In the sediment-flux-dependent model , the incision rate E [m s⁻¹] is obtained by: 2E=ViIr(1-Pc) in which Vi is the average volume of bedrock detached per particle impact, Ir is the rate of particle impacts per unit area per unit time, and Pc is the fraction of the covered riverbed.

Assuming the bedrock to be an elastic brittle material, Vi can be rewritten by the classic impact wear model of as: 3Vi=1/2Mp(Uisin⁡α)2εv where Mp [kg] denotes particle mass. Ui [m s⁻¹] indicates the impact velocity of the particle, and α is the saltation angle. The parameter εv [J] denotes the total energy required to erode a unit volume of rock . The threshold energy εv is calculated by : 4εv=kvσt22Y where σt [MPa] is bedrock tensile strength, and kv denotes rock resistance coefficient. Y [MPa] represents Young's modulus. By substituting Eq. () into Eq. (), we obtain: 5Vi=πρsDs3wsi2Y6kvσt2 where ρs [kg m⁻³] and Ds [m] denote the density of sediment and the diameter of a spherical sediment grain, respectively. Mp=ρsπDs3/6 is the mass of a spherical grain, and wsi [m s⁻¹] is the vertical component of the particle velocity on impact (i.e., wsi=Uisin⁡α).

Here, the number of particle impacts per unit time and area Ir is proportional to the flux of the bedload particles and inversely proportional to the downstream distance between the impacts. Using the sediment flux volume per unit width qs [m² s⁻¹] and the saltation hop length Ls [m], Ir is expressed as: 6Ir=ρsqsMpLs. Substituting Mp=ρsπDs3/6 into Eq. (), 7Ir=6πDs3qs1Ls.

Substituting Eqs. () and  () into Eq. (), bedrock incision rate E is recast as: 8E=ρswsi2YLskvσt2qs(1-Pc).

proposed that the abrasion coefficient can be defined as: 9β=ρswsi2YLskvσt2. Using this formulation, Eq. () can be rewritten as: 10E=βqs(1-Pc).

Based on the flume experiments in the field scale, pointed out that the erosion rate is proportional to the square root of the grain size rather than the shear stress and that it does not significantly depend on the Young's modulus. From this experimental result, proposed the following relation: 11E=β0σt2Dsks0.5qs(1-Pc) where β0 is an empirical coefficient (=0.0001) [kg² m⁻³ s⁻⁴], ks [m] represents the height of the hydraulic roughness, which is defined by the equation ks=κaPc + κb(1-Pc). Here, κa is a roughness coefficient that is linear to the grain diameter Ds, and κb is a constant representing the bedrock roughness. In , κa=2.5Ds and κb=0.0032. We adopted these relations and values in the model calculation.

The bedrock covered ratio Pc can be expressed in various ways; however, in this study, it is defined as: 12Pc=qsqt following the work of . The sediment transport capacity qt [m² s⁻¹] takes the form : 13qt=5.7RbgDs31/2τ∗-τc∗32 where Rb denotes the nondimensional buoyant density of the sediment (Rb=(ρs-ρw)/ρw). The parameters ρw [kg m⁻³] and g [m s⁻²] denote the water density and gravity acceleration, respectively. The Shields stress τ∗, which is the nondimensional bed shear stress, is defined as: 14τ∗=τbρs-ρwgDs where τb is the bed shear stress. The critical Shields number τc∗ is the value of τ∗ at the threshold of particle motion, which was regarded as constant (0.03) for simplicity.

Assuming that the stream flows in a uniform steady condition, the bed shear stress τb is calculated as : 15τb=ρwCf1/3g2/3QwW2/3S2/3 where Cf denotes the bed friction coefficient. Qw [m³ s⁻¹] and W [m] represent the water discharge and the width of the river, respectively. S indicates the bed slope.

