**5. Discussion**

#### *5.1. Value and Extension of the Proposed Method*

The present paper provides a convenient, flexible, and reliable method of obtaining the annual peak discharge of an ephemeral river in arid ungauged regions. In empirical studies, statistical methods, hydrological models, and multi-source remote sensing methods are most commonly used to estimate peak discharge (Table 4). In the application of these methods, a variety of ways have been conducted for verification. Most studies used records from nearby gauging stations, such as the water level, the flow rate, and the discharge, to validate discharge estimations [47–53]. A few studies were verified directly by field observations [16,46,54]. Furthermore, flood marks on the deposits, places, cliffs and other facilities were also commonly used to calculate the flood peak discharge with reliable accuracy in regions lacking flood gauging measurements or to reconstruct historical floods [48,55–58]. In arid ungauged regions, flood processes of ephemeral rivers are few and continue to be difficult to directly monitor. Therefore, the identification of clear flood watermarks on construction facilities can be used to verify the existence of flood processes effectively and to validate the maximum peak discharge.


**Table 4.** Overview of studies to calculate peak discharge.

Daily precipitation data of the study area were also collected from the nearest meteorological station during the study period to reveal the characteristics of the precipitation in the study area. According to empirical studies, the average rainfall intensity in the study area is around 22.14 mm/h [60]. Generally, it will take more than one hour to generate surface runoff for continuous precipitation with this rainfall intensity in arid regions [61]. In the study area, considerable precipitation was rare, and the top 10 precipitations (>30 mm) during the study period are presented (Figure 10). Only during concentrated precipitations would the surface flow probably yield and then lead to flooding in the channel. Stones in the ephemeral river channel could have been moved by only one or two flood peaks in September of 2017 or July of 2018. Therefore, the stone movement by flooding in the channel could be used to estimate the discharge of the flood peak.

**Figure 10.** Top 10 precipitations in the study area in 2017.8–2018.8.

The proposed method takes full advantage of the UAV data and the long-term water-free characteristic of ephemeral rivers to estimate peak discharge of ephemeral rivers in ungauged regions. Considering both geographical variables and hydraulic variables, the proposed method is similar to the classic slope-area method, but with the high-resolution UAV data playing a dominant role in the process. On the one hand, the UAV data help to identify terrain information and stone movement. On the other hand, the UAV data are used to derive significant hydraulic variables, such as hydraulic slope, hydraulic radius, and underwater cross-sectional area. The proposed method is vital to the effective long-term monitoring of ephemeral rivers in vast ungauged regions and has grea<sup>t</sup> potential to promote at spatial and temporal scales. The primary data required for this method is high-resolution imagery in the dry season. In addition to the UAV data used in this study, other centimeter-level high-resolution images can also be used as data sources. Selecting a typical cross section during the dry season is only needed for in situ measurements, and the water level at the time of the maximum peak flood flow can be determined by comparing the distribution of the soil layer, vegetation, and cobbles. The method established in this paper can also be extended to calculate the maximum peak floods of ephemeral rivers in other typical arid and semi-arid regions with determined dry seasons in these regions.

#### *5.2. Performance Evaluation of the Estimated Velocities*

Due to the short and variable flowing time of ephemeral rivers during the year, it remains hard to capture the flooding process in time and to obtain field measurements for validation. According to empirical studies, the culvert peak discharge estimated by flood watermarks was regarded as reliable and could be used to evaluate the estimated peak discharge. In addition to the validation of the peak discharge, we also examined the performance of cross-sectional velocity estimations by the incipient motion of stones. The culvert peak discharge was regarded as the real flood peak discharge and was used to calculate cross-sectional velocities (Equation (15)) for evaluating the estimated velocities by logarithmic and exponential velocity distribution methods:

$$v\_i = \frac{Q\_{\mathcal{E}}}{A\_i}'\tag{15}$$

where *vi* represents the cross-sectional velocities calculated by the culvert peak discharge (i = 1,2,3,4,5), Qc is the culvert peak discharge through each river channel, and *Ai* is the cross-sectional area (i = 1,2,3,4,5).

Results indicate that the performance of velocity estimation was best using the exponential method in channel H (Figure 11). The overestimation by the logarithmic method was mostly due to the poor performance of original parameters of large grain sediment [27,40]. In contrast, the exponential velocity equation used in this study was the classic formula proposed by Shamov. The formula is concise and generalized, only using K as a comprehensive coe fficient to represent three major forces exerted on the stones and the e ffects of the irregular shape of stones on the incipient motion. The K value used in this study, also proposed by Shamov, has been verified by many experiments and popularization, and is most suitable for inland arid areas [35]. Furthermore, the river channel S only has a small peak discharge of 0.74 m<sup>3</sup>/s, which could move only small stones during the flood. Due to the current spatial resolution of UAV data, enough small stones moved by water could not be identified. Therefore, the estimated velocities by both methods in channel S significantly deviated from the velocities calculated by culvert peak discharge.

