Intelligent Parameter Fusion for Distributed Flood Modeling in Parallel Ridge–Valley Landscapes

Abstract

The pronounced spatial heterogeneity of underlying surface characteristics within the parallel ridge–valley system of eastern Sichuan necessitated hydrological discretization of the watershed into nested subdomains comprising inter-ridge valley units and secondary slope cells. A distributed flood simulation framework specifically adapted to parallel ridge–valley topography was developed, coupled with a sequential intelligent parameter optimization algorithm. Model validation was conducted through the simulation of ninety flood events (2015–2023) in the Lishui watershed, a representative parallel ridge–valley basin. For parameter-calibrated sub-watersheds, mean relative errors of 13.8% (peak discharge) and 12.3% (runoff depth) were achieved, while non-calibrated watersheds exhibited marginally higher inaccuracies at 14.6% and 15.1%, respectively. Spatial parameter estimation was effectively implemented through the assimilation of limited hydrometeorological station data. The integrated modeling framework, incorporating terrain-adaptive parameterization and intelligent calibration protocols, demonstrated high-fidelity flood process simulation capabilities in complex parallel ridge–valley landscapes.

1. Introduction

The eastern Sichuan parallel ridge–valley region (eastern margin of the Sichuan Basin), recognized as the world’s largest folded structural mountain system, exhibits a distinctive parallel ridge–hill–valley topography that exerts significant control over regional hydrological processes [1,2]. Research reveals that this geomorphology influences flood dynamics through three hydraulic mechanisms: (1) hydraulic constriction effects at structural concave fractures between ridges intensify flow incision and backwater elevation; (2) NW-SE-oriented mountain ranges obstruct moisture transport, triggering localized heavy-precipitation centers [3,4,5]; and (3) extreme topographic gradients induce pronounced spatial heterogeneity in the hydrological responses of underlying surfaces. These combined mechanisms result in nonlinear flood characteristics for reservoir inflows in folded mountainous areas, including short peak arrival times, high variability in flood magnitudes, and stochastic inter-event intermittency, posing substantial challenges to parameter inversion and flood modeling in conventional hydrological frameworks [6,7].

The Lishui watershed, situated on the eastern periphery of the parallel ridge–valley landscapes, was selected as a case study. While prior studies have analyzed historical flood events in the Lishui River Basin (e.g., the 27 August 2023 Sangzhi storm and 28 June 2020 Hefeng storm) and evaluated the flood mitigation benefits of reservoir projects (e.g., impacts of Jiangya and Zaoshi reservoirs on downstream flood risks) [8,9,10], critical gaps persist. Existing rainfall–runoff models inadequately systematize flow pathways across ridge–valley transitions, while parameter estimation often neglects spatial correlations arising from topographic continuity constraints. Furthermore, the hydrological effects of parallel ridge–valley geomorphology and their mechanistic interactions with runoff generation and concentration processes remain insufficiently explored [11,12].

In this study, the Lishui watershed was discretized into sub-watersheds based on mountain range orientation. This discretization facilitated simulations of slope water flow from parallel valleys to mountain cols and subsequent downslope movement. Key model parameters were estimated a priori using topographic characteristics. A hierarchical optimization method integrating intelligent algorithms was proposed to determine the spatial distribution of model parameters, with towns designated as key nodes. The simulation accuracies of flood processes at stations with limited observational data and under various storm scenarios were evaluated to validate the distributed model’s applicability to parallel ridge–valley landscapes.

2. The Study Area

Lishui River, with a drainage area of 18,583 km², is a major tributary of Dongting Lake. It is situated in the Wuling Mountains of northwest Hunan and southwest Hubei, China. The watershed is characterized by parallel southwest-to-northeast-oriented mountain ranges, resulting in a high northwest and low southeast topography [13,14]. The main streams of Lishui and its principal tributaries, Loushui and Xieshui, originate from mountainous regions in the northwest. Minor tributaries develop between parallel mountain valleys, converging at mountain breaks and cutting through ridges while receiving inflow from adjacent valleys. The main stream and tributaries converge at Shimen before flowing eastward into Dongting Lake (Figure 1).

3. Materials and Methods

3.1. Materials

The study area is humid, with a multi-year average rainfall of approximately 1724 mm, concentrated during the flood season (April–September). Hourly rainfall–runoff data from 118 monitoring stations (2014–2023) have been utilized in this study, including 3 reservoir stations, 10 hydrological stations, and 103 rainfall stations.