Water discharge Qw is determined by the following equation : 16Qw=kSFDMP‾A where kSFDM is a discharge coefficient representing rainfall variability, and P‾ [m s⁻¹] denotes the average precipitation per unit area. A [m²] represents the drainage area. Because the amounts of precipitation causing bedrock erosion are expected to be significantly larger than the average condition, the discharge coefficient kSFDM is also expected to be considerably greater than unity.

Assuming that the bedload sediment supply from the tributaries of the stream is proportional to the volume of the eroded material in the drainage area , the bedload discharge per unit width qs domain is obtained as: 17qs(x)=1W(x)qs(0)W(0)+a∫dAdxE(x)dx where x [m] is the streamwise distance from the upstream end of the calculation domain, and qs(0) is the bedload sediment supply at the upstream end. The ratio of the bedload to the total sediment supply is represented by a. In this study, we set a=0.14 based on a previous study of .

The sediment supply per unit width at the upstream end is assumed to be in a steady state, where the erosion rate Eeq is equal to the uplift rate of the bedrock. The bedload sediment discharge qs(0) at the upstream end (x=0) is written as: 18qs(0)=aA(0)EeqW(0) where A(0) and W(0) denote the drainage area and the channel width at the upstream end, respectively.

The channel width W is estimated by the empirical formulation using the river discharge as follows: 19W=kwQw0.5 where kw is the uniquely determined coefficient for each tributary.

To calculate the steady-state (E=U) channel profiles, Eq. () was recast to solve for the channel slope S using Eqs. (), (), (),  (), and (), which takes the form: 20Seq,SFDM=Rbg1/3Dskw2/3Cf1/3Qw1/3{τc∗21+β0qs25.7(Rbg)1/2Dsβ0Ds1/2qs-ks1/2σt2U2/3}3/2.

We applied two types of the stream power model. One is the Area-based Stream Power Model (ASPM), which considers the lithologic strength of bedrock according to its lithologic types . This model does not account for the effect of the sediment cover ratio on the river bed. The other is the Stream Power with Alluvium Conservation and Entrainment Model (SPACEM) . SPACEM incorporates both lithologic strength and sediment cover ratio.

The ASPM is represented by the following equation: 22E=kaLESnAm where ka (m^1−2 m yr⁻¹) is the erosional efficiency parameter excluding the influence of the erodibility of bedrock, and LE is the relative erodibility of bedrock index. The positive exponents m and n are empirical parameters depending on lithology, rainfall variability, and sediment load . The erosional efficiency ka was determined by a Bayesian optimization in this study. LE was determined using the value proposed in .

On the other hand, SPACEM considers lithology and sediment cover ratio but does not explicitly include sediment tool effects. This model considers the rates of erosion Er [m yr⁻¹] and entrainment Es [m yr⁻¹], which relate to the sediment thickness on the riverbed. The following equation represents the erosion rate Er: 23Er=Krqw,yrSne-H/H∗ where Kr [m^1−2 m yr⁻¹] is the bedrock erodibility, qw,yr [m² yr⁻¹] is the mean annual water discharge per unit width, H [m] is the thickness of the sediment cover, and H∗ [m] is the bedrock roughness scale. Kr reflects the bedrock strength. The entrainment rate of the sediment from the bed Es is represented as: 24Es=Ksedqw,yrSn1-e-H/H∗ where Ksed [m^1−2 m yr⁻¹] denotes the sediment erodibility. In this model, the cover ratio Pc is determined as H/H∗. The sediment thickness H depends on the rate of sediment entrainment Es and deposition Ds, so that the relationship is calculated as: 25(1-ϕ)∂H∂t=Ds-Es=qsqw,yrWs-Es where ϕ and qs denote the sediment porosity and the sediment flux per unit width, respectively. Ws is the grain settling velocity determined by the grain size Ds. In this study, qs was calculated using Eq. (). The mean annual water discharge qw,yr was calculated using mean annual precipitation rate P‾yr as follows: 26qw,yr=kSPACEMP‾yrAW where kSPACEM denotes a discharge coefficient representing rainfall variability.

