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Colored noise in river level
oscillations as triggering factor for
unstable dynamics in a landslide
model with displacement delay
Srđan Kostić
1
,
2
* and Milan Stojković
3
1
Jaroslav Černi Water Institute, Geology Department, Belgrade, Serbia,
2
Faculty of Technical Sciences,
University of Novi Sad, Novi Sad, Serbia,
3
The Institute for Artificial Intelligence Research and Development
of Serbia, Belgrade, Serbia
In the present paper we examine the effect of the noise in river level oscillation on
the landslide dynamics. The analysis is conducted in several phases. In the first
phase, we analyze the multi-annual level oscillation of the Kolubara and the Ibar
river (Serbia). Based on the observed dataset, we suggest a deterministic model for
the river level oscillation with the additional contribution of the noise part, which
we confirm to have the properties of colored noise. In the second phase of the
research, we introduce the influence of the river-level oscillation, with the
included effect of colored noise in the spring-block delay model of landslide
dynamics. Results of the research indicate conditions under which the effect of
river noise has both stabilizing and destabilizing effects on the landslide dynamics.
The effect of noise intensity Dand correlation time εis systematically analyzed in
interaction with delayed interaction, spring stiffness and friction parameters. It is
determined that the landslide dynamics is sensitive to the change of noise intensity
and that the increase of noise intensity leads to onset of unstable landslide
dynamics. On the other hand, results obtained indicate that the examined
model of landslide dynamics is rather robust towards the change of correlation
time ε. Interaction of this parameter and some of the friction parameters leads to
stabilization of landslide dynamics, which confirms the importance of the
influence of the noise color in river level oscillations on the landslide dynamics.
KEYWORDS
landslide, spring-block model, colored noise, noise intensity, correlation time, delayed
interaction, bifurcation
1 Introduction
The main external triggering factors of a slope instability (e.g., landslides) commonly
depend on the corresponding surroundings of the potentially unstable feature itself. For
example, in active seismic regions, (re)activation of landslides often occurs due to the
dynamic impact; in wet and rainy areas, the rainfall is the primary contributing factor of the
slope instability. In case when slope actually represents a river bank, activation of slope
instability is caused by the regime of river level oscillations. In any of these cases, it is
assumed that the geological composition and slope geometry provide initial conditions for
the possible occurrence of landslides, i.e., existence of weak rock masses, with impermeable
bedrock at a reasonable depth and steep inclination of the slope. Depending on the particular
values of these slope properties, initial conditions could be close or far from the instability
point. If initial conditions are far from the instability point, then the impact of any of the
OPEN ACCESS
EDITED BY
Yi Xue,
Xi’an University of Technology, China
REVIEWED BY
Annette Witt,
Max Planck Society, Germany
Mohammad Hadi Fattahi,
Islamic Azad University, Iran
*CORRESPONDENCE
Srđan Kostić,
srdjan.kostic@jcerni.rs
RECEIVED 26 July 2023
ACCEPTED 24 October 2023
PUBLISHED 08 November 2023
CITATION
KostićS and StojkovićM (2023), Colored
noise in river level oscillations as
triggering factor for unstable dynamics in
a landslide model with
displacement delay.
Front. Earth Sci. 11:1267225.
doi: 10.3389/feart.2023.1267225
COPYRIGHT
© 2023 Kostićand Stojković. This is an
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Frontiers in Earth Science frontiersin.org01
TYPE Original Research
PUBLISHED 08 November 2023
DOI 10.3389/feart.2023.1267225
aforementioned external triggering factors should be strong enough
to push the slope over the stability limit. In such cases, the strength
of the external factor could be measured by the intensity, frequency
or duration of the event. On the other hand, if initial conditions of
the slope are close to instability point, then even a small external
force could be sufficient to trigger the landslide. Both from the
theoretical and practical viewpoint, this case is more interesting,
since small disturbing external effect is ubiquitous and ever-present,
so one may ask two questions: 1) what is the minimum threshold of
this external force required to activate the instability? 2) in what state
is the slope most sensitive to this small external effect? In the present
paper, we try to provide answers to these questions for the following
scenario: influence of the noise in river level oscillations on the
occurrence of unstable landslide dynamics. In accordance with this
main topic of the paper, we postulate the starting hypotheses of our
research: i) river level oscillations could induce slope instability; ii)
noise is the main constituent part of the river level oscillations; iii)
type of noise and noise properties have significant effect on landslide
dynamics.
Regarding the first starting hypothesis, there are many previous
studies that confirmed the river level oscillation as the main
contributing factor of landslide triggering. In particular, Abam
(1993) determined several main mechanisms of the riverbank
failures: rotational, translational, overhang/toppling and flow
mechanism, which commonly occur at the early stages of lowering
of channel water level. Dapporto et al. (2001) determined two
dominant mechanisms of riverbank failure, namely, slab-type and
alcove-shaped sliding failures, depending on different river stages.
Ujvari et al. (2009) proposed a model for slope failure evolution based
on the investigation of the Danube riverbank stability at Dunaszekcso.
