Forecasting of the first hour aftershocks by means of the perceived magnitude

Abstract

The majority of strong earthquakes takes place a few hours after a mainshock, promoting the interest for a real time post-seismic forecasting, which is, however, very inefficient because of the incompleteness of available catalogs. Here we present a novel method that uses, as only information, the ground velocity recorded during the first 30 min after the mainshock and does not require that signals are transferred and elaborated by operational units. The method considers the logarithm of the mainshock ground velocity, its peak value defined as the perceived magnitude and the subsequent temporal decay. We conduct a forecast test on the nine M  ≥ 6 mainshocks that have occurred since 2013 in the Aegean area. We are able to forecast the number of aftershocks recorded during the first 3 days after each mainshock with an accuracy smaller than 18% in all cases but one with an accuracy of 36%.

Introduction

Short-term aftershock forecasting (STAF) models^1,2,3 are based on the Omori–Utsu (OU) law⁴ which states that the rate of aftershocks with magnitude M above a threshold M_c, at time t after the mainshock, is given by

$$\lambda (t) = \frac{{K10^{b{\mathrm{\Delta }}M}}}{{(t + c)^p}},$$

(1)

where ΔM = M_m − M_c and M_m is the mainshock magnitude. Equation (1) may be used in STAF models once the four model parameters b, p, K, and c are known. The parameters b and p are similar for different sequences and setting b ≃ p ≃ 1 almost always provides reasonable results. Conversely the parameters K and c exhibit huge fluctuations from one sequence to another⁵ and, therefore, they must be fitted as soon as the ongoing earthquake sequence produces a sufficient number of aftershocks³. This fitting procedure requires the identification of all aftershocks above a sufficiently small magnitude threshold. This, however, is a very difficult task because of the overlapping of coda-waves, among close in time aftershocks, which obscures the recordings of smaller events. As a consequence small aftershocks are not recorded in the first part of seismic sequences and the evaluation of K and c is strongly biased^{6,7,8,9,10,11,12,13,14,15,16,17,18}. Some fitting procedures^{11,13,16,17,18,19,20} have been recently developed to take into account this incompleteness of data sets, but they become efficient only when a sufficient number of aftershocks has been recorded. Another problem of STAF methods is that, once λ(t) has been estimated, the predicted aftershock rate must be converted in terms of the probability of ground shaking at a given site. This requires accurate empirical attenuation functions^21,22 which are usually only approximately known and often ignore site effects caused by local site conditions. Here we present a novel method based on a fitting procedure applied to the ground velocity recorded at a site of interest. The method does not require the identification of aftershocks and provides the probability of strong ground shaking without attenuation relations. This idea is, to a certain extent, similar to the proposal²³ of extrapolating the long-term probability of ground shaking directly from the frequency distribution of the maximum amplitude of seismograms²⁴. Also in our method, indeed, it is not necessary to locate earthquakes and to rely on assumptions on seismicity.

Results

Data set and the envelope function

We apply the method to nine M ≥ 6 mainshocks (Table 1) occurred in the Aegean area in the last 5 years^25,26,27 (see Fig. 1) including the M = 6.8 Zakynthos earthquake occurred on 25 October 2018, after the first manuscript submission. The study area is characterized by remarkably high seismic activity, with frequent M > 6 earthquakes that have caused severe casualties and damage during the last centuries, since historical information is available. Each mainshock was followed by a significant increase of the seismic rate in areas of tens of kilometers from the mainshock epicenter. The largest aftershocks are typically observed in the first part of the aftershock sequence and, in some cases, aftershocks as large as M ≥ 5 were recorded just a few hours after the mainshock.

