# Instability Induced by Random Background Noise in a Delay Model of Landslide Dynamics

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Analysis of the Recorded Background Seismic Noise

_{0}is within ε for all the tested surrogates.

## 3. Effect of Noise on Landslide Dynamics

_{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, τ is a time delay in the position between two neighboring blocks, while a, b, and c are parameters of the cubic friction force, which according to Morales et al. [24] has the following form:

^{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}) = δ

_{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.

^{−4}to 10

^{−2}, the Hopf bifurcation curve ”moves” further, i.e., higher values of time delay or spring stiffness are needed for the instability to occur. This effect is pronounced both for the low and high values of time delay and spring stiffness, as shown in Figure 9. Such an effect for high values of time delay (3.5–5.0) and relatively low values of spring stiffness is shown in Figure 9a. These conditions are expected in strongly weathered rock masses, with loose connections between the actual moving parts of the unstable slope, where higher values of delayed interactions are expected. It is clearly seen that for constant spring stiffness and time delay, solely by increasing the noise intensity, the system under study becomes stable. The effect of noise for low values of time delay (0.75–0.80) and high values of spring stiffness is shown in Figure 9b. Such a regime corresponds to rock mass which is just slightly weathered with low deformability, where low values of delayed interaction are expected. One could see that increase of D for constant k and τ leads to the stabilization of landslide dynamics. Additionally, vice versa, the reduction of D destabilizes the dynamics of the landslide. Moreover, this effect is pronounced between 10

^{−4}and 10

^{−3}, while the further effect of D is significantly lower. This is a rather unexpected result implying a positive role of noise on slope stability compared to the influence of delay interaction or coupling stiffness.

## 4. Discussion and Conclusions

- Creeping slopes composed of highly weathered rocks, where individual unstable parts have loose interaction with high delay, are sensitive to the change of the background noise intensity. In this case, the interaction of time delay and noise intensity is particularly emphasized: an increase of noise intensity for two units of order leads to the significant stabilization of landslide activity, with the increase of time delay for 1. The interaction of noise and spring stiffness is not so significant, and approximately one order of units is smaller compared to the interaction of noise and time delay. In particular, an increase of noise intensity for two orders of units leads to the change in critical values of spring stiffness for approximately 0.1.
- Creeping landslides composed of slightly weathered rock masses, where individual unstable parts have strong interactions, with low values of time delay, are more sensitive to the interaction of delay and spring stiffness compared to the interaction of D and τ. With the increase of D for two orders of units, critical values of k (for which bifurcation occurs) change by approximately 0.5, while the critical value of τ changes by approximately 0.03.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

_{i}= 0.001, y

_{i}= 0.001, are initial conditions.

_{1}N, and $\u2329x\left(t-\tau \right)\u232a=\underset{N\to \mathrm{\infty}}{\mathrm{lim}}\frac{1}{N}{\sum}_{i=1}^{N}{x}_{i}(t-\tau )$.

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**Figure 2.**Recorded background seismic noise, accelerograph of ETNA type on 8 April 2022 (

**a**), 17 April 2022 (

**b**), and 18 August 2022 (

**c**) [21] at Zavoj dam after WWSSN-SP filter is applied.

**Figure 3.**Results of surrogate data testing—testing of I 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 ε

_{0}for the original dataset, while the black lines denote prediction error ε 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).

**Figure 4.**Results of surrogate data testing—testing of II 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 ε

_{0}for the original dataset, while the black lines denote prediction error ε 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).

**Figure 5.**Examined an infinite chain of blocks on an inclined slope as a landslide model: k stands for the spring stiffness, while V denotes the constant background velocity.

**Figure 6.**Characteristic engineering–geological cross-section along the left coast of the Zavoj dam.

**Figure 7.**Bifurcation diagram for the interaction of time delay τ and spring stiffness k for model (3). While τ and k are varied, other parameters are being held constant: V = 0.2, a = 3.2, b = −7.2, c = 4.8, initial conditions: m

_{x}= 0.001, m

_{y}= 0.0001; s

_{x}= s

_{y}= 0.05, D = 0.0001. Velocity time series for points a (k = 10, τ = 1) and b (k = 10, τ = 3), both for the models (1) and (3), are given in Figure 8.

**Figure 8.**Time series of displacements before and after the bifurcation curve in Figure 7. Upper diagrams show velocity time series for the parameters denoted by point a in Figure 7 (k = 10, τ = 1), (

**a**) for the starting model (1), and (

**b**) for the corresponding mean-field model. Lower diagrams show velocity time series for the parameters denoted by point b in Figure 7 (k = 10, τ = 3). (

**c**) for the starting model (1), and (

**d**) for the corresponding mean-field model. It is clear that both models exhibit qualitatively the same dynamics. While τ and k are varied, other parameters are being held constant: V = 0.2, a = 3.2, b = −7.2, c = 4.8, initial conditions: m

_{x}= 0.001, m

_{y}= 0.0001; s

_{x}= s

_{y}= 0.05, D = 0.0001, x = 0.001, y = 0.001.

**Figure 9.**Effect of noise intensity on dynamics of the observed system (3): (

**a**) for high values of time delay τ and low values of spring stiffness k and (

**b**) for low values of time delay τ and high values of spring stiffness k. The constant values of other parameters are the same as in Figure 7.

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**MDPI and ACS Style**

Kostić, S.; Vasović, N.; Todorović, K.; Prekrat, D.
Instability Induced by Random Background Noise in a Delay Model of Landslide Dynamics. *Appl. Sci.* **2023**, *13*, 6112.
https://doi.org/10.3390/app13106112

**AMA Style**

Kostić S, Vasović N, Todorović K, Prekrat D.
Instability Induced by Random Background Noise in a Delay Model of Landslide Dynamics. *Applied Sciences*. 2023; 13(10):6112.
https://doi.org/10.3390/app13106112

**Chicago/Turabian Style**

Kostić, Srđan, Nebojša Vasović, Kristina Todorović, and Dragan Prekrat.
2023. "Instability Induced by Random Background Noise in a Delay Model of Landslide Dynamics" *Applied Sciences* 13, no. 10: 6112.
https://doi.org/10.3390/app13106112