In their work they compared geodetic datasets and field observations
with the timing of rainfall and water level changes of the Danube and
concluded that perched water table, among others, was responsible for
landslide triggering. Duong et al. (2014) examined the variations of
the riverbank stability with the river level fluctuations, for the case
study of the Red River of Hanoi (Vietnam). Based on the research
conducted, they concluded that the pore water pressure and the rate of
the river level change are the most important factors affecting the
riverbank stability when hydraulic conductivity of the soil is greater
than 10
−6
m/s. Liang et al. (2015) examined the influence and
sensitivity of river level fluctuations and climatic factors on
riverbank stability. They determined that river level fluctuations
dominate riverbank collapse in the Lower River Murray (South
Australia). Chen et al. (2017) investigated the influences of
drawdown rate of river stage, initial water elevation, and riverbank
slope angle on riverbank stability due to the fall of river water level. As
a result, they suggested a model for riverbank stability which includes
the integral effect of all forces acting upon the failure plane and
tension crack. Duong and Do (2019) investigated the influence of the
fluctuation of the river water level on the shape of the riverbank
cantilever failure, i.e., the low-raterise of the river level could lead to a
mass failure, while the high-rate rise of river level may lead to the
overhanging riverbank failure. Mentes (2019) examined the
relationship between riverbank stability and hydrological processes
using in situ measurement data. As a result, an early warning system
was developed, combining the effect of the groundwater level and river
level fluctuations with the displacements measured by borehole
tiltmeters.
As could be seen from the aforementioned, the impact of the
river level fluctuations on the riverbank stability is well studied and
could be considered as expected. Nevertheless, in the present paper,
we investigate the conditions of the existing landslides along the
riverbank for which the effect of the noise in the river level
fluctuations is significant. As far as we are aware, this has not
been examined before. Motivation for this research topic comes
from two main sources. First, we have previously shown that noise
could have significant impact to the dynamics of certain dynamical
systems, under the condition that such systems are in “critical”state.
For instance, we previously showed that the background-colored
noise with low correlation time could trigger different dynamical
regimes of seismogenic fault motion, namely, steady stationary state,
aseismic creep and seismic fault motion (Kostićet al., 2020). On the
other hand, we also showed that noise could have “stabilizing’” effect
on the system’s dynamics: the increase of the intensity of the random
background noise could lead to stabilization of the landslide
dynamics (Kostićet al., 2023). Secondly, some previous studies
indicated the existence of noise in river flow fluctuations. In
particular, Vračar and Mijić(2011) investigated hydroacoustic
noise in the Danube, the Sava and the Tisza River. Results of
their research indicated that noise spectra are characterized by
wide maximums at frequencies between 20 and 30 Hz, while
spectral level of noise changes in wide limits. Tu et al. (2023)
analyzed timescale-dependence of the noise color in streamflow
across the US hydrography, using streamflow time series from
7,504 gauges. As a result, they found that daily and annual flows
are dominated by red and white spectra, respectively. In contrast to
these previous studies, in present paper we focus on the river level
fluctuations rather than flow fluctuations, since the effect of river
flow on the riverbank stability (which also includes the erosion) is
not studied in current research.
It should be emphasized that in present study we start from the
main assumption that landslide already exists along the riverbank,
but it is in a creep regime, i.e., slope is moving constantly, but with
very low velocity and, consequently, small constant displacements.
Therefore, we do not analyze conditions for the occurrence of
landslide displacement, but the conditions for the occurrence of
unstable landslide dynamics. Such scenario is very common in real
conditions: for example, the whole right bank of the Danube River,
from Belgrade (Serbia) to the HPP Iron Gate 2 is at many points
under the impact of long-lasting creeping landslides, whose activity
is for many decades controlled by the regime of HPP Iron Gate 1 and
Iron Gate 2.
The presented research is structured as follows. Firstly, we analyze
the river level fluctuations for two case studies: the Kolubara River
(Beli Brod hydrological station) and the Ibar River (Lopatnica Lakat
hydrological station). For these examples, we derive deterministic
oscillatory model, with the added noise effect. Secondly, we confirm
the existence of colored noise in river level oscillations. In the third
part of the research, we incorporate the effect of the colored noise in
the previously suggested general model of landslide dynamics.
Dynamics of such model is further examined, and three main
points are addressed: 1) the conditions of the existing landslide
along the riverbank for which the effect of the colored noise is
relevant, 2) the nature of the effect of the colored noise, and 3) the
size of the colored noise which is relevant forthe significant change in
landslide dynamics.
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Kostićand Stojković10.3389/feart.2023.1267225
2 Applied methodology
In the first phase of the research, we examined previously
recorded river water levels. In general, the river water levels
display considerable variations both annually and over long-term
periods. Consequently, a reliable model for simulating water levels,
which accounts for seasonal and monthly changes, is suggested. This
model consists of two parts: seasonal (ZS)and residual part (ε). The
following equation represents the proposed model for water level
simulation:
Zt
()ZSt
()
+rt
() t1,2,...,N(1)
where Zdenotes the simulated water levels, tsymbolizes a time step,
and Nsignifies the final time step during the simulation period.
Initially, the seasonal component is deduced by utilizing the
LOWESS (Locally Weighted Scatterplot Smoothing) regression
method (Kostićet al., 2019), with the aim of filtering out noise
(residual part) from the observed water levels. Subsequently, the
filtered water levels are represented as a timely dependent function
comprising stationary signals by the usage of spectral analysis
(Stojkovićet al., 2017). The following equation is employed to
mimic the seasonal component of the water level time series:
ZSt
()n
i1aisin 2πfit
+bicos 2πfit
+residuals noise
()
t1,...,N,(2)
where aand brepresent the wave amplitudes associated with the
seasonal component. The frequency of the seasonal harmonics is
denoted by f, and nrepresents the total number of significant
harmonics.
