Lộc Lương

Predicting bedload transport rate has been an active research over an century. Many bedload transport models and equations have been proposed, however, we are still far from having a complete and effective model that is capable to capture the underlying physics of fluvial bedload transport, or having good prediction capability (e.g., under 1 order of magnitude). The reasons are partly because of the turbulence or stochastic manner of bedload transport.

Recently, we have a tool, or to be more precise, we borrow the seismic tool from seismology to listen to the noise generated by the movement of sediment particles that move and impact riverbed. And seismic monitoring of fluvial sediment transport has been utilized in several regions around the world to quantify bedload transport rate. Not only in fluvial research, this technique can also be applied to quantify a variety of geophysical or surface processes (e.g., landslides or rockfalls, debris flow), and many other natural hazards and environmental changes.

Theoretical models have suggested that bedload flux can be estimated based on seismic energy within appropriate frequency bands. Here, in this note, I’d like to summary what we did in our experiment and share some interesting results. We first applied a physics-based model of seismic noise generated by bedload transport to an ephemeral channel, I mean we applied this model to invert back to how much bedload is moving given an amount of seismic noise we recorded.

It turns out that the result does not quite match what we measure in our natural lab. I want to use the term “nature lab” which is essentially a small river/channel located in Socorro, New Mexico because I think Earth science, in general, use the Earth as the a dynamic lab to understanding more about it. It is sure that measurement taken from the fields have a lot of uncertainties and variability due to the nature of the Earth (e.e., heterogeneity).

The first physics-based model by Tsai et al. (2021) only assumed the particle impact vertically and elastically onto the riverbed. We then modified this model by adding rolling and sliding mechanisms, and consider inelastic and oblique impacts of bedload particles on the riverbed. Our new results show that the modified model better estimate bedload flux for ephemeral channel, which we think maybe due to the inclusion of all transport mechanisms, and a hop time distribution that represent the three transport modes at once. However, this model still has limitation such as: the parameter space is too high (too many parameters), or it is very sensitive to certain model parameters that, in practice, we need to have a good constraints in order to obtain a useful inversion for bedload flux. 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:

$$ P_v(f, D) = \frac{C_1 W q_b D \bar{w}_s}{V_p U_b H_b} \cdot \frac{\pi^2 f^3 m^2 w_i^2}{\rho_s^2 v_c^3 v_u^2} \cdot \chi(\beta) $$

A derivation from scratch

I wanted to give a shot to explain how we come up with a modified model of Tsai et al., 2012. In seismology, we have a solid theory to describe the propagation of seismic waves traveling though the Earth interior and layers cause by distant earthquakes. This theory can be transferred to describe how seismic noise generated by bedload movement in river settings. Bedload particles move and impact onto the riverbed can be described by a source function $F$, and resulted seismic waves travel through the ground can be described by a Green’s function $G$. We can use a convolution operator between the source function $F(t,\mathbf{x}_0$ and the Green’s function $G(t,\mathbf{x},\mathbf{x}_0)$ to describe the ground displacement in space at specific time:

$$ s(t,\mathbf{x}) = F(t,\mathbf{x}_0) \ast G(t,\mathbf{x},\mathbf{x}_0) $$

In frequency space (using Fourier transform), the ground velocity can be expressed as:

$$ \mathbf{u}(f,\mathbf{x}) = \frac{\partial s(t,\mathbf{x})}{\partial t} = 2\pi f \mathbf{F}(t,\mathbf{x}_0) \ast \mathbf{G}(t,\mathbf{x},\mathbf{x}_0) $$

where $\mathbf{F}(t,\mathbf{x}_0)$ is the Fourier’s transform of $F(t,\mathbf{x}_0$, and $\mathbf{G}(t,\mathbf{x},\mathbf{x}_0)$ is the Fourier’s transform of $G(t,\mathbf{x},\mathbf{x}_0)$.

The key assumption from Tsai et al. (2012) is that impacts by sediment particle onto the riverbed are assumed to be random in time, therefore, the sum of impacts does not affect the resulting force spectrum. This can be explained using a 1D random walk analogy. For example, expected value of the distance squared is the number of steps in a 1D random walk process.

$$ Pv(f, D) = \int{\mathbb{R}} R_D \, \left| \mathbf{u}(f, x) \right|^2 \, dx $$

where $\mathbb{R}$ is the length of the river, $R_D$ is is the total rate of impacts per unit river length, $R_D$ = \frac{n_D}{t_D}$ where $n_D$ is the number of impacts (of grain $D$) per unit river length and $t_D$ is the time between impacts. Lamb et al., 2008 relate $n_D/t_D$ to:

$$ \frac{n_D}{t_D} = \frac{q_{bD}W}{V_p s} $$

where $q_{bD}$ is the sediment flux per unit river length per unit size $D$, $W$ is river width, $V_p$ is particle volume and $s$ is hop length which is the length it jumps, hops, slides on the riverbed during $t_D$.

A force history for a single impact of grain size $D$ can be expressed as:

$$ F(t) = I\delta t $$

where $\delta t$ is the Dirac delta function. To account for oblique impacts (in stead of purely vertical impacts), Farin et al. (2019) introduced a coefficient $f_j$, and using a restitution coefficient $\gamma$:

$$ \mathbf{F}{j} = (1+\gamma)m U{b} f_{j} $$

where $m$ is mass of sediment particle and $U_b$ is the sediment velocity.

Aki and Richards (2002) came up with an analytical expression of ground displacement Green’s function as:

$$ G_{jz}(f, x;\, x_0) \approx \frac{N_{jz}}{8\rho_s v_c v_u} \sqrt{e^{-\frac{\pi f r}{v_u Q}}} $$

Substituting these equation to the definition of power spectral density function (PSD), we can arrive at our modified model.

Modified model

In this model, we account for 3 transport mechanisms: rolling, hopping, and sliding 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 snapshot of the comparison when applying both model to field observation.

References

[1] Tsai, V.C., Minchew, B., Lamb, M.P. and Ampuero, J.P., 2012. A physical model for seismic noise generation from sediment transport in rivers. Geophysical research letters, 39(2).
[2] Luong, L., Cadol, D., Bilek, S., McLaughlin, J.M., Laronne, J.B. and Turowski, J.M., 2024. Seismic modeling of bedload transport in a gravel‐bed alluvial channel. Journal of Geophysical Research: Earth Surface, 129(9), p.e2024JF007761.