Earthquake Forecast Testing Experiment in Japan (II)
 Article
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Earthquake forecast models for inland Japan based on the GR law and the modified GR law
Earth, Planets and Space volume 63, Article number: 8 (2011)
Abstract
The frequencymagnitude distribution expressed by the GutenbergRichter (GR) law is the basis of a simple method to forecast earthquakes. The frequencymagnitude distribution is sometimes approximated by the modified GR law, which imposes a maximum magnitude. In this study we tested three earthquake forecast models: Cbv (the Constant bvalue model) based on only the GR law with a spatially constant bvalue, Vbv (the Variable bvalue model) based on only the GR law with regionally variable bvalues, and MGR (the Modified GR model) based on the modified GR or GR law (chosen according to Akaike Information Criterion) with regionally variable bvalues. 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 shortterm (one year) forecasts; little difference between MGR and Cbv for shortterm forecasts; and MGR and Vbv tended to be better than Cbv for longterm (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.
1. Introduction
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 physicsbased 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 GutenbergRichter (GR) 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 GR law with a spatiotemporally constant bvalue, which we call the Cbv (constant bvalue) model here. However, because the bvalue varies spatially (e.g., Wiemer and Wyss, 1997; Hirose et al., 2002a, b; Schorlemmer et al., 2005), we made an earthquake forecast model using bvalues estimated for each region to capture the regionality of seismicity, which we call the Vbv (variable bvalue) model. The frequencymagnitude distribution is also sometimes approximated by the modified GR 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 GR law for regions where it fits the data better than the original GR law, which we call the MGR (modified GR) model.
This paper reports the results of comparison of retrospective forecasts made using the MGR, Vbv, and Cbv models.
2. The GR and Modified GR Laws for FrequencyMagnitude Distributions
2.1 The GR law
The general property of the size distribution of earthquakes, that large earthquakes occur in small numbers and small earthquakes occur in large numbers, is well known. When the number of earthquakes with magnitudes from M to M + dM in a given region and a given period is defined as n(M)d M, their size distribution is approximated by the GR law (Gutenberg and Richter, 1944) given by
where a and b are constants. Furthermore, cumulative frequencymagnitude distribution is given by
where N(M) is the number of earthquakes with greater or equal to M, and A is a constant expressed as A = a − log(b ln 10). The bvalue, which indicates the slope of the linear curve in the frequencymagnitude distribution, is an especially important parameter. Many researchers report that bvalue varies spatiotemporally (e.g., Suyehiro, 1966; Anderson et al., 1980; Wyss, 1990; Wiemer and Benoit, 1996; Wiemer and McNutt, 1997; Wiemer and Wyss, 1997; Murru et al., 1999; öncel and Wyss, 2000; Wyss et al., 2000; Gerstenberger et al., 2001; Hirose et al., 2002a, b; Schorlemmer et al., 2005), and the results of laboratory experiments by Scholz (1968) are often quoted to explain the spatiotemporal variation of bvalues. Scholz (1968) conducted a rock fracturing experiment and showed that bvalues decrease with an increase of the shear stress acting on a medium. Schorlemmer et al. (2005) also suggested the dependency of bvalues on stress by showing the relationship of bvalues and the type of focal mechanisms. All of these results support the spatial variation of bvalues.
2.2 The modified GR law
It is commonly found that a frequencymagnitude distribution is a convexupward curve rather than a straight line (e.g., Utsu, 1974) and departs from the GR law. That is, the number of larger or smaller earthquakes is fewer than expected from the GR law. This sometimes happens when the catalog is incomplete because of magnitude saturation for large earthquakes or nondetection of small earthquakes.
However, a frequencymagnitude distribution can depart greatly from the GR law even if the catalog is perfect (e.g., Utsu, 1974; Umino and Sacks, 1993). Umino and Sacks (1993) researched frequencymagnitude 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 frequencymagnitude distributions departed from the GR 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 frequencymagnitude distributions based on seismic wave duration magnitudes and amplitude magnitudes. Therefore, they suggested that this result is robust.