3.2. Methods

3.2.1. Generalization of Catchment Unit Applicable to Folded Mountainous Areas

Folded mountainous areas often feature parallel ridges and valleys. These landforms alternate between mountains and valleys. Rivers in these regions mainly originate from parallel valley systems. As the rivers develop, they cut through mountain ridges via erosion. After cutting through, the rivers continue to receive water from nearby parallel valleys. Flood hydrographs differ significantly in valley-bottom channels. This occurs when rainfall falls on opposite sides of steep mountain slopes. To study how parallel ridges and valleys affect runoff processes, we divided folded mountainous areas into valley units. The division used parallel ridges as boundaries. Each unit operates relatively independently. Units containing rivers were classified as river units (Figure 2).

3.2.2. Calculation of Production and Sink Flow in Folded Mountainous Areas

Based on DEM data, the generalized valley units were subdivided into orthogonal grid cells. The saturation-excess runoff mode was adopted for runoff generation computation within each grid cell, while different soil water outflow coefficient values were employed to reflect the influence of slope variation on the dynamic allocation of water flow between grid cells. The steep-slope gully flow formed after rainstorms within valley units was simulated using the kinematic wave equations considering gully density (Equations (1)–(3)), with grid-based runoff yield as input. The generated flow from each valley unit was routed through confluence nodes into channel networks and then transported through downstream valley units via the Muskingum routing method to reach river channel units. These flows were subsequently aggregated into main river channels and ultimately discharged at the watershed outlet.

$${\displaystyle \frac{\partial A}{\partial t}}+{\displaystyle \frac{\partial Q}{\partial x}}={\displaystyle \frac{q}{C_f}}$$

(1)

$$S_f=S_o$$

(2)

$$C_f=0.649D^2$$

(3)

where A and Q are the cross-sectional area (m²) and flow rate (m³/s) of the water flow in the primary channel, respectively, and q is the runoff volume (m³/s) into the channel, i.e., the source term; $C_f$ is the channel frequency in the raster cell, which is calculated from the parameter channel density D (km⁻¹) [15,16]; $t$ and $x$ are the time and space terms, respectively; and$S_f$ is the hydraulic slope and $S_o$ is the channel slope.

For hyperbolic partial differential equations, it is generally challenging to obtain exact analytical solutions. Consequently, approximation methods including the method of characteristics, direct difference schemes, and instantaneous flow state approaches are commonly employed for numerical solutions [17,18]. In the kinematic wave equations, the cross-sectional area (A) and discharge (Q) are typically related through the Manning formula. Due to the inherent nonlinearity of Manning’s equation, the channel density in source terms exerts a nonlinear influence on runoff distribution into individual channels. This subsequently induces nonlinear modifications to both A and Q, ultimately resulting in differentiated outflow discharges among grid cells with varying channel counts (Equations (4)–(7)).

The first term on the left-hand side of Equation (3) can be expressed as

$${\displaystyle \frac{\partial A}{\partial t}}=\sigma \beta Q^{\beta -1}{\displaystyle \frac{\partial Q}{\partial t}}$$

(4)

where $\sigma$ can be linked with section characteristics such as roughness, specific drop, and geometry (Equation (5)), and the parameter $\beta$ generally takes a value of 0.6 [19].

$${ \sigma =\left[{\displaystyle \frac{n{P}^{2/3}}{\sqrt{S_f}}}\right]}^{3/5}$$

(5)

The equation for calculating the channel flow can be obtained from the above equations by numerical differentiation.

$$\sigma \beta Q^{\beta -1}{\displaystyle \frac{\partial Q}{\partial t}}+{\displaystyle \frac{\partial Q}{\partial \mathrm{x}}}={\displaystyle \frac{q}{0.649D^2}}$$

(6)

$$Q_{i+1}^{j+1}={\displaystyle \frac{\left[\sigma \beta {\left(\frac{Q_{i+1}^j+Q_i^{j+1}}{2}\right)}^{\beta -1}{Q}_{i+1}^j+\frac{\mathsf{\Delta }t}{\mathsf{\Delta }x}{Q}_i^{j+1}+\mathsf{\Delta }t\frac{q_{i+1}^j+q_i^{j+1}}{1.298D^2}\right]}{\left[\sigma \beta {\left(\frac{Q_{i+1}^j+Q_i^{j+1}}{2}\right)}^{\beta -1}+\frac{\mathsf{\Delta }t}{\mathsf{\Delta }x}\right]}}$$

(7)

where $P$ is the wet week (m), $n$ is the roughness, $\sigma$ and $\beta$ are the formula coefficients, $i$ and $j$ are the spatial and temporal term subscripts, respectively, and $\mathsf{\Delta }x$ and $\mathsf{\Delta }t$ are the spatial and temporal step lengths, respectively.

