Assessment of earthquake localization uncertainties for the design of local seismic networks

doi:10.21203/rs.3.rs-2563775/v1

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Assessment of earthquake localization uncertainties for the design of local seismic networks

https://doi.org/10.21203/rs.3.rs-2563775/v1

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The capability of estimating earthquake source locations, together with the appraisal of the relevant uncertainties, plays a crucial role in monitoring and managing both underground anthropogenic activities as well as the natural (micro)seismicity. This is especially true in the close proximity of hydrocarbon production or storage sites, geothermal fields and in general all activities that involve injection/production of fluid or gases in the subsurface. To this end, a monitoring network must be carefully designed to minimize the location errors introduced by geometrically unbalanced networks.

In this study, we first review the different sources of errors that are relevant to the localization of seismic events, how they propagate through the localization algorithms, and their impact on the outcome. We then propose a quantitative methodology, based on a Monte-Carlo approach, to estimate the accuracy of earthquake localization, and particularly suited to the design, optimization, and assessment of the performances of a local seismic monitoring network. This work is an effort to propose a more realistic and reliable way to evaluate the location uncertainty of seismic events, going beyond simplified approaches that tend to under- and over-estimate this metric.

To illustrate the performance of the proposed approach, we have analyzed the distribution of the localization errors and their related dispersion on a very dense grid of theoretical hypocenters, in both horizontal and vertical directions, by using a real monitoring network layout. The results expand in a quantitative fashion the qualitative indications drawn from purely geometrical parameters (the azimuthal gap), and from classical detectability maps.

network design

earthquake localization

detectability

localization uncertainties

1. We implement a Monte-Carlo approach to characterize the uncertainty associated to earthquake locations.

2. We examine how this approach enhances the performance assessment of seismic monitoring networks.

3. This method can be used to optimize the geometry of local seismic networks to better locate small earthquakes.

The production and underground storage of hydrocarbons, the mix of technologies aiming at reducing the GHG emissions such as carbon capture and storage (CCUS), as well as the proposed hydrogen and compressed air storage have a common trait as their management requires an adequate knowledge of the structure and conditions of the subsurface.

Operators are generally interested in monitoring their fields to ensure that they are operated safely, and that the integrity of the structures is not at risk. However, a number of concerns have been raised in recent years concerning the safety of underground operations particularly in terms of seismic risk. Consequently, across Europe, operators are also asked by national authorities to install and maintain seismic monitoring networks to detect the natural and the possibly induced or triggered micro-seismicity in the proximity of the exploited resource (the reservoir). National guidelines (e.g., in Italy, Switzerland, France, UK etc. see – see Braun et al., 2020 and Kraft et al., 2020) set some minimum requirements that the networks should have, in terms of both detection capabilities (the weakest events that the network must be able to identify) and location accuracy. When a system to manage seismicity is in use (for example a Traffic Light System, TLS, based e.g. on specific thresholds for event magnitudes, e.g. Ader et al., 2019; Verdon and Bommer, 2021) for the operators it is also important to be able to detect the onset of very-low intensity seismicity, so that operational measures can be put in place to prevent reaching the critical thresholds (i.e. the earthquake intensity above which operations should be reduced/stopped). To meet these requirements, a monitoring network must be carefully designed and optimized. A wave of approaches and applications, published in the past few years, testifies of the importance of the topic (e.g., Gesret et al., 2014; Wuestefeld et al., 2018; Hassani et al., 2018; Salinas et al., 2019; Shaheen et al., 2021; Nolte et al., 2022; Liu et al., 2022; Parolai and Oth, 2022; Vera Rodriguez and Myklebust, 2022).

