## Probability of successive earthquakes based on global catalogues

On the basis of global earthquake statistics, we first inferred the probabilities of successive occurrence of an earthquake of Mw 7.0 or larger and an earthquake of Mw 8.0 or larger. This was conducted by independent verification and elaboration of the findings of a previous study 23, which were reflected in the official guidelines for disaster management measures in response to warnings 24.

We referred to earthquakes of Mw larger than or equal to 8.0 as “M8 + earthquakes”. Similarly, we referred to earthquakes of Mw larger than or equal to 7.0 and smaller than 8.0 as “M7-class earthquakes”. We calculated the probabilities of an M7-class or M8 + earthquake, followed by another M8 + earthquake nearby, alongside their probability gains and confidence intervals (see Methods for details).

We used two global earthquake catalogues, namely the International Seismological Centre-Global Earthquake Model (ISC-GEM ver 6.0) Global Instrumental Earthquake Catalogue 25,26 and the Advanced National Seismic System (ANSS) Comprehensive Earthquake Catalog 27. Inclusion criteria for these data sets were as follows: 1) earthquakes in all regions without area restrictions and 2) subduction zone earthquakes only. A previously reported approach was adopted 23 with modifications. The previous study used only the ISC-GEM ver 6.0 Global Instrumental Earthquake Catalogue, did not assess restricting the area to subduction zones, and did not evaluate the uncertainties.

The probabilities calculated using the ISC-GEM ver 6.0 Global Instrumental Earthquake Catalogue and ANSS Comprehensive Earthquake Catalog data sets for all the regions and subduction zones were consistent (Supplementary Tables S1 and S3). Subsequently, the results obtained using the ISC-GEM ver 6.0 Global Instrumental Earthquake Catalogue without area restrictions were used.

Among the 105 M8 + events recorded from 1904 to 2015, the number of subsequent M8 + earthquakes that occurred in the vicinity within 1 day, 3 days, 1 week, 2 weeks, and 3 years were 2, 3, 3, 5, and 11 (1.9%, 2.9%, 2.9%, 4.8%, and 10%), respectively. The confidence intervals were usually wide (0.23–6.7% within 1 day, 1.6–11% within 1 week, and 5.3–18% within 3 years) owing to the small sample size (Supplementary Table S1).

Among the 1,354 M7-class events recorded from 1904 to 2015, the number of subsequent M8 + earthquakes that occurred in the vicinity within 1 day, 3 days, 1 week, 2 weeks, and 3 years were 3, 5, 8, 9, and 23 (0.22%, 0.37%, 0.59%, 0.66%, and 1.7%), respectively (Supplementary Table S3). The confidence intervals were narrower than those of successive M8 + cases, ascribed to the relatively large sample size. Moreover, the probability of an “M7 class–M8 + successive occurrence” was smaller by one order than that of an M8 + successive occurrence.

Considering the confidence intervals, the results were consistent with the claim of the official guidelines of countermeasures in response to a warning of Nankai megathrust earthquakes 24. The guidelines state that, “the frequency of an M8-class or larger earthquake occurring within 7 days after an earthquake of M8.0 or larger is once per just over a dozen times”, and “the frequency of an M8-class or larger earthquake occurring within 7 days after an earthquake of M7.0 or larger is once per a few hundred times.”

The probability gains, obtained by dividing the probability of successive occurrence by the base rate, are listed in Supplementary Tables S4 and S5. Here, we calculated the base rates of probability by assuming a Poisson model, with an average recurrence interval of 90 years (Supplementary Table S2), which applies specifically to the Nankai Trough region (see Methods for details of the calculation).

As expected, we observed a sharp rise in the probability gains just after the occurrence of the first earthquake. For example, the probability of a successive M8 + occurrence within a day was 76 to 2,200 times higher than usual, and the probability gain dropped to 1.6 to 5.5 in 3 years (Supplementary Table S4). Slight differences were observed between the probability gains in the current study and those stipulated in the guidelines 24. However, we do not discuss the differences in detail as they do not have crucial implications for our conclusions.

The cumulative count of earthquakes with respect to time aligned with the predictions obtained using the Omori–Utsu law used for modelling the occurrence rate of aftershocks of a single earthquake (Fig. 2). In the Omori–Utsu law, the aftershock rate \(n\left(t\right)\) and the cumulative number of aftershocks \(N\left(t\right)\) are expressed by

$$\begin{array}{c}n\left(t\right)= {K\left(t+c\right)}^{-p} ,\ \left(1\right)\end{array}$$

and

$$\begin{array}{c}N\left(t\right)= {K\{{c}^{1-p}-\left( t+c\right)}^{1-p}\}/(p-1) , (p\ne 1)\ \left(2\right)\end{array}$$

where \(K, c, p\) are constants. The best-fit values of the constants are \(K=0.65\), \(c=1.0\times {10}^{-3}\), and \(p=0.90\). As our findings are consistent with the Omori–Utsu law, we considered the aftershock law suitable for modelling the probability of the successive occurrence of similar-size earthquakes.

