Improving the knowledge of seismogenic faults requires the integration of geological, seismological, and geophysical information. Among several analyses, the definition of earthquake focal mechanisms plays an essential role in providing information about the geometry of individual faults and the stress regime acting in a region. Fault plane solutions can be retrieved by several techniques operating in specific magnitude ranges, both in the time and frequency domain and using different data.

For earthquakes of low magnitude, the limited number of available data and
their uncertainties can compromise the stability of fault plane solutions.
In this work, we propose a useful methodology to evaluate how well a seismic
network, used to monitor natural and/or induced micro-seismicity, estimates
focal mechanisms as a function of magnitude, location, and kinematics of
seismic source and consequently their reliability in defining seismotectonic
models. To study the consistency of focal mechanism solutions, we use a
Bayesian approach that jointly inverts the P

Fault plane solutions represent primary information to describe earthquakes. The assessment of earthquake location, magnitude, and focal mechanism are the fundamental operations to characterize the earthquake source using a point source approximation. The focal mechanism describes the basic geometry and kinematics of a point source in terms of strike, dip, and rake of the fault plane along which the earthquake occurred. So, the focal mechanism is the most important marker of the geometry of the seismogenic faults and their style of faulting. Moreover, seismicity and focal mechanisms of events are often used to constrain seismotectonic models, individual seismogenic sources, the regional strain, and stress fields, also for small magnitudes. Consequently, an evaluation of their effective reliability becomes a fundamental issue in seismotectonic studies.

Nevertheless, focal mechanisms cannot be calculated and constrained every time an earthquake occurs. Although the calculation of focal mechanisms represents a routine analysis for seismological agencies, the solutions are calculated only for a specific range of magnitudes, usually greater than 4. In fact, constraining the solution for earthquakes with small magnitude is still a challenge, despite the advancement in the technological process and the use of increasingly performing seismic networks. This is due to several factors that we will analyse in detail. The techniques used to define the focal mechanism of large to moderate earthquakes are based on the inversion of the moment tensor, which corresponds to a stable and robust procedure, so much that it is the most common method for this type of analysis (Dreger, 2003; Delouis, 2014; Sokos and Zahradnik, 2013; Cesca et al., 2010). This technique requires accurate knowledge of the propagation medium in relation to the range of frequencies used for the modelling waveforms recorded during an earthquake. The smaller an earthquake, the higher the frequency range of the signal to be modelled, the more detailed the knowledge and scale of the Earth's interior must be. Several methods have been proposed to achieve a stable inversion of the moment tensor for earthquakes with a magnitude of less than 3. Hybrid approaches that invert both amplitude and waveform moment tensor use the principal component analysis of seismograms (Vavrycuk et al., 2017) or moment tensor refinement techniques (Kwiatek et al., 2016; Bentz et al., 2018) to facilitate a robust determination of the source type and its kinematics. In particular, the retrieved moment tensor is typically decomposed into volumetric and deviatoric components. Constraining the earthquake as a double-couple source can erroneously affect the retrieved fault plane solutions, especially in the case of induced seismicity where the volumetric or non-double couple component must be considered (Kwiatek et al., 2016).

Other analytical techniques are based on the recognition of the source
radiation pattern. According to the position of seismic stations relative to
the source, seismic waves on seismograms show different amplitudes and
polarities. These features can constrain the geometry of the earthquake
faulting through estimating the angular parameters strike, dip, and rake.
The classical method (Raesenberg and Oppenheimer, 1985) uses the P-wave
polarities; more advanced approaches better constrain the focal mechanism of
small earthquakes using P- or S-wave amplitudes or amplitude ratios
together with first motions (Snoke, 2003). In fact, the use of polarities
alone is inappropriate, especially if we consider micro-seismicity (

Two kinds of errors generally influence the goodness of the solution and retrieved model (Michele et al., 2014): the perturbation errors that are related to how the uncertainty on data affects the model, and the resolution errors that are referred to the capability to retrieve a correct model, given a dataset as input or how accurate the model that we can recover could be, even with error-free data. The sum of perturbation and resolution errors corresponds to the final errors on the model obtained by solving an inverse problem, as the solution of the focal mechanism. In particular, the resolution errors depend on the available data and so on the initial condition of the inverse problem. In the case of the focal mechanism, the number of seismic stations, as well as the seismic network geometry, and the velocity structure of the crust influence the resolution and the reliability of the retrieved model.

