**4. Results**

#### *4.1. Stone Movement and Velocity Distribution in the River Channel*

Comparing the derived orthoimages in two years, stone movement by water flow was identified in 10 m river reaches over each selected cross section in two river channels (Figure 6). In the magnified view in the white rectangular area in each river reach over the two years, it is possible to distinguish the movement or non-movement of stones in each river channel. Stones found in the 2017 image but that had disappeared in the 2018 image were identified as moving with the flow throughout the year, while stable stones were regarded as unmoved stones. The nominal particle size of the identified stones was at least 8 cm due to the resolution of the UAV images, and most of these stones can be described as cobbles. The particle sizes of some large boulders in the river section are even more than the average water depth of the selected cross section during the large peak flow event. These boulders were excluded from the circle selection in Figure 6 and the following calculation.

**Figure 6.** Moved and unmoved cobbles in four river reaches of river channel H and S throughout the year. Magnified views of the moved cobbles are presented in each white rectangular area. (**<sup>a</sup>**–**d**) identified stones in river channel H in 2017, (**<sup>e</sup>**–**h**) identified stones in river channel H in 2018, (**i**–**l**) identified stones in river channel S in 2017, and (**<sup>m</sup>**–**p**) identified stones in river channel S in 2018.

In both river channel H and S, the movement of cobbles was visible, and cobbles apparently moved by water flow were mainly distributed in the main channel. The main channel was more specific in river channel H with more moved cobbles (8–15) found in red circles (Figure 6a–h). In river channel S (Figure 6i–p), cobbles of various sizes were found, and the main channel was not obviously washed

by flow over one year. Consequently, more unmoved cobbles (3-25) in blue circles were identified in river channel S.

Considering that all moving cobbles were mainly moved by water flow, the initial velocities of all identified moving cobbles calculated by the logarithmic or the exponential velocity distribution are presented in Figure 7. In four river reaches of river channel H (Figure 7a–h), the critical initial velocity of the same moving cobble calculated using the logarithmic method is appreciably larger than the velocity calculated using the exponential method. In river reach A, the critical initial velocities of moving cobbles greatly varied, 1.96–2.65 m/s for the exponential method and 3.40–4.39 m/s for the logarithmic method. In addition, there were more cobbles in the higher velocity grading (shown with the longer arrow) for the logarithmic method (Figure 7a) than for the exponential method (Figure 7e). In river reach B, the critical initial velocities of moving cobbles were low, and there were more cobbles with the lowest velocity grading for the logarithmic method (Figure 7b) than for the exponential method (Figure 7f). The channel in river reach C was slowly washed by the water flow and the critical initial velocities of moving cobbles were high (above 4 m/s by the logarithmic method and above 2.4 m/s by the exponential method). There were more cobbles with the highest velocity for the logarithmic method (Figure 7c) than for the exponential method (Figure 7g), with there being more red arrows in Figure 7c.

**Figure 7.** The critical initial velocities of all identified moving cobbles in four river reaches of river channel H and S are calculated separately by the logarithmic or the exponential velocity distribution. Each arrow indicates the direction of the water flow when each moving cobble starts to move, and a longer length of the arrow indicates a higher initial velocity. (**<sup>a</sup>**–**d**) initial velocities calculated by the logarithmic velocity distribution in river channel H, (**<sup>e</sup>**–**h**) initial velocities calculated by the exponential velocity distribution in river channel H, (**i**–**l**) initial velocities calculated by the logarithmic velocity distribution in river channel S, and (**<sup>m</sup>**–**p**) initial velocities calculated by the exponential velocity distribution in river channel S.