In general, the performances of a seismic monitoring network composed of several surface sensors (i.e. sensors installed at the soil surface or in shallow wells) is assessed by looking at the quality of its spatial coverage (network geometry) and its sensitivity, defined as the minimum earthquake magnitude that can be detected in the areas of interest (e.g. Carannante et al., 2017). The classical approach for event localization is based upon the approach of Geiger (1910) who proposed the minimization of differences between measured and theoretical phase arrival times. While migration-style location schemes (e.g., Gharti et al. 2010) use a different approach, the sources of uncertainty remain essentially the same as for the classical approach, where the localization uncertainties of seismic events are the result of three key factors (Pavlis, 1986; Husen and Hardebeck, 2010):

Uncertainty of arrival-time estimation (picking error – e.g., Phillips et al. 2000);

Travel-time modelling errors, arising from the imperfect knowledge of the compressional and shear velocity models (e.g., Usher et al., 2013);

Non-linear nature of the localization problem, which must be accounted for (e.g. Lomax et al., 2000).

to which it is necessary to add the obvious effect of the

(d) Geometry of the station network, constrained by the limited availability of observation points (e.g., Eisner et al., 2010).

The quality of the spatial coverage, which influences the accuracy of the localization, depends on the positions of the stations and it is traditionally evaluated by building the azimuthal gap maps. The azimuthal gap is a purely geometrical parameter that quantifies the largest angle separating two adjacent seismic stations, both measured from the epicenter of an earthquake and represents a quantitative measure of how well the epicenter is surrounded by the stations. The seismic location accuracy is badly affected, in fact, if the azimuthal gap is greater than 180° (Bondár, et al, 2004; Tiira et. al, 2016).

In addition to purely geometrical considerations, sufficient signal to noise ratio must be ensured in order to warrant sufficient first arrival picking accuracy. An essential step in assessing the sensitivity of a seismic monitoring network is the characterization of the environmental noise at the sites where the seismic stations are or must be installed. A classic way to describe the environmental noise, and its variability with respect to frequency, is based on the recording of few days of data that are subsequently used for the computation of the Probabilistic Power Spectral Density (i.e., PPSD) following the approach described in McNamara and Buland (2004). The PPSDs produced for the various candidate of actual measurement locations are then compared against the New Low Noise Model (NLNM) and the New High Noise Model (NHNM) curves as presented by Peterson (1993). The intervals of data used to compute the PPDSs can be combined in different manners serving different goals, for instance, to highlight day versus night (or working days versus weekend days) variability of the noise where anthropic activity is particularly intense. Another way to characterize the noise variability is computing the RMS amplitude of the signal within a certain frequency band using segments of data of fixed duration (e.g., 1 hour of recordings) and then plotting the RMS amplitude versus time.

Once the noise at each site has been analyzed, the next step is earthquake modeling. The methodology for the simulation stage is well-known and widely described in literature (e.g. D’Alessandro et al., 2011, Kraft et al., 2013, Stabile et al., 2013) and commonly adopted for sensitivity analysis and design of monitoring network at different scales. The workflow is articulated into three distinct phases. In the first phase, a grid of hypocentral locations is defined and the traveltimes and the ray-paths pertaining to each hypocenter are computed. In the second phase, for each hypocenter, the displacement spectra are computed at all receivers using the Brune source model (Brune, 1979) in combination with the local spreading and attenuation models. In the third and final phase, the displacement spectrum at each station is compared with the measured local noise levels in order to assess the expected signal to noise ratios.

Based on the considerations above, it is possible, given the hypocentral depth, to create a sensitivity map showing the smallest event magnitude detectable for a given event signal to noise ratio (SNR) threshold (or, alternatively, for a given PPSD percentile). The sensitivity varies, of course, with the depth of the hypocenter and with epicentral distance from the center of the network.

Despite being very useful and informative, the approaches above that provide azimuthal gaps and detectability maps do not offer a quantitative measure of the uncertainty of the seismic event localization as they do not take into considerations the mechanics of the hypocentral estimation process and the associated uncertainties.

