1. A new prior distribution of a Bayesian forecast model for small repeating earthquakes in the subduction zone along the Japan Trench Masami Okada (MRI, JMA), Naoki Uchida (RCPEV, Tohoku Univ.), and Shigeki Aoki (MRI, JMA) CSEP Workshop, 2010.11.1

2. SREs near the east coast of Japan is considered to occur on the same small asperity surrounded by creeping zone on the plate boundary. They are identified objectively by very high correlation in waveform. The recurrence times for most sequences are fairly regular and short enough to evaluate the forecast. Small repeating earthquakes (SRE)

3. New Results: Forecasts and observations, Jan - June, 2010 Forecast probabilities (left) and observation (right) of 163 SRE sequences in the forecast period, January to June, 2010. Mean log-likelihood score and Brier score are better than those in former experiments.

4. Models for probabilities Current model: A Bayesian approach with log- normal distribution, for time interval. The prior distribution is uniform for mean parameter of, and inverse gamma,, for variance. Two parameters, in inverse gamma are currently common for all sequences. New model: We are now developing a new model for better forecast used in near future in which the prior distributions are different from zone to zone.

5. Parameter determination for : It is important to determine the prior distribution for Bayesian approach. We applied the maximum likelihood method (LLH) for. The pdf of unbiased variance,, from data of is given as follow, where. Thus it is easy to define the likelihood for. Calculation of forecasting probability: Conditional probability on a renewal process

6. SRE data for this study (1) SRE Catalog: We use the catalog for present study compiled by Tohoku University for the period from Jan. 1993 to June 2010. Declustering: Some sequences contain doublets or triplets shown in upper figure. If time interval, between two events is less than 50 days, or is less than, the smaller event of two is removed from the sequence. After declustering SREs, the distribution for the deviation of from its sequence average (red bars) is fairly symmetric (lower figure).

7. SRE data for this study (2) Check of aftershocks: The number of possible aftershocks are less than one third of the events in the sequence. We regarded SREs as possible aftershocks of the 2003 Off Tokachi earthquake (M8.0, Sept. 26, 2003) and the 1994 Off Sanriku earthquake (M7.6, Dec. 28, 1994) that occurred in the areas of “Off Tokachi” and “Off Sanriku” in the period from the main shocks through March 31, 2005 and Jan. 31, 1996, respectively. Magnitude: The averaged magnitude for each sequence is 2.75 or larger which determined from upper figure. The largest averaged magnitude is 4.3.

8. SRE data for this study (3) Number of SREs: The SRE sequences have five or more events. Right figure shows the locations of sequences for present study. The frequency of sequences in upper figure indicates that many of sequences have rather few events, 5 to 7. Therefore we adopted a Bayesian approach with log-normal distribution. Total SRE data are 168 sequences with 1298 events and are much more than our previous study in 2007, 70 sequences and 448 SREs. 5 13 20 Number of SREs

9. Prior distribution for variance of Prior distribution for variance is inverse gamma,, a natural conjugate prior distribution. Here, the and are called “shape” and “scale” parameters, respectively. Upper figure shows the distribution of LLH for. Horizontal axis is not but, a kind of standard deviation. The peak is fairly sharp and high. The lower is LLH for previous dataset (Okada et al., 2007). Two patterns are similar and two peaks locate near position. Therefore we can not expect the new parameters for whole region to improve the forecast so much. H H

10. Regional prior distributions for (1) Left figure shows locations of clusters and the sequence standard deviation of x=ln(Ti) indicated with colors, from which we preliminarily partitioned the SRE generating region into three sub-regions. In the coast zone many SRE clusters crowded along the coast line and most of them have smaller standard deviation. The LLH map in upper figure for the coast zone has a fairly sharp peak. In the south region SRE clusters distribute widely in space. The LLH maps for “East region” and “South Region” have duller and lower peaks. Sum of three maximum heights is 54.4 and close to 53.2 of the whole region type in the previous slide. Let us inspect the distribution of clusters in south region on the cross section at a line of AB. H

11. Regional prior distributions for (2) Cross section Right figures are standard deviations of x=ln(Ti) in south region projected to AB line and vertical distribution on the cross section. We can see clear difference in the west and east parts. In the west part the standard deviation is generally small and SRE clusters locate near the plate boundary. In the east part the deviations and sequence averaged depths distribute widely due to larger error in hypocenter determination. Sum of two maximum LLHs is 12.0, which is surely larger than 7.0 for all south region, therefore we consider this partition might be effective.

12. Regional prior distributions for (3) Left figure is a final partition of the SRE generating region. Right three figures show the distributions of LLH. Interval of contours for Trench region (right lower) is one fourth of those for other two. Locations of LLH peak are different on map to map, and sum of their heights is 58.5 and 5.3 higher than the whole region type. Therefore we expect that the partition will improve our SRE forecasts.

13. Conclusions and a future plan We developed a new prior distributions for a Bayesian approach with log-normal distribution models to estimate the probability for the small repeating earthquake on the plate boundary east off NE Japan. Parameters in inverse gamma prior distribution for are different from zone to zone. Regional models may improve the SRE forecast. Preliminary values are as follow: North Coast zone: South Coast zone: Trench zone: We wish to apply the new partition and parameters for forecasting the small repeating events in 2011. I didn’t talk about the prior distribution for parameter, for which we are trying to apply a normal prior distribution for SREs in the North Coast zone.