Characterizing the Surface Grain Size Distribution in a Gravel-Bed River Using UAV Optical Imagery and SfM Photogrammetry

Highlights

What are the main findings?

• Surface roughness metrics derived from UAV-SfM point clouds effectively characterize grain-size distributions in gravel-bed rivers.
• A reach-scale grain size–roughness relation was established for riverbeds with wide grain-size variability.

What is the implication of the main finding?

• The integrated relation enables rapid estimation of riverbed grain-size distributions using UAV-SfM-derived roughness.
• Applicability tests indicate more reliable grain-size estimation for coarser grains than for finer grains in heterogeneous gravel beds.

Abstract

Understanding the sediment grain size distribution in riverbeds is essential for analyzing sediment transport, riverbed morphology, and ecological habitats. Previous studies have shown that riverbed grain size can be inferred from surface roughness using linear relations between manually sampled grain sizes and percentile roughness derived from point-cloud data. However, these relations are often established within narrow grain-size ranges, causing regression coefficients to vary across percentiles and limiting their applicability to broader grain-size variability. This study conducted field investigations and UAV (Unmanned Aerial Vehicle) surveys to examine grain size–roughness relations across four coarse-grained mountainous river reaches in Taiwan, characterized by a wide grain-size distribution (D₁₆–D₈₄: 2.3–525 mm). High-resolution 3D point clouds were generated using UAV-SfM (Structure-from-Motion) techniques for roughness metric computation. Linear relations between grain size D_i (i = 16, 25, 50, 75, and 84) and their corresponding percentile roughness RH_i were developed and evaluated. Results indicate that D_i-RH_i relations exhibit moderate to strong correlations (R² = 0.60–0.94), and the regression slope increases exponentially with grain size. To address cross-percentile variability, an integrated power-law relation was proposed by pooling all paired D_i-RH_i data from Reach R1, yielding a single, continuous reach-scale grain size–roughness correlation. Applicability tests using data from the remaining three reaches show that the integrated relation performs better for coarser grains (D₅₀–D₈₄) than for finer grains. Future work incorporating more sampling sites across diverse river types will help further refine the integrated relation and improve its cross-reach applicability.

1. Introduction

The distribution of sediment grain sizes in riverbeds is an important characteristic for analyzing sediment transport, riverbed morphology, and ecological habitats [1,2]. Grain sizes influence flow resistance and the magnitude of critical shear stress against erosion [3,4,5], thereby affecting sediment transport dynamics during floods and debris-flow events [6,7]. The comprehension of grain size distribution and its variations is essential for fluvial dynamics analysis and riverbed erosion and deposition estimation [8,9]. Grain size distribution is also a crucial parameter in hydraulic modelling and sediment transport simulations [10].

Conventional field-based grain size sampling methods (e.g., Wolman pebble counts [11]), though reliable, are labor-intensive and time-consuming, especially for the rivers in steep mountainous regions [12,13]. To improve survey efficiency, researchers have developed grain-size estimation methods based on either image analysis or topographic data [14]. Image-based approaches, including photosieving [13,15,16] and texture-based metrics such as image entropy [17,18], can characterize resolvable grains, though their performance may be affected by lighting conditions, image resolution, and grain texture [19].

Parallel developments for estimating grain sizes were conducted using topographic data analysis. Researchers had established the reach-scale correlation between local grain sizes (such as D₅₀ and D₈₄, the size at which 50% or 84% of measured b-axes are finer, b-axes represent an intermediate dimension of a grain, see Figure 1) obtained through manual sampling and the roughness metrics derived from topographic data [18,20,21,22,23,24,25] (as shown in

Table 1. Summary of grain size-roughness relations obtained by previous researchers (modified from [16]).

Researchers	Sediment Description (Grain Size Range)	Grain Size Sampling	Data	Roughness Metric * (Grid Size)	Grain Size-Roughness Relation (in mm)	R2
Heritage and Milan [21]	Gravel-bed river with discs dominating (D50 = 30–95 mm)	Pebble counts	TLS	2σ (0.05 m)	D50 = 0.73(2σ50) + 37.0	0.37
Hodge et al. [22]	Tabular form and rounded edges (D50 = 18–63 mm)	Pebble counts	TLS	σd (1.0 m)	D50 = 1.42σd50 + 8.51	0.65
Brasington et al. [20]	Schistose, cobble-sized grains (D50 = 30–117 mm)	Pebble counts	TLS	σd (1–2 m)	D50 = 2.59σd50 + 11.9	0.92
Woodget and Austrums [25]	Cobbles and boulders (D84 = 10–160 mm)	Areal sample & Photosieving	SfM	RH (0.4 m)	D84 = 12.35RH50 − 2.90	0.80
Vázquez-Tarrío et al. [24]	Well-rounded and subspherical grains (D50 = 28–65 mm)	Pebble counts	SfM	RH (1.0 m), 2σ (1.0 m) & σd (1.0 m)	D16 = 0.73RH16 + 7.26	0.64
					D50 = 0.89RH50 + 7.95	0.89
					D84 = 0.78RH84 + 18.9	0.83
Pearson et al. [23]	Oblate (53%), prolate (24%), and sphere (23%) shaped particles. (D50 = 13–72 mm)	Areal sample	SfM	σ (0.2 m) & RH (0.5 m)	Poor sorting D50 = −0.29RH50 + 50.0	0.02
					Moderately well-sorted D50 = 1.85 RH50 + 22.0	0.69
Wong et al. [18]	Sands, gravels, and cobbles (D50 = 13–16 mm)	Photosieving	SfM	σ (0.03 m) RH (0.08 m)	D50 = 1.07RH50 + 11.6	0.42
					D84= 3.87RH50 + 13.7	0.49

* σ: Standard deviation, σ_d: Detrended standard deviation, RH: Roughness height.

). This method offers the advantage of spatially continuous characterization of grain size, benefiting from the continuous nature of topographic data [24] and has become increasingly feasible with recent improvements in UAV (Unmanned Aerial Vehicle)-based Structure-from-Motion (SfM) photogrammetry [26], which provides high-quality terrain data with greater accessibility, flexibility and cost-efficiency compared to the TLS (terrestrial laser scanning) [18,24,25].

