Possibility of short-term probabilistic forecasts for large earthquakes making good use of the limitations of existing catalogs

Earthquakes are quite hard to predict. One of the possible reasons can be the fact that the existing catalogs of past earthquakes are limited at most to the order of 100 years, while their characteristic time scale is sometimes greater than that time span. Here we rather use these limitations positively and characterize some large earthquake events as abnormal events that are not included there. When we constructed probabilistic forecasts for large earthquakes in Japan based on similarity and difference to their past patterns—which we call known and unknown abnormalities, respectively—our forecast achieved probabilistic gains of 5.7 and 2.4 against a time-independent model for main shocks with the magnitudes of 7 or above. Moreover, the two abnormal conditions covered 70% of days whose maximum magnitude was 7 or above.

Figure 1

Calculation of edit distances. (a) Examples of two marked point processes. Circles and crosses indicate event series of $H_w\left(u\right)=\left\{h_1\left(\mathrm{u}\right),h_2\left(\mathrm{u}\right)\right\}$ and $H_w\left(u^{\prime }\right)=\left\{h_1\left(u^{\prime }\right),h_2\left(u^{\prime }\right),h_3\left(u^{\prime }\right)\right\}$, respectively. (b) The bipartite graph representing two marked point processes in (a). The black square indicates a dummy node. The numbers shown by the arrows represent the costs required for editing. If the numbers are 1, then it means deletion or insertion of events. If “min” is shown, then its first arguments are the costs for deletion and insertion of the corresponding events, and the second arguments are the costs required for a shift when ${\lambda }_1={\lambda }_2=0.4$. In addition, the numbers after $\text{“}=\text{”}$ show the costs chosen after taking the minimum of the two costs. When a cost for a shift is larger than 2, which is the total cost of an insertion and a deletion, we choose the insertion and deletion of these events rather than the shift. For example, because the cost of a shift from $h_2\left(u\right)$ to $h_2\left(u^{\prime }\right)$ is 2.3 and larger than 2, the cost of the edge between these nodes is 2 which is required for the deletion of $h_2\left(u\right)$ and insertion of $h_2\left(u^{\prime }\right)$. Solid lines indicate the editing procedure which minimizes the total cost, while broken lines show the other potential edges. The edit distance between these two marked point processes is 2.6.

Figure 2

Schematic diagram for persistence prediction and nearest neighbor prediction. Panel (a) corresponds to an original marked point process. Panel (b) shows how we generate the persistence prediction, which is just using the current window as the prediction for the following window. Panel (c) shows how we generate nearest neighbor prediction, by which we first find the closest match in the past and use the window next to the closest match as the prediction for the following window.

Figure 3

The window size vs the normalized sum of mean prediction errors $e_w$, for the case of the integrate-and-fire neuron forced by the Lorenz model.

Figure 4

The window size vs the normalized sum of mean prediction errors $e_w$, for the case of a local maxima series of the Rössler model.

Figure 5

The original time series (the solid line) of the Lorenz model and the extracted events (the crosses) for the third example. Here we extracted the times and values for the local maxima on the upper lobe as well as the times and values for the local minima on the lower lobe.

Figure 6

The size of time window vs the normalized sum of mean prediction errors $e_w$, for the third example of the Lorenz model.

Figure 7

The original time series (the solid line) of the Lorenz model and the extracted events (the crosses) for the fourth example. Here we extracted the times and values for the local maxima for the upper lobe.

Figure 8

The size of time window vs the normalized sum of mean prediction errors $e_w$, for the fourth example of the Lorenz model.

Figure 9

The window size vs the normalized sum of mean prediction errors $e_w$, for the earthquake series around Japan.

Figure 10

Distribution of earthquakes in the ETAS model.

Figure 11

Magnitudes, longitudes, and latitudes of earthquakes with the largest magnitudes that were 7 or above, depending on the probabilistic forecasts. Black diamonds, blue upward triangles, and red downward triangles show the days with forecasts with normal conditions, known abnormal conditions, and unknown abnormal conditions, respectively. Panels (a), (b), and (c) show magnitudes, longitudes, and latitudes, respectively.

Figure 12

Locations for earthquakes whose magnitudes were 7 or above. Black diamonds, blue upward triangles, and red downward triangles show the days with forecasts with normal conditions, known abnormal conditions, and unknown abnormal conditions, respectively.

Figure 13

Magnitudes, longitudes, and latitudes of earthquakes with the largest magnitudes that were 6 or above, depending on the probabilistic forecasts in the case of New Zealand. See the caption of Fig. 11 to interpret the results.

Figure 14

Locations for earthquakes whose magnitudes were 6 or above in the case of New Zealand. See the caption of Fig. 12 to interpret the results.

Figure 15

The comparison between the original time series (the solid line) of the Lorenz model and its reconstruction (the dash dotted line) from the simple point process generated by the integrate-and-fire neuron. The reconstruction shown here is the first principal component obtained after drawing a recurrence plot by plotting points on 10% of places, making the plot continuous by the method of Ref. [51], and reproducing the original time series by the method of Refs. [52, 53].

Figure 16

The comparison between the subsampled time series (the solid line) of the Rössler model and its reconstruction (the dash dotted line) from the local maxima series of the Rössler model. Here, the reconstruction shown is the first principal component obtained after drawing a recurrence plot, making the plot continuous in time by the method of Ref. [51], and reproducing its time series by the method of Refs. [52, 53].

Figure 17

The comparison between the subsampled time series (the solid line) of the Lorenz model and its reconstruction (the dash dotted line) for the third example. Here, the reconstruction shown is the third principal component after obtaining a recurrence plot, making the plot continuous by the method of Ref. [51], and reproducing the encoded time series by the method of Refs. [52, 53].

Figure 18

The comparison between the subsampled dataset (the solid line) of the Lorenz model and its reconstruction (the dash dotted line) from the local maxima series of the upper lobe of $x$. Here, the reconstruction shown is the fourth principal component obtained after obtaining a recurrence plot, making the plot continuous by the method of Ref. [51], and reproducing the time series by the method of Refs. [52, 53].

Figure 19

The comparison between the top three components underlying the earthquake activity around Japan obtained by the edit distances and the maximum magnitude for the next days. We excluded the days when the maximum magnitudes were less than 4.

Figure 20

The comparison between the top three components for the earthquake activity around New Zealand obtained by the edit distance, and the maximum magnitude for the next days. We excluded the days whose maximum magnitudes were less than 3.5.