Assuming the steady-state (E=U), the channel slopes can be calculated in the same manner as SFDM: 27Seq,SPACEM=qsWsKsedqw,yr2+Uqw,yrKr1/n In this calculation, the grain settling velocity Ws was assumed to be spatially constant in this model.

The value of Kr for the most fragile rock type (i.e., tuff) was set to 1.0×10-5 according to . Assuming that this coefficient is proportional to the rock tensile strength, the bedrock erodibility Kr,i for the ith rock type was determined as follows: 28Kr,i=σt,iKr,tuffσt,tuff where σt,i and σt,tuff denote the tensile strengths of the ith rock type and tuff, respectively.

In this study, the discharge coefficient kSFDM, kSPACEM, channel width coefficient kw, and erosional coefficient (ka) were optimized to minimize the elevation differences between the actual river profile and the result of the model calculation. The objective function was defined as the root mean square (RMS) of the total elevation differences summed over the 5 tributaries. 29RMS=1NM∑n,m=1NMzn,mO-zn,mC zn,mO and zn,mC represent the observed and calculated elevation, respectively. The optimal parameters were determined using Optuna , which is an optimization framework based on Bayesian optimization, a method that efficiently explores the optimal solution by sequentially updating the posterior distribution based on the evaluation results of a probabilistic model.

This study performed the Bayesian optimization with 10 000 trials to fit the model outputs to the observed river longitudinal profiles. As a result, the optimal parameters were obtained: kSFDM and kw for SFDM, ka for ASPM, and kSPACEM for SPACEM. In the case of SFDM, kw was individually optimized for each tributary, while the remaining parameters were treated as common values across all tributaries. The search ranges for these parameters are summarized in Table . Note that the uncertainties associated with the optimized parameters were not explicitly evaluated in this analysis.

In addition, we performed a sensitivity analysis of the parameters used in SFDM, ASPM, and SPACEM that were adopted from the literature. Each parameter was varied within a reasonable range (Table ), and the model was re-optimized for each parameter set using Bayesian optimization with Optuna to minimize the RMS error between the observed and calculated topography of the Gohyaku River.

The topographic elevation, slope, and drainage area along the channels of the surveyed rivers were extracted from the digital elevation model (DEM) of the Geospatial Information Authority of Japan. First, 10-m-mesh DEM data of the surveyed tributaries were utilized to calculate the flow accumulation. The Deterministic 8 algorithm, in which water at each pixel is assumed to flow in the direction of maximum downslope gradient, was employed to obtain the drainage areas . Flow paths draining more than 7×104 km² were regarded as river channels. The channel slope was then calculated along the channel path from the elevation data. The geographical information system software SAGA GIS was used for these procedures .

The normalized steepness index ksn was used to clarify the effect of the bedrock strength on the channel steepness in all the investigated tributaries. This parameter is defined as the upstream area-weighted channel gradient : 30ksn=SAθ The exponent θ is the concavity index. Empirical studies show that concavity indices typically range from 0.3 to 0.7 . In this study, we adopted a value of 0.36, which was optimized for the study area.

Chi plots are commonly used to assess whether uplift rate and bedrock erodibility are spatially uniform . Under steady-state conditions with uniform erodibility, river profiles are linear in chi space; deviations from linearity indicate spatial variations in uplift rate or erodibility. It is a coordinate transformation to linearize the river profile determined by: 31χ=∫XbXA0A(x)m/ndX where X is the distance from downstream, and xb is a base level. A0 denotes a reference drainage area, and the drainage area at the downstream end is set in this study.

The lithologies of the bedrock were distinguished in the field survey and were recorded in the geological route map of the researched tributaries. The 1:50 000 and 1:200 000 geologic maps published by the Geological Survey of Japan were also used to distinguish the lithologies where the bedrock was not exposed on the channels.

