Ilaria Spassiani

The b value, the scaling exponent of the Gutenberg–Richter law, is a widely investigated parameter in statistical seismology, quantifying the relative proportion of small and large earthquakes in a seismic catalog. Despite its simple definition, an accurate and reliable estimation is not straightforward due to some methodological limitations (e.g., assumptions behind their derivation) and to catalog incompleteness. Different methods have been proposed during the last six decades for measuring the b value. Among them, the b‐positive approach, using the positive difference of consecutive magnitudes, enables the mitigation of the so‐called short‐term incompleteness. In this article, we review these methods, showing that all the approaches converge to a universal formula if the magnitudes of earthquakes are properly standardized and selected. We also review b value estimations by establishing methodological consistency through comparison with extensive simulations. Moreover, we provide bEST, a novel MATLAB/Python tool for a multimethodological assessment of the b value.

The Lilliefors test is commonly applied to assess the exponentiality of earthquake magnitudes and, consequently, to estimate the minimum threshold above which seismic events are completely recorded (the completeness magnitude). In theory, the test assumes continuously distributed exponential data; however, real earthquake catalogs typically report magnitudes with finite resolution, resulting in a discrete (geometric) distribution. To address this mismatch, standard practice adds uniform noise to the data prior to testing for exponentiality. In this work, we analytically demonstrate that uniform dithering cannot recover the exponential distribution from its geometric counterpart. Instead, it produces a piecewise-constant residual lifetime distribution, whose deviation from the exponential model becomes increasingly detectable as the catalog size or bin width increases, as confirmed also by numerical experiments. We further prove that an exponential distribution truncated over the bin interval is the exact noise distribution required to correctly restore the continuous exponential distribution over the whole magnitude range. Numerical tests also show that this correction yields Lilliefors rejection rates consistent with the significance level for all bin widths and catalog sizes. Correcting the exponentiality test for binned magnitudes according to these results ensures a more reliable estimation of the completeness threshold, particularly in the case of high-resolution earthquake catalogs.

Above the magnitude of completeness - the minimum threshold for which a 100% detection rate is assumed - earthquake magnitudes are typically modeled as a continuous exponential distribution. In practice, however, earthquake catalogs report magnitudes with finite resolution, resulting in a discrete (geometric) distribution. To determine the magnitude of completeness, the Lilliefors test is commonly applied. Because this test assumes continuous data, it is standard practice to add uniform noise to binned magnitudes prior to testing exponentiality.Here we show analytically that uniform dithering does not recover the underlying continuous exponential distribution from its discretized (geometric) form. It instead returns a piecewise-constant residual lifetime distribution, whose deviation from the exponential model becomes detectable as catalog size or bin width increases. Through numerical experiments, we demonstrate that this deviation yields a systematic overestimation of the magnitude of completeness, with biases exceeding one magnitude unit in large, high-resolution catalogs. We derive the exact noise distribution - a truncated exponential within each magnitude bin - that correctly restores the continuous exponential distribution over the whole magnitude range. Numerical tests show that this correction yields Lilliefors rejection probabilities that are consistent with the significance level across a wide range of bin widths and catalog sizes. Although illustrated for the Lilliefors test, the identified bias and the proposed correction are independent of the specific statistical test and apply generally to exponentiality testing of discretized magnitude data.

The back‐stripping technique is widely used in geological modeling to quantify basin subsidence history, sedimentation rates, and tectonic subsidence. Recent applications involve reconstructing paleo‐water depths, especially in oceanic and Arctic studies. Despite the availability of open‐source Matlab codes based on this procedure, comprehensive investigations including errors from data acquisition remain lacking. Many studies address the errors related to model parameters, neglecting a systematic approach crucial for result accuracy. To enhance the reliability in subsidence rate calculations via back‐stripping, we propose a method to analyze errors introduced during the pre‐processing of input data. Our approach starts with a qualitative identification of key error sources and proceeds with a quantitative estimation of each of them, using appropriate mathematical techniques such as linear interpolation and combinatorics. The proposed method is applied to the Po Delta in northern Italy, a region historically influenced by anthropogenic and natural subsidence. Analyzing a 2D geological section characterized by thin Holocene sedimentary successions, we identified 12 error sources, grouped into four basic categories: geometry of the model layers, distribution of lithologies, petrophysical properties, and factors related to depositional environments and geodynamics. We then assessed the error ranges and their probability of occurrence. The results show that errors can vary significantly—from the meter to millimeter‐scale—defining the magnitude and distribution of each error source along line, which is essential for accurately interpreting model results and assessing related uncertainties. The study also establishes a replicable workflow for future uncertainty management, contributing to enhance open‐source tools based on the back‐stripping procedure.

Earthquake forecasting at different time scales is one of the most challenging goal for seismologists and geophysicists. The delivery of reliable forecasts is crucial to reduce seismic risk and establish rational operational strategies, but the task is demanding due to the complex nature of the earthquake phenomenon. Seismic events are characterized by self-organized criticality, and can be labeled as extreme, rare events, making the probability theory a fundamental tool to resort to.Retrospective studies in statistical seismology mainly focus on the evolution of aftershocks following a large event, the results being then used for analyses of prospective type. However, much attention is also paid to the largest event in the earthquake sequence, because the forecasting of extreme events might be crucial to prevent significant damage or casualties.In this study, we derive the probability of extreme events in any seismic cluster generated by the Epidemic Type Aftershock Sequence (ETAS) model, a benchmark in statistical seismology for any probabilistic earthquake forecasting application. Specifically, we compute the probability for the largest event within any ETAS cluster to occur at a specific space, time, and magnitude point, considering both the temporal and the spatial components of the process. The results obtained shed light on understanding the distinguishing features between mainshocks and foreshocks, and may actively contribute to operational forecasting in assigning, in real-time, the probability for any new event to be the largest of an ongoing seismic sequence.

