A Physics-Based Method for Delineating Homogeneous Channel Units in Debris Flow Channels

Abstract

For runoff-generated debris flow continuum mechanics-based early warning models, the computational unit must satisfy the homogeneity assumption of continuum mechanics. Although traditional grid cells meet the homogeneity assumption as computational units, they segment channel geomorphological functional reaches, weaken the clustered mobilization of sediment sources, and constrain efficiency due to grid-by-grid calculations. To address these limitations, we construct a Froude number ($Fr$) calculation model constrained by key factors such as the channel cross-sectional geometry and topographic parameters. The absolute deviation of $Fr$ is used as a criterion for homogeneity within the computational unit. By combining critical shear stress theory and velocity perturbation, physical thresholds for the criteria are derived. A physical model-based method for automatically delineating homogeneous channel units (${CU}_j$) is proposed, ensuring that the geometric features and hydrodynamic parameters within ${CU}_j$ are homogeneous, while ensuring heterogeneity between adjacent ${CU}_j$. Comprehensive multi-scale validation in Yeniu Gully, a typical debris flow catchment in Wenchuan County, demonstrates that parameters such as longitudinal gradient, cross-sectional area, flow depth, and shear stress remain relatively homogeneous within each ${CU}_j$ but differ significantly between adjacent ${CU}_j$. Furthermore, the proposed method can stably characterize key channel geomorphological functional units, such as bends, confluences, abrupt width changes, longitudinal gradient changes, erosion segments, and deposition segments. Sensitivity analysis demonstrates that the method satisfies both robustness and universality under various conditions of rainfall intensity, runoff coefficient, and Manning’s roughness coefficient. Even under the most unfavorable extreme conditions, the accuracy of ${CU}_j$ delineation exceeds 88.64%, indicating high reliability and suitability for deployment in various debris flow catchments. The proposed framework for defining ${CU}_j$ resolves the conflict in traditional computational units between the “continuum model homogeneity requirement” and “geomorphological functional unit continuity,” providing a more rational and efficient computational environment for runoff-generated debris flow continuum mechanics-based early warning models.

1. Introduction

Debris flow poses persistent and widespread threats to lives and property in mountainous regions and is widely distributed across steep terrain on all continents except Antarctica [1,2,3,4,5,6,7]. Engineering countermeasures are direct and effective means of risk reduction [8,9,10,11], but their construction and maintenance demand substantial human, material, and financial resources. Moreover, the sheer number and wide distribution of debris flow gullies in mountainous areas make comprehensive engineering measures infeasible everywhere [12,13,14]. In contrast, debris flow early warning and forecasting—an economical and efficient non-structural measure—can provide advance information on the evolution of debris flow events, reducing casualties and economic losses, and has played a key role in practice across many regions [15,16,17,18,19].

Existing debris flow early warning/forecasting approaches primarily include statistical models [20,21,22,23], machine learning/deep learning models [6,24,25,26], empirical causative models [12,27,28], and physics-based mechanistic models [29,30,31,32]. The first three categories of models heavily rely on long time series of historical debris flow disaster records, which are difficult to obtain in many mountainous regions prone to debris flow; consequently, their applicability is limited in data-scarce areas [19,33,34,35,36]. In contrast, physics-based mechanistic forecasting can operate with only basic parameters of the study area and exhibits very low dependence on historical event data; underpinned by sound physical principles, such models often achieve higher warning accuracy with fewer false and missed alarms and show stronger generalizability [32,37]. Therefore, deductively derived mechanistic approaches are frequently used as a benchmark to evaluate the accuracy of inductive approaches (statistical, AI-driven, and empirical causative models) [38,39].

According to assumptions about the flowing medium, physics-based models for mechanistic debris flow early warning can be categorized as continuum models (dominated by the liquid phase) [8,32,40,41], discrete models (dominated by the solid phase) [42,43], and mixture/two-phase models (co-evolving solid and fluid phases) [44,45,46]. Discrete and mixture models often suffer from high computational complexity and difficulties in verification/validation [43,47,48,49], whereas continuum models—owing to simplifying assumptions and computational tractability—have become the mainstream physical framework for mechanistic early warning of debris flow [8,50]. Substantial progress has been made for soil mechanics-type debris flow (for which initial motion is primarily driven by the gravitational component of the soil–debris mass) in mechanistic early warning studies [37,51]. However, for runoff-generated debris flow, research on mechanistic early warning remains comparatively weak, and a number of scientific challenges still need to be addressed [52,53,54]. Most mechanistic early warning models for runoff-generated debris flow adopt continuum formulations [29,31,55,56]. Under continuum mechanics assumptions, the governing equations require that, within each computational unit, mechanical and topographic conditions are relatively uniform—without abrupt discontinuities—to ensure convergence and numerical stability [8,32,57,58]. Consequently, the computational unit plays a critical, irreplaceable role in mechanistic early warning (and forecasting) models for runoff-generated debris flow.

At present, grid cells are the computational unit most commonly used in mechanistic early warning of runoff-generated debris flow [59,60,61,62,63]. Although grid cells satisfy the local homogeneity requirement of continuum models [64,65,66], natural channel networks often contain homogeneous channel reaches—tens to hundreds of meters in length—where geomorphic geometry and hydrodynamic conditions are relatively uniform. Prior studies have shown that debris flow channel networks commonly develop typical geomorphological functional units, such as erosional reaches, depositional reaches, and transport (erosion–deposition balance) reaches [63,67,68,69]. These units exhibit spatial self-similarity in morphology, hydraulic gradients, and sedimentological parameters, and they meet the homogeneity assumption of continuum models. In addition, initiation of channel-stored debris by runoff often occurs as clustered mobilization within a homogeneous channel reach (rather than in an isolated grid cell), and erosion and deposition processes frequently occur coherently at the channel reach (i.e., cluster-of-grids) scale. Thus, a purely grid-based spatial discretization is limited in representing the continuity of natural geomorphic functional units; it segments the spatial linkage of the erosion–transport–deposition sequence, weakens the clustering effect of sediment source mobilization, and reduces computational efficiency due to grid-by-grid calculations. Therefore, the entire channel network can be conceptualized as an assembly of discrete channel reaches, each characterized by relatively uniform geomorphic geometry features and hydrodynamic parameters. These properties are relatively homogeneous within any given channel reach but differ markedly between adjacent channel reaches. Scientifically extracting such homogeneous channel reaches from debris flow gullies and using them as computational units would satisfy the homogeneity requirement of continuum models while preserving the morphological continuity and functional structure of geomorphic units, thereby improving the efficiency and applicability of mechanistic early warning models.

This study integrates geomorphic-process theory and hydrodynamics to propose a physics-constrained division theory and method for a new homogeneous computational unit—the “channel unit”—achieving the unification of computational-unit homogeneity and geomorphic functional-segment continuity. The approach calculates segment-scale (channel-discretized segment) Froude numbers and applies a physically derived homogeneity threshold, based on critical shear stress and velocity-perturbation analysis, to merge adjacent channel-discretized segments into coherent channel units. The method can automatically delineate channel units on the ArcGIS 9.3 platform and is expected to provide a more rigorous and efficient computational environment for continuum mechanics-based early warning models of runoff-generated debris flow, as well as a practical reference for researchers and engineers.

2. Methods

This study uses the Digital Elevation Model (DEM) as the data carrier. First, the channel network is discretized into a sequence of channel-discretized segments (a differential perspective), each with a step length following the D8 flow-direction path: $a_{gc}$ for orthogonal moves and $\sqrt{2}{a}_{gc}$ for diagonal moves, where $a_{gc}$ denotes the DEM cell size (pixel side length). Next, using the absolute deviation of the Froude number ($Fr$) as the homogeneity criterion, an along-channel rolling merge/split iterative algorithm is applied to aggregate adjacent channel-discretized segments into homogeneous channel units that satisfy hydraulic–geometric consistency (an integral perspective). The threshold is determined from the critical shear stress for incipient motion together with an adaptive velocity-perturbation analysis, thereby ensuring that the physical state within each channel unit can be regarded as locally homogeneous under the continuum approximation, while preserving the morphological continuity and structural integrity of geomorphic functional segments.

2.1. Definition and Mathematical Description of Channel Units (${CU}_j$)

From the DEM perspective, a debris flow channel network can be regarded as an assemblage of grid cells distributed longitudinally and laterally along the channel. In this study, a channel unit is defined as a relatively homogeneous channel reach composed of contiguous grid cells with consistent geomorphic geometry and hydrodynamic conditions. Channel unit conforms to the basic requirement imposed on the computational unit by continuum-based physical models [70,71]: parameters within a unit are approximately uniform, and significant differences exist between adjacent units. A channel unit thus simultaneously satisfies the local homogeneity assumption of continuum equations and the morphological continuity of a geomorphic functional segment, and serves as the computational unit for continuum mechanics-based early warning of runoff-generated debris flow.

Let the entire channel network ($CN$) be the set consisting of $n$ grid cells, each denoted $gc$. The $CN$ can also be expressed as the proper union of $m$ channel units, each denoted ${CU}_j$; any ${CU}_j$ comprises $w$ grid cells (see Figure 1d,e). Then the mathematical relationship among the total set $CN$, the proper subsets ${CU}_j$, and the elements $gc$ is

$$\left\{\begin{array}{c}CN=\left\{{gc}_1,{gc}_2,{gc}_3,\dots ,{gc}_n\right\}\\ {CU}_j=\left\{{gc}_1,\dots ,{gc}_w\right\},n\gg w\ge 1\\ {CU}_j⊊CN, j=\mathrm{1,2},3,\dots ,m; 1\ll m\ll n\\ {CU}_j \cap { CU}_{j+1}=\varnothing \\ CN={CU}_1 \cup {CU}_2 \cup \dots \cup {CU}_m\end{array}\right.$$

(1)

From Equation (1), the ${CU}_j$ are mutually exclusive and unique in space, and their union reconstructs the entire channel network.

To constrain the geometric scale of ${CU}_j$, boundary conditions are defined for the length $L_{CUj}$ and width $b$ as follows:

$$\left\{\begin{array}{c}{1\times a_{gc}\le L}_{CUj}\le x\times a_{gc}\\ 1\times a_{gc}\le b\le y\times a_{gc}\\ {\sqrt{2}\times a_{gc}\le L}_{CUj}\le \sqrt{2}x\times a_{gc}\\ {\displaystyle \frac{\sqrt{2}}{2}}\times a_{gc}\le b\le {\displaystyle \frac{\sqrt{2}}{2}}y\times a_{gc}\end{array}\right.$$

(2)

where $a_{gc}$ is the side length of a square grid cell (DEM cell size; Figure 1a,c), and $\sqrt{2}{a}_{gc}$ is its diagonal (Figure 1b,c). Here, $x$ and $y$ are positive integers: $x$ scales the along-channel length as an integer multiple of $a_{gc}$ or $\sqrt{2}{a}_{gc}$ (Figure 1c), and $y$ scales the cross-channel width as an integer multiple of $a_{gc}$ or $(\sqrt{2}/2)a_{gc}$ (Figure 1d,e). The first two expressions in Equation (2) apply when the channel centerline grid cells are aligned with the flow direction (orthogonal moves; Figure 1a,c,d), while the last two apply when the centerline grid cells are arranged at 45° to the flow direction (diagonal moves; Figure 1b,c,e). The distinction between the orthogonal-move and diagonal-move cases is made using the D8 flow-direction algorithm; after the channel network has been extracted, each channel segment along the flow path (i.e., each potential ${CU}_j$) is thus classified as either an orthogonal-move case or a diagonal-move case. Accordingly, $L_{CUj}$ is at least one integer multiple of $a_{gc}$ or $\sqrt{2}{a}_{gc}$, and $b$ is at least one integer multiple of $a_{gc}$ or $(\sqrt{2}/2)a_{gc}$.

