## 2 Vegetation Model
### Validation of the Combined Vegetated Model  
To validate the discussed three-layer vegetated model, various reported measurements in the literature have been used in this study. In the first validation, [PERSON] et al. [3] obtained measured data from a vegetated field (created using vegetation measuring 0.6 cm in diameter and 19 cm in height), and calculations were made using the presented model. The utilised experimental data were obtained from two pieces of vegetation situated parallelly, i.e., one after another longitudinally, as this location has been identified as a zone that is significantly affected by vegetation-induced flow turbulence. Also, in this first test, [PERSON] et al. [3] proposed a model for the inner vegetated region that has been used to compare with the proposed modified log law model, benchmarking against the measured data. [PERSON] and [PERSON] [22] measurements were chosen as the second validation test. Different from [PERSON] et al. [3], [PERSON] and [PERSON] [22] used truncated cone-shaped vegetation with 0.4 cm lower and 0.2 cm upper diameters at 24 cm height. Their measured data can further examine the capability of the proposed model to represent the unique resulting flow field. Finally, [PERSON] et al. 's [23] laboratory data have also been used for validation, as itpresented a rough sand bed where the vegetation was seated on. A total of two substantially varying k\({}_{\rm s}\) of the rough bed from their study have been employed for our testing, 0.0001 m (for Case 1.1.3) and 0.0158 m (for Case 4.1.1), where the utilised vegetations had elliptical cross-sections with a longer diameter of 0.095 cm and a shorter diameter of 0.07 cm at a height of 16.5 cm. All parameters from the literature-reported data are presented in Table 1, in which they are used as the input conditions for the presented model's calculation. Figure 2 shows the comparison of the theoretical models with measurements from [PERSON] et al. 's [3] experimental study. Away from the bed at the free-water zone, the presented model presents some discrepancies relative to the measurements, which can be a result of the error range in measurements in this zone by [PERSON] et al. [3]--as observed from the oscillating measured data. Apart from this discrepancy, the proposed model captures the rest of the measured regions reasonably. At the near-bed region (i.e., at the inner vegetation region), both [PERSON] et al. [3] and the modified log law models have been presented and compared. Using the benchmark of measurements, the modified log law model calculated the velocity profile with better accuracy compared to [PERSON] et al. 's [3] model in the inner vegetation region, where the modified log law model improved velocity estimation accuracy by nearly 70% compared to the latter. This closer correspondence has been consistently shown for both Cases 1 and 2, which demonstrated the capability of the modified log law model. More specifically, [PERSON] et al. 's [3] three-layer model underestimated the point of vegetation's inner-outer layer interface, which has been improved by the employment of \(\xi_{1}\) in the modified log law approach. To further explore the capability of the proposed model with the modified log law, the experimental runs by [PERSON] and [PERSON] [22] have been utilised in this study (refer to the comparison in Figure 3). Their test conditions spread across 1.7 to 52.0 l/s of flow, and this can further investigate the model's adaptability to fast and slow flows. Additionally, the utilised vegetation exhibited truncated cone shapes, which can assess the competence of the bespoke model to represent the flow field via different vegetation shapes. As observed in Figure 3, the calculation presents good correspondence to the measured data in all runs. It can be further observed that the computed result shows a higher discrepancy relative to the laboratory measurement at Run B's free-water layer.