Table 1 Information on the mainshocks with M > 6.0 that occurred in the Aegean area since 2013 and on the Ischia earthquake

Fig. 1

Epicentral distribution of considered mainshocks. The spatial distribution of the M ≥ 6.0 mainshocks that occurred in the Aegean area since October 2013 along with their fault plane solutions shown as lower hemisphere equal area projections. On the top of each ball the occurrence year is given, whereas the numbers inside the yellow stars representing the mainshock epicenters designate their sequential occurrence in time: (1) Crete, (2) Lixouri1, (3) Lixouri2, (4) North Aegean, (5) Karpathos, (6) Lefkada, (7) Lesvos, (8) Kos, (9) Zakynthos. The map is generated using the Generic Mapping Tool (http://www.soest.hawaii.edu/gmt;). Fault plane solutions were taken from http://Ideo.columbia.edu/gcmt

For each mainshock we have investigated the ground velocity V(t) recorded at the closest seismic station during the first 3 days. In all cases we consider the vertical component of the seismogram but very similar results are found using the other two components. We identify the time t_max corresponding to the maximum peak ground velocity V_max and arbitrarily chose the origin time of the mainshock at the time t₀ < t_max such that V(t₀) = qV_max. Different choices of q ∈ [0.25,0.75] do not alter our results. We then define the envelope function μ(t) as

$$\mu (t) = {\mathrm{log}}_{10}\left( {{\cal{H}}\left( {\hat V(t - t_0)} \right)} \right).$$

(2)

Here $\hat V(t - t_0)$ is the filtered signal, in the range [2,10] Hz, of the peak ground velocity V(t) shifted by t₀ and ${\cal{H}}(x)$ is the Hilbert transformation. Finally, we obtain the average envelope $\bar \mu (t)$ via a smoothing procedure (see Methods section).

We plot in Fig. 2b the envelope $\bar \mu (t)$ of the two earthquakes (Lesvos and Kos) occurred in 2017 in the Aegean region. We include for comparison the M = 3.9 Ischia earthquake also occurred in 2017 in Italy. For this earthquake, because of technical problems with the seismic station, only the ground acceleration was available. A common feature of these three earthquakes is their location in or very close to three famous Mediterranean Isles during the summer touristic season and, consequently, their huge economic impact caused by the cancellation of many holiday bookings after their occurrence. In the main panel of Fig. 2 we plot the envelope $\bar \mu (t)$ rescaling the time scale by a fitting parameter τ_M, different for each mainshock. We find that the three envelopes exhibit an excellent collapse up to the time (t − t₀)/τ_M ≃ 10 (Fig. 2c), suggesting the same functional behavior

$$\bar \mu (t) = \mu _M + F\left( {(t - t_0)/\tau _M} \right).$$

(3)

Fig. 2

The average envelope function of the Kos, Lesvos, and Ischia earthquakes. a The average envelope function $\bar \mu (t)$ is plotted as a function of the time divided by the characteristic time τ_M= 4.3 s for Ischia, τ_M= 22.6 s for Kos, and τ_M= 25.7 s for Lesvos. Different open symbols and colors are used for the three mainshocks (see legend). Filled symbols represent the theoretical envelope $\overline {\mu _{\mathrm{{th}}}} (t)$ obtained setting b = 1, p = 1.1 and implementing different choices of K and c for the three different mainshocks. b The same data of panel (a) without time rescaling. c A zoom inside the orange dotted rectangle of the main panel. The green line is the scaling function F(x) = 6 + log₁₀(x) − 3.5log₁₀(x + 0.43)