Regarding the procedure of noise separation from the main part
of the time series - in the present paper, we examine finite dataset
comprised of discrete measurements of river level. In order to
simulate the continuous river level oscillations, we firstly develop
a deterministic model of river level oscillations, as a limited
combination of sine and cosine waves, where residuals are
treated as noise.
In the second phase of the research, we introduced the river level
variations, with the included noise into the previously suggested
landslide model. Local dynamics of such model is then examined
numerically using Runge-Kutta fourth order method. In all
examined cases, initial conditions are set near the equilibrium
state: x
1
=x
2
=1.001, y
1
=y
2
=0.002, z
1
=z
2
=0.001, θ=0.001.
3 Analysis of river level oscillations
This study focuses on the analysis of monthly water levels in the
Kolubara and Ibar rivers (Figure 1). The Ibar River, a major tributary
of the Western Morava River, merges with it near Kraljevo in
Central Serbia. With a length of approximately 272 km and
drainage area of 13.059 km
2
, the Ibar is among Serbia’s longest
rivers. Conversely, the Kolubara River, a key tributary of the Sava
River, stretches over 123 km and drains 3,600 km
2
. Originating from
the western region of Serbia, the Kolubara flows through the Valjevo
valley and passes the Kolubara coal basin, a crucial energy source in
Serbia, as it moves northward.
Lopatnica Lakat and Beli Brod hydrological stations, located on
the Ibar and Kolubara rivers respectively, are selected for analysis as
they exhibit minimal anthropogenic impacts on the hydrological
FIGURE 1
Hydrological map of Serbia (A) with location of hydrological stations (B).
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Kostićand Stojković10.3389/feart.2023.1267225
regime, such as flow regulation from upstream dams. Hydrological
data are obtained from the Water Management Study of Serbia
(WMSS, 2010), which includes records up to 2007. Table 1 provides
the most important physical characteristics and available recording
periods of hydrological stations analysed.
Water levels are smoothed using the local-regression LOESS
method to extract the seasonal component (ZS) from the time series
analyzed (Z). This seasonal component envelops the wide range of
frequencies corresponding to stationary seasonal and long-term
signals. However, it considers harmonics with different
frequencies resulting in the unique seasonal signal characterized
as non-stationary time series. Time series filtering is carried out over
various durations. Initially, a 12-month time window is utilized to
capture the seasonal periodicity of the time series. To assess the
sensitivity of the applied LOESS method, time windows of 6 and
30 months have also been employed.
Once the seasonal component is extracted, its characteristics are
estimated in the frequency domain using Spectral analysis. The
significant seasonal harmonics are identified for both hydrological
stations (Beli Brod at the Kolubara River and Lopatnica Lakat at the
Ibar River) using the Relative Cumulative Periodogram. For this
purpose, a significant level of 95% is selected as it represents a
substantial part of seasonal variance. The main properties of
statistically significant harmonics for a 12-month time window
filtering are provided in Table 2 for hydrological station considered.
It is important to highlight that the most significant periodic
component for both stations expectedly has an oscillation period of
12 months. It accounts for 8.2% and 4.6% of the total variance of the
seasonal component for Beli Brod and Lopatnica Lakat hydrological
stations, respectively. Additionally, the subsequent five significant
harmonics cumulatively explain 22.6% of the variance in the time
series of Beli Brod station (Kolubara River). At the Lopatnica Lakat
site of the Ibar River, the next five harmonics account for 10.0% of
the overall seasonal variation.
The remaining part of the time series modeling (Eq. 1)
represents residuals (r) that are characterized by a mean value of
zero for both hydrological stations. Furthermore, the minimum and
maximum values of these residuals at the Beli Brod hydrological
station and 12-months time period filtering are recorded
as −0.512 and 0.999 m, respectively. Please note that residuals
explain significant share of water level variance reaching the
value of 37.2%. In the case of the Ibar river (Lopatnica Lakat
hydrological station), the total variance share of residuals is equal
to 37.8%, while the highest and lowest values of residuals stand at
0.844 and −0.671 m, respectively.
The seasonal components of the water levels for the Kolubara
(Beli Brod) and Ibar River (Lopatnica Lakat) are modeled and
depicted in Figure 2. It considers harmonics that reach a
significant level of 95% for three smoothing/filtering window
lengths: 12, 30 and 6 months. Furthermore, Figure 2B displays
the residual of the modeled seasonal component for both stations
and filtering window lengths considered.
The results obtained suggest that the minimal seasonal
component is achieved during the autumn months, specifically in
the ninth and eighth months for the Kolubara and Ibar rivers,
respectively. The water levels peak in spring, with the Ibar river
recording the highest levels in the third month, while the Kolubara
water levels peak in the fourth month. These variations in the
seasonal water level components signify differences in the hydro-
climatic regimes and morphological factors.
The modeled seasonal component and water levels for
hydrological stations considered are also illustrated in
Figure 2.Thegivenfigures illustrate the reliability of the
proposed modeling approach (Eqs 1,2). They reveal a
stronger correlation between the observed and predicted
values when a 12-month (Figures 2A, D)anda6-month
(Figures 2C, F)filtering window is employed. However, the
30-month window length suggests a weaker correspondence.