Based on the DEM, each channel is first partitioned—under the differential perspective—into channel-discretized segments, denoted ${CDS}_i$, whose length equals $a_{gc}$ (orthogonal moves) or $\sqrt{2}{a}_{gc}$ (diagonal moves) and whose width equals an integer multiple of $a_{gc}$ (orthogonal moves) or $(\sqrt{2}/2)a_{gc}$ (diagonal moves) (Figure 1a,b,d,e). We assume that geomorphic–geometric attributes and hydrodynamic conditions can be regarded as uniform within each ${CDS}_i$ at this length scale. Then, based on whether the topographic–geometric features and hydrodynamic conditions between ${CDS}_i$ and its adjacent segments ${CDS}_{i \pm 1}$ are sufficiently similar (i.e., homogeneous), we apply a merge/split rule to delineate ${CU}_j$ (integral perspective). The mathematical relationship among ${CDS}_i$, ${CU}_j$, and $gc$ is

$$\left\{\begin{array}{c}{CU}_j=\left\{{gc}_1,\dots ,{gc}_w\right\},n\gg w\ge 1\\ {CDS}_i=\left\{{gc}_1,\dots ,{gc}_y\right\},w\ge y\ge 1\\ {CDS}_i\subseteq {CU}_j , {CDS}_i\cap {CDS}_{i+1}=\varnothing \\ {CU}_j{=CDS}_1\cup {CDS}_2\cup \dots \cup {CDS}_x,\left[w/y\right]=x\ge 1\end{array}\right.$$

(3)

where $\left[w/y\right]$ denotes rounding to the nearest integer. The resulting homogeneous ${CU}_j$ satisfy the geometric bounds in Equation (2) for $L_{CUj}$ and $b$ (see Figure 1c–e). Field surveys further indicate that most relatively homogeneous channel reaches have lengths of several tens to several hundreds of meters and widths of several to several tens of meters (Figure 1a,b,e), supporting the reasonableness of the bounds in Equation (2). In what follows, Equation (2) is used as the geometric constraint on $L_{CUj}$ and $b$. Combined with the physical criterion in Section 2.2, the entire channel network is automatically delineated to obtain $m$ ${CU}_j$ and their associated $L_{CUj}$ and $b$.

2.2. Physical Identification Model for Delineating ${CU}_j$

2.2.1. Physical Criterion and Its Computation

The Froude number ($Fr$) is widely used in hydraulics, sediment transport, and debris flow dynamics [72,73,74,75,76]. Its magnitude is directly controlled jointly by the cross-sectional mean flow velocity $v$ and flow depth $h$ [77,78,79]. In channels, $v$ and $h$ are, in turn, governed by channel geometry and runoff generation–concentration conditions, including bottom width $b$, bank-slope coefficient $m$, longitudinal gradient $J$, drainage area $A_0$, Manning’s roughness $n$, runoff coefficient $k$, and peak hourly rainfall intensity $I$. Hence, apart from $I$ and the material-property coefficients ($n$, $k$), the remaining predictors influencing the $Fr$ at a given ${CDS}_i$ are the relatively fixed geometric parameters associated with that ${CDS}_i$. Mechanically, $Fr$ is the ratio of inertial to gravitational forces, and it can also indicate the order of magnitude of impact pressure for both dilute and viscous debris flow [72,80]. Accordingly, $Fr$ serves as a bridge linking geomorphic geometry and hydrodynamics [81,82,83], and is thus suitable as the homogeneity criterion in this study.

Adjacent ${CDS}_i$ and ${CDS}_{i \pm 1}$ are spatially continuous, with centroids separated by only $a_{gc}$ or $\sqrt{2}{a}_{gc}$. They can normally be regarded as sharing the same rainfall intensity and material-property coefficients, so the difference in $Fr$ between them is driven mainly by differences in geometric parameters ($b$, $m$, $A_0$, $J$). If $\left|{Fr}_i-{Fr}_{i \pm 1}\right|$ is sufficiently small, the two adjacent segments (${CDS}_i$ and ${CDS}_{i \pm 1}$) are deemed hydraulically–geometrically similar and are merged; otherwise, they are split.

The original definition of the Froude number is as follows [77,78,79]:

$$Fr={\displaystyle \frac{v}{\sqrt{gh}}}$$

(4)

where $v$ is the cross-sectional mean velocity ($\mathrm{m}/\mathrm{s}$), $h$ is the mean flow depth (m), and $g$ is gravitational acceleration ($\mathrm{m}/{\mathrm{s}}^2$).

To compute $Fr$ at the ${CDS}_i$ scale, we couple the continuity equation with Manning’s formula, explicitly introducing $A_0$ (drainage area, ${\mathrm{m}}^2$), $J$ (longitudinal gradient), $A=\left(b+mh\right)h$ (cross-sectional area, ${\mathrm{m}}^2$; $b$ is channel bottom width, m; $m=cot\alpha$ is the average bank-slope coefficient), the material-property coefficients $n$ (Manning’s roughness, $\mathrm{s}·{\mathrm{m}}^{-{\displaystyle \frac{1}{3}}}$) and $k$ (runoff coefficient), as well as $I$ (peak hourly rainfall intensity, mm/h, converted to m/s as I/3,600,000). Other quantities are as defined above, and all inputs are taken locally at the position of ${CDS}_i$. This yields

$$v={\displaystyle \frac{kI{A}_0}{\left[\left(b+mh\right)h\right]}}$$

(5)

$$v={\displaystyle \frac{1}{n}}{[{\displaystyle \frac{\left(b+mh\right)h}{b+2h\sqrt{1+m^2}}}]}^{{\displaystyle \frac{2}{3}}}{J}^{{\displaystyle \frac{1}{2}}}$$

(6)

Combining Equations (5) and (6) gives a transcendental implicit function for $h$ at the ${CDS}_i$ scale. All parameters are first non-dimensionalized and then transformed using the natural logarithm (trial calculations with field-based channel parameters show numerical solutions equivalent to the non-log forms with negligible residuals), ensuring the strict validity of the logarithmic operations:

$$f\left(h\right)={\displaystyle \frac{5}{3}}\mathit{ln}\left(bh+m{h}^2\right)-{\displaystyle \frac{2}{3}}\mathit{ln}\left(b+2\sqrt{1+m^2}h\right)-\mathit{ln}\left(nkI{A}_0\right)+{\displaystyle \frac{1}{2}}\mathit{ln}J$$

(7)

Substituting Equations (4) and (5) yields the ${CDS}_i$-scale expression for $Fr$, which characterizes both geometric and hydraulic conditions and is constrained by cross-sectional geometry and other key topographic factors:

$$Fr={\displaystyle \frac{kI{A}_0}{(b{h}^{\frac{3}{2}}+m{h}^{\frac{5}{2}})\sqrt{g}}}$$

(8)

Mass conservation is enforced via $Q=kI{A}_0$ and the section continuity $Q=vA$ (obtain Equation (5)). Momentum is not solved using the full Saint–Venant equation; instead, a steady, gradually varied one-dimensional closure is adopted via Manning’s law $v={\displaystyle \frac{1}{n}}{R}^{{\displaystyle \frac{2}{3}}}{J}^{{\displaystyle \frac{1}{2}}}$ (i.e., Equation (6), with $S_f\approx J$), which combined with continuity yields Equation (7) and hence $Fr$ in Equation (8). The depth $h$ appearing in Equation (8) is obtained from Equation (7). By setting $f\left(h\right)=0$, the transcendental implicit function is cast as an implicit equation, which is then solved for $h$. We set the numerical tolerance for $h$ to 10⁻⁶ m (0.001 mm), which ensures the relative error in $Fr$ is far smaller than the threshold magnitude required by the decision rule in Section 2.2.2. Accordingly, for any single ${CDS}_i$, the variation of its $Fr$ is driven by changes in the inputs ($b, m, A_0, J$, $n, k, I$); in contrast, the difference in $Fr$ between adjacent segments ${CDS}_i$ and ${CDS}_{i \pm 1}$ is governed primarily by differences in the geometric parameters ($b,m,A_0,J$), whose changes also induce corresponding changes in hydraulic variables. This demonstrates that the $Fr$ computation model formed by Equations (7) and (8) effectively represents the geomorphic geometry and hydraulic state of any ${CDS}_i$, and that the inter-segment change in $Fr$ effectively reflects the degree of difference in geometric and hydrodynamic conditions between adjacent segments.

2.2.2. Determination of the Critical Threshold for the Physical Criterion

This study requires that the $Fr$ among all ${CDS}_i$ within a given ${CU}_j$ vary only within a reasonable range (i.e., small permissible deviations). If one were to use only the difference $|{Fr}_i-{Fr}_{i \pm 1}|$ between adjacent ${CDS}_i$ and ${CDS}_{i \pm 1}$ as the merging criterion (i.e., merge when the difference is below a threshold), sequentially merging adjacent ${CDS}_i$ and ${CDS}_{i \pm 1}$ may accumulate deviations, leading to an overly long ${CU}_j$. This would enlarge the difference between ${Fr}_{max}$ and ${Fr}_{min}$ within the ${CU}_j$, and ultimately increase intra-${CU}_j$ differences in geomorphic–geometric and hydrodynamic parameters. To avoid this, every ${CDS}_i$ inside a given ${CU}_j$ is required to satisfy the following:

$$│{Fr}_i-\overline{Fr}│\le C$$

(9)

where $C$ is a small constant threshold, ${Fr}_i$ is the Froude number of any ${CDS}_i$ within a given ${CU}_j$, and $\overline{Fr}$ is the mean of ${Fr}_i$ over all ${CDS}_i$ in that ${CU}_j$. Each ${CU}_j$ is uniquely labeled by its $\overline{Fr}$, and adjacent ${CU}_j$ and ${CU}_{j\pm 1}$ generally exhibit large differences in $\overline{Fr}$. Introducing $\overline{Fr}$ effectively suppresses the progressive accumulation of errors when consecutively merging adjacent ${CDS}_i$ and ${CDS}_{i \pm 1}$, because $│{Fr}_i-\overline{Fr}│$ controls the overall aggregated deviation among multiple consecutive $|{Fr}_i-{Fr}_{i \pm 1}|$ along the channel. At the same time, the magnitude of $│{Fr}_i-\overline{Fr}│$ is directly governed by the magnitudes of the individual $|{Fr}_i-{Fr}_{i \pm 1}|$. For an already delineated ${CU}_j$, $\overline{Fr}$ is the average over all ${CDS}_i$ within that ${CU}_j$, whereas at the onset of a merge/split decision for a pair of adjacent ${CDS}_i$ and ${CDS}_{i \pm 1}$, $\overline{Fr}$ is taken as the average of their two values, ${Fr}_i$ and ${Fr}_{i \pm 1}$ (see Figure 2).