Comparisons of the grain size and roughness relations established by previous researchers discovered that the linear coefficients of the relations vary for the results obtained by different researchers from different river reaches, as shown in Table 1. These variations could be ascribed to multiple factors affecting the grain size-roughness relations, including the difference between sampling methods (pebble counts or areal sample, [12]), grains (size, composition, stacking structure, [23]), and roughness metrics (method and grid size, [17,23,24]). Experimental evidence shows that surface roughness is primarily governed by vertical height contrasts among grains, with spherical or protruding grains producing higher roughness than flat grains of comparable size [23]. Vázquez-Tarrío et al. [24] revealed that the correlations in larger grain (D₅₀ and D₈₄) and their corresponding percentile roughness were higher than those of finer grain (D₁₆). Wong et al. [18] similarly noted that flatter or smaller grains reduce roughness variability and weaken the resulting correlations. Together, these studies highlight that grain size and composition can influence the form and strength of grain size–roughness relations.

Previous studies also showed that most existing correlations between grain size and roughness were developed in gravel-bed rivers with D₈₄ typically below 160 mm (Table 1). The grain size–roughness relations for riverbeds containing larger boulders and exhibiting broad, highly heterogeneous grain size distributions remain understudied. Moreover, previous research mainly focused on exploring the linear relationship between local grain size and corresponding roughness. Such linear relations are constrained by the limited grain-size ranges from which they were derived, and because each percentile pair generally yields distinct regression coefficients, they cannot consistently estimate grain sizes across a wide range of grain-size variability, which often results in fragmented solutions for different local grain sizes.

To address this gap, we conducted manual samplings in four mountainous reaches (R1, R2, R3, and R4) across two watersheds in Taiwan, where riverbeds are characterized by coarse grains, poorly sorted (highly heterogeneous) composition, and a broad grain size distribution that includes boulders (>256 mm). High-precision 3D point cloud data were generated using UAV-SfM techniques for roughness metric calculation. First, we tested different kernel radii to determine an appropriate roughness height (RH) calculation size. Next, the reach-scale linear D_i-RH_i (i = 16, 25, 50, 75, and 84, respectively) relations were established by comparing manually sampled grain sizes (D_i) with their corresponding percentile roughness (RH_i) at eight sites in Reach R1. Then, all paired D_i-RH_i data were applied to derive an integrated power-law relation, thereby providing a single, continuous relation capable of characterizing a wider range of grain sizes. Finally, the applicability of the integrated relation was tested using the data taken from Reaches R2–R4.

2. Materials and Methods

2.1. Study Area

This study was conducted in four river reaches (R1, R2, R3, and R4) within two mountainous watersheds in Taiwan: The Heshe River watershed in central Taiwan and the Laishe River watershed in southern Taiwan (Figure 2). The Heshe River drains a 92.3 km² catchment, ranging in elevation from 758 to 2859 m a.s.l. (above sea level), while the Laishe River drains a 44.2 km² catchment with elevations between 125 and 2286 m a.s.l. The mean basin slopes are 34.6° and 36.2°, respectively. Both watersheds are located in a subtropical monsoon climate, with average annual rainfall of 3100 mm in the Heshe watershed (Shemu Village gauge) and 3600 mm in the Laishe watershed (Xinlaiyi gauge). Approximately 75% of annual rainfall occurs between May and September, primarily during typhoons and frontal rainstorms, resulting in pronounced seasonal variations in streamflow. The geological distribution in the Heshe River watershed is primarily dominated by the Nanjuang Formation, characterized by interbedded sandstone and shale. The Laishe River watershed is mainly composed of the Chaozhou Formation, featuring predominantly hard shale or slate [27].

The studied Reach R1 and Reach R2 are located in the middle section of the Heshe River watershed. Reach R1 is situated downstream of the confluence of three upstream tributaries and is characterized by a relatively straight, single-thread channel with lateral sediment bars. The channel width is approximately 100 m with a mean channel slope of approximately 0.075. Reach R2 is located approximately 2 km downstream of Reach R1 and displays a braided channel morphology, with multiple flow paths and mid-channel bars. Its channel width is approximately 150 m, and its channel slope is about 0.053.

The studied Reach R3 and Reach R4 are located in the mid-lower section of the Laishe River watershed, immediately upstream of the confluence between the main stream and a major tributary. Reach R3 lies on the tributary side and exhibits a relatively narrow, single-thread channel with a slope of approximately 0.054 and a width of about 80 m. In contrast, Reach R4 lies along the main stream, with a wider channel of about 150 m and a gentler slope of 0.033.

Landslides and debris flow have introduced coarse grains into the study reaches [28,29]. Consequently, boulders (>256 mm) are commonly scattered across the riverbed, producing a wide grain size distribution. The exposed bed surfaces are heterogeneous and dominated by coarse grains. Sediments are mainly composed of irregularly shaped gravels and cobbles, which are generally arranged in a random packing state, with interstitial spaces partly filled by discontinuous patches of sand and fine gravel.

2.2. Field Surveys

Seven field surveys, including the utilization of UAV for topographic data collection and Wolman pebble counts samplings, were conducted in this study. Each UAV survey was paired with corresponding sites of grain-size sampling conducted during the same field campaign, as summarized in

Table 2. Seven field surveys in the studied reaches for UAV photography and grain size samplings.

Set	Date	Reach	Surveyed Area (ha)	Point Cloud Density (pts/m2)	GSD (mm/px)	Georeferencing Errors (cm)	Sampling Sites
1	15 December 2021	R1	0.77	2803.1	5.8	0.8	H01
2	24 October 2022	R1	4.20	2908.8	7.2	1.8	H02, H03, H04, & H05
3	5 January 2023	R2	1.58	4948.3	4.7	2.1	H09
4	6 July 2023	R1	4.50	4176.3	7.2	2.0	H06, H7, &H 08
5	24 November 2023	R3	2.37	3380.1	6.3	3.2	N01 & N02
6	18 January 2024	R4	7.75	2944.9	6.5	2.4	L01 & L02
7	21 February 2024	R2	2.64	3345.2	7.1	1.6	H10

. To ensure smooth progress in field investigations, all activities were conducted on sunny days to obtain high-quality UAV images and reliable grain-size samples. The subsequent section delineates the implementation approach of the field surveys.

2.2.1. UAV Photography and Point Cloud Data

This study employed a DJI Phantom 4 Pro (DJI, Shenzhen, China) quadrotor UAV equipped with a 20-megapixel camera to conduct aerial surveys over mountain river channels. The vertical takeoff and landing capability of the quadrotor UAV provided operational flexibility in terrains with significant elevation changes. Prior to each flight, several ground control points (GCPs) were strategically deployed throughout the survey area. The coordinates of these GCPs were obtained using real-time kinematic (RTK) Global Navigation Satellite System (GNSS) receiver (SatLab Geosolutions AB, Askim, Sweden) to ensure georeferencing accuracy.