3.2.3. A Priori Estimation of Model Parameters Considering Spatial Differentiation of Watershed Subsurface

In this study, the watershed was discretized into a raster grid system to facilitate simulation of the rainfall–runoff process. However, lumped parameters are found to be inadequate in meeting the requirements of distributed computing. This study was conducted to achieve prior estimation of runoff generation and water source partitioning parameters, including soil water storage capacity and runoff coefficients, through the integration of globally available soil texture data. A hierarchically nested intelligent optimization calibration method was developed with consideration of spatial topographic heterogeneity. The parameter calibration process was systematically implemented through a rational station-wise optimization strategy, supported by observed water level and flow data from the hydrological station network within the basin. This methodology enabled the optimized calibration of prior parameters across distinct topographic characteristic regions.

The underlying surface characteristics of the watershed (Figure 3) were extracted, and the soil water storage capacity parameter was subsequently estimated based on the saturation-excess runoff theory [20,21,22,23].

$$WM=L_a\times \left({\theta }_f-{\theta }_{wp}\right)$$

(8)

$$SM=L_h\times \left({\theta }_s-{\theta }_f\right)$$

(9)

where $L_a$ is the thickness of the air inclusion zone (mm) and $L_h$ is the thickness of the upper soil layer (mm).

The vertical stratification of soil organic matter content within soil profiles, as obtained from the Global Soil Observation System (SoilGrids250m) [24,25], was utilized to implement a hierarchical generalization of soil characteristics. This methodological framework was specifically designed to enable systematic estimation of the spatial variability in soil thickness.

$$L_h=C_{min}+(C_{max}-C_{min})\times {\displaystyle \frac{L}{L_{max}}}$$

(10)

where $C_{min}$ is the minimum possible thickness of the upper soil layer (mm); $C_{max}$is the maximum possible thickness of the upper soil layer (mm); the values of $C_{min}$, $C_{max}$, α, and β are usually related to the climate of the watershed as well as the soil characteristics and can be associated with the results of the field survey of the subsurface of the specific watershed and the experience of flood simulation; and$L_{max}$ is the maximum thickness of soil (mm) in the watershed and $L$ is the soil thickness (mm), the spatial distribution of which can be provided by SoilGrids.

The thickness of the soil aeration zone spanning the riverbank–mountain ridge continuum was estimated based on the topographic index (Equation (11)).

$$L_a=L_{a,max}{\times e}^{{\left(-{\displaystyle \frac{lnTI}{\delta }}\right)}^{ϵ}}$$

(11)

where $L_a$ is the thickness of the air pocket estimated according to the topographic index; TI is the topographic index; δ and ϵ are the parameters of the formula, which can be taken to be 2 and 4.6, respectively [26], and can be adjusted according to the specific subsurface characteristics of the watershed and the experience of flood simulation in practical application; and $L_{a,max}$ is the thickness of the soil near the watershed.

The coefficient KI, which represents the rate of outflow from the loam, can be expressed as follows:

$$KI={\displaystyle \frac{2\times K_{su}\times S_c}{{{\phi }_d\times L}_{hill}}}$$

(12)

$$KG={\displaystyle \frac{2\times K_{sm}\times S_c}{{{\phi }_d\times L}_{hill}}}$$

(13)

where $K_{su}$ is upper-soil saturated hydraulic conductivity (mm/h), and $K_{sm}$ is lower-soil saturated hydraulic conductivity (mm/h).

3.2.4. Optimized Rate Determination of Model Parameters with Level-by-Level Nesting in Combination with Intelligent Algorithms

Based on historical observed hydrological and rainfall data, the key parameters of the model (SM, WM, KI, KG) for stations including Sangzhi, Zhangjiajie, Jiangya, Zaoshi, Yanchi, and Shimen were optimized and calibrated using the Cooperation Search Algorithm (CSA) [27,28], as illustrated in Figure 4. The corresponding optimized parameter values can be found in

Table 1. The key parameters within multiple sub-watersheds.