By constraining the “departure from the ${CU}_j$ mean,” Equation (9) suppresses along-channel error accumulation while ensuring that variations in hydraulic–geometric parameters within a ${CU}_j$ remain sufficiently small so as not to significantly change the mobilized amount of stored debris. This, in turn, keeps the variation in the mixture density within a ${CU}_j$ small, because the amount of mobilized debris directly affects mixture density, and classical studies have demonstrated that mixture density/concentration strongly influences rheological and dynamic behavior [74,84,85]. Adjacent ${CU}_j$ and ${CU}_{j \pm 1}$, in contrast, are separated where hydraulic–geometric parameters differ markedly. Accordingly, we merge adjacent ${CDS}_i$ and ${CDS}_{i \pm 1}$ into the same ${CU}_j$ when $│{Fr}_i-\overline{Fr}│\le C$, and we split (do not merge) them when $│{Fr}_i-\overline{Fr}│>C$, assigning the two adjacent segments to different ${CU}_j$. Applying this rule across the entire channel network yields the ${CU}_j$ partitioning for the whole catchment. Therefore, determining the threshold $C$ is crucial. The numerical value of $C$ governs the allowable variations in geomorphic–geometric and hydrodynamic attributes within each ${CU}_j$, thereby indirectly affecting the permissible variation in mixture density inside each ${CU}_j$, as well as the final ${CU}_j$ delineation.

The dynamics of debris flow are strongly modulated by the proportion of solid and liquid phases [32,74,86]. Among them, the coarse solid fraction primarily controls mixture density, with an influence far exceeding that of the fine liquid-phase fraction [74,84,86]. Pronounced changes in mixture density substantially affect rheology and flow dynamics [74,84,85]. Therefore, to quantify the “minimum admissible dynamic perturbation” as an operational threshold, we employ the critical shear (friction) velocity for incipient motion, $u_{*}$, associated with the solid–liquid boundary grain size $D$ [8,84], to construct an adaptive velocity perturbation: $∆v=({\displaystyle \frac{v}{v+u_{*}}})u_{*}$. Its properties are as follows: when $v\ll u_{*}$, $∆v\to 0$, avoiding excessive disturbance to the system, as $v$ increases, $∆v$ increases monotonically, but its upper bound is limited by $u_{*}$, so the perturbation remains “small and controllable” ($0\le ∆v<u_{*}$). Classical initiation theory indicates that, when no perturbation is applied and $v\approx u_{*}$, only particles with diameter ≤ D (the boundary size) are likely to be entrained—smaller particles have higher entrainment probability—whereas particles with diameter > D (coarse solids) remain stable [30,87,88,89]. Hence, under the foregoing “small and controllable” perturbation, the tiny $∆v$ primarily affects liquid-phase dynamics, with negligible impact on the entrainment statistics of coarse solids, thereby keeping mixture density and rheology approximately unchanged (i.e., relatively stable). Because the upper bound of $∆v$ is constrained by $u_{*}$, the perturbation will not significantly alter the entrainment statistics of coarse grains. This preserves the stability of the mixture density within each ${CDS}_i$, so the tolerated fluctuation of $Fr$ can be regarded as a small local variation under the “same mechanical state.” We then define the allowable change in $Fr$ under this small perturbation as $C_i$ for each ${CDS}_i$. Taking the average of the $N$ values of $C_i$ over all ${CDS}_i$ in the channel network yields $\overline{C}$, which serves as the final threshold $C$ in Equation (9). In other words, the small-scale (per-${CDS}_i$) admissible $Fr$ variations of $C_i$ are averaged to impose a strict constraint on the large-scale (per-${CU}_j$) dispersion of $Fr$, thereby enforcing stronger homogeneity within each ${CU}_j$. Thus, $\overline{C}$ reflects the average allowable range of $Fr$ within any ${CU}_j$ and corresponds to the average permissible variations in geomorphic geometry and hydrodynamics within that ${CU}_j$. The quantities $C_i$ and $\overline{C}$ are computed by Equation (10) (with $\overline{C}$ used as $C$ in Equation (9)); all physical parameters in the formula are as defined previously:

$$\left\{\begin{array}{c}\overline{C}={\displaystyle \frac{{\sum }_{i=1}^N{C}_i}{N}}\\ \left|{\displaystyle \frac{v_i}{\sqrt{g{h}_i}}}-{\displaystyle \frac{v_i^{*}}{\sqrt{g{h}_i^{*}}}}\right|=C_i\\ v_i^{*}=v_i+∆v\\ ∆v=({\displaystyle \frac{v_i}{v_i+u_{*}}})u_{*}\\ h_i^{*}={\displaystyle \frac{(\sqrt{b^2+4mkI{A}_0/v_i^{*}}-b)}{2m}}\end{array}\right.$$

(10)

As Equation (10) indicates, the critical shear (friction) velocity $u_{*}$ required to initiate motion of grains at the solid–liquid boundary size $D$ is essential for computing $C_i$ and hence $\overline{C}$.

Derivation of $u_{*}$ (mechanics):

Under hydrostatic conditions and the bed slope $\theta =0^0$, the maximum stable angle of a spherical particle (volume $V=\pi D^3/6$, contact area $S=\pi D^2/4$) is the subaqueous angle of repose $\mathrm{\varnothing }$. Under flowing conditions with $\theta <\mathrm{\varnothing }$, the critical shear stress for incipient motion of a spherical sediment particle with a diameter $D$ is as follows:

$${\tau }_c={\displaystyle \frac{2}{3}}{(\rho }_s-\rho )gDcos\theta ·tan\mathrm{\varnothing }$$

(11)

where ${\rho }_s$ is sediment density (kg/m³) and $\rho$ is water density (kg/m³).

The bed shear stress exerted by the flow is as follows [90,91]:

$$\tau =\rho {u_{*}}^2$$

(12)

Combining Equations (11) and (12) with force balance gives the initiation condition:

$$\left\{\begin{array}{c}{\displaystyle \frac{2}{3}}({\rho }_s-\rho )gDsin\theta +\rho {u_{*}}^2\ge {\displaystyle \frac{2}{3}} {(\rho }_s-\rho )gDcos\theta ·tan\varnothing ; (\theta <\varnothing )\hfill \\ \theta \ge \varnothing (\mathrm{a}\mathrm{u}\mathrm{t}\mathrm{o}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{i}\mathrm{c} \mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n})\end{array}\right.$$

(13)

from which

$$u_{*}\ge \sqrt{{\displaystyle \frac{2}{3}}\left({\displaystyle \frac{{\rho }_s-\rho }{\rho }}\right)gD(cos\theta ·tan\mathrm{\varnothing }-sin\theta )};\left(\theta <\mathrm{\varnothing }\right)$$

(14)

For a horizontal bed ($\theta =0°$) the maximum is as follows:

$$u_{*}=\sqrt{{\displaystyle \frac{2}{3}}\left({\displaystyle \frac{{\rho }_s-\rho }{\rho }}\right)gDtan\mathrm{\varnothing }}$$

(15)

The empirical formula for the subaqueous angle of repose from Tianjin University flume experiments [92] is as follows:

$$\mathrm{\varnothing }=32.5+1.27D\hspace{1em}\hspace{1em}\left(0.2 \mathrm{m}\mathrm{m}\le D\le 4.37 \mathrm{m}\mathrm{m}\right)$$

(16)

We take the solid–liquid boundary grain size as $D=2 \mathrm{m}\mathrm{m}$, a commonly adopted threshold across sediment, soil, and debris flow literature [93,94,95,96,97]. Using Equations (15) and (16) yields $u_{*}$, which is then substituted into Equation (10) to compute $\overline{C}$. Finally, $\overline{C}$ is used as the threshold $C$ in Equation (9).

2.2.3. Identification Model for Delineating ${CU}_j$

Using $\overline{C}$ as the threshold for the admissible variation $│{Fr}_i-\overline{Fr}│$ within a ${CU}_j$:

$$\left\{\begin{array}{c}│{Fr}_i-\overline{Fr}│\le \overline{C} \mathrm{m}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{e} \mathrm{a}\mathrm{d}\mathrm{j}\mathrm{a}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t} {CDS}_i \mathrm{a}\mathrm{n}\mathrm{d} {CDS}_{i \pm 1}\\ │{Fr}_i-\overline{Fr}│>\overline{C} \mathrm{s}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{t} \mathrm{a}\mathrm{d}\mathrm{j}\mathrm{a}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t} {CDS}_i \mathrm{a}\mathrm{n}\mathrm{d} {CDS}_{i \pm 1}\end{array}\right.$$

(17)

We adopt a cyclic iterative merge/split strategy (see Figure 2): starting from the headwater ${CDS}_i$ of each channel, we proceed downstream segment by segment, compute ${Fr}_i$ for each ${CDS}_i$, then compute $\overline{Fr}$, and evaluate $│{Fr}_i-\overline{Fr}│$. According to Equation (17) and the threshold $\overline{C}$, we merge or split adjacent ${CDS}_i$ and ${CDS}_{i \pm 1}$ to delineate ${CU}_j$, until the entire channel network of the debris flow catchment is partitioned. The algorithm is implemented in Python 3.11.9 (see Section 2.2.4 for details) and outputs relatively homogeneous ${CU}_j$, with heterogeneity preserved between adjacent ${CU}_j$ and ${CU}_{j \pm 1}$.

2.2.4. Software and Implementation

DEM pre-processing and channel-network extraction were performed in ArcGIS 9.3 software (Spatial Analyst/3D Analyst): Fill → Flow Direction (D8) → Flow Accumulation → Raster Calculator → Stream Link → Stream to Feature, followed by raster-to-point conversion to obtain step lengths of $a_{gc}$ (orthogonal moves, $a_{gc}$ is the length of ${CDS}_i$ in the along-channel direction) and $\sqrt{2}{a}_{gc}$ (diagonal moves, $\sqrt{2}{a}_{gc}$ is the length of ${CDS}_i$ in the along-channel direction).