The UAV was flown along pre-planned flight paths to ensure full coverage of the study area. High image overlaps were adopted to enhance image quality and reconstruction reliability [30], with front and side overlaps set to 85% and 75%, respectively. The flights were conducted at 20 m above ground level, providing a relatively high spatial resolution. Across the seven UAV campaigns, the surveyed areas ranged from 0.77 ha to 7.75 ha. Each surveyed area was covered by one to five individual flight missions, depending on its size and terrain complexity. Each single flight mission typically covered approximately 0.77–1.55 ha and lasted about 10–18 min. To minimize the influence of lighting and shadow variations, UAV surveys were conducted between 10:00 and 14:00 under stable daylight conditions. The flight speed was maintained at approximately 10 m/s. Aerial images and GCP data were subsequently processed using Pix4Dmapper (version 4.4.12) software, applying the Structure from Motion (SfM) algorithm to generate dense 3D point clouds of the study reaches. Detailed survey information was summarized in Table 2, including survey dates, reaches, surveyed area, point cloud density, ground sampling distances (GSD), and georeferencing errors. Point cloud densities across the seven UAV campaigns ranged from approximately 2803 to 4948 points/m². GSD ranged from 4.7 mm/px to 7.2 mm/px. The root-mean-square georeferencing errors for the SfM outputs ranged between 0.8 cm and 3.2 cm, comparable to those reported in previous studies, such as ±5.3 cm in [24] and 1.0–5.0 cm in [18].

2.2.2. Wolman Pebble Counts Sampling Method

This study employed the Wolman pebble counts sampling method [11] to investigate riverbed grain size distribution. The sampling area was set in 10 m × 10 m (100 m² in area) square grids, designated as sampling patches. A rope, tapped with markings at 1 m intervals, was stationed, and the b-axis of grains was measured along the rope at these specified intervals [31]. For each patch, a total of 121 grains (11 samples per row) were sampled and measured for size, following the non-repetitive sampling principle [12]. The sampling templates used in this study had different opening sizes, including 8 mm, 16 mm, 32 mm, 45.3 mm, 64 mm, 90.5 mm, 128 mm, 181 mm, 256 mm, and 512 mm. The grains having sizes larger than 512 mm within the sampling area were measured using a caliper. In subsequent analyses, grains with a b-axis smaller than 8 mm will be treated as 4 mm in the analysis of grain size distribution.

Following the recommendation of [24], 8 to 10 Wolman samples would be sufficient to obtain reliable reach-scale grain size-roughness relations. Accordingly, we conducted eight field samples in Reach R1 to develop the relations. To evaluate the broader applicability of the approach, we further carried out two field samplings at each of three other reaches (Reach R2, R3, and R4). Therefore, a total of 14 manual grain size samplings were performed throughout the study area (Figure 2). The D₅₀ of the 8 sampling sites in the Reach R1 ranged from approximately 33.8 to 175.8 mm, with an average size of 102.8 mm. The coefficient of uniformity $C_u$, and sorting coefficient $S_c$ for the grain size distribution, as defined in Equations (1) and (2), varied from 5.2 to 43.6, and 1.8 to 6.1, respectively, for the eight sampling sites in the Reach R1. The D₅₀, $C_u$ and $S_c$ of the 6 sampling sites distributed in Reach R2, R3, and R4 ranged from 45.8 to 97.0 mm, from 6.3 to 63.2, and from 1.9 to 5.0, respectively. The proportion of boulders (>256 mm) at each sampling site varied between 7.0% and 35%. These results indicate that the riverbed exhibited a wide range of grain size distribution in the studied reaches. Detailed results of the manually sampled local grain sizes are presented in Appendix A,

Table A1. The results of local grain sizes by manual sampling at 14 sites in this study.

Site	D10	D16	D25	D30	D50	D60	D75	D84	D90	Dmax	${\mathit{C}}_{\mathit{u}}$	${\mathit{S}}_{\mathit{c}}$	${\mathit{\sigma }}_{\mathit{g}}$
H01	3.3	14.7	31.5	40.9	84.3	104.4	167.6	205.5	303.3	1050.0	31.6	2.3	3.7
H02	1.4	2.3	4.5	5.8	33.8	62.4	167.8	231.0	265.8	1350.0	43.6	6.1	10.0
H03	2.9	9.3	22.7	30.1	78.9	104.5	193.0	246.1	356.9	1470.0	36.6	2.9	5.1
H04	4.0	31.3	73.9	97.5	175.8	206.3	315.5	525.3	1134.6	2150.0	51.6	2.5	4.4
H05	3.6	7.6	19.0	25.3	74.5	97.5	193.8	251.6	327.3	860.0	26.8	3.2	5.7
H06	39.0	71.0	94.7	107.8	174.6	203.3	310.4	374.6	471.2	850.0	5.2	1.8	2.3
H07	4.0	5.8	27.4	39.5	120.6	161.0	298.0	380.3	825.0	1750.0	40.3	3.3	8.1
H08	4.0	6.3	9.8	11.7	79.5	124.3	247.1	320.8	467.2	800.0	31.1	5.0	7.1
H09	6.9	11.3	17.3	20.7	45.8	76.7	144.0	184.4	211.0	930.0	11.1	2.9	4.0
H10	16.7	28.3	46.3	55.1	85.6	104.1	164.9	205.9	284.0	520.0	6.3	1.9	2.7
N01	13.7	24.6	43.2	55.4	97.0	112.2	157.6	217.0	265.7	960.0	8.2	1.9	3.0
N02	2.3	3.6	12.8	18.7	49.5	68.9	166.5	227.8	265.8	1030.0	30.4	3.6	7.9
L01	2.0	3.2	14.5	25.7	63.7	108.0	170.5	273.4	312.0	950.0	53.5	3.4	9.2
L02	1.7	2.8	8.5	17.7	93.4	109.3	212.3	307.1	343.5	1000.0	63.2	5.0	10.5

The unit of local grain sizes: mm; The symbols: $C_u={\mathrm{D}}_{60}/{\mathrm{D}}_{10}$, $S_c=\sqrt{{\mathrm{D}}_{75}/{\mathrm{D}}_{25}}$, ${\sigma }_g=\sqrt{{\mathrm{D}}_{84}/{\mathrm{D}}_{16}}.$

