## 2 Vegetation Model
### Modified Log Law Implementation at the Vegetated Bed Layer  
At the near-bed region of a submerged vegetation flow, the velocity profile's estimation is complicated, and finite numerical models (i.e., finite volume, finite element, or finite difference models) have usually been employed for its calculation [3,4]. However, the numerical calculation is usually time-consuming as compared to the analytical model and can defeat the purpose of timely vegetated flow prediction. Since [PERSON]'s investigation, the log law and its various modified approaches have been proven to represent a wide range of flow conditions well, including those over smooth and rough beds [19].  
In this study, the treatment of the near-bed region of vegetated flow using the modified log law has been proposed to mimic the flow via a representative roughness depth caused by vegetation. This representation method is rarely performed for the vegetation case in the literature, but it can provide a cost-effective method to represent the complex flow condition at the vegetated bed. In summary, to simplify the computation consideration, this study modifies the \"rough-bed\" representation to the inner layer of vegetation, where the basic model is presented for a fixed rough-bed flow as follows [20]:  
\[\mathrm{U}=\frac{\mathrm{u}\cdot\mathrm{z}}{\mathrm{k}}\mathrm{ln}\left( \frac{\mathrm{z}-\mathrm{z}_{\mathrm{o}}}{\mathrm{k}_{\mathrm{s}}}\right)+Z \tag{10}\]  
where \(\mathrm{k}_{\mathrm{s}}\) is the Nikuradse roughness, \(\mathrm{z}\) is the depth up to velocity dip position, \(\mathrm{z}_{\mathrm{o}}\) is the physical zero-velocity position (which is regarded at channel fixed bed for vegetated flow), and \(Z\) is the logarithmic integration constant. The vegetation boundary condition is found in [21]:  
\[\mathrm{U}(\xi)=\left[\frac{\xi_{1}^{2}/2+\xi_{1}+\mathrm{C}_{\mathrm{Ar}}}{ \xi^{2}/2+\xi_{+}+\mathrm{C}_{\mathrm{Ar}}}\right]\left[\frac{\mathrm{g} \mathrm{S}_{\mathrm{o}}\left[\left(\xi^{2}/4+\xi\right)-\left(\xi_{1}^{2}/4+ \xi_{1}\right)+\mathrm{C}_{\mathrm{Ar}}\mathrm{ln}(\xi/\xi_{1})\right]}{ \mathrm{u}\cdot\mathrm{k}\left(\xi_{1}^{2}/2+\xi_{1}+\mathrm{C}_{\mathrm{Ar}} \right)}+\mathrm{U}(\xi_{1})\right] \tag{11}\]  
in which \(\xi=\mathrm{z}/\mathrm{h}\) (\(\mathrm{h}\) is full water depth), and it is regarded as the characteristic depth. \(\xi_{1}\) is the characteristic depth for the inner flow region where the vegetated flow velocity profile changes from being log-based to exhibiting linear characteristics. \(\mathrm{C}_{\mathrm{Ar}}\) is the constant calibrated by [21] based on the aspect ratio of flows.  
Similarly to all proposed analytical models in the literature, this study's model also needs assumptions to function. One such assumption is steady-state flow. A lot of previous studies have employed the same assumption since a non-steady, time-varying approach is too difficult to model using an analytical approach. The examples of the analytical model that used the steady flow assumption include those presented by [PERSON] et al. [3], [PERSON] et al. [15], [PERSON] [18], [PERSON] et al. [19], and [PERSON] [21]. In the coming sections, this paper will fully validate the proposed model by comparing it with actual measurements in the literature, in order to prove its functionality and accuracy.