Considering Figure 2, it is evident that the seasonal component
aligns more closely with the lower values of the recorded water
levels. A noticeable deviation between the modeled and recorded
water levels is observed in the higher water level range, leading to
higher residual values in this domain. However, the modeling
efficiency metrics for a 12-month filtering window indicate
satisfactory results at the site of Beli Brod hydrological station,
with an RMSE of 0.0169 and a r value of 0.798. For Lopatnica Lakat
hydrological station, the corresponding values for the same matrices
are 0.0120 for RMSE and 0.881 for r, respectively.
Analysis of autocorrelation of residuals (Figure 3) indicates that
residuals are not independent and that their influence could be
TABLE 1 The main characteristic of Beli Brod (Kolubara River) and Lopatnica Lakat (Ibar River) hydrological stations.
Hydrological station River stream Drainage area (km
2
) Available recording period Average flows (m
3
/s) X/Y Coordinates
Beli Brod Kolubara 1896 1959–2007 15.78 7,436,625/4,914,375
Lopatnica Lakat Ibar 7,818 1948–2007 56.72 7,465,225/4,835,150
TABLE 2 Statistical properties of seasonal components at Beli Brod (Kolubara
River) and Lopatnica Lakat (Ibar River) hydrological stations: a–amplitudes of
sine functions; b- amplitudes of cosine functions; f–frequency of both
amplitudes.
a (m) b (m) f (−)
Beli Brod hydrological station (Kolubara River)
Minimal value −0.0378 −0.0318 0.0017
Mean value 0.0015 0.0006 0.0909
Maximal value 0.0743 0.1474 0.2347
Standard deviation 0.0169 0.0219 0.0588
Lopatnica Lakat hydrological station (Ibar River)
Minimal value −0.0859 −0.0592 0.0014
Mean value 0.0020 0.0019 0.2279
Maximal value 0.1289 0.2569 0.4986
Standard deviation 0.0155 0.0196 0.1421
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Kostićand Stojković10.3389/feart.2023.1267225
modelled as the effect of colored noise. Figure 3 also shows the
results of the further analysis of the type of colored noise, indicating
existence of brown noise in all examined cases. One should note that
type of the colored noise is not in the focus of present paper, but
merely to make a distinction between the white (random) and
colored (correlated) noise.
FIGURE 2
Upper row: recorded water levels Zand the seasonal component ZSfor Beli Brod hydrological station (Kolubara River): smoothing window is
respectively equal to 12 months (A), 30 months (B), and 6 months (C); lower row: recorded water levels Zand the seasonal component ZSfor Lopatnica
Lakat hydrological station (Ibar River): smoothing window is respectively equal to 12 months (D), 30 months (E), and 6 months (F).
FIGURE 3
Upper row: (A) autocorrelation function for residuals of the derived models for Beli Brod station; determination of the type of colored noise for (B) 6,
(C) 12 and (D) 30 months residuals (Beli Brod); Lower row: (E) autocorrelation function for residuals of the derived models for Lopatnica station;
determination of the type of colored noise for 6 (F),12(G) and 30 (H) months residuals (Lopatnica).
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Kostićand Stojković10.3389/feart.2023.1267225
4 Dynamics of landslide model
We assume that landslide dynamics could be modeled as the
movement of the ninterconnected blocks:
dxit
()yit
()
dt
dyit
()− aV+yit
()
3−bV+yit
()
2+cV+yit
()
dt +
aV3−bV2+cV
dt +
N
j1
kxjt−τ
()
−xit
()
dt +Zi t
()
dZit
()ξ1sin θ+ξ2cos θ
()
−Zi
εdt +
2D
ε
dWi
dθωdt
(3)
where x
i
and y
i
represent displacement and velocity of the ith
block, respectively, Kis the constant of spring connecting the
blocks, τis time delay and Vis the nondimensional pulling
background velocity. Z
i
(t) stands for the river level oscillations
represented as an Ornstein-Uhlenbeck process, and terms
(2D/ε)
dWi represent stochastic increments of independent
Wiener process, i.e., dW
i
satisfy: E(dWi)=0,E(dW
i
dW
j
)=δ
ij
dt,
where E( ) denotes the expectation over many realizations of the
stochastic process. The noise correlation time εand the intensity
of noise Dare parameters that can be varied independently.
Colored noise generated by Ornstein-Uhlenbeck process with
this parametrization is referred to as power-limited colored
noise, since the total power of the noise (the integral over the
spectral density of the process) is conserved upon varying the
noise correlation time. One should note that, although river level
fluctuations are usually modelled as fractional noises (Hurst,
1951;Koscielny-Bunde et al., 2006;Ghil et al., 2011), modelling
colored noise as Ornsten-Uhlenbeck process represents a
standard approach for the analysis of the noise effect in
nonlinear dynamics. Parameter ωis the angular frequency,
θ=ωt, while parameters ξ
1
and ξ
2
denote the amplitudes of
deterministic oscillatory part of the model. All parameters and
variables in the examined model (3) are dimensionless.
Parameters a,band care parameters of the cubic friction force,
which according to Morales et al. (2017) has the following form:
Fc V
()
3.2V3−7.2V2+4.8V(4)
Results of the numerical analysis of the model (3) indicate the
existence of two dynamical regimes.
FIGURE 4
Bifurcation diagram k-τfor variable values of noise intensity D. While k,τand Dare varied, other parameters of the model (1) are being held constant:
ε=1, ω=1, ξ
1
=ξ
2
=0.001, a=3.2, b=7.2, c=4.8. Initial conditions are set: x
1
=x
2
=1.001, y
1
=y
2
=0.002, z
1
=z
2
=0.001, θ=0.001.
