Estimation of velocity profile through mud depth is a long-standing and
essential problem in debris-flow dynamics. Until now, various velocity
profiles have been proposed based on the regression of experimental
measurements, but these are often limited by the observation conditions,
such as the number of the configured sensors. Therefore, the resulting
linear velocity profiles exhibit limitations in reproducing the
nonlinear behavior and its temporal variation during the debris-flow
process. In this study, we present a novel approach to explore
debris-flow velocity profile in detail upon our previous 3D-HBP-SPH
numerical model, i.e., the three-dimensional Smoothed Particle
Hydrodynamic model incorporating with the Herschel-Bulkley-Papanastasiou
rheology. Specifically, we propose a stratification statistical
algorithm for interpreting the details of SPH particles, which enables
the recording of temporal velocities of debris flow at different mud
depths. To regress the velocity profile, we introduce a
logarithmic-based nonlinear function with two empirical parameters, that
controlling the shape of velocity profile and concerning its temporal
evolution. We verify the proposed velocity profile and explore its
sensitivity using 34 sets of velocity data from three individual flume
experiments in previous literatures. Our results demonstrate that the
proposed temporal-varying and depth-nonlinear velocity profile
outperforms the previous ones.