Research on the Threshold of the Transverse Gradient of the Floodplain in the Lower Yellow River Based on a Flood Risk Assessment Model

Abstract

Due to the influence of water and sediment conditions, engineering projects, channel erosion and siltation, river-related factors, and human activities (such as adjustments in floodplain production structures and village construction), there have been significant variations in the transverse gradient of the floodplain in the lower Yellow River. An irrational transverse gradient can lead to the rapid conversion of gravitational potential energy into kinetic energy during the flood evolution process, resulting in increased flow velocity and inundated areas. Exploring reasonable transverse gradients can provide technical support for floodplain management. Using “flood risk assessment” as a keyword, research papers from the Web of Science core database and CNKI published in the past five years were collected. Through a VOS viewer analysis of indicators, a flood risk assessment model based on the “Source–Path–Receptor–Consequence–Resilience” framework was established. A two-dimensional water and sediment model was used to simulate flood inundation scenarios with different transverse gradients in the same flood event, evaluate flood risks in the floodplain, and determine the optimal transverse gradient based on flood risk levels. The results indicate that, compared to low transverse gradients, moderate and high transverse gradients have a more significant driving effect on flood inundation, increasing flood risk opportunities for floodplains. Lower transverse gradients (i.e., TG = 10LG = 1.25‰) are the most favorable for flood protection in the floodplain after flood inundation.

1. Introduction

The lower Yellow River is a “hanging river” crossing the heartland of China, and flood safety is crucial for the stability of people’s lives in the floodplain [1]. The floodplain within the river channel serves as a place for major floods to flow, be detained, and deposit sediment while also being the homeland for millions of people, playing a dual role in flood safety along the Yellow River and social and economic development in the region [2]. The floodplains in the lower Yellow River are widespread in meandering and transitional river sections. Transitional sections have highly curved channels, poor connectivity, and significant changes in sediment transport with imbalanced water–sediment relationships due to a large sediment load, leading to massive sediment deposition and severe channel infilling. It has been nearly two decades since large-scale flooding occurred in the downstream floodplain, allowing residents to live and work in peace. However, human activities have significantly altered the conditions of the floodplain [3]. The downstream Yellow River represents a typical compound river channel, where the production embankment areas on both sides enclose the river channel, with vast floodplains between the main embankments [4]. Since the beginning of this century, changes in water and sediment flow and the regulation of the Xiaolangdi Reservoir have caused new alterations in the water and sediment characteristics entering the downstream channel. There has been a substantial decrease in sediment loads and a noticeable increase in low-flow processes [5]. Erosion in the downstream channel has intensified, with maximum average erosion depths reaching around 3 m. Inadequate alignment between channel management projects and water–sediment processes has led to substantial adjustments in local river stretches, resulting in deformed river profiles and instances of bank collapses and village destruction [6]. Despite not experiencing significant floods in over twenty years, changes are occurring on the floodplain due to human activities [7].

The transverse gradient of the floodplain surface is an important indicator reflecting the development of a “secondary hanging river” [8]. Similar to the water surface transverse gradient, the transverse gradient of the floodplain refers to the ratio of the elevation difference between the floodplain lip and the embankment toe to the horizontal distance between them [9]. The variation in transverse gradient largely depends on water and sediment conditions, channel and floodplain management projects, cross-sectional morphology, river-related factors, erosion and silting changes due to channel flooding, changes in floodplain land use, and flood routing along embankment toes. Influenced by water and sediment conditions, engineering projects, channel erosion and siltation, river-related parameters, and human activities, significant changes also occur in the transverse gradient of the floodplain [10]. This has a major impact on flood inundation, where a higher transverse gradient can lead to the rapid conversion of gravitational potential energy into kinetic energy during the flood evolution process, resulting in increased flow velocity and inundated areas. An appropriate transverse gradient not only facilitates material and energy exchange between the floodplain and the river channel and promotes vegetation growth in the floodplain but also helps avoid flood routing along the embankment during flood events. Chinese scholars have utilized measured cross-sectional data to analyze trends in transverse gradient variations, conducting spatial and temporal studies on the degree of change in transverse gradients. Zhang Jinliang [11] conducted a temporal and spatial analysis of changes in the transverse gradient on both sides of the High Cun-Taochengpu river section. They found that since 2000, there has been an increase in the transverse gradient, with the ratios concentrated between 0.5‰ and 2‰ along the course, where the transverse gradient of the left bank was slightly larger than that of the right bank. Liu Yan weighted the difference in height between the floodplain and channel along the Dongbatou-Gaocun section and found that the “secondary hanging river” appeared in 1972 and has persisted since then. It is widely recognized among domestic scholars that the development of secondary hanging rivers in the Gaocun-Taochengpu river section is severe [12]. Many scholars have investigated the impact of various factors on transverse gradients. Xu Linjuan studied flood processes and the development of secondary hanging rivers in meandering river sections, finding that a sand influx coefficient less than 0.1 and a flooding coefficient greater than 0.29 can effectively restrict the formation of secondary hanging rivers [13]. Yang Jishan, analyzing longitudinal cross-sectional data and embankment conditions, highlighted how embankments limit sediment accumulation areas and play a significant role in increasing the transverse gradient. In general, an increase in the transverse gradient is the result of adverse water and sediment conditions combined with boundary conditions [14]. Changes in water and sediment conditions are the primary reasons for a significant increase in the transverse gradient, while human activities within the river channel, particularly the construction of embankments, alter flow conditions, leading to a drastic rise in the transverse gradient, thus exacerbating the development of secondary hanging rivers [15]. The specific manifestations of transverse gradients include deep channels, low floodplains, and depressions at the base of embankments. During floods, this can easily trigger the formation of transverse flows, oblique flows, and rolling currents, resulting in flood overtopping of embankments and causing flood routing along embankments, posing a serious threat to flood defenses [14]. Moreover, the swift flow and deep water near embankment toes can lead to dangerous situations, such as seepage and piping at the back of large embankments. Sun Dongpo established a two-dimensional mathematical model for water and sediment transport, studying the effects of different suspended sediment loads on flood evolution in the floodplain, flow structures in the flood channel, and flood routing along embankments [16]. The results indicated that increasing suspended sediment loads could heighten the risk of embankment erosion. Once they exceeded critical values, they could significantly affect flow velocities along embankments. The results of an experimental project for controlling secondary hanging rivers in 2003 revealed that reducing the transverse gradient to some extent decreased the likelihood of flood inundation, flood routing along embankments, transverse and oblique flows, and rolling currents occurring in the study river section [17].