.

$$C_u={\displaystyle \frac{{\mathrm{D}}_{60}}{{\mathrm{D}}_{10}}}$$

(1)

$$S_c=\sqrt{{\displaystyle \frac{{\mathrm{D}}_{75}}{{\mathrm{D}}_{25}}}}$$

(2)

2.3. Roughness Metric

2.3.1. Concept of Roughness Height

Roughness metric describes the degree of surface undulation represented in point cloud data [20]. Various roughness metrics have been adopted for grain size estimation, including roughness height (RH), standard deviation (σ), and detrended standard deviation (σ_d) (Table 1). Among them, RH has been widely applied in recent years [18,24,25] and was shown by [24] to yield stronger correlations with grain size than other metrics. Therefore, RH was selected for analysis in this study. Unlike the conventional hydraulic roughness height (i.e., the equivalent sand roughness k_s) used in flow resistance and sediment transport formulations [3,4,5], where roughness height is typically defined as a representative grain size (such as αD₈₄, α = 1.0−3.5), RH here is a statistical measure of local surface variability derived from UAV-SfM point cloud data. It is calculated as the distance between this point and the best-fitting plane computed on its nearest neighbors within a specific kernel radius [32]. As such, RH provides a remote sensing–based proxy for bed-surface texture that can be related to characteristic grain sizes, as shown in Table 1.

We utilized CloudCompare (version 2.14), an open-source point cloud processing software, to compute RH. Prior to roughness computation, areas affected by vegetation or other non-bed features were manually filtered out from the point-cloud data to prevent local distortion. The roughness value for each point was calculated using a fixed kernel radius (described in Section 2.3.2), which defines the neighborhood size for the local plane fitting. This process yields a spatially distributed roughness field across the study area ([32] & Figure 3).

We followed the procedure outlined by [24] to extract representative roughness values to enable us to make relations between surface roughness and grain size sampling data (as in previous studies, summarized in Table 1). Specifically, the median roughness value within each 1 m $\times$ 1 m area, which corresponds to the manual sampling interval, was computed and used as the representative roughness for that cell. The representative roughness values within the area corresponding to the sampling site were extracted and sorted in ascending order. From these values, the 16th, 25th, 50th, 75th, and 84th percentiles were determined and designated as the RH₁₆, RH₂₅, RH₅₀, RH₇₅, and RH₈₄, respectively. These percentile roughness values were then paired with the corresponding local grain sizes (D₁₆, D₂₅, D₅₀, D₇₅, and D₈₄) to perform correlation analyses for riverbeds containing larger boulders and exhibiting broad grain size distribution (described in Section 2.3.3).

2.3.2. Grid Size for Computing Roughness Metrics

The selection of grid size (or kernel radius in RH) is a critical factor in computing roughness metrics, as it defines the neighborhood of point-cloud data used for calculation [32]. A larger kernel radius produces a smoother fitting plane and thus larger RH values. This effect is illustrated in Figure 4, where three RH distributions were evaluated by three different kernel radii for the same sampling site. Such variations in RH with kernel size can, in turn, influence the correlation between grain size and roughness [17].

To determine a suitable kernel radius, we tested eight kernel radii (0.03125, 0.0625, 0.1, 0.125, 0.25, 0.5, 0.75, and 1 m) in Reach R1. For each radius, RH was computed, and the representative RH₅₀ at the eight sampling sites (extracted following the procedure in Section 2.3.1) was regressed against the corresponding D₅₀. The coefficient of determination (R²) of the linear D₅₀-RH₅₀ relation in different kernel radii was evaluated. A suitable kernel radius was chosen for subsequent investigations by identifying a value that produced a high R² (>0.8), without necessarily being the highest R².

Since grain-size characteristics varied across study sites in different studies, we also introduced a dimensionless grid size (S) to facilitate cross-study comparisons. S is defined as the grid size divided by the average D₅₀ of the datasets, as illustrated in Equation (3).

$$S={\displaystyle \frac{\mathrm{g}\mathrm{r}\mathrm{i}\mathrm{d} \mathrm{s}\mathrm{i}\mathrm{z}\mathrm{e}}{{\mathrm{D}}_{50}}}={\displaystyle \frac{2\times \mathrm{k}\mathrm{e}\mathrm{r}\mathrm{n}\mathrm{e}\mathrm{l} \mathrm{r}\mathrm{a}\mathrm{d}\mathrm{i}\mathrm{u}\mathrm{s}}{{\mathrm{D}}_{50}}}$$

(3)

2.3.3. Correlation Analysis of Grain Size-Roughness Relationship

The reach-scale correlations between grain size and roughness were established based on eight sampling patches in the R1 reach, following the recommendation of [24]. Due to the time-consuming nature of fieldwork, data collection was carried out at different times, but always under sunny conditions to secure reliable grain-size sampling and high-quality UAV imagery. As long as data quality is maintained, temporal differences in sampling generally have little effect on grain-size and roughness correlations, as noted by [24]. Therefore, the datasets obtained at different times were jointly used to establish the grain size–roughness relations.

The linear regression (Equation (4)) analyses were first conducted between the grain size (D_i, where i = 16, 25, 50, 75, and 84, respectively) and the corresponding percentile roughness (RH_i) in Reach R1.

D_i=a_i × RH_i + b_i (D_i and RH_i in mm)

(4)

Subsequently, all paired D_i-RH_i data in Reach R1 were pooled together for integrated analysis. A power-law regression was then applied using the MATLAB Regression Learner App (R2022b) to establish an integrated relation between grain size and roughness.

To evaluate estimation accuracy, we first examined the integrated grain size–roughness relations using the original eight sampling sites in Reach R1. Its applicability was further tested with six additional sites in Reach R2–R4 and compared with the results estimated by individual linear D_i-RH_i relations. The relative errors (RE) of local grain sizes (${\mathrm{D}}_{\mathrm{i}}$) were calculated following Equation (5):

$$\mathrm{R}\mathrm{E}({\mathrm{D}}_{\mathrm{i}})=\left|{\displaystyle \frac{{\mathrm{D}}_{\mathrm{i}}^{\mathrm{*}}-{\mathrm{D}}_{\mathrm{i}}}{{\mathrm{D}}_{\mathrm{i}}}}\right|$$

(5)

where ${\mathrm{D}}_{\mathrm{i}}$ (mm) denotes the grain sizes obtained from manual samplings, ${\mathrm{D}}_{\mathrm{i}}^{\mathrm{*}}$ (mm): denotes the grain sizes estimated from the integrated D-RH relation or individual linear D_i-RH_i relations based on roughness data. Figure 5 depicts the research workflow and the software utilized at each stage.

