Wireless sensor networks (WSNs) enable solutions in multiple fields, and they are adopted in environmental, health, urban, and military applications [1, 2]. A problem commonly solved within WSNs is the detection and localization in space of relevant events or targets [3–7].

This study focuses on earthquake early warning systems (EEWSs) [8–10], which are deployed in seismic areas for the real-time detection of earthquakes, with the ultimate goal of sending alerts to citizens and stopping critical processes before ground shaking begins.

Classic EEWSs are based on a dense network of scientific-grade instruments, with construction and operating costs on the order of millions of euros [11]. This largely limited their implementation, especially in seismic developing countries.

Due to smartphone technology, low-cost EEWSs have been recently implemented at the global level [12]. Smartphones are used to detect ground shaking using the on-board accelerometer, and a warning is issued to the population as soon as the earthquake is detected. This path has been explored by the Earthquake Network (EQN), a citizen science initiative [13, 14], that, since 2013, implements the first smartphone-based EEWS.

Within the EQN EEWS, nodes of the WSN are the smartphones voluntarily made available by citizens. This poses many challenges because personal smartphones mainly sense the “anthropic noise” connected with human activities.

The primary challenge faced by the EQN is to control the probability of false alarms and the probability to miss an earthquake. Alerts may be triggered by events unrelated to earthquakes and some (possibly strong) earthquakes may be missed, especially if the number of monitoring smartphones is small. Both false alarms and missed detections may undermine people's trust in the EQN.

In the pivotal study by Finazzi and Fassò [15], a statistical methodology is developed for identifying in real-time earthquake occurrence. The study, however, does not take into account the spatial dimension of the smartphone network, making the detection algorithm prone to false alarms. Moreover, the methodology does not allow to estimate important earthquake parameters such as epicenter and depth. In Finazzi et al. [16], instead, the EQN detection capabilities are modeled within a probabilistic framework. It is discovered that the EQN missed some relatively strong earthquakes that were supposed to be detected by the smartphone network. These considerations and findings suggest that there is room to improve EQN's methods and algorithms.

This study proposes a statistical methodology for 1) controlling the probability of false alarms, 2) controlling the probability of missed detection, 3) classifying a detection between true and false earthquake, and 4) estimating earthquake epicenter and depth (if the detection is classified as a true earthquake).

The methodology is based on a statistical parametric model, statistical hypothesis testing, and Monte Carlo simulation. Contrary to model-less approaches (see for instance [3]), the methodology exploits the fact that the spatio-temporal dynamic of seismic waves is well-known. This information is retained by the statistical model, and it helps to both classify the EQN detection and to estimate the earthquake parameters.

Due to the peculiarity of the specific application, real-time is a constraint. Ideally, classification and earthquake parameter estimation should not exceed 1 or 2 s of computing time.

The smartphone-based EQN is used to test the statistical methodology, which is then applied to some true and false EQN detections.

Before formalizing the classification and the earthquake parameter estimation problems, it is useful to detail the output of the earthquake detection algorithm currently implemented by the EQN [15]. For any given area of radius 30 km, the algorithm compares the number of triggering smartphones in the last 10 s with the number of active smartphones. A triggering smartphone is a smartphone that detected an acceleration above a threshold, while an active smartphone is a smartphone known to monitor earthquakes. If the ratio between triggering smartphones and active smartphones exceeds a threshold, an earthquake is claimed to be detected. The output of the detection algorithm consists of the detection location and the list of the triggering smartphones (triggers for short), which are identified by their spatial coordinates (latitude and longitude) and the triggering time.

An earthquake detection made by an EQN is defined in terms of k_j>0 triggers, where j is the index of the generic detection. In general, k_j is not a constant, meaning that each detection is characterized by a different number of triggers. Each trigger is described by the feature vector as follows:

where t_i∈ℝ is the triggering time, while (lati,loni)∈S2 are the smartphone coordinates, with S2 being the sphere embedded in ℝ³. The k_j×3 matrix X=(X1′,...,Xkj′)′ is the data point, and the feature space is X=∪k=1∞Xk, with Xk=ℝk×(S2)k and k>0 is the generic number of triggers.