FIGURE 5
Bifurcation diagram a-τfor variable values of noise intensity.
While τand Dare varied, other parameters of the model (1) are being
held constant: k=3, ε=1, ω=1, ξ
1
=ξ
2
=0.001, b=7.2, c=4.8. Initial
conditions are set: x
1
=x
2
=1.001, y
1
=y
2
=0.002, z
1
=z
2
=0.001,
θ=0.001.
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- Equilibrium state, which manifests as steady stationary
movement (corresponding to the landslide creep regime);
- Small-amplitude regular periodic oscillations (corresponding
to the active phase of landslide displacement).
4.1 Effect of noise intensity D
Results of the performed research indicate that the increase of
noise intensity induces destabilization of the landslide creeping and
leads to landslide activation. Such qualitative impact of the intensity
of colored noise is recorded for variable values of D(in the range
10
−2
–10
−5
). One should note that no significant change in dynamics
of the model (1) is observed for the change of D from 10
−5
to 10
−4
.
It should be noted that, regarding the noise level determination,
in the present paper, since model is dimensionless, noise level is also
dimensionless and it is defined in two ways.
•compared to the amplitude of periodic oscillations after the
bifurcation curve is crossed. In the present case, for the chosen
fixed parameter values ε=1, ω=1, ξ
1
=ξ
2
=0.001, a=3.2, b=7.2,
c=4.8 and set initial conditions: x
1
=x
2
=1.001, y
1
=y
2
=0.002,
z
1
=z
2
=0.001, θ=0.001, amplitudes of oscillations are of order
1.5 (k=4, τ=4). When the highest assumed value of noise is
introduced D=10
−2
, differences of oscillations with and without
the assumed noise is of the order 10–4. Hence, the assumed
range of noise level D=10
−5
to 10
−2
is significantly smaller than
the compared oscillations above the bifurcation curve.
•compared to the amplitude of river level periodic
oscillations. In the present case, for the chosen fixed
parameter values ε=1, ω=1, ξ
1
=ξ
2
=0.001, a=3.2, b=7.2,
c=4.8 and set initial conditions: x
1
=x
2
=1.001,
y
1
=y2=0.002, z
1
=z
2
=0.001, θ=0.001, amplitudes of river
level oscillations are of order 10–4 below the bifurcation
curve (k=1, τ=1). From this point of view, relevant noise
FIGURE 6
Bifurcation diagrams displacement delay-friction parameters: (A) b-τand (B) c-τ, for variable values of noise intensity. While τand Dare varied, other
parameters of the model (1) are being held constant: k=3, ε=1, ω=1, ξ
1
=ξ
2
=0.001, a=3.2, b=7.2, c=4.8. Initial conditions are set: x
1
=x
2
=1.001, y
1
=y
2
=0.002,
z
1
=z
2
=0.001, θ=0.001.
FIGURE 7
Bifurcation diagrams slope stifness-friction parameters: (A) a-k and (B) c-k, for variable values of noise intensity. While kand Dare varied, other
parameters of the model (1) are being held constant: τ=3, ε=1, ω=1, ξ
1
=ξ
2
=0.001, a=3.2, b=7.2, c=4.8. Initial conditions are set: x
1
=x
2
=1.001, y
1
=y
2
=0.002,
z
1
=z
2
=0.001, θ=0.001.
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levels are in the range 10
−4
–10
−5
, when differences in the
amplitude of river level oscillations with and without noise
are of the order 10
−8
–10
−4
. Therefore, from the viewpoint of
the amplitude of river level oscillations, relevant noise levels
are of the order 10
−5
–10
−4
.Highervaluesofnoiseleadtothe
extremecasewhennoiselevelisapproximatelythesameas
the “main signal”.
For the fixed values of friction parameters, it appears that for each
increase of D(for one order of unit), critical values of displacement
delay and slope stiffness for which bifurcation occurs reduce by 0.1. As
shown in Figure 4,bifurcationline“moves”parallel to the lower values
of τand kwiththeincreaseofnoiseintensity.Oneshouldnotethatthe
observed changes are happening for rather large values of displacement
delay and slope stiffness, of the order 0.1, so one can interpret the
recorded features as very significant, since changes induced by the
change of noise intensity occur at the level of observation.
From the geodynamical viewpoint, such findings indicate that
noise intensity reduces the critical values of slope stiffness and
displacement delay for which instability occurs. In other words,
in the presence of colored noise landslide hazard increases.
In case when friction parameters are being varied, noise intensity
has qualitatively the same effect on stability of landslide dynamics as in
thepreviouscase(Figure 5). It seems that friction parameter ais the
most sensitive to the change of the noise intensity; similarly to the co-
effect of delay and slope stiffness, the change of Dfrom 10
−5
to 10
−2
leads
to the transition of bifurcation curve down by 0.1 concerning the
friction parameter a(Figure 5). It should be emphasized that the
increase of noise intensity leads to destabilization of landslide dynamics.
From the geodynamical viewpoint, parameter acontrols the
brittle-ductile transition; in particular, the increase of parameter a
in the examined range (and further) indicates a change in the slope
behavior from ductile, with pronounced peak and residual shear
strength, to plastic, where no clear drop of shear strength is observed.
