Statistical forecast for interplate repeating earthquakes at east off NE Japan [Japanese]
[Uploaded on June 2011, 2008, 2009 and 2010 pages]

Masami Okada (MRI, JMA), Naoki Uchida (RCPEV, Tohoku Univ.), Shigeki Aoki (MRI, JMA)

Introduction

A large number of sequences of small repeating events with identical waveform have been found near the east coast of NE Japan. The repeating earthquakes are thought to be occurring on the same small asperity surrounded by creeping area on the plate boundary [Igarashi et al., 2003; Uchida et al., 2003]. These repeating earthquakes are unique data to test predictability of statistical models because the earthquakes are identified objectively by waveform correlation and the recurrence intervals (1-4 years) are short enough to evaluate the forecast. In this page we introduce the result of statistical forecast based on a Bayesian approach.


Statistical forecast for 2011

We use time interval of successive small repeating earthquakes. Based on a Bayesian approach with lognormal distribution, we calculated probabilities of earthquake occurrence for the period from January 1 to December 31, 2011 (Fig.1). We applied two conditions for the repeating earthquake sequences to study : (1) Five or more earthquakes are occurred from 1993 to 2010, (2) Averaged magnitude is 2.75 or larger. This time, we made a forecast taking account of the regional difference of earthquake activity and divided the study region into three when calculating parameters used for the forecast(Fig. 1).


Fig. 1. The probabilities of earthquake occurrence for 183 small repeating earthquake sequences in 2011. Black line show three study regions.


Verification of statistical forecast for 2010

Figure 2 shows the forecast (left) and actual occurrence (right) of 163 repeating earthquake sequences for the period from January 1, 2010 to December 31, 2010. There are many groups with relatively high probability near the coast line of northern part of northeastern Japan (left figure), and most of repeating earthquake sequences with high probability had an earthquake during the forecasting period. On the other hand, there are also many earthquakes at the southern area (35- 38 N) where the estimated probabilities were low.


Fig. 2. The comparison of statistical forecast (left) and actual occurrence (right) of repeating earthquakes for the period from January 1, 2010 to December 31, 2010.

To check the performance of the forecast, we show the frequency distribution of sequences for every 10% (Fig. 3). Left yellow bar shows the number of sequences for every 10% range and the green show the expected number of earthquakes for the probability range. Right red bar show the actual number of earthquakes for the sequences that fall in the probability ranges. The number of sequences with earthquake (red) is comparable with that of the forecast (green). However, the expected number of earthquakes (green) tend to smaller than the actual number of earthquakes.

Fig. 3. The frequency distribution of forecasted (green) and actual (red) number of sequences that had a earthquake in 2010 for every 10% probability. The yellow bars show the total number of sequences.

Next we try to verify the performance of the forecast quantitatively. The "Brier score" and "Mean log-likelihood" described below were used to score the results.

Brier score : Average of (Pq-Ev)^2

Mean log-likelihood : Average of Ev*ln(Pq)+(1-Ev)*ln(1-Pq)

Pq means forecasted earthquake occurrence probability and Ev means presence (Ev=1) or absence (Ev=0) of the forecasted earthquake. The model is considered to be superior to the alternative one, if the Brier score is smaller or mean log-likelihood is larger than those of the alternative.

The scores for the forecast of 2010 were 0.196 for the Brier score and -0.581 for the mean log-likelihood. This values are comparable with the results for previous years (0.197 for the Brier score and -0.574 for the mean log-likelihood for 2006-2007, 2008 and 2009). We compare the Mean log-likelihood of our results with the result of probability forecast of precipitation at Tokyo from 2006 to 2010. Here we used precipitation probability data from http://homepage3.nifty.com/i_sawaki/WeatherForecast/index.htm. The score for 2010 show that the earthquake predictability is lower than that of the precipitation forecast of the 7 days later.

We also performed statistical test of the probability forecast. We compare the theoretical distribution of scores that assumes the forecasted probability is correct and the score that is estimated from observation (data). If the score is bad, the model will be rejected [Schorlemmer et al., 2007]. For example, if the earthquakes are occurring randomly following the present probability, the number of sequences with event and log likelihood follows blue and red line in Fig. 4 and 5, respectively. The number of sequences with event (72) is located 90% probability and it is not rejected by the 95% significance level. The result of log likelihood is larger than 5% and it is also not rejected (Fig. 5).


Fig. 4. The test using the number of earthquakes (N-test)

Fig. 5. The test using the log likelihood (L-test)


In addition to these tests, if we use two models for the same dataset, we can test the difference of score for the two model. The Brier score and mean log-likelihood for the exponential distribution model that assume the earthquakes are randomly occurring is 0.228 and -0.648, respectively. We show in Fig. 6 the result of dBS-test which use the difference of the Brier score. If the exponential distribution model is true, the distribution of 'dBS' become blue line (H0:correct). Since the observed 'dBS' (-0.032) is smaller than 5% point, the exponential distribution model is rejected. If our model is correct, the distribution of 'dBS' become red line (H1:correct) and this is not rejected. This shows our 2010 model is significantly superior to the exponential distribution model.


Fig. 6. The test of the difference in Brier score (dBS) for exponential distribution model (H0) and probability model in this study (H1).


Feature of the 2010 forecast and future study

The statistical test for the 2010 forecast by Brier score and mean log-likelihood are not rejected but the performance was relatively bad. The cause of relatively bad performance is the occurrence of large number of events in the latter half of the year at the southern part of the study area. We are planning to investigate the detail of the activity change especially its relation with the 2011 off the Pacific coast of Tohoku earthquake (M9.0). We also expect the performance of the 2011 forecast will be very bad due to the aftershock activity of the 2011 earthquake. However this forecast will be a good reference when studying the aftershock activity.


Acknowledgement: We used waveform data not only from Tohoku university's seismic stations but also from Hokkaido university and university of Tokyo to identify small repeating earthquakes. We also used earthquake catalogue by Japan meteorological agency. We thank Hiroyuki Takayama for the drawing program used for Fig. 1 and 2.


Supplemental material

1. The location, magnitude, probability and occurred earthquakes for the probability forecast for 2010 (data plotted in Fig. 2) probobs10.csv
2. The location, magnitude and probability for 2011 (data plotted in Fig. 1) prob11.csv


Reference
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