The evolution of floods refers to the process of flood waves propagating and evolving along river channels (or diversion areas or detention areas) [18]. After flooding on the floodplain, there are roughly two modes of evolution on the floodplain surface: the triangular floodplain exchange mode or the elongated floodplain exchange mode. The former indicates that water first enters from the triangular floodplain area, undergoes sediment deposition and accumulation on the floodplain, and then returns to the river channel. The latter mode involves water flowing into the floodplain at the breach connecting the river engineering structures and embankments, gradually advancing toward the near-bank floodplain, and resulting in flood routing along the embankment or longitudinal flooding along the floodplain. It then returns to the main river at the next weak point in the engineering structure or embankment. Conducting research on floodplain evolution requires calculations and studies on water and sediment transport processes in compound river channels [19]. To determine the characteristics of floodplain water flow evolution, scholars both domestically and internationally have conducted in-depth research. Wang Shudong, starting from the turbulent Reynolds transport equation, combining turbulence theory with floodplain water flow characteristics, derived analytical solutions for the two-dimensional distribution of point flow velocities [20]. Shinon and Knight proposed the SKM method, which integrates the turbulent Reynolds transport equation along the water depth. They derived the motion equations of floodplain water flow based on the Boussinesq assumption and the Darcy–Weisbach formula [21,22]. Zong Hucheng, in a study on solving floodplain water flow velocities, incorporated the factor of the transverse gradient into the model, validating the driving role of the transverse gradient in the floodplain evolution process [23]. Li Xizhi simulated the floodplain evolution process at a specific location in Langyuan using the Delft3D model. They observed that there was a buffer zone near the river channel before widespread floodplain inundation. Floodwaters first gathered in the vicinity of the river channel before advancing extensively onto the floodplain, entering a converging state prior to flood occurrence. At the middle stage of flood evolution, floodwaters spread along a certain trajectory within the floodplain. In the later stages of floodplain inundation, the floodwaters were insufficient to support the expansion of the flooded area [24]. Zhang Xiaolei constructed a two-dimensional mathematical model to explore the evolution of floodwaters from breached embankments on the floodplain with different roughness values. The results indicated that the time taken for the flood wave front to reach various measurement points in the floodplain increased gradually with higher roughness values and decreased gradually with increasing flood levels. Flow velocities at various measurement points in the floodplain showed an initial increase followed by a decrease, with the rate of increase in velocity being significantly greater than the rate of decrease [25]. Guo Peng established a two-dimensional mathematical model to simulate the progress of floodwaters resulting from embankment breaches on the floodplain [26]. The study demonstrated that, driven by the transverse gradient, water discharged from breaches spread laterally across the floodplain, eventually reaching the main levee of the Yellow River, posing a threat to its safety. When foreign scholars simulate flood evolution in floodplains, they generally do not consider sediment transport and bed deformation processes, with most of them utilizing diffusive wave models [27,28]. Caleffi used a two-dimensional shallow-water equation to simulate flood evolution in the Toce River [29]. Bates, based on high-precision digital elevation data, calculated the flood evolution process in the Severn River floodplain using a diffusive wave model [30]. Currently, research on floodplain evolution driven by the transverse gradient mainly focuses on theoretical derivation and experimental simulation. In terms of theory, studies have deduced the movement of floodwaters on the floodplain under the influence of the transverse gradient and analyzed flow velocity distributions.

Previous research has made some progress in understanding the lateral gradient of the floodplain and the flood evolution within the floodplain driven by this gradient. However, the terrain and geomorphology of the Yellow River’s downstream floodplain are heavily influenced by human activities, leading to complex variations. Most existing studies focus on the spatiotemporal distribution characteristics of the lateral gradient, its driving factors, and its impacts on flood management. Few scholars have explored how changes in the lateral gradient affect the progression of overbank flooding. Further research is needed to determine how to select an appropriate lateral gradient to accommodate changes in the floodplain during flood events. Flood risk assessment involves establishing an evaluation system with a multi-level, multi-indicator structure. This system standardizes each indicator value, assigns weights to the indicators, and performs superposition operations on them. Evaluating the differences in flood risk levels in different areas of the study region after these processes becomes a crucial method for assessing the extent of losses caused by floods in disaster-affected areas [31,32]. This study established a comprehensive and reliable flood risk assessment model to explore the impact of changes in the lateral gradient of the floodplain on the progression of overbank flooding. By utilizing the flood inundation results within the floodplain, this study provides theoretical support and a decision-making basis for optimizing the threshold values of the lateral gradient. The objective of this study is as follows: (1) establish a two-dimensional water and sediment model for the Gaocun–Sunkou river section to simulate flood evolution with different transverse gradients; (2) develop a flood risk assessment model suitable for floodplains and apply it to the Gaocun–Sunkou river section to calculate the spatial distribution of flood risk under different transverse gradients; (3) based on the results of the flood risk assessment, select a more reasonable transverse gradient.

