![Relationship between the average classification uncertainty and its variance. The three vertical lines correspond to the three risk thresholds in time; wt0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_{t_{0}}$$\end{document}: time of regime change, tR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_R$$\end{document}: time of emergent risk, and tI\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_I$$\end{document}: time of imminent risk](https://www.researchgate.net/publication/381881369/figure/fig3/AS:11431281269998920@1722995847354/Relationship-between-the-average-classification-uncertainty-and-its-variance-The-three_Q320.jpg)
Access to this full-text is provided by Springer Nature.
Content available from Stochastic Environmental Research and Risk Assessment
This content is subject to copyright. Terms and conditions apply.
https://doi.org/10.1007/s00477-024-02713-3
ORIGINAL PAPER
Monitoring the risk of a tailings dam collapse through spectral analysis
of satellite InSAR time-series data
Sourav Das1,2 ·Anuradha Priyadarshana3·Stephen Grebby4
Accepted: 19 March 2024
© The Author(s) 2024
Abstract
Slope failures possess destructive power that can cause significant damage to both life and infrastructure. Monitoring slopes
prone to instabilities is therefore critical in mitigating the risk posed by their failure. The purpose of slope monitoring is to
detect precursory signs of stability issues, such as changes in the rate of displacement with which a slope is deforming. This
information can then be used to predict the timing or probability of an imminent failure in order to provide an early warning.
Most approaches to predicting slope failures, such as the inverse velocity method, focus on predicting the timing of a potential
failure. However, such approaches are deterministic and require some subjective analysis of displacement monitoring data
to generate reliable timing predictions. In this study, a more objective, probabilistic-learning algorithm is proposed to detect
and characterise the risk of a slope failure, based on spectral analysis of serially correlated displacement time-series data. The
algorithm is applied to satellite-based interferometric synthetic radar (InSAR) displacement time-series data to retrospectively
analyse the risk of the 2019 Brumadinho tailings dam collapse in Brazil. Two potential risk milestones are identified and
signs of a definitive but emergent risk (27 February 2018-26 August 2018) and imminent risk of collapse of the tailings dam
(27 June 2018-24 December 2018) are detected by the algorithm as the empirical points of inflection and maximum on a risk
trajectory, respectively. Importantly, this precursory indication of risk of failure is detected as early as at least five months
prior to the dam collapse on 25 January 2019. The results of this study demonstrate that the combination of spectral methods
and second order statistical properties of InSAR displacement time-series data can reveal signs of a transition into an unstable
deformation regime, and that this algorithm can provide sufficient early-warning that could help mitigate catastrophic slope
failures.
Keywords Landslide monitoring ·InSAR ·Periodogram ·Non-stationarity
Sourav Das, Anuradha Priyadarshana and Stephen Grebby contributed
equally to this work.
BSourav Das
sourav.das@curtin.edu.au
Anuradha Priyadarshana
anupriyadarsh@gmail.com
Stephen Grebby
stephen.grebby@nottingham.ac.uk
1EECMS, Curtin University, Perth, WA 6102, Australia
2Adjunct Senior Lecturer, College of Science and Engineering,
James Cook University Smithfield, Cairns, QLD 4878,
Australia
3Department of Statistics, Faculty of Applied Sciences,
University of Sri Jayewardenepura, Nugegoda, Sri Lanka
4Nottingham Geospatial Institute, Faculty of Engineering,
University of Nottingham, Nottingham NG7 2TU, UK
1 Introduction
Slope failures in the form of landslides or the collapse of engi-
neered structures (e.g., tailings dams) pose a considerable
risk to both life and infrastructure. Mitigating the risk they
pose through providing an early warning relies critically upon
monitoring slopes for precursory signs of instability (Intrieri
et al. 2013). Conventionally, this has involved using survey
monuments, inclinometers, piezometers, extensometers and
ground-based radar to monitor how a rock mass is deform-
ing, with changes in the displacement and velocity providing
the most reliable indication of the stability of a slope (Carlà
et al. 2017). Accordingly, several empirical approaches have
been developed to predict the timing of a slope failure based
on displacement monitoring data (Intrieri and Gigli 2016).
The majority of failure timing prediction approaches
utilise the concept of accelerating (tertiary) creep theory
123
Stochastic Environmental Research and Risk Assessment (2024) 38:2911–2926
/ Published online:1
July 2024
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
(Saito 1969; Fukuzuno 1985), under which the behaviour
of a material in the terminal stages of failure under constant
stress and temperature conditions is governed by the empir-
ical power-law (Voight 1988,1989):
v(t)−αdv(t)
dt =A(1)
where vis the creep velocity, tis the time, and αand Aare
constants. It has been observed that for a wide range of slope
failures α≈2 (Voight 1989). Consequently, for α=2,Eq.1
has the solution:
1/v(t)=A(tf−t)(2)
where tfis the time of failure. Equation 2implies that
approaching the time of failure, the inverse of the velocity
scales as a linear function of time. Consequently, the inter-
cept point on the time axis of an inverse velocity (1/v(t))
vs. time (t) plot corresponds to the time of failure (tf). This
approach, often referred to as the inverse velocity method, is
commonly applied to surface displacement monitoring data
to predict the time of slope failures, by using linear regres-
sion to extrapolate the inverse velocity trend to the point of
intersection on the time axis (Carlà et al. 2018).
Although predicting the timing of slope failures can be
crucial in mitigating the risk they pose, there are several
factors that affect the ability to do so effectively. Firstly,
the conventional ground-based monitoring techniques often
provide measurements that are low density, have limited
coverage and have a low or irregular temporal sampling fre-
quency, which can make it difficult to detect the precursory
tertiary creep (Carlà et al. 2019). Secondly, empirical pre-
diction approaches like the inverse velocity method require
expert user-intervention in order to derive reliable estimates
of the slope failure timing. This includes filtering the velocity
time-series measurements to remove instrumental and ran-
dom "noise" to help determine the onset of tertiary creep
(i.e., Onset Of Acceleration) and increase the fit of the lin-
ear regression line on the inverse velocity-time plot (Dick
et al. 2015; Carlà et al. 2017). However, in the absence of a
general rule regarding filtering, the selection of the optimal
method is subjective and typically determined through trial-
and-error (Rose and Hungr 2007). Furthermore, the inverse
velocity trend often only approaches linearity in the final few
weeks prior to a slope failure (Rose and Hungr 2007), there-
fore only enabling reliable short-term predictions that leave
little time for risk mitigation measures to be implemented.
The use of satellite Interferometric Synthetic Aperture
Radar (InSAR) has been increasingly recognised as an effec-
tive solution in overcoming the issue of limited surface
displacement measurement coverage. The InSAR method
is based upon the concept of interferometry, with precise
changes in the surface elevation being derived from the
interference of electromagnetic waves from two Synthetic
Aperture Radar (SAR) acquisitions (Rosen et al. 2000).
Accordingly, satellite InSAR has been used to measured
ground motion for a wide range of applications, including
volcanology (Massonnet and Feigl 1995; Pinel et al. 2014),
landslides (Fruneau et al. 1996; Colesanti and Wasowski
2006; Song et al. 2022), seismology (Massonnet et al. 1993;
Peltzer and Rosen 1995; Cheloni et al. 2024) and various
other types of surface deformation monitoring (Gee et al.
2019; Raspini et al. 2022). More recently, satellite InSAR
has also been applied to slope failure prediction in relation
to landslides, tailings dams and open-pit mines (Intrieri et al.
