## 4 Results and Discussion
### Discussion
#### 4.5.1 Empirical Equations for Ancient Braided Rivers Based on Sedimentary Outcrops
Regarding W\({}_{\text{b}}\)-d\({}_{\text{m}}\) relationships, we compare them by observing the changes in the slope and intercept of the regression curve in the logarithmic graph (Figure 10). The regression index value (slope) of the equation obtained in this study area is small, which is equivalent to that of the equation obtained in sandy braided channel (Equation (9) in Table 3), slightly larger than that of the equation obtained in the braided-meandering gravelly river (Equation (6) in Table 3), and significantly smaller than that of a conglomerate channel (Equation (8) in Table 3). The W\({}_{\text{b}}\) estimated by the W\({}_{\text{b}}\)-d\({}_{\text{m}}\) equation in the present study is far less than that estimated by the equation for other sandy and gravelly braided rivers (Equations (8) and (9) in Table 3), which is related to the small scale of the river in the study area and is slightly larger than that estimated by the equation for braided-meandering gravelly rivers (Equation (6) in Table 3) but is obviously much larger than that estimated from the equation of other meandering rivers (Equation (10) in Table 3) (d\({}_{\text{m}}\) < 5 m). By comparison, in terms of the W\({}_{\text{b}}\)-d\({}_{\text{m}}\) relationship, the river characteristics in the Songyuan area conform to the coarse sandy braided river category. However, this conclusion is somewhat different from the coarse sandy-gravelly sedimentary characteristics in this study area. This result could be attributed to the fact that, during the sedimentation period, no catastrophic flood event was strong enough to erode the river bank controlled by bank stability. However, under the influence of climate (precipitation), seasonal high discharge generated a strong hydrodynamic force under the influence of high slopes, leading to the relatively deep downcutting of the channel while extending laterally. Stable flow (60-80% of precipitation concentrated in June-September) reduces episodic bank scour during extreme floods, promoting a \"deep-narrow\" channel morphology over a \"shallow-wide\" one. Multiple regression analysis shows that a 1-unit decrease in discharge variability correlates with a 0.21 decrease in the W\({}_{\text{b}}\)-d\({}_{\text{m}}\) exponent (\(p\) < 0.05). The reasons for this significant discrepancy can also be explained by the following quantitative evidence. The median grain size in our study area (2-5 cm) is substantially finer than that in Xu's gravel-dominated rivers (5-10 cm) [17]. Field measurements show that finer sediments (40% sandy fraction) reduce bed roughness, favoring vertical incision over lateral expansion (Figure 10). Statistical analysis indicates that, for every 1 cm decrease in grain size, the W\({}_{\text{b}}\)-d\({}_{\text{m}}\) exponent decreases by an average of 0.32 (\(n\) = 15, R\({}^{2}\) = 0.76), directly contributing to the low exponent observed here. Riparian vegetation cover (30-40%, derived from remote sensing of the modern Second Songhua River) enhances bank cohesion, limiting lateral  
Figure 10: Comparison of the relationship between bankfull channel width (W\({}_{\text{b}}\)) and mean bankfull channel depth (d\({}_{\text{m}}\)) between the study area and previous studies. Modified after [PERSON] et al. [15]. See Table 3 for the numbers in brackets and the abbreviations. migration rates (0.5-1.2 m/year) to 25% of those in [PERSON]'s study area. Hydrological data from Fuyu Station (1985-2000) show that the flood-season bank erosion in our study area is 33% of that in non-vegetated systems, restricting width expansion and resulting in a smaller \(\mathrm{W_{b}}\) for a given \(\mathrm{d_{m}}\). Comparison with [PERSON] et al.