2. Materials and Methods

2.1. An Overview of the Study Area

The Gaosun River section (located between 115°39′ to 116°08′ E and 35°43′ to 36°02′ N) spans the region between the Gaocun and Sunkou cross-sections, covering parts of Henan and Shandong provinces (Figure 1). This river segment is 118.2 km long with a channel slope of 1.11‰. It represents a transitional river section from a meandering to a sinuous morphology and is characterized by wide floodplains. The river consists of a compound channel formed by both the main channel and the floodplain. The evolution of the river shows dual characteristics [33], preserving the basic features of a meandering river, including scattered flow, significant swing amplitudes, rapid changes, and relatively straight channels. Simultaneously, it exhibits characteristics of sinuous rivers with single-channel bends and minor variations in the main channel’s curvature in the plan view. The river’s planform alternates between straight and curved sections, featuring mostly straight and elongated river segments such as Liuzhuang-Susizhuang and Yingfang-Penglou, each exceeding a length of 8 km. Due to its typical compound channel features, moderate to small flows pass through the main channel, while during major flood events, floodwaters need to inundate the floodplain, with overland flow occupying around 20% of the entire cross-section due to the presence of broad floodplains.

In the Gaosun River section, the channel is close to the main embankment, and there is a dense population of channel engineering structures [34]. The average distance between embankments on both sides is 4.5 km. The floodplain area between the production embankment and the main embankment is approximately 540 km², with the left bank and right bank floodplain areas covering 321 km² and 139 km², respectively. The floodplain at the base of the embankments is low-lying, contains numerous sandbars, and has uneven surfaces with many depressions (Figure 2). Some areas of the floodplain have difficulty draining, leading to the potential formation of stagnation zones during inundation events, showcasing significant water storage capacity [2]. The average transverse gradients of the left and right bank floodplains are 9.80‰ and 10.39‰, respectively [35].

2.2. Data Processing

2.2.1. Natural Geographic Data

The Digital Elevation Model (DEM) data are sourced from the Shuttle Radar Topography Mission (SRTM) data by the National Aeronautics and Space Administration (NASA). It can be downloaded from the Chinese Academy of Sciences Geographic Spatial Data Cloud website (http://www.csdb.cn/index.html#, accessed on 1 September 2023), with a grid size of 30 m × 30 m. The data type is raster data.

2.2.2. Land Use Data

The land use data used in this study are based on the National Resource and Environment Database of the Institute of Geographic Sciences and Natural Resources Research, Chinese Academy of Sciences. It utilizes Landsat satellite remote sensing image data from the United States as the primary information source and establishes a multi-period land use/land cover remote sensing monitoring database (CNLUCC) through manual visual interpretation.

2.2.3. Remote Sensing Image Data

The Landsat series data are from the joint program by NASA and the U.S. Geological Survey, providing the longest continuous space-based observation records globally. The spatial resolution is 30 m, and it can be downloaded from the U.S. Geological Survey website (https://earthexplorer.usgs.gov/, accessed on 1 September 2023).

2.2.4. Hydrometeorological Data

To monitor changes in riverbeds, the Yellow River Conservancy Commission established fixed cross-sections along the lower reaches of the Yellow River in 1950. Professional personnel conduct cross-sectional measurements before and after the flood season each year. Additionally, they regularly monitor the water and sediment conditions of the river channel during both the flood and non-flood seasons according to different standards, documenting and organizing the data. The Yellow River Conservancy Commission publishes technical standards and checks the data to ensure measurement quality, focusing mainly on water level, flow rate, and sediment concentration data at the Gaocun and Sunkou cross-sections.

2.2.5. Cross-Sectional Data

A total of 38 hydrological cross-sections are established within the study area. Actual measured cross-sectional data before and after the flood season for each section are collected to reflect changes in the cross-sectional morphology at different times. The method of cross-section analysis was employed to calculate erosion and deposition within the river section.

2.3. Calculation of Transverse Gradient

In order to facilitate calculations, the river channel cross-sections were generalized for the Gaosun River, a compound channel characterized by a narrow and deep main channel with broad floodplains and a significant transverse gradient. Referencing the longitudinal gradient formula for rivers, the transverse gradient of the floodplain was computed as shown in Figure 3. In Figure 3, b1 represents the width of the river, and b2,L and b2,R are the widths of the left and right bank floodplains, respectively, while HL and b2,R indicate the elevation differences between the lip of the floodplain and the base of the embankment on the left and right banks. To minimize errors in transverse gradient calculations near the lip of the floodplain and the base of the embankment due to local variations, an average elevation of 100m was used for these locations. Therefore, the transverse gradient of the floodplain can be calculated based on Equations (1) and (2):

$$TGL={\displaystyle \frac{h_L}{b_{2,L}}}$$

(1)

$$TGR={\displaystyle \frac{hR}{b_{2,L}}}$$

(2)