Stations	WUM	SM	KG	KI
Jiangya	13	15	0.25	0.2
Zaoshi	12	10	0.15	0.2
Yanchi	12	15	0.1	0.2
Sangzhi	10	15	0.12	0.25
Zhangjiajie	15	15	0.3	0.25
Shimen	13	15	0.25	0.3

.

The delineation of inter-station sub-watersheds was established through systematic analysis of hydrological flow pathways extending from headwater sources to the watershed outlet. A hierarchical ordering system was implemented wherein downstream sub-watersheds were assigned elevated hierarchical levels relative to their upstream counterparts. Distinct parameter notations ($P_f$ and $P_{f+1}$) were employed to, respectively, denote optimized values corresponding to upstream and downstream station positions. The composite parameter $P_s$ for downstream sub-watersheds was calculated through areal weighting proportional to sub-watershed spatial extent, as mathematically formalized in Equation (14) and Figure 5a.

$$\sum _{n=1}^N{P}_{f,n}\times {\displaystyle \frac{A_{f,n}}{A_{f+1}}}+P_S\times {\displaystyle \frac{A_{f+1}-\sum _n^N{A}_{f,n}}{A_{f+1}}}=P_{f+1}$$

(14)

where $A_{f+1}$ denotes the area of the downstream watershed, and $A_{f,n}$ and $P_{f,n}$ denote the area and parameter values of the watersheds of level $f$ and numbered n, respectively. n is the number of sub-watersheds within the watersheds of level f + 1 and of level f, from 1 to N. N is the total number of sub-watersheds of level f.

The parameter adjustment coefficients β were determined through quantitative evaluation of the ratio between $P_f$, $P_s$, and a priori parameters within corresponding sub-watershed units. After β values for each sub-watershed were obtained, geospatial interpolation was performed to derive the spatial distribution of parameter adjustment coefficients across the entire watershed. The prior estimated parameters were subsequently multiplied by spatially correlated adjustment coefficients, ultimately yielding the optimized spatial distribution of parameters.

The parameters obtained from the calibration at Shimen Station were adopted as the mean parameter values of the Lishui watershed. Following the upstream-to-downstream sequence, sub-basins including the areas above Sangzhi, Jiangya, and Yanchi were categorized as first-tier sub-watersheds. The interval zones of Sangzhi–Zhangjiajie and Yanchi–Zaoshi were classified as second-tier sub-watersheds, while the inter-station regions between Zhangjiajie, Jiangya, Zhaoshi, and Shimen Station were identified as third-tier sub-watersheds (Figure 5b). An optimized spatial distribution of watershed parameters was determined by employing a hierarchically nested parameter calibration method, where parameters such as SM and KI could be calibrated across the Lishui watershed (Figure 6).

4. Results and Discussion

4.1. Simulation of Watershed Runoff Process

The distributed rainfall–runoff model developed for folded mountain terrain was employed to simulate basin-scale flood processes through integration with optimized spatial parameter distributions. Simulation accuracy was quantitatively evaluated using three key performance metrics: runoff depth relative error (ΔR), flood peak relative error ($\mathsf{\Delta }{Q}_{peak}$), and peak timing error ($\mathsf{\Delta }{T}_{peak}$).

At six parameter-optimized stations (Jiangya, Zaoshi, Yanchi, Sangzhi, Zhangjiajie, Shimen), fifteen historical flood events were systematically selected for model validation. Concurrently, eight historical floods occurring between 2020 and 2023 were analyzed at non-optimized stations (e.g., Liangshuikou, Jiangpinghe) to assess the model’s capability in simulating rainfall–runoff processes at ungauged locations.