2018; Carlà et al. 2019). A particularly pertinent example
concerns the 25 January 2019 catastrophic collapse of the
upstream-constructed tailings Dam I at the Córrego do Feijão
iron ore mine complex in Brumadinho, Brazil. The dam, stor-
ing 11.7 million m3of tailings, collapsed suddenly releasing
of flow of material that resulted in the death of approximately
270 people, without any apparent risk indicators from the
array of conventional methods (e.g., inclinometers, piezome-
ters, ground-based radar) being used to monitor its stability
(Robertson et al. 2019). However, several studies have ret-
rospectively analysed the stability of the dam preceding the
collapse using satellite InSAR (Du et al. 2020), with some
of these deriving inverse velocity-based failure timing pre-
dictions from the precursory deformation (Gama et al. 2020;
Grebby et al. 2021). While successful predictions close to the
actual time of failure were possible using the inverse velocity
method, there remains some degree of uncertainty over the
reliability of these predictions, due largely to the fitting of
the linear regression line and the amount of prior warning of
a potential failure that can be provided.
There have been attempts to generalise the inverse veloc-
ity method in Eq. 1by extending the temporal dynamics of
the governing mechanisms of failure to consider locally (spa-
tially) varying dynamics. Recent examples include studies by
Tordesillas et al. (2018); Niu and Zhou (2021) and references
therein. Furthermore, there is a growing body of litera-
ture exploring the potential to offer probabilistic predictions
of slope failure by modifying the inverse velocity method
in various ways, such as through the application of linear
regression (Zhang et al. 2020), stochastic differential equa-
tions (Bevilacqua et al. 2019), quantile-regression (Das and
Tordesillas 2019; Wang et al. 2020), and mixture-distribution
(Zhou et al. 2020). However, with the exception of Bevilac-
qua et al. (2019), none of these studies explicitly consider
the temporal serial correlation of the displacement or veloc-
ity fields. Additionally, almost all of these studies assume a
parametric statistical distribution on the observed response
(i.e., displacement or velocity). Nevertheless, rapid develop-
123
Stochastic Environmental Research and Risk Assessment (2024) 38:2911–2926
2912
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
ments in technology for monitoring ground displacements,
such as ground-based (Casagli et al. 2010) and satellite-based
InSAR (Wasowski and Bovenga 2014) present the opportu-
nity to develop much faster and sensitive precursory warnings
using algorithms that are based on streaming data and require
less stringent statistical assumptions about the displacement
mechanism. Das and Tordesillas (2019) proposed a data-
driven, non-parametric multivariate statistical methodology
for near real-time monitoring of slope failures. However,
Das and Tordesillas (2019) acknowledged that serial correla-
tion inherent in such deformation signals was not modelled.
Nonetheless, their framework led to a much earlier predic-
tion of the failure than that offered by the inverse velocity
method.
Accordingly, the aim of this study is to further develop
the framework of Das and Tordesillas (2019) to propose a
new generalised algorithm that can accommodate temporal
correlation whilst retaining its non-parametric nature. One
key advantage of this statistical approach is that often regu-
lar sampling may reveal a change in the slope stability regime
much earlier than that observed using the mechanistic failure
law. The failure dynamics obtained by the algorithm is driven
entirely by properties of the data and a fundamental statisti-
cal property underpinning conventional theory of time-series
analysis called second order stationarity. Consequently, the
statistical nature of this approach also makes it more objec-
tive in comparison to the inverse velocity method because the
subjectivity associated with manually identifying the Onset
of Acceleration and optimally filtering the time-series data
is circumvented.
Heuristically, a second order stationary time-series has
time-invariant statistical properties of second order. It is
therefore postulated that the displacement time-series of a
slope, sampled during a stable epoch, must be second order
stationary. However, gradual deviations from second order
stationary should occur as the slope transitions into an unsta-
ble epoch. Here, it is this principle that is utilised in order
develop the new algorithm, which can be broadly defined as
two phases:
1. Phase 1: Detecting a change in regime – this involves
first characterising (estimating) the second order statis-
tical features of an area during a stable regime based on
regularly sampled InSAR displacement time-series data.
This is achieved through estimating statistical features
of multiple time-series in terms of their dynamic peri-
odograms. The stable regime (stationarity) is then defined
as the entire history of the observations up until the detec-
tion of statistical variation in periodogram features; this
point marks the time of regime change.
2. Phase 2: Risk classification – this involves continuously
tracking the deviation of the time-series from second
order stationarity after the time of regime change detect
in Phase 1. Using a combination of supervised and unsu-
pervised learning algorithms, this deviation is described
as a concave probabilistic function of time. Risk of fail-
ure warnings are then generated based on estimates of
inflection points on the trajectory of this function.
In this study, the proposed algorithm is applied to satellite
InSAR time-series data to retrospectively analyse the risk of
the Dam I tailings dam collapse. Section 2describes the study
area and the InSAR time-series data used in the analysis. The
proposed algorithm is then presented in Section 3, before
the results and discussion for its application are provide in
Section 4. Finally, the key conclusions, limitations and future
research associated with this study are outlined in Section 5.
2 Study area and data
Dam I (Fig. 1a), at the Córrego do Feijão iron ore mine
complex in Brumadinho, Minas Gerais State (southeastern
Brazil), was an upstream tailings dam (active during 1976–
2015) that underwent a sudden and complete failure across its
full 720 m width and 86 m height at 12:28pm on 25 January
2019 as a result of a flow (static) liquefaction mechanism
(de Lima et al. 2020). This was initially attributed to a com-
bination of internal creep and a loss of suction induced by
a period of heavy rainfall from about October 2018 to the
time of the failure (Robertson et al. 2019), although recent
modelling has suggested that creep alone would have been
sufficient to initiate the failure of the dam (Zhu et al. 2024).
The catastrophic failure of Dam I caused a debris- and mudl-
fow of approximately 11 million m3of mining waste to
flow rapidly downstream, destroying properties, infrastruc-
ture and agricultural land and entering the Paraopeba River
(Gama et al. 2020). Within days of the collapse, the mud
coveredanareaof3.13x10
6m3(Rotta et al. 2020), ulti-
mately claiming approximately 270 lives and affecting the
whole region’s ecosystem (Porsani et al. 2019).
The data used to analyse the precursory deformation and
risk in this study comprises Sentinel-1 InSAR displacement
time-series data for two overlapping descending orbit tracks
(tracks 53 and 155), for the 17 months preceding the dam
collapse. The relative line-of-sight displacement (RLOSD)
time-series data covering the extent of Dam I at 20-m spa-
tial resolution and a 12-day interval (sampling frequency)
were generated using the Intermittent Small Baseline Sub-
set (ISBAS) differential InSAR technique (Sowter et al.
123
Stochastic Environmental Research and Risk Assessment (2024) 38:2911–2926 2913
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
Fig. 1 The Brumadinho (Brazil) study area. a) Satellite image of Dam I pre- and post-collapse, b) Relative LOS displacement maps for tracks 155
and 53 prior to the collapse, with locations 1-6 indicated (after (Grebby et al. 2021))
2013) and made publicly available by Grebby et al. (2021).
Representative displacement time-series for areas exhibiting
distinctive deformation within each of the main structural
elements of the dam (i.e., dam wall, tailings beach) were
obtained by extracting the time-series of a subset of contigu-
ous pixels at each locality (Grebby et al. 2021). Figure 1b)
shows the locations of the six areas (labelled locations 1-6)
identified as exhibiting distinctive deformation over the dam.
The subsets of pixels were delineated manually based on the
uniformity of the respective deformation patterns observed
at each location, and vary in size according to the area over
which the patterns extend. Accordingly, the number of con-
tiguous pixels for which the time-series were extracted at
locations 1 through to 6 is 30,44,46,25,38 and 37, respec-
tively. Both the multiple individual displacement time-series
and the average time-series at each of the six locations were
used to construct the machine-based algorithm for character-
ising and precursory monitoring of the tailings dam failure.
3 Methods
The method can be summarised in the following three stages:
(i) Monitoring of the RLOSD to identify the time of regime
change; (ii) Estimating state-of-the-system at the time of
regime change; (iii) Sequential risk estimation and hazard
classification. All the above steps build on monitoring the
second order statistical properties of a time-series signal.
