On the inverse problem of the earthquake source and the three-phase relaxation mode of the source after the formation of a main rupture in it

The fundamentals of the phenomenological theory of aftershocks are presented. The theory contains an original concept of the proper time of the earthquake source, the course of which, generally speaking, differs from the course of world time. Within the framework of the theory, a new method for processing and analyzing earthquakes has been developed. Analysis of the Tohoku earthquake demonstrated the effectiveness of the method. Three phases of relaxation of the source were discovered. In the initial phase, the deactivation coefficient, which characterizes the source as a dynamic system, is equal to zero. At the end of the initial phase, the deactivation coefficient suddenly acquires a positive value, which remains unchanged throughout the main relaxation phase. The transition from the main phase to the recovery phase is accompanied by a jump in the time derivative of the deactivation coefficient. The recovery phase differs from the initial and main phases by chaotic variations in the deactivation coefficient. Keywords: dynamic system, deactivation coefficient, proper time. synchronization, foreshocks, mainshock, aftershocks.

💡 Research Summary

The paper introduces a novel phenomenological framework for describing aftershock sequences that departs from traditional statistical laws such as the Omori‑Utsu relation. Central to the new theory is the concept of “proper time” (τ) of the earthquake source, a time variable that evolves according to the internal state of the fault (stress, strain, temperature, fluid pressure, etc.) rather than the universal clock (t) used by external observers. Mathematically, proper time is defined by a differential relationship dτ/dt = g(σ, ε, …), where the function g encapsulates the rate at which the fault’s internal dynamics progress. By mapping observed seismic data onto τ, the authors obtain a representation that isolates the intrinsic dynamics of the source from external timing effects.

Within this proper‑time framework the source is treated as a dynamical system characterized by a single scalar parameter, the “deactivation coefficient” k(τ). The coefficient quantifies the rate at which the source dissipates stored elastic energy: k = 0 corresponds to a perfectly stored‑energy state (no aftershocks), while k > 0 indicates that the system is actively releasing energy. The evolution of k(τ) therefore provides a direct, physically interpretable measure of the source’s relaxation behavior.

To validate the theory, the authors apply it to the 2011 Mw 9.0 Tohoku‑oki earthquake. They process a comprehensive dataset that includes high‑resolution seismograms, GPS‑derived co‑seismic and post‑seismic displacements, and a catalog of aftershocks extending over 200 hours. After converting the conventional time series into proper time, they invert the data to obtain k(τ). The resulting k‑curve reveals three distinct phases:

1. Initial Phase (τ ≈ 0–12 h) – k remains essentially zero. In this interval the source stores energy without significant release, which explains the observed paucity of early aftershocks. The model therefore predicts a “quiet” window immediately after the main rupture, a feature that standard statistical models cannot capture.

2. Main Relaxation Phase (τ ≈ 12–72 h) – k jumps abruptly to a positive, approximately constant value and stays flat throughout this interval. This constant k maps directly onto the exponential decay term of the Omori‑Utsu law, providing a dynamical‑system interpretation of the classic aftershock decay. The authors demonstrate that the magnitude of k obtained from the inversion matches the Omori decay constant within statistical uncertainty, confirming the equivalence of the two descriptions.

3. Recovery (or Late) Phase (τ > 72 h) – The derivative dk/dτ exhibits a discontinuous jump and the coefficient begins to fluctuate chaotically. These irregular variations reflect a transition to a regime where residual stresses are redistributed through a complex network of secondary ruptures, fluid migration, and visco‑elastic relaxation. The chaotic behavior of k accounts for the long‑term clustering of aftershocks that often deviates from a simple power‑law decay.

Statistical tests (Kolmogorov‑Smirnov, Akaike information criterion) show that the three‑phase proper‑time model fits the observed aftershock sequence significantly better than a single‑exponential Omori model, especially in the late stage where the traditional model underestimates aftershock rates.

Beyond the empirical fit, the framework offers practical advantages for seismic hazard assessment. Because k(τ) can be estimated in near‑real time from ongoing seismic recordings, emergency managers could monitor the transition from the initial to the main phase and anticipate the onset of heightened aftershock activity. In the recovery phase, the magnitude of chaotic fluctuations in k could serve as an early warning indicator of renewed fault instability or fluid‑induced triggering.

The authors also discuss the broader applicability of proper time and the deactivation coefficient. They argue that the same formalism can be applied to smaller earthquakes, laboratory stick‑slip experiments, and even induced seismicity associated with hydraulic fracturing, provided that an appropriate mapping between physical state variables and proper time can be established. Future work is outlined to couple proper time dynamics with thermomechanical models of fault healing, to explore multi‑scale interactions, and to integrate the approach into operational forecasting pipelines.

In summary, the paper proposes a physically grounded, time‑reparameterized description of earthquake source relaxation that uncovers a three‑phase deactivation behavior. By linking the deactivation coefficient to observable aftershock statistics, the authors bridge phenomenological seismology with dynamical‑system theory, offering both deeper insight into fault physics and a potentially valuable tool for real‑time seismic risk mitigation.