3. Results

3.1. Grid Size for Computing Roughness Metrics

The grid size (or kernel radius) used for computing the roughness metrics would affect the resulting roughness values and subsequently influence the correlation between grain size and roughness. In this study, eight dimensionless grid sizes ($S$, ranged from 0.6 to 19.5) were used to compute corresponding roughness heights in Reach R1, and the R² of D₅₀-RH₅₀ relation in various kernel radii were presented in Figure 6. When $S$ was less than 2.0, the correlation between D₅₀ and RH₅₀ was weak, with R² less than 0.7. As the $S$ increased, the D₅₀-RH₅₀ correlation improved steadily, and their R² exceeded 0.88 for S ranging from 2.4 to 14.6, reaching a peak value of 0.945 at S = 14.6. The R² dropped to 0.81 when $S$ = 19.5. Although the highest R² of the D₅₀-RH₅₀ relation occurred at a kernel radius of 0.75 m ($S$ = 14.6), we selected a kernel radius of 0.5 m ($S$ = 9.5) for subsequent roughness computations because it achieved a similarly high R² (0.936, a difference of only 0.009 with R² at kernel radius of 0.75 m) while matching the 1.0 m interval used in our manual field sampling and is also consistent with the kernel radius used by [24].

We used a consistent kernel radius (0.5 m) for roughness analysis across Reach R2–R4 to avoid introducing additional scale-dependent effects associated with varying kernel radii. The adopted 0.5 m radius corresponds to dimensionless grid sizes of 15.2, 13.7, and 12.7 for these reaches—within the S range associated with stronger correlations in Reach R1 (R² > 0.8 for S = 2.4−19.5).

3.2. Linear Correlation Between Grain Size and Roughness Height

The reach-scale linear D_i-RH_i (where i = 16, 25, 50, 75, and 84, respectively) correlations at 8 sites in Reach R1 were displayed in Figure 7a–e. The coefficients of determination (R²), slope ($a_{\mathrm{i}}$), and intercept ($b_{\mathrm{i}}$) of the D_i-RH_i relations were summarized in

Table 3. The linear D_i-RH_i relations and their coefficients of determination (R²), slope ($a_i$), and intercept ($b_{\mathrm{i}}$).

Di-RHi	R2	${\mathit{a}}_{\mathbf{i}}$	${\mathit{b}}_{\mathbf{i}}$
D16-RH16	0.79	2.32	−46.67
D25-RH25	0.92	3.41	−73.90
D50-RH50	0.94	5.71	−142.58
D75-RH75	0.70	9.45	−265.02
D84-RH84	0.60	15.99	−605.11

. Detailed results of the percentile roughness are provided in Appendix A (

Table A2. The results of the percentile roughness at 14 sites in this study.

Site	RH10	RH16	RH25	RH30	RH50	RH60	RH75	RH84	RH90	RHmax
H01	22.5	25.2	27.5	29.3	37.9	40.9	45.5	49.6	53.6	71.2
H02	14.5	16.6	20.4	23.5	31.0	37.9	49.6	58.4	61.8	70.3
H03	22.0	24.4	28.6	32.4	42.0	47.3	49.9	54.7	61.5	71.7
H04	36.1	38.6	42.7	46.0	54.0	58.9	60.9	65.0	70.6	82.1
H05	24.0	26.0	31.0	31.9	41.1	47.7	53.7	57.3	64.8	82.8
H06	41.1	43.5	47.3	49.2	55.5	57.7	61.2	64.5	68.9	84.4
H07	26.4	28.5	33.4	37.7	46.1	49.0	54.6	57.8	63.7	88.6
H08	20.2	21.8	25.6	28.8	36.0	42.8	49.0	54.2	66.3	88.6
H09	18.9	21.7	25.7	27.4	32.9	39.6	47.6	51.3	54.8	70.2
H10	28.4	31.3	33.5	35.4	42.0	45.1	50.4	56.9	62.1	77.8
N01	24.9	30.2	33.0	36.2	43.5	46.2	52.5	56.6	60.5	71.0
N02	20.4	22.7	27.5	29.0	35.2	40.7	46.7	51.0	57.7	76.9
L01	21.8	24.6	27.2	28.2	34.3	38.0	47.4	50.6	58.8	78.5
L02	25.5	26.6	30.4	32.5	39.8	42.3	44.8	50.5	57.0	84.6

The unit of RH: mm.

).

The results displayed moderate to strong correlations (R² = 0.60–0.94, Table 3) between grain size and roughness in variations in sizes in Reach R1. The D₅₀-RH₅₀ relation (Figure 7c) exhibited the highest correlation (R² = 0.94), with D₅₀ ranging from 33.8 mm to 175.8 mm and RH₅₀ ranging from 31.0 mm to 55.5mm. The correlation weakened as the grain size deviated from D₅₀, with R² equal to 0.79, 0.92, 0.70, and 0.60 in D₁₆-RH₁₆, D₂₅-RH₂₅, D₇₅-RH₇₅, and D₈₄-RH₈₄ relations, respectively.

Figure 8 illustrates the variation in the regression slope ($a_{\mathrm{i}}$) and intercept ($b_{\mathrm{i}}$) with respect to different D_i-RH_i relations. As the grain size increases from D₁₆ to D₈₄, the value of $a_{\mathrm{i}}$ increased consistently, exhibiting a near-exponential trend: 2.32 (D₁₆-RH₁₆), 3.41 (D₂₅-RH₂₅), 5.71 (D₅₀-RH₅₀), 9.45 (D₇₅-RH₇₅), and 15.99 (D₈₄-RH₈₄), while the value of $b_{\mathrm{i}}$ became increasingly negative: −46.67 (D₁₆-RH₁₆), −73.9 (D₂₅-RH₂₅), −142.58 (D₅₀-RH₅₀), −265.02 (D₇₅-RH₇₅), and −605.11 (D₈₄-RH₈₄). The result suggests that for coarser grains, the fitted linear regression has a steeper slope and a larger (more negative) intercept.