The parameter μ_M in Eq. (3) represents the maximum value of $\bar \mu (t)$ and we define it the perceived magnitude since it provides the magnitude of ground shaking felt close to the station. We find that the three mainshocks present roughly the same value of μ_M, a result expected for the Kos and Lesvos earthquakes with comparable magnitudes and similar epicentral distance from the recording station. In the case of the Ischia earthquake, this result can be primarily attributed to the smaller epicentral distance from the recording station (see Table 1) but also to soil conditions at the seismic station location²⁸. Concerning the functional dependence of the scaling function F(x) in Eq. (3), the shape of the envelope function is dominated by different mechanisms such as source properties, attenuation, scattering effects, etc.²⁹. The scaling collapse of Fig. 2c suggests that these features are, at a first approximation, incorporated in two seismic source-dependent parameters: the perceived magnitude μ_M and the horizontal time rescaling τ_M. This result is confirmed by the scaling collapse also obtained for the other mainshocks, when plotting $\bar \mu (t) - \mu _M$ as a function of (t − t₀)/τ_M (Fig. 3). In all cases we find that, for x < 10, F(x) ≃ log₁₀(x) − qlog₁₀(x + x₀) with q ≃ 3.5 (continuous curve in Fig. 2c) and x₀ = (q − 1)q^q/(1−q) ≃ 0.43 such that the maximum value of F(x) is equal to zero. We assume the same functional dependence of F(x) for all mainshocks in order to reduce the number of fitting parameters in our method. This fit corresponds to a power law decay of the amplitude of coda-waves of V(t)³⁰ which anticipates the exponential decay at longer timescales²⁹. We wish to emphasize that our results are weakly affected by the specific choice of F(x) and we expect that other choices for F(x) lead to similar results.

Fig. 3

The average envelope function of the recent Aegean earthquakes. The quantity $\bar \mu _{\mathrm{{th}}}(t) - \mu _M$ is plotted versus (t − t₀)/τ_M with: τ_M= 16.5 s for Crete, τ_M= 39.0 s for Lixouri1, τ_M= 27.3 s for Lixouri2, τ_M= 25.5 s for North Aegean, τ_M= 12.9 s for Karpathos, τ_M= 12.1 s for Lefkada, τ_M= 25.7 s for Lesvos, τ_M= 22.6 s for Kos, and τ_M= 11.4 s for Zakynthos. The green continuous line is the scaling function F(x) = log₁₀(x) − 3.5log₁₀(x + 0.43)

Figure 4 shows $\bar \mu (t)$ for the same mainshock recorded at seismic stations at different epicentral distances δr. We found that, as expected, attenuation effects lead to a perceived magnitude μ_M which is a decreasing function of δr. At the same time, we observe that τ_M increases for increasing δr and this can be probably attributed to scattering effects. Nevertheless, the plot of $\bar \mu (t)$ − μ_M as a function of (t − t₀)/τ_M (right panels of Fig. 4) still exhibits a good scaling collapse which is consistent with Eq. (3) for (t − t₀)/τ_M < 10. The collapse extends up to (t − t₀)/τ_M = x_B > 10, whereas for (t − t₀)/τ_M > x_B the envelope presents an almost flat behavior $\bar \mu (t)$ = μ_B related to the contribution of background seismicity which is expected to be quite stationary in time. Figure 4 presents results for only three earthquakes: The more recent one (Zakynthos), the one with the largest $\bar \mu (t)$ − μ_M for t > τ_M (Kos) and the one with the smallest one (Crete). A similar behavior is observed in the other earthquakes.

Fig. 4

The average envelope function at different epicentral distances. a–c The average envelope function $\bar \mu (t)$ obtained from the signal recorded at seismic stations at different epicentral distances δr from the mainshock after the Zakynthos earthquake (a), Crete earthquake (b), and Kos earthquake (c). The station name and epicentral distances are reported in the figure legend. d–f The same quantity of the corresponding left panel plotted after the rescaling $\bar \mu (t)$ − μ_M and (t − t₀)/τ_M, with τ_M= 11.4, 27.7, 16.1, 57.7, and 50.8 s for the stations RTZL, ANKY, LKD2, ALN, and CHOS, respectively, for the Zakynthos earthquake (d), τ_M= 16.5, 57.7, 55.4, and 18.5 s for the stations IMMV, HORT, AGG, and GVD, respectively, for the Crete earthquake (e), and τ_M= 22.6, 23.1, 25.6, and 41.8 s for the stations DAT, ARG, KARP, and PRK, respectively, for the KOS earthquake (f). The green continuous line is the scaling function F(x) = log₁₀(x) − 3.5log₁₀(x + 0.43)