We represent an earthquake catalog as a complex network, where each node corresponds to a seismic event, and the links between nodes represent their distances in the space-time-energy domain. This approach, commonly known as the nearest-neighbor (NN) method, was introduced by Baiesi and Paczuski in 2004 to analyze the Californian earthquake catalog. Unlike the traditional fixed window-based techniques, which group into a cluster all the events within a predefined space-time domain centered at a selected mainshock, the nearest-neighbor method offers greater flexibility and precision. In fact, the resulting weighted network is able to capture extensive information about the space-time seismicity of a given region.In our study, we use network theory to analyze the Italian earthquake catalog from 2010 to 2020, compiled by the National Institute of Geophysics and Volcanology (INGV). Specifically, we first apply the NN method to derive the seismic Italian structure as a weighted network. Then, we focus our analysis on its topological properties to detect communities, that are groups of nodes more tightly interconnected with each other than with the rest of the network.To achieve this, we use the Louvain and Leiden algorithms: they are based on the maximization of modularity, a metric that quantifies the strength of a network’s division into modules (also referred to as communities or clusters). For implementation, we rely on IT tools new to seismology, but largely used in complex network analysis. Specifically, we use Radatools, an Ada library designed for analyzing complex networks, alongside the NetworkX Python package, which facilitates the creation, manipulation, and study of network structures, dynamics, and functions.This approach enables us to identify communities in the Italian network, and to compare them with clusters derived using traditional window-based techniques commonly employed in the literature, such as the Gardner-Knopoff technique. By investigating these differences and similarities, we aim to provide a robust and comprehensive analysis of the Italian earthquake catalog, leveraging high-performance tools for studying complex seismic networks.

Short-term seismic clustering, a crucial aspect of seismicity, has been extensively studied in literature. Existing techniques for cluster identification are predominantly deterministic, relying on specific constitutive equations to define spatiotemporal extents. Conversely, probabilistic models, such as the Epidemic Type Aftershock Sequence (ETAS) model, dominate short-term earthquake forecasting. The ETAS model, known for its stochastic nature, has been employed to decluster earthquake catalogs probabilistically. However, the challenge arises when selecting a probability threshold for cluster identification, potentially distorting the model's underlying hypothesis.This study aims to assess the consistency between seismic clusters identified by deterministic window-based techniques specifically, Gardner-Knopoff and Uhrhammer-Lolli-Gasperini and the associated probabilities predicted by the ETAS model for events within these clusters. Both deterministic techniques are implemented in the NESTOREv1.0 package and applied to the Italian earthquake catalog spanning from 2005 to 2021.The comparison involves evaluating, for each event within an identified cluster, both the probability of independence and the expected number of triggered events according to the ETAS model. Results demonstrate overall agreement between the two cluster identification methods, with identified clusters exhibiting consistency with corresponding ETAS probabilities. Any minor discrepancies observed can be attributed to the fundamentally different nature of the deterministic and probabilistic approaches.This research is supported by a grant from the Italian Ministry of Foreign Affairs and International Cooperation.

Seismic hazard can be quantified by using probabilities. Modern seismic forecasting models, such as Operational Earthquake Forecasting (OEF) systems, allow us to quantify short-term variations of such probabilities. These probabilities change indeed with time and space, in particular after strong seismic events. However, the short-term seismic hazard could also change during seismic swarms, e.g. sequences with several small/medium size events. The goal of this work is to quantify these changes by both using the Italian OEF system, and estimating the variations of the b-value parameter in Gutenberg-Richter frequency-magnitude distribution. We focus our attention on three seismic swarms that occurred in Central Italy during October-November 2023. Our results indicate that the short-term variations of seismic hazard are limited, less than an order of magnitude. The b-value variations are also not significant. Our conclusion is then that, with the currently available models and catalogs, occurrence of seismic swarms does not significantly affect the short-term seismic hazard.

While deterministically predicting the time and location of earthquakes remains impossible, earthquake forecasting models can provide estimates of the probabilities of earthquakes occurring within some region over time. To enable informed decision‐making of civil protection, governmental agencies, or the public, Operational Earthquake Forecasting (OEF) systems aim to provide authoritative earthquake forecasts based on current earthquake activity in near‐real time. Establishing OEF systems involves several nontrivial choices. This review captures the current state of OEF worldwide and analyzes expert recommendations on the development, testing, and communication of earthquake forecasts. An introductory summary of OEF‐related research is followed by a description of OEF systems in Italy, New Zealand, and the United States. Combined, these two parts provide an informative and transparent snapshot of today's OEF landscape. In Section 4, we analyze the results of an expert elicitation that was conducted to seek guidance for the establishment of OEF systems. The elicitation identifies consensus and dissent on OEF issues among a non‐representative group of 20 international earthquake forecasting experts. While the experts agree that communication products should be developed in collaboration with the forecast user groups, they disagree on whether forecasting models and testing methods should be user‐dependent. No recommendations of strict model requirements could be elicited, but benchmark comparisons, prospective testing, reproducibility, and transparency are encouraged. Section 5 gives an outlook on the future of OEF. Besides covering recent research on earthquake forecasting model development and testing, upcoming OEF initiatives are described in the context of the expert elicitation findings.