystyle \sum }{({\mathrm{Q}}_{\mathrm{m}}-\overline{{\mathrm{Q}}_{\mathrm{m}}})}^2}$$

(7)

where Q_m is the measured discharge, Q_s is the simulated discharge, and $\overline{Q_m}$ is the averaged measured discharge.

2.3. Study Area

The Misai basin has an area of 797 km² and is located in south China. The 30 m ASTER GDEM V2 elevation data is used in this study (Figure 1). As shown, mountains in the northwest part of the basin rise to 1242 m and drop steeply towards the plain in the southwest. The elevation of the whole basin ranges from 128 m to 1242 m. There are six precipitation gauging stations and one of them at the outlet of the basin also serves as a stream gauging station (shown in Figure 1). The annual rainfall ranges between 1500 and 2000 mm. Based on 30 m Landsat TM image data and in consideration of the impact of land characteristics on water cycle, the land use type is classified into seven categories—forest, heavy brush, cultivated land, grassland, pond/river, bare land, and urban land (Figure 2). The area proportion and roughness coefficients (Manning’s n) of different land use types are presented in Table 1.

Figure 1. Map of Misai basin.

Figure 2. Land use type of Misai basin.

Table 1. The area proportion and Manning’s n of different land use types in the Misai basin.

Land Use Type	Area Proportion (%)	Manning’s n 1
Forest	45.5	0.15
Heavy brush	37.3	0.11
Cultivated land	2.2	0.035
Grassland	7.3	0.03
Pond/river	5.8	0.027
Bare land	0.7	0.025
Urban land	1.2	0.016

Note: ¹ summarized from [29,30,31].

2.4. Typical Flood Processes and Surface Runoff Coefficients

(1)

Three different types of flood processes (FPs) were chosen to represent big, medium, and small floods, respectively, as shown in Table 2.

Table 2. The three typical flood processes.

Name	Start Time	End Time	Rainfall	Peak Flow
FP1 (big)	29 May 1983, 8:00	30 May 1983, 13:00	222 mm	1820 m3/s
FP2 (medium)	5 May 1985, 8:00	7 May 1985, 12:00	106 mm	708 m3/s
FP3 (small)	26 May 1987, 8:00	27 May 1987, 20:00	65 mm	202 m3/s

(2)

Four types of surface runoff coefficient (SRC) were assumed a priori as below: SRC1 = 0.55, for all grid cells; SRC2 = 0.65, for all grid cells; SRC3 = 0.75, for all grid cells; SRC4: each grid cell was given a SRC based on its land use type—1.0 for urban/pond/river, 0.85 for bared land, 0.7 for farmland/grassland, and 0.5 for forest/heavy brush, as shown in Figure 3.

Figure 3. The spatial distribution of surface runoff coefficient 4 (SRC4).

(3)

As such, the above-mentioned FP1, FP2, FP3 and SRC1, SRC2, SRC3, SRC4 then formed twelve combinations and were used for the sensitivity analysis in the following sections. The combination is denoted FPi-SRCj hereafter, e.g., FP1-SRC1 representing FP1 under SRC1.

2.5. Modelling Set

The considered basin was divided into 909,824 grid cells upon a regular uniform grid of resolution 30 m and the rainfall input was given at 1 h intervals based on the historic records from the six gauging stations over the basin. The rainfall distribution was calculated by an inverse distance weighted grid interpolation algorithm with the observed data. The accumulated rainfalls corresponding to the three FPs are shown in Figure 4. The Manning’s n is given in Table 1. The employed hardware included a GTX980ti GPU and an Intel Core I7 4790K CPU desktop. The CFL number is set to be 0.35 in the present work.

Figure 4. The spatial distribution of accumulated rainfall of the three flood processes (FPs): (a) FP1; (b) FP2; (c) FP3.

3. Results and Discussion

In the present work, the simulations of the twelve combinations using HiPIMS were assessed by comparing the calculated and recorded discharge time series at the outlet gauging station. As shown in Figure 5, Figure 6 and Figure 7, the simulations of the three FPs presented an acceptable agreement with the measurements. However, obvious discrepancies were found among the various combinations.

Figure 5. Comparison of the measured and simulated FP1 under the four SRCs.

Figure 6. Comparison of the measured and simulated FP2 under the four SRCs.

Figure 7. Comparison of the measured and simulated FP3 under the four SRCs.

Further study has been explored through statistical analysis on ARED, DPAT, and NSE. As shown in Table 3, FP1-SRC2, FP2-SRC4, and FP3-SRC1 gave better flood peak simulation in terms of ARED; The simulated arrival time of the flood peak occurred before/after the measured one by no more than 1 h and showed no difference under different SRCs according to DPAT; FP1-SRC4, FP2-SRC4, and FP3-SRC4 obtained better simulated flood processes in the light of NSE. The above results show the potential of HiPIMS in providing satisfactory flood process simulation at the large basin level. Furthermore, it is found that (1) NSE under the four SRCs varies between 0.90 and 0.95 with an average of 0.92 for FP1, between 0.70 and 0.94 with an average of 0.85 for FP2, and between 0.76 and 0.88 with an average of 0.83 for FP3, meaning that the bigger is the FP, the better is the average NSE; (2) the best ARED appears with FP3-SRC1; (3) if only considering SRC1, SRC2, and SRC3, all of which are evenly distributed over the basin, the best NSE comes from FP1-SRC4; And (4) if considering all SRCs, among which SRC4 is uneven over the basin, the best NSE appears in FP1-SRC4. These findings imply that (1) SRC can greatly affect the HiPIMS simulation accuracy and is more sensitive for medium and small floods; (2) SRC varies with flood scale; and (3) SRC is uneven over the basin and is a function of land use type.

Table 3. The statistical analysis of the measured and simulated discharge (Q).

Combination	Measured Q (m3/s)	Simulated Q (m3/s)	ARED (%)	DPAT (h)	NSE
FP1-SRC1	1820	1599	12.1	−1	0.91
FP1-SRC2	1820	1732	4.8	−1	0.93
FP1-SRC3	1820	1999	9.8	−1	0.90
FP1-SRC4	1820	1665	8.5	−1	0.95
FP2-SRC1	708	744	5.0	0	0.90
FP2-SRC2	708	789	11.4	0	0.85
FP2-SRC3	708	874	23.4	0	0.70
FP2-SRC4	708	721	1.8	0	0.94
FP3-SRC1	202	201	0.5	1	0.81
FP3-SRC2	202	238	17.8	1	0.86
FP3-SRC3	202	274	35.6	1	0.76
FP3-SRC4	202	212	5.0	1	0.88

4. Conclusions

The sensitivity of the surface runoff coefficient in the HiPIMS source term was tested in the Misai basin with an area of 797 km² in south China. To this end, the basin was divided into 909,824 grid cells, for each of which its land use type was interpreted from remote sensing data. Considering the twelve combinations composed of three typical flood processes representing big, medium, and small floods, respectively, and four types of surface runoff coefficient assumed a priori, the analysis was conducted upon the three error functions, i.e., the Absolute Relative Error of flood-peak Discharge, the Difference of Peak Arrival Time, and the Nash–Sutcliffe Efficiency coefficient.

The results from the presented work are as follows: (1) demonstrated that HiPIMS has the potential to provide acceptable flood simulation at the large basin level; (2) revealed that the surface runoff coefficient constitutes a remarkable limit in using HiPIMS, and this is especially true for medium and small floods; (3) indicated that this coefficient varies with flood scale and is uneven over the basin; (4) highlighted the need to develop a proper methodology to deal with this source term. In this study, the simulation can be improved by integrating land use information into the application of HiPIMS.