All physics-based computations were implemented in Python 3.11.9: (i) for each ${CDS}_i$, solve Equation (7) for $h$ (bisection; tolerance 10⁻⁶ m); (ii) compute $Fr$ by Equation (8); (iii) compute $\overline{C}$ using Equations (10)–(16); (iv) apply the along-channel iterative merge/split rule in Equation (17) to aggregate adjacent ${CDS}_i$ into ${CU}_j$; and (v) write the resulting ${CU}_j$ identifiers back to the existing DEM–channel network layer as an attribute field for visualization. Software versions and numerical settings are reported here to facilitate reproduction.

2.3. Accuracy Evaluation of the ${CU}_j$ Delineation—Identification Model

2.3.1. Practical Application Test

We select a typical debris flow catchment and extract its channel network. The network is discretized into ${CDS}_i$ with lengths $a_{gc}$ or $\sqrt{2}{a}_{gc}$. For each ${CDS}_i$, we compute $b$, $J$, $m$, $A_0$, $n$, $k$ and $I$. Using Equations (7) and (8), we obtain ${Fr}_i$ for each segment (each ${CDS}_i$), compute $\overline{C}$ using Equation (10), and then delineate ${CU}_j$ according to Equation (17) (see Section 2.2.4 for details). The evaluation includes the following:

(i)

Testing the homogeneity of key geomorphic–geometric parameters ($J$, $h$) among ${CDS}_i$ within a ${CU}_j$, and the contrasts in $J$ and $h$ between adjacent ${CU}_j$ and ${CU}_{j \pm 1}$;

(ii)

Testing the homogeneity of key mechanical factors ($\tau$) and the similarity of kinematic trends ($v$) among ${CDS}_i$ within a ${CU}_j$, as well as the dissimilarity in $v$ and the contrasts in $\tau$ between adjacent ${CU}_j$ and ${CU}_{j \pm 1}$;

(iii)

Preliminarily assessing the ability to identify geomorphic functional units (e.g., large bends, confluences, abrupt width changes, and abrupt changes in longitudinal gradient).

Accordingly, we conduct validation at both catchment and tributary channel scales.

2.3.2. Accuracy Assessment at Field-Measured Homogeneous Reaches

At field-measured homogeneous reaches representing typical geomorphic functional units (erosional, depositional, and transport/balance reaches), we perform detailed validation. The ${CU}_j$ length $L_{CUj}$, longitudinal gradient $J$, and cross-sectional area $A=\left(b+mh\right)h$ jointly characterize the 3-D channel geometry and are compared with observations. We evaluate the model prediction errors (for $L_{CUj}$ and mean $A$) using Equation (18), and analyze the distributions of $\tau$, $J$ and $v$ within the ${CU}_j$ at the field-measured locations.

$$\xi ={\displaystyle \frac{\left|X_M-X_P\right|}{X_P}}\times 100\%$$

(18)

where $\xi$ represents the absolute error rate, $X_P$ represents the original observed values, and $X_M$ represents the predicted values calculated by the model.

Validation follows a multi-scale scheme of “point (single measured reach)—line (tributary channel)—area (catchment)” to assess the reliability of the ${CU}_j$ delineation.

3. Study Area and Data Sets

3.1. Study Area

Yeniu Gully is a small catchment located in Wenchuan County, Sichuan Province, China (103°24′45″ E–103°29′16″ E; 31°9′25″ N–31°12′4″ N; see Figure 3a). It is a first-order tributary on the right bank of the Min River, with a drainage area of approximately 24.2 km² and a main-channel length of about 8.6 km. The main channel is 30–90 m wide in the downstream section, reaching a maximum width of ~165 m at the alluvial fan. The maximum relative relief of the catchment is 2974 m, and the overall mean slope is ~28°. The main channel is gentler than the tributaries; in the channel network, slopes range from ~0° to ~64.5°, with a mean of ~19°. Several major tributaries display a venation-like (dendritic) planform, which promotes runoff convergence and downcutting of channel deposits, thereby providing both the hydrological and sediment-source conditions for runoff-generated debris flow [98].

Along-channel topography in the Yeniu Gully network varies markedly from upstream tributaries to the downstream main stem, exhibiting alternating widths, undulating slopes, and repeated curvature changes (Figure 3b). Overall, the channel’s geometric attributes are highly heterogeneous; nevertheless, multiple locally homogeneous reaches (${CU}_j$) exist. This geomorphic pattern—macroscopic complexity coexisting with local homogeneity—leads to high-frequency fluctuations of key hydraulic parameters along the profile where geometric conditions change rapidly, whereas locally homogeneous geometry maintains uniformity in hydraulic parameters and similarity in kinematic trends; both effects regulate $Fr$. Yeniu Gully is a natural debris flow gully without artificial check structures. Its sinuous planform and complex, spatially variable hydraulics provide an ideal setting for testing the ${CU}_j$ delineation method. In addition, seven channel cross-sections were surveyed in the catchment (Figure 3a) to obtain geometric parameters for model validation.

3.2. Data Sets

3.2.1. Data Set for Method Implementation and Testing

To obtain the parameters of each ${CDS}_i$ required by Equations (7), (8), and (10), we used a clipped DEM of Yeniu Gully with 12.5 m × 12.5 m spatial resolution (DEM cell size) (Figure 3a). The inputs were derived and processed as summarized in

Table 1. Sources and processing of model input parameters.

Parameter	Processing Tools	Processing Methods	Outputs	Data Sources	Sample Size
$L_{{CDS}_i}$	ArcGIS 9.3	Spatial Analyst Tools → Hydrology	$L_{{CDS}_i}$ = 12.5 m $\mathrm{Or} L_{{CDS}_i}$ = 17.7 m	DEM-based extraction, DEM data: https://asf.alaska.edu (accessed on 24 November 2025)	N = 1030
$b$	ArcGIS 9.3	Longitudinal channel centerline +DEM	$8.8 \mathrm{m}\le b\le 159 \mathrm{m}$	DEM-based extraction	N = 1030
$J$	ArcGIS 9.3	3D Analyst Tools → Slope	$0<tan\theta <2.1$	DEM-based extraction	N = 1030
$m$	ArcGIS 9.3	3D Analyst Tools → Slope	${\displaystyle \frac{cot{\alpha }_L+cot{\alpha }_R}{2}}$	DEM-based extraction	N = 1030
$A_0$	ArcGIS 9.3	Spatial Analyst Tools → Flow Accumulation	$0.75\times {10}^6 {\mathrm{m}}^2\le A_0\le 24.2\times {10}^6 {\mathrm{m}}^2$	DEM-based extraction	N = 1030
$n$	References	Computing the mean	$0.096 \mathrm{s}.{\mathrm{m}}^{-{\displaystyle \frac{1}{3}}}$	[99,100,101,102]	N = 1
$k$	References	Computing the mean	0.4	[103,104,105,106]	N = 1
$I$	ArcGIS 9.3+ Python 3.11.9	Multi-year mean of peak hourly rainfall intensity	36.5 mm/h	Rainfall intensity data: http://data.cma.cn (accessed on 24 November 2025)	N = 1

.

We set $I=36.5 \mathrm{m}\mathrm{m}/\mathrm{h}$, the mean of peak hourly intensities from multiple extreme rainfall events in Yeniu Gully. Given the strong influence of $I$ on $h$ and $Fr$, Section 5 quantitatively evaluates how changes in $I$ affect ${Fr}_i$ for any ${CDS}_i$, the catchment-wide $\overline{C}$, $│{Fr}_i-{Fr}_{i \pm 1}│$ between adjacent ${CDS}_i$ and ${CDS}_{i \pm 1}$, $│{Fr}_i-\overline{Fr}│$ within each ${CU}_j$, and the final ${CU}_j$ delineation; the impacts of the material property coefficients $n$ and $k$ on these outcomes are assessed in parallel.

All parameters above were substituted into Equations (7)–(10), and ${CU}_j$ were delineated according to Equation (17). All computations were implemented in Python 3.11.9. In addition, based on channel-network imagery of Yeniu Gully, representative geomorphic functional reaches—major bends, confluences, abrupt width changes, and abrupt changes in longitudinal gradient—were compiled (remote-sensing perspective) to preliminarily assess the method’s ability to identify geomorphic functional units.

3.2.2. Field-Surveyed Reach Evaluation Data Set

To assess the applicability of the proposed ${CU}_j$ delineation at specific localities, seven channel reaches (1–1 to 7–7 in Figure 3a) with relatively homogeneous geometric characteristics were selected in Yeniu Gully. In the field, we measured their length $L_{{CU}_j}$, mean width $b$, bank-slope coefficient $m$, and longitudinal gradient $J$. Field identification (in situ perspective) indicates that these relatively homogeneous reaches include transport reaches (1–1, 6–6), depositional reaches (2–2, 3–3), and erosional reaches (4–4, 5–5, 7–7). Within these geomorphic functional units, geometric form, hydraulic gradient, and the roughness and thickness of deposits all exhibit pronounced spatial self-similarity and relative homogeneity.

4. Results

All results in this section are computed under the baseline scenario ($I=$ 36.5 mm/h, $n=$ 0.096 $\mathrm{s}.{\mathrm{m}}^{-{\displaystyle \frac{1}{3}}}$, $k=$ 0.4). Sensitivity and robustness analyses for variations in $I$, $n$, and $k$ are provided in Section 5.

4.1. Threshold of the Physical Criterion ($\overline{C}$)

With the solid–liquid cutoff grain size set to $D$ = 2 mm, Equation (16) gives the underwater angle of repose $\mathrm{\varnothing }=35.04°$. Substituting into Equation (15) yields the corresponding critical friction velocity $u_{*}=0.123 \mathrm{m}/\mathrm{s}$. We then use Equation (10) to compute, for the entire channel network in Yeniu Gully, the permissible perturbation thresholds $C_i$ for all $N=$ 1030 ${CDS}_i$ (small scale), and take their arithmetic mean to obtain the final threshold $\overline{C}=0.134$ (Figure 4) for delineating ${CU}_j$ (large scale). (A demonstration of the representativeness and consistency of using the mean $\overline{C}$ of all $C_i$ as the final threshold is provided in Section 5.) Accordingly, the constant $C$ in Equation (9) and the threshold $\overline{C}$ in Equation (17) both take the value 0.134, which is used to decide whether adjacent ${CDS}_i$ are merged or split to form ${CU}_j$. Spatially, $C_i$ values generally decrease from upstream to downstream (Figure 4): narrow, steep headwater tributaries are more sensitive to small perturbations, whereas the wider, gentler main stem is more stable. This pattern is consistent with the general mechanical understanding that larger physical systems are more disturbance-resistant and also matches the geomorphic contrast of “narrow–steep upstream vs. wide–gentle downstream” in Yeniu Gully.