Utsu (1974) proposed a modification of the magnitude distribution, employing an upper limit to model convexupward curves:
where a, b, and c are constants, and c represents an upper magnitude limit. This frequencymagnitude distribution approaches zero asymptotically as M approaches c. Constants a and b in Eq. (21) cannot be treated as equivalents of those in Eq. (1), although Eq. (21) adds only the logarithmic function of c M to Eq. (1). The bvalueof theGRlaw indicates the inclination of the frequencymagnitude distribution whereas the inclination in Utsu’s modified GR law is the function of b, c, and M. The avalue of the GR law indicates the number of earthquakes of M = 0, but this is not necessarily true in the modified GR law. To avoid confusion, in this paper we rewrite a, b, and c in Eqs. (2) as a_{ m }, b_{ m }, and c_{ m }, respectively:
Furthermore, cumulative frequencymagnitude distribution is given by
It is obvious from Eqs. (32) and (34) that earthquakes with M ≥ c_{ m } are not expected to occur by definition in the modified GR 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 GR 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 bvalue of the GR law and the b_{ m } and c_{ m }values of the modified GR law. However, as the parameters of the modified GR law, unlike the GR 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 GR and modified GR laws to observed frequencymagnitude 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 GR 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 modeling, in consideration of the detectability of earthquakes at different times in the past (K. Z. Nanjo, private communication), we selected earthquakes of M ≥ 5.0, M ≥ 4.0, M ≥ 3.0, and M ≥ 2.0 for the periods 1965–2007, 1980–2007, 1990–2007, and 2000–2007, respectively. To investigate the stability of the models, we prepared virtual catalogs of eight different time spans, extrapolating the number of events in each magnitude bin from the shorter record periods. The virtual catalogs cover periods of 36, 37,…, 43 years, corresponding to 1965–2000, 1965–2001, …, 1965–2007, respectively. For the 43year catalog (1965–2007), for example, the numbers of earthquakes with M ≥ 5.0, M ≥ 4.0, M ≥ 3.0, and M ≥ 2.0 were multiplied by 43/43 (= 1), 43/28, 43/18, and 43/8, respectively, according to the number of years in their respective periods (43, 28, 18, and 8 years) of record (Fig. 1).
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 GR 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 (km^{2}) 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” M_{th} 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 M_{th} is not 2.1 but 2.05.

Step 3.
We derived parameters for both the GR and modified GR law from the frequencymagnitude distribution in regions where the number of earthquakes in the virtual catalog with M ≥ M_{th} was at least 200. In regions with fewer than 200 such events, the bvalue was assumed to be the mean bvalue estimated by the GR 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 tectonically different, as discussed in Section 7.6.

Step 4.
We compared AIC values for the GR and modified GR laws for the frequencymagnitude 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 GR 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 GR law and two free parameters for the modified GR law, because a and a_{ m }values are treated as fixed values determined by the total number of earthquakes with M ≥ M_{th}.

Step 5.
To estimate the expected rate of earthquakes with M ≥ M_{th} in the testing period, we used the data from the final year in the modeling periods and estimated the avalue for the GR law and a_{ m }value for the modified GR 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 avalue in the GR law and a_{ m } value in the modified GR 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 GR law and the bvalue 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 GR law with the mean bvalue.
5. Statistical Evaluation of Earthquake Forecast Models
We evaluated the proposed earthquake forecast models statistically by using a loglikelihood test and an Ntest (Kagan and Jackson, 1995; Schorlemmer et al., 2007).
5.1 Loglikelihood evaluation
We evaluated the models by comparing the loglikelihood of the models for the observed target earthquake distribution. We divided the target area and the earthquake magnitudes into a threedimensional grid of K cells in which the third dimension is magnitude. When earthquakes occur independently in each cell, the probability P_{ ij } that an earthquake, whose average occurrence rate (Poisson rate) is λ in the cell, occurs just k times is expressed by the Poisson process in the form
where the subscript i indicates grid number (1 to 5483) and j indicates the index of the target magnitude (1 to 41, corresponding to steps of 0.1 magnitude from M 5.0 to M 9.0), for a total number of cells K = 5483 × 41 = 224,803. The loglikelihood (lnL) for all cells is obtained by
As P_{ ij } is not larger than 1, the closer to zero that ln L approaches, the more accurate the model becomes.
