Entropy-based depth-averaged velocity assessment from surface flow velocity

Discharge estimation at a river site depends on local hydraulic conditions identified by recording water levels. In fact, stage monitoring is straightforward and relatively inexpensive compared with the cost necessary to carry out flow velocity measurements which are, however, limited to low flows and constrained by the accessibility of the site. In this context, the mean flow velocity is hard to estimate for high flow, affecting de-facto the reliability of discharge assessment for extreme events. On the other hand, the surface flow velocity can be easily monitored by using radar sensors allowing to achieve a good estimate of discharge by exploiting the entropy theory applied to rivers hydraulic (Chiu,1987). The growing interest towards the use of no-contact methods to estimate discharge (Tauro et al., 2018) in field applications has shown that the cross-track velocity distribution can be inferred with sufficient accuracy using the surface velocities, u_surf, sampled using Surface Velocity Radars (SVR) (Fulton and Ostrowski, 2008; Moramarco et al., 2017, Alimenti et al. 2020), the quantitative imaging techniques as LSPIV (Fujita et al., 1998) or PTV (Tauro et al., 2019). In this context, overall the velocity-area method is applied to estimate the mean flow velocity starting from the depth-averaged velocity, u_vert, which is inferred through the velocity index, k=u_vert/u_surf.. For many river gage sites configurations, k has been set to 0.85. However, considering k refers to a monotonous velocity profile, not taking account of dip phenomena, the application may fail in estimating the depth-averaged velocity (Moramarco et al., 2017; Koussis et al., 2022, Pumo et al., 2025). Based on that, this work proposes a new entropy-based approach to estimate the depth-averaged velocity starting from the measured surface velocity retrieved by conventional and/or no-contact measurements. The approach exploits the dependence of the entropy parameter M with the hydraulic and geometric characteristics of channel (Moramarco and Dingmann, 2017), allowing to derive formulations on Manning’s roughness, shear velocity and water surface slope. Based on these features, the entropy-based method by using the measured surface velocity and the geometry of the river site is able to turn u_surf  into u_vert considering for each  u_surf  an index which depends on the local water surface slope. The application to river sites along the Tiber River, Po River and Amazon River has shown the effectiveness of the approach in estimating the depth-averaged velocities with a fair accuracy along all verticals. Therefore, the method well lends itself to be integrated in the field of no-contact streamflow measurements.