1 Introduction

Landslides occur across a wide range of Earth surface environments, posing severe threats to life and property (Froude & Petley, 2018; Lacroix et al., 2020; Pánek et al., 2024; Petley, 2012). Therefore, forecasting catastrophic slope failures is a fundamental goal of landslide hazard analysis. Over the past decades, great efforts have been dedicated to developing and deploying high-precision monitoring technologies to observe unstable slope movements (Casagli et al., 2023; Crosta et al., 2017), aiming to detect precursory features of imminent catastrophic failure events.

Several mechanisms and models have been proposed for understanding the approximate power law time-to-failure dynamics (Bell et al., 2011; Bufe & Varnes, 1993; Fukuzono, 1985; Kilburn & Petley, 2003; Voight, 1988, 1989) often observed in landslides. One mechanism builds on the analogy between catastrophic failures and critical points, based on the intrinsic scaling symmetry of power laws which is reminiscence of the scaling invariance symmetry enjoyed by critical phase transitions (Ausloos, 1986; Sornette, 2006). Other mechanisms have been proposed, such as the slider-block friction model, which attributes accelerating displacements to frictional instabilities along sliding surfaces (Handwerger et al., 2016; Helmstetter et al., 2004; Paul et al., 2024; Poli, 2017; Yamada et al., 2016) as well as stress corrosion-driven damage accumulation (Cornelius & Scott, 1993; Kilburn & Voight, 1998; Main, 2000; Sammis & Sornette, 2002). Modeling the time-to-failure dynamics as a finite-time power law singularity (PLS) has been used to forecast the timing of landslide failures and develop early warning systems (Bell, 2018; Crosta & Agliardi, 2002, 2003; Intrieri et al., 2012, 2019; Leinauer et al., 2023). However, significant uncertainties have been found in applying the PLS model for time-to-failure prediction, primarily due to the sporadic nature of slope rupture phenomena, which challenges the assumption of a smooth, monotonic power law acceleration.

The Log-Periodic Power Law Singularity (LPPLS) model, which incorporates log-periodic corrections to the power law trend, has been developed to capture the intermittent oscillations of heterogeneous systems approaching global breakdown (Anifrani et al., 1995; Johansen & Sornette, 1998; Sornette & Sammis, 1995). Recently, this model was shown to provide superior fits compared to the conventional PLS model, based on the thorough analysis of a comprehensive global data set of historical geohazard events, including landslides, rockbursts, glacier breakoffs, and volcanic eruptions (Lei & Sornette, 2025c) (see Section 4 for further discussion). By leveraging the irregular and intermittent patterns in rupture dynamics, the LPPLS model transforms unsteady non-monotonous signals, traditionally perceived as noise, into essential components of the predictive framework, offering a promising tool for forecasting catastrophic events.

This study aims to provide further empirical evidence and theoretical arguments on the presence of log-periodicity in landslides, which is of both fundamental and practical interest. From a fundamental perspective, log-periodicity signals a spontaneous hierarchical organization of damage in heterogenous systems, offering insights into the mechanisms that drive rupture dynamics during landslide crises. From a practical perspective, log-periodicity can enhance the reliability of failure time forecast by “locking” the model fit into the accelerating oscillatory pattern of landslide motions. In this study, our primary objective is to demonstrate that the LPPLS model provides an excellent description of the observed data. Establishing this descriptive power is a critical prerequisite that lays the foundation for the next step: pseudo-prospective and prospective forecasting. By focusing on a rigorous retrospective analysis, we highlight the significance of log-periodicity and set the stage for meaningful and robust future research on prediction.