3.3. Integrated D_i-RH_i Relations by a Power Law

According to the findings presented in Section 3.2, the relations between grain size and corresponding roughness vary with grain size. Applying a single linear regression to a riverbed with highly heterogeneous and wide-ranging grain sizes would result in limited accuracy, as it can only reliably predict grain sizes within a specific range, rather than across the full distribution. Moreover, the analysis revealed that the slope ($a_i$) of the grain size–roughness relations tend to increase with grain size, exhibiting a near-exponential trend (Figure 8). To address this variability, we compiled all paired D_i-RH_i data (where i = 16, 25, 50, 75, and 84, respectively) in Reach R1 for an integrated analysis. A power-law regression was then proposed to establish an integrated relation between grain size and surface roughness, as shown in Equation (6) and Figure 9 (${\mathrm{D}}_i$ and ${\mathrm{R}\mathrm{H}}_i$ in mm), with a coefficient of determination (R²) of 0.89.

D = 0.00003 × RH^3.97 (D and RH in mm)

(6)

The integrated relation represents the average tendency across the grain size–roughness correlation in Reach R1. It provides a single, continuous relation capable of estimating grain size distributions ranging over a broad range, from 8 mm—the smallest sampling sieve size—to approximately 500 mm, provided that RH is known. According to Equation (6), a greater surface roughness is typically associated with a larger grain size. Furthermore, differentiating Equation (6) with respect to D yields Equation (7), which illustrates the sensitivity of roughness to changes in grain size.

$${\displaystyle \frac{d\mathrm{R}\mathrm{H}}{d\mathrm{D}}}={\displaystyle \frac{\mathrm{R}\mathrm{H}}{3.97\times \mathrm{D}}}$$

(7)

Equation (7) shows that a unit change in D for smaller grain sizes leads to a relatively larger change in RH, indicating that the variation in surface roughness is more sensitive to fine grains. For instance, ${\displaystyle \frac{d\mathrm{R}\mathrm{H}}{d\mathrm{D}}}$ is approximately 0.186 for D = 50 mm (RH ≈ 36.9 mm) but only about 0.033 for D = 500 mm (RH ≈ 66.0 mm), a difference of about 5.6 times. This indicates that sensitivity decreases progressively with coarser grains.

To further examine the scaling behavior embedded in the integrated relation, Equation (6) was differentiated with respect to RH, yielding Equation (8):

$${\displaystyle \frac{d\mathrm{D}}{d\mathrm{R}\mathrm{H}}}=0.0001191\times {\mathrm{R}\mathrm{H}}^{2.97}$$

(8)

Using the mean RH₁₆, RH₂₅, RH₅₀, RH₇₅, and RH₈₄ from the eight sites in Reach R1, the resulting slopes from Equation (8) are 2.38, 3.54, 8.43, 15.78, and 20.25, respectively. These values show consistent scaling trends observed in individual linear D_i-RH_i relations (Table 3), indicating that the integrated power-law model effectively captures the scaling behavior exhibited across different grain-size percentiles.

3.4. Examination of the Integrated Grain Size-Roughness Relation

We examined the integrated grain size–roughness relation using the data of the original eight sampling sites in Reach R1, where the relations were established. Relative errors (RE, Equation (5)) were computed by comparing grain sizes from manual sampling with those estimated from the relation, and the results were summarized in

Table 4. The relative errors of local grain sizes estimated from the integrated grain size-roughness relation with those from manual samplings at 8 sites in R1 reach (unit: %).

	Sites	Reach R1 (%)								Mean (%)
Di		H01	H02	H03	H04	H05	H06	H07	H08
D16		21.2	15.0	20.8	115.0	85.5	36.0	362.0	79.6	91.9
D25		50.0	7.6	4.2	26.1	32.8	39.1	52.4	72.1	35.6
D50		32.9	26.4	11.8	30.2	13.6	45.0	2.1	27.9	23.7
D75		31.4	0.1	0.6	20.2	27.8	22.9	16.2	4.2	15.4
D84		20.6	32.3	8.0	6.5	30.7	25.9	17.5	12.7	19.3
Mean		31.2	16.3	9.1	39.6	38.1	33.8	90.0	39.3	37.2

. For D₁₆, REs ranged from 15.0% (H02) to 362.0% (H07), with a mean relative error (MRE) of 91.9%. For D₂₅, REs ranged from 4.2% (H03) to 72.1% (H08), with an MRE of 35.6%. For D₅₀, REs ranged from 2.1% (H07) to 45.0% (H06), with an MRE of 23.7%. For D₇₅, REs ranged from 0.1% (H02) to 31.4% (H01), with an MRE of 15.4%. For D₈₄, REs ranged from 6.5% (H04) to 32.3% (H02), with an MRE of 19.3%. As the integrated relation reflects an average grain size–roughness correlation tendency in Reach R1, the RE of local grain sizes may be varied in individual sites. Overall, the results indicated that the integrated relation yielded better consistency for coarser grains (D₅₀–D₈₄) than for finer grains (D₁₆ and D₂₅).

3.5. Applicability of the Integrated Grain Size-Roughness Relation

We applied the integrated grain size-roughness relation to six additional sampling sites to examine its applicability. These included two sites in Reach R2, located about 2 km downstream of Reach R1 within the same watershed, and four sites in Reaches R3 and R4, situated in a different watershed. Their RH values were derived from their topographic data. The grain-size distributions estimated from the integrated relation were first compared with those obtained from manual sampling (Figure 10a–f). The comparisons showed that the estimated grain size distribution had better agreement at sites H09 and H10 in Reach R2 and at sites N01 and N02 in Reach R3, compared to that at sites L01 and L02 in Reach R4.

We further compared the relative errors of local grain sizes from manual samplings with those estimated from the integrated R-D relation (

Table 5. The relative errors of local grain sizes estimated from the integrated grain size-roughness relation with those from manual samplings at 6 sites in Reach R2–R4 (unit: %).

	Sites	Reach R2 (%)			Reach R3 (%)			Reach R4 (%)
Di		H09	H10	Mean	N01	N02	Mean	L01	L02	Mean
D16		46.3	8.4	27.3	8.2	100.9	54.5	210.6	390.7	300.7
D25		31.9	20.8	26.3	25.4	21.5	23.4	2.9	171.6	87.2
D50		31.1	2.3	16.7	1.0	16.3	8.7	41.3	27.6	34.4
D75		4.8	4.6	4.7	28.5	23.5	26.0	21.1	49.1	35.1
D84		0.3	35.3	17.8	26.1	20.7	23.4	36.2	43.3	39.7
Mean		22.9	14.3	18.6	17.8	36.6	27.2	62.4	136.4	99.4

) and individual linear D_i-RH_i relations (i = 16, 25, 50, 75, and 84; Table 3), with the results of the latter summarized in

Table 6. The relative errors of local grain sizes estimated from the individual linear grain size-roughness relations with those from manual samplings at 6 sites in Reach R2–R4 (unit: %).

