## 2 Previous Studies for Estimation of Stability Number  
A breakwater is a port structure that is constructed to provide a calm basin for ships and to protect port facilities from rough seas. It is also used to protect the port area from intrusion of littoral drift. There are two basic types of breakwater: rubble mound breakwater and vertical breakwater. The cross section of a typical rubble mound breakwater is illustrated in Figure 2. To protect the rubble mound structure from severe erosion due to wave attack, an armor layer is placed on the seaward side of the structure. The stability of the armor units is measured by a dimensionless number, so-called stability number, which is defined as  
\[N_{s}=\frac{H_{s}}{\Delta D_{n50}}, \tag{1}\]  
where \(H_{s}\) is the significant wave height in front of the structure, \(\Delta=\rho_{s}/\rho_{w}-1\) is the relative mass density, \(\rho_{s}\) and \(\rho_{w}\) are the mass densities of armor unit and water, respectively, and \(D_{n50}\) is the nominal size of the armor unit. As shown in Equation (1), the stability number is defined as the ratio of the significant wave height to the size of armor units. A larger stability number, therefore, signifies that the armor unit with that size is stable against higher waves, that is, the larger the stability number, the more stable the armor units against waves. To estimate the stability number, it is required to determine the relationship between the stability number and other variables which would describe the characteristics of waves and structure. [PERSON] (1954) proposed an empirical formula:  
\[N_{s}=(K_{D}\text{cot}\alpha)^{1/3}, \tag{2}\]  
where \(\alpha\) is the angle of structure slope measured from horizontal, and \(K_{D}\) is the stability coefficient which depends on the shape of the armor unit, the location at the structure (_i.e._, trunk or head), placement method, and whether the structure is subject to breaking wave or non-breaking wave. The [PERSON] formula is simple, but it has been found to have a lot of shortcomings. To solve the problems of the [PERSON] formula, [PERSON] (1954) conducted an extensive series of experiments including the parameters which have significant effects on armor stability. Based on the experimental data, empirical formulas were proposed by [PERSON] (1954); [PERSON] (1954) as follows:  
Figure 2: Typical cross-section of rubble mound breakwater. \[N_{s}=\frac{1}{\sqrt{\xi_{m}}}\left[6.2P^{0.18}\left(\frac{S}{\sqrt{N_{w}}}\right)^ {0.2}\right]\text{ for }\xi_{m}<\xi_{c}, \tag{3a}\] \[N_{s}=1.0P^{-0.13}\left(\frac{S}{\sqrt{N_{w}}}\right)^{0.2}\sqrt{\text{ cot}\alpha}{\xi_{m}}^{p}\text{ for }\xi_{m}\geqslant\xi_{c}, \tag{3b}\]  
where \(\xi_{m}=\text{tan}\alpha/\sqrt{2\pi{H_{s}}/{gT_{m}}^{2}}\) is the surf similarity parameter based on the average wave period, \(T_{m}\), \(\xi_{c}=\left(6.2P^{0.31}\sqrt{\text{tan}\alpha}\right)^{1/(P+0.5)}\) is the critical surf similarity parameter indicating the transition from plunging waves to surging waves, \(P\) is the permeability of the core of the structure, \(N_{w}\) is the number of waves during a storm, and \(S=A/D_{n50}^{2}\) (where \(A\) is the eroded cross-sectional area of the armor layer) is the damage level which is given depending on the degree of damage, e.g., onset of damage or failure. On the other hand, with the developments in machine learning, various data-driven models have been developed based on the experimental data of [PERSON] [26]. A brief summary is given here only for the ANN models. [PERSON] _et al_. [3] constructed an ANN by using randomly selected 100 experimental data of [PERSON] [26] for training the network. The total number of experimental data excluding the data of low-crested structures was 579. In the test of the ANN, they additionally used the 30 data of [PERSON] _et al_. [28]. They employed six input variables: \(P\), \(N_{w}\), \(S\), \(\xi_{m}\), \(h/H_{s}\), and the spectral shape parameter \(SS\), where \(h\) is the water depth in front of the structure.