Earthquake localization algorithms range from standard linearized techniques (e.g., Hypoinverse, Hypoellipse, etc. – Lahr, 1999; Klein, 2002) to global optimization methods and probabilistic techniques employing direct searches– Lahr solution space (e.g., NonLinLoc - Lomax, et al., 2000). Although these approaches explore and exploit the solution space in different ways, they share the same goal: find a set of model parameters which maximizes the match, or equivalently minimizes the misfit, between observed arrival times of seismic phases at the seismic stations and the corresponding predicted arrival times using a given model of seismic velocities (essentially, Geiger’s 1910 approach). Hence the elements entering the localization process are essentially three: the arrival times of the P and S phases at the seismic stations, the velocity model used for focusing the hypocenter location and the algorithm governing the localization. Each element carries, or produces, imperfect information that translates into an uncertainty around the resulting position of the hypocenter.

The goals of this paper are

to describe and analyze the diverse types of errors and uncertainties involved in the earthquake localization process, and

to propose a workflow to evaluate how these errors are propagated throughout this procedure and how they affect the accuracy and the reliability of the result.

Compared to standard methods used to evaluate location uncertainty, in this work (1) we quantify in a well-defined manner the uncertainties that enter the location process, and (2) the resulting uncertainty distribution does not rely on a linearized solution, hence it does not necessarily assume that the results are normally distributed (as inherent in considering the uncertainty ellipsoids).

The output of the proposed workflow is a set of maps showing, for different depths, the spatial distribution of the localization error and uncertainty thus providing an additional tool for the performance assessment of the seismic monitoring network.

The approach will be exemplified using synthetic examples, and a practical application of the workflow is proved using the layout of a real monitoring network installed in Northern Italy by STOGIT, one of the leading gas storage operators in Europe.

In the following we describe and analyze, using synthetic examples, the two main types of errors affecting the localization process, namely (i) the uncertainty of arrival-time estimation (picking error) and (ii) the imperfect knowledge of the compressional and shear velocity models. We assume that the non-linear nature of the localization problem is properly accounted for by the inversion algorithm (localization software) thus implying a negligible contribution to the whole error. The geometry of the monitoring network, obviously, influences the error propagation in the localization process.

2.1 Uncertainty of arrival-time estimation

There is a quite vast body of literature discussing the factors influencing the identification accuracy of the P and S phases on a seismogram (e.g., Husen and Hardebeck, 2010; Abakumov et al. 2020). The consensus is that there are subjective factors affecting the picking, such as the experience of the operator performing it, as well objective ones such as the strength of the seismic event and dominant frequency.

Assuming that the subjective factors can be at least limited, the signal-to-noise ratio (SNR) becomes the main factor controlling the quality and the accuracy of the picking. Hence, the improvement of the quality of the picking is directly connected to the pre-processing applied to the seismograms with the aim of improving their SNR.

Picking errors define the precision of the hypocenter location since they lead to a relative location scattered around the true earthquake point location (Husen and Hardebeck, 2010). The pick is commonly assumed to be a random variable symmetrically and normally distributed around the true arrival time, even though some authors have demonstrated that the distribution can be highly asymmetrical (Diehl et al. 2012, Abakumov et al. 2020).

Following these considerations, we assume that the estimation error follows a Gaussian distribution with zero mean and variance ${\sigma }^{2}$. This latter can be quantified using the information theory and the theorem of Shannon-Hartley (Aki, 1976), and described by the equation

${\sigma }^{2} = {\left({\text{log}}_{2}\left(1+\frac{SNR}{K}\right)\cdot {2\cdot f}_{max}\right)}^{-2}$

(1)

where ${f}_{max}$ is the maximum frequency of the recorded signal, SNR the signal-to-noise ratio, and $K$ represents an empirical factor equal to $K=20$ (Aki, 1976). The derived picking errors, for different values of the parameter ${f}_{max}$, are shown in Fig. 1 as a function of the SNR. It is worth noting that for good quality events (i.e., SNR > 10) the picking error curves become progressively flat with increasing frequency and ultimately nearly independent of ${f}_{max}$. In practical applications, f_max can be taken as the maximum of the simulated spectra of the seismic event (using, for example, Brune’s 1976 model) whereas SNR is a design parameter. It is important to note that both the seismic spectrum and the SNR are also used in the computation of network sensitivity, hence a link between picking uncertainty and event detectability can be established. In other words, the picking error distribution computed from Eq. (1) is directly associated to the minimum magnitude that the seismic network will be able to detect for a given SNR.