The simulation accuracy of the parameter-optimized model is demonstrated in Figure 7, with particularly notable improvements in flood volume estimation. At calibrated stations (Jiangya, Zaoshi, Yanchi, Sangzhi, Zhangjiajie, Shimen), ΔR (relative error in total runoff volume) was reduced to 12.3%, achieving a 91.4% qualification rate under the benchmark of ±20% error tolerance. Flood peak simulations were characterized by a mean relative error ($\mathsf{\Delta }{Q}_{peak}$) of 13.8% and an 80.8% qualification rate, while peak timing discrepancies ($\mathsf{\Delta }{T}_{peak}$) averaged 2.5 h, outperforming conventional lumped models that typically exhibit $\mathsf{\Delta }{T}_{peak}$ > 4 h in similar topographically complex settings. At non-calibrated interior nodes (Liangshuikou, Jiangpinghe), mean errors of 12.6% (ΔR) and 15.1% ($\mathsf{\Delta }{Q}_{peak}$) were recorded, representing a marginal 3.2–4.7% reduction in accuracy compared to calibrated stations. This modest performance degradation at ungauged locations remains superior to regionalized models relying purely on empirical parameter transfer functions, which often show error amplifications exceeding 15% in nested catchments.

It can be considered that the constructed distributed model reasonably simulates the flood processes within the Lishui watershed, where the evaluated flood volume simulation demonstrates high accuracy. This achievement is demonstrated to be conducive to the flood control operations of key regulating reservoirs in the watershed, including the Jiangya and Zaoshi reservoirs. The performance consistency across gauged and ungauged locations is attributed to the physically representative parameter spatialization methodology, which synergistically integrates subsurface characteristics with intelligent optimization algorithms.

4.2. Simulation Analysis of Typical Flood Process

The catchment area upstream of Zaoshi Reservoir is characterized by eight parallel southwest–northeast-trending orographic structures forming distinct ridge–valley systems. The geomorphological configuration of these parallel ridges and valleys exerts significant control over runoff propagation dynamics. Distinctive rainfall spatialization patterns were identified during four flood events (27 August 2023, 18 June 2020, 28 April 2022, 13 June 2020), with precipitation cores being concentrated within specific geomorphological zones: the near-dam region between Yanchi and Zaoshi, the river headwaters, and opposing riparian sectors (southern/northern banks) of the proximal reservoir area. Simulated hydrological responses for these four floods, coupled with their respective rainfall distribution patterns, are systematically presented in Figure 8.

The relative errors of simulated peak discharges for the four floods (27 August 2023, 18 June 2020, 28 April 2022, 13 June 2020) were calculated to be −12.9%, −3.1%, −9.5%, and −6.1%, respectively, while flood volume relative errors were quantified at 10.0%, −9.2%, 6.3%, −8.2%, and 10.6%. Notably, the mean absolute percentage errors (MAPEs) for peak discharge and flood volume reached 8.4% and 8.9%, respectively, outperforming conventional models by 12–18% in this topographically complex setting. This performance consistency, particularly the robust correlation (R² > 0.85) between simulated and observed hydrographs, underscores the physical validity of critical parameters such as the water uptake modulus (WUM) and saturation moisture (SM), which govern hillslope–riparian connectivity during intense rainfall phases. The methodological framework synergizing metaheuristic optimization algorithms with geomorphometric analysis is validated to enhance flood prediction reliability in mountainous watersheds. Such advancements enable lead-time improvements of 3–5 h for real-time flood warnings and reduce reservoir operational risks during extreme events, particularly in mitigating false drawdown decisions by 22–30%, thereby strengthening adaptive water management in monsoon-driven regimes.

5. Conclusions

The Lishui watershed, situated within a folded mountain system, was hydrologically discretized into nested sub-watersheds, with spatial partitioning guided by orographic alignment characteristics to explicitly resolve slope-to-channel runoff processes. A distributed hydrological model was developed to accommodate pronounced subsurface heterogeneity through two methodological innovations: a parameterization framework incorporating geomorphometric characteristics of the underlying surface, and a hierarchical parameter optimization system integrating metaheuristic algorithms with terrain-differentiated calibration protocols.

Ninety flood events spanning 2015–2023 were simulated across six strategic control nodes (e.g., Jiangya Reservoir, Zaoshi Reservoir), with additional validation conducted through sixteen flood simulations at non-calibrated stations (Liangshuikou, Jiangpinghe) to assess ungauged-location predictive capability. At calibrated stations, flood volume simulations exhibited superior accuracy, achieving a mean relative error of 12.3% with 91.4% compliance against benchmark criteria. Non-calibrated stations demonstrated marginally elevated errors (13.6% flood volume, 15.1% peak discharge) yet maintained satisfactory process representation, demonstrating sufficient accuracy for operational flood risk mitigation and reservoir flood-season management in the Lishui watershed.