An unbiased estimation of the b-value and of its variability is essential to verify empirically its physical contribution to the earthquake generation process, and the capability to improve earthquake forecasting and seismic hazard. Notwithstanding the vast literature on the b-value estimation, we notice that some potential sources of bias that may lead to non-physical b-value variations are too often ignored in seismological common practice. The aim of this paper is to discuss some of them in detail, when the b-value is estimated through the popular Aki’s formula. Specifically, we describe how a finite dataset can lead to biased evaluations of the b-value and its uncertainty, which are caused by the correlation between the b-value and the maximum magnitude of the dataset; we quantify analytically the bias on the b-value caused by the magnitude binning; we show how departures from the exponential distribution of the magnitude, caused by a truncated Gutenberg-Richter law and by catalogue incompleteness, can affect the b-value estimation and the search for statistically significant variations; we derive explicitly the statistical distribution of the magnitude affected by random symmetrical error, showing that the magnitude error does not induce any further significant bias, at least for reasonable amplitude of the measurement error. Finally, we provide some recipes to minimise the impact of these potential sources of bias.

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.

In this paper we gather and take stock of the results produced by the Operational Earthquake Forecasting (OEF) system in Italy, during its first 10 years of operativity. The system is run in real-time: every midnight and after each M_L 3.5+ event, it produces the weekly forecast of earthquakes expected by an ensemble model in each cell of a spatial grid covering the entire Italian territory. To evaluate the performance skill of the OEF-Italy forecasts we consider here standard tests of the Collaboratory for the Study of Earthquake Predictability, which have been opportunely adapted to the case of the overlapped weekly OEF forecasts; then we also adopt new performance measures borrowed from other research fields, like meteorology, specific to validate alarm-based systems by a binary criterion (forecast: yes/no; occurrence: yes/no). Our final aim is to: i) investigate possible weaknesses and room for improvements in the OEF-Italy stochastic modelling, ii) provide performance measures that could be helpful for stakeholders who act through a boolean logic (making an action or not), iii) highlight possible features in the Italian tectonic seismic activity.

We estimate the b-value parameter of the Gutenberg-Richter law for earthquake magnitudes in the early stage of the Costa Marchigiana (Italy) seismic sequence, starting on November 9, 2022, with an ML 5.7 event in the Adriatic sea. In particular, we estimate both the completeness magnitude M_c and the b-value within the first 4 and 7 days after the initial strong event in the sequence. Our work represents a practical example of b-value estimation in “true” real time, i.e. during the seismic sequence, and its possible interpretation in terms of short-term forecasting. We highlight some critical issues to consider both in estimating/intepreting the b-value, and in evaluating the real time estimation of M_c. These issues are mainly due to the fact that preliminary catalogs available in real time are quite different from the revised ones, which are usually delivered after a few months. The criticalities are linked to the raw data recorded at an early-stage, an unreliable evaluation of the M_c with statistical approaches, the short term aftershock incompleteness entailed after the initial strong event, and the magnitude binning. Our results show that real time estimation of the b-value can give insights into the evolution of an ongoing seismic sequence, when attention is paid to data quality and quantity.

(1) Background: A better understanding of COVID-19 dynamics in terms of interactions among individuals would be of paramount importance to increase the effectiveness of containment measures. Despite this, the research lacks spatiotemporal statistical and mathematical analysis based on large datasets. We describe a novel methodology to extract useful spatiotemporal information from COVID-19 pandemic data. (2) Methods: We perform specific analyses based on mathematical and statistical tools, like mathematical morphology, hierarchical clustering, parametric data modeling and non-parametric statistics. These analyses are here applied to the large dataset consisting of about 19,000 COVID-19 patients in the Veneto region (Italy) during the entire Italian national lockdown. (3) Results: We estimate the COVID-19 cumulative incidence spatial distribution, significantly reducing image noise. We identify four clusters of connected provinces based on the temporal evolution of the incidence. Surprisingly, while one cluster consists of three neighboring provinces, another one contains two provinces more than 210 km apart by highway. The survival function of the local spatial incidence values is modeled here by a tapered Pareto model, also used in other applied fields like seismology and economy in connection to networks. Model’s parameters could be relevant to describe quantitatively the epidemic. (4) Conclusion: The proposed methodology can be applied to a general situation, potentially helping to adopt strategic decisions such as the restriction of mobility and gatherings.

The magnitude–frequency distribution (MFD) of many earthquake catalogs is well described by the Gutenberg–Richter (GR) law or its tapered version (TGR). This distribution is usually extrapolated to any subsets of the space–time window covered by the catalog. However, some empirical observations and logical thoughts may raise doubts about the validity of this extrapolation. For example, according to the elastic rebound theory, we may assert that the probability of a strong shock nucleating within a short‐time interval in a small area $A$ just ruptured by another strong event should be lower than that expected by GR (or TGR): a lot of energy has already been released, and it takes time to recover to the previous state. Here, we put forward a space–time modification of the TGR, named energy‐dependent TGR (TGRE) in which the corner seismic moment becomes a time‐varying energy function depending on (1) the conceivable strongest shock that may nucleate in $A$⁠; (2) the time elapsed since the last strong earthquake that reset the elastic energy in $A$ to a residual value; and (3) the rate of the energy recovery, linked to the recurrence time of the fault(s) involved. The model also verifies an invariance condition: for large space–time windows, the occurrence of a strong shock does not affect significantly the whole elastic energy available, that is, the TGRE becomes the TGR. The model is simple and rooted in clearly stated assumptions. To evaluate its reliability and applicability, we apply it to the 1992 Landers sequence. As expected by TGRE, we find that the MFD close to the fault system interested by the mainshock (⁠ $Mw$ 7.3) differs from that of earthquakes off‐fault, showing a lower corner magnitude. We speculate that TGRE may be profitably used in operational earthquake forecasting and that it explains the empirical observation that the strongest aftershocks nucleate always outside the mainshock fault.