# Safety Evaluation of a Water-Immersed Bridge Against Multiple Hazards via Machine Learning
## 4 Simulation of Water Depth, Stream Velocity, and Scouring Depth  
When calculating bridge scouring depth, two parameters must be obtained: water depth and stream velocity. Because the water depth and the stream velocity should be observed over time, normally they are calculated through simulation. Presuming that the mean value of water depth and stream velocity in different return years can be regarded as a log-normal distribution with three parameters, the water depth and stream velocity of the same return year can also be regarded as a log-normal distribution. Based on the above assumptions and the information revealed in the design drawing (i.e., the discharge rate for return periods of 50 and 100 years), the mean value of water depth \(y_{200}\) and stream velocity \(V_{200}\) for a 200-year return period were obtained. Furthermore, the coefficients of variation of water depth and stream velocity were presumed identical under different return periods. In this way, we may acquire the probability distributions of water depth and stream velocity for 50, 100, and 200 year return periods, as per the detailed calculation method. Provided below is the hydrological frequency equation:  
\[\hat{x}=\overline{x}+K_{T}\cdot S,\ K_{T}=\frac{\exp(\sigma_{y}\cdot t-\frac{ \sigma_{y}^{2}}{2})-1}{\left[\exp(\sigma_{y}^{2})-1\right]^{1/2}}, \tag{4}\]  
where \(\overline{x}\) and \(S\) represent mean value and standard deviation of the sample, respectively. \(K_{T}\) represents frequency factor and is calculated below;  
\[\sigma_{y}=\left[\ln(z^{2}+1)\right]^{1/2}; \tag{5}\]  
Figure 9: Illustration of the calculated added mass (kg). \[z=\frac{1-w^{2/3}}{w^{1/3}}; \tag{6}\]  
\[w=\frac{-C_{s}+\sqrt{C_{s}{}^{2}+4}}{2}; \tag{7}\]  
where \(C_{s}\) means the skewness. Parameter \(t\) in Equation (4) can be acquired by the formula below:  
\[\left\{ \begin{array}{l}t=-W+\frac{C_{0}+C_{1}W+C_{2}W^{2}}{1+d_{1}W+d_{2}W^{2 }+d_{3}W^{3}},\ W=\sqrt{-2\ln(1-P)},\ \text{if}\ P>0.5\\ t=W-\frac{C_{0}+C_{1}W+C_{2}W^{2}}{1+d_{1}W+d_{2}W^{2}+d_{3}W^{3}},\ W=\sqrt{-2 \ln(P)},\ \text{if}\ P\leq 0.5\end{array} \right., \tag{8}\]  
where \(P\) is the inverse of the return period, \(C_{0}=2.515517\), \(C_{1}=0.802853\), \(C_{2}=0.010328\), \(d_{1}=1.432788\), \(d_{2}=0.189269\), and \(d_{3}=0.001308\). Through calculations with the aforesaid equation, the average water depth \(y_{200}\) as 5.9525 m and flowing speed \(V_{200}\) as 2.407 m/s for the 200 year regression period were obtained. To calculate scour depth, especially to obtain the uncertainty of scour, a simulation of water level and stream velocity was necessary. [PERSON] et al. (2019) indicated that the correlation coefficient between water depth and stream velocity was about 0.92. Based on this, a Copula function was used in this research for generating random samples for water depth and stream velocity. After acquiring the water depths and stream velocities for 50, 100, and 200 year return periods, they were revolved in the formula proposed by [PERSON] and [PERSON] (2018), respectively, in order to acquire the corresponding distribution of scouring depth, as shown below. \[d_{s}=K_{s}K_{t}K_{G}K_{gb}K_{t}K_{d}. \tag{9}\]  
In the formula, \(d_{s}\) represents the scour depth (m), \(K_{s}\) means foundation shape factor, \(K_{t}\) means time factor, \(K_{G}\) means approach channel geometry factor, \(K_{\theta}\) means foundation alignment factor, \(K_{gb}\) means foundation size factor, \(K_{I}\) means flow intensity factor, and \(K_{d}\) means sediment size factor. Among these factors, \(K_{gb}\) poses the most significant impact. Detailed calculations of \(K_{gb}\) in this article is provided below. Based on the classification of caisson width and elevation of river bed, a pier-equivalent width was derived for calculating the bridge scour (2018). [PERSON] (2018) utilized an optimization approach to update the parameter weights for the pier-equivalent width formula proposed by [PERSON] and [PERSON] (2018), as displayed below.