2.4. Flood Risk Assessment Model

2.4.1. Selection of Model Indicators

In order to construct a flood risk assessment model specifically for the Gaosun River section’s floodplain, the Web of Science (WOS) core collection database was used as the platform for collecting English literature research data, while the CNKI database served as the platform for collecting Chinese literature research data. The search term “Flood Risk Assessment” was employed, and the time span from 2018 to 2023 was selected to ensure the accuracy of the chosen indicators. Academic papers published since 2018 were filtered, with “Article” as the document type. Through manual screening and the elimination of less relevant subject entries, a total of 183 articles were gathered as the data source for knowledge graph analysis. Utilizing the professional literature analysis tool VOSviewer v1.6.21, this study conducted a visual analysis of the selected indicators from the retrieved literature to identify evaluation indicators used in flood risk assessment studies. By integrating the results of clustering and co-occurrence analyses of hotspots in the literature, appropriate evaluation indicators were selected to establish an assessment system tailored to the floodplain of the Gaosun River section. The data content related to the evaluation indicators was derived from the measured data of the Yellow River Institute of Hydraulic Research. It was preliminarily screened and deemed relatively detailed and reliable.

Based on the indicators drawn in the heat map shown in Figure 4, it is evident that current research on flood assessment is mainly focused on urban flood disasters, watershed floods, and coastal storm surges. These indicators can generally be categorized into four classes: source, path, acceptor, and consequence. The indicators related to flood propagation paths often pertain to natural attributes, such as topography and landforms, while those linked to flood acceptors mainly revolve around societal attributes like population structure. However, the Tan District of Gaosun River involves three major socioeconomic industries, and its post-disaster resilience, to some extent, reflects the level of flood risk in the district. Considering that the study area is the Tan District of Gaosun River, relevant indicators of resilience were incorporated into the aspects of source, flood propagation paths, flood acceptors, and consequences. Building upon the indicators presented in the heat map, additional indicators with characteristics unique to the district were included to establish an indicator system. A total of 14 indicators were selected, and the constructed indicator system is shown in

Table 1. Construction table of evaluation indicators.

Criteria	Indicator	Explanation	Attributes
Source	Waterbody	Identifies whether the water body within the area is classified as a water body to determine whether it would serve as a source of flooding during inundation. The value can be either 1 or 0.	Positive
	Distance from channel	By calculating the distance of each point within the area to the river channel, the likelihood of inundation at each point can be determined based on the proximity to the river. Generally, the farther a point is from the river channel, the less likely it is to be inundated.	Negative
Path	Elevation	By assessing the elevation of each point within the area, it can be determined whether it is susceptible to flooding. Generally, lower-elevation points are more prone to inundation.	Negative
	Roughness	A dimensionless parameter reflecting the influence on water flow resistance. The rougher the boundary surface, the higher the roughness coefficient, resulting in slower water flow; conversely, the smoother the boundary surface, the lower the roughness coefficient, leading to faster water flow.	Positive
Acceptor	NDVI	A standardized index used to generate images displaying the vegetation amount (relative biomass). It can reflect losses in inundated areas during floods.	Positive
	NDWI	The NDWI is typically used to extract water body information from images, reflecting water bodies within inundated areas and displaying flood losses.	Positive
	Population density	Population density is the number of people per unit area of land and is an important indicator for measuring the distribution of population in inundated areas.	Positive
	Imperviousness	A crucial indicator for identifying impermeable surfaces and can reflect the infiltration situation in inundated areas after floodplain inundation.	Positive
	GDP per unit	Gross Domestic Product (GDP) per unit area in the study area represents the economic status of the inundated area.	Positive
	Nighttime light index	The nighttime light index is based on the image data of human nighttime activities and production extracted using satellite remote sensing and data analysis techniques. Economically developed and densely populated areas often shine brightly at night, resulting in a high nighttime light index. Conversely, economically underdeveloped and sparsely populated areas exhibit dim or no nighttime lights, leading to a low nighttime light index.	Positive
Consequence	Floodwater depth	The data derived from the results of a two-dimensional hydro-sediment model can output the water depth across the entire computation area. In reality, the water depth is the difference between the calculated water surface elevation and the elevation in the Digital Elevation Model (DEM) below it. The greater the submerged water depth, the more significant the resulting damages.	Positive
	Submergence duration	Data exported from the simulation results of a two-dimensional water–sediment model can output the submergence duration of the flooded area. The longer the submergence duration, the greater the resulting damage.	Positive
	Flood flow velocity	Data exported from the simulation results of a two-dimensional water–sediment model can output the flood flow velocity of the inundated area. The higher the flow velocity, the more dangerous the inundated area becomes, resulting in greater losses.	Positive
Resilience	NDBI	A remote sensing index used to identify the distribution of buildings in urban areas, capable of identifying the distribution of buildings.	Negative

.

2.4.2. Determination of Indicator Weights

Currently, there are three main types of comprehensive quantitative evaluation methods commonly used in domestic and international contexts [36]. These include methods based on deterministic and uncertain information for quantitative or qualitative evaluations [37,38], such as the Analytic Hierarchy Process (AHP) and Gray Relational Analysis. Statistical-law-based methods [39,40] include Principal Component Analysis and Cluster Analysis, among others. Methods based on goal programming [41,42] include the Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS). Ultimately, the three types can be classified, according to objective and subjective factors, into subjective weighting methods, objective weighting methods, and combined weighting methods. The AHP is selected for subjective weighting as the subjective approach, and the CRITIC method (Criteria Importance Though Inter criteria Correlation) is chosen for objective weighting as the objective approach, followed by using game theory methods for combined weighting [43]. Utilizing this comprehensive weighting method not only leverages the information inherent in the data but also integrates the expertise of industry specialists, resulting in more credible weights [44].