Regarding river channel S (Figure 7i–p), velocities obtained by the logarithmic method and exponential method were similar, but the former method produced a slightly broader range of velocities (1.66–2.31 m/s) than the latter method (1.68–1.90 m/s). The logarithmic velocities of moving cobbles in river reach A (Figure 7i) were the lowest among four reaches, all graded at the minimum velocity level, whereas the calculated exponential velocities (Figure 7m) were graded at the average level for the four sections. The critical initial velocities of the moving cobbles in river reach C were similar for the two methods. The condition of river reaches B and D was alike. The maximum velocities calculated using the logarithmic method were concentrated in river reaches B and D (Figure 7j,l) while the maximum and minimum velocities obtained using the exponential method were both in sections B and D (Figure 7n,p).

#### *4.2. Peak Discharge of the River Cross Section*

The profiles of each selected cross section at the highest water level are presented in Figure 8. In two river channels, the elevation of the riverbed and riverbank in each cross section continuously decreased downstream. The underwater area of each cross section of river channel H was relatively large, and the section shape was likely rectangular. In contrast, the underwater area of river channel S was relatively small, and the section shape was irregular.

**Figure 8.** Profile of four selected cross sections in river channel H and S.

When the maximum peak flood flowed through cross sections of the two rivers, the maximum water depth and average water depth of the section greatly varied, while the submerged underwater area was similar (Table 2). At the highest peak water level, the average critical initial velocity of all identified moving cobbles in each river reach was regarded as the cross-sectional flow velocity, and the cross-sectional flow rate and peak discharge calculated using the logarithmic and exponential methods were obtained, as shown in Table 2. The cross-section velocities of the river channel H are different using the two methods, with all logarithmic velocities exceeding 3 m/s and all exponential velocities being around 2 m/s. The difference in the cross-sectional velocity in river channel S is not apparent between the two calculation methods, with all velocities being about 2 m/s. Additionally, the peak discharge calculated by the logarithmic method is larger than that by the exponential method in the case of river channel H. For river channel S, the calculated peak discharges are also similar in four cross sections using the logarithmic and exponential method, with the biggest peak discharges both in section C and the smallest both in section D.

**Table 2.** Results of the average velocity and discharge through four cross sections in river channels H and S.


#### *4.3. Validation of the Estimated River Discharges*

The specific profile and key hydraulic variables of each culvert at the time of the maximum peak flood within one year were determined from the topographic information from the DSM and the field measurement (Figure 9). Culvert H is a wide and shallow box-type culvert with a maximum water depth of 1.25 m when the flood peak passed through. The abundant water in river channel H was almost close to the maximum design discharge of the culvert, while culvert S was square with a smaller water flow. The maximum water depth of culvert S was only 0.3 m. The bases of the two culverts were relatively flat laying a large amount of sediment mixed with sporadically distributed gravel stones. The culvert roughness nc was confirmed as 0.033, a practical value from local hydrological work experience according to conditions of the culvert.

$$v\_{\mathfrak{c}} = \frac{k}{n\_{\mathfrak{c}}} \mathcal{R}\_{\mathfrak{c}}^{2/3} l^{1/2} \tag{14}$$

The validation of the peak flood in each river channel is given in Table 3. The flooding in river channel H is evident with a larger peak discharge through culvert H than culvert S, while in river channel S there is little water with larger relative accuracies. The relative accuracies of the logarithmic method and in river channel S all exceed the threshold of 20%. Only the exponential method applied in river channel H indicates an accurate estimation, with the relative accuracies within 10%. The RMSE and MAPE help to further evaluate the performance of different velocity distributions used in the proposed method. RMSE and MAPE are both lower for the exponential method than for the logarithmic method in two river channels, indicating that the former method is more accurate in terms of the incipient motion of moving cobbles regardless of the amount of water in the river channel.

**Figure 9.** (**a**) and (**c**) are the magnified view of bridge culverts H and S. (**b**) and (**d**) present cross-section profiles and hydraulic parameters used to calculate average velocity and peak discharge through culverts H and S. Ac is the underwater cross-sectional area of the culvert, Xc is the wetted perimeter, J is the hydraulic gradient, Rc is the hydraulic radius, and nc is the roughness coefficient.


**Table 3.** Validation of the peak discharge estimation.