Let Y={-1,1}∋y be the label space. For each earthquake detection, y = 1 if the detection is false while y = −1 if the detection is related to a true earthquake.

The aim is to learn a hypothesis map h:X→Y such that y≈h(X) for any data point X (i.e., for any future EQN detection). The map h is highly non-linear since the information content of X is determined by the spatio-temporal dynamics of the seismic waves and spatial distribution of the smartphones at the time of the earthquake.

A statistical parametric model f:X→Θ is adopted to understand if X is generated by a true earthquake. The unknown model parameter vector is θ∈Θ = ℝ^s, with s≪k_j as the vector size. The hypothesis map is then h(X) = g(f(X)) = g(θ). Note that s is constant, and it does not depend on the dimension of X.

When dealing with EEW systems, it is required to control two parameters: the probability α of missed detections (true earthquakes which are not detected by the system) and the probability β of false detections (detections which are not related to any occurred earthquake). It is thus reasonable to adopt a 0/1 loss function as follows:

and to learn a g that minimized the Bayes risk

As discussed by Jung [17], solving (Equation 1) requires knowing the joint probability distribution p(X, y). Instead, we rely on the fact that it is relatively easy to simulate EQN detections under different smartphone geometries and different earthquake parameters. This induces a variability on X and on the number of triggers k_j. Assuming to have a data set D=(X(1),y(1)),...,(X(m),y(m)) and that D is a representative sample of p(X, y), we define the empirical risk as follows:

and g is learned from the following minimization problem:

Note that solving (Equation 2) is equivalent to solve

where it is made explicit that the probabilities of missed and false detections depend on g.

From an EEW perspective, the solution provided by Equation (3) is not necessarily the best. In some contexts, a missed detection has a larger negative impact than a false detection, while in other contexts, it is the opposite. In this case, one probability is fixed to the desired level, and the other probability is minimized. Two other minimization problems for learning g are the following:

In this section, we propose a statistical parametric model for the generic data point X. The observed triggering time for a smartphone sensing an earthquake is modeled as

where ti* is the expected triggering time, while ϵi~N(0,σϵ2) is a random component. More in detail

with

as the distance between the hypocentre and the smartphone location, v is the seismic wave speed, and t_O∈ℝ is the earthquake origin time.

In Equation (8), D_{i, E} is the distance between the epicenter (latE,lonE)∈S2 and the smartphone location, d_E∈[0, 500] is the earthquake depth, and R is the earth radius (6, 371 km). Here, it is assumed that all smartphones either detect the primary seismic wave (v = 7.8 km/s) or they all detect the secondary wave (v = 4.5 km/s). This assumption is justified by the fact that earthquake detection is based on smartphones within a radius of 30 km, which is a relatively small area.

The role of the random component ϵ_i is to model the difference between the expected and the observed triggering time. This difference is mainly due to the smartphone detection delay and a seismic wave velocity that may differ from the expected value.

Equations (6–8) fully define the statistical model f and the model parameter vector is θ=(latE,lonE,dE,tO,σϵ2)∈Θ=S2×[0,500]×ℝ×ℝ+⊂ℝ6.

The minimization problem in Equation (17) has no closed-form solution. For this reason, we implement a Monte Carlo simulation that aims to simulate a data set D and to minimize Equation (17).

A total of 1,000 true EQN detections and 1,000 false EQN detections are simulated considering the true locations of 1,000 smartphones of the EQN in Lima (Peru).

The probability of missed detection is fixed to α = 0.01 while δ is made varying from 0.1 to 1.5 with step 0.1. For each value of δ, β(δ) is computed by estimating the model f and by implementing the hypothesis test (Equation 13) overall data points X^(j) in D. Finally, δ^ is the value of δ that minimizes β(δ).