5.2 Ntest
The Ntest is a statistical method to check the consistency in occurrence numbers between observed earthquakes and predicted ones (Kagan and Jackson, 1995; Schorlemmer et al., 2007). Assuming the Poisson rate (average seismicity rate) for the J th cell in the forecast period to be λ_{ J } (where J = 1,2,…, K), the expected number of earthquakes in the cell, E[n_{ J }], is given by
Thus the total expected number in all cells, E[n], is given by summation of the Poisson rate in each cell:
As the total expected number follows the Poisson distribution with the Poisson rate E[n], the confidence interval corresponding to an arbitrary significance level can be estimated easily. We used a significance level of 95% in this study. It should be noted that the Ntest does not evaluate the spatial distribution of the number yielded by a model. For example, when a model forecasts five earthquakes in western Japan and none in eastern Japan, and five actual earthquakes occurred only in eastern Japan, the Ntest does not reject the model. Therefore, we used the Ntest results only for reference in this study.
6. Results
6.1 MGR model
Figure 5 shows the expected number of earthquakes with 5.0 ≤ M ≤ 9.0 for each year forecasted by the MGR model on the basis of the catalog from 1965 to the end of the previous year. The results for 2008 (based on the 1965–2007 catalog), for example, are separated into three regions in Fig. 6 according to the GR law that was applied: the GR law with variable b, the modified GR law, and the GR law with constant mean b. Table 1 lists the number of grid cells for each GR law and the results of the loglikelihood and Ntests. For forecast year 2008, for example (Fig. 5(h) and Fig. 6), the GR and modified GR laws could be distinguished and applied in 2038 of the 5483 grid cells because the number of earthquakes in these cells was atleast 200 (step 3 in Section 4.1). The frequencymagnitude distribution followed the GR law in 1653 of these cells and the modified GR law in the other 385 cells through step 4 in Section 4.1. The number of target earthquakes expected by the MGR model was large in Niigata prefecture, NaganoGifu prefecture, eastern Izu peninsula, Kinki district, and central Kyushu district (Fig. 5(h) and Fig. 6(a) and 6(b); see Fig. 2(c) for place names) because the seismicity rate is high or the bvalue is small in these regions. Similar features are seen for the other forecast years (Fig. 5).
Four earthquakes with M ≥ 5.0 occurred in 2008: the mainshock (M 7.2) called the IwateMiyagi Nairiku earthquake and three of its aftershocks (M 5.7, M 5.3, M 5.2). Frequencymagnitude distributions in the grid cells where the M 7.2, M 5.3, and M 5.2 events occurred followed the GR law and yielded bvalues of 0.81, 0.63, and 0.87, respectively, which are smaller than or equal to the nationwide mean bvalue of 0.87. For the grid cell of the M 5.7 event, the frequencymagnitude distribution followed the modified GR law, and c_{ m } was estimated at 6.3. The expected seismicity rates for that year were 0.4594 × 10^{−5}, 0.3518 × 10^{−3}, 0.5159 × 10^{−3}, and 0.1707 × 10^{−3} for the cells of the M 7.2, M 5.7, M 5.3, and M 5.2 events, respectively. Figure 6(d) shows the distribution of c_{ m } in 385 cells. The estimated upper limits of c_{ m } were lower in southern Hokkaido, southern Kinki district, and southern Kyushu district than in other regions. Regions where c_{ m } was relatively high correspond to places where large earthquakes occurred during the modeling periods.
6.2 MGR model vs. Vbv model
Table 1 shows that the MGR model was generally superior to the Vbv model because the loglikelihood for the MGR model, which combines the GR and modified GR laws, was greater than that for the Vbv model, which uses only the GR law and assumes the bvalue will vary regionally. The difference between these models is in the use of the modified GR law. The numbers of earthquakes expected by the MGR model in cells where the modified GR law is applied are lower than those expected by the Vbv model (Fig. 7). Note that no target earthquakes with 5.0 ≤ M ≤ 9.0 occurred in those cells. As the expected seismicity rate for earthquakes larger than c_{ m } was constrained to a minimum limit by the modified GR law, the MGR model was superior to the Vbv model when target earthquakes did not occur in the grid cell.