2.2 Estimation of the traveltime modelling errors

The errors in modelling of the traveltimes of P and S phases arise from the inexact knowledge of the compressional and shear waves velocities. This uncertainty is even more pronounced at locations where a new monitoring network must be designed and installed, because the knowledge of the subsurface conditions is generally rather limited at that stage of development.

Seismic velocity can be derived from a number of information sources, ranging from very localized, high frequency, seismic logs in boreholes to estimates derived from distant seismic events and recorded by the local network. Active seismic acquisition methods (based on refraction, reflection, or surface waves) can also be utilized and cover the intermediate scale range. The variety of scale, frequency range, acquisition geometry and nature of the propagating waves (P, S, Rayleigh) pose a severe challenge in integrating the data into a single, potentially three-dimensional, model of P and S wave seismic velocities that is also, potentially, anisotropic. This is particularly true for the depth range between the scope of near-surface active acquisitions (~ 100 m) and deep investigations relevant to the reservoir zone (~ 1000 m). Whatever the intricacies of this data integration process, a reliable P and S velocity model is essential to provide reliable localizations of seismic events, and possibly not enough attention is traditionally paid to this aspect that proves absolutely critical in the case of small-scale, high-resolution monitoring networks focused on a specific area of interest.

Although it is feasible to use three-dimensional velocity models for localizing earthquakes, the use of layered models (i.e., one-dimensional velocity models as a function of depth) is a very common and a widely accepted practice in both global and local seismology (e.g. Husen et al., 2011; Theunissen et al. 2018), in particular during the network design phase. For local networks, this approach is valid when lateral variations of elastic properties are negligible. This may depend on the local geological situation. In addition, it is also useful to recall that during the design of a new seismic monitoring network a 3D velocity model at the scale of interest is seldom or never available, therefore this situation has clear practical implications.

Considering that the velocities of each layer are affected by a certain degree of uncertainty, to exemplify how the uncertainty in the velocity model affects travel-times in the following we will assume that both P- and S-wave velocities are distributed symmetrically and uniformly in the range of ± 10% about the estimated layer velocities.

Based on the assumption above of a uniformly distributed velocity error of ± 10%, ${v}_{i}$ (the velocity of the $i$-th layer) is a random variable uniformly distributed between $0.9{v}_{i}$ and $1.1{v}_{i}$, while ${p}_{i}$, the slowness of each layer, is a random variable with inverse uniform distribution. Consequently, traveltime, defined as the integral of slowness along the ray-path, is distributed as the weighted sum of inverse uniform random variables. The weights involved in the computation of the traveltime depend on the lengths of the ray-paths in each layer which in turns depend on the position of the hypocenter as schematically shown in Fig. 2a for two different hypocentral locations. The position of the hypocenter with respect to the station (the black triangle in Fig. 2a) also controls the number of random variables to be summed.

As the velocity, and consequently the slowness, changes from layer to layer, the corresponding random variables span different ranges and thus the probabilistic distribution of the traveltime errors cannot, in general, be derived analytically. The effect of the velocity errors on traveltime can be evaluated numerically by a stochastic approach. For the sake of an example, we generated an ensemble of 1000 1D velocity models (500 P- and 500 S-wave velocity models) by perturbating the velocity value of each layer in a range of $\pm 10\%$ using a uniform distribution. Figure 3b shows the average one-dimensional P and S wave velocity models, red and blue curve respectively, with their corresponding velocity ranges. The modelling is performed for 24 hypocenter locations per depth slice (at depths of 2, 3, 4 and 5 Km, hence a total of 96 hypocenter locations) on a regular coarse grid with 2 km spacing in both $x$ (Easting) and $y$ (Northing) directions, ranging from 2 to 12 km Easting and from 2 to 8 km Northing (see Fig. 5a as reference). Exploiting the one-dimensional nature of the velocity models, thus in a cylindrical coordinate system, the epicenters locate at distances ranging from 1.2 to 12 Km for a reference station (see Fig. 4d). A schematic view of the hypocenter positions with respect to the reference station is shown in Fig. 2b.