4.2. ${CU}_j$ Delineation for the Yeniu Gully Channel Network

4.2.1. Basin-Wide (“Areal”) Scale

Applying the along-channel rolling, iterative rule of Equation (17) to the channel network (which contains 2893 $gc$, each 12.5 m × 12.5 m, yields a total of 409 ${CU}_j$ (see the count of light-blue spheres in Figure 5a and Figure 6a). Measured by element count, using ${CU}_j$ as the computational unit compresses the computational scale by a factor of ∼7.1 relative to grid cells ($gc$), which can substantially improve efficiency when running continuum-based mechanistic models within the channel network.

In terms of geometric scales, the minimum/maximum ${CU}_j$ length $L_{{CU}_j}$ is 12.5/177 m, and the minimum/maximum width $b$ is 8.8/159 m; most ${CU}_j$ fall within $L_{{CU}_j}\in (35,138)$ m and $b\in [12.5,63)$ m, consistent with field observations of relatively homogeneous reaches (lengths of tens to hundreds of meters and widths of several to several tens of meters).

Spatially, the main channel contains more ${CU}_j$ in total because it is longer, yet its ${CU}_j$ density per unit channel length (250 m) is lower than in tributaries (Figure 6a); tributaries exhibit more frequent fluctuations of geometric parameters along the profile, leading to denser ${CU}_j$ distributions over the same length (Figure 6a). This contrast is visually apparent from the densities of light-blue markers in Figure 5a and red markers in Figure 5b: at greater upstream distances (mostly tributaries), the ${CU}_j$ “spheres” are more densely spaced.

Regarding $Fr$, which jointly reflects channel geometry and hydraulics: within each ${CU}_j$, all ${CDS}_i$ satisfy $│{Fr}_i-\overline{Fr}│\le 0.134$. Basin-wide, ${Fr}_i$ decreases from upstream to downstream (Figure 5a), consistent with the overall setting: narrow, steep upstream reaches with larger $J$ (and energy slope), smaller $b$, and higher $Fr$, versus wide, gentle downstream reaches with reduced $J$, larger $b$, and lower $Fr$. This indicates that $Fr$ is effective for representing both geometric and hydraulic conditions.

At the adjacent-scale contrast: within any given ${CU}_j$, $|{Fr}_i-{Fr}_{i+1}|$ between adjacent ${CDS}_i$ is generally small (blue spheres in Figure 5b). Although a local peak max $\left|{Fr}_i-{Fr}_{i+1}\right|=0.235$ occurs, the corresponding ${CU}_j$ still satisfies $│{Fr}_i-\overline{Fr}│\le 0.134$ (green spheres in Figure 5a). In contrast, $|\overline{{Fr}_j}-\overline{{Fr}_{j+1}}|$ between adjacent ${CU}_j$ is clearly larger (mean/maximum/minimum of 0.337/1.628/0.099; red spheres in Figure 5b). These results show that $Fr$ is homogeneous within ${CU}_j$ and heterogeneous between ${CU}_j$, and that the “relative-to-mean” constraint $│{Fr}_i-\overline{Fr}│\le 0.134$ effectively suppresses along-reach accumulation of deviation, thereby maintaining intra-${CU}_j$ homogeneity.

4.2.2. Representative Single Tributaries (“Linear” Scale)

To present the delineation more clearly, we zoom in on the two shortest tributaries, TC1 and TC3 (Figure 5c,d). Each is partitioned into 8 ${CU}_j$ (not all tributaries necessarily have 8 ${CU}_j$), namely ${CU}_{118}~{CU}_{125}$ and ${CU}_{35}~{CU}_{42}$, respectively.

For both tributaries, $│{Fr}_i-\overline{Fr}│$ and $|{Fr}_i-{Fr}_{i+1}|$ within each ${CU}_j$ are very small (blue and green dotted lines in Figure 5c,d), whereas $|\overline{{Fr}_j}-\overline{{Fr}_{j+1}}|$ between adjacent ${CU}_j$ is markedly larger (yellow pentagrams in Figure 5c,d). In TC3 (Figure 5d), at $L$ = 3627 m and $L$ = 3778 m the values of $|\overline{{Fr}_j}-\overline{{Fr}_{j+1}}|$ between adjacent ${CU}_j$ are slightly smaller (0.136 and 0.148), close to the critical value 0.134, and $|{Fr}_i-{Fr}_{i+1}|$ at those locations is < 0.1. However, forcibly merging across these locations would produce a ${CU}_j$ with $│{Fr}_i-\overline{Fr}│>0.134$, and the merged ${CU}_j$ would have a large spread between the maximum and minimum ${Fr}_i$. Hence, “split” is more appropriate here. This again verifies that constraining intra-${CU}_j$ dispersion by $│{Fr}_i-\overline{Fr}│\le 0.134$ prevents along-reach accumulation of deviations and avoids pseudo-homogeneous units, ensuring that $Fr$ remains homogeneous within ${CU}_j$ and heterogeneous between ${CU}_j$.

Explicit presentations of key geometric and hydraulic parameters (Figure 7a–d) show that, for both tributaries, the mean differences in longitudinal gradient $J$, flow depth $h$, flow shear stress $\tau$, and the velocity-ratio trend $v_i/v_{i+1}$ between adjacent ${CDS}_i$ within each ${CU}_j$ (i.e., $\overline{\left|J_i-J_{i+1}\right|}$, $\overline{\left|h_i-h_{i+1}\right|}$, $\overline{\left|{\tau }_i-{\tau }_{i+1}\right|}$, and $\overline{\left|v_i/v_{i+1}-v_{i+1}/v_{i+2}\right|}$) are all small (black error bars and the internal dispersion of each stacked bar are small), indicating relative homogeneity of key geometric and hydraulic parameters within any ${CU}_j$. In contrast, the mean differences in these parameters between adjacent ${CU}_j$ (i.e., $\left|\overline{J_{{CU}_j}}-\overline{J_{{CU}_{j+1}}}\right|$, $\left|\overline{h_{{CU}_j}}-\overline{h_{{CU}_{j+1}}}\right|$, $\left|\overline{{\tau }_{{CU}_j}}-\overline{{\tau }_{{CU}_{j+1}}}\right|$, and $\left|\overline{{{(v}_i/v_{i+1})}_{{CU}_j}}-\overline{({v_i/v_{i+1})}_{{CU}_{j+1}}}\right|$) are all pronounced (larger green error bars and greater separation between stacked bars), demonstrating that significant heterogeneity exists in the key geomorphological and hydrodynamic parameters between adjacent ${CU}_j$.

At typical geomorphic functional units (Figure 6b,c; remote-sensing perspective), the delineated ${CU}_j$ in both tributaries are highly sensitive to major bends (black circles) and confluences (red circles), separating them into different ${CU}_j$ according to hydraulic–geometric contrasts. Although this treatment breaks the morphological continuity of a bend or confluence considered as a single geomorphic unit, subdividing a bend into an inlet homogeneous segment (${CU}_j$) and an outlet homogeneous segment (${CU}_j$) better satisfies the homogeneity requirement of continuum models than treating the whole bend as one unit; similarly, subdividing a confluence (“Y”-shaped junction) into the two pre-confluence homogeneous branches (${CU}_j$) and the post-confluence homogeneous segment (${CU}_j$) better meets the homogeneity condition. This is consistent with our “dual requirement”: prioritize homogeneity for continuum modeling and, subject to that, pursue morphological continuity. Where bends or confluences are absent, the two can be satisfied simultaneously. In other tributaries (not TC1/TC3), the method also performs well at detecting abrupt changes in $b$ and $J$ (Figure 6d,e; remote-sensing perspective): locations with strong width changes (e.g., ${CU}_{315}~{CU}_{316}$ in Figure 6d) and with abrupt longitudinal gradient changes (horizontally arranged ${CU}_{85}~{ CU}_{89}$ in Figure 6e) are accurately partitioned into different ${CU}_j$, further showing that the method can identify geomorphic functional units at geometric breakpoints and thus reconcile “homogeneity vs. morphological continuity.”

4.3. Independent Validation at Field-Identified Homogeneous Reaches (“Point” Scale)

4.3.1. Fr and Its Dispersion

As shown in Figure 8a,b, for the seven ${CU}_j$ delineated at the seven known, field-identified homogeneous reaches in Yeniu Gully, the ${Fr}_i$ of all ${CDS}_i$ within the seven ${CU}_j$ cluster around $\overline{Fr}$ (small dispersion), and all satisfy $│{Fr}_i-\overline{Fr}│\le 0.134$. The maximum of ${|Fr}_i-{Fr}_{i+1}|$ between adjacent ${CDS}_i$ and ${CDS}_{i+1}$ within the seven ${CU}_j$ is 0.191 (at the depositional reach 2–2), yet $│{Fr}_i-\overline{Fr}│\le 0.134$ still holds there. At 6–6 (a transport reach), the ${CU}_j$ contains only two ${CDS}_i$; hence the “mean = min = max” situation arises for ${|Fr}_i-{Fr}_{i+1}|$, which slightly exceeds 0.134, but $│{Fr}_i-\overline{Fr}│$ for the two ${CDS}_i$ remains <0.134, so the delineation is still justified. For the other five reaches, ${|Fr}_i-{Fr}_{i+1}|$ is consistently very small and $│{Fr}_i-\overline{Fr}│\le 0.134$, indicating relative homogeneity of $Fr$ within each ${CU}_j$ at these sites.

4.3.2. Comparison of Geometric Quantities

Figure 9a,b shows that, relative to the observed values at the seven homogeneous reaches, the mean absolute error rates of model-predicted $L_{{CU}_j}$ and cross-sectional area $A$ are 14.475% and 5.577%, respectively. The largest error in $L_{{CU}_j}$ (35.86%) occurs at the erosional reach 5–5 because the long “measured homogeneous reach” is reasonably split: the cumulative extreme deviation of ${Fr}_i$ would exceed the threshold if it were forced into a single ${CU}_j$, producing $│{Fr}_i-\overline{Fr}│>0.134$. The model, therefore, splits it into two or more ${CU}_j$, reflecting the method’s sensitivity and the strict control on intra-unit dispersion. As seen from the overall distribution of $L_{{CU}_j}$ in Section 4.2.1, this sensitivity does not lead to universally short ${CU}_j$.

The maximum error rate for $A$ is 16.086% (at depositional reach 3–3), likely due to inherent discrepancies between remote-sensing imagery and actual field object scales. From the perspective of ${|A}_i-A_{i+1}|$ between adjacent ${CDS}_i$ and ${CDS}_{i+1}$ within each ${CU}_j$ (Figure 9d), aside from the relatively larger dispersion of $A$ within the ${CU}_j$ at 2–2 (deposition), the other six ${CU}_j$ show fairly uniform $A$. Across the seven ${CU}_j$, the overall mean absolute adjacent difference ${|A}_i-A_{i+1}|$ is 2.5976 ${\mathrm{m}}^2$; the maximum ${|A}_i-A_{i+1}|$ is 9.4367 ${\mathrm{m}}^2$ (at 2–2). Although $A$ is more dispersed within the ${CU}_j$ at 2–2, the adjacent difference in longitudinal gradient $|J_i-J_{i+1}|$ there has both very small maximum and mean values, and this ${CU}_j$ shows the tightest clustering of $J$ around $\overline{J}$ among the seven (Figure 9c). Because the proposed delineation considers overall three-dimensional geometric homogeneity (with $b$, $m$, $J$, $h$ jointly characterizing the cross section, and $b$, $m$, $h$ directly determining $A$), the ${CU}_j$ at 2–2 can still be judged geometrically homogeneous at the reach scale.