How will the geometry of a seismic network determine the accuracy of focal mechanism solutions? The answer to this question requires a deep knowledge of the geophysical and geological characteristics of the region, often unavailable. Moreover, the theoretical relationships that predict the focal mechanism solutions for an earthquake scenario could be very complicated if several factors, such as network configuration, noise level, source magnitude, or source kinematics are taken into account. A network configuration may be optimal for earthquake locations but not for retrieving fault plane solutions (Hardt and Scherbaum, 1994). In fact, a given geometry may resolve some fault kinematics better than others.

A seismic network layout is strictly associated with the goals of the network and the available funds; according to these features, a network operator decides how many stations are required and where they should be located (Havskov et al., 2012). So, the number of seismic stations, the size, and geometry of the network are defined after a preliminary phase based on the evaluation of the specific seismological target (Trnkoczy et al., 2009; Hardt and Scherbaum, 1994; Steinberg et al., 1995; Bartal et al., 2000). In the case of small earthquakes, the available recordings come from only a portion of the total network, while the distant stations show a seismic signal buried inside the noise. In order to detect and locate low-magnitude earthquakes, we must increase the number of seismic stations for area units by building a dense seismic network.

In this study, we propose a useful tool to evaluate both (1) the reliability of focal mechanism solutions inferred by the inversion of different seismological data and (2) the performance of the seismic network to assess focal mechanism solutions and their errors. We evaluate the network capability to solve focal mechanisms as a function of magnitude, location, and kinematics of seismic source. We consider three synthetic datasets: P-wave polarities, P- to S-wave amplitude spectral ratios, and polarities and amplitude ratios together. Moreover, different levels of noise are considered in order to simulate more realistic conditions.

As a target, we selected the Irpinia Seismic Network (ISNet), a local seismic
network that monitors the Irpinia complex normal fault system (southern
Italy), activated during the

Epicentral map of the earthquakes (green circles)
recorded by the Irpinia Seismic Network (ISNet, red triangles) from 2008 to 2020
(

Fault plane solutions used for earthquake simulations.

Regular grid of epicentres (yellow stars) used for
simulating earthquakes. The area is

With the main aim to define the reliability of focal mechanisms retrieved by specific seismic networks, we propose a methodology based on an empirical approach that consists of different steps.

For a double-couple seismic source, the radiation pattern depends on fault
kinematics and relative source–station position. In fact, it can be
represented as a function of (1) strike, dip, and rake angles (

BISTROP jointly inverts the spectral amplitude ratios with the observed
P-wave polarities to infer the parameters

Assuming that the observables have the same finite variance, for the

For the

As a test case for our methodology, we choose the area of the

The 1980,

KAM (Kagan angle misfit) map for retrieved focal
mechanisms with the D1 dataset as input data and simulating earthquakes with M3 magnitude and FM1

KAM (Kagan angle misfit) map for retrieved focal
mechanisms with D2

We applied the method we proposed and evaluated the capability of the ISNet
local network to resolve fault plane solutions using different observables
as input data: (a) P-wave polarities, (b) P

FMM (focal mechanism parameter misfit) maps for retrieved
focal mechanisms with D3 datasets as input data and simulating earthquakes
with M1

KAA (Kagan angle average) maps for retrieved focal
mechanisms with D2

For each grid node and according to the earthquake magnitude to be tested,
we have to select the ISNet stations for simulations. The number of seismic
stations that record an event depends on earthquake magnitude,
source–station distance, crustal medium properties, and the noise level.
Theoretical relationships that link the seismic source to the signal
recorded at every single station are quite complicated (Kwiatek et al.,
2016, 2020) and are based on the accurate knowledge of crustal volumes in
which the seismic waves propagated, such as the three-dimensional wave
velocity structure, anelastic attenuation, and/or site conditions of a single
receiver. To overcome this limitation, we used an empirical approach to
define the number and the distance of the seismic stations that record a
seismic signal as a function of magnitude, once its epicentral location
(grid node) and depth are fixed. Using the bulletin data retrieved by INFO
at ISNet during the last 2 years (January 2019–March 2021;

KAS (Kagan angle standard deviation) maps for retrieved
focal mechanisms with D2

Maximum distance of the farthest triggered seismic station and number of P-wave polarities as function of earthquake magnitude and depth. The values, empirically derived from the ISNet bulletin, are used for the earthquake simulations.

