Stefano Tinti

The estimation of the slope (b-value) of the frequency–magnitude distribution of earthquakes is based on a formula derived by Aki decades ago, assuming a continuous exponential distribution. However, as the magnitude is usually provided with a limited resolution, its distribution is not continuous but discrete. In the literature, this problem was initially solved by an empirical correction (due to Utsu) to the minimum magnitude, and later by providing an exact formula such as that by Tinti and Mulargia, based on the geometric distribution theory. A recent paper by van der Elst showed that the b-value can be estimated also by considering the magnitude differences (which are proven to follow an exponential discrete Laplace distribution) and that in this case the estimator is more resilient to the incompleteness of the magnitude data set. In this work, we provide the complete theoretical formulation including (i) the derivation of the means and standard deviations of the discrete exponential and Laplace distributions; (ii) the estimators of the decay parameter of the discrete exponential and trimmed Laplace distributions and (iii) the corresponding formulas for the parameter b. We deduce (iv) the standard 1σ intervals for the estimated b. Moreover, we are able (v) to quantify the error associated with the Utsu minimum-magnitude correction. Furthermore, we have discussed the formulas to produce statistically independent magnitude differences. We tested extensively the b-value estimators on simulated synthetic data sets including complete catalogues as well as catalogues affected by a strong incompleteness degree such as aftershock sequences where the incompleteness is made to vary from one event to the next. We have also analysed the real aftershock sequence of the 30/10/2016 Norcia (central Italy) to integrate the finding of the simulations. To judge the performance of the various estimators we have introduced an index p that can be seen as a non-parametric extension of the Student's t index. The main outcomes of this paper are that (1) the b-value estimators devised for continuous magnitude data are not adequate for binned magnitudes, (2) for complete data sets, estimators based on magnitudes and on magnitude differences provide substantially equivalent results, (3) for incomplete magnitude data sets, estimators based on magnitude differences provide better results and (4) for incomplete aftershock sequences there is no evidence that methods based on positive magnitude differences are superior than other methods using differences. This conclusion is further confirmed by our analysis of the above-mentioned Norcia seismic sequence. This last finding contrasts with the van der Elst's claim that the so called |${{b}_ + }$| method is the most adequate to treat real aftershock sequences.

The estimation of the slope (b-value) of the frequency magnitude distribution of earthquakes is usually based on a formula derived decades ago under the hypothesis of continuous exponential distribution of magnitudes. However, as the magnitude is provided with a limited resolution (one decimal digit usually), its distribution is not continuous but discrete. In the literature this problem is solved mostly by applying an empirical correction to the minimum magnitude of the dataset depending on the binning size, but a recent paper recalled that this solution is only approximate and proposed an exact formula. The same paper further showed that the b-value can be estimated also by considering the positive magnitude differences (which are proven to follow an exponential discrete Laplace distribution) and that in this case the estimator is more resilient to the incompleteness of the magnitude dataset. In this work we provide the complete theoretical formulation including the derivation of i) the means and standard deviations of the discrete exponential and Laplace distributions; ii) the estimators of the decay parameter of the discrete exponential and trimmed Laplace distributions by the methods of the mean as well as of the maximum likelihood; and iii) the corresponding formulas for the parameter b. We further deduce iv) the standard confidence limits for the estimated b. Moreover, we are able v) to quantify the error associated with the formula including the Utsu minimum-magnitude correction. We tested such formulas on simulated synthetic datasets including cases with a certain amount of incompleteness.