Regarding the interaction of noise intensity with other two
friction parameters, b(Figure 6A) and c(Figure 6B), results
obtained indicate that these parameters are less sensitive,
meaning that with the change of noise intensity only the critical
value of the displacement delay changes, while the relevant values of
parameter band cremain almost the same. One could note that
parameter chas qualitatively the same effect to landslide dynamics as
parameter a: the increase of parameter cleads to stabilization of
landslide dynamics, while noise intensity acts as a destabilizing
factor. On the other hand, the increase of parameter bleads to
destabilization of the landslide dynamics, with the same effect of D.
Considering the observed interaction of friction parameters and
noise intensity with displacement delay, one could conclude that
noise intensity, although small compared to the relevant scales of
other control parameters, alters the governing friction low and, for
FIGURE 8
Bifurcation diagram b-k for variable values of noise intensity.
While kand Dare varied, other parameters of the model (1) are being
held constant: τ=3, ε=1, ω=1, ξ
1
=ξ
2
=0.001, a=3.2, c=4.8. Initial
conditions are set: x
1
=x
2
=1.001,y
1
=y
2
=0.002, z
1
=z
2
=0.001,
θ=0.001.
FIGURE 9
Bifurcation diagram k-τfor variable values of correlation time ε. While k, τand εare varied, other parameters of the model (1) are being held constant:
D=10
−4
,ω=1, ξ
1
=ξ
2
=0.001, a=3.2, b=7.2, c=4.8. Initial conditions are set: x
1
=x
2
=1.001, y
1
=y
2
=0.002, z
1
=z
2
=0.001, θ=0.001.
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Kostićand Stojković10.3389/feart.2023.1267225
certain values of friction parameters, makes slope more susceptible
to the onset of unstable landslide dynamics.
Such effect of the change of noise intensity is also observed for the
case when displacement delay is held constant. It appears that for high
values of Dbifurcation occurs for lower values of slope stiffness and for
higher values of friction parameters a(Figure 7A)andc(Figure 7B).
Comparedtothecaseoffixed friction parameters (Figure 4), the
amount of change of kis almost the same for parameters band c,
whileitis2timessmallerforthecasewhenparameterais being varied.
As for the interaction of parameter band noise intensity when
displacement delay has a constant value (Figure 8), it seems that, in
such case, parameter bis more sensitive to the change in D. Change
of Dfor 4 orders of units leads to the onset of bifurcation for values
of parameter bwhich are smaller for 0.2 units, compared to
0.05 units, in case when slope stiffness has a constant value.
4.2 Effect of correlation time ε
Compared to the influence of noise intensity, the effect of
correlation time is much less emphasized, e.g., displacement
delay, spring stiffness and friction parameters are rather robust to
the effect of correlation time. Nevertheless, this effect is clearly
captured and confirms the significance of influence of the noise color
on the landslide dynamics.
From Figure 9 one could identify the effect of correlation time ε
on the dynamics of model (1). In particular, increase of correlation
time εfrom 0.1 to 10 leads to destabilization of landslide dynamics,
but only for certain values of delay displacement (3.2 ≤τ≤3.4) and
slope stiffness (3.3 ≤τ≤3.5). This further means the effect of
correlation time is not generic, but could occur only for certain
values of delay and stiffness.
Regarding the interaction of correlation time and friction
parameters, results obtained indicate that friction parameter bis
robust to the change of correlation time in the range from 0.1 to
10. Some weak sensitivity to change of correlation time is captured for
friction parameter a, where the increase of correlation time actually
leads to stabilization of landslide dynamics (Figure 10A). Such effect
of correlation time is opposite to the effect of noise intensity. From the
geodynamical viewpoint, it means that degree of noise correlation
could be crucial for the stability of landslide dynamics. As for the
friction parameter c, it seems that nature of the effect of correlation
time change also changes with the increase of parameter c(Figure 10).
For low values of parameter c, increase of correlation time leads to
stabilization of landslide dynamics, i.e., such effect is opposite to the
effect of the noise intensity. For higher values of c, increase of
correlation time leads to destabilization of landslide dynamics,
which is qualitatively the same effect as the noise intensity.
Stabilizing effect of the correlation time is more clearly seen for the
constant value of displacement delay (Figure 11). As for the
interaction of friction parameters band cand the change of
correlation time when displacement delay is fixed, the same
destabilizing effect is captured both for parameters band c(Figure 12).
FIGURE 10
Bifurcation diagrams time delay-friction parameters: (A) a-τ,(B) b-τand (C) c-τ, for variable values of correlation time ε. While τand εare varied, other
parameters of the model (1) are being held constant: D=10
−4
,k=3, ω=1, ξ
1
=ξ
2
=0.001, a=3.2, b=7.2, c=4.8. Initial conditions are set: x
1
=x
2
=1.001,
y
1
=y
2
=0.002, z
1
=z
2
=0.001, θ=0.001.
FIGURE 11
Bifurcation diagrams spring stifness-friction parameter k-a for
variable values of correlation time ε. While k,aand εare varied, other
parameters of the model (1) are being held constant: D=10
−4
,k=3, ω=1,
ξ
1
=ξ
2
=0.001, a=3.2, b=7.2, c=4.8. Initial conditions are set:
x
1
=x
2
=1.001, y
1
=y
2
=0.002, z
1
=z
2
=0.001, θ=0.001.
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Kostićand Stojković10.3389/feart.2023.1267225
5 Conclusion
In present paper, we examine the impact of colored noise in river
level oscillations on the landslide dynamics. Firstly, we prove, by
analyzing the measurement of the river oscillation recordings, that
noise in river level oscillations could be treated as the colored noise.