This study employed a normalization method to standardize the raw data, aiming to reduce the differences between various indicator variables [45].

If $x_{\mathrm{j}}$ is a positive indicator (“the larger, the better” type of indicator), then

$$x_{\mathrm{i}\mathrm{j}}={\displaystyle \frac{n_{\mathrm{i}\mathrm{j}}-\mathrm{m}\mathrm{i}\mathrm{n}\left(n_{\mathrm{j}}\right)}{\mathrm{m}\mathrm{a}\mathrm{x}\left(n_{\mathrm{j}}\right)-\mathrm{m}\mathrm{i}\mathrm{n}\left(n_{\mathrm{j}}\right)}}$$

(3)

If $x_{\mathrm{j}}$ is a negative indicator (“the smaller, the better” type of indicator), then

$$x_{\mathrm{i}\mathrm{j}}=\frac{\max \left(n_{\mathrm{j}}\right)-n_{\mathrm{i}\mathrm{j}}}{\max \left(n_{\mathrm{j}}\right)-\min \left(n_{\mathrm{j}}\right)}$$

(4)

The formula for the AHP is as follows:

$$\mathrm{A}={\left({\mathrm{a}}_{\mathrm{i}\mathrm{j}}\right)}_{\mathrm{n}\times \mathrm{n}}=\left[\begin{array}{ccc}{\mathrm{a}}_{11}& \cdots & {\mathrm{a}}_{1\mathrm{n}}\\ ⋮& \ddots & ⋮\\ {\mathrm{a}}_{\mathrm{n}1}& \cdots & {\mathrm{a}}_{\mathrm{n}\mathrm{n}}\end{array}\right]$$

(5)

$$\overline{{\mathrm{W}}_{\mathrm{i}}}=\sqrt[\mathrm{n}]{\prod _{\mathrm{j}=1}^{\mathrm{n}}{\mathrm{a}}_{\mathrm{i}\mathrm{j}}}\left(\mathrm{i},\mathrm{j}=\mathrm{1,2},\cdots \mathrm{n}\right)$$

(6)

$${\mathrm{W}}_{\mathrm{i}}={\displaystyle \frac{\overline{{\mathrm{W}}_{\mathrm{i}}}}{{\sum }_{\mathrm{i}=1}^{\mathrm{n}}\overline{{\mathrm{W}}_{\mathrm{i}}}}}\left(\mathrm{i}=\mathrm{1,2},\cdots \mathrm{n}\right)$$

(7)

where ${\mathrm{a}}_{\mathrm{i}\mathrm{j}}$ is the ratio scale, indicating the relative importance of element i compared to element j at the same level; $\overline{{\mathrm{W}}_{\mathrm{i}}}$ represents the n-th root of the product of the ratios; and W_i is the weight value.

The AHP consistency test is performed as follows:

$$\mathrm{C}\mathrm{I}={\displaystyle \frac{{\mathsf{\lambda }}_{\mathrm{m}\mathrm{a}\mathrm{x}}}{\mathrm{n}-1}}$$

(8)

$$\mathrm{C}\mathrm{R}={\displaystyle \frac{\mathrm{C}\mathrm{I}}{\mathrm{R}\mathrm{I}}}$$

(9)

where ${\mathsf{\lambda }}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ denotes the maximum eigenvalue of the judgment matrix; CI is the consistency index; CR is the consistency ratio; and RI is the average random consistency index. When CR < 0.1, the judgment matrix A is considered to have satisfactory agreement.

The formula for CRITIC is as follows:

$${\sigma }_j={\displaystyle \frac{\sqrt{{\sum }_{i=1}^e(x_{i{j}^{’}}-\overline{x_{j’}})}}{e-1}}$$

(10)

$$f_j={\sum }_{i=1}^e(1-r_{ij})$$

(11)

$$C_j={\sigma }_j{f}_j$$

(12)

$$w_j={\displaystyle \frac{C_j}{{\sum }_{j=1}^f{C}_j}}$$

(13)

In the formulas, ${\sigma }_j$ represents the distinctiveness of the j-th criterion, $f_j$ denotes the discordance of criterion j with the other criteria, $r_{ij}$ is the Pearson correlation coefficient between criteria i and j, $C_j$ signifies the information load of the j-th criterion, and $w_j$ denotes the weight.

The process of the integrated weighting calculation based on game theory is as follows:

$$\overline{W}=a_1{w}_1+a_2{w}_2, { a}_1+a_2=1,{ a}_1>0,{ a}_2>0$$

(14)

$$min‖{\sum }_{i=1}^2{a}_i{w}_i^T-w_j^T{\left|\right|}_2$$

(15)

$$\left[\begin{array}{cc}{w}_1{w}_1^T& w_1{w}_2^T\\ w_2{w}_1^T& w_2{w}_2^T\end{array}\right]\left[\begin{array}{c}{a}_1\\ a_2\end{array}\right]=\left[\begin{array}{c}{w}_1{w}_1^T\\ w_2{w}_2^T\end{array}\right]$$

(16)

where $a_1$ and $a_2$ represent the combination coefficients, and W denotes the comprehensive weight of the indicators.

