# Scouring Depth Assessment Downstream of Weirs Using Hybrid Intelligence Models
## 2 Materials and Methods  
### Laboratory Data for Estimating Scour Depth  
Sour depth downstream of weirs depends on bed material, flow conditions, weir geometry and tailwater depth [7; 11; 42]. Hence, the following relation can be used for the scour depth estimation:  
\[d_{s}=f\Big{(}\rho,\
u,h_{0},g,h_{t},U_{0},\rho_{s},d_{50},\sigma_{g},U_{c},b,z \Big{)} \tag{1}\]  
where, \(\rho\) is the density of water, \(\
u\) stands for kinematic viscosity of fluid, \(h_{0}\) refers to average approach flow depth, \(g\) is the acceleration of gravity, \(h_{t}\) embodies tailwater depth, \(U_{0}\) refers to average approach flow velocity, \(\rho_{s}\) is bed particle density, \(d_{50}\) denotes mean bed particle size, \(\sigma_{g}\) is the standard deviation of bed particle size, \(U_{c}\) identifies critical average approach flow velocity, \(b\) indicates weir width, \(z\) stands for weir height and \(d_{s}\) is scouring depth. To estimate the scour depth, the non-dimensional parameters provide more performance in comparison with dimensional ones [43; 44]. A number of investigations assessed the impact of the non-dimensional parameters (e.g., \(\frac{d_{50}}{h_{t}},\frac{z}{h_{t}},\frac{U_{0}}{U_{c}}\)) on scouring depth downstream of weirs. Several studies indicated that the changes in the magnitude of approach flow and tailwater have significant impact on scour depth [1; 45; 46]. [PERSON] et al. [2] found that downstream slope and flow intensity have a considerable effect on downstream scouring depth. [PERSON] et al. [1] attained that scour depth increased with decreasing tailwater depth (\(\frac{z}{h_{t}}\)) and increasing flow intensity (\(\frac{U_{0}}{U_{c}}\)). Otherstudies showed that the weir width (\(\frac{b}{h_{t}}\)) also has significant effect on scouring depth downstream of weirs [7; 8]. Hence, the following equation is used based in dimensional analysis [1]:  
\[\frac{d_{s}}{h_{t}}=f\Big{(}\frac{d_{50}}{h_{t}},\frac{z}{h_{t}},\frac{U_{0}}{U_ {C}}\Big{)} \tag{2}\]  
where, \(\frac{d_{50}}{h_{t}},\frac{U_{0}}{U_{C}},\frac{z}{h_{t}}\), respectively quantify the effect of bed particle size, flow condition, and tailwater depth on scouring depth. A dataset with 186 experimental data is gathered from four published papers [15; 47; 48; 49] with different conditions of surveys to assess the capability of the new hybrid ANFIS techniques. In the following, the summary of gathered datasets is briefly discussed: [PERSON] [47] experimentally examined the scour depth downstream of weirs using a rectangular flume, where the width of weir and flume was 0.5 m. In all experimental runs, the difference between flow depth upstream and downstream of weir was kept as 1 m. Different bed properties with \(d_{50}\) values of 9.1, 14.2, 21 and 36.2 mm were used. Additionally, the \(\frac{Q}{b}\) was changed in the range of 0.001 and 0.083 \(\frac{m^{2}}{s}\). The observed scour depths were reported between 0.055 and 0.22 m. [PERSON] and [PERSON] [49] performed other laboratory experiments to assess the scour depth downstream of weir. They varied the specific discharge between 1.2 and 13.4 \(\frac{m^{2}}{s}\). Several widths of rectangular channel were considered in the range of 0.4 to 3.5 m. The \(d_{50}\) values were in the range of 0.019-0.1 m. The scour depth was observed within 0.4-3.5 m. [PERSON] [15] conducted laboratory experiments to investigate local scour downstream of weirs. They designed two types of weirs with different \(\frac{b}{B}\) ratio of 0.3 and 0.6 (\(B\) is the flume width). The median diameters of bed particle were in the range 4.1 to 17.6 mm. In all experimental runs, discharge values were considered between 0.0167 and 0.167 \(\frac{m^{2}}{s}\) which caused the scour depth within 0.045-0.280 m. Similar investigation has also been carried out by [PERSON] and [PERSON] [48] to assess scour depth downstream of weirs. The \(\frac{b}{B}\) values were similar to previous study although \(d_{50}\) values were in the range of 9.1 to 11.5 mm. [PERSON] et al.