The estimation of the slope (b-value) of the frequency magnitude distribution of earthquakes is based on a formula derived by Aki decades ago, assuming a continuous exponential distribution. However, as the magnitude is usually provided with a limited resolution, its distribution is not continuous but discrete. In the literature this problem was initially solved by an empirical correction (due to Utsu) to the minimum magnitude, and later by providing an exact formula such as that by Tinti and Mulargia, based on the geometric distribution theory. A recent paper by van der Elst showed that the b-value can be estimated also by considering the magnitude differences (which are proven to follow an exponential discrete Laplace distribution) and that in this case the estimator is more resilient to the incompleteness of the magnitude dataset. In this work we provide the complete theoretical formulation including i) the derivation of the means and standard deviations of the discrete exponential and Laplace distributions; ii) the estimators of the decay parameter of the discrete exponential and trimmed Laplace distributions; and iii) the corresponding formulas for the parameter b. We further deduce iv) the standard 1-sigma confidence limits for the estimated b. Moreover, we are able v) to quantify the error associated with the Utsu minimum-magnitude correction. We tested extensively such formulas on simulated synthetic datasets including complete catalogues as well as catalogues affected by a strong incompleteness degree such as aftershock sequences where the incompleteness is made to vary from one event to the next.

The Hunga Tonga-Hunga Ha’apai eruption of January 15 2022 was the culminating event of a sequence of seismic and volcanic events starting back in December 2021. The January 15 eruption manifested itself above the sea level with a number of phenomena, including the generation of a convective column ascending well into the stratosphere, pyroclastic flows travelling over the sea surface, an atmospheric pressure wave recorded by several instruments around the globe, and a tsunami, that represents the main focus of this study.The tsunami that followed the eruption was observed both in the near-field and in the far-field, propagating across the entire Pacific Ocean and causing damage and loss of lives as far as Peru. In the near-field (Tonga archipelago) it is trickier to distinguish the damage induced by the impact of the eruption and the tsunami waves.It is still not clear what the main generating mechanism for the ensuing tsunami was. In this contribution, several different hypotheses are investigated, adopting simplified models ranging from the submerged volcanic edifice collapse to the phreatomagmatic explosion and to the atmospheric pressure wave that was recorded across the entire globe. The propagation of the tsunami is simulated numerically with both non-dispersive and dispersive codes. Different spatial scales and resolutions are adopted to check the relative weight of the different generating mechanisms in the near- and in the far-field. Tentative conclusions are drawn by comparing the simulated results with the available experimental data in terms of tide-gauge records and near-field coastal impact.

Ischia Island has been repeatedly affected by mass collapses, which are mainly caused by the steepness of the main peak (Mt Epomeo) and by phenomena related to its volcanic activity. The most relevant cases of mass failure studied in the literature and postulated to be tsunamigenic cover a wide spectrum of sizes, from sector collapse to small-volume mass transports. Tsunamis generated by landslides in Ischia may affect the coast of the Campania mainland, including the Gulf of Naples. The focus of this work is an evaluation of the pattern of the maximum tsunami energy. To this purpose, we perform a series of numerical simulations by moving the same landslide source in different hypothetical positions around the island. The landslide dynamics are computed through the code UBO-BLOCK, and the tsunami propagation by employing the code UBO-TSUFD, both developed in-house. The final goal is to characterize the coastal areas of the Campania mainland most exposed to tsunami attack from Ischia sources. It is found that the position of the landslide deeply influences the distribution of the tsunami elevation in the coastal stretch north of the Procida Mt, while, remarkably, it is irrelevant inside the Gulf of Naples where the bathymetric effect prevails.