Then, the tensile strengths of the various bedrock materials of the representative lithologies were sampled and measured using the Brazilian tension splitting test . Before the analysis, the specimens were submerged in the water under a vacuum for 12 weeks until their weight did not change. This procedure was selected because the goal was to measure the bedrock strength in a wet condition. We justify this approach because it has been shown that the water content in the pore space of bedrock can affect the rock tensile strength .

In this test, the specimens had a diameter of 50 mm and a length of 25 mm . The load was applied to the specimen at 1 µm s⁻¹ until a crack formed, and the maximum load Pmax [kN] at the moment of failure was measured.

The tensile strength σt was then determined using the following Eq. (). 32σt=2PmaxπDL×1000 where D (mm) and L denote the diameter and length of the specimen, respectively.

The gneissosity planes in gneiss can result in large variability depending on the surface on which the failure occurred. We assumed that the riverbed failure occurred along the weak plane, so we adopted not the average value, but the value when the specimen was cropped along the weak plane.

The grain sizes of the riverbeds were measured from drone-derived 3D point cloud data. In recent years, the methods for automated grain-size analysis have been rapidly developing e.g.,. Among those studies, the method proposed by can measure the three axes of the grains from 3D point cloud data obtained by drone photogrammetry, allowing the grain axes lengths and the volume of grains to be measured. In this study, we adopted the volume-based metric for the representative grain size, so that the G3Point method was employed.

The measurement procedures were as follows. (1) The drone (DJI Air-2s) photographed approximately 100–200 m² areas of the riverbeds. (2) 3D point clouds exhibiting riverbed surface morphologies were produced by the Structure from Motion algorithm using Agisoft Metashape. (3) Triaxial ellipsoids fitted the morphology of each gravel to measure the diameters of the grains using the software G3Point .

In this study, we defined the representative grain diameter D‾s as the mean of the diameter of the spheres in the weighted arithmetic mean (D50), which are equal to the fitted tri-axial ellipsoids in volume.

Since the G3Point program can analyze a maximum of 1 million points at a time, we split the analysis area into two or three non-overlapping rectangular regions with a width of 5 m or less. The mean values of these subareas were then used as the results of the measurements in the surveyed area. We cropped the water surface contained in the 3D point cloud manually because G3Point sometimes misidentified the water surface as the grain surface. In the G3Point method, 10 points are required to fit an ellipsoid to a grain, so the measurable minimal grain size is 1–2 cm.

To estimate the mean diameter Ds of the riverbed gravels along each stream, the measured values were interpolated with Sternberg's law Eq. () : 33Dx=D0exp⁡-αdx where D0 is the grain diameter at the origin, and αd is the change rate of the mean diameter. Here, we assumed that the river gravels fine downstream , so that αd should be a positive value. These parameters D0 and αd were estimated from the mean of the measured values using the least-squares method.

The grain sizes of the riverbed gravels were also measured manually at two locations (Sakura River and Gohyaku River) to evaluate the accuracy of the automated measurements (Fig. ). In this manual measurement, the longest (a), intermediate (b), and shortest (c) axes were measured. First, we delineated a several-square-meter area on a river bar using spray paint. Within this area, we measured the sizes of clasts that were visually identifiable from a vertical viewpoint, had exposed surfaces, and possessed a short axis of at least 1 cm. Then the results were compared with those obtained by the G3Point program.

We identified ten types of sedimentary, pyroclastic, gneiss, and igneous rocks exposed on the riverbed of five tributaries of the Abukuma River (Takinosawa, Hisawa, Fukazawa, Sangasawa, and Gohyaku Rivers) (Fig. ). The sedimentary rock is subdivided into conglomerate, massive sandstone, and siltstone. The pyroclastic rocks include volcanic breccia, lapilli tuff, and tuff. The igneous rocks are granodiorite and andesite. The volcanic and pyroclastic rocks are distributed in the northeastern region where Takinosawa and Fukazawa exist. The granodiorite is exposed in the northwestern region (Sangasawa). A major fault exists in the upstream region of Gohyaku River, while no significant topographic change at the fault was observed in the field. A gentle synclinal structure with an NNE-SSW trending fold axis was observed in the study area, whereas an anticlinal structure was also observed upstream of Takinosawa and a further one in a south tributary. These clinal structures were all trending NNE-SSW.