Theoretical interpretation

The main observation at the basis of our method concerns the clear separation of the envelope $\bar \mu (t)$ for the three mainshocks at times t > 10τ_M (Fig. 2), which we attribute to a different aftershock productivity, i.e. different K and c values. This temporal domain, as already observed in refs. ^15,31, is indeed controlled by aftershock occurrence. If an aftershock with perceived magnitude μ_i has occurred at time t_i, the envelope $\bar \mu (t)$ cannot become smaller than $\overline {\mu (t)} \simeq \mu _i$ at times t ≃ t_i. Therefore, larger values of $\bar \mu (t)$, for equal μ_M, indicate a higher occurrence rate of big aftershocks and, consequently, a greater aftershock productivity. In particular, the low level of μ(t) for t ∈ [50, 120] s in the case of the Ischia earthquake can be interpreted as a clear indication of a very small occurrence probability of μ_i ≥ 4 aftershocks in Ischia, in the hours following the mainshock. Indeed, only aftershocks with μ_i ≤ 3 have been subsequently recorded. Accordingly, from the low levels of $\bar \mu (t)$ recorded few minutes after the Ischia earthquake it would have been already possible to inform the authorities and issue a confirmatory announcement for mitigating the economic impact of the earthquake. The situation, conversely, was different after the Kos earthquake where $\bar \mu (t)$ has been stably above $\bar \mu (t) > 3.5$ in the first minutes and indeed tens of μ_i > 4 aftershocks occurred in the next 2 days (see Table 2). Similar indications can be obtained from the envelope function $\bar \mu (t)$ for all the M ≥ 6 mainshocks occurred in the Aegean area after 2013. Figure 3, indeed, confirms that dividing the time by the appropriate τ_M, $\bar \mu (t) - \mu _M$ reveals a quite universal behavior up to t/τ_M < 10. For longer times the split of different envelopes is evident with important consequences for aftershock forecasting. In the following we present a forecasting tool which provides the quantitative estimation of the above considerations.

Table 2 The comparison between the observed n_obs and predicted n_pred (T) aftershocks

The forecasting method

Since aftershocks and mainshock mostly occur in the same tectonic environment, we expect that they present very similar attenuation and scattering properties. This is confirmed by the envelope function of the largest aftershocks, with occurrence time t_i and perceived magnitude μ_i, which is well described by $\bar \mu _i(t - t_i) = \mu _i + F\left( {(t - t_i)/\tau _M} \right)$ with the same scaling function F(x) and the same τ_M of the mainshock. Accordingly, given a mainshock with perceived magnitude μ₀ = μ_M occurred at time t₀ = 0 and an aftershock sequence {t_i, μ_i} (i ≥ 1) we expect that the envelope function is given by

$$\bar \mu _{\mathrm{{th}}}(t) = \mathop {{\max }}\limits_{i:t_i < t} \bar \mu _i(t - t_i) + \mu _{\mathrm{B}},$$

(4)

where the maximum must be evaluated over all events, including the mainshock, occurred at times t_i < t, and μ_B corresponds to the background level of ground velocity during stationary periods. Equation (4) just states that $\bar \mu _{\mathrm{{th}}}(t)$ initially follows the mainshock envelope but as soon as a sufficiently strong aftershock occurs, $\bar \mu _{\mathrm{{th}}}(t)$ increases up to $\bar \mu _{\mathrm{{th}}}(t) = \mu _i$ and decays according to F((t − t_i)/τ_M) unless an other aftershock occurs and the signal can increase again. Equation (4) is supported by the scaling collapse (Fig. 4) of the envelope of the signal recorded at different stations which indicates that, even for distant stations, $\bar \mu (t)$ is dominated by the aftershock occurrence up to the time t_B. Our method is based on the idea to infer the statistical features of aftershock occurrence from $\bar \mu (t)$ in the time interval [τ_M, t_B]. In order to reduce the contamination of background seismicity, in the following we restrict to the signal recorded at the closest station.