It is well known that for earthquake-generated tsunamis impacting near-field coastlines the focal mechanism, the position of the fault with respect to the coastline and the on fault slip distribution are key factors in determining the efficiency of the generation process and the distribution of the maximum run-up and inundation along the nearby coasts. The time needed to obtain the aforementioned information from the analysis of seismic records is usually too long compared to the time required to issue a timely tsunami warning/alert to the nearest coastlines. In the context of tsunami early warning systems, a big challenge is hence to be able to define 1) the relative position of the hypocenter and of the fault and 2) the earthquake focal mechanism, based only on the preliminary earthquake localization and magnitude estimation, which are made available by seismic networks soon after the earthquake occurs.In this study, the intrinsic unpredictability of the position of the hypocenter on the fault plane is studied through a probabilistic approach based on the analysis of two finite fault model datasets (SRCMOD and USGS) and by limiting the analysis to moderate-to-large shallow earthquakes (Mw  6 and depth  50 km). After a proper homogenization procedure needed to define a common geometry for all samples in the two datasets, the hypocentral positions are fitted with different probability density functions (PDFs) separately in the along-dip and along-strike directions.Regarding the focal mechanism determination, different approaches have been tested: the most successful is restricted to subduction-type earthquakes. It defines average values and uncertainties for strike, dip and rake angles based on a combination of a proper zonation of the main tsunamigenic subduction areas worldwide and of subduction zone geometries available from publicdatabases.The general workflow that we propose can be schematically outlined as follows. Once an earthquake occurs and the magnitude and hypocentral solutions are made available by seismic networks, it is possible to assign the focal mechanism by selecting the characteristic values for strike, dip and rake of the zone where the hypocenter falls into. Fault length and width, as well as the slip distribution on the fault plane, are computed through regression laws against magnitude proposed by previous studies. The resulting rectangular fault plane can be discretized into a matrix of subfaults: the position of the center of each subfault can be considered as a “realization” of the hypocenter position, which can then be assigned a probability. In this way, we can define a number of earthquake fault scenarios, each of which is assigned a probability, and we can run tsunami numerical simulations for each scenario to quantify the classical observables, such as water elevation time series in selected offshore/coastal tide-gauges, flow depth, run-up, inundation distance. The final results can be provided as probabilistic distributions of the different observables.The general approach, which is still in a proof-of-concept stage, is applied to the 16 September 2015 Illapel (Chile) tsunamigenic earthquake (Mw = 8.2). The comparison with the available tsunami observations is discussed with special attention devoted to the early-warning perspective.

Siracusa is an important historical city of Greek origin, located on the southern part of the eastern coast of Sicily. The old town developed on the island of Ortigia, and expanded on the near mainland, but later it declined and during the Middle Ages it occupied only the island. The development of the built areas on the mainland restarted at the end of the nineteenth century, with the construction of a number of new quarters. Nowadays the island of Ortigia is connected to the rest of the town by two drive-over bridges. The history of Siracusa as well as of the eastern coast of Sicily is marked by destructive earthquake events that caused significant damage and many fatalities, and also by lethal tsunamis (occurred in the years 1169, 1693 and 1908). Indeed, this region is one of the coastal areas most prone to tsunami attacks in the Mediterranean Sea, being affected by local-source tsunamis and also by those generated by earthquakes in the Western Hellenic Arc.For these reasons, in the last decade the need has developed to prepare adequate evacuation measures to respond to tsunami hazardous events. This work, using the method proposed by Pagnoni et al. (2020) and applied to the near town of Augusta, studies the tsunami risk for different inundation levels. The results are provided in terms of the Human Damage (HD), which is the number of people involved and the number of fatalities, and of the Economic Loss (EL), which returns the loss of economic value of buildings affected by tsunamis. Maps of HD and EL per each inundation scenario allow one to understand which areas of Siracusa are most involved and also to identify evacuation paths to potential safe collection areas and/or buildings for efficient emergency plans.

The Palinuro volcanic chain is located nearly 80 km offshore the Campania coasts (Italy), in the southern sector of the Tyrrhenian Sea. As many as 15 distinct volcanic edifices have been recently detected that covers a 90 km long and 20 km wide belt. The associated volcanism is still poorly understood but the presence of shallow seismicity and active hydrothermal activity suggest that this large volcanic complex is still active. Specific sectors of the chain show the presence of ongoing slope instability and thus the chance of mass movements cannot be ruled out in case of seismic or volcanic activity. In this work, a stability analysis for typical seismic loads in such a volcanic area has been performed through a revised limit equilibrium approach. In the revealed weaker sections, three mass failures of different scales have been reconstructed and their motion has been calculated by means of numerical models. The tsunami produced by each slide has been simulated, and considerable waves have been found in two of the three hypothesized scenarios. For the biggest slide of 2.4 km3, waves as high as 10 m could reach portions of the Calabria coasts with consequent hazardous impact.This study belongs to a series of works focused on the volcanoes of the Tyrrhenian Sea that are very many and still poorly investigated. Considering scenarios involving mass movements of different sizes with distinct characteristics and based on geomorphological features seems to be a viable strategy to evaluate the tsunami hazard in the region.