**Figure 11.** Error analysis of cross-sectional velocity estimation by the incipient motion of moving stones.

We also added onemore cross section (cross section E) to each river channel to evaluate the performance of velocity estimation by the proposed method when smaller stones were used. The identified stone movements of cross section E are presented in Figure 12. The main difference of the new cross section E is that moving stones were much smaller than the other four cross sections, with an average nominal diameter of around 8 cm (Table 5). When smaller stones were identified and used, estimated velocities by the exponential method decreased. In channel S, using smaller stones significantly lead to a decrease of the error of cross-sectional velocity, and the error of cross section E is the minimum in channel S. In channel H, with evident flooding, only using small stones in the calculation could not reveal the real peak flood and led to underestimations. Regarding the logarithmic method, using smaller stones also reduced the error of velocity estimations in cross section E of channel H. Regardless of using smaller stones, the logarithmic method still performed poorly in channel S with only a little flow.


**Table 5.** Comparison of the size of moving stones.

**Figure 12.** Stone movement in section E in two river channels.

#### *5.3. The E*ff*ects of the Selection of Large Boulders on the Estimation of Peak Discharge*

Some of the identified moving stones in the river channel are irregular and appreciably larger than other moving stones in the riverbed. They cannot be entirely submerged during the flood peak events, and major forces exerted on them are different when they start to move [62]. Therefore, whether large boulders are selected in the calculation may affect the estimation of peak discharge and is worth exploring [63,64]. Moved boulders were only identified in cross sections A and D of river channel H, and were found in all four cross sections of river channel S. The results of the critical initial velocity and peak discharge in two river channels using two velocity distribution methods are compared in Figure 13.

**Figure 13.** Cross section velocity and peak discharge results on condition of boulder selection: (**a**) logarithmic velocity distribution of river channel H, (**b**) logarithmic velocity distribution of river channel S, (**c**) exponential velocity distribution of river channel H, (**d**) exponential velocity distribution of river channel S.

In the case of river channel H with a large amount of water flow, the selection of large boulders apparently increases the cross-sectional velocity and discharge for both logarithmic and exponential methods (Figure 13a,c). The peak discharge in river channel H obtained using the logarithmic method is initially greater than the culvert flow, and the peak discharge thus deviates further from the culvert peak discharge considering the large boulders. In the case of the exponential method, the calculated peak discharge is similar to the culvert peak discharge, irrespective of whether large boulders are included in the calculation. In the case of river channel S with only a small amount of water, there are many stones with an equal nominal diameter more substantial than the average water depth (around 0.1 m). When large boulders are included, the velocity and peak discharge calculated by the logarithmic method are relatively stable in three cross sections A, C, and D, and only decrease in cross section B with a corresponding decline in the deviation of peak discharge (Figure 13b). Instead, the velocity and peak discharge obtained by the exponential method in four sections of channel S increase, and the deviation from the peak discharge of the culvert further increases (Figure 13d). In all, whether large boulders are considered does a ffect the discharge estimation, and the e ffect greatly varies in two river channels with di fferent scales of water flow. With a large amount of water flow, the inclusion of large moved boulders in the calculation significantly increases the estimation of peak discharge in the river channel. Nevertheless, the inclusion of large moved boulders does not significantly a ffect the results of peak discharge if there is only a small amount of water in the river channel.

#### *5.4. Limitations and Uncertainties of the Present Research*

The proposed method of calculating the peak discharge of ephemeral rivers using the critical initial velocity still has some weaknesses and uncertainties. First of all, the resolution of the UAV dramatically affects the number of stones identified moving, and the measurement accuracy of the length and width of the cobbles. With the current image resolution, we failed to distinguish enough small moving stones, which leads to less accurate estimations in river channel S with a small discharge. Topographic data and orthoimages with higher resolution could e ffectively enhance the performance of the method in the future, especially in ephemeral rivers with a small discharge [65]. Secondly, stone properties, such as density and shape, would also generate uncertainties [66–68]. In the study area, most of the river stones are irregular granite stones. The constant value of density and the generalized size of stones used in critical initial velocity estimations influenced the accuracies of the peak flood discharge. In addition, only classic incipient motion theories were adopted in this study, while the equation to calculate the critical initial velocity of each moved cobble could be modified with local empirical studies for future extension.