Figure 3a and 3c show examples of the perturbation distribution for a single hypocenter position and for S-velocity and P-velocity, respectively. The histograms of the velocity values in each layer are juxtaposed vertically, with the shallowest layer at the top of the display. It can be noted that at deeper layers the velocity distributions are no longer uniform and are no longer symmetrical about the true velocity values. This is due to a limitation of the used ray-tracing method that only accepts velocity models that increase monotonically with depth. At deeper layers (e.g., below 5 km in Fig. 3b) the differential velocity between two adjacent layers decreases. This leads to a narrower and asymmetric interval about the mean value where random values are drawn.

For each hypocenter and for each perturbed model in the ensemble, we computed the traveltime using the forward model of Hypoinverse (Klein, 2002). Each run is then compared against the reference traveltime provided by the unperturbed model. This yields an ensemble of traveltime realizations that shows the uncertainty associated with traveltimes for the reference station and for the different hypocentral distances and depths considered.

Figure 4a and Fig. 4b show the histograms of the traveltime errors for a set of hypocenters at 2km and 5km depth, respectively. The Gaussian distribution overlaid to the histograms has mean and standard deviation computed from the population. Figure 4c displays the histogram of the traveltime errors, overlaid by the Gaussian curve estimated from the entire model ensemble, for all the hypocenters. Figure 4d shows, for each modeled hypocenter, the relative error associated to the traveltime (ratio between the absolute traveltime error (${\Delta }t$) and the reference time ${t}_{ref}$) expressed as a percentage. The points are plotted in the distance-depth plane together with a map obtained via interpolation of the points. Interestingly the percentage ratio is between 2% and 3% and shows a clear decreasing trend with depth. Figure 4 considers only the effect on P-traveltime errors, but a similar behavior is observed using S velocities.

2.3 Localization uncertainty assessment

Based on the considerations and the results presented in the previous sections, here we evaluate the performances of a monitoring network by computing the spatial uncertainty of the localization. To this end we consider a real monitoring network installed by STOGIT in Italy. The network is composed of 7 surface stations installed with the spatial configuration shown in Fig. 5a. Figure 5b shows the corresponding azimuthal coverage (AG, Azimuthal Gap) for this network.

We estimate the earthquake hypocentral locations on a fine grid in both $x$ (Easting) and $y$ (Northing) directions (with 150 m spacing in both directions) at 4 different depths: 2, 3, 4 and 5 Km. The resulting grid has more than 24000 hypocenter locations.

The traveltime for each of these hypocenters is perturbed, for each realization $r$, according to the equation

$${\left({T}_{P,S\left(pert\right)}^{i,j,k, s}\right)}_{r} ={\left({T}_{P,S \left(true\right)}^{i,j,k,s}+{\epsilon }_{pick}+{\epsilon }_{tt}\right)}_{r}$$

where ${T}_{P,S \left(true\right)}^{i,j,k,s}$ is the true traveltime (for both P and S waves) from the hypocenter at the coordinates $i,j,k$ along the axes $x, y, z$, respectively, to the station $s,$ ${\epsilon }_{pick}$ is the picking error, and ${\epsilon }_{tt}$is the error caused by the uncertainty associated to the velocity models. In this work the total number or realizations is $r=500$.

The picking error ${\epsilon }_{pick}$ is calculated from Eq. (1) with parameters ${f}_{max}=25$Hz, and SNR=5 and SNR=2 for shear and compressional wave traveltimes, respectively. These SNR are realistic for a situation when the picking is manually performed by an analyst.

The traveltime modelling errors ${\epsilon }_{tt}$ are estimated, according to the results obtained in the previous section, assuming a relative error ${\Delta }\text{t}/{t}_{ref}$ constant and equal to 3%. For each hypocenter location, each traveltime realization is inverted using the software package Hypoinverse (Klein, 2002).