2 Methodology

By introducing three additional parameters, that is, a constant C, an angular log-frequency ω, and a phase shift ϕ, Equation 4 contains a log-periodic correction with a relative amplitude of C/B (typically on the order of 10⁻¹) to the power law trend with the pre-factor B. Here, the continuous scale invariance is partially broken into a discrete scale invariance (Saleur et al., 1996a; Sornette, 1998), where the observable obeys scale invariance under scaling of t_c ‒ t by specific factors that are integer powers of a specific fundamental scaling ratio λ = exp(2π/ω) > 1. The local maxima of the log-periodic term in the LPPLS formula occur at times converging to t_c according to a geometric time series {t₁, t₂, …, t_k, …} with t_c ‒ t_k = λ^−kexp(ϕ/ω) and k being an integer. This geometric time series is formed by time points where the argument of the cosine function in Equation 4 is an integer multiple of 2π. With this embedded discrete hierarchy of time scales, the LPPLS model can capture the intermittent rupture dynamics with a geometric increase in burst frequency on the approach to t_c, arising from the localized and threshold nature of rupture in heterogeneous materials (Johansen & Sornette, 2000; Sornette, 2002). Here, Equation 4 includes only the first correction term, while higher-order terms with decreasing amplitudes also exist but are in general relatively less significant (Zhou & Sornette, 2002a).

We implement a stable and robust parametric calibration scheme, briefly described as follows (see Text S1 in Supporting Information S1 for more details). First, the Lagrange regularization approach (Demos & Sornette, 2019) is employed to detect the onset time t₀ of an acceleration crisis, based on which the optimal time window is defined; here, the end of this time window is fixed either at the last available data point (if the crisis culminates in a catastrophic failure) or at the time stamp of peak velocity during the crisis (if the landslide self-stabilizes afterward). Then, the optimal parameter values for the LPPLS model are determined by minimizing the sum of the squares of the residuals, which quantifies the difference between the model and data (Filimonov & Sornette, 2013). The data used for LPPLS calibration are cumulative rather than rate-based, so as to suppress high-frequency noise and enhance signal clarity. This acts as a low-pass filter (Huang et al., 2000), reducing noise-dominated fluctuations while preserving low-frequency, physically meaningful signals.

3 Results

The first case study is the Veslemannen landslide, composed of high-grade metamorphic rocks and situated on a north-facing slope in Romsdalen, western Norway. A ground-based interferometric synthetic-aperture radar system has been installed since October 2014 to continuously monitor this actively moving landslide (Kristensen et al., 2021). Major acceleration events, accompanied by substantial surface displacements and pronounced velocity spikes, were recorded in 2017, 2018, and 2019 (Figure S1 in Supporting Information S1). This instability complex was mainly active during summer/autumn months, likely due to rainwater infiltration into the slope through the thawed upper frost zone. Eventually, on 5 September 2019, ∼54,000 m³ of unstable rock collapsed. Over the 5-year monitoring period, the slope cumulatively displaced ∼19 and ∼4 m in the upper and lower regions, respectively.

We perform the LPPLS calibration and Lomb periodogram analysis on the slope displacement time series during the three major acceleration crises in 2017, 2018, and 2019. This landslide exhibits a superimposition of acceleration and oscillations during all the three crises, as observed across all radar points (Figure 1 and Figure S2–S4 in Supporting Information S1). It is evident that the LPPLS model gives an excellent fit to the data, with the intermittency well captured. The duration of this log-periodic behavior increased progressively over the years. Log-periodic oscillations are evident in the ϵ-lnτ plot that closely follows a cosine function (insets of Figures 1a, 1c, and 1e), where some discrepancy in the small lnτ region in Figure 1e may arise from the effects of higher-order harmonics of the oscillations. From the Lomb periodograms (Figures 1b, 1d, and 1f as well as Figures S2–S4 and Table S2 in Supporting Information S1), we can clearly identify a dominant peak at the log-frequency f ≈ 1.1 (with the angular log-frequency ω ≈ 7.0 and the scaling ratio λ ≈ 2.5) for the 2017 and 2019 crises across all the radar points, while the dominant peak for the 2018 crisis occurs at f ≈ 0.8 (with ω ≈ 5.0 and λ ≈ 3.5). Interestingly, for the 2017 and 2018 crises, a harmonic can be respectively found at the log-frequency of around 2.2–2.5 and 1.5–2.0 among most radar points (Figures 1b, 1d, and 1f and Figures S2–S4 in Supporting Information S1), which corresponds roughly to the second harmonic 2f. It is also evident that the maximum Lomb peak height P_max increases over time, from 17.4 ± 1.9 in 2017 to 22.8 ± 2.1 in 2018, and then to 31.6 ± 6.1 in 2019 (Figures 1b, 1d, and 1f as well as Figures S2–S4 and Table S2 in Supporting Information S1), confirming increasing statistical significance.