2.5. Two-Dimensional Water–Sediment Model

In a flood risk assessment system, indicator data related to the source, the path, the acceptor, and resilience can be obtained from remote sensing images and statistical yearbooks. These data can be processed using spatial analysis techniques to convert them into raster data for flood risk assessment. Indicator data related to the consequences of flooding require conducting flood inundation simulations through the construction of a two-dimensional water–sediment model [46]. Below are the basic control equations and solution methods for the two-dimensional water–sediment model, along with an explanation of how key issues in the model are addressed, aiming to provide technical support for flood risk assessment in the Tan District of Gaosun River.

2.5.1. Hydrological Data

The Yellow River has not witnessed a large-scale flooding event in the floodplain for nearly 20 years. In recent years, a significant flooding incident occurred during the autumn flood of 2003, mainly affecting the Landon floodplain area. Detailed records are available for this event. In 1996, a major flood inundated the downstream floodplain areas of the Yellow River, impacting as many as 804 villages, with 830,000 affected individuals and 1.46 million mu (approximately 243,333 acres) of flooded farmland, resulting in significant economic losses to the floodplain region. The hydrological data collected during the 1996 flood event are quite comprehensive, with complete flow, water level, and sediment concentration data available at the Gao Village and Sunkou sections. These datasets meet the requirements for conducting two-dimensional water–sediment modeling. Additionally, Landsat 5 satellite imagery offers high-precision remote sensing images that vividly depict the inundation situation in the Gaosun River section of the floodplain region during this flood event. This imagery can be used to validate the model accuracy. Please refer to Figure 5 for the satellite image data.

2.5.2. Roughness Settings

According to previous research findings and the “Hydrological Yearbook of the People’s Republic of China” (hydrological data for the Yellow River Basin),

Table 2. The roughness coefficients for different land use types.

Type	Range
Farm	0.02–0.06
Forest	0.03–0.2
Grassland	0.02–0.05
Water	0.02–0.035
Floodplain	0.02–0.038
Building	0.025–0.07
Unuse	0.02–0.06

lists the roughness coefficients for various land use types.

2.5.3. Grid Division

In numerical simulations, the computational grid typically represents the discretization of irregular physical calculation regions. The number of grids, geometric shapes, grid quality, density, etc., all directly affect the accuracy and efficiency of numerical analysis. Based on the shape generated by generating 2D planar grids, they can be divided into quadrilateral grids and triangular grids. Triangular grids are suitable for complex regions. In the model, structured grids or unstructured grids, or quadrilateral grids or triangular grids, can be used based on the complexity of the physical region’s boundaries. Considering the morphology of the Gao Sun River section, triangular grids were selected. The determination of the grid size needs to be comprehensively considered from the perspective of both model accuracy and computational performance. Smaller grids result in higher model accuracy but longer simulation times, requiring higher computational performance. Due to different simulation focuses, different grid sizes were chosen for local refinement for the river channel and floodplain in the study area. The grid is shown in Figure 6.

2.6. Optimal Cross-Slope Selection Scheme

In order to express the severity of the transverse gradient of the beach surface, experts and scholars often use multiple relationships between the transverse gradient and longitudinal gradient. To differentiate between the impacts of changes in the transverse gradient on hydrological connectivity in the Gao Sun River section, this study categorizes the transverse gradient into three levels, namely, a low transverse gradient, a medium transverse gradient, and a high transverse gradient, based on the ratio relationship between the longitudinal gradient and transverse gradient of the river section. The classification criteria are shown in

Table 3. Criteria for dividing transverse gradient levels.

	Low TG	Medium TG	High TG
Relationship	TG$\le 12$LG	12LG < TG $\le$ 17LG	17LG$<$TG

Note: TG represents the transverse gradient, while LG indicates the longitudinal gradient.

.

Calculations showed that the longitudinal gradient of the Gao Sun River section is 0.125‰. Based on the calculated results of the transverse gradient in the Gao Sun River section, the ratio relationship between the transverse gradient and the river channel’s longitudinal gradient was computed. Different simulation scenarios were then set up based on this ratio relationship. As shown in

Table 4. Flood simulation scenario settings.

Number	Type of Flood	TG	Explanation
1	Flood of 1996	TG of 2000	Simulating a real flood to validate the model’s feasibility for calculating the inundation of floodplains.
2		Low TG	Setting the TG to be 10 times the LG of the river channel, simulating the flooding and free evolution process toward a two-dimensional plane. This helps determine the driving effect of low TG on the flood evolution process.
3		Medium TG	Setting the TG of the floodplain to be 14 times the LG of the river channel, simulating the flooding and free evolution process toward a two-dimensional plane. This helps determine the driving effect of medium TG on the flood evolution process.
4		High TG	Setting the TG of the floodplain to be 18 times the LG of the river channel, simulating the flooding and free evolution process toward a two-dimensional plane. This helps determine the driving effect of high TG on the flood evolution process.

, during the flood inundation simulation process in the Gao Sun River section, the transverse gradient was sequentially set to low, medium, and high levels according to the ratio relationship between the transverse gradient and the longitudinal gradient for flood simulations.

