Earthquake forecast models for inland Japan based on the G-R law and the modified G-R law
© The Society of Geomagnetism and Earth, Planetary and Space Sciences (SGEPSS); The Seismological Society of Japan; The Volcanological Society of Japan; The Geodetic Society of Japan; The Japanese Society for Planetary Sciences; TERRAPUB. 2011
Received: 7 May 2010
Accepted: 15 October 2010
Published: 4 March 2011
The frequency-magnitude distribution expressed by the Gutenberg-Richter (G-R) law is the basis of a simple method to forecast earthquakes. The frequency-magnitude distribution is sometimes approximated by the modified G-R law, which imposes a maximum magnitude. In this study we tested three earthquake forecast models: Cbv (the Constant b-value model) based on only the G-R law with a spatially constant b-value, Vbv (the Variable b-value model) based on only the G-R law with regionally variable b-values, and MGR (the Modified G-R model) based on the modified G-R or G-R law (chosen according to Akaike Information Criterion) with regionally variable b-values. We also incorporated aftershock decay and minimum limits of expected seismicity in these models. Comparing the results of retrospective forecasts by the three models, we found that MGR was almost always better than Vbv; Cbv was better than Vbv for short-term (one year) forecasts; little difference between MGR and Cbv for short-term forecasts; and MGR and Vbv tended to be better than Cbv for long-term (three years or longer) forecasts. We propose the use of MGR in the earthquake forecast testing experiment by the Collaboratory for the Study of Earthquake Predictability for Japan.
An earthquake forecast testing experiment was started by the Collaboratory for the Study of Earthquake Predictability (CSEP) for Japan on November 1, 2009 (Nanjo et al., 2009). The main purposes of the experiment are to elicit the submission of statistical and physics-based models, to evaluate the performance of these earthquake forecast models, and to better understand the physics and statistics of earthquake occurrence. The target of the forecasts is to predict a seismicity rate (number of earthquakes in a predefined time window) for each magnitude bin at each predefined grid node within a predefined testing region.
The Gutenberg-Richter (G-R) law (Gutenberg and Richter, 1944) is the basis of a simple method to predict earthquakes (e.g., Earthquake Research Committee, 2006; Wiemer and Schorlemmer, 2007). The Earthquake Research Committee (2006), for example, estimated nationwide occurrence probabilities for earthquakes in Japan whose location cannot be predefined. They used the G-R law with a spatiotemporally constant b-value, which we call the Cbv (constant b-value) model here. However, because the b-value varies spatially (e.g., Wiemer and Wyss, 1997; Hirose et al., 2002a, b; Schorlemmer et al., 2005), we made an earthquake forecast model using b-values estimated for each region to capture the regionality of seis-micity, which we call the Vbv (variable b-value) model. The frequency-magnitude distribution is also sometimes approximated by the modified G-R law of Utsu (1974), which imposes a maximum magnitude and assumes that no earthquakes larger than that magnitude occur. Accordingly, we also made a model that uses the modified G-R law for regions where it fits the data better than the original G-R law, which we call the MGR (modified G-R) model.
This paper reports the results of comparison of retrospective forecasts made using the MGR, Vbv, and Cbv models.
2. The G-R and Modified G-R Laws for Frequency-Magnitude Distributions
2.1 The G-R law
2.2 The modified G-R law
It is commonly found that a frequency-magnitude distribution is a convex-upward curve rather than a straight line (e.g., Utsu, 1974) and departs from the G-R law. That is, the number of larger or smaller earthquakes is fewer than expected from the G-R law. This sometimes happens when the catalog is incomplete because of magnitude saturation for large earthquakes or nondetection of small earthquakes.
However, a frequency-magnitude distribution can depart greatly from the G-R law even if the catalog is perfect (e.g., Utsu, 1974; Umino and Sacks, 1993). Umino and Sacks (1993) researched frequency-magnitude distributions of earthquakes in the crust and in the upper plane of the double seismic zone in the Pacific slab beneath northeast Japan, using the Tohoku University and JMA catalogs with aftershocks removed. They found that both frequency-magnitude distributions departed from the G-R law, to a slight degree for earthquakes in the crust and to a greater degree for those in the slab. They confirmed that the catalogs were complete for M ≥ 2.0 in the crust and M ≥ 2.1 in the upper plane in the slab by the method of Rydelek and Sacks (1989). Furthermore, they found no difference between frequency-magnitude distributions based on seismic wave duration magnitudes and amplitude magnitudes. Therefore, they suggested that this result is robust.