The results of topographic analysis and geological survey indicated that the river profiles did not vary significantly in slope at the lithologic boundaries. The river course does not show significant changes at fault or fold locations, suggesting that tectonic structures exert limited control on channel morphology in this area. Figure  exhibits the channel longitudinal profiles of the surveyed tributaries extracted from the DEM, where the lithologies identified through the field surveys were plotted as symbols and colors. Most of the rivers have weakly concave-upward smooth profiles. The channel slopes in the surveyed region range from 0.02 to 0.05.

Several knickpoints are associated with lithological boundaries. In Takinosawa, a knickpoint occurs at the boundary between sandstone and conglomerate, with a thin tuff layer in between. The average channel slope in the conglomerate reach is 0.056, whereas that in the upstream sandstone reach is 0.052, yielding a slope difference of only 0.004 across the lithologic boundary. Other minor knickpoints that coincide with lithological boundaries exhibit a trend opposite to the commonly expected pattern, in which channel gradients are steeper in harder bedrock and gentler in weaker lithologies. In Fukazawa, knickpoints occur at both the sandstone-gneiss and breccia-tuff boundaries. However, in both cases, softer rocks (sandstone and breccia) are located upstream, where channel slopes are steeper, whereas harder rocks (gneiss and lapilli tuff) occur downstream, where slopes are lower.

There are also knickpoints that do not correspond to lithologic boundaries. The knickpoint in the middle Sangasawa corresponds to a significant change in the drainage area of the river. The lithologic boundary between the granodiorite and sandstone is located about 120 m downstream of this knickpoint. The knickpoint downstream of Hisawa coincides with the location of the check dam. In Sangasawa, a check dam is located 10 m downstream of the confluence, whereas the step observed in the topography within the granite occurs upstream of the dam. Several knickpoints are located within the same lithology, such as gneiss in the Fukazawa and Gohyaku rivers, even where no significant change in drainage area or check dam is observed.

The chi-plot results (Fig. ) did not show knickpoints that correspond to all tributaries. The Takinosawa River has a knickpoint that is absent in other tributaries. The steepness of Takinosawa, Hisawa, and Gohyaku River over χ=20 000 was almost the same; on the other hand, that of Fukazawa and Sangasawa was much steeper.

The measured results of the bedrock strengths indicated that igneous and metamorphic rocks exhibited significantly larger tensile strength than sedimentary rocks (Fig. ; Table ). The strengths of igneous and metamorphic rocks ranged from 4.3 to 8.2 MPa. Strength measurements accomplished on the dacite at Fukazawa yielded a maximum value of 8.2 MPa, which was more than 18 times larger than the minimum tensile strength of tuffs (0.46 MPa). The sedimentary rocks ranged in tensile strength from 0.4 to 2.4 MPa. They demonstrate variation in their strength depending on the location. The sandstones along Takinosawa were harder than those along Sangasawa.

The relationship between channel slope and bedrock tensile strength exhibited a weak correlation, with a correlation coefficient r of 0.23 and an R2 value of 0.054 (Fig. ). The p-value was 0.0025, indicating that the correlation is statistically significant at the level of α=0.05.

The numerical experiments using the sediment-flux-dependent model reproduced well the actual river profiles (Fig. , red line). The channel width coefficient kw, which was optimized for the calculation, ranged from 1.0 to 2.2, and the optimized value for the discharge coefficient kSFDM was 120.4.