This is done for the recordings made at two different hydrological
stations in Serbia: Beli Brod station (the Kolubara river) and
Lopatnica Lakat station (the Ibar river). We examined available
water level recordings for the Kolubara river and the Ibar river for
the periods 1959–2007 and 1948–2007, respectively. By applying
LOESS method and spectral analysis, we derived deterministic
models for the recorded time series, which are proved to be
statistically significant. At the end of the first phase of the
research, it is shown that residuals of derived models could be
characterized as an example of colored noise. In that way, we
justified the introduction of both the deterministic water level
oscillation and low-magnitude color noise into the model of
landslide dynamics.
In the second phase of the research, we investigate the landslide
dynamics by analyzing the model of two coupled blocks, with
displacement delay and with the assumed additive colored noise.
The results obtained indicate the existence of two different
dynamical regimes, all of which could have its correspondence
with the real observed regimes of landslide dynamics: (1) steady
stationary state (creep regime); and (2) active landslide dynamics.
The results indicate significant effect of noise intensity and
correlation on the onset of unstable landslide dynamics. In
particular, increase of noise intensity leads to destabilization of
landslide dynamics, in the following way.
- noise intensity reduces the critical values of slope stiffness and
displacement delay for which instability occurs,
- it seems that friction parameter ais the most sensitive to the
change of the noise intensity; since parameter acontrols the
ductile-plastic transition, it could be said that the change in the
colored noise actually alters deformable properties of the slope,
- effect of correlation time is much less emphasized, in a way
that displacement delay, spring stiffness and friction
parameters are rather robust to the effect of correlation time,
- some weak sensitivity to change of correlation time is captured
for friction parameter a, where the increase of correlation time
actually leads to stabilization of landslide dynamics. Such effect
of correlation time is opposite to the effect of noise intensity.
If one compares the effect of colored noise, analyzed in this
paper, and random noise, analyzed in our previous paper (Kostić
et al., 2023), the difference lies in the following. For random
background noise, we confirmed its stabilizing effect on the
landslide dynamics. In particular, in our previous paper (Kostić
et al., 2023) noise has the same effect for highly weathered rock
masses, with rather low values of spring stiffness and high values of
time delay, and for slightly weathered rock masses, with high values
of spring stiffness and low values of time delay. On the other hand, in
present paper we show that noise intensity, although small
compared to the relevant scales of other control parameters,
alters the governing friction low and, for certain values of friction
parameters, makes slope more susceptible to the onset of unstable
landslide dynamics. Also, we show that correlation time could also
have significant effect, even opposite to the effect of noise intensity,
which further confirms the significance of the color of noise on
landslide dynamics.
It could be interesting to briefly discuss the possible natural roots
of the increased noise intensity in the river level fluctuations. Root of
noise in river level oscillations in general could be considered as the
product by a combination of geographic, hydroclimatic and
anthropogenic variables, which was also previously suggested by
Tu et al. (2023) as a possible cause of spatial variation in noise colour
of daily and annual river flow. Increase of noise in river water levels
could be attributed to both seasonal and multi-annual climate
variabilities. While seasonal variability consistently affects annual
water level patterns, it is the prolonged climate variability,
particularly arising from atmosphere-ocean oscillation, that
profoundly influences flood-related dynamics (Kundzewicz et al.,
FIGURE 12
Bifurcation diagrams slope stiffness-friction parameters: (A) b-k and (B) c-k, for variable values of correlation time ε. While b,c,kand εare varied,
other parameters of the model (1) are being held constant: τ=3, D=10
−4
,ω=1, ξ
1
=ξ
2
=0.001, b=7.2, c=4.8. Initial conditions are set: x
1
=x
2
=1.001,
y
1
=y
2
=0.002, z
1
=z
2
=0.001, θ=0.001.
Frontiers in Earth Science frontiersin.org10
Kostićand Stojković10.3389/feart.2023.1267225
2019). Such variability is postulated to be a fundamental factor of
spatial and temporal shifts in hydrometeorological variables across
Europe (Nobre et al., 2017). Also, torrential floods could be the cause
of the increase of noise level. In particular, low-frequency climate
variations have been linked to genesis of flash floods (Archer et al.,
2019), where swift of flood events is recognized to be a consequence
of morphological, pedological, geological, and climatic attributes of
specific torrential basins. In the present paper, goal of the research
was to indicate that when slope is in critical state (near the
bifurcation), even small increase in noise amplitude within the
river level oscillations could lead to occurrence of instability.
It should be noted that the performed analysis is conducted for the
initial conditions near the equilibrium state; hence, only the local
dynamics of the landslide model is examined. Possible occurrence of
global bifurcations when initial conditions are away from the
equilibrium state (possible occurrence of global bifurcations) was not
investigated in the present paper. Further work on this topic could
include the analysis of other local bifurcation curves, or the possible
existence of global bifurcation, which could eventually lead to definition
of conditions for the occurrence of stick-slip dynamics, which is
qualitatively the most resemblant to the real unstable dynamics.
Data availability statement
The raw data supporting the conclusion of this article will be
made available by the authors, without undue reservation.
Author contributions
SK: Formal Analysis, Investigation, Writing–original draft.
MS: Formal Analysis, Investigation, Resources, Writing–original
draft.
Funding
The author(s) declare that no financial support was received for
the research, authorship, and/or publication of this article.
Conflict of interest
The authors declare that the research was conducted in the
absence of any commercial or financial relationships that could be
construed as a potential conflict of interest.