It is obvious from Eqs. (3-2) and (3-4) that earthquakes with M ≥ c m are not expected to occur by definition in the modified G-R law. However, there is a possibility that earthquakes with M ≥ c m might occur in the real activity during the predicting period, especially when the modeling period is not long enough to include a large earthquake or the predicting period is very long. Therefore, we introduced a minimum limit of seismicity rate to cover the disadvantage of using the modified G-R law, which is discussed in Section 4.2.
2.3 Parameter estimation
We applied the maximum likelihood estimation method (Aki, 1965; Utsu, 1974) to estimate the b-value of the G-R law and the b m - and c m -values of the modified G-R law. However, as the parameters of the modified G-R law, unlike the G-R law, cannot be obtained analytically, we estimated them numerically using the Newton method (Mabuchi et al., 2002). The MGR model compares values of the Akaike Information Criterion or AIC (Akaike, 1974) calculated by applying the G-R and modified G-R laws to observed frequency-magnitude distributions for each region, adopting the law that yields the smaller AIC for each region, unless the difference in AIC was less than 1, in which case we used the G-R law (see Section 4.1).
3. Data and Target Earthquakes
We used the Japan Meteorological Agency (JMA) unified hypocenter catalog and selected inland earthquakes with depths of 30 km or less, then divided this data base into two groups, of which one was used for modeling and the other for testing.
As for data for testing, we defined target earthquakes as those that were with depth ≥ 30 km and 5.0 ≥ M ≥ 9.0 occurring during the testing period just after the modeling period. Testing period lengths were set at one, three, five, and seven years.
4. Construction of the Modified G-R Earthquake Forecast Model
4.1 Procedure for calculating model parameters
- Step 1.We set 5483 grids with a spacing of 0.1° in latitude and longitude over Japan and selected earthquakes occurring within a region with a radius of 20 km from each grid (Fig. 2). A circular area with a radius of 20 km is almost equal to the area of the source region S (km2) of an M 7.0 event given by an empirical relation log S = 1.02M − 4.0 (Utsu and Seki, 1955).
- Step 2.We treated the “threshold magnitude” Mth in each region to be the magnitude bin that includes the largest number of events (Fig. 3). Note that because the JMA catalog rounds off magnitudes to the nearest tenth, M 2.1, for example, means the bin 2.05 ≤ M < 2.15, and Mth is not 2.1 but 2.05.
- Step 3.
We derived parameters for both the G-R and modified G-R law from the frequency-magnitude distribution in regions where the number of earthquakes in the virtual catalog with M ≥ Mth was at least 200. In regions with fewer than 200 such events, the b-value was assumed to be the mean b-value estimated by the G-R law for all earthquakes in the study region, excluding the area around Miyake Island (enclosed area in Fig. 2(a)) because events in that region were tecton-ically different, as discussed in Section 7.6.
- Step 4.
We compared AIC values for the G-R and modified G-R laws for the frequency-magnitude distribution in each region and selected the law that yielded the smaller value. If the difference between the AIC values was less than 1, we selected the G-R law to reduce the risk of underestimating the probability (discussed further in Section 7.3). When calculating AIC values, we imposed conditions of one free parameter for the G-R law and two free parameters for the modified G-R law, because a- and a m -values are treated as fixed values determined by the total number of earthquakes with M ≥ Mth.
- Step 5.To estimate the expected rate of earthquakes with M ≥ Mth in the testing period, we used the data from the final year in the modeling periods and estimated the a-value for the G-R law and a m -value for the modified G-R law by taking the testing period length into account (Fig. 4(a)). When a large earthquake closely preceded the testing period (earthquakes with M ≥ 5.0 within one year or M ≥ 7.0 within five years), we forecasted the expected seismicity by applying the modified Omori formula (Utsu, 1957) to the modeling data (Fig. 4(b)). When more than one such large earthquake occurred in a region, the largest (and the last if there are multiple largest events) one was assumed to be the mainshock. There were also cases in which a region included aftershocks but not a mainshock. However, if the maximum earthquake extracted in each region satisfied the above magnitude and period condition, aftershock activity was estimated the same way even if the maximum earthquake was not a true mainshock.
- Step 6.
Finally, we used the obtained parameters to estimate the seismicity rate for a range of M(5.0 ≤ M ≤ 9.0) in the testing periods. The forecast area corresponding to each grid cell was defined as an outline of 0.1° × 0.1° in latitude and longitude centered at each grid (inset map A in Fig. 2(a)). As a region for calculating model parameters was circular while a forecast area was rectangular, the expected seismicity rate for a forecast area was corrected by taking into account the ratio of a rectangular area to circular one. The locations of target earthquakes were represented by epicenters without considering the size of their source area.