Figure 2 shows that it is possible to generate a synthetic aftershock sequence $\{ t_i,\mu _i\}$, for each one of the three mainshocks, which presents a $\bar \mu _{\mathrm{{th}}}(t)$ such that $\bar \mu _{{\mathrm{{th}}}}(t) \simeq \bar \mu (t)$ at different times $t$. More specifically, the aftershock sequence is simulated assuming that aftershock occurrence follows a generalized Poisson process with occurrence rate $\lambda (t)$ given by the OU law (Eq. 1) (see Methods section). The agreement between the theoretical $\bar \mu _{\mathrm{{th}}}(t)$ and the instrumental $\bar \mu (t)$ represents the key ingredient of our method. Indeed, since $\bar \mu _{\mathrm{{th}}}(t)$ strongly depends on $\lambda (t)$, we observe that different choices of the parameters (b, p, c, K) in the synthetic sequence lead to a very different $\bar \mu _{{\mathrm{{th}}}}(t)$. On the other hand, different synthetic sequences implementing the same value of the parameters (b, p, c, K), correspond to a similar average envelope $\bar \mu _{{\mathrm{{th}}}}(t)$. As a consequence, the values of the model parameters (b, p, c, K), which represent the best description of the aftershock decay rate, are expected to correspond to those values which provide the best agreement between $\bar \mu _{\mathrm{{th}}}(t)$ and $\bar \mu (t)$.

The possibility to extrapolate the parameters (b, p, c, K) from the decay of the envelope function has been already recognized in Lippiello et al.¹⁵. The central assumption was that $\bar \mu (t)$, for times t − t₀ > 10τ_M, obeys a logarithmic decay, $\bar \mu (t)$ = μ_M − Δμ − ϕ log (t) and that the parameters (Δμ, ϕ) are somehow related to K and c. The novel and more efficient method developed and applied in this study does not rely on assumptions on the functional form of $\bar \mu (t)$. In particular, it can be implemented in an automatic and almost real-time procedure which provides the occurrence probability of strong ground shaking in a few minutes after the mainshock. The procedure is detailed in the Methods section and here we outline the main steps. In the first step we obtain the best fit of μ_M and τ_M considering only data for t ≤ 100 s. In the second step we consider the learning period t ∈ (0, T) and, for a given set of parameters K, c, b, p, we simulate an aftershock sequence {t_i, μ_i} assuming that the aftershock occurrence probability in time is given by Eq. (1) with ΔM  = μ_M − μ_i. We then assume the Ishimoto and Iida law²⁴ for the peak ground velocity which corresponds to an exponential law for the frequency distribution of the aftershock perceived magnitude $P(\mu _i) \propto 10^{ - \beta \mu _i}$. This is related²³ to the Gutenberg–Richter law for the frequency distribution of earthquake magnitude and, in order to reduce the number of fitting parameters, we always consider β = b = 1 which is a quite realistic value for all the considered mainshocks. For the same reason we also consider a fixed p value, p = 1.1. We have explicitly verified that routines with variable b and p values do not produce any improvement.

The subsequent step is the evaluation of $\bar \mu _{\mathrm{{th}}}(t)$ along with the root mean square deviation $\chi (T) = {\int}_0^T {\mathrm{d}} t\left( {\bar \mu (t) - \bar \mu _{\mathrm{{th}}}(t)} \right)^2$ between the instrumental and theoretical envelope functions. By means of a simulated annealing procedure (see Methods section) we find the best parameter values which correspond to the minimum value of χ(T). This is a fast procedure, of the order of few minutes on standard CPU, which can be repeated as soon as time goes on and new data of $\bar \mu (t)$ become available. In this way we can obtain the best set of parameters in the OU law at different times T after the mainshock.