Publisher’s note
All claims expressed in this article are solely those of the authors
and do not necessarily represent those of their affiliated organizations,
or those of the publisher, the editors and the reviewers. Any product
that may be evaluated in this article, or claim that may be made by its
manufacturer, is not guaranteed or endorsed by the publisher.
References
Abam, T. K. S. (1993). Factors affecting distribution of instability of river banks in the
Niger delta. Eng. Geol. 35, 123–133. doi:10.1016/0013-7952(93)90074-m
Archer, D., O’donnell, G., Lamb, R., Warren, S., and Fowler, H. J. (2019). Historical
flash floods in England: new regional chronologies and database. J- Flood Risk Manag.
12, e12526. doi:10.1111/jfr3.12526
Chen, C. H., Hsieh, T. Y., and Yang, J. C. (2017). Investigating effect of water level
variation and surface tension crack on riverbank stability. J. Hydro-Environ. Res. 15,
41–53. doi:10.1016/j.jher.2017.02.002
Dapporto, S., Rinaldi, M., and Casagli, N. (2001). Failure mechanisms and pore water
pressure conditions: analysis of a riverbank along the Arno River (Central Italy). Eng.
Geol. 61, 221–242. doi:10.1016/s0013-7952(01)00026-6
Duong, T. T., and Do, M. D. (2019). Riverbank stability assessment under river water
level changes and hydraulic erosion. Water 11, 2598. doi:10.3390/w11122598
Duong, T. T., Komine, H., Do, M. D., and Murakami, S. (2014). Riverbank stability
assessment under flooding conditions in the Red River of Hanoi, vietnam. Comput.
Geotechnics 61, 178–189. doi:10.1016/j.compgeo.2014.05.016
Ghil, M., Yiou, P., Hallegatte, S., Malamud, B. D., Naveau, P., Soloviev, A., et al.
(2011). Extreme events: dynamics, statistics and prediction. Nonlinear Process Geophys
18, 295–350. doi:10.5194/npg-18-295-2011
Hurst, H. E. (1951). Long-term storage capacity of reservoirs. Trans. Am. Soc. Civ.
Eng. 116, 770–799. doi:10.1061/taceat.0006518
Koscielny-Bunde, E., Kantelhardt, J. W., Braun, P., Bunde, A., and Havlin, S. (2006).
Long-term persistence and multifractality of river runoff records: detrended fluctuation
studies. J. Hydrol. 322, 120–137. doi:10.1016/j.jhydrol.2005.03.004
Kostić, S., Stojković, M., Guranov, I., and Vasović, N. (2019). Revealing the background
of groundwater level dynamics: contributing factors, complex modeling and engineering
applications. Chaos, Solit. Fract. 127, 408–421. doi:10.1016/j.chaos.2019.07.007
Kostić,S.,Vasović,N.,Todorović,K.,andFranović,I.(2020).Effectofcolored
noise on the generation of seismic fault movement: analogy with spring-block
model dynamics. Chaos Solit. Fract. 135, 109726. doi:10.1016/j.chaos.2020.
109726
Kostić, S., Vasović, N., Todorović, K., and Prekrat, D. (2023). Instability induced by
random background noise in a delay model of landslide dynamics. Appl. Sci. 13, 6112.
doi:10.3390/app13106112
Kundzewicz, Z. W., Szwed, M., and Pińskwar, I. (2019). Climate variability and
floods—a global review. Water 11 (7), 1399. doi:10.3390/w11071399
Liang, C., Jaksa, M., Ostendorf, B., and Kuo, Y. (2015). Influence of river level
fluctuations and climate on riverbank stability. Comput. Geotech. 63, 83–98. doi:10.
1016/j.compgeo.2014.08.012
Mentes, G. (2019). Relationship between river bank stability and hydrological processes
using in situ measurement data. Cent. Eur. Geol. 62, 83–99. doi:10.1556/24.62.2019.01
Morales, J. E. M., James, G., and Tonnelier,A. (2017). Traveling waves in a spring-block
chain sliding down a slope. Phys. Rev.E 96, 012227. doi:10.1103/physreve.96.012227
Nobre, G. G., Jongman, B., Aerts, J. C. J. H., and Ward, P. J. (2017). The role of climate
variability in extreme floods in Europe. Environ. Res. Lett. 12 (8), 084012. doi:10.1088/
1748-9326/aa7c22
Stojković, M., Kostić, S., Plavšić, J., and Prohaska, S. (2017). A joint stochastic-
deterministic approach for long-term and short-term modelling of monthly flow rates.
J. Hydrol. 544, 555–566. doi:10.1016/j.jhydrol.2016.11.025
Tu, T., Comte, L., and Ruhi, A. (2023). The color of environmental noise in river
networks. Nat. Commun. 14, 1728. doi:10.1038/s41467-023-37062-2
Újvári, G., Mentes, G., Bányai, L., Kraft, J., Gyimóthy, A., and Kovács, J. (2009).
Evolution of a bank failure along the river Danube at dunaszekcső, Hungary. Geomorph
109, 197–209. doi:10.1016/j.geomorph.2009.03.002
Vračar, M. S., and Mijić, M. (2011). Ambient noise in large rivers (L). J. Acoust. Soc.
Am. 130, 1787–1791. doi:10.1121/1.3628666
WMSS (2010). Water management study of Serbia. Jaroslav Černi Water institute.
Belgrade, Serbia.
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