3. Results

3.1. The Trend of TG Variation in the Floodplain

Based on the measured cross-sectional data of the Gaosun River before and after the flood, the TG of the river section from 1986 to 2021 was calculated. The calculation results are shown in Figure 7. The average transverse gradients before and after the flood over the past 35 years were 1.271‰ and 1.272‰, respectively. The long-term average transverse gradient before the flood is slightly lower than after the flood. Before the flood, the long-term average transverse gradients in the early, middle, and late stages of the Xiaolangdi Reservoir construction were 1.255‰, 1.057‰, and 1.369‰, respectively. After the dam was built, the average transverse gradient increased by 9.08% compared to before the dam construction. After the flood, the average transverse gradients in the early, middle, and late stages of the Xiaolangdi Reservoir construction were 1.221‰, 1.083‰, and 1.384‰ respectively, with an increase of 13.35% in the transverse gradient after dam construction compared to before. The operation of the Xiaolangdi Reservoir changed the water and sediment characteristics of the river section, affecting the erosion and sedimentation evolution of the river section and reducing the probability of flooding in the floodplain. This results in more sediment deposition at the flood lip, leading to an increase in the transverse gradient of the floodplain.

The operation of the Xiaolangdi Dam has increased the transverse gradient of the Gaosun River section. The change cycles and trends of the transverse gradient before and after the flood are quite consistent. Before 1997, it was in a state of fluctuating decrease, but it rapidly increased and stabilized thereafter. The main difference between the transverse gradients before and after the flood is that the interannual variability of the transverse gradient after the flood is more unstable. This is because the transverse gradient after the flood is more easily influenced by flood erosion and sediment deposition.

3.2. Optimal Selection of TG of the Floodplain

3.2.1. Flood Inundation Results

Based on the low, medium, and high transverse gradient scenarios described in Table 4, a two-dimensional water–sediment model was constructed to simulate flood inundation in different scenarios. Figure 8, Figure 9 and Figure 10 show the distribution of the total water depth during the evolution of flood inundation under different transverse gradient conditions. For comparison purposes, only the flooded areas are displayed.

Based on the simulation results, compared to the low transverse gradient scenario, the inundation area with a higher water depth slightly increases in the medium transverse gradient scenario, while the inundation area with a lower water depth decreases. Additionally, the inundation water depth at the embankment toe is significantly larger in the medium transverse gradient scenario compared to the floodplain results in the low transverse gradient scenario. On the other hand, the increase in the inundation area is more pronounced in the high transverse gradient scenario, especially in the area where the water depth exceeds 1.8 m. The areas with higher inundation water depths driven by medium and high transverse gradients are mainly concentrated at the embankment toe. It is easier for flooding to occur along the embankment in the medium and high transverse gradient scenarios compared to the low transverse gradient scenario. This is because, after floodplain inundation, the gravitational driving force accelerates the flow due to topographical effects, leading to higher flow velocities.

3.2.2. Determination of Indicator Weights

The combined weights of each evaluation indicator were calculated based on the calculation steps described earlier, as shown in

Table 5. Combined weights of each evaluation indicator.

Criteria Source	Indicator Waterbody	Subjective Weights (%)	Objective Weights (%)	Weight Coefficients		Combined Weights (%)
				α1	α2
Path	Distance from channel	11.542	9.637	0.428	0.572	10.453
	Elevation	5.157	6.933			6.173
Acceptor	Roughness	2.558	5.213			4.077
	NDVI	2.218	6.727			4.797
Consequence	NDWI	1.352	5.341			3.634
	Population density	9.94	3.034			5.990
	Imperviousness	12.466	1.552			6.223
	GDP per unit	3.328	10.184			7.250
	Nighttime light index	3.87	2.782			3.247
	Floodwater depth	3.305	2.227			2.688
Resilience	Submergence duration	13.254	10.734			11.813
	Flood flow velocity	13.254	14.836			14.159
	NDBI	13.254	6.520			9.402
Criteria	Indicator	4.503	14.280			10.095

.

3.2.3. Flood Risk Zoning

For different transverse gradients, the flood risk is mainly analyzed by changing the transverse gradient to analyze flood inundation. This involves altering the inundation depth, duration, and extent in the results, achieving the flood risk assessment of the floodplain with different transverse gradients for the same event (i.e., the autumn flood of 2003). Visualizing the flood risk level in the floodplain provides a basis for optimizing the threshold of the transverse gradient. Based on the comprehensive weights mentioned above and actual data, the flood risk assessment results were calculated using a raster calculator, as shown in Figure 11, Figure 12 and Figure 13.

To evaluate the flood risk in low, medium, and high scenarios, it is essential to assess the area proportions of different risk levels and analyze the changes in the areas of medium- to high-risk zones to determine the optimal transverse gradient for the three scenarios. By considering these two perspectives, one can make informed decisions on the most suitable transverse gradient for each scenario based on the distribution of risk levels and changes in medium- to high-risk areas.

(1) Assessment of flood risk with low TG

With a low transverse gradient, medium- and high-risk levels are mainly distributed in the river channel, while the flood risk level in the floodplain is mostly low or below. According to the statistical results, in the Gao Sun River segment, there are 43 blocks classified as high risk and 3 blocks as very high risk, accounting for only 7.64% of the total river segment area. These high-risk areas are primarily located within the river channel, with lower flood risks near the embankments, resulting in relatively low exposure to flood risks. The majority of the study area falls within the medium risk level or below, with the largest area classified as low risk, covering 55.98% of the river segment area. During the autumn flood of 2003, the flood risk faced by the floodplain with a low transverse gradient was relatively low. Combined with the land use conditions mentioned in Section 3.2.4, areas where residents live and work, such as developed land and farmland, face minimal threats from floods, which is beneficial for the economic development of the floodplain region.