Following the same procedure, we perform the LPPLS calibration and Lomb periodogram analysis on various landslides based on a compiled global data set of 52 landslides including 114 displacement time series (Figures S5–S15 in Supporting Information S1). This data set covers different types of landslides including rockfalls, rockslides, clayslides, and embankment slopes, monitored by different instruments (e.g., extensometers, reflectors, distometers, inclinometers, satellites, LiDAR, and synthetic aperture radar) (Table S1 in Supporting Information S1). Figure 2 shows typical examples analyzed.

Figure 2a presents the surface displacement of the Ruinon rockslide in Italy monitored by a distometer (Crosta & Agliardi, 2002, 2003). This rockslide consisting of 13 million m³ phyllite exhibited significant episodic movements during 1997–2001, but no collapse occurred. This rockslide showed strong seasonal patterns in its displacement behavior, periodically accelerating during rainy seasons (summer and autumn) and then decelerating during dry seasons (winter and spring). However, after 2000, one can observe a clear log-periodic pattern, which is well described by the LPPLS model (Figure 2a, left). The presence of log-periodicity is also evident in the ϵ-lnτ plot (inset of Figure 2a, left), with the Lomb periodogram yielding f = 2.15, ω = 13.49, and λ = 1.59 (Figure 2a, right).

Figure 2b shows the displacement data recorded by a bench mark on the La Clapière rockslide in France, which develops on a slope of metamorphic rocks mainly consisting of gneiss, amphibolites, and migmatites (El Bedoui et al., 2009). This rockslide experienced a major crisis between 1985 and 1987, before restabilizing after late 1987 (Helmstetter et al., 2004). During this acceleration crisis, the landslide movements were found to correlate well with the flow rate of the Tinée river running along the slope toe (Sornette et al., 2004). The existence of log-periodicity is demonstrated by the LPPLS fit and the ϵ-lnτ pattern (though the amplitude of cyclical signals diminishes on the approach to t_c) (Figure 2b, left). On the Lomb periodogram, a peak is observed at f = 1.02 corresponding to ω = 6.41 and λ = 2.67 (Figure 2b, right).

Figure 2c displays the data monitored by a Terrestrial LiDAR for the Puigcercós scarp in Spain, which experienced a major rockfall on 3 December 2013 (Royán et al., 2015). This rock face primarily consists of alternating layers of marl, sandstone, silt, and clay, capped by limestone. A clear log-periodic oscillating behavior decorating an overall power law acceleration can be seen in the displacement data which is well captured by the LPPLS model (Figure 2c). The ϵ-lnτ plot also confirms the presence of log-periodicity (inset of Figure 2c, left), and the Lomb spectral analysis indicates f = 0.73, ω = 4.58, and λ = 3.94 (Figure 2c, right). Notably, the Lomb periodogram highlights a series of harmonics occurring at integer multiples of this fundamental frequency f—another signature of log-periodicity. The existence of log-periodicity is also found for other monitored areas of this scarp (Figure S13 in Supporting Information S1).

Figure 2d shows the displacement time series reconstructed based on synthetic-aperture radar images acquired by Sentinel-1 satellites for the Maoxian rockslide in China, which catastrophically failed on 24 June 2017 (Intrieri et al., 2018). This rockslide with a volume of ∼13 million m³ mainly consists of metamorphic sandstone, marbleized limestone, and phyllite (Fan et al., 2017). Before the final collapse, a pronounced accelerating and oscillating behavior is evident and well captured by the LPPLS model (Figure 2d, left). The existence of log-periodic oscillations is indicated by the sinusoidal-like signals in the ϵ-lnτ plot (inset of Figure 2d, left) and the emergence of a major Lomb peak (Figure 2d, right), pointing to f = 0.67, ω = 4.23, and λ = 4.42. Similar log-periodic characteristics have been observed at two other measurement points (Figures S9 in Supporting Information S1).