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 GR 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
Comparing the loglikelihoods of the Vbv model, which used variable bvalues in each region, and the Cbv model, which used a constant mean bvalue for the whole study area, we found that the Cbv model was better than the Vbv model in five cases out of eight (Table 1). Although we expected the use of regional bvalues to favor the Vbv over the Cbv model, the results did not show a clear tendency. Figure 8 shows an example of the forecasts for 2002 by the Vbv and Cbv models using data from 1965 through 2001. As seen in Table 1, the Cbv model was better than the Vbv model for 2002, mainly because an earthquake occurred in western Tottori prefecture where the expected number of events was relatively small (Fig. 8(c)) a result of the high bvalue in that region (Fig. 8(d)).
The Vbv model was superior to the Cbv model for 2008 (Fig. 9). The expected number distribution in Fig. 9 is similar to that in Fig. 8. The IwateMiyagi Nairiku earthquake occurred in the forecast period. The Vbv model was better than the Cbv model because the number of earthquakes expected by the Vbv model was larger in the region where the M 5.7 aftershock occurred, because the estimated regional bvalue was 0.50, which is smaller than the mean bvalue (0.87), although the difference of expected numbers in the grid cell in the two models was very small (0.01/year).
6.4 MGR model vs. Cbv model
The loglikelihoods 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. Discussion
7.1 The effect of the radius of each region
To simplify the procedure, we adopted a constant region radius in this study. As the median target magnitude was M 7.0, we selected a region radius of 20 km, with an area almost equivalent to the focal size of an M 7.0 earthquake (Utsu and Seki, 1955). Because the number of earthquakes extracted from a region is small if its radius is small, the number of regions where the GR and modified GR laws could be discriminated (step 3 in Section 4.1) decreases at small radii. In addition, when the number of earthquakes is small, the estimation error of the parameters for the frequencymagnitude distribution becomes large and we cannot obtain stable results. On the other hand, if a large radius is selected, the estimated parameters represent spatially smoothed features that provide less information about the regional variation of seismicity. Tables 2 to 4 show results for radii of 10 km, 30 km, and 40 km, corresponding to M 6.4, M 7.3, and M 7.6 earthquakes, respectively. For a radius of 10 km, less than 1/10 of the grid cells (524 of 5483) are suitable for discriminating between the GR and modified GR laws (step 3 in Section 4.1). This proportion is about 1/4, 1/2, and 2/3 for radii of 20 km, 30 km, and 40 km, respectively. The results of the forecast change to some extent by changing a radius. For example, with a 10km radius (Table 2), the MGR model is better than the Cbv model for 2001 but worse for 2006 based on the loglikelihood value, whereas the results are the opposite with a 20km radius (Table 1).
In addition, in the forecast for 2008, the MGR model is better than the Vbv model for a 20km radius (Table 1), but the results are reversed for a 30km and 40km radius (Tables 3 and 4). The reason is as follows. The M 7.2 IwateMiyagi 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 southsoutheast 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 loglikelihood for the MGR model becomes small because the modified GR 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 loglikelihood 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 GR law
The loglikelihood values for the MGR model, which applies the modified GR law to some regions, tend to be larger than those for the Vbv model, which uses only the GR law (e.g. Table 1). Therefore the effect of introducing the modified GR 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 GR 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 longterm 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 uppertruncated power law, which is equivalent to the modified GR law, applied to earthquake cumulative frequencymagnitude distributions yields a timeindependent scaling parameter called the αvalue. By analyzing several types of seismicity, they showed the αvalue for the short time intervals is equal to the bvalue obtained by applying the GR law to the entire record. This suggests that the modified GR law is not necessarily applicable for a long term data and the GR law using αvalue instead of bvalue derived from short term data might be useful for forecasting. The method using the αvalue can be an alternative in the future work for a longer term prediction. However, from our result mentioned above, the introduction of the modified GR 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 frequencymagnitude distribution exhibits a convexupward curve rather than a straight line and departs from the GR law by nature, or our forecasting period is short enough for the modified GR law to become superior to the GR law in regions where the modified GR law is selected in the modeling period.