4.2 Minimum limit of seismicity rate
When no earthquake with M ≥ M th occurred in the modeling data in a region, the expected seismicity rate of target earthquakes could not be obtained because the a-value in the G-R law and a m -value in the modified G-R law could not be estimated. To estimate such low seismicity rates in some regions, we adopted the following assumption. The Earthquake Research Committee (2006) proposed that the average displacement rate for active faults of grade C, the lowest class of activity, is 0.024 mm/y. We assumed that the average displacement rate is 1/10 of this rate, 0.0024 mm/y, in grid cells where seismicity is too low to be estimated from seismicity data. Matsuda (1975) derived an empirical relation of displacement D (m) and M given by log D = 0.6M−4.0, and by extrapolation, displacement D for M 5.0 is 0.1 m. Thus, an earthquake of M 5.0 would occur once in about 42,000 years (2.4 × 10−5/y) for an average displacement rate of 0.0024 mm/y. Then we estimated minimum limits of seismicity rate for each M bin between 5.0 and 9.0 from the G-R law and the b-value for each grid cell.
Note that we assumed that the minimum limit of seismicity rate mentioned above is applicable to earthquakes with M ≥ c m , and we estimated the minimum rate for each M bin from the G-R law with the mean b-value.
5. Statistical Evaluation of Earthquake Forecast Models
5.1 Log-likelihood evaluation
6.1 MGR model
6.2 MGR model vs. Vbv model
There was one exceptional case in 2007 in which the Vbv model was superior to the MGR model for target earthquakes. An earthquake with M 5.2 occurred beneath the Boso Peninsula on August 18, 2007, where the modified G-R law was used in accordance with step 4 in Section 4.1. The estimated value of the upper magnitude limit c m was 4.6, thus the occurrence of the M 5.2 earthquake made the MGR model perform worse than the Vbv model.
6.3 Vbv model vs. Cbv model
6.4 MGR model vs. Cbv model
The log-likelihoods in Table 1 show that the MGR model was better than the Cbv model in four of the eight cases, which means there is little difference between the performance of the models. From the other model comparisons in Sections 6.2 and 6.3, it might have been expected that the MGR model would be the best of the three models, but the superiority of the MGR model was not clear. We discuss ways to clarify this situation below.
7.1 The effect of the radius of each region
In addition, in the forecast for 2008, the MGR model is better than the Vbv model for a 20-km radius (Table 1), but the results are reversed for a 30-km and 40-km radius (Tables 3 and 4). The reason is as follows. The M 7.2 Iwate-Miyagi Nairiku earthquake occurred on June 14, 2008. However, during the modeling period, seismicity was relatively high at locations 30 km southwest and 30–40 km south-southeast of the mainshock. Thus the region centered at lat 39.0°N, long 140.9°E includes both this seismicity and the mainshock if the radius is 30 km or 40 km, but not if the radius is 20 km. When the region includes this seismicity, the log-likelihood for the MGR model becomes small because the modified G-R law is selected and c m is estimated at 6.9, which is less than the mainshock magnitude of 7.2.
When a radius of 40 km is selected, the total log-likelihood for the MGR model is largest, which indicates that we might have a choice to select a radius of 40 km instead of 20 km. The selection of radius is a difficult problem. It may be advantageous in future work to treat radius as a variable parameter, depending on the magnitude of the target earthquake or the location of the regions.
7.2 The effect of the modified G-R law
The log-likelihood values for the MGR model, which applies the modified G-R law to some regions, tend to be larger than those for the Vbv model, which uses only the G-R law (e.g. Table 1). Therefore the effect of introducing the modified G-R law into the forecast model is evident. The MGR model was worse only in the forecast for 2007 because the magnitude of the M 5.2 earthquake beneath the Boso Peninsula on August 18, 2007, exceeded the value of c m , which was estimated at 4.6. In this region an M 4.9 earthquake that occurred on October 8, 1966, was excluded from our virtual catalog used for modeling because events less than M 5 were rejected during that period. Thus, introducing the modified G-R law may be risky for estimating c m if the data period is not long enough. This problem may be avoidable in future work by setting a lower limit for c m based on the largest earthquake in a long-term catalog. In addition, a forecast model incorporating the estimation error of c m (Mabuchi et al., 2002) may yield better results.