Figure 2e presents the displacement data recorded by a reflector installed on the Preonzo slope in Switzerland, where a volume of ∼210,000 m³ rock collapsed on 15 May 2012 (Gschwind et al., 2019; Loew et al., 2017). Similar displacement patterns are captured by other extensometers and reflectors instrumented on this rockslide (Figures S11–S12 in Supporting Information S1). This instability complex, predominantly composed of augen gneiss, exhibited a clear precursory acceleration phase, aligning well with the LPPLS model (Figure 2e and Figures S11–S12 in Supporting Information S1). Log-periodicity is evidenced by the cyclical pattern in the ϵ-lnτ plot (inset of Figure 2e, left) and the Lomb periodogram (Figure 2e, right), which reveals f = 0.67, ω = 4.23, and λ = 4.42, along with a series of harmonics.

Figure 2f displays the displacement time series of the Achoma landslide in Peru, derived from the images of high-frequency PlanetScope satellites (Lacroix et al., 2023). This landslide, situated in lacustrine deposits consisting of soils and weak rocks, experienced a catastrophic failure on 18 June 2020, prior to which a clear precursory accelerating motion was observed (Lacroix et al., 2023). Overall, the LPPLS model provides a good fit to the data (Figure 2f, left), though some scatter is evident, likely due to uncertainties associated with satellite-based measurements. However, log-periodicity is still identifiable in the ϵ-lnτ plot (inset of Figure 2f, left) and the Lomb periodogram, revealing f = 0.79, ω = 4.99, and λ = 3.52 (Figure 2f, right).

We compile and analyze the parameters derived from the Lomb analysis of 52 landslides, presenting histograms of selected key parameters in Figure 3 (see Table S2 in Supporting Information S1 for the complete list). The angular log-frequency ω ranges from 3 to 15, with a concentration around 5 (Figure 3a). Correspondingly, the scaling ratio λ varies from 1.5 to 5, with a median value at around 3.5 and a peak around 4 (Figure 3b). Frequency distributions of ω and λ derived from the Lomb method are in general compatible with those obtained from the LPPLS calibration (see Figure S16 in Supporting Information S1; note that the discrepancy in the low ω and high λ regions is attributed to the filter imposed for ω in the LPPLS calibration, as described in Text S1 in Supporting Information S1). The histogram of maximum Lomb peak heights P_max of these landslides indicates that 94% exceed 5, 76% are beyond 10, 50% surpass 15%, and 42% reach over 20 (Figure 3c), highlighting the significance of log-periodicity. The first-to-second peak ratio η, representing the ratio between the two highest peaks in each Lomb periodogram, shows that 72% of these landslides exceed 2% and 46% surpass 4 (Figure 3d), further strengthening the evidence of log-periodicity by the presence of both a fundamental log-frequency and its first harmonic.

4 Discussion

Our analysis reveals log-periodicity as a pervasive feature of landslide crises, indicated by the excellent LPPLS fit to the monitoring data of various landslides (Figures 1 and 2 and Figures S2–S15 in Supporting Information S1). By employing a comprehensive suite of well-established evaluation metrics, we show that the LPPLS model outperforms the conventional PLS model in describing landslide behavior during crises (Text S3, Figure S17, and Table S3 in Supporting Information S1). In particular, the potential for overfitting by the LPPLS model has been excluded based on evaluation metrics like the normalized Akaike and Bayesian information criteria.

