Predicting bedload transport rate has been an active research over an century. Many bedload transport models have been proposed, yet we are far from a complete and effective modeling of the underlying physics of fluvial bedload transport, or having good prediction capability, particularly capturing the turbulence and stochasticity bedload transport. Recently, seismic monitoring of fluvial sediment transport has been utilized in several regions around the world to quantify bedload transport rate. Theoretical models have suggested that bedload flux can be estimated based on seismic energy within appropriate frequency bands. Here, we first applied a physics-based model of seismic noise generated by bedload transport to an ephemeral channel. We then modified this model by adding rolling and sliding mechanisms, and consider inelastic and oblique impacts of bedload particles on the riverbed. Our results show that the modified model better estimate bedload flux for ephemeral channel. You can read the full paper here.
Concise summary
Assuming that bedload particles vertically and elastically impact onto the riverbed (only hoping or satation mechanism), and the impact is instantaneous (Hertzian impact), and Rayleigh waves are mainly generated by the impact, Tsai et al., 2012 developed a relationship between seismic power spectral density (PSD) per unit grain size $D$ to the amount of bedload flux $q_b$ as follows:
$$ P_v(f, D) = \frac{C_1 W q_b D \bar{w}_s}{V_p U_b H_b} \cdot \frac{\pi^2 f^3 m^2 w_i^2}{\rho_s^2 v_c^3 v_u^2} \cdot \chi(\beta) $$
We modified this model to also account for the rolling and sliding mechanism of bedload movement. We also assume inelastic and oblique impacts between particles and riverbed.
$$ P_T(f, x) = \int_D \int_t p_t(t_D) \cdot \frac{p(D) q_b W}{V_p U_b t_D} \cdot \left| N_{11} (1 + \gamma) U_b f_z \right|^2 \cdot \frac{\pi^2 f^3 m^2}{4 \rho_s^2 v_c^3 v_u^2} \cdot \chi(\beta) \, dt_D \, dD $$
And here is the comparison when applying both model to field observation.
