Abstract

Earthquake occurrences are often described by self-exciting point process models, with the Epidemic Type Aftershock Sequence (ETAS) model representing a widely used framework for space--time seismicity. Modern seismic networks provide locations, times and magnitudes, together with waveform records, treated as functional marks attached to each event. Motivated by the potential of functional data analysis, we consider waveform information as a main information to study the triggering mechanism. In this work, based on FPCA waveform clustering approaches such as Adelfio et al. (2011), we represent each waveform via its FPCA scores and include these scores in the triggering component of an ETAS model. The estimation is performed through the Forward Likelihood for prediction (FLP) method proposed by Chiodi and Adelfio (2011).
Each event's waveform, recorded by seismic stations, is preprocessed into a smooth functional curve on a common time grid, then decomposed via Functional Principal Component Analysis (FPCA). The retained FPCA scores summarize key waveform shape variability and serve as covariates in the triggering function, modulating event productivity beyond traditional time, space, and magnitude dependencies.
The approach is applied to the 2009 L'Aquila mainshock sequence, comprising origin times, hypocenters, magnitudes, and associated waveforms in a selected spacetime window.
The waveforms are then aligned, filtered and transformed into functional observations on a common time grid. Using a suitable functional basis (e.g., B-splines or Fourier basis), each waveform is expressed as a smooth curve and subjected to Functional Principal Component Analysis (FPCA). A reduced number of principal components is retained to capture most of the variability, and the corresponding FPCA scores are used as numerical summaries of the waveform shapes. These scores define the functional marks attached to each point in the space--time point pattern. Exploratory summaries of times, locations, magnitudes and FPCA scores are provided, together with histograms of occurrence times indicating that the first day of the sequence exhibits a particularly high level of activity.
Exploratory analysis reveals high aftershock activity on the first day. A local random labelling test for functional marked point processes identifies significant spatial dependence in FPCA scores founding that some of the vents have a deviant waveform-location, clustered spatially and temporally near the mainshock.
Model estimation uses the Forward Likelihood for Prediction (FLP) method, which sequentially maximizes conditional likelihoods for predictive suitability. It alternates nonparametric background intensity estimation with parametric fitting of triggering parameters, including log-linear regression coefficients for FPCA scores.
A baseline ETAS (no marks) is compared to the functional version. Parameter estimates shift notably, reflecting waveform influence. Diagnostics confirm improvement; spatial residuals for background and total intensity show homogeneous patterns, with the functional model better resolving early clustering.
Results affirm waveform features' non-negligible role in aftershock generation, as suggested by source physics. The framework, combining functional data analysis with point process modeling, can be the basis for improving seismicity forecasts, using the covariates information.