To prove how the location accuracy is sensitive to the Azimuthal Gap (AG), we show the analysis of the inverted hypocenters for all model realizations in two localization cases. The first case comprises the localization of seismic events with a low AG (approximately ${115}^{\circ }$), whereas the second scenario shows the localization of events with a high AG (approximately ${175}^{\circ }$).

The first case, displayed in Fig. 6, shows the inverted hypocenters at 4 different depths (2 Km, 3 Km, 4 Km, ad 5 Km) but with the same epicentral location and with an AG of approximately ${115}^{\circ }$. It can be observed that the inverted hypocenters have a low dispersion (a low variance) about the true hypocenter locations both in the horizontal and in the vertical directions and picture with a high accuracy the true hypocenter location. Also note that the variance of the scattered inverted hypocentral locations increases with depth.

The second case (Fig. 7), although the events’ epicenters are near one seismic station (ST6), has an AG approximately equal to ${175}^{\circ }$, that is close to the 180° threshold as indicated in the literature for a bad network performance (Bondár, et al, 2004; Tiira et. al, 2016). In fact, the localization of the hypocenters has a wider distribution both in the vertical and horizontal directions, i.e., the variance of the inverted hypothetical increases as function of the AG.

Extending the localization analysis to all epicentral locations on the fine grid and for all depths, we have an ensemble of 500 locations for each hypocenter location. Each of these ensembles is then used to derive statistical measures such as:

the horizontal localization error, given by the ${L}_{2}$-norm of the true and median of the estimated epicenters, which will be indicated as ${\delta }_{h}$.
the vertical localization error, given by the ${L}_{1}$-norm of the true and median values of the estimated hypocenter depths, indicated as${\delta }_{z}$
the standard deviation of the horizontal localization error, indicated as ${\sigma }_{h}$ and
the standard deviation of the vertical localization error, indicated as ${\sigma }_{z}$.

Since these quantities are evaluated on a fine grid, it is convenient to display them in form of maps as done in Fig. 8–11, where each panel represents one of the aforementioned quantities for each given hypocentral depth.

The horizontal localization error at all depth slices, as shown in Fig. 8, consistently follows the AG values for this network configuration. Therefore, the minimum error is found inside the seismic station network where the AG has lower values (see Fig. 5b). Although the error shape map remains the same at different depths, on average it increases with depth.

Inside the network the error that we make in the localization of epicenters is smaller than 200 m and increases up to values greater than 600 m at the edges of the model. This is observed for all hypocentral depths.

Differently from the horizontal localization error, the vertical error (Fig. 9) shows a different pattern with respect to the hypocentral depths.

To a hypocentral depth of 2 km the error shows lower values inside the seismic network with absolute values smaller than 150 m. Conversely, for deeper hypocenters the minimum values of the error progressively move towards the peripheral regions of the model. At 3 km depth the localization uncertainty shows a narrow region of minima that surrounds the seismic network, while at 4 km and 5 km the region of minimum error moves progressively away from the seismic network, showing consequently an increase of the localization uncertainties inside it. In fact, the uncertainty associated with a seismic event at 5 km depth, with an epicenter falling inside the seismic network, reaches values greater than 400 m (Fig. 9d).

The maps showing the statistical dispersion of the inverted hypocenters at all evaluated locations, quantified with the measure of the standard deviation, for the horizontal and vertical directions are displayed in Fig. 10 and Fig. 11, respectively. In all cases, for both vertical and horizontal errors and for all evaluated depths, the standard deviation increases from the center of the network to the peripheral regions of the model. This evidence agrees with the AG map where it reaches values greater than 180° outside the seismic network.

In this work we review the different types of errors and uncertainties that affect the seismic event localization process, how they are propagated through the workflow and their effects on the outcome of the localization. A Monte-Carlo approach has been used to correctly characterize the resulting uncertainties, with no reliance, thus, on linearized solutions (confidence ellipsoid). The approach is particularly suitable to local monitoring networks, and some of the results are specific to this type of system. The approach and methods discussed in this work can therefore be used to both design local seismic monitoring networks, to optimize the layout of the stations (i.e., the spatial distribution) in case a network is being updated and also to assess quantitatively the network performances for a system already in the production phase.