(2) Assessment of flood risk with medium TG

Compared to the floodplain conditions with a low transverse gradient, the areas classified as extremely low risk and low risk decrease with a moderate gradient. Meanwhile, the areas classified as medium risk and high risk increase by 18.73%. With a moderate transverse gradient, the low-risk and medium-risk areas constitute the largest proportions in the total river segment area, accounting for 44.11% and 42.16%, respectively, while the high-risk area proportion is 11.27%, indicating an increase of 4.23% compared to the low transverse gradient scenario. Apart from the river channel, significant areas of high-risk and very high-risk zones are observed in the Gao Cun section and near the Sun Kou section. The area near the Sun Kou section has densely developed land, making it susceptible to flood threats. Under the influence of a moderate transverse gradient, after flooding overflows into the floodplain, some areas experience increased flood risks or even transition to higher-risk categories. This situation impacts flood control measures in the vicinity of the Gao Cun section’s and the Sun Kou section’s floodplain areas.

(3) Assessment of flood risk with high TG

With the high transverse gradient, the area of high-risk zones in the river segment dramatically increases, accounting for 26.35% of the total area. The high-risk areas are 3.74 times greater with high transverse gradients compared to those with low transverse gradients. High-risk zones and very high-risk zones also appear in the mid-section floodplain of the river segment. These high-risk areas are characterized by flat terrain, close proximity to the river channel, high population density, developed socioeconomic conditions, and predominantly agricultural land use. As these areas serve as flood buffer zones, they are highly susceptible to adverse effects from flooding. Consequently, they face significant flood control pressures.

3.2.4. The Optimization of the TG for the Floodplain

The flood risk assessment results in different scenarios indicate that, compared to a low transverse gradient, the gravitational driving force for flood inundation is more pronounced with moderate and high transverse gradients, increasing the flood risk faced by the floodplain. For floods of similar magnitudes, the areas classified as low risk and below with moderate and high transverse gradients decreased by 18.73% and 27.66%, respectively, compared to those with a low transverse gradient. The areas classified as medium risk increased by 14.50% and 8.02% with moderate transverse gradients compared to those with low transverse gradients. Additionally, the high-risk and above areas increased by 4.23% and 19.64% with high transverse gradients compared to those with low transverse gradients. Therefore, considering the different scenarios analyzed during the research process, it is most advantageous for floodplain flood safety to have a low transverse gradient (i.e., TG = 10LG = 1.25‰) in place when flash flooding occurs.

4. Conclusions

Based on historical cross-sectional data from the downstream Gao Sun River segment of the Yellow River, calculations and analyses of the transverse gradient trends on the floodplain were conducted. A flood risk assessment model for the Gao Sun River segment was constructed with a two-dimensional hydro-sediment model as the core. Different scenarios of transverse gradients were established to simulate floods and evaluate flood risk levels in the floodplain areas. A comparative analysis of flood risk areas with low, moderate, and high transverse gradients was carried out to determine the most favorable transverse gradient for flood protection in the floodplain. The results indicate the following:

(a) Over the past 35 years, the average pre-flood season and post-flood season transverse gradients on the floodplain were 1.271‰ and 1.272‰, respectively. Due to the operation of the Xiaolangdi Reservoir, the transverse gradient increased by 9.08% before the flood season and increased by 13.35% after the flood season.

(b) The mechanism by which water–sediment conditions and sedimentation influence the transverse gradient differs from the impact of river channel sedimentation control projects. Water overflows onto the floodplain, causing a rapid decrease in sediment-carrying capacity, leading to significant deposition near the flood lip. Subsequently, the difference between the sediment concentration and the sediment-carrying capacity gradually decreases, resulting in reduced sediment deposition. Water–sediment conditions and channel sedimentation directly impede sediment exchange on the floodplain, while sediment control projects alter the path of water–sediment evolution on the floodplain, consequently changing local sedimentation patterns and influencing transverse gradient changes.

(c) With moderate to high transverse gradients, there is a higher likelihood of floodwaters flowing along embankments after overflowing onto the floodplain, posing a threat to levee safety. For floods of similar magnitude, the areas classified as low risk and below decreased by 18.73% and 27.66% with moderate and high transverse gradients compared to those with low transverse gradients. The medium-risk areas increased by 14.50% and 8.02%, while high-risk and above areas increased by 4.23% and 19.64% with moderate and high transverse gradients. A low transverse gradient (TG = 10LG = 1.25‰) is deemed more favorable for flood protection.

This paper focuses on the Gaosun River section of the lower Yellow River, analyzing the trends in lateral gradient variation in the river’s floodplains and their driving factors. Subsequently, considering flood prevention factors in the floodplain area, a threshold optimization scheme for the lateral gradient based on a flood risk assessment model is proposed. The scheme is evaluated by comparing the methodology itself and the optimization results, which appear to be relatively reasonable. However, due to limitations in time and resources, there are certain constraints in the current research. Future studies could improve on these results by incorporating flood risk assessments for different flood events and lateral gradient scenarios, thereby enhancing the continuity of scenario settings in the research process. In practical applications, it is necessary to conduct threshold studies of the lateral gradient along different sections of the lower Yellow River. In future research, the lateral gradient of major cross-sections can replace the average lateral gradient of the river sections. Assessing floodplain spillover risk using different flood frequencies allows for timely forecasting and warning. Flood risk assessments across various ranges of the lateral gradient can more precisely identify the most effective gradients conducive to flood prevention in floodplain areas.