## Probability Of Successive Occurrence From Past Nankai Megathrust Earthquakes

Subsequently, we focused on the “twin earthquake” scenario along the Nankai megathrust, where an M8-class earthquake is followed by an event of the same scale, which has the potential to have the most momentous impact on the society.

The history of Nankai megathrust earthquakes has been extensively studied using historical documents and archaeological and geological surveys 2–4,6,10,28−30. Here, we first elucidated the rupture segments of past earthquakes by reviewing previous studies on past earthquakes (see Methods). We focused on the earthquakes that occurred in 1361 and later, which were understood relatively well compared with earlier earthquakes. All the earthquakes discussed below were inferred as larger than Mw 8.0 5 and were categorized as M8 + earthquakes.

The results of our re-evaluation of the historical events are summarized in Fig. 3. In association with the warnings for Nankai megathrust earthquakes, the focus here is on the percentage of twin earthquake cases, where two M8 + earthquakes occur successively within a short time interval. Among the earthquake sequence of 1361, 1498, 1605, 1707, 1854, and 1944–1946, the evident successive M8 + earthquakes (twin earthquakes) occurred in 1854 and 1944–1946. Possible successive M8 + earthquake cases occurred in 1361 and 1498. A single M8 + earthquake was observed in 1605 and 1707. The probability (more precisely, the maximum likelihood estimates of the probability) of successive M8 + earthquake occurrences within 3 years was 33% (two out of six) or 67% (four out of six), depending on how the relevant cases were counted. The 95% confidence intervals for the two cases were 4.3–77% and 22–96%, respectively. Combining the two cases, the confidence interval was 4.3–96%. The 3-year probability of successive M8 + earthquakes for the Nankai megathrust was higher than that of the global average (10%, with 95% confidence interval of 5.3–18%), although the confidence intervals overlapped.

**Probability Curve For Occurrence Of Successive Nankai Megathrust Earthquakes**

As shown earlier, the occurrence of an M8 + earthquake followed by a subsequent M8 + event is well characterized by the Omori–Utsu law. Here, we assumed that the successive occurrence of Nankai megathrust earthquakes follow the Omori–Utsu law, with \(c=1.0\times {10}^{-3}\) and \(p=0.90,\) derived from global statistics. The cumulative probability for a successive occurrence of Nankai megathrust earthquakes can be written as follows:

$$\begin{array}{c}P\left(t\right)=1-\text{exp}\left(-N\left(t\right)\right),\ \left(3\right)\end{array}$$

assuming a non-stationary Poisson process and considering that the cumulative number \(N\left(t\right)\) (Eq. 2) is the transformed time 31. The constant \(K\) was scaled such that the 3-year probability was 4.3–96% in a 95% confidence interval, consistent with the history of M8 + earthquake occurrence along the Nankai megathrust.

The resulting probability curves and 95% confidence intervals for the Nankai megathrust are shown in Fig. 4. The probability curve based on the global statistics was near the minimum limit of the confidence interval, implying that the Nankai subduction likely hosts successive M8 + earthquakes more often than the global average indicates. Deriving the confidence interval of the probability curve aided the deduction of the probabilities in arbitrary time frames (Table 1). For example, the probability of an M8 + earthquake occurring within 1 day (24 h) and 1 week after an M8 + earthquake along the Nankai megathrust was 1.4–65% and 2.1–77%, respectively.

Table 1

Probability and probability gain for successive occurrence of great Nankai megathrust earthquakes within different time frames.

Time frame | Probability (%) | Probability gain |
---|

6 hours | 1.0–53 | 1.3 × 103–7.0 × 104 |

12 hours | 1.3–60 | 8.6 × 102–4.0 × 104 |

24 hours | 1.4–64 | 4.6 × 102–2.1 × 104 |

3 days | 1.8–72 | 2.0 × 102–7.9 × 103 |

1 week | 2.1–77 | 99–3.6 × 103 |

2 weeks | 2.3–85 | 54–2.0 × 103 |

1 month | 2.6–85 | 28–9.1 × 102 |

3 years | 4.3–96 | 1.3–29 |

The probability gains corresponding to the estimated probabilities indicated a sharp rise in probability compared with the norm (Table 1). The probability gain for 6 h was 1,000–70,000-fold, decreasing to 100–3,600-fold for 1 week and, eventually, to values comparable with the norm (1.3–29) for 3 years.