Log-periodic signatures prior to volcanic eruptions: evidence from 34 events

Forecasting volcanic eruptions is highly challenging owing to the inherent complexity and variability of volcanic processes. A key source of uncertainty stems from the sporadic nature of volcanic unrest, which is often marked by alternating phases of deflation and inflation, rather than a steady, predictable build-up toward eruption. This non-monotonic evolutionary pattern complicates eruption forecasting as it challenges conventional time-to-failure models that typically assume a simple smooth, monotonic power law acceleration. We develop a log-periodic power law singularity model that effectively captures the intermittent and non-monotonic rupture behaviour characteristic of reawakening volcanoes at the site scale. Mathematically, by generalising the power law exponent by extending it from real to complex numbers, this model captures the partial break of continuous scale invariance into discrete scale invariance, which underlies the sporadic nature of damage and rupture processes in heterogeneous crustal systems. Through both parametric and nonparametric analyses of a large dataset of 34 historical eruptions worldwide, we provide empirical evidence and theoretical reasoning that support the statistical significance of log-periodic oscillations superimposed on power law finite-time singularities during pre-eruptive volcanic unrest. Log-periodicity in volcanoes may originate from various mechanisms, including diffusion-dominated magma flow, magma-driven propagation of subparallel dykes, interaction between stress drop and stress corrosion, and/or interplay of inertia, damage, and healing within volcanic systems. Our findings have profound implications for volcano forecasting, as understanding and characterising log-periodic signatures could transform the intermittent nature of volcanic activity from a challenge into a key asset for improving predictability.