	Sites	Reach R2 (%)			Reach R3 (%)			Reach R4 (%)
Di		H09	H10	Mean	N01	N02	Mean	L01	L02	Mean
D16		67.4	8.4	37.9	4.7	67.0	35.8	224.3	442.2	333.3
D25		21.5	13.1	17.3	10.1	55.3	32.7	30.5	249.6	140.0
D50		1.5	13.8	7.6	9.1	18.1	13.6	16.2	9.1	12.7
D75		22.0	31.4	26.7	56.7	2.1	29.4	1.1	36.2	18.6
D84		10.8	52.1	31.5	42.0	12.9	27.4	30.5	38.3	34.4
Mean		24.6	23.8	24.2	24.5	31.1	27.8	60.5	155.1	107.8

. In Reach R2, the MREs from the integrated relation ranged from 4.7% (D₇₅) to 27.3% (D₁₆), with an average of 18.6%. By comparison, the individual linear relations produced MREs ranging from 7.6% (D₅₀) to 37.9% (D₁₆), with an average of 24.2%. In Reach R3, the MREs from the integrated relation varied between 8.7% (D₅₀) to 54.5% (D₁₆), with an average of 27.2%, whereas those from the linear relations ranged from 13.6% (D₅₀) to 35.8% (D₁₆), with an average of 27.8%. Substantially larger errors were observed in Reach R4, where the integrated relation yielded MREs from 34.4% (D₅₀) to 300.7% (D₁₆), with an average of 99.4%, while the linear relations produced errors from 12.7% (D₅₀) to 333.3% (D₁₆), with an average of 107.8%. Overall, these results indicate that the integrated relation performs comparably to individual linear regressions across Reaches R2–R4, while offering the practical advantage of a single equation applicable to multiple grain-size percentiles.

Consistent with the examination results, the integrated relation demonstrated higher accuracy for coarser grains (D₅₀–D₈₄) than for finer grains, such as D₁₆, in the applicability tests. In Reaches R2 and R3, estimation errors were generally within acceptable ranges and comparable to those in the examination stage (Section 3.4). In Reach R4, estimations of D₅₀, D₇₅, and D₈₄ remained within acceptable error ranges, but those of D₁₆ and D₂₅ exhibited substantially larger discrepancies, with relative errors exceeding 60%. Notably, the linear relations also produced large errors for D₁₆ and D₂₅. The common features among sites L01 and L02 were their relatively high proportion of fine grains (<8 mm: 19.8% in L01, and 23.1% in L02) and high coefficients of uniformity ($C_u$ = 53.5 and 63.2). To further illustrate this effect, Figure 11 compares orthophotos of two representative sampling sites. Site H10, characterized by smaller estimated errors, featured a well-sorted surface dominated by relatively closely packed pebbles and cobbles, creating a uniform texture. Conversely, site L02, with the largest errors, displayed a poorly sorted surface where the bed is interspersed with distinct patches of sand and fine gravel.

4. Discussion

4.1. Grid Size Effect on Roughness Evaluation

The selection of grid size (or kernel radius) for computing roughness metrics directly influences the resulting roughness values and consequently affects the correlation with grain size [17,33]. As illustrated in Figure 6, Woodget et al. [17] tested S ranging from 6.3 to 11.3 (kernel radius from 0.1 m to 0.5 m) and reported the highest correlation (R² = 0.31 for D₈₄-RH₈₄) at $S$ ≈ 8.8. Wong [33] evaluated the S between 4.3 and 7.1 (kernel radius between 0.01 and 0.06 m) and found the highest correlation (R² = 0.49 for D₈₄-RH₈₄) at $S$ ≈ 5.7. In the present study, S ranged from 0.6 to 19.5 (kernel radius between 0.03125 and 1.0 m), and the highest correlation (R² = 0.945 for D₅₀-RH₅₀) was observed at S = 14.6. Although the optimal S varies among studies, the variation of R² with changes in S revealed a consistent pattern in Figure 6: correlation strength generally increased with S, peaked at intermediate specific values, and then gradually declined as S became larger. Notably, our results showed relatively high correlations (R² > 0.88 for D₅₀-RH₅₀) across a broader interval of S = 2.4−14.6 compared to previous studies [17,33]. It suggested some degree of flexibility in kernel radius used in the riverbed with coarse grains and a broad grain size distribution (D₁₆–D₈₄: 2.3–525 mm in our study).

Earlier studies commonly selected grid sizes larger than the dominant clast dimension to ensure that the roughness calculation size fully encompassed individual grains [24,25]. For example, Vázquez-Tarrío et al. [24] used a 0.5 m kernel radius (S ≈ 25.6), roughly two to three times the maximum grain size in their study reach. In our case, some boulders exceed the adopted 0.5 m radius, meaning that their full geometric influence may not be captured. Although using a larger radius would better encompass such clasts, our results show that increasing the radius reduces correlation strength (e.g., R² = 0.81 at S = 19.5 for a 1.0 m radius). This highlights an inherent trade-off in coarse, heterogeneous gravel beds: the radius must be large enough to capture meaningful surface variability, yet not so large that it over-smooths topography and weakens the D-RH relation.

4.2. Linear Correlation Between Grain Size and Roughness Height

The linear D₅₀-RH₅₀ relation derived from mountainous rivers in our study was compared with those reported in previous studies using the RH metric (Table 1; Figure 12). The ranges of the correlations were plotted based on the extent of data provided in each study. The slope ($a_i$) of these relations ranges from 0.89 to 5.71, and the intercept ($b_i$) ranges from −142.58 to 22.0 (unit of the relation in mm). Such considerable variability in coefficients likely reflects differences in the grain-size characteristics and the kernel radius used in roughness calculations across studies (Table 1). For instance, although both [24] and the present study adopted the same kernel radius of 0.5 m, their D₅₀ ranges differed (28–65 mm in [24] vs. 34–176 mm in this study), resulting in an $a_i$ value in this study that is approximately 6.4 times larger. Regarding kernel radius, Wong et al. [18] used 0.04 m, Pearson et al. [23] applied 0.2 m, while [24] and this study adopted 0.5 m. As shown in Figure 12, larger kernel radii consistently yielded higher roughness values, underscoring the strong influence of scale on roughness metrics.

