A probabilistic method for the estimation of earthquake source parameters from spectral inversion : application to the 2016-2017 Central Italy seismic sequence

We develop a probabilistic framework based on the conjunction of states of information between data and model, to jointly retrieve earthquake source parameters and anelastic attenuation factor from inversion of displacement amplitude spectra. The evaluation of the joint probability density functions (PDFs) enables us to take into account between-parameter correlations in the final estimates of the parameters and related uncertainties. Following this approach, we first search for the maximum of the a-posteriori PDF through the basin hopping technique that couples a global exploration built on a Markov chain with a local deterministic maximization. Then we compute statistical indicators (mean, variance and correlation coefficients) on source parameters and anelastic attenuation through integration of the PDF in the vicinity of the maximum likelihood solution. Definition of quality criteria based on the signal to noise ratio and the similarity of the marginal PDFs with a Gaussian function enable us to define the frequency domain for the inversion and to get rid of unconstrained solutions. We perform synthetic tests to assess theoretical correlations as a function of the signal to noise ratio and to define the minimum bandwidth around the corner frequency for consistent parameter resolution. As an application, we finally estimate the source parameters for the 2016-2017 Central Italy seismic sequence. We found that the classical scaling between the seismic moment and the corner frequency holds, with an average stress drop of $\Delta\sigma$ = 2.1 +- 0.3 MPa. However, the main events in the sequence exhibit a stress drop larger than the average value. Finally, the small seismic efficiency indicates a stress overshoot, possibly due to dynamic effects or large frictional efficiency.

💡 Research Summary

The paper presents a comprehensive probabilistic framework for jointly estimating earthquake source parameters—seismic moment (M), corner frequency (f_c), high‑frequency decay exponent (γ), and the anelastic attenuation factor (Q)—from displacement amplitude spectra. Building on the generalized Brune model, the authors express the observed displacement spectrum as the product of a source term and a propagation term, incorporating geometrical spreading, travel time, and a frequency‑independent quality factor. By taking the logarithm of the spectral amplitude, the forward operator becomes linear in the logarithms of the unknown parameters, facilitating the formulation of a Bayesian inverse problem.

A uniform prior is assumed for the model parameters, while both measurement errors and model uncertainties are modeled as Gaussian. This leads to a posterior probability density function (PDF) proportional to the product of a Gaussian likelihood and the prior. The maximum‑a‑posteriori (MAP) estimate corresponds to the point that maximizes this posterior PDF. To locate the MAP efficiently in the four‑dimensional parameter space, the authors adopt the Basin‑Hopping algorithm, which combines a global stochastic exploration based on a Markov chain (controlled by a “temperature” parameter) with a local deterministic quasi‑Newton optimizer. This hybrid approach mitigates the risk of becoming trapped in local minima while keeping computational costs low.

After identifying the MAP solution, the authors sample the posterior PDF in its vicinity to compute marginal means, variances, and correlation coefficients for each parameter pair. They introduce two quality criteria: (1) a Gaussianity test on the marginal PDFs to ensure that the posterior is well‑behaved, and (2) a signal‑to‑noise ratio (SNR) threshold (SNR > 10) to define the frequency band used in the inversion. Synthetic tests reveal that low SNR dramatically inflates the correlation between f_c and γ, and that a minimum bandwidth of roughly one octave around the corner frequency is required for reliable, independent parameter recovery.

The methodology is applied to the 2016‑2017 Central Italy seismic sequence, comprising about thirty events ranging from magnitude 3.5 to 5.7. The results confirm the classic Brune scaling between seismic moment and corner frequency, with an average stress drop of Δσ = 2.1 ± 0.3 MPa. However, the larger events (Mw ≥ 5.5) exhibit stress drops of 3–4 MPa, significantly above the mean, suggesting size‑dependent rupture dynamics. The estimated quality factor Q lies between 150 and 300, showing limited spatial variation. Moreover, the calculated seismic efficiency (the ratio of radiated energy to total energy release) is low, indicating a stress overshoot that may be caused by dynamic effects or high frictional efficiency during rupture.

Overall, the study demonstrates that a fully probabilistic inversion, coupled with robust global‑local optimization and objective quality controls, can reliably retrieve earthquake source parameters while explicitly accounting for inter‑parameter correlations and uncertainties. This approach is well suited for large‑scale automated processing of seismic catalogs and can enhance the accuracy of seismic hazard assessments and the physical interpretation of rupture processes.