# Instability Induced by Random Background Noise in a Delay Model of Landslide Dynamics
## 3 Effect of Noise on Landslide Dynamics  
We assume that landslide dynamics could be modeled as the movement of the n interconnected blocks (Figure 5):  
\[ \begin{array}{l}dx_{i}(t)=y_{i}(t)dt\\ dy_{i}(t)=-\Big{[}a(V+y_{i}(t))^{3}-b(V+y_{i}(t))^{2}+c(V+y_{i}(t))\Big{]}dt+\\ (aV^{3}-bV^{2}+cV)dt+\sum_{j=1}^{N}k_{1}\big{(}x_{j}(t-\tau)-x_{i}(t)\big{)}dt+ \sqrt{2D}dW_{i}\end{array}  \tag{1}\]  
where \(x_{i}\) and \(y_{i}\) are displacement and velocity of the i-th block, respectively, \(k\) is the spring stiffness (could be considered to qualitatively correspond to the deformable properties of the surrounding soil), \(V\) is the background velocity, \(\tau\) is a time delay in the position  
Figure 4: Results of surrogate data testing—testing of \(\Pi\) hypothesis. The set of curves represents the distribution of prediction errors for the original and 20 surrogate datasets (for 100 prediction steps). The red line denotes prediction error \(\varepsilon_{0}\) for the original dataset, while the black lines denote prediction error \(\varepsilon\) for the surrogate datasets. The first row corresponds to the results of surrogate data testing of the recorded background noise before (**a**) and after (**b**) the event on 17 April 2022 (Figure 2a), the second row corresponds to the results of surrogate data testing of the recorded background noise before (**c**) and after (**d**) the event recorded on 8 April 2022 (Figure 2b), while the third row denotes the results of the analysis of the noise before (**e**) and after (**f**) the event recorded on 18 August 2022 (Figure 2c). between two neighboring blocks, while \(a\), \(b\), and \(c\) are parameters of the cubic friction force, which according to [PERSON] et al. [24] has the following form:  
\[Fc(V)=3.2V^{3}-7.2V^{2}+4.8V \tag{2}\]  
One should note that previous studies suggested that landslides exhibit frictional velocity-weakening [25; 26]. Additionally, the assumed cubic friction law qualitatively corresponds to the Coulomb-Mohr nonlinear \(\tau\)-u relationship. Terms \((2D)^{1/2}dW_{i}\) represent stochastic increments of the independent Wiener process, i.e., \(dW_{i}\) satisfy: \(E(dW_{i})=0\), \(E\) (\(dW_{i}dW_{j})=\delta_{ij}dt\), where \(E()\) denotes the expectation over many realizations of the stochastic process and \(D\) is the intensity of additive local noise. In the present paper, we examine the effect of the persistent background seismic noise in order to determine whether this continuous factor has any effect on the landslide dynamics. The effect of noise on the stability of the model dynamics is examined through the effect of its intensity as the most important property of random noise. It should be noted that the suggested model (1) represents a phenomenological model of landslide dynamics, with friction, displacement delay, spring stiffness, and random background noise as the main affecting factors. The effect of soil density is not explicitly included in the suggested model and could be considered constant for the purpose of the performed analysis. One should note that model presented in Figure 5 is just an illustration; blocks within a model are globally (all-to-all) coupled. The idea of representing landslide dynamics as a dynamics of a spring-block model sliding down the slope came from [PERSON] [27], and it was subsequently examined by [PERSON] et al. [24]. Hence, model (1) is inspired by the original model by [PERSON] [25] but with the explicitly included time delay and the effect of background noise. The combined effect of time delay and noise has been previously analyzed and reported as justified for different dynamical systems [28]. Additionally, some previous studies have indicated that vibrations could lead to frictional weakening [29]. We consider that the proposed model is close to natural conditions since it includes two main properties of the natural slope: delayed interaction among different parts of the slope and the influence of the background seismic noise.