Bankfull widths predicted using Eq. () with the optimized values of kw and kSFDM are consistent with field observations and with estimates derived from 3D point-cloud datasets of the riverbeds. For Takinosawa, the predicted bankfull width is approximately 5–9 m, whereas the observed width is about 7 m. Similarly, the predicted widths for Hisawa and the Gohyaku River are approximately 5–7 and 22–24 m, respectively, which are comparable to the observed widths of about 10 and 15–20 m. For Fukazawa and Sangasawa, the predicted bankfull widths are approximately 13–16 and 4–9 m, respectively. Direct measurements of bankfull width were not available for these rivers.

Both the observed and modeled river profiles exhibited consistently smooth longitudinal trends, regardless of underlying bedrock strength (Fig. ). The mean squared error between the SFDM and actual profile was 3.5 m, which was the best among the three models (SFDM, ASPM, and SPACEM) examined in this study.

The SFDM prediction for Sangasawa reproduced the knickpoint due to the remarkable change in drainage area. Takinosawa has a concave-upward profile, and the model simulated a similar profile due to the downstream fining grain size distribution. However, the model failed to reconstruct the step-like knickpoints of the river profile observed in Fukazawa and Gohyaku Rivers. This discrepancy was particularly pronounced in areas underlain by hard bedrocks such as granodiorite and gneiss.

Thus, although the SFDM did not capture the fine topographical changes on a scale of a few hundred meters, it accurately reproduced the overall characteristics of the bedrock river profiles.

In contrast to the SDFM, the profiles predicted by the stream power models (ASPM and SPACEM) did not agree with the actual profiles. Their predictions clearly reflected the controls of variations in bedrock strength through the slope profiles (Fig. , red dashed-dot pattern and dashed line). The optimized erosional coefficient for ASPM ka was 5.8 [yr⁻¹], and the optimized discharge coefficient kSPACEM for SPACEM was 3.1. The ASPM profile exhibited a remarkable change in slope at the lithologic boundary of the river bedrock types. Especially in Sangasawa, where the contrast in bedrock strength was most pronounced, the ASPM predicted that upstream steepness in granodiorite was 9 times greater than the downstream steepness in sandstone. The actual Sangasawa slope change was very small, so that this ASPM result was significantly different from the actual profile (Fig. ).

The profiles predicted by the SPACEM still deviated from the actual profiles (Fig. ), although they were smoother than those of the ASPM. In Sangasawa, the modeled slope of SPACEM changed at the lithological boundary, with the upstream slope being about four times steeper than the downstream slope. The mean squared error between the ASPM and the actual profiles was 21.7, and that of the SPACEM was 13.2.

The results of sensitivity analysis exhibited that in the SFDM, variations in the nondimensional critical shear stress, friction coefficient, and bedload ratio have only minor effects on river profiles, whereas changes in erosional efficiency produce slight gradient variations associated with lithological contrasts but overall stable results (Figs. S1–S8). In contrast, the ASPM exhibited strong parameter sensitivity: river profiles deviate markedly from observations when the slope exponent n exceeds 1.5 or when the drainage area exponent m is below 0.5, indicating that the adopted values (n=1.0, m=0.5) lie within a stable range (Figs. S9–S12). In the SPACEM, channel gradients were less sensitive to n than in the ASPM and remained stable for n≥0.7, while variations in sediment erodibility Ksed within reported ranges caused only minor changes in gradients and RMS (Figs. S13–S18). Overall, the sensitivity analysis confirms that the parameter values used in this study fall within stable regimes and yield results consistent with the observed topography.

The field measurements in this study indicated that the bedrock strength has a limited influence on the river longitudinal profiles. , , and also suggested that the influence of lithological erodibility on the river profile was small. The measurement of bedrock tensile strength included in the present analysis serves to justify this observation.