We have, in particular, discussed two types of errors: the error affecting the picking of the P and of the S phase arrivals and the errors in the one-dimensional velocity models used in the localization process. We have presented a workflow based on stochastic modelling that allows us to translate the error of the velocity model into errors of traveltime modelling.

Finally, we have evaluated the performances of a real local monitoring network exploiting the stochastic perturbation of reference traveltimes and assuming both picking and velocity model errors. We applied the stochastic technique to a very dense grid of hypocenters, and we produced maps showing the distribution of the average localization errors and the related dispersion (i.e., standard deviation) for both the horizontal and vertical directions.

The two most relevant conclusions that are drawn from the stochastic analysis of the traveltime error estimate are that

the distribution of the traveltimes for all realizations is nearly Gaussian (the distribution is not normal due to the limitations of the ray-tracing model that forces the stochastic population to always have increasing velocities with depth), and

the relative error is nearly constant and equal to 2–3% if we assume that the velocity is uniformly distributed in the range of $\pm 10\%$. This result allows us to easily model the traveltime error for all realizations.

The error and uncertainty maps produced with the presented workflow allow us to assess with a very fine spatial resolution the performance of the local monitoring network, in terms of accuracy and reliability of the localization. These maps enrich and expand in a quantitative fashion the qualitative indications that can be drawn from azimuthal gap maps and from detectability maps.

The quantitative evaluation of the localization accuracy is of paramount importance during the design of a local monitoring network. In fact, the knowledge of the spatial distribution of errors allows the designer to proceed with an iterative process aiming at optimizing the positioning of the monitoring stations and minimizing the costs of installation provided the target level of accuracy over the monitored area.

During the execution of the study, we have heavily relied on HYPONVERSE as a tool for translating the velocity uncertainties into traveltime uncertainties as well as a hypocenter localization engine. As already mentioned above, one limitation of HYPONVERSE is the need of having layers with increasing velocities that introduces a bias into the distribution of the velocity perturbation of the deepest layers which is no longer distributed as a uniform random variable.

Another drawback of the approach used in this work is related to the execution time required by the stochastic modelling exercise. In view of the iterative nature of the network design exercise based on the evaluation of multiple scenarios, an excessive runtime can become a severe limiting factor.

As a further development of this work, future efforts will be expended to finding alternative ways to perform the stochastic modelling and the corresponding localization, free the limitations highlighted above. In addition, the evaluation of the error propagation in the localization procedure, as here evaluated, does not free the user from making the best effort particularly to estimate at best the P and S local velocity fields, which is ultimately crucial in order to the achieve the precise localization of events required by this type of local (micro-) seismological applications.

Ethical Approval

Not applicable.

Competing interests

The authors have no relevant financial or non-financial interests to disclose.

Authors' contributions

Antonio Fuggi performed the analytic calculations, the numerical simulations and wrote the main manuscript text. Simone Re wrote the main manuscript text and contributed to the analytic calculations. Alessandro Brovelli conceived the original idea, contributed to the analysis and interpretation of data and wrote the main manuscript text. Giorgio Cassiani, Giorgio Tango, and Sergio del Giudice contributed to the drafting of the manuscript, aided in interpreting the results and revised the article critically. All authors provided critical feedback and helped shape the research, analysis and manuscript.

Funding

No funding was obtained for this study.

Availability of data and materials

Due to confidentiality agreements, neither data nor source code can be made available for the current study.

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No competing interests reported.

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Assessment of earthquake localization uncertainties for the design of local seismic networks

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1. Introduction

2. Errors And Uncertainties In Seismic Event Localization

2.1 Uncertainty of arrival-time estimation

2.2 Estimation of the traveltime modelling errors

2.3 Localization uncertainty assessment

3. Discussion And Conclusions

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