Our results of parametric fitting reveal that the singularity exponent m, which characterizes the nonlinearity of the overall power law acceleration, predominantly ranges from −1.5 to 0.5 (Figure S16 in Supporting Information S1), with the corresponding exponent α = 1 + 1/(1‒m) primarily varying between 1.4 and 3.0, consistent with previously reported α values for landslides (Intrieri et al., 2019). However, the m values are found to concentrate around −0.5, with the corresponding α value near 1.7, which deviates from the commonly assumed α = 2.0 in the inverse velocity method (Carlà et al., 2017; Fukuzono, 1985; Leinauer et al., 2023; Voight, 1988, 1989). The presence of log-periodic structures is qualified by the sinusoidal temporal evolution of residuals as a function of log normalized time lnτ = ln[(t_c ‒ t)/(t_c ‒ t₀)] to singularity, after removing the general power law trend, across almost all the landslides studied (see insets of Figures 1–2 and Figures S2–S15 in Supporting Information S1). Discrepancies observed in a few landslides stem from either data scarcity or the influence of harmonics not accounted for in Equation 4. Indeed, Equation 4 expresses just the first-order log-periodic correction to the pure power law and we refer to Saleur et al. (1996a, 1996b) and Gluzman and Sornette (2002) for the general derivation of the full log-periodic solutions. Incorporating higher-order harmonics could enhance model fit (Johansen et al., 2000; Sornette & Johansen, 1997); however, this would increase model complexity and the risk of overfitting. Consequently, we adopt Equation 4 as a robust first-order correction and reserve the exploration of higher harmonics for future work. The relative amplitude of log-periodic components is found to range between 0.005 and 0.15, with a concentration around 0.05, aligning with the conjecture of being on the order of 10⁻¹ for systems undergoing failures (Sornette, 1998).

Our nonparametric tests provide further evidence of log-periodicity in landslide crises. We emphasize that noise, although inevitably present in the data, is unlikely to be the source of the detected log-periodic signatures. This is supported by two complementary diagnostics of log-periodic signals: (a) their amplitude, evaluated using the Lomb peak height, false-alarm probability, and signal-to-noise ratio; and (b) their log-frequency, assessed against known artifactual values. Regarding amplitude, the consistently high Lomb peaks for various landslides, with the majority exceeding 10 (Figure 3c), highlight the statistical significance of log-periodic components (Zhou & Sornette, 2002b). If the data were due to Gaussian noise, we can analytically derive the false-alarm probability quantifying the likelihood that random noise is incorrectly identified as a valid log-periodic signal (Text S2 in Supporting Information S1). Out of the 114 time series analyzed, only seven exhibit a false-alarm probability greater than 0.05 (Table S2 in Supporting Information S1). It is possible that the noise in the data does not follow a Gaussian distribution; however, the high first-to-second Lomb peak ratios, with the majority exceeding 2 and about half surpassing 4, still offer strong evidence of log-periodicity (Zhou & Sornette, 2002b). This is further supported by the high signal-to-noise ratios—a metric broadly applicable to various types of noise (Zhou & Sornette, 2002a, 2002b)—for most landslides (Table S2 in Supporting Information S1), with 96% greater than 1, 76% exceeding 1.5%, and 58% surpassing 2. Regarding log-frequency, noise-induced artifactual log-periodicity typically corresponds to ∼1.5 oscillations over the observation window (Huang et al., 2000). In most cases analyzed, the detected log-frequency differs significantly from this likely artifactual value (Table S2 in Supporting Information S1). Even in cases where the two are similar, the high signal-to-noise ratios suggest that the detected log-periodicity is unlikely to originate from noise (Table S2 in Supporting Information S1). Together, these results indicate that noise has a minor impact and the observed log-periodicity reflects genuine features of landslide dynamics. Our framework exhibits robustness to variations in the choice of the initial point of monitoring data. When data are available before the crisis onset, the Lagrange regularization method can identify the onset with high reliability and small associated uncertainty (Demos & Sornette, 2017, 2019). If monitoring begins later, persistent log-periodic oscillations still enable reliable signal extraction, and even a small number of cycles can be detected by the Lomb method. This adaptability ensures consistent diagnostics across the data set, despite noise or missing early-stage data. While some landslides may exhibit periodic signals driven by external seasonal, tidal, or climatic factors, our analysis suggests that such signals have minimal impact on the detection of log-periodic signatures (see Text S4 in Supporting Information S1), which arise from internal hierarchical failure processes and dominate during landslide crises.