Testing procedure

To test the method efficiency, the parameters K, c, and μ_B obtained as best fit in the learning period (0, T) are used to forecast the occurrence of aftershocks with perceived magnitude μ_i > μ_M − 3 in the testing period [t_in = 2 h, t_f = 72 h]. The choice μ_i > μ_M − 3 represents a compromise between a sufficient number of aftershocks for robust statistical tests and considerable ground shaking induced by each aftershock. Aftershocks are identified according to the procedure outlined by Peng et al.⁸ selecting among all peaks of the envelope μ(t) with μ_peak ≥ μ_M − 3. This procedure allows us to identify many more aftershocks than those reported in the official Greek catalog. Figure 5 presents with filled symbols the hourly rate λ(t) of μ_i > μ_M − 3 aftershocks identified starting from 1 h after the mainshock. Different colors are used for the different mainshocks. The Kos sequence produces roughly 30 times more aftershocks than the Ischia one. In Kos and Lesvos sequences, λ(t) is consistent with the OU law with p ≃ 1, whereas the Ischia sequence comprises only a few aftershocks and the OU decay is not observed. In the same figure open symbols are used for the hourly rate of events with μ_i > μ_M − 3 predicted by implementing in the OU law (Eq. (1)) the best-fit parameters K, c obtained in the learning period (0, T). Different colors correspond to different mainshock sequences whereas different symbols, for each sequence, correspond to different values of T. Results show that for the Kos and Lesvos mainshocks an accurate prediction of subsequent aftershock occurrence could be available only ten minutes after the mainshock. In the Ischia earthquake, the forecasting after 10 min is not so efficient but for T ≥ 15 min the agreement between predicted and recorded λ(t) is satisfactory.

Fig. 5

Test of the method for the Kos, Lesvos, and Ischia earthquakes. We use filled large symbols for the hourly rate λ(t) of aftershocks with μ_i > μ_M − 3 with different symbols and colors for different mainshocks (see legend). Small empty symbols are used for the predicted aftershock rate obtained using in the OU law Eq. (1) the values of K and c provided by our model. Different colors correspond to different mainshocks whereas different symbols (circles, squares, diamonds, triangles up, triangles left) correspond to different learning periods (T = 1/6, 1/4, 1/2, 1, and 6 h), respectively

In order to better verify the efficiency of our method, we apply the same procedure to estimate the best OU parameters K and c for all the M ≥ 6 mainshocks occurred after 2013 in the Aegean area (see Table 1). The best values of K and c obtained by our procedure for different learning periods T and different mainshocks are listed in Table 3. In Fig. 6 we plot with continuous lines the number of aftershocks n_obs with perceived magnitude μ_i > μ_M − 3 identified in the testing interval [2, 72 h] after each mainshock. Results show huge differences (up to 6000%) in the number of observed aftershocks, among different sequences, even for similar magnitude mainshocks. The number of identified aftershocks is compared with the number of μ_i > μ_M − 3 aftershocks predicted by our model n_pred(T) which, according to Eq. (1), is given by $n_{\mathrm{{pred}}}(T) = \frac{{K10^{b{\mathrm{\Delta }}M}}}{{p - 1}}\left( {\left( {t_{\mathrm{{in}}} + c} \right)^{1 - p} - \left( {t_f + c} \right)^{1 - p}} \right)$, with ΔM = 3, b = 1 and p = 1.1 fixed. We use t_in = 2 h and t_f = 72 h and plot results for different testing periods T with open symbols in Fig. 6. Results have been also reported in Table 2 and a zoom can be found in Supplementary Fig. 1. For all mainshocks we found out that our procedure efficiently forecasts the number of observed aftershocks with a discrepancy typically smaller than 20%. Results also show that the best agreement is obtained for a learning period T ∈ (0.5 h, 2 h). The agreement tends to become worse for increasing T probably because of the contribution of higher order generation aftershocks, not considered in our study. Indeed, the possibility of each aftershock to trigger its own aftershocks leads to deviations from the pure OU law and the estimate of parameters (K, c) according to Eq. (1) becomes less and less accurate for increasing time^32,33,34. Notice that for testing periods with T > 2 h, the testing and learning periods overlap. We make this choice in order to have a single value of n_obs in Fig. 6 and in Table 2 and this simplifies the presentation. At the same time the overlap is null in the more relevant temporal window T ≤ 2 h where our method produces the best agreement. Similar results are also found for a testing period with t_in = 1 h, as illustrated in the Supplementary Figs. 2 and 3 and in the Supplementary Table 1.