7.3 Model selection by AIC
Generally, a model having a smaller AIC should be selected when models are chosen on the basis of AIC. However, we selected the GR law when the difference of AIC between the GR and modified GR laws was less than 1, as mentioned in Section 4.1, to reduce the risk of underestimating the probability. We assumed that no earthquake with M ≥ c_{ m } will occur when the modified GR law is applied to the region, and this assumption is a severe constraint. For example, AIC for the modified GR law is smaller than that for the GR law by 0.54 in the region where the IwateMiyagi Nairiku earthquake occurred (Fig. 10). Note that the data period in Fig. 10 is from 1930 through 2007. The value of c_{ m } was estimated to be 5.7 by the modified GR law, but a mainshock with M 7.2 occurred on June 14, 2008. In this case, the modified GR law had a negative effect. Therefore, we decided to set a bias of AIC by 1 when selecting the GR law, but it was a decision made for trial purposes rather than one arising from a rigorous analysis.
7.4 The effect of aftershocks
As mentioned in Section 4.1, when a large earthquake occurred closely preceding the forecast period, we forecasted the expected number of earthquakes with M ≥ M_{ th } in the forecast period by using the modified Omori formula
where t is time after the mainshock, n(t) is the number of aftershocks per unit time, and K_{ a }, c_{ a }, and p are constants (Utsu, 1957). In this study, to simplify the procedure we took into account only mainshocks with M ≥ 5.0 within the preceding year or M ≥ 7.0 within the preceding five years. When there were no such mainshocks, we estimated the expected number in the forecast period from the seismicity rate in the year just before the forecast period. As shown in Fig. 4(b), models that make adjustments on the basis of the aftershock decay rate can avoid overestimating the seismicity in forecast periods.
Generally it may be better to account for aftershock activity added to the background seismicity in the form
where S_{b} is the background seismicity rate. Theoretically, if we apply Eq. (9) to the modeling data, we can estimate the background seismicity as well as the aftershock activity and expect to forecast future seismicity more appropriately. However, if the data are insufficient, Eq. (9) is not necessarily a good selection. For example, when we apply Eqs. (8) and (9) to aftershocks of the IwateMiyagi Nairiku earthquake (M 7.2) of June 14, 2008, and estimate parameters using data between the mainshock and the end of 2008, the result of Eq. (8) is a better match to the actual activity for 2009 than the result of Eq. (9). Therefore, in this study we adopted Eq. (8) to evaluate the effect of aftershock activity.
7.5 The effect of forecast periods
Although forecast periods were fixed at one year or three years in accordance with the rule of CSEP for Japan, we tested the models for a wider range of periods. Comparing total loglikelihood values of three models in Tables 1, 5, 6, and 7, we found that the MGR model shows a tendency to improve for longer forecast periods such as three, five, or seven years. The effect of setting regionally variable bvalues in the MGR and Vbv models tends to become evident in longer term forecasts because the number of target earthquakes increase and we can get statistically more stable results. Note that there are some cases that result in worse Ntest results for the three models if the forecast period includes a large number of earthquakes, as in the case of the Mid Niigata prefecture earthquake (M 6.8) of October 23, 2004, and its many aftershocks. We discuss this further in Section 7.9.
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 onland events that were our main target. Therefore, we excluded the Miyake Island region when we estimated a nationwide mean bvalue from seismicity in the study area (Fig. 2(b)). This bvalue was also used as the default in regions where the GR and modified GR laws could not be distinguished (step 3 in Section 4.1), including near Miyake Island.
7.7 Effect of minimum seismicity rate
As mentioned in Section 4.2, we assumed that target grid cells have a minimum rate of seismicity for earthquakes with M > c_{ m }. We presumed that an earthquake of M 5.0 will occur once in about 42,000 years (2.4 × 10^{−5}/y) as the minimum rate. But we also examined cases in which an earthquake of M 5.0 will occur once in 4,200 years (2.4 × 10^{−4}/y) or 420,000 years (2.4 × 10^{−6}/y). The results are shown in Tables 8 and 9 for a radius of 20 km and forecast period of one year. The loglikelihood changes to a small extent, but only for the 2006 forecast did it change the rank order of the different models.
7.8 Longterm or shortterm 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, longterm 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 longterm data and line S for shortterm data, and the latter is a better fit in prediction. Therefore, for shortterm 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
Aftershocks cannot be forecasted by our models because they assume that earthquakes occur independently and do not consider triggering effects. However, as aftershocks are also targeted in the forecast experiment by CSEP for Japan, the results listed in Tables 1 to 9 are for target earthquakes including aftershocks, which makes results worse when many aftershocks occur. For example, the Mid Niigata prefecture earthquake (M 6.8) occurred on October 23, 2