For $J$ (Figure 9c), across the seven ${CU}_j$, the maximum absolute adjacent difference max $|J_i-J_{i+1}|$ is 0.086 (roughly corresponding to a slope difference < 5°), and the mean $|J_i-J_{i+1}|$ is 0.041 (slope difference < 2.4°), which is compatible with intra-unit homogeneity. Overall, the delineated ${CU}_j$ at the seven field-identified homogeneous reaches satisfy relative homogeneity of geometric conditions.

From Section 3.2.2, reaches 1–1 and 6–6 are transport segments; 2–2 and 3–3 are depositional segments; and 4–4, 5–5, and 7–7 are erosional segments. Field observations show marked spatial self-similarity within these geomorphic functional units and relative homogeneity of geometric and hydraulic parameters. Correspondingly, the delineation yields relatively homogeneous ${CU}_j$ at all seven sites, except that 5–5 is split into two ${CU}_j$ due to its length (Figure 9a–d), the other sites preserve both morphological continuity of the functional unit and intra-unit homogeneity.

4.3.3. Hydraulic and Kinematic Indicators

As shown in Figure 10a, the bed shear stress $\tau$ values within each of the seven ${CU}_j$ at the seven field-identified homogeneous reaches cluster near the mean $\overline{\tau }$ (small dispersion). This indicates similar $\tau$ across ${CDS}_i$ within each ${CU}_j$ because the maximum adjacent difference max ${|\tau }_i-{\tau }_{i+1}|$ is only 0.36578 kN/m² at erosional reach 5–5, i.e., 0.36578 kPa (equivalent head ∼3.7 cm), which is small and acceptable; moreover, the mean $\overline{{|\tau }_i-{\tau }_{i+1}|}$ between adjacent ${CDS}_i$ and ${CDS}_{i+1}$ within each ${CU}_j$ is very small at all seven sites. Thus, the hydraulic parameters are similar and relatively homogeneous within each ${CU}_j$.

From Figure 10b, it is evident that the velocity ratios $v_i/v_{i+1}$ between adjacent ${CDS}_i$ within the seven ${CU}_j$ at the seven measured homogeneous reach locations are mostly clustered around the mean value $\overline{v_i/v_{i+1}}$, with the majority of $v_i/v_{i+1}$ values close to 1. Furthermore, the absolute difference $\left|{\displaystyle \frac{v_i}{v_{i+1}}}-{\displaystyle \frac{v_{i+1}}{v_{i+2}}}\right|$ between successive pairs of adjacent ${CDS}_i$ (${CDS}_i$, ${CDS}_{i+1}$, and ${CDS}_{i+2}$) is generally small within each ${CU}_j$, indicating that $v_i/v_{i+1}$ and $v_{i+1}/v_{i+2}$ are similar, i.e., the kinematic variation is consistent within each ${CU}_j$.

Taken together (Figure 8, Figure 9 and Figure 10), at the “point” scale, the delineated ${CU}_j$ are self-consistent with the field-identified homogeneous reaches along four evidence chains: $Fr$–geometry–mechanics–kinematics. Statistically, $Fr$ and cross-sectional area $A$ vary inversely (cf. Figure 8a and Figure 9d), whereas $Fr$ correlates positively with the longitudinal gradient $J$ and shear stress $\tau$ (cf. Figure 8a, Figure 9c and Figure 10a). ${CU}_j$ with smaller $|{Fr}_i-{Fr}_{i+1}|$ and $│{Fr}_i-\overline{Fr}│$ exhibit greater intra-unit homogeneity in geometric and hydraulic parameters and more similar kinematic behavior, indicating that local hydraulics are primarily controlled by local near-field geometry; basin-wide hydraulic contrasts are jointly modulated by upstream contributing area and rainfall conditions. The resulting ${CU}_j$ simultaneously satisfy the morphological continuity of geomorphic functional segments and the homogeneity requirement within computational units.

In summary, multi-scale (areal–linear–point) evidence shows that, with $│{Fr}_i-\overline{Fr}│\le \overline{C}$ as the core constraint, the physical framework formed by Equations (4)–(8) and (10)–(17)—i.e., an “$Fr$-driven homogeneity criterion + along-channel rolling merge/split”—stably yields ${CU}_j$ that are “relatively homogeneous within units and significantly heterogeneous between units.” These ${CU}_j$ meet the homogeneity assumption for computational units in continuum models while preserving and depicting the morphological continuity of geomorphic functional segments (by representing relatively homogeneous reaches as grid clusters). They also avoid the issues of grid cells segmenting functional-unit continuity and overlooking clustered sediment-source mobilization, thereby providing a more rational and efficient computational unit for subsequent continuum-based mechanistic early warning models.

5. Discussion

This study proposes an integrated theoretical–computational framework for delineating ${CU}_j$. Under satellite remote-sensing DEM and GIS conditions, we first develop, from hydraulics and geomorphic-process theory, an $Fr$ computation model that jointly characterizes channel geometric and hydraulic conditions, enabling $Fr$ to directly quantify the hydro-geomorphic state of each ${CDS}_i$. We then describe inter-$CDS$ contrasts via $\Delta Fr$, including the two metrics $|{Fr}_i-{Fr}_{i+1}|$ and $│{Fr}_i-\overline{Fr}│$. Next, drawing on the critical bed-shear theory at the solid-liquid cutoff grain size and a small-velocity-perturbation analysis, we determine the threshold $\overline{C}$. With the “relative-to-mean constraint” $│{Fr}_i-\overline{Fr}│\le \overline{C}$ and within the framework of Equations (4)–(8) and (10)–(17), an along-channel rolling merge/split procedure yields ${CU}_j$ as clusters of grid cells that are relatively homogeneous within units but markedly heterogeneous between units. Compared with conventional grid cells, ${CU}_j$ not only satisfy the homogeneity hypothesis required of computational units in continuum mechanics models, but also preserve the morphological continuity of geomorphic functional segments, thereby avoiding the fragmentation of functional units by single pixels and the neglect of clustered source-mobilization effects. Section 4 (“Results”) has validated the method across “point–line–areal” scales along four evidence chains: $Fr$, geometry, mechanics, and kinematics. Building on this, we now discuss robustness, generality, advantages, and limitations (Section 5.1 and Section 5.2), closing the loop with the prior statements in Section 3.2.1 and Section 4.1.

5.1. Robustness, Generality, and Advantages of the ${CU}_j$ Delineation

In Section 3.2 (Data Sets), the material-property coefficients (runoff coefficient $k$ and Manning roughness $n$) were assigned recommended values within commonly used ranges [99,100,101,102,103,104,105,106], and the peak hourly rainfall intensity $I$ was set to a computed baseline. In Equations (7), (8), and (10), parameters other than $I$, $k$, and $n$—namely $b$, $m$, $J$, $A_0$, and $g$—are fixed or can be uniquely determined from the DEM via ArcGIS 9.3. Because $I$, $k$, and $n$ have some empirical latitude and, when entering Equations (7), (8), and (10), act through $h$ and $v$ to influence $Fr$, $\Delta Fr$, and $C_i$ (with $\overline{C}$ defined as the arithmetic mean of all $C_i$ basin-wide), it is necessary to examine whether changes in $I$, $k$, and $n$ materially affect $Fr$, $\Delta Fr$, $\overline{C}$, and thus the ${CU}_j$ delineation. To this end, we select tributary TC1 and, under multiple combinations of $I/k/n$, compute ${Fr}_i$, $|{Fr}_i-{Fr}_{i+1}|$, $│{Fr}_i-\overline{Fr}│$, $\overline{C}$, and the resulting ${CU}_j$. We also chose a representative ${CDS}_i$ to test how their local $C_i$ respond to changes in $k$ and $n$, and whether those trends accord with the basin-wide $\overline{C}$, thereby assessing representativeness and consistency.

5.1.1. Effect of $I$ on $Fr$, $\Delta Fr$, $\overline{C}$, and ${CU}_j$

As shown in Figure 11a,b and Figure 12a,b, we keep $k$ and $n$ fixed (Table 1) and set $I$ = 11.26, 14.11, 16.88, 18.47, 20.49, 23.28, 30, 36.5, 42.5, 49, and 55 mm/h, corresponding to return periods of 5-, 10-, 20-, 30-, 50-, and 100-year events, plus several random values (increments of 6–7 mm/h) around the 100-year peak. Following our method, for each $I$ we obtain ${Fr}_i$ for all ${CDS}_i$ in TC1, $|{Fr}_i-{Fr}_{i+1}|$ for every adjacent pair, and $│{Fr}_i-\overline{Fr}│$ within each ${CU}_j$. We also compute $\overline{C}$ and the new ${CU}_j$ delineation for the whole basin. The main findings are:

(1)

Increasing $I$ markedly elevates ${Fr}_i$ (Figure 11a), whereas $∆Fr\equiv |{Fr}_i-{Fr}_{i+1}|$ shows very weak sensitivity to $I$ (Figure 11b). At the location with the strongest response ($L$ = 2128 m), the difference between the minimum $\Delta Fr$ at $I$ = 11.26 and the maximum $\Delta Fr$ at $I$ = 55 mm/h is only 0.055. Where adjacent ${CDS}_i$ have similar geometry (e.g., at $L$ = 2030, 2057, 2087, 2098, 2102, and 2191 m), $\Delta Fr$ is essentially insensitive to $I$.

(2)

As $I$ increases, the growth rates of both ${Fr}_i$ and $\Delta Fr$ diminish (convergent behavior), indicating that $I$ is not the sole control on these quantities, and supports the need for a multi-parameter $Fr$ model.

Using $│{Fr}_i-\overline{Fr}│$ and $\overline{C}$ to decide merge/split (Equation (17)), the delineation matches the baseline ($I=$ 36.5 mm/h, $n=$ 0.096 $\mathrm{s}.{\mathrm{m}}^{-{\displaystyle \frac{1}{3}}}$, $k=$ 0.4) exactly for $I$ ≥ 30 mm/h (TC1: ${CU}_{118}-{CU}_{125}$); the ${CU}_j$ division error rate is 0 (

Table 2. Robustness of the ${CU}_j$ delineation in tributary TC1 under various operating conditions.