The numerical experiments imply that the smoothness of the longitudinal profiles of these rivers, in which clear breaks corresponding to a change in lithology are not seen, is attained by the sediment-cover effect (Fig. ). The sediment-flux-dependent model predicted that the sediment cover ratio mitigated the effect of bedrock strength. Through the cover ratio being higher in soft rocks and lower in hard rocks, erosion is suppressed in soft rock areas, and erosion is promoted in hard rock areas (Fig. ). Equation () indicates that the bedrock strength appears only in the denominator of the first term on the right-hand side of the equation. However, in this equation, the product of the square root of hydraulic roughness ks1/2, the squared rock tensile strength σt2, and the uplift rate U (375 m Myr⁻¹ in this study) has a value in the range the order of 10-11 to 10-13 [kg² m^−1∕2 s⁻⁵], while the term of the product of coefficient β0, the grain size Ds1/2, and sediment supply rate qs is the order of 10-10 [kg² m^−1∕2 s⁻⁵]. Thus, it is clear from the equation that the value of rock tensile strength has little effect on the resulting slope. The mechanical explanation for controlling the sediment cover ratio in relation to lithology is as follows.

Considering the cover ratio, even minor changes in river slope can substantially alter erosion rates, effectively mitigating the influence of the bedrock strength. The sediment cover ratio on the bedrock surface increases as the sediment transport capacity approaches the actual sediment supply. Conversely, the cover ratio Pc decreases when the sediment supply is limited or when transport capacity increases. This dynamics is explicitly considered in the sediment-flux-dependent model, in which Pc is formulated as the ratio of sediment supply qs to the transport capacity qt . Here, the sediment supply is primarily governed by upstream conditions, while the transport capacity qt is controlled by the bed shear stress, which increases with the channel slope S. Therefore, under constant sediment supply, the cover ratio Pc is slope-dependent (Fig. ). This dependence is particularly pronounced at low gradients, where the bed shear stress is near the threshold for the gravel mobilization, making Pc highly sensitive to small changes in slope (Fig. ). Because the sediment cover shields the bedrock from direct impact by bedload particles, even slight variations in slope – and thus in cover ratio – can lead to significant differences in erosion rates (Fig. ).

As a result, river longitudinal profiles tend not to exhibit marked changes across lithologic boundaries unless the slope becomes exceptionally steep. Although channel gradients over resistant lithologies are marginally steeper than those over weaker ones, the difference is often too subtle to be discerned either in numerical simulations or natural river profiles (Figs. , , and ). reported that the slope can vary by as much as 0.08 depending on lithology in hillslope areas. In comparison with their results, such abrupt lithology-driven slope changes do not occur in the alluvial-bedrock reaches investigated here.

This finding aligns with previous studies. demonstrated that the rock strength exerts only a limited influence on channel profiles in models that incorporate the sediment cover effect. Similarly, showed that the variability in channel slope due to differences in rock erodibility is smoothed out when sediment cover dynamics are included in numerical simulations. The present study further supports these findings by demonstrating, through numerical experiments using field-derived datasets, that sediment cover effectively offsets the influence of bedrock strength on river profile morphology.

This study proposes that the relationship between the cover ratio and river slope is a critical factor, and the SFDM well reproduces this relationship. The ASPM does not consider the sediment cover ratio, and therefore, it fails to predict the actual river longitudinal profiles. While the SPACEM also considers the cover ratio, its reproducibility is inferior to that of the SFDM (Fig. ). The reason for this discrepancy lies in the formulation of the relationship between cover ratio Pc and channel slope S in the SPACEM. Figure a, b illustrates cover ratio and erosional rates against slopes for both SPACEM and SFDM. Both models assume that the erosional rate is proportional to the ratio of bedrock exposure. In the case of SFDM, the bedrock exposure ratio is defined as 1-Pc. In contrast, SPACEM adopts the log-scale definition e-H/H∗, resulting in a weaker topographic response to variations in the ratio of sediment cover. At present, the former formulation is more suitable for representing actual river profiles. However, there is no physical basis to suggest that the cover ratio is linearly correlated with the sediment supply/transport capacity ratio; therefore, further investigation is necessary to improve the cover (or exposure) ratio to better fit real-world conditions in future research.