Table 3 The estimated parameters K and c

Fig. 6

Short-term test of the method for the recent Aegean earthquakes. We compare the number of observed and predicted aftershocks with μ_i > μ_M − 3 in the testing period [t_in, t_f], with t_in=2 h and t_f= 72 h except for Crete where t_f= 48 h and for Karpathos where t_f= 54 h. Open symbols represent the value of n_pred(T) for different values of the testing period T. The error bars correspond to statistical fluctuations in the number of expected aftershocks for fixed K and c values. Dashed horizontal lines represent n_obs, i.e. the number of identified aftershocks according to the method described in ref. ⁸. Different symbols and colors are used for different mainshocks (see legend)

The same comparison between observed and predicted aftershock number is performed in Fig. 7. In this case we plot the number n_pred(T) of μ_i > μ_M − 3 aftershocks predicted by our model in the temporal range [t_in = 7, t_f = 90] days when short-term catalog incompleteness is not relevant. We therefore directly compare with the number n_aft of m_i > M_m − 3 aftershocks included in the Greek earthquake catalog in the same temporal period. We still found a good agreement between prediction and observation, which shows that in only a few minutes after the mainshock it is already feasible to provide insights in the evolution of seismicity in the following months. There is an exception related to the 2014 North Aegean sequence since the model predicts about 70 aftershocks with m_i > M_m − 3 in the interval [7, 90] days whereas no aftershock is included in the Greek catalog. In this case we observe an abrupt change of the seismic rate, in the mainshock area, which is not consistent with the OU law and, therefore, it is not predicted by our procedure. On the other hand, in the case of the Zakynthos earthquake we find n_pred(T = 1 h) = 220 ± 30 and we cannot compare with n_aft in the Greek catalog, since the testing period is not yet finished. The same analysis of Fig. 7 for a testing period of [30, 90] days is presented in Supplementary Fig. 4.

Fig. 7

Long-term test of the method for the recent Aegean earthquakes. The same plot as in Fig. 6 for the number of predicted aftershocks n_pred(T) (open symbols) with μ_i > μ_M − 3 in the testing period [7, 90] days. Dashed horizontal lines is the number of aftershocks with m_i > M_m − 3 recorded in the Greek catalog in the interval [7, 90] days. Aftershocks are defined as events occurring within a radius of size $L_M = 0.02 \times 10^{0.5m_M}$ from each mainshock epicenter

Discussion

Our results have shown that, in only 30 min after a mainshock, it is possible to have an accurate forecasting of the aftershock occurrence rate in the subsequent days. We wish to emphasize that, to our knowledge, existing methods are not capable to provide such an efficient forecasting in a very short learning period of T ~ 1 h, like the one here considered. The most refined way to take explicitly into account aftershock incompleteness is the Omi et al. method^11,16,20. This method is based on a Bayesian estimate of the completeness magnitude which, however, requires a sufficient number of identified aftershocks. In Omi et al. (2018) the method has been applied only to sequences with at least 30 events in the learning period. In the case of the mainshocks considered in our study, the number of identified aftershocks, at temporal distances smaller than T ~ 2 h reported in the Greek earthquake catalog, is typically smaller than 20 and this makes the prediction according to the Omi et al. method very unstable.

In conclusion, we have presented a novel method for aftershock forecasting where the parameters of the OU law are extracted from the ground velocity recorded some minutes after the mainshock. The accuracy of the method has been verified by means of retrospective tests on recent Greek mainshocks and future developments correspond to the validation by means of prospective tests. Refinements of the method can be obtained by incorporating higher order generation aftershocks which should contribute to improve the agreement for learning periods larger than 1h.