Additionally, we simulated the uncertainty on the measure of spectral level
ratios or the effect of seismic noise adding a zero mean, Gaussian noise to
the synthetic data with a standard deviation equal to two different
percentage levels, as 5 % and 30 %. With this configuration, we
simulated

three datasets of seismic observables: P-wave polarities (D1), P

two hypocentre depths: 5 and 10 km;

three magnitude bins:

three focal mechanism solutions: FM1 (317, 59,

Summary of the Figs. 4–12 with parameters used for earthquake simulations whose results are represented as a specific map.

Kagan angle misfit map (KAM)

map of the focal mechanism parameter misfit (FMM)

strike, dip, and rake error map (FME)

Kagan angle average map (KAA)

Kagan angle standard deviation map (KAS).

The Kagan angle (KA) measures the difference between the orientations of two seismic moment tensors or two double couples. It is the smallest angle needed to rotate the principal axes of one moment tensor to the corresponding principal axes of the other (Kagan et al., 1991; Tape and Tape, 2012). The smaller the KA between two focal mechanisms, the more similar they are. In the KAM map, for each node the value of KA between the theoretical and retrieved solution is reported, while in the FMM map, the absolute value of the misfit between the strike, dip, and rake angles of the retrieved and theoretical solution is indicated. FME is defined as the error map of strike, dip, and rake in which the uncertainties (standard deviations) are calculated considering all the solutions with a probability larger than the 90 % (S90) of the maximum probability, corresponding to the best solution retrieved. Additionally, these solutions are used to study how constrained the FM solution is. The KA is calculated between each FM of S90 solutions and the retrieved best solution. The mean and the standard deviation of the resulting KA distribution are plotted in the KAA and KAS maps, respectively. The smaller the KA mean and standard deviation (SD), the more constrained the obtained fault plane solution is (Table 2).

Fault plane solutions of instrumental seismicity that occurred in the Irpinia region in 2005–2008 and calculated by De Matteis et al. (2012). The solutions are classified according to a quality code based on the resolution of fault plane kinematics as derived in this study. The result of our simulations suggests a quality as follows: FM1: C; FM2: B; FM3: A.

We consider FM1, i.e. the focal mechanism of the 1980 Irpinia earthquake
located at 10 km depth, first. Looking at Figs. 4 and 5, we see the effect
of using the three different datasets. Considering D1, we can calculate the
FM only for earthquakes with a magnitude of 2.0–2.5 for which at least six polarities are available. As shown by the KAM map in Fig. 4a, the retrieved
solutions are characterized by high KA (

Looking at Fig. 6, using the D3 dataset, the dip angle is the best
resolved compared with strike and rake angles. For the M2 and M3 focal
mechanisms, the misfit of dip is very low (

The KAA and KAS maps (Figs. 7 and 8) show how the network constrains the
fault plane solution as a function of the epicentral location. Moreover,
Figs. 7d–f and 8d–f indicate that the areas with KA mean and standard
deviation greater than 30 and 20

Looking at the uncertainties of FM parameters, obtained by using the D3
dataset, Fig. 9 shows that the dip is the better-constrained parameter
with an error

FME (strike, dip, and rake error) maps for retrieved focal
mechanisms with D3 datasets as input data and simulating earthquakes with M1

KAM (Kagan angle misfit) maps for retrieved focal
mechanisms with D3 datasets as input data and simulating earthquakes with M1

FMM (focal mechanism parameter misfit) maps for
retrieved focal mechanisms with D3 datasets as input data and simulating
earthquakes with M1