Log-periodic signals span timescales from days to years, reflecting multiscale rupture processes in heterogeneous geomaterials. They tend to also scale with landslide volume, with larger events exhibiting relatively longer-term log-periodicity. The log-periodic characteristics offer valuable insights into the underlying mechanisms driving the intermittent rupture behavior of geomaterial masses during landslide crises. It reflects the presence of discrete scale invariance (Saleur et al., 1996a; Sornette, 1998), which is associated with complex critical exponents typically arising in nonunitary (dissipative) systems with out-of-equilibrium dynamics and quenched disorder (Saleur & Sornette, 1996). One possible mechanism for discrete scale invariance is the cascade of ultraviolet Mullins-Sekerka instabilities during crack growth, where larger cracks are less affected by screening and propagate faster, while smaller cracks are suppressed due to stress shadowing and crack interactions (Huang et al., 1997). This theory developed for a regular array of pre-existing cracks, supported by geological evidence found in natural outcrops (Ouillon, Sornette, et al., 1996), predicts λ = 2, which lies at the lower end of the range of λ values reported in our current study. The variability of λ values found here may reflect the presence of a heterogenous spatial and length distribution of pre-existing cracks and sliding surfaces. Another possible mechanism is the interplay between stress drop associated with rupture dynamics and stress corrosion during inter-rupture phases, as demonstrated in sandpile models with a mean-field prediction of λ = 3.6 (Lee & Sornette, 2000), aligning closely with our median λ value of ∼3.5 (Figure 3b). Log-periodicity could also arise from the interplay of inertia, damage, and healing (Ide & Sornette, 2002; Sornette & Ide, 2003), which may account for the significant variability in λ values observed for different landslides, due to the possible variations in their healing properties. Furthermore, the interplay of heterogeneities and stress concentrations may also produce discrete scale invariance (Sahimi & Arbabi, 1996), while the presence of hierarchical fracture networks in geological media (Bonnet et al., 2001; Lei & Wang, 2016; Ouillon, Castaing, et al., 1996) might contribute as well, though discrete scale invariance can arise spontaneously in the absence of predefined hierarchical structures (Sornette, 1998). It is likely that these different mechanisms coexist and interact in real landslides, with the dominant mechanism varying both across different sites and over time for the same site. For instance, the Veslemannen landslide exhibits λ ≈ 2.5 during the 2017 and 2019 crises, indicating a possible dominance of crack growth and interaction, while λ ≈ 3.5 during the 2018 crisis may suggest a dominant interaction between frictional stress drop along geological structures and stress corrosion damage in rock bridges.

In addition to shedding light on landslide mechanisms, log-periodic signals could be practically useful for forecasting impending slope failures. More specifically, by “locking” into the oscillatory structure of rupture dynamics, time-to-failure predictions employing the LPPLS model can transform intermittency—traditionally viewed as a nuisance—into valuable information, thereby enhancing the precision of critical time of failure estimations (Lei & Sornette, 2025a, 2025b; Sornette, 2002). Its potential for prospective forecasting will be explored in our future work. The LPPLS model adapts to system-specific characteristics and idiosyncrasies, making it well-suited for prospective forecasting, as each landslide evolves under unique conditions and path-dependent failure dynamics. A Bayesian inference framework—akin to those used in earthquake hazard assessments (Page et al., 2016; Reasenberg & Jones, 1989; Werner et al., 2011)—could assimilate prior knowledge of landslide behavior into LPPLS calibration, enabling the integration of real-time monitoring data with prior information to continuously update probabilistic failure forecasts in an adaptive manner. Moreover, log-periodic signatures could serve as indicators for early warning, a concept proven effective for forecasting financial crises (Demirer et al., 2019; Jiang et al., 2010; Sornette & Zhou, 2006; Zhang et al., 2016; Zhou & Sornette, 2003) and to be explored for landslides in the future. Our analysis of the Veslemannen landslide suggests that the significance of log-periodic oscillations tends to increase as rupture approaches, with Lomb peak heights rising over the years (Figure 1 and Figures S2–S4 in Supporting Information S1). However, the Lomb nonparametric analysis may be less robust for prediction than the LPPLS parametric fit (Zhou & Sornette, 2002a).