## 3 Materials and Methods
### Empirical Formulae  
Five empirical formulae (Table 2) were applied to estimate bedload transport at Taquri River cross-section: [PERSON] [19], [PERSON] [20], [PERSON] [17], Einstein-Brown [18] and [PERSON] [21]. The equations presented at the end of Table 2 are general formulae used to compute some of the parameters needed in the empirical transport equations. From now on, the mentioned methods are referred to by the following acronyms, respectively: K, MPM, E, EB and VR. All the equations developed by these researchers use information about hydraulic parameters and bed material characteristics to evaluate bedload transport capacity. The \begin{table}
\begin{tabular}{c c} \hline \hline
**Reference** & **Equations** \\ \hline [PERSON] [19] & \(\frac{\theta_{b}}{u_{*},s\lambda}=10\times\Psi^{*2}\) \\ \hline [PERSON] [20] & \(\phi=(4\times\Psi^{\prime}-0.188)^{3/2}\) \\ \hline [PERSON] [17] & \(\Psi_{E}=\frac{(s-1)\times d}{R^{\prime}\times l}\) \\  & \(R=(ks/ks\pi)^{3/2}\times R\phi\) _vs_\(\Psi_{E}\to see\) _chart in the reference_ \\ \hline [PERSON] [18] & \(\phi=40\times\Psi^{3}\) \\ \hline [PERSON] [21] & \(q_{b}=0.015\times\
u\times h\times(d/h)^{1.2}\times M_{e}^{1.5}\) \\  & \(M_{e}=\frac{\frac{\psi-q_{r}}{\sqrt{(s-1)\times g\times d}}}{\sqrt{(s-1)\times g \times d}}\) \\ \hline General formulae & \(\psi=\frac{\theta_{b}}{\frac{\theta^{2/2}\times\sqrt{(s-1)\times g}}{(s-1) \times g}}\) \\  & \(\Psi^{\prime}=\frac{\overline{\psi}_{b}}{(s-1)\times g\times g\times d}=(ks/ks^ {\prime})^{3/2}\times\Psi=\frac{1}{\Psi_{E}}\) \\  & \(q_{b1}=q_{b}\times\rho_{s}\) \\  & \(ks=\frac{1}{\pi}\) \\  & \(\tau_{0}=\rho\times g\times R\times l\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Empirical formulae applied to estimate bedload transport rates. values applied in the equations are space and time averaged, showing a more general and simplified approach than other methods that account for local effects. The presented formulae were selected for this study because they have been found to provide accurate bedload transport estimates in large sand rivers with dunes. In Table 2, \(q_{b}\) is the unit bedload transport rate in volume (m\({}^{3}\)/s\(\cdot\)m); \(q_{b1}\) is the unit bedload transport rate in mass (kg/s\(\cdot\)m); \(\tau_{0}\) is the bed shear stress (N/m\({}^{2}\)); \(u_{*}\) is the shear velocity, also called friction velocity (m/s); \(\Psi\) is the shields parameter, also called flow parameter (dimensionless); \(\Psi^{\prime}\) is the flow parameter related to grains, not influenced by bedforms roughness (dimensionless); \(\Psi_{E}\) is the flow parameter used by [PERSON], inverse of \(\Psi^{\prime}\) (dimensionless); \(\phi\) is called transport parameter (dimensionless); the apostrophe \(\prime\) refers to the influence of grains, (disregarding bedforms); \(g\) is the acceleration due to gravity (m/s\({}^{2}\)); \(d\) is the sediment diameter (m); \(\rho\) is the density of water (kg/m\({}^{3}\)); \(\rho_{s}\) is the density of sediments (kg/m\({}^{3}\)); \(I\) is the water surface slope of the study reach (m/m); \(h\) is the flow depth (m); \(R\) is the hydraulic radius (m) of the cross-section (ratio of the cross-sectional area to the wetted perimeter); \(M_{e}\) is a mobility parameter which represents the excess mobility of sediments (dimensionless); \(s\) is the specific gravity of sediments (dimensionless); \(v\) is the depth-averaged flow velocity (m/s); \(v_{\sigma}\) is the critical water velocity for currents based on shields (initiation of motion) (m/s); \(ks\) is a roughness coefficient of the bed (m\({}^{1/3}\)/s), equivalent to the inverse of [PERSON]'s \(n\) roughness coefficient (s\(\cdot\)m\({}^{-1/3}\)). [PERSON] [19] developed a simple theory of bedload transport where the main driver was particle velocity. He assumed that particle velocity adapted instantaneously to fluid velocity (linear relationship), which is the main shortcoming of his approach because of the different processes of particle motion (sliding, rolling and saltating).