These comparisons point to an important implication: no single, universal linear relation exists between grain size and surface roughness, as also noted by [23,24]. Differences in dataset grain-size ranges and the grid size used in roughness calculations appear to exert a strong influence, making direct cross-study comparisons and applications challenging. In this context, the integrated relation proposed in this study offers a practical alternative. Rather than being tied to a narrow dataset or a specific D_i-RH_i relation, it provides a single, continuous D-RH correlation applicable across a broader grain-size variability (≈8–500 mm).

4.3. Integrated Grain Size—Roughness Relation

4.3.1. Examination and Applicability of the Integrated Relation

This study established a reach-scale integrated power-law grain size-roughness relation derived from multiple sets of grain sizes and their corresponding percentile roughness in Reach R1. This relation captures the average tendency of grain size-roughness correlations within Reach R1. It provides a single, continuous relation capable of efficiently estimating grain sizes between 8 and 500 mm in riverbeds characterized by a broad grain size distribution when applied to UAV-SfM–derived roughness metrics. Both the examination and applicability results consistently showed that the integrated relation achieved better accuracy for coarser grains (D₅₀–D₈₄) than for finer grains such as D₁₆.

Both individual linear and integrated relations yielded relatively large MREs for D₁₆. The reduced performance for finer grains may be explained by the higher sensitivity of small roughness values to topographic resolution and georeferencing errors, which amplify estimation uncertainty. This pattern is consistent with [24], who also reported weaker correlations for finer grains. The relatively large errors in D₁₆ at sites H07, N02, L01, and L02 may have been further amplified by their relatively high proportions (15.0–23.1%) of fine grains (<8 mm). These observations suggest that the integrated relation may have limited applicability at sites with substantial fine-grain content, reinforcing that the use of surface roughness as a proxy for finer grains remains constrained and requires further investigation.

Performance also varied across different reaches. The integrated relation performed relatively well at Reach R2, located 2 km downstream of Reach R1, and at Reach R3 in another watershed, while higher biases were observed at Reach R4. Geomorphic settings appear to influence these outcomes, partly through their effect on bed material composition (Figure 11). For example, reach-scale slope is often linked to sediment composition [34]. Reaches R2 and R3, with slopes of 0.053 and 0.054, were more comparable to Reach R1 (0.075) and showed better agreement, whereas the gentler slope of Reach R4 (0.033) coincided with poorer performance. Although these interpretations remain tentative, given the limited number of sites, the observed trend suggests that geomorphic similarity, such as slope, may help indicate where the integrated relation is more applicable.

4.3.2. Cross-Reach Validation of the Integrated Relation

The integrated grain-size–roughness relation was originally developed using the eight sampling sites in Reach R1. To evaluate the cross-reach robustness under the limited number of sites available in R2–R4, we performed an approach using all six sites from Reaches R2–R4 with two randomly selected sites from Reach R1 to fit a cross-reach integrated relation, which was then validated using the remaining six sites in Reach R1 (Figure 13). The optimal result is expressed in Equation (9), derived using sites H02 and H04 from Reach R1 together with the six sites from Reach R2–R4:

D = 0.00005 × RH^3.85 (D and RH in mm)

(9)

The R² of the cross-reach relation is approximately 0.86 for the fitting data, and the RMSE (root mean square error) for the validation data is 42.9 mm. For reference, applying the original R1-based integrated relation to six sites in Reach R2–R4 produced an RMSE of 44.1 mm. These results suggest that data from multiple reaches can jointly support a power-law relation, and that its performance remains within a reasonably bounded error range even under cross-reach variability. Future work incorporating more sampling sites across diverse river reaches will help better constrain the error bounds of the integrated relation and clarify the extent to which its cross-reach applicability holds.

4.3.3. Limitations and Future Directions

Although the integrated grain size–roughness relation was developed in riverbeds characterized by coarse grains and a broad grain-size distribution and showed applicability in Reach R2 and R3, further validation will help the applicability to other river systems since the gravel-bed rivers are highly heterogeneous in grain-size composition and such variability can influence grain size–roughness relations [23]. Additionally, the integrated relation was constructed using a fixed kernel radius of 0.5 m, evaluating how integrated relations constructed under different kernel radii would provide further insight into the sensitivity of the method.

Currently, the roughness metrics used in this study were derived from high-density UAV-based point clouds (2803–4948 points/m²) collected from flights at approximately 20 m altitude. The influence of UAV flight altitude, and the associated changes in point-cloud density, on the stability of the derived relations was not systematically examined and should be further investigated. Future advances in UAV endurance and higher-resolution imaging may enhance operational scalability and point-cloud quality, thereby improving the efficiency and applicability of roughness-based grain-size estimation.

5. Conclusions

This study investigated the reach-scale relations between grain size from manual samplings and surface roughness derived from high-resolution UAV-SfM point clouds in mountainous river reaches characterized by coarse grains and a broad grain size distribution. Moderate to strong correlations (R² = 0.60−0.94) were observed for linear D_i-RH_i (i = 16, 25, 50, 75, and 84, respectively) relations, and the regression slope increases exponentially with grain size. To address cross-percentile variability, an integrated power-law grain size and roughness relation (R² = 0.89) was then developed using all paired D_i-RH_i data on Reach R1. This relation provides a single, continuous relation capable of efficiently estimating grain sizes between 8 and 500 mm in riverbeds characterized by a broad grain size distribution when applied to UAV-SfM–derived roughness metrics. It captured the average tendency of the D-RH correlations in Reach R1, with mean relative errors equal to 91.9%, 35.6%, 23.7%, 15.4%, and 19.3% for D₁₆, D₂₅, D₅₀, D₇₅, and D₈₄, respectively. The applicability tests at six additional sites in Reach R2–R4 showed that the integrated relation produced more accurate estimates for coarser grains (D₅₀–D₈₄) than for finer grains such as D₁₆.

The results demonstrate that surface roughness computed from UAV-SfM point clouds can serve as a proxy for estimating grain sizes in exposed riverbeds. The integrated relation developed here offers a practical empirical tool for characterizing surface grain-size distributions over a broad variability of sizes in gravel-bed mountain rivers. However, its performance varied among different reaches, with weaker accuracy observed in lower-slope reaches (Reach R4) that contained relatively large proportions of fine grains (<8 mm; 19.8–23.1%). Future work incorporating more sampling sites across diverse river types will help further refine the integrated relation and improve its cross-reach applicability.