Parameters (Unit)	Variables	$\mathbf{No}. \mathbf{of} {\mathit{C}\mathit{D}\mathit{S}}_{\mathit{i}}$ Classified as Unstable	$\mathbf{Total} \mathbf{No}. \mathbf{of} {\mathit{C}\mathit{D}\mathit{S}}_{\mathit{i}}$	$\mathbf{Percentage} \mathbf{of} {\mathit{C}\mathit{D}\mathit{S}}_{\mathit{i}}$ Classified as Stable	$\mathbf{Worst}\text{-}\mathbf{Case} \mathbf{Performance} \mathbf{of} \mathbf{Stable}\text{-}\mathbf{Classified} {\mathit{C}\mathit{D}\mathit{S}}_{\mathit{i}}$	$\mathbf{Total} \mathbf{No}. \mathbf{of} {\mathit{C}\mathit{U}}_{\mathit{j}}$ Divisions Under the Current Operating Condition	$\mathbf{Total} \mathbf{No}. \mathbf{of} {\mathit{C}\mathit{U}}_{\mathit{j}}$$\mathbf{Divisions} \mathbf{Under} \mathit{I}$ = $36.5 \mathbf{mm}/\mathbf{h}, \mathit{k}$ = $0.4, \mathit{n}$ = $0.096 \mathbf{s}.{\mathbf{m}}^{-\frac{1}{3}}$	$\mathbf{Division} \mathbf{Error} \mathbf{Rate} \mathbf{of} {\mathit{C}\mathit{U}}_{\mathit{j}}$ Under the Current Condition	$\mathbf{Mean} \mathbf{Accuracy} \mathbf{of} {\mathit{C}\mathit{U}}_{\mathit{j}}$ Divisions Across All Conditions (Excluding Outliers)	$\mathbf{Mean} \mathbf{Accuracy} \mathbf{of} {\mathit{C}\mathit{U}}_{\mathit{j}}$ Divisions Across All Conditions (Including Outliers)
$I$$(\mathrm{mm}/\mathrm{h}), k$ = 0.4, $n$ = 0.096 $\mathrm{s}.{\mathrm{m}}^{-{\displaystyle \frac{1}{3}}}$	11.26	2	17	88.24%	88.24%	5	8	37.5%	Outliers	88.64%
	14.11	2		88.24%		5		37.5%	Outliers
	16.88	1		94.12%		7		12.5%	94.44%
	18.47	1		94.12%		7		12.5%
	20.49	1		94.12%		7		12.5%
	23.28	1		94.12%		7		12.5%
	30	0		100%		8		0
	36.5	0		100%		8		0
	42.5	0		100%		8		0
	49	0		100%		8		0
	55	0		100%		8		0
$k$$, I$ = 36.5 $\mathrm{mm}/\mathrm{h}, n$ = 0.096 $\mathrm{s}.{\mathrm{m}}^{-{\displaystyle \frac{1}{3}}}$	0.2	1	17	94.12%	94.12%	7	8	12.5%	98.21%	98.21%
	0.3	0		100%		8		0
	0.4	0		100%		8		0
	0.5	0		100%		8		0
	0.6	0		100%		8		0
	0.7	0		100%		8		0
	0.8	0		100%		8		0
$n$ $(\mathrm{s}.{\mathrm{m}}^{-{\displaystyle \frac{1}{3}}}),$ $I$ = 36.5 mm/h, $k$ = 0.4	0.02	2	17	88.24%	88.24%	10	8	25%	Outliers	94%
	0.04	0		100%		8		0	96.4%
	0.06	0		100%		8		0
	0.08	0		100%		8		0
	0.096	0		100%		8		0
	0.11	0		100%		8		0
	0.13	1		94.12%		7		12.5%
	0.15	1		94.12%		7		12.5%

Notes: ${CU}_j$ division error rates ≥ 25% are treated as outliers (highlighted in bold italics). The baseline scenario ($I=$ 36.5 mm/h, $n=$ 0.096 $\mathrm{s}.{\mathrm{m}}^{-{\displaystyle \frac{1}{3}}}$, $k=$ 0.4) is marked in bold. “No.” denotes “Number”.

; Figure 12a,b). For 16.88 $\le I\le$ 23.28 mm/h, only one ${CDS}_i$ is over-merged into ${CU}_{122}$ (i.e., the would-be one-segment ${CU}_{123}$ collapses into ${CU}_{122}$), leaving 94.12% of ${CDS}_i$ stably classified and a ${CU}_j$ division error rate of 12.5% (Table 2; Figure 12a,b). For 11.26 $\le I\le$14.11 mm/h, two over-merges cause ${CU}_{122}-{CU}_{125}$ to collapse into a single ${CU}_j$, yielding two unstable ${CDS}_i$ and 88.24% stable classification but a ${CU}_j$ division error rate of 37.5% (flagged as an outlier in Table 2; Figure 12a,b). The outlier arises from a thin-sheet flow situation at very low $I$: the actual wetted width may not fully span the bed, whereas our DEM+GIS workflow assumes bed-wide coverage, producing smaller $v$ and $h$ and slightly larger $|{Fr}_i-{Fr}_i^{*}|=C_i$, hence a modestly inflated $\overline{C}$. Apart from that low-$I$ outlier range, $\overline{C}$ decreases mildly with $I$ and remains close to the baseline value 0.134, without materially altering the delineation (Table 2; Figure 12a,b).

Excluding outliers, the mean ${CU}_j$-division accuracy across $I$ scenarios is 94.44%; including outliers it is still 88.64%. Thus, small $I$ can slightly perturb the result, but robustness remains controllable; large $I$ scarcely affects delineation. We recommend using larger $I$ (e.g., $I\ge$ 30 mm/h) when delineating ${CU}_j$, which stabilizes results and avoids thin-sheet-flow biases.

5.1.2. Effect of $k$ on $Fr$, $\Delta Fr$, $\overline{C}$, and ${CU}_j$

With $I$ and $n$ fixed (Table 1), we set $k=$ 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, and 0.8 (Figure 11c,d and Figure 12c,d). Because both $k$ and $I$ affect discharge and hence $h$ and $v$, their impacts on ${Fr}_i$, $|{Fr}_i-{Fr}_{i+1}|$, $│{Fr}_i-\overline{Fr}│$, and $\overline{C}$ are analogous to those in Section 5.1.1 and are not repeated here (see Figure 11c,d and Figure 12c,d).

Using $│{Fr}_i-\overline{Fr}│$ and $\overline{C}$ as the decision rule, the delineation matches the baseline exactly for $k\ge$ 0.3 (TC1: ${CU}_{118}-{CU}_{125}$); the ${CU}_j$ division error rate is 0 (Table 2; Figure 12c,d). For 0.2 $\le k<$ 0.3, only one ${CDS}_i$ is over-merged into ${CU}_{122}$ (the one-segment ${CU}_{123}$ collapses into ${CU}_{122}$), leaving 94.12% of ${CDS}_i$ stably classified and a ${CU}_j$ division error rate of 12.5% (Table 2; Figure 12c,d). Across all $k$ scenarios, the mean ${CU}_j$-division accuracy reaches 98.21% (Table 2). We therefore recommend 0.3 $\le k\le$ 0.5: robustness is excellent, and the range is consistent with values commonly reported for debris flow gullies [103,104,105,106]. Overall, the delineation is insensitive to $k$.

5.1.3. Effect of $n$ on $Fr$, $\Delta Fr$, $\overline{C}$, and ${CU}_j$

With $I$ and $k$ fixed (Table 1), we set $n=$ 0.02, 0.04, 0.06, 0.08, 0.096, 0.11, 0.13, and 0.15 $\mathrm{s}.{\mathrm{m}}^{-{\displaystyle \frac{1}{3}}}$ (Figure 11e,f and Figure 12e,f). The main results are:

(1)

Increasing $n$ (rougher bed) typically decreases $v$ and increases $h$, thereby reducing $Fr$; this effect is stronger than those of $I$ and $k$ (Figure 11e).

(2)

The impact of $n$ on $\Delta Fr$ is selective: where adjacent ${CDS}_i$ differ strongly in geometry, $\Delta Fr$ is more sensitive to $n$. Near $L=$ 2128 m, the difference between $\Delta Fr$ at $n=$ 0.02 and at $n=$ 0.15 reaches 1.161. Where adjacent ${CDS}_i$ have similar geometry (e.g., $L=$ 2030, 2087, 2098, 2102, and 2191 m), $\Delta Fr$ is barely affected (Figure 11f). Both $Fr$ and $\Delta Fr$ show convergent decreases with increasing $n$.

Using $│{Fr}_i-\overline{Fr}│$ and $\overline{C}$ as the decision rule, the delineation matches the baseline exactly for 0.04 $\le n\le$0.11 $\mathrm{s}.{\mathrm{m}}^{-{\displaystyle \frac{1}{3}}}$ (TC1: ${CU}_{118}-{CU}_{125}$); the ${CU}_j$ division error rate is 0 (Table 2; Figure 12e,f). For 0.02 $\le n<$0.04 $\mathrm{s}.{\mathrm{m}}^{-{\displaystyle \frac{1}{3}}}$, two ${CDS}_i$ are split into separate ${CU}_j$ (one at $L=$ 2164 m detaches from the original ${CU}_{118}$, passively causing another at $L=$ 2146 m to detach as well), increasing the ${CU}_j$ count by two; 88.24% of ${CDS}_i$ remain stably classified, but the ${CU}_j$ division error rate is 25% (flagged as an outlier; Table 2; Figure 12e,f). This outlier likely reflects unrealistically small $n$: debris flow gullies commonly have very rough beds (vegetation, cobbles, and boulders), often rougher than mountain streams, for which recommended $n$ is already 0.04–0.07 $\mathrm{s}.{\mathrm{m}}^{-{\displaystyle \frac{1}{3}}}$ [107]. Hence $n<$ 0.04 is typically too small for debris flow settings. For 0.11 $<n\le$ 0.15 $\mathrm{s}.{\mathrm{m}}^{-{\displaystyle \frac{1}{3}}}$, only one ${CDS}_i$ is over-merged into ${CU}_{122}$ (${CU}_{123}$ collapses into ${CU}_{122}$); 94.12% of ${CDS}_i$ remain stable and the ${CU}_j$ division error rate is 12.5% (Table 2; Figure 12e,f). As $n$ increases, $\overline{C}$ declines mildly and converges toward the baseline value, without materially affecting delineation (except for the $n<$ 0.04 outlier range).

Excluding outliers, the mean ${CU}_j$-division accuracy over $n$ scenarios is 96.4%; including outliers it is 94%. Given typical roughness in debris flow gullies (cobbles/boulders and rough vegetation, often no smoother than mountain streams), we recommend 0.04 $\le n\le$ 0.11 $\mathrm{s}.{\mathrm{m}}^{-{\displaystyle \frac{1}{3}}}$: use 0.04–0.07 for less-rough perennial gullies and 0.07–0.11 for rough or ephemeral (rain-season) gullies. These ranges agree with prior studies [99,100,101,102]; delineations are highly robust within them. Field calibration of $n$ is advisable when possible.