The methodology developed in this study is applied to true and false detections made by the EQN. As a true earthquake, the event occurred near Genova (Italy) on 4 October 2022 at 21:41:10.5 UTC is considered. Figure 5 depicts the triggering smartphones (n = 21), while estimation and classification results are reported in Table 1 for v = 7.8 and v = 4.5 km/s, respectively.

For both seismic wave velocities, we can observe that latitude and longitude are accurately estimated, while the error in depth is not negligible. Nonetheless, the true values are within the 99% confidence intervals evaluated from the standard errors on the model parameters. In addition, the earthquake is classified as true under both velocities since both observed test statistics are lower than the test critical value. This happens because triggers are close to the epicenter, and primary and secondary seismic waves are nearly concurrent.

The estimation and classification results were obtained in less than 1 s using an Intel(R) Core(TM) i7-9750H CPU @2.60GHz, suggesting that the approach can be adopted for real-time applications.

Figure 6 shows the n = 108 triggers of a false detection occurred near Acapulco (Mexico) on 25 September 2022, at 09:55:45 UTC. In this case, the computed test statistics are 1039.7 and 1026.0 for v = 4.5 and 7.8 km/s, respectively, while the critical value is 141.62. H₀ is rejected in both cases and the detection is claimed as false. In this particular case, the detection was caused by a strong lightning bolt. The speed of sound, however, is around 0.3 km/s, a value much smaller than the speed of primary and secondary seismic waves.

The methodology developed in this study allows to classify detections made by smartphone-based earthquake early warning systems between true (related to a real earthquake) and false. This is done analyzing the information content of the smartphone triggers that contributed to the detection.

With respect to classic classification problems, the data point describing the triggers has a varying dimension which depends on the smartphone network geometry. The proposed solution is based on two steps. First, a statistical parametric model is used to convert the data point into a parameter vector with a fixed (and small) dimension. Second, a hypothesis test is implemented for classification.

While we do not claim our choices of f and g to be optimal, both steps are based on well-established statistical methods. With respect to the specific choice of g, it is worth discussing that a simpler alternative is the linear map g^* = δ′ϕ, with δ = (δ, 1)′ and ϕ=(1,-σ^ϵ2)′. In this case, the classification is based on the more intuitive comparison σ^ϵ2⋛δ. This simpler solution, however, does not take into account neither the actual number of triggers for the specific detection (10 or 1,000 makes a difference in the uncertainty of σ^ϵ2) nor the fact that the distribution of σϵ2 is known under the null hypothesis (that the detection is related to a true earthquake). Using hypothesis testing, we are thus able to retain a part of the information which is lost when X is synthesized with θ.

Classification and earthquake parameter estimation are performed in near real time, making the statistical methodology suitable to be implemented in operational systems. On the contrary, the methodology does not fully exploit the information available on the EQN system. Specifically, the modeling is only on the triggering smartphones, while the active non-triggering smartphones are ignored. Knowing, at the EQN detection time, which smartphones have not (yet) triggered may better constraint epicenter and depth, thus improving their estimates.

In addition, for an EEWS like EQN that works globally, it would be important to study if the data set D generated by the Monte Carlo simulation is a representative sample of p(X, y). If not, the observed α and β probabilities might deviate from the expected ones.

Finally, a limit of the approach proposed by this study is that the statistical methodology is applied downstream of EQN detections. Ideally, the detection, the classification, and the earthquake parameter estimation problems should be jointly addressed in a unified approach. In this regard, the vast literature on wireless sensor networks may help propose a solution under the real-time constraint.

These open problems, along with the estimation of the earthquake magnitude, will be the focus of future works.

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

FF: conceptualization, writing–review, and editing. FM: investigation, methodology, validation, and writing–original draft preparation. All authors contributed to the article and approved the submitted version.