## 3 Analysis of Datasets: Dimensional Analysis and Experimental Tests
### Dimensional Analysis  
Three-dimensional scour rates depend on several factors, namely pipeline geometry, wave properties, embedded depth of pipeline compared to the initial level of seabed, and physical properties of seabed sediments (e.g., [PERSON] et al. (2018), [PERSON] and [PERSON] (2018), [PERSON] et al. (2018), [PERSON] and [PERSON] (2018)). Hence, the following functional relationship can be formulated:  
\[\chi(V_{H},V_{R},V_{L},e,\ D,U_{w},T,d_{50},\phi,\alpha,\rho,\rho_{s},\mu,g)=0 \tag{1}\]  
in which \(V_{H}\) is the scour propagation velocity along the longitudinal (axial) direction of the pipeline; \(V_{L}\) and \(V_{R}\) are the scour propagation velocities at the left- and right-hand shoulders of the pipeline, respectively; \(e\) is the embedment depth; \(D\) is the pipe diameter; \(U_{w}\) is the wave orbital velocity at the seabed; \(d_{50}\) is the median grain size of the seabed sediment; \(T\) is the wave period; \(\phi\) is the angle of repose of seabed sediments, \(\alpha\) is the flow incident angle to the pipeline (angle of attack); \(\rho\) is the density of water; \(\rho_{s}\) is the density of the seabed sediment; \(\mu\) is the dynamic viscosity of water; and \(g\) is the acceleration due to gravity. Previous studies revealed that the implementation of various AI models for the prediction of scour rates below pipelines by considering dimensionless variables would lead to results with highly convincing performance in comparison to those where dimensional variables are used. What is more, the use of non-dimensional parameters can ameliorate scale effects from experimental data on the performance of AI techniques (e.g., [PERSON] and [PERSON] (2018), [PERSON] (2018), [PERSON] and [PERSON] (2018), [PERSON] et al. (2018)). In this way, by using the Buckingham theorem, the non-dimensional analysis was performed. At first, \(\rho\), \(D\), and \(U_{w}\) were assigned as repeating variables, and then \(11\) non-dimensional parameters (\(\prod_{1}\), \(\prod_{2}\), \(\prod_{3}\), \(\cdots\), \(\prod_{10}\), \(\prod_{11}\)) were obtained:  
\[\chi_{1}\left(\frac{V_{H}}{U_{w}},\frac{V_{R}}{U_{w}},\frac{V_{L}}{U_{w}}, \frac{e}{D},\ \frac{U_{w}T}{D},\frac{d_{50}}{D},\phi,\alpha,\frac{\rho_{s}}{\rho},\frac{ \rho U_{w}D}{\mu},\frac{\wp D}{U_{w}^{2}}\right)=0 \tag{2}\]  
where \(\prod_{5}\) and \(\prod_{10}\) are the Kelugan-Carpenter (KC) and Reynolds numbers of the pipeline due to the regular wave (\(Re_{w}\)), respectively. According to [PERSON] et al. (2018), some \(\prod\)-variables in Equation (2) could be rearranged and clustered as follows:  
\[\Pi^{\prime}_{7}=tan\phi;\ \Pi^{\prime}_{8}=sina;\ \Pi^{\prime}_{9}=\frac{\rho_{s} }{\rho}-1 \tag{3}\]  
\[\Pi^{\prime}_{11}=\frac{1}{\prod_{11\cdots\Pi^{\prime}_{9}\cdot\Pi_{6}}}=\frac {U_{w}^{2}}{g[(\rho_{s}/\rho-1)]d_{50}} \tag{4}\]  
\[\Pi^{\prime}_{1}=\Pi^{\prime}_{7}\cdot\sqrt{\Pi^{\prime}_{11}\cdot\left(\frac {\Pi_{1}}{\Pi_{6}}\right)^{2}}=\frac{V_{H}\cdot D\cdot tan\phi}{\sqrt{g[(\rho_ {s}/\rho-1)d_{50}^{3}]}} \tag{5}\]\[\Pi_{2}^{\prime}=\Pi_{7}^{\prime}\cdot\sqrt{\Pi_{11}^{\prime}\cdot\left(\frac{\Pi_{ 2}}{\Pi_{6}}\right)^{2}}=\frac{V_{R}\cdot D\cdot tan\phi}{\sqrt{g[(\rho_{s}/ \rho-1)d_{50}^{3}]}} \tag{6}\]  
\[\Pi_{3}^{\prime}=\Pi_{7}^{\prime}\cdot\sqrt{\Pi_{11}^{\prime}\cdot\left(\frac{ \Pi_{3}}{\Pi_{6}}\right)^{2}}=\frac{V_{L}\cdot D\cdot tan\phi}{\sqrt{g[(\rho_ {s}/\rho-1)d_{50}^{3}]}} \tag{7}\]  
in which \(\Pi_{1}^{\prime}\), \(\Pi_{2}^{\prime}\), and \(\Pi_{3}^{\prime}\) can be indicated with \(\mathrm{V}_{H}^{*}\), \(\mathrm{V}_{R}^{*}\), and \(\mathrm{V}_{L}^{*}\), respectively, and \(\Pi_{11}^{\prime}\) is the Shields' parameter \(\theta_{w}\), which plays a key role in scour propagation below pipelines. Therefore, the functional relationship (2) can be reduced as:  
\[\chi_{2}\left(V_{H}^{*},V_{R}^{*},V_{L}^{*},\frac{e}{D},\ Kc,sina,Re_{w}, \theta_{w}\right)=0 \tag{8}\]  
In fact, [PERSON] et al. [9] found that the Reynolds number of pipeline did not influence the scouring process, and \(Re_{w}\) can be removed from Equation (8).