Modeling long-term in GNSS time series: a comparative study with exponential, power-law, and stretched exponential decay

Major earthquakes can induce significant offsets in geodetic time series tens of kilometres far from their epicenters. Additionally, viscoelastic relaxation along faults and within surrounding crustal volumes generates slow postseismic transients, known as afterslip, which persist for months to years and are typically modelled using exponential decay (Montillet & Bos, 2020). However, growing evidence suggests that seismic cycle perturbations may influence GNSS time series for decades (e.g., Carafa et al., 2024), challenging the assumption of stationarity over human timescales beyond periodic modulations (e.g., seasonal loading, tides, and anthropogenic effects). Here, we evaluate three competing models for long-term afterslip in pre-processed GNSS daily time series: exponential, power-law, and stretched exponential decay. While exponential relaxation reflects a purely viscoelastic fault response, power-law attenuation may emerge from a superposition of exponential processes or a power-law rheology (Ingleby & Wright, 2017). The stretched exponential, conversely, arises from heterogeneous crustal rheology. We demonstrate that stretched exponentials outperform or match exponentials and power laws in fitting both short- and long-term afterslip trends, enabling more reliable forecasts. Furthermore, we provide a physical interpretation linking their efficacy to stress redistribution in fault-adjacent crustal volumes. References Carafa M. M., Bird P., Verdecchia A., Taroni M. and Doglioni C.; 2024: Empirical evidence for multi-decadal transients affecting geodetic velocity fields and derived seismicity forecasts in Italy. Sci. Rep., 14(1), 19941. Ingleby, T., & Wright, T. J. (2017). Omori‐like decay of postseismic velocities following continental earthquakes. Geophysical Research Letters, 44(7), 3119-3130. Montillet, J. P., & Bos, M. S. (Eds.). (2019). Geodetic time series analysis in earth sciences. Springer