The accuracy of fault plane solutions evaluated using the KA misfit and D3
dataset is similar for FM1, FM2, and FM3, mostly with values lesser than
8

KAM (Kagan angle misfit) map for retrieved focal
mechanisms with D3

Three-dimensional histograms of the test results in terms of the number of
stations

Considering the effect of hypocentre depth, the results achieved for
earthquakes at 5 km depth, by using the D3 dataset, are overall unchanged
(Fig. 11). We note that the fault plane solutions are slightly worse
resolved due to a smaller number of P-wave polarities available for M2 and
M3. The KA misfit is generally less than 10

Previous analyses are carried out considering data affected by 5 %
Gaussian error. In the last test, we simulated synthetic data adding a
30 % Gaussian error. As illustrated in Fig. 12, FM solutions show an
overall larger misfit, in particular, the KA inside the seismic network is
less than 20

As a last analysis, we carried out a test in a more general framework, without
a fixed network configuration. We explored the reliability of focal
mechanism estimation as a function of the uniformity of the focal sphere
coverage, defined by the number of recording seismic stations and the azimuthal
gap. We simulated 10 400 earthquakes fixing the fault plane solution and
varying (1) the number of seismic stations (6–30), (2) the take-off angle, and (3) the azimuth of each single station. For each possible number of seismic
stations, we run about 400 simulations, and we randomly sampled the focal
sphere varying the azimuth and take-off of the stations, thus changing the
geometrical configuration of our virtual network of each simulation. We
computed the KA between the theoretical and retrieved focal mechanism (best)
solutions using only P polarities for each simulation. We show the results
in Figs. 13 and S7 in the Supplement, as 3-D histograms and a 3-D scatter plot, respectively.
In Figs. 13a, as expected, the number of stations increases, while the KA
and its range of variation decrease. If the number of stations is less than
nine, only few solutions have

We studied the focal mechanism reliability retrieved by the inversion of
data recorded by ISNet, a local dense seismic network that monitors the
Irpinia fault system in southern Italy. Three different datasets of
seismological observables are used as input data for focal mechanism
determination: (a) P-wave polarities, (b) P

Our results show the following:

The joint inversion of P-wave polarities and P

The spatial resolution analysis of ISNet shows that the most accurate FM
solutions are obtained for earthquakes located inside the network with
strike, dip, and rake misfit

The geometry of the network allows us to resolve well fault plane solutions
varying between a normal and normal strike focal mechanism with strike, dip, and rake misfit generally less than 10

Among the FM parameters, the dip angle shows the lowest uncertainty. Strike
and rake angles have higher errors especially for

Adding a 30 % Gaussian error worsens the accuracy of the retrieved FMs.
Despite the higher uncertainty fault plane solutions (KA misfit

The methodology described in this work can be a valid tool to design and test the performance of local seismic networks, aimed at monitoring natural or induced seismicity. Moreover, given a network configuration, it can be used to evaluate the reliability of FMs or to classify fault plane solutions that represent fundamental information in seismotectonic studies. Although this is a theoretical study, many earthquake scenarios with several magnitudes, locations, and noise conditions can be simulated to mimic the real seismicity.

The tool developed in this work can be provided by the corresponding author upon request.

Catalogue data of earthquakes recorded by ISNet are available at

The supplement related to this article is available online at:

GMA and RDM contributed to the paper conceptualization. GMA carried out the analysis, wrote the paper, and prepared the figures. GMA, RDM, RdN, and AZ reviewed and edited the paper. All authors approved the final version.

The contact author has declared that neither they nor their co-authors have any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This article is part of the special issue “Tools, data and models for 3-D seismotectonics: Italy as a key natural laboratory”. It is not associated with a conference.

We sincerely thank the executive editor Federico Rossetti and topical editor
Luca de Siena and two anonymous reviewers for their constructive suggestions, which contributed to the improvement of our paper. The paper was carried out
within the framework of the Interuniversity Center for 3D Seismotectonics
with territorial applications – CRUST (

This research was supported by the PRIN-2017 MATISSE project, no. 20177EPPN2, funded by the Italian Ministry of Education, University and Research.

This paper was edited by Luca De Siena and reviewed by two anonymous referees.