## 2 Material and Methods
### Landslide Prediction Models
#### 2.4.2 Whale Optimization Algorithm (WOA)  
The WOA algorithm is a meta-heuristic optimization process that mimics humpback whales [103], whose brain has the same spindle cells in the cortex as humans' [114]. WOA is inspired from the sole hunting method of colossal humpback whales [103] and is known as the bubble-net feeding method [115]. This method is based on the unique pattern for catching far more fishes at once. A group of whales come together and dive beneath the school of fishes by producing high pitch calls. At that moment, the fishes flee to the surface, where the whales release the distinctive bubbles along a circle of 9-shaped trail in an upward shrinking spiral around the fishes as an obstacle so that the fishes cannot swim, as shown in Figure 5. Finally, the whales ascend to the surface with their mouths open by the helix-shaped movement when the whale leader emits a hunting call [114]. The WOA can be formulated as following steps:  
(1) Encircling prey Encircling prey in the best position is the initial step in hunting prey. The humpback whales look for the suitable position of prey and renew their positions based on the optimal known solution. The solution can be represented by Equations (5) and (6):  
\[\overset{\rightarrow}{D}=|\overset{\rightarrow}{CX}^{\ast}(t)-\overset{ \rightarrow}{X}(t)| \tag{5}\]  
\[\overset{\rightarrow}{X}(t+1)=\overset{\rightarrow}{X}^{\ast}(t)-\overset{ \rightarrow}{A.D} \tag{6}\]  
where \(\overset{\rightarrow}{X}^{\ast}\) and \(\overset{\rightarrow}{X}\) are two position vectors and \(\overset{\rightarrow}{X}^{\ast}\) indicates the optimal solution obtained so far, \(t\) is the current iteration. Furthermore, \(\overset{\rightarrow}{A}\) and \(\overset{\rightarrow}{C}\) specify the coefficient vectors that are given by:  
\[\overset{\rightarrow}{A}=2\overset{\rightarrow}{a.}\overset{\rightarrow}{ r}-\overset{\rightarrow}{a} \tag{7}\]  
\[\overset{\rightarrow}{C}=2\overset{\rightarrow}{r} \tag{8}\]  
where \(a\) is a linearly decreasing variable and \(r\) is a random vector in the range from 0 to 1. (2) Bubble-net attacking method (exploitation phase). This exploitation phase includes two processes given by:  
(a) Shrinking encircling prey  
In this mechanism, the values of \(\overset{\rightarrow}{a}\) and \(\overset{\rightarrow}{A}\) decrease in Equations (7) and (8). The vector \(\overset{\rightarrow}{A}\) has a random value between [-a, a] and declines from 2 to 0. The new position can be reached using the original and current optimal agent positions. (b) Spiral position updating  
Let \((X,Y)\) and \((X^{\ast},Y^{\ast})\) indicate the positions of the whale and prey and \(\overset{\rightarrow}{D}^{\prime}\) the distance between them; the whales mimic the helix-shaped movement according to spiral equation as follows:  
\[\overset{\rightarrow}{X}(t+1)=\overset{\rightarrow}{D}^{\prime}.e^{bl}.\cos( 2\pi l)+\overset{\rightarrow}{X}^{\ast}(t) \tag{9}\]  
where \(\overset{\rightarrow}{D}^{\prime}=|\overset{\rightarrow}{X}^{\ast}(t)- \overset{\rightarrow}{X}(t)|\) is constant, and \(l\) is a random variable between [-1, 1], and b is constant. A mathematical model is required for the updating of the whales' position in optimization process  
Figure 5: Bubble-net feeding behavior of humpback whales [103]. because they swim towards the prey in two mechanisms of shrinking circle and spiral-shaped path. The updating of the whales' position is formulated as follows:  
\[\overrightarrow{X}(t+1)=\left\{ \begin{array}{cc}\overrightarrow{X}^{*}(t)- \overrightarrow{A}.\overrightarrow{D}&if\ p<0.5\\ \overrightarrow{D}.\overrightarrow{e^{bl}}.\cos(2\pi l)+\overrightarrow{X}(t)& if\ p\geq\ 0.5\end{array} \right. \tag{10}\]  
where \(p\) is a random variable ranging from 0 to 1. (3) Search for prey (exploration phase)  
This phase can perform a global search when the whales search a randomly chosen agent. This mechanism is used when \(\overrightarrow{A}\) has a random value larger than 1 or less than \(-1\).