It should be noted that several small-scale knickpoints are present in the rivers of the study area (Fig. ) that cannot be explained by any of the models considered in this study. Most of these knickpoints occur independently of bedrock strength, and some display trends opposite to those expected from bedrock strength, such as occurring in relatively hard rock reaches with gentle channel gradients (Fig. ). All models used in this study assume that rivers are in a steady-state balance between rock uplift and erosion. The presence of these minor knickpoints therefore suggests that parts of the river profiles may locally deviate from equilibrium. Knickpoints generated by the processes, such as past sea-level fluctuations or localized fault activity in bedrock rivers, are known to migrate upstream over time and eventually dissipate in upstream reaches . Such transient processes are not represented in equilibrium-based models. Despite these limitations, the models employed here – particularly the SFDM – successfully reproduce the overall longitudinal profile characteristics of the rivers. This indicates that, at a macroscopic scale, the river systems in the study area can reasonably be regarded as being close to equilibrium.

Although bedrock strength may not directly control local channel gradients, lithology can still exert an indirect influence on the entire morphology of channel longitudinal profiles by shaping the grain size distribution and sediment flux as mentioned in . As shown in Eq. (), the primary factors determining channel profiles are the water discharge per unit width qw, grain size Ds, and sediment flux qs. Among these, the latter two parameters are potentially regulated by the lithologic characteristics of the upstream drainage areas.

Grain size and distributions of sediment are known to vary significantly depending on the lithology of their drainage basins . Sediment grains are initially produced on hillslopes through physical and chemical weathering processes, where lithology, along with tectonics and climate, plays a critical role in determining grain sizes . Field measurements have consistently highlighted the importance of lithology, owing to its control on the physical and chemical properties of parent rocks, in influencing grain size distribution. For instance, demonstrated a clear correlation between rock strength and the sizes of rock fragments measured in soils in California. Similarly, conducted field measurements of clast sizes, indicating that gravels derived from massive granitic plutons exhibited the largest, those from surficial basalt flows had intermediate sizes, and those from marine ribbon chert were the smallest. Since tool size undergoes downstream fining through abrasion, local bedrock lithology alone does not uniquely determine channel gradient. In contrast, basin-integrated lithologic controls can still shape longitudinal profiles through their influence on tool production and size distributions. Collectively, these studies underscore the significant control exerted by hillslope lithology on grain size.

Bedrock lithology in hillslope regions also controls the ratio of bedloads to the eroded materials , which affects the sediment flux qs provided to streams. The sediment supply rate depends on the erosional rate, the drainage area, and the mode of mechanical weathering that determines the ratio of bedload materials in sediment (Eq. ). For example, examined the frequency and size of landslides in southern Italy, focusing on their relationship with the local slope steepness and rock strength. They discovered that the frequency of landslides increases as the slope is composed of softer rock, which promotes the larger sediment production rates.

Thus, the lithologic composition of bedrock across the drainage basin from hillslopes to fluvial zones can play a critical role in determining the overall longitudinal profile of river channels. Supporting this notion, the field-based analysis by in the Tsugaru region of Japan demonstrated that variations in channel morphology are strongly linked to differences in sediment load and grain size, both of which are influenced by upstream lithology. The findings of this study are consistent with this interpretation. The bedrock types in the hillslope regions of the studied tributaries can be broadly divided into two categories: granodiorites and sedimentary rocks. Granodioritic bedrocks are predominantly exposed in the upstream catchments of Fukazawa and Sangasawa, whereas sedimentary rocks dominate the upstream areas of Takinosawa, Hisawa, and Gohyaku River . Chi-plots analysis (Fig. ) revealed that the channel steepness indices of Fukazawa and Sangasawa tended to be notably higher in their upper reaches compared to the other tributaries.

This pattern suggests that upstream lithology – specifically, the presence of more resistant granodiorite – may enhance channel steepness by influencing sediment grain size and transport rates. These observations underscore the indirect yet significant role of lithologic variation in controlling longitudinal channel profile. Further quantitative predictions of such lithologic controls remain a key challenge for future research.