Methods

The average envelope

The quantity $\bar \mu (t)$ is obtained by means of a logarithmic smoothing procedure that corresponds to the average value of μ(t) over windows of increasing duration Δt_k = Δt₀(1 + ζ)^k with Δt₀ = 0.1 s and ζ = 0.005.

Algorithm to invert the OU parameters

The algorithm is composed of four separate routines.

The first routine considers $\bar \mu (t)$ restricted to the time interval [t₀, t₀ + 100 s] and looks for the parameters μ_M and τ_M which minimize the quantity

$$\chi = \left( {\bar \mu (t) - \mu _M - {\mathrm{log}}_{10}((t - t_0)/\tau _M) + q{\mathrm{log}}_{10}((t + \tau _0)/\tau _m + x_0)} \right)^2$$

with q = 3.5.

The second routine generates, for a given set of parameters (b, p, c, K), an aftershock sequence. More precisely, we assume that the number of aftershocks follows a Poisson distribution with average value K10^bΔM with ΔM = 3. We randomly extract the occurrence time of each aftershock t_i from the probability distribution $P(t_i) = \frac{{p - 1}}{{c^{1 - p}}}(t_i + c)^{ - p}$. To each aftershock we associate a perceived magnitude μ_i = μ_M − ΔM + m_i with m_i is extracted from the GR law $P(m_i) = b{\mathrm{log}}(10)10^{ - bm_i}$, without any extra constraint, i.e. μ_i can be greater than μ_M.

The third routine uses the information on τ_M and μ_M, provided by the first routine, and starting from an aftershock sequence {t_i, μ_i} generated by the second routine, gives in output $\bar \mu _{\mathrm{{th}}}(t)$ according to Eq. (4). More precisely, the time is discretized in steps of 0.05 s and at each time t, we evaluate the quantity $\bar \mu _i(t - t_i) = \mu _i + {\mathrm{log}}_{10}\left( {(t - t_i)/\tau _M} \right) - 3.5{\mathrm{log}}_{10}\left( {(t - t_i)/\tau _M + x_0} \right)$, for all aftershocks with t_i < t. The quantity μ_th(t) is given by the maximum value of $\bar \mu _i(t - t_i)$ and we use the same logarithmic smoothing applied to the instrumental envelope to obtain $\bar \mu _{\mathrm{{th}}}(t)$.

The fourth routines finds the best OU parameters (b, p, c, K) via the minimization of the discrepancy between the theoretical and the instrumental envelope, up to a learning time T. This is achieved by means of the algorithm developed by Bottiglieri et al.³⁵ for the log-likelihood maximization. More precisely, for a given parameter set (b, p, c, K), we use n_real times the third routine to obtain n_real independent realizations of the envelope $\bar \mu _{\mathrm{{th}}}^{(j)}(t)$ with j = 1,..., n_real. For each realization we evaluate the quantity $\chi ^{(j)}(T) = {\int}_0^T {\mathrm{d}} t\left( {\bar \mu (t) - \bar \mu _{\mathrm{{th}}}^{(j)}(t)} \right)^2$ and then define χ(T) as the minimum value of χ^(j)(T) for j = 1, ..., n_real. The parameters (b, p, c, K) are then updated according to the algorithm of Bottiglieri et al.³⁵ until we find the best values which correspond to the minimum of χ(T). To reduce the number of parameters we fix b = 1 and p = 1.1. The code is available from the authors upon request.

In the limit n_real → ∞ the procedure identifies the synthetic aftershock sequence which, according to Eq. (4), gives the best description of the instrumental signal and therefore the corresponding (b, p, c, K) is considered the best parameter set. We have verified (see Supplementary Fig. 5) that the procedure appears quite stable for n_real ≥ 10. In order to reduce the computational time, results presented in this manuscript are obtained for n_real = 10. We have also explored two alternative definitions of χ(T): The average value of χ^(j)(T) for j = 1, ..., n_real and its median value. We have found that the three different definitions lead, for n_real ≥ 10, to a similar number of predicted aftershocks n_pred(T) within an uncertainty typically smaller than the 20%.