On the other hand, Burroughs and Tebbens (2002, BSSA) found that an upper-truncated power law, which is equivalent to the modified G-R law, applied to earthquake cumulative frequency-magnitude distributions yields a time-independent scaling parameter called the α-value. By analyzing several types of seismicity, they showed the α-value for the short time intervals is equal to the b-value obtained by applying the G-R law to the entire record. This suggests that the modified G-R law is not necessarily applicable for a long term data and the G-R law using α-value instead of b-value derived from short term data might be useful for forecasting. The method using the α-value can be an alternative in the future work for a longer term prediction. However, from our result mentioned above, the introduction of the modified G-R law into the forecast model is obviously effective to improve the prediction as far as our tested forecasting period is concerned (seven years at most in Section 7.5). We consider that there exist some regions in inland Japan where a frequency-magnitude distribution exhibits a convex-upward curve rather than a straight line and departs from the G-R law by nature, or our forecasting period is short enough for the modified G-R law to become superior to the G-R law in regions where the modified G-R law is selected in the modeling period.
7.3 Model selection by AIC
7.4 The effect of aftershocks
7.5 The effect of forecast periods
7.6 Seismicity near Miyake Island
Most of the target earthquakes in this study occurred in the continental crust. However, regions from the Izu Peninsula to Miyake Island belong to a volcanic arc on the Philippine Sea plate, and earthquakes in these regions occur in oceanic crust or mantle. Near Miyake Island, volcanic earthquakes began June 26, 2000, and in the next two months, a notable swarm of activity occurred, including more than 7200 events with M ≥ 2.8 (= M th ), more than 870 with M ≥ 4.0, 78 with M ≥ 5.0, and 6 with M ≥ 6.0 (of which the two largest were M 6.5). This activity is equivalent to the aftershock sequence of an M 8 class mainshock. However, it damped rapidly, and the average number of earthquakes with M ≥ 2.8 was around 20 per year from 2003 through 2008.
As this swarm was related to magmatic activity in oceanic crust and mantle, its characteristics might be very different from the on-land events that were our main target. Therefore, we excluded the Miyake Island region when we estimated a nationwide mean b-value from seismicity in the study area (Fig. 2(b)). This b-value was also used as the default in regions where the G-R and modified G-R laws could not be distinguished (step 3 in Section 4.1), including near Miyake Island.
7.7 Effect of minimum seismicity rate
7.8 Long-term or short-term data for seismicity estimations
As mentioned in Section 4.1, we estimated the seismicity rates from the data for the year just before the testing period. There is an advantage in using a long data record for estimating parameters b, b m , and c m , because the estimation error is expected to decrease as the number of earthquakes increases. On the other hand, long-term data may be a disadvantage for estimating parameters a and a m because seismicity rates fluctuate over relatively short periods. For example, in Fig. 4(a), the expected seismicity is expressed by line L for long-term data and line S for short-term data, and the latter is a better fit in prediction. Therefore, for short-term forecasts such as one or three years, as proposed in CSEP for Japan, we estimated the expected seismicity rate from the last year of data in the modeling period.
7.9 Effect of definition of target earthquakes
7.10 Future problems
The models in this study were designed to forecast target earthquakes with 5.0 ≤ M ≤ 9.0 and depth ≤ 30 km by using only seismicity data after January 1965, according to the framework of CSEP for Japan. We ignored all information about tectonic settings (except in the case of Miyake Island), geodetic crustal movements, velocity structure in the crust, spatial distribution of active faults, physical mechanism of earthquakes, and so on, all of which are related to seismicity. To improve the models, it is important to bring this information into them. For example, considering tectonic information, in the Kanto district both continental and oceanic crust interact beneath the Boso Peninsula, thus it is reasonable to treat the crustal types separately by considering the distribution of hypocenters rather than epicenters.
For another example, recently it has been suggested that earthquakes in the continental crust are related to melts in the lower crust (Okada et al., 2008). Our model might be improved by clarifying the relationship between seismicity and velocity structure in the lower crust. We consider the models proposed here to be basic models that can be improved by adding more information.
The MGR model, using both the modified G-R and G-R laws, was better than the Vbv model, using only the G-R law.
The Cbv model, based on a spatially constant b-value, was better than the Vbv model, based on regionally variable b-values for short-term (one year) forecasts.
The difference between the MGR and Cbv models was not clear for short-term forecasts.
The MGR and the Vbv models, using regionally variable b-values, tended to become better than the Cbv model for long-term (three years or longer) forecasts.
On the basis of our results, we propose the use of the MGR model for CSEP for Japan.
We thank all the institutions, universities, and JMA for providing the unified hypocenter catalog. We also thank T. Utsu and Y. Ogata for the use of the SASeis program (Utsu and Ogata, 1997) to estimate parameters of aftershocks. This manuscript was greatly improved by careful reviews of two anonymous reviewers. Figures were prepared using GMT (Wessel and Smith, 1991).
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