These steps and the related concepts are described in the
following subsections.
3.1 Statistical preliminaries
Conventional statistical modelling of a time-series, {Yt},
often involves modelling its second order statistical prop-
erties, its trend – slowly varying mean effect (μt)–the
autocovariance (Cov(Yt,Yt+u), and variance (σ2
t), which are
defined as:
E(Yt)=μt
var(Yt)=E(Yt−μt)2=σ2
t
Cov(Yt,Yt+u)=E{(Yt+u−μt)(Yt−μt)}
=c(t,t+u)
(3)
Key to such modelling is the notion of second order sta-
tionarity.
3.1.1 Second order stationarity
Let {Yt}be a time-series that is discretely sampled at nregu-
lar time intervals {t=1,2,...,n}, such that its population
mean, variance ( var (.)) and auto-covariance (Cov(.))are
123
Stochastic Environmental Research and Risk Assessment (2024) 38:2911–2926
2914
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
defined as in Eq. 3. Then, Ytis second order stationary if:
E(Yt)=μ(a constant)
var(Yt)=σ2(a constant)
Cov(Yt)=E(Yt+u−μ)(Yt−μ) =c(u),
a function of time interval, (t,t+u).
(4)
A second order stationary time-series is a characteristic
property that can be observed in the Fourier domain.
3.1.2 Spectral density and periodogram
If Ytis a second-order stationary time-series with the auto-
covariance function, c(u), satisfying u|u|c(|u|)<∞,
then Ythas a continuous spectrum f(ω) that is defined as:
f(ω) =
u
e−iωuc(u), −1/2<ω<1/2,
admitting the following inversion
c(u)=1/2
−1/2
eiωuf(ω)d(ω).
(5)
The spectrum f(ω) is a unique signature of a second order
stationary time-series marked by invariance over time. It par-
titions the total variance (power) of the time-series among
sinusoidal event frequencies. Physicist Schuster (1906)pro-
posed the periodogram to estimate the spectrum of a second
order stationary time-series. The discrete Fourier transform
(DFT) and periodogram of observed time-series data yt,t=
1,2,...,nare defined, respectively, by d(ωk) and I(ωk)as:
d(ωk)=1
√n
n
t=1
e−iωktyt,
I(ωk)=1
n
n
t=1
e−iωktˆc(u),
ˆc(u)=1
n
n−u
t=1{(yt−¯y)(yt+u−¯y)},where
ωk=2πk/n,k=0,1,2...,n−1.
(6)
3.2 Non-stationarity: detecting geological regime
change
The algorithm assesses instability based on non-stationary
stochastic (statistical) transition in the time series. It is well-
known that trend, seasonality or other cyclical signals - that
are interpreted as deterministic, as opposed to stochastic -
can show (unwanted) non-stationarity. Hence, the algorithm
begins by filtering them out at the start of the process, using
methods such as regression, to ensure that they do not influ-
ence the detection of a regime change.
A non-stationary time-series is characterized by a spec-
trum that varies over time. The empirical methods for
monitoring non-stationarity are described below.
3.2.1 Moving window mean and variance
Exploratory investigations into deviations from second order
stationarity of a time-series Ytcan be performed by esti-
mating local sample means and variances of the time-series
within subsets of time windows. If partitioning the length of
the time-series ninto Loverlapping time windows wtl,l=
1,2,...,Lof equal length, nL, the local sample mean and
variances of Ytin window wlare defined as:
ˆmwl=1
nL
t:t∈wl
yt
ˆvwl=1
nL−1
t:t∈wl{yt−ˆmwl}2.
(7)
3.2.2 Dynamic periodogram: evolving features
of a time-series
Is(ωk,l)is defined as the local periodogram (see Eq. 6)ofthe
time-series Ytpartitioned in each of the wtl,l:l=1,2..., L
overlapping time windows of length nL, for all sampling
locations, {s:s=1,2, ..., v}, across the Fourier fre-
quencies, ωk=2πk/nL,k=1,2, ..., nL. Accordingly, Is
denotes the complete periodogram matrix at any arbitrary
location, s, defined as:
Is=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
Is(ω1,1)Is(ω1,2)...Is(ω1,L)
Is(ω2,1)Is(ω2,2)... Is(ω2,L)
.....
.....
.....
Is(ωnL,1)Is(ωnL,2)...Is(ωnL,L)
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
(8)
where, for simplicity, the local time window Is(ω1,w
tl)is
denoted by Is(ωk,l). Reordering the periodogram matrix in
Eq. 8for the periodograms at time window wtlacross all
sampling locations {s:s=1,2, ..., v}:
Iwtl=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
Iwtl,s1(ω1)Iwtl,s1(ω2)...Iwtl,s1(ωnL)
Iwtl,s2(ω1)Iwtl,s2(ω2)...Iwtl,s2(ωnL)
.....
.....
.....
Iwtl,sν(ω1)Iwtl,sν(ω2)...Iwtl,sν(ωnL)
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
(9)
For a second order stationary time-series, Iwtldefines the
signature of multiple time-series, Yt={Ys1(t), Ys2(t), . . . ,
123
Stochastic Environmental Research and Risk Assessment (2024) 38:2911–2926 2915
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
Ysv(t)}tin time window wtl.IfYtis composed of all sec-
ond order stationary time-series, Il1would be time invariant.
However, statistically evolving time-series would lead to a
time varying periodogram matrix, Il1.
3.2.3 Dynamic feature space: PCA periodograms
Principal components analysis (PCA) was introduced by Karl
Pearson and has since become a standard unsupervised statis-
tical learning method (Hastie et al. 2009). A common use of
PCA is for reducing the dimensionality of correlated primary
features in multivariable dataset. If a dataset consists of n
— potentially correlated — features, the application of PCA
leads to a smaller number (m<n)of uncorrelated secondary
features {Z1,Z2,...,Zm}that comprise linear combinations
of the primary (n) features.
It is well known that periodogram ordinates I(ωk)
of second order stationary time-series are asymptotically
uncorrelated at Fourier frequencies, ωk=2πk/n,k=
0,1,2, ..., n−1 (Shumway et al. 2000). However, a non-
stationary time-series has correlated periodograms. Here,
PCA is used for two purposes:
•to reduce the dimension of periodogram vectors (i.e.,
spectral features in each time window, wl) retaining most
of the variation in the signal;
•to obtain uncorrelated secondary spectral features (see
columns of Eq. 9,Iwtl) for multiple RLOSD time-series.
Mathematically, this is achieved by solving the following
optimization problem using a linear combination of peri-
odogram columns of Iwtl:
min Amvar(Zsj,m)where Zsj,m=
nl
i=1
αiIwtl,s1(ωi),
such that
nl
i=1
α2
i=1.
(10)
Subsequently, the obtained uncorrelated secondary fea-
tures are used to characterize the evolution of the state-of-
the-system, using multivariate statistical learning methods,
cluster analysis and classification.
3.2.4 Characterizing the system at regime change
The first stage in the stability monitoring process is to deter-
mine a time window within which the statistical features of
the ground displacement signal undergo a structural change
from a period of second order stationarity – known as the time
of regime change (wt0). A variety of statistical methods can
be used to detect the transition of the system from a second
order stationary to a non-stationary regime (e.g., frequentist,
Bayesian). However, owing to the relatively short time-
series in this retrospective analysis, an alternative approach
is adopted here. Defining:
Var(wtl)=
nl/2
i=0
Iwtl,s1(ωi), l=1,2,...,n.(11)
as the local variance of a time-series in window, wtl. Under
second order stationarity, Var(wtl)remains approximately
the same in all windows. However, a substantial departure
indicates a change in the statistical regime, which can be
identified by plotting Var(wtl)– for the average RLOSD
time-series for each of the six locations on the dam – against
time windows, wtl,l=1,2,...,L.
