# Wavelet Packet Method for Locating Critical Slip Surface Using the Strength Reduction Method
## 1 Introduction  
The finite element method is the most widely used method for slope stability analysis (SSA). Its strong ability in nonlinear analysis gives the finite element method (FEM) great potential in geotechnical engineering. It was first introduced to geotechnical engineering by [PERSON] and [PERSON] [1]. In SSA, the factor of safety (FOS) is defined as the ratio of the actual soil shear strength to the minimum shear strength required to prevent failure [2]. As [PERSON] [3] pointed out, the FOS is the factor that continuously reduces the soil shear strength to bring the slope to the verge of failure. This method, which brings slope into a limit equilibrium state (LES) by reducing the shear strength of the soil, is called the strength reduction method (SRM), was first used in SSA by [PERSON] et al. [4], and has been studied by [PERSON] [5], [PERSON] [6], [PERSON] et al. [7], [PERSON] et al. [8], [PERSON], and others. [PERSON] et al. systematically summarized the advantages of the finite element method-strength reduction method compared with limit equilibrium methods (LEM) [9] in SSA; more detail can be found in [10]. In the past few decades, various other effective numerical methods have also been developed for SSA, such as the mesh-free methods [11], the virtual element method [12], the boundary element method [13; 14], the smoothed particle hydrodynamics [15], the material point method [16; 17], the discrete element method [18], the numerical manifold method [19] and other numerical methods [20; 21]. Great progress also has been made in the research of three-dimensional SSA [22; 23; 24; 25; 26; 27; 28], but two-dimensional SSA is still a hot academic issue [12; 29; 30; 31; 32; 33; 34; 35; 36; 37]. In practice, there are two crucial issues in the SSA: calculation of the FOS and location of the critical slip surface (CSS). However, research into SRM is mainly concentrated on thecomputation of the FOS rather than locating the CSS. By bringing the slope to LES, the FOS agreeing with the LEM can be easily obtained. In contrast, the CSS of a slope is usually determined approximately through visualization techniques, such as the distribution of maximum shear-strain rates [4], shear-strain increment distributions [8], yielding regions of the slope [38], the velocity field at collapse [39], or other technical measures. However, to this day, there is no rigorous mathematical method to locate the CSS. For two-dimensional SSA, [PERSON] et al. [40] developed a new approach to search the CSS for any yield criterion by using the equivalent plastic strain field in the LES, in which the least-square method is used to fit the CSS. However, for a complicated slope composed of multiple materials, its CSS shape may contain several turning points [41], and the least-square method will not be applicable in this case. For this reason, the wavelet method [42] was proposed and applied to the analysis of homogeneous and multi-layer slopes, in which the maximum equivalent plastic strain point is regarded as a signal function, and the high-frequency part of the signal is filtered to obtain a smooth CSS. The results show that the wavelet method has good applicability. Wavelet analysis is a rapidly developing new field in applied mathematics and engineering [43]. It can be used for damage detection [44], time-frequency analysis [45], signal-to-noise separation [46], multi-scale edge detection [47], and so on. Wavelet packet analysis (WPA) divides the time-frequency more carefully, and its resolution of the high-frequency part of the signal is higher than that of the dyadic wavelet. In other words, the dyadic wavelet only further decomposes the low-frequency part into low-frequency and high-frequency parts, while wavelet packet decomposition further decomposes both low-frequency and high-frequency parts as shown in Figure 1. Hence, the wavelet packet has a wide range of application values [48].