5.1.4. Representativeness of $\overline{C}$: Consistency Between $C_i$ and $\overline{C}$

To test whether the arithmetic mean $\overline{C}$ is representative basin-wide, we select three ${CDS}_i$: one whose $C_i\approx \overline{C}$ under the baseline (TC5, ${CU}_{27}$, $L=$ 5152 m; Figure 13a,b), one with the basin-minimum velocity $v_{min}$ (main channel MC, ${CU}_{48}$, $L=$ 4403 m; Figure 13c,d), and one with the basin-maximum velocity $v_{max}$ (MC, ${CU}_{104}$, $L=$ 2809 m; Figure 13e,f). We then examine how their hydraulic variables and $C_i=|{Fr}_i-{Fr}_i^{*}|$ respond to changes in $k$ and $n$, and whether the trends match those of $\overline{C}$. Given that Figure 11 and Figure 12 demonstrate identical effects of $I$ and $k$ on the relevant physical quantities and their variation patterns, a representative analysis of either parameter is deemed sufficient. Accordingly, $k$ and $n$ were selected from among the three parameters ($I$, $k$, $n$) for subsequent analysis.

Results show that increasing $k$ raises $v$ (or $v_{*}$), $h$ (or $h_{*}$), ${Fr}_i$ and ${Fr}_i^{*}$ with diminishing increments, while $C_i$ decreases monotonically and converges (Figure 13a,c,e), consistent with $\overline{C}$ decreasing and converging as $k$ increases (Figure 12c,d). Increasing $n$ lowers $v$ (or $v_{*}$) and raises $h$ (or $h_{*}$) via discharge continuity and quasi-hydrostatic depth effects; both ${Fr}_i$ and ${Fr}_i^{*}$ decrease with diminishing decrements, and $C_i$ decreases monotonically and converges (Figure 13b,d,f), consistent with $\overline{C}$ decreasing and converging as $n$ increases (Figure 12e,f). Therefore, $\overline{C}$ reflects the basin’s overall resistance to flow-field perturbations and is consistent with local $C_i$ responses, demonstrating representativeness and robustness. Together with Section 5.1.1, Section 5.1.2 and Section 5.1.3, the method exhibits high consistency under parameter changes and stably outputs ${CU}_j$.

5.1.5. Generality and Methodological Advantages

Although the application here focuses on Yeniu Gully (Wenchuan County), the method is physics-based and DEM+GIS-driven, requiring only three data types: DEM, peak hourly rainfall intensity $I$, and material-property coefficients $n$ and $k$. Hence, it has good engineering portability and practical convenience. The parameters $I$, $n$, and $k$ can be assigned from regional information or field calibration, as is common for theoretical models in practice. Thus, the proposed ${CU}_j$ delineation is both robust and general, facilitating deployment across different debris flow gullies and regions.

Compared with grid cells, ${CU}_j$ can satisfy the computational-unit requirements of continuum models (homogeneous within, heterogeneous between) and more faithfully express structural and functional attributes of channel segments. By analogy—as chords vs. isolated notes in music, organs vs. single cells in biology, phrases vs. single characters in linguistics, or molecules/atoms vs. sub-particles in physics—${CU}_j$ capture structure and function more realistically than grid cells, thereby reflecting overall network function and process continuity. This also implies that the smallest computational scale is not always “best”; rather, subject to homogeneity, moderately enlarging the scale can better represent functional structure and clustered hydro-dynamics. The idea is consistent with the continuum mechanics “Representative Volume Element (RVE)” concept: macroscopically infinitesimal yet microscopically large (it must be macroscopically infinitesimal to preserve field continuity, yet microscopically sufficiently large to encompass the underlying microstructure, a manifestation of the inherent scale dependence between “macro” and “micro” domains). Practically, ${CU}_j$ provide a scientifically sound computational environment for convergence and stability of continuum models, reduce the computational burden at the network scale, and improve efficiency when simulating with continuum mechanics-based models. They also avoid breaking the morphological continuity of functional segments inherent to pixel-wise discretization and account for clustered source-mobilization effects. Overall, ${CU}_j$ constitute a more rational and efficient computational unit for continuum mechanics-based early warning models of runoff-generated debris flow.

5.2. Limitations and Future Directions

First, the method benefits from higher-resolution DEMs; lower resolutions can degrade ${CU}_j$ delineation. In headwater sectors where actual channel width is narrower than the DEM pixel size, the delineated ${CU}_j$ width may exceed reality, reflecting the resolution limit. Systematic geolocation bias between remote-sensing imagery and ground objects is also hard to eliminate. We therefore recommend high-resolution DEMs and multiple ground control points for geospatial calibration to reduce system error. Second, extracting the channel network via the ArcGIS 9.3 D8 algorithm (with prior depression filling) tends to smooth sharp knickpoints and local rises to enforce outlet drainage, which makes delineation at these sites conservative, a known challenge of DEM–GIS toolchains in this domain. In debris flow channels that contain artificial check dams or other hydraulic control structures, the delineation of ${CU}_j$ can be locally biased in the vicinity of the dams. Channel segments immediately upstream and downstream may be merged into a single, excessively long unit, or two hydraulically distinct reaches on either side of a dam may not be separated. This bias stems from the DEM preprocessing and hydraulic closure. First, channel networks are derived from a DEM subjected to depression filling, which enforces continuous flow paths and tends to represent the area immediately upstream and downstream of a dam as a single continuous surface, forming an artificially unified channel segment. Second, in the hydrodynamic module, the depth-averaged momentum balance is approximated by a one-dimensional Manning formulation closed with $S_f\approx J$, where $J$ is the bed slope estimated from the filled DEM. Depression filling smooths local adverse slopes upstream of dams, so the geometric slope $J$ may differ from the true energy slope $S_f$ while the model still uses $J$ as its surrogate. This limitation is confined to the immediate vicinity of artificial check dams; channel segments located sufficiently far from such structures are not affected, and the delineation of ${CU}_j$ remains reliable for the majority of the channel network. Nevertheless, the ${CU}_j$ delineated around check-dam sites should be interpreted with caution. Overall, the method remains applicable to debris flow channels containing artificial check dams and other hydraulic control structures, provided that the delineated ${CU}_j$ in the vicinity of the dams are treated with appropriate caution. Fortunately, continuing advances in remote sensing and GIS (higher resolution, better algorithms) are expected to improve ${CU}_j$ delineation accuracy with our method. Regarding the trade-off between unit homogeneity and morphological continuity: while the method performs well at depositional, erosional, and transport segments, at major bends, confluences, and strong jumps in width or slope, we deliberately prioritize homogeneity first, and morphological continuity where possible. For critical engineering sites, manual review or field checks are advisable to balance physical consistency and engineering interpretability.

Future work should compare using ${CU}_j$ versus grid cells as computational units in continuum mechanics-based early warning/forecasting of runoff-generated debris flow, evaluating accuracy, false-alarm/miss rates, and timeliness. We will also extend validation across basins and sediment-source settings, and under different DEM resolutions, to comprehensively assess performance. Finally, delineated ${CU}_j$ can be directly coupled with mobile-bed shallow-water equations or viscoplastic rheological models for integrated initiation–propagation hazard mapping and reach-based engineering layout; related efforts are underway.

6. Conclusions

In response to the needs of continuum mechanics-based early warning and forecasting of runoff-generated debris flow, this study proposes and validates a physics-based method for delineating homogeneous channel units (${CU}_j$). The method integrates hydraulic theory with geomorphic-process theory and develops an identification model centered on the Froude number ($Fr$) as the core criterion. It aims to unify the relative homogeneity of computational units (in both hydraulic and geometric parameters) with the morphological functionality of channel segments, thereby overcoming the structural limitations of conventional grid cells, which cannot concurrently ensure homogeneity in hydraulic and geometric parameters and preserve the morphological continuity of geomorphic functional segments. The main conclusions are as follows:

(1)

We construct an $Fr$ computation model constrained by cross-sectional geometry, topographic parameters, and material property coefficients, enabling $Fr$ to jointly characterize the geometric features and hydraulic parameters of channel-discretized segments (${CDS}_i$). The spatial variability of $Fr$ between adjacent ${CDS}_i$ is shown to be controlled primarily by local differences in geometric parameters; statistically, $Fr$ is positively correlated with longitudinal gradient and bed shear stress, and negatively correlated with cross-sectional area. Using the critical bed-shear stress for incipient motion at the solid–liquid cutoff grain size together with a small velocity-perturbation analysis, we determine a physical identification threshold. Constraining the absolute deviation of $Fr$ within each ${CU}_j$ by this threshold yields an automated, physics-based framework for delineating homogeneous ${CU}_j$, ensuring the physical self-consistency and transferability of the units.

(2)

Multi-scale (point–line–areal) validation in the Yeniu Gully catchment demonstrates that the proposed framework produces computational units that are relatively homogeneous within (in longitudinal gradient, cross-sectional area, flow depth, shear stress, etc.) and heterogeneous between ${CU}_j$, while preserving the morphological continuity of geomorphic functional segments. A total of 409 ${CU}_j$ are delineated across the basin; their spatial pattern accords with terrain complexity—denser ${CU}_j$ in tributaries and sparser ${CU}_j$ in the main channel—consistent with actual geomorphic complexity and along-channel hydraulic attenuation. The ${CU}_j$ stably capture key geomorphic functional units, including bends, confluences, slope and width breaks, erosional reaches, and depositional reaches. Under varying rainfall intensity, runoff coefficient, and Manning roughness scenarios, the overall delineation accuracy remains 94.44–98.21%, and even under the most adverse extreme condition it exceeds 88.64%, indicating strong robustness and generality.

(3)

The method requires only three types of input data—DEM, peak hourly rainfall intensity, and material property coefficients—and the DEM+GIS workflow affords strong engineering portability and operational convenience, enabling deployment in diverse debris flow gullies. While the delineation accuracy is constrained by DEM resolution and the inherent behavior of the D8 flow-routing algorithm in ArcGIS 9.3, these limitations are common to the field; anticipated advances in remote sensing and GIS will continuously improve performance. Future work will (i) extrapolate and validate the proposed ${CU}_j$ delineation across basins with different sediment sources and under DEMs of varying resolution; (ii) compare the performance of ${CU}_j$ versus grid cells as computational units in continuum mechanics-based early warning/forecasting of runoff-generated debris flow (hit rate, false-alarm rate, miss rate, and timeliness). We also expect ${CU}_j$ to couple directly with other continuum models (e.g., mobile-bed shallow-water equations or viscoplastic rheologies) for integrated initiation–propagation hazard mapping and reach-scale engineering design.

In sum, the proposed physics-constrained delineation of homogeneous ${CU}_j$ is well-founded in terms of theoretical completeness, computational reproducibility, robustness, and engineering portability, and shows promise as a new-generation computational unit within continuum models for runoff-generated debris flow. It unifies the computational unit with the geomorphic functional unit, and is expected to enhance the accuracy, timeliness, and geomorphic functional relevance of continuum mechanics-based early warning and forecasting.