Inflection points on the the Var(wtl)plot are chosen as
potential candidates for the time of regime change. From a
pool of candidate inflection points, the most likely candidate
for the time window of regime change (wt0) is determined
using the principle of maximum inter-cluster variance (Das
and Tordesillas 2019). To achieve this, the local periodogram
in the window of regime change, Iwt0, for all locations is
estimated, as shown in Eq. 9, before using PCA to obtain a
matrix of secondary spectral features, Zwt0, for each loca-
tion s=,1,2,...,v. The feature matrix Zwt0encapsulates
the signature (i.e., slope stability) of the state-of-the-system
(i.e., the dam) at wt0, and is derived from the average RLOSD
time-series of all pixels at each location. This feature matrix is
then used to partition the (six) sampling locations of the dam
into a finite number (k) of clusters, C={C1,C2,...,Ck}.
Cisassumedtobethebaseline stable state-of-the-system
immediately preceding a dynamic transition into an unsta-
ble deformation regime. In the final stage, the evolution of
the system is measured using a statistic that sequentially
compares all subsequent spectral features Zwt0against the
baseline, C.
3.3 Classification and risk of failure
In this final phase we derive risk thresholds based on a
sequential classification methodology against the baseline
feature matrix C.
3.3.1 Classification
The following statistical law underpinning classification
error was a key observation by Das and Tordesillas (2019).
Consider a classification problem where at any given point
in time, t, it is desired to classify a group of observations
{1,2,...,n}based on features {I1,I2,...,In}into a finite
123
Stochastic Environmental Research and Risk Assessment (2024) 38:2911–2926
2916
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
number of classes, C={C1,C2,...,Ck}.Then,
ptj
Il =Prob.( j∈Cl|Ij), l=1,2,...,k
k
l=1
ptj
Il =1,j=1,2,...,n,
(12)
which denotes the probability of classification of a randomly
selected observation j=1,2,...,ninto class Clat time t.
For simplicity, the superscript, t, is ignored henceforth.
3.3.2 Risk trajectory
The classifier allocates the jth observation to class Cz;z∈
{1,2,...,k}if the class posterior probability pj, given fea-
tures Ij, satisfies:
pj=k
max
l=1pj
Il =Prob(Ij∈Cz), and
qj=1−pj,that is, qj=Prob(Ij/∈Cz)
z=arg max
lpIl,z∈{1,2,...,k},
the estimated class.
(13)
Defining M(pj)=pjqj,j=1,2,...,nto be the
theoretical classification variation and M(ˆpj)as the cor-
responding maximum likelihood estimator, then Das and
Tordesillas (2019) showed that expectation and variance of
ˆpjshare a parabolic relationship:
E{M(ˆpj)}=pjqj,and its variance,
var {M(ˆpj)}=1
n{pjqj(1−4pjqj)}(14)
as shown in Fig. 2. The algorithm can then estimate two
sequential slope instability risk warning thresholds – emer-
gent risk (tR) and imminent risk (tI) – based on the progres-
sion of this trajectory (Fig. 2):
tR=min
t0≤t≤tn
tsuch that var {M(ˆpj)}∼
=0.125,
tI=min
t0≤t≤tn
tsuch that E{M(ˆpj)}∼
=0.25 (15)
The first milestone, tR, is the time of an emergent insta-
bility risk, which corresponds to the theoretical maximum of
the trajectory at pjqj=0.125 (i.e., the point of inflection),
indicating a time of monotonic progression towards failure.
This is followed by tI, the time of imminent failure. The risk
warning tIis estimated as the time window when the clas-
sification variance reaches close to its theoretical maximum
that is, pjqj=0.25 (i.e., the maximum point of the trajec-
tory). Intuitively, tRcorresponds to the time point of highest
spatial (pixel) volatility of mis-classification (misalignment
Fig. 2 Relationship between the average classification uncertainty and
its variance. The three vertical lines correspond to the three risk thresh-
olds in time; wt0: time of regime change, tR: time of emergent risk, and
tI: time of imminent risk
with equilibrium classification) for the estimated features,
whereas tIcorresponds to the time block of highest spatial
average of mis-classification for the estimated features. On
this basis, retrospective empirical estimates of tRand tIfor
the Brumadinho tailings dam collapse are generated.
Note that estimates of population parameters (e.g. p)
based on nsamples p1,p2,..., pnare formally denoted by
ˆp. Accordingly, functions of parameters such as M(p)are
formally denoted by M(ˆp). However, for brevity we drop hat
notation from the estimates in the remainder of the article.
3.4 Algorithm implementation
Algorithm 1 summarises the methodology for generating
warnings of an emergent risk of failure (tR) and an immi-
nent risk of failure (tI). A primary difference of this new
algorithm compared with Das and Tordesillas (2019)isin
the features used in monitoring. While (Das and Tordesillas
2019) focused on first order properties of time-series, here
the framework is extended to include serially correlated and
non-stationary time-series. The algorithm is implemented
as a two-phase process using the statistical software R(R
Core Team 2021). For visualization, the package ggplot2
(Wickham 2016) is used.
123
Stochastic Environmental Research and Risk Assessment (2024) 38:2911–2926 2917
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
Algorithm 1 Algorithm for slope stability risk monitoring.
while wt<w
t0do
Estimate, I(ωk,w
t)
Estimate features, S(wt)from I(ωk,w
t)
Repeat if S(wt)is not an inflection point
end while
if S(wt0)is inflection point then
At Baseline: wt=wt0
Apply PCA on I(ωk,w
t0)
Cluster PCA features at wt0into Cclasses
if Inter-cluster variance ≥80% then
Move to classification phase
else
Continue feature estimation
end if
end if
if Baseline =wt0then
At wj,j=t0+1
Classify all locations into Cclasses of t0
pl(t)=class probability for class l∈C
ql(t)=1−pl(t)
Uj=med.L
lpl(j)ql(j)
if Uj<0.25 then
Repeat sequential classification at wj,j=
t0+2,t0+3,...,t0+m...
else
Ensure: Uj≈0.25
Declare j=tI, imminent risk
end if
end if
4 Results and discussion
4.1 Estimating the time of regime change
The first phase of the algorithm is the estimation of the time of
regime change, wt0, as described in Section 3.2.4. Figure 3a)
and b) show the trend and moving window variance of the
time-series of the mean RLOSD observed at each location
of the six locations, respectively. As previously outlined, the
mean RLOSD was computed as the spatial mean of all the
individual time-series extracted for the subsets of contiguous
pixels at the six locations (see Fig. 1), as defined by Grebby
et al. (2021). The time-series for locations 1, 2 and 3 cover
the time period (sampling window) from 19 August 2017
until 17 January 2019 (extracted from track 155), while those
for locations 4, 5 and 6 covered 24 August 2017 until 22
January 2019 (extracted from track 53). With the exception
of location 1, all locations exhibit a monotonically decreasing
(downward and/or westward) displacement trend (Fig. 3a).
Furthermore, for locations 2–6, it is also apparent that both
mean displacement and the moving window variances vary
over time, which indicates non-stationary in second order
statistical properties.
A key time-varying second order property is that of sam-
ple variance. An examination of Fig. 3b) shows that the
Brumadinho tailings dam underwent a deformation phase
transition. Prior to 2 July 2018, the trajectory of spatial vari-
ance decreased in magnitude at each location, while after this
date, increases in variance can be seen. The trend-adjusted
moving window variances shown in Fig. 3c) retain this prop-
erty. This pattern is consistent across all six locations, albeit
in varying magnitude.
As discussed in Section 3, the temporal invariance of the
spectral density function (Eq. 5) is the unique signature of
a secondary order stationary time-series, whereas the peri-
odogram (Eq. 6) is the sample estimator of the spectrum.
Thus, a common approach of assessing non-stationarity in
a time-series is to estimate local periodograms in moving
overlapping time windows and to then analyse the tempo-
ral variation of local periodograms. Periodograms of the
mean adjusted RLOSD in local windows of several dif-
ferent lengths (nL)were estimated. In each local window
wtl,l=1,2,...,Lwe add one time unit (12 days) and
remove the first time point of the previous window. The
resulting local periodograms for each location for a local
window length of 16 and overlaps of 15 time units are shown
in Fig. 4, where the x−axis on the plots represents the
Fourier frequencies, ωk. For a time-series of length 16, it is
only obtain possibly to obtain distinct estimates of spectra
at 8 Fourier frequencies due to aliasing (Nason et al. 2017).
Figure 4clearly shows clear temporal variation in the local
periodogram curves for each location. The evolving nature
of the time-series is evident in that the periodogram ordinates
appear more similar in some time windows – in terms of both
intercept and slope – than in other windows. It is the 8 peri-
odogram ordinates that are used as features in the sequential
slope stability monitoring algorithm.
The next key step in Phase I of the algorithm is to
then utilise the periodograms to identify a time (window)
of regime change – i.e., a phase transition of the system
from a stationary to non-stationary paradigm. To do this,
Das and Tordesillas (2019) suggested implementing the non-
stationarity metric on second order time domain properties
proposed by Das and Nason (2016) to assess deviation from
stationarity. Here, however, a simpler approach that is appli-
cable to spectral methods was adopted. It is well-known that
the spectrum is a orthogonal partition of the variance of a
time-series over Fourier frequencies (Shumway et al. 2000).
Equivalently, the moving window spectra partitions the local
variance – within a particular time window – among the
Fourier frequencies. When a time-series is second order sta-
tionary, the spectrum is its unique signature and does not
vary over time. This is observed as invariance in periodogram
plots, over local time windows. In other words, the local vari-
ance – obtained as the sum of local periodograms at Fourier
frequencies – must be uniform across all time windows. How-
ever, non-stationary time-series would have time varying (or
evolving) local variance as shown in Fig. 5a).
123
Stochastic Environmental Research and Risk Assessment (2024) 38:2911–2926
2918
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
Fig. 3 Spatial trend and moving window variances at six sampling locations. a) Time-series of spatial sample mean of relative LOS displacement
(RLOSD). b) Moving window spatial sample variance (MWV) of mean RLOSD. c) Moving window spatial sample variance (MWV) of the
detrended mean RLOSD
For all locations (1-6), the spatial median local variance
across all contiguous pixels across moving time blocks were
estimated to identify the points of inflection on the local
variance trajectory (Fig. 5b). Each point of inflection is con-
sidered to be a hypothetical point of regime change, wt0.
Accordingly, as shown in Fig. 5b), three prospective time
windows of regime change were identified, corresponding
to windows 7 (2017-10-30 : 2018-04-28), 9 (2017-11-23 :
2018-05-22), and 16 (2018-02-15 : 2018-08-14). At each
prospective wt0, medoid clustering (Hu et al. 2021)ofall
pixels into a finite number of clustering with periodograms
as features was performed, as described in Section 3.2.4.The
time window that led to the highest proportion of explained
inter-cluster variation (with a threshold of ≈80%) for the
fewest number of cluster partitions was chosen as wt0(Das
and Tordesillas 2019). The corresponding variations for the
different numbers of cluster partitions are given in Table 1.
With inter-cluster variation of 81% for 4 cluster partitions,
wt0=16 (2018-02-15 : 2018-08-14) was selected as the
optimal time of regime change.
4.2 Risk of failure warnings
Using local periodograms for a window length of 16 as
features, all time-series were subsequently classified into
the class labels of wt0at all subsequent points of time
wt0+1,w
t0+2,.... Mis-classification ( pq) is treated as an indi-
cation of the evolution of the state-of-the-system. At each
time window, the spatial median of mis-classification and
the corresponding interquartile range (IQR) for pq were cal-
culated. The time-series of Median(pq)and IQR(pq)for
wt0=16 (2018-02-15 : 2018-08-14) are shown in Fig. 6.
Based on the relationship between pq and its variability, ret-
rospective warnings for the time windows of emergent (tR)
and imminent risk of failure (tI) for the Brumadinho tail-
ings dam were 27 February 2018–26 August 2018 and 27
123
Stochastic Environmental Research and Risk Assessment (2024) 38:2911–2926 2919
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
Fig. 4 Local periodogram (window length of 16) of detrended mean RLOSD in successive time windows for each location
June 2018–24 December 2018, respectively. Consequently,
the algorithm identifies a risk of imminent failure of the dam
at least a month prior to the actual event. This is compa-
rable with the 40 days of advanced warning based on the
inverse velocity method by Grebby et al. (2021). If moni-
tored using this approach in near real-time, this would have
provided adequate warning for detailed site investigations or
risk mitigation measures to have been implemented. More-
over, the warning of an emergent risk is even earlier, at least
five months prior to the collapse, coinciding with an initial
phase of accelerating deformation highlighted by Gama et al.
(2020). This is considerably earlier than the earliest reliable
indication of an emerging failure 51 days prior based on the
inverse velocity method (Grebby et al.2021). Although, with
hindsight, this deformation could correspond to a period of
secondary creep rather than tertiary, at the time, such a warn-
ing would still have been useful in initiating crucial follow-up
assessments of the stability of the dam.
4.3 Effect of time windows on risk of failure
warnings
4.3.1 Local periodograms
The discussion thus far has focused on the results associated
with the estimation of local periodograms over time windows
Table 1 Explained intra- and inter-cluster variations for different numbers of cluster partitions, 2,3,4,5, at various hypothetical times of regime
change, wt0=7 (2017-10-30 : 2018-04-28), wt0=9 (2017-11-23 : 2018-05-22), and wt0=16 (2018-02-15 : 2018-08-14)
Time window No. of clusters Within cluster variation Inter-cluster variation(%)
7 2 0.304 0.153 54%
3 0.117 0.071 0.113 70%
4 0.041 0.034 0.078 0.071 78%
5 0.033 0.027 0.041 0.014 0.062 82%
9 2 0.182 0.245 57%
3 0.115 0.097 0.06 73%
4 0.012 0.075 0.058 0.038 82%
5 0.040 0.048 0.020 0.012 0.021 86%
16 2 0.148 0.32 53%
3 0.106 0.072 0.071 75%
4 0.018 0.042 0.06 0.072 81%
5 0.060 0.005 0.018 0.024 0.042 85%
123
Stochastic Environmental Research and Risk Assessment (2024) 38:2911–2926
2920
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
Fig. 5 Local variance as sum of
periodogram ordinates. a)
Pixel-wise local variance of
mean RLOSD at successive time
windows. b) Median local
variance of mean RLOSD at
successive time windows
of length 16. In order to assess the effect of the length of the
moving window on the regime change and risk estimation,
several other window lengths were used in the computations.
These included periodogram estimates for the same time-
series over moving windowsof length 24 and 32 , with varying
overlap lengths.
Comparing Figs. 7and 8with Fig. 4, it can be observed
that all three window estimates show temporal variation (and
Fig. 6 Risk time-series and risk thresholds tR(in gold) and tI(in red) for wt0=16 (2018-02-15 : 2018-08-14). a)Riskoffailure,Median(pq).
b)Variabilityofriskoffailure,IQR(pq )
123
Stochastic Environmental Research and Risk Assessment (2024) 38:2911–2926 2921
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
clustering) of local periodogram curves. The primary differ-
ences are two-fold:
•longer windows allow for a broader spectrum of vari-
ation. That is, when choosing local periodograms for
time-series of length 32 it is possible to estimate (low
frequency) periodic components at twice the length of
that allowed by time window of length 16 (1 time unit =
12 days).
•Accordingly, the local window (numbers) of clustering
vary.
Overall, the analysis reveals that all time window lengths
estimated the maximum variance to coincide with the Fourier
frequency that corresponds to the period of July–August
2018. This confirms that the window length over which the
local periodograms are estimated have negligible effect on
the prospective time windows of regime change and, hence,
the subsequent timing of the risk of failure warnings. How-
ever, this may not necessarily always be true, and so a more
objective window selection approach would be beneficial in
generalising the algorithm for application to a broader range
of cases.
The key to objectively optimising window selection is a
methodology that can estimate periodograms at all frequen-
cies that have significant variance components, for given a
slope failure process. In other words, such a method must
allow estimation of power, inherent in the RLOSD time-
series, at the smallest typical frequency or longest time period
typical of that type of slope failure, conditional on the sam-
pling frequency. Examples of window width estimation (as
also a window kernel) include those from within the statisti-
cal sciences and signal processing (equivalent to bandwidth),
such as density estimation (Green and Silverman 1993;Bow-
man 1984), trend estimation (Fried 2004), and periodogram
estimation (Ombao et al. 2001). As demonstrated by many
of these examples, a rigorous and automatic approach can
be developed using the resampling methodology known as
cross-validation. One potential solution, based on mecha-
nistic apriori knowledge of prevalent Fourier spectra of the
RLOSD time-series for a given type of slope failure, would be
to use a cross-validation-based decision criteria to select the
optimal window length by minimizing a loss function which
will ensure that the spectra at the lowest relevant frequency
can be estimated.
4.3.2 Time of regime change windows
As previously shown in Fig. 5b), three inflection points were
identified and then subjected to cluster analysis (Table 1)to
estimate the optimal time window of regime change ( ˆwt0). As
a result, wt0=16 (2018-02-15 : 2018-08-14) was selected.
However, to assess the effect of the time window of regime
change on the risk of failure warnings, estimates of the risk
thresholds, tRand tI, were computed for the two alternative
prospective time windows of wt0=7 (2017-10-30 : 2018-
04-28) and 9 (2017-11-23 : 2018-05-22). Figures 9and 10
show the time-series of the median and interquartile range
(IQR) for the risk statistic ( pq) and risk thresholds tRand tI
for wt0=7 and wt0=9, respectively. As for wt0=16, it
can be seen that both alternative candidates for the time of
regime change also predict an imminent risk of failure (tI)
well in advance of the actual collapse of the dam. For wt0=7,
an imminent risk of failure is detected in the time window
11 March 2018–7 September 2018, while for wt0=9this
risk is detected during the period 16 April 2018–13 Octo-
ber 2018. These risk warnings precede that for the optimal
window for the time of regime change (wt0=16) by 2–3
months and the actual collapse by several months. However,
as also observed by Das and Tordesillas (2019), these earlier
time windows of regime change (wt0=7 and wt0=9) pro-
duce maximum values for the risk statistic (pq) that are less
than that for both wt0=16 and the theoretical maximum
of 0.25. Consequently, the risk warnings generated for the
earlier alternative time windows are less accurate than those
for the optimal time of regime change window of 16.
Although in this case the cluster analysis was able to
identify the optimal time of regime change window from
three prospective candidates, a more robust approach could
be more effective in addressing the issue of identifying a
unique solution. The problem of detecting a time of regime
change is similar in essence to that of detecting a structural
change point in a time-series. With regards to change point
detection, there are numerous potential approaches from both
the frequentist and Bayesian perspectives that require future
investigation (Killick et al. 2012; Cho and Fryzlewicz 2012,
2015; Ghassempour et al. 2014). Alternately, detecting ˆwt0
can also be considered analogous to detecting anomalies in
streaming data, as in Hill et al. (2009); Dereszynski and Diet-
terich (2011).
5 Conclusions
Conventional approaches to predicting slope failures are
based on the inverse velocity method. However, predictions
made using this method can be unreliable for risk manage-
ment owing to its highly subjective nature and the fact that
the deterministic relationships of this exponential law often
materialises close to the occurrence of a failure, therefore
offering limited prior warning of a potential catastrophe. In
this study, an alternative risk identification approach is pro-
posed, which incorporates theories from spectral analysis
of time-series to extend a data-driven sequential monitor-
ing algorithm developed by Das and Tordesillas (2019).
This statistical learning algorithm is more objective and can
123
Stochastic Environmental Research and Risk Assessment (2024) 38:2911–2926
2922
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
Fig. 7 Local periodogram (window length of 24) of detrended mean RLOSD in successive time windows
be applied to any displacement (or velocity) time-series,
sampled at regular time intervals, accounting for serial cor-
relation.
The efficacy of this new algorithm has been demonstrated
through application to satellite InSAR displacement monitor-
ing data associated with the 2019 Dam I Brumadinho tailings
dam collapse. The algorithm analyses second order proper-
ties to show that the tailings dam transitioned into an unstable
epoch around July 2018, several months before the actual col-
lapse. This was evident in the moving window variance plots
and further corroborated using the periodogram-based local
variance plots; the latter of which accounts for serial corre-
lation within each displacement time-series. Whereas most
slope stability monitoring analysis only considers the trend
or first order moments of the deformation signal, the results
of this study shows that investigating second order statistical
properties is critical.
Fig. 8 Local periodogram (window length of 32) of detrended mean RLOSD in successive time windows
123
Stochastic Environmental Research and Risk Assessment (2024) 38:2911–2926 2923
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
Fig. 9 Panels show the trajectories of risk statistic with time and risk thresholds tR(in gold) and tI(in red) for wt0=7 (2017-10-30 : 2018-04-28).
a) Risk of failure, Median(pq). b) Variability of risk of failure, IQR(pq )
The optimum time of regime change window, wt0=16
(2018-02-15 : 2018-08-14), was identified based on cluster
analysis of dynamic spectral features and then subsequently
used to retrospectively estimate risk of failure warnings for
Dam I. The algorithm detected an emerging risk of failure
during the period 27 February 2018–26 August 2018 and an
imminent (maximum) risk of collapse of the tailings dam
during 27 June 2018–24 December 2018. The latter warning
comes at least several weeks prior to the actual dam failure
on 25 January 2019, while the first sign of an emerging risk is
evident at least five months prior to the failure. The amount of
advanced warning for the risk of imminent failure provided
by the algorithm is comparable to previous analysis based on
the inverse velocity method, although considerably longer in
terms of reliably detecting signs of an emerging failure. This
is ultimately due to the statistical nature of the algorithm mak-
ing it more objective and robust than empirical approaches,
which also means that it can be readily applied to other slope
failures. Although the risk warnings generated here are based
on retrospective analysis, application of the algorithm on
near real-time monitoring data could have provided sufficient
early-warning to enable detailed site investigations or more
urgent risk mitigation measures to have been implemented in
order to advert a humanitarian and environmental catastro-
phe. Overall, the results of this study further attests the role
that satellite InSAR can have in integrated dam monitoring
and early warning systems.
This study has demonstrated the efficacy of the algorithm
on deformation time-series acquired by at a sampling fre-
quency of 12-day intervals through the Sentinel-1 satellite
mission. Nevertheless, it is readily adaptable to any regularly
sampled deformation time-series, as demonstrated in Das and
Tordesillas (2019) with ground-based radar data acquired at
a significantly higher frequency of 6-minute intervals. How-
ever, whilst a higher sampling frequency would likely allow a
faster lead time for the detection of the time of regime change,
it could also introduce more false-positive risk warnings.
Although this can be addressed by determining the opti-
mum frequency at which to sample the specific process being
studied (Nason et al. 2017), the algorithm itself could be
made more robust and extensible to analysing the risk asso-
ciated with any potential slope failure scenario. This could
Fig. 10 Panels show the trajectories of risk statistic with time and risk thresholds tR(in gold) and tI(in red) for wt0=9 (2017-11-23 : 2018-05-22).
a) Risk of failure, Median(pq).b) Variability of risk of failure, IQR(pq )
123
Stochastic Environmental Research and Risk Assessment (2024) 38:2911–2926
2924
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
be primarily achieved through the implementation of more
objective or automated approaches for selecting the optimal
time window length and time window of regime change.
Additionally, the algorithm may also be further augmented
by considering both spatial and temporal auto-correlation, in
order to develop a spatio-temporal risk statistic. This would
enable not just a temporal warning of a risk of failure, but
also highlight the specific location of the instability.
Author Contributions S.D.: Conceptualization, Formal Analysis,
Methodology, Software, Supervision, Visualization, Writing - Origi-
nal Draft, S.G.: Data curation, Supervision, Writing- Original draft,
Reviewing and Editing. A.P.: Conceptualization, Formal Analysis,
Methodology, Software, Supervision, Visualization, Writing - Original
Draft
Funding Open Access funding enabled and organized by CAUL and
its Member Institutions.
Data Availability Please contact authors for availability of data, if the
manuscript is accepted for publication.
Declarations
Competing interests The authors declare no competing interests.
Open Access This article is licensed under a Creative Commons
Attribution 4.0 International License, which permits use, sharing, adap-
tation, distribution and reproduction in any medium or format, as
long as you give appropriate credit to the original author(s) and the
source, provide a link to the Creative Commons licence, and indi-
cate if changes were made. The images or other third party material
in this article are included in the article’s Creative Commons licence,
unless indicated otherwise in a credit line to the material. If material
is not included in the article’s Creative Commons licence and your
intended use is not permitted by statutory regulation or exceeds the
permitted use, you will need to obtain permission directly from the copy-
right holder. To view a copy of this licence, visit http://creativecomm
ons.org/licenses/by/4.0/.
References
Bevilacqua A et al (2019) Probabilistic enhancement of the failure fore-
cast method using a stochastic differential equation and application
to volcanic eruption forecasts. Front Earth Sci 7. https://doi.org/
10.3389/feart.2019.00135
Bowman AW (1984) An alternative method of cross-validation for the
smoothing of density estimates. Biometrika 71(2):353–360
Carlà T et al (2017) Guidelines on the use of inverse velocity method as
a tool for setting alarm thresholds and forecasting landslides and
structure collapses. Landslides 14(2):517–534
Carlà T et al (2019) Perspectives on the prediction of catastrophic slope
failures from satellite insar. Sci Rep 9(1):1–9
Carlà T, Farina P, Intrieri E, Botsialas K, Casagli N (2017) On the
monitoring and early-warning of brittle slope failures in hard rock
masses: examples from an open-pit mine. Eng Geol 228:71–81
Carlà T, Farina P, Intrieri E, Ketizmen H, Casagli N (2018) Integration
of ground-based radar and satellite insar data for the analysis of an
unexpected slope failure in an open-pit mine. Eng Geol 235:39–52
Casagli N, Catani F,Del Ventisette C, Luzi G (2010) Monitoring, predic-
tion, and early warning using ground-based radar interferometry.
Landslides 7(3):291–301
Cheloni D, Famiglietti NA, Tolomei C, Caputo R, Vicari A (2024) The
8 september 2023, mw6.8, Morocco earthquake: a deep transpres-
sive faulting along the active high atlas mountain belt. Geophys
Res Lett 51(2):e2023GL106992
Cho H, Fryzlewicz P (2012) Multiscale and multilevel technique for
consistent segmentation of nonstationary time series. Stat Sin 207–
229
Cho H, Fryzlewicz P (2015) Multiple-change-point detection for high
dimensional time series via sparsified binary segmentation. J R
Stat Soc: Ser B: Stat Methodol 475–507
Colesanti C, Wasowski J (2006) Investigating landslides with space-
borne synthetic aperture radar (sar) interferometry.Eng Geol 88(3–
4):173–199
Das S, Nason GP (2016) Measuring the degree of non-stationarity of a
time series. Stat 5(1):295–305
Das S, Tordesillas A (2019) Near real-time characterization of spatio-
temporal precursory evolution of a rockslide from radar data:
integrating statistical and machine learning with dynamics of
granular failure. Remote Sens 11(23). https://doi.org/10.3390/
rs11232777
de Lima RE, de Lima Picanço J, da Silva AF, Acordes FA (2020) An
anthropogenic flow type gravitational mass movement: the cór-
rego do feijão tailings dam disaster. brumadinho brazil. Landslides
17:2895–2906
Dereszynski EW, Dietterich TG (2011) Spatiotemporal models for
data-anomaly detection in dynamic environmental monitoring
campaigns. ACM Trans Sens Netw (TOSN) 8(1):1–36
Dick GJ, Eberhardt E, Cabrejo-Liévano AG, Stead D, Rose ND
(2015) Development of an early-warning time-of-failure analy-
sis methodology for open-pit mine slopes utilizing ground-based
slope stability radar monitoring data. Can Geotech J 52(4):515–
529
Du Z et al (2020) Risk assessment for tailings dams in brumadinho
of brazil using insar time series approach. Sci Total Environ
717:137125
Fried R (2004) Robust filtering of time series with trends. J Nonpara-
metric Stat 16(3–4):313–328
Fruneau B, Achache J, Delacourt C (1996) Observation and modelling
of the saint-etienne-de-tinée landslide using sar interferometry.
Tectonophysics 265(3–4):181–190
Fukuzuno T (1985) A method to predict the time of slope failure caused
by rainfall using the inverse number of velocityof surface displace-
ment. Landslides 22(2):8–13
Gama FF,Mura JC, Paradella WR, de Oliveira CG (2020) Deformations
prior to the brumadinho dam collapse revealed by sentinel-1 insar
data using sbas and psi techniques. Remote Sens 12(21):3664
Gee D et al (2019) National geohazards mapping in Europe: Interfero-
metric analysis of the Netherlands. Eng Geol 256:1–22
Ghassempour S, Girosi F, Maeder A (2014) Clustering multivariate
time series using hidden markov models. Int J Environ Res Public
Health 11(3):2741–2763
Grebby S et al (2021) Advanced analysis of satellite data reveals ground
deformation precursors to the brumadinho tailings dam collapse.
Commun Earth Environ 2(1):1–9
Green PJ, Silverman BW (1993) Nonparametric regression and gener-
alized linear models: a roughness penalty approach (Crc Press)
Hastie T, Tibshirani R, Friedman JH, Friedman JH (2009) The elements
of statistical learning: data mining, inference, and prediction, vol
2. Springer
Hill DJ, Minsker BS, Amir E (2009) Real-time bayesian anomaly detec-
tion in streaming environmental data. Water Resour Res 45(4)
123
Stochastic Environmental Research and Risk Assessment (2024) 38:2911–2926 2925
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
Hu J et al (2021) A novel landslide susceptibility mapping portrayed by
oa-hd and k-medoids clustering algorithms. Bull Eng Geol Environ
80(2):765–779
Intrieri E et al (2018) The maoxian landslide as seen from space:
detecting precursors of failure with sentinel-1 data. Landslides
15(1):123–133
Intrieri E, Gigli G (2016) Landslide forecasting and factors influencing
predictability. Nat Hazards Earth Syst Sci 16(12):2501–2510
Intrieri E, Gigli G, Casagli N, Nadim F (2013) Brief communication"
landslide early warning system: toolbox and general concepts".
Nat Hazards Earth Syst Sci 13(1):85–90
Killick R, Fearnhead P, Eckley IA (2012) Optimal detection of
changepoints with a linear computational cost. J Am Stat Assoc
107(500):1590–1598
Massonnet D et al (1993) The displacement field of the landers earth-
quake mapped by radar interferometry.Nature 364(6433):138–142
Massonnet D, Feigl KL (1995) Discrimination of geophysical phe-
nomena in satellite radar interferograms. Geophys Res Lett
22(12):1537–1540
Nason GP, Powell B, Elliott D, Smith PA (2017) Should we sample a
time series more frequently?: decision support via multirate spec-
trum estimation. J R Stat Soc: Ser A (Stat Soc) 180(2):353–407
Niu Y, Zhou X-P (2021) Forecast of time-of-instability in rocks under
complex stress conditions using spatial precursory ae response
rate. Int J Rock Mech Min Sci 147:104908
Ombao HC, Raz JA, Strawderman RL, Von Sachs R (2001) A simple
generalised crossvalidation method of span selection for peri-
odogram smoothing. Biometrika 88(4):1186–1192
Peltzer G, Rosen P (1995) Surface displacement of the 17 May 1993
Eureka Valley, California, earthquake observed by sar interferom-
etry. Science 268(5215):1333–1336
Pinel V, Poland MP, Hooper A (2014) Volcanology: lessons learned
from synthetic aperture radar imagery. J Volcanol Geotherm Res
289:81–113
Porsani JL, Jesus FANd, Stangari MC (2019) Gpr survey on an iron
mining area after the collapse of the tailings dam i at the córrego
do feijão mine in Brumadinho-mg, Brazil. Remote Sens 11(7):860
R Core Team (2021) R: A language and environment for statistical com-
puting. R Foundation for Statistical Computing, Vienna, Austria.
https://www.R-project.org/
Raspini F et al (2022) Review of satellite radar interferometry for sub-
sidence analysis. Earth-Sci Rev 235:104239
Robertson P, de Melo L, Williams D, Wilson G (2019) Report of the
expert panel on the technical causes of the failure of feijão dam i.
Commissioned by Vale
Rose ND, Hungr O (2007) Forecasting potential rock slope failure in
open pit mines using the inverse-velocity method. Int J Rock Mech
Min Sci 44(2):308–320
Rosen PA et al (2000) Synthetic aperture radar interferometry. Proc
IEEE 88(3):333–382
Rotta LHS et al (2020) The 2019 brumadinho tailings dam collapse:
possible cause and impacts of the worst human and environmental
disaster in brazil. Int J Appl Earth Obs Geoinf 90:102119
Saito T (1969) Geomagnetic pulsations. Space Sci Rev 10(3):319–412
Schuster A (1906) Ii. On the periodicities of sunspots. Philos Trans R
Soc Lond. Ser A, Contain Paper Math Phys Character 206(402–
412):69–100
Shumway RH, Stoffer DS, Stoffer DS (2000) Time series analysis and
its applications, vol 3. Springer
Song C et al (2022) Triggering and recovery of earthquake accelerated
landslides in central italy revealed by satellite radar observations.
Nat Commun 13:7278
Sowter A, Bateson L, Strange P, Ambrose K, Syafiudin MF (2013)
Dinsar estimation of land motion using intermittent coherence with
application to the south derbyshire and leicestershire coalfields.
Remote Sens Lett 4(10):979–987
Tordesillas A, Zhou Z, Batterham R (2018) A data-driven complex
systems approach to early prediction of landslides. Mech Res Com-
mun 92:137–141
Voight B (1988) A method for prediction of volcanic eruptions. Nature
332(6160):125–130
Voight B (1989) A relation to describe rate-dependent material failure.
Science 243(4888):200–203
Wang H, Qian G, Tordesillas A (2020) Modeling big spatio-temporal
geo-hazards data for forecasting by error-correction cointegration
and dimension-reduction. Spat Stat 36. https://doi.org/10.1016/j.
spasta.2020.100432
Wasowski J, Bovenga F (2014) Investigating landslides and unstable
slopes with satellite multi temporal interferometry: current issues
and future perspectives. Eng Geol 174:103–138
Wickham H (2016) ggplot2: elegant graphics for data analysis.
Springer-Verlag New York. https://ggplot2.tidyverse.org
Zhang J, Wang Z, Zhang G, Xue Y (2020) Probabilistic prediction of
slope failure time. Eng Geol 271:105586
Zhou S, Bondell H, Tordesillas A, Rubinstein BI, Bailey J (2020) Early
identification of an impending rockslide location via a spatially-
aided gaussian mixture model. Ann Appl Stat 14(2):977–992
Zhu F, Zhang W, Puzrin AM (2024) The slip surface mechanism of
delayed failure of the Brumadinho tailings dam in 2019. Commun
Earth Environ 5:33
Publisher’s Note Springer Nature remains neutral with regard to juris-
dictional claims in published maps and institutional affiliations.
123
Stochastic Environmental Research and Risk Assessment (2024) 38:2911–2926
2926
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
1.
2.
3.
4.
5.
6.
Terms and Conditions
Springer Nature journal content, brought to you courtesy of Springer Nature Customer Service Center GmbH (“Springer Nature”).
Springer Nature supports a reasonable amount of sharing of research papers by authors, subscribers and authorised users (“Users”), for small-
scale personal, non-commercial use provided that all copyright, trade and service marks and other proprietary notices are maintained. By
accessing, sharing, receiving or otherwise using the Springer Nature journal content you agree to these terms of use (“Terms”). For these
purposes, Springer Nature considers academic use (by researchers and students) to be non-commercial.
These Terms are supplementary and will apply in addition to any applicable website terms and conditions, a relevant site licence or a personal
subscription. These Terms will prevail over any conflict or ambiguity with regards to the relevant terms, a site licence or a personal subscription
(to the extent of the conflict or ambiguity only). For Creative Commons-licensed articles, the terms of the Creative Commons license used will
apply.
We collect and use personal data to provide access to the Springer Nature journal content. We may also use these personal data internally within
ResearchGate and Springer Nature and as agreed share it, in an anonymised way, for purposes of tracking, analysis and reporting. We will not
otherwise disclose your personal data outside the ResearchGate or the Springer Nature group of companies unless we have your permission as
detailed in the Privacy Policy.
While Users may use the Springer Nature journal content for small scale, personal non-commercial use, it is important to note that Users may
not:
use such content for the purpose of providing other users with access on a regular or large scale basis or as a means to circumvent access
control;
use such content where to do so would be considered a criminal or statutory offence in any jurisdiction, or gives rise to civil liability, or is
otherwise unlawful;
falsely or misleadingly imply or suggest endorsement, approval , sponsorship, or association unless explicitly agreed to by Springer Nature in
writing;
use bots or other automated methods to access the content or redirect messages
override any security feature or exclusionary protocol; or
share the content in order to create substitute for Springer Nature products or services or a systematic database of Springer Nature journal
content.
In line with the restriction against commercial use, Springer Nature does not permit the creation of a product or service that creates revenue,
royalties, rent or income from our content or its inclusion as part of a paid for service or for other commercial gain. Springer Nature journal
content cannot be used for inter-library loans and librarians may not upload Springer Nature journal content on a large scale into their, or any
other, institutional repository.
These terms of use are reviewed regularly and may be amended at any time. Springer Nature is not obligated to publish any information or
content on this website and may remove it or features or functionality at our sole discretion, at any time with or without notice. Springer Nature
may revoke this licence to you at any time and remove access to any copies of the Springer Nature journal content which have been saved.
To the fullest extent permitted by law, Springer Nature makes no warranties, representations or guarantees to Users, either express or implied
with respect to the Springer nature journal content and all parties disclaim and waive any implied warranties or warranties imposed by law,
including merchantability or fitness for any particular purpose.
Please note that these rights do not automatically extend to content, data or other material published by Springer Nature that may be licensed
from third parties.
If you would like to use or distribute our Springer Nature journal content to a wider audience or on a regular basis or in any other manner not
expressly permitted by these Terms, please contact Springer Nature at
onlineservice@springernature.com
