3.3 Modeling and applying the V–L relationship
In 2021, the total volume of 2,312 ephemeral gullies was 19,175.95 m3 (per ephemeral gully average 8.29 ± 6.34 m3), with 82.70% of the single ephemeral gully volume concentrated within 13.0 m3, and only 3.46% of the single ephemeral gully volume exceeding 20 m3. The regression model fit the following equation for ephemeral gully volume and length (Fig. 6(a)):
\(V=0.0842L^{1.1932}\ (R^{2}=0.80,\ n=1,594)\) (8)
where \(V\) is ephemeral gully volume (m3) and \(L\)is ephemeral gully length (m).
The above regression function indicated a power function relationship between ephemeral gully volume and ephemeral gully length (R2 = 0.80). Previous studies also used a power function of the form V=aLb to reveal the relationship between ephemeral gully volume (V ) and ephemeral gully length (L ), where the values of constant a ranged from 0.0082–2.94 and exponent b ranged from 1.12–2.1622 (Capra et al., 2005; Frankl et al., 2013; Li et al., 2017; Muñoz-Robles et al., 2010; Zucca et al., 2006). In this study, constant a (0.0842) and exponent b (1.1932) were within the corresponding intervals reported in previous studies, and ephemeral gully length can be accurately and conveniently extracted based on large-scale DOM or DEM. Moreover, the volumes of the ephemeral gullies were modeled using Eq. (8) and compared to measured ephemeral gully volumes. Linear regression between the predicted and measured ephemeral gully volumes produced R2 and RMSE values of 0.97 and 0.85 m3, respectively (Fig. 5(b)). There was no observable systematic bias between the predicted and measured ephemeral gully volumes using the V–L relationship. Therefore, the V–Lmodel can be used to assess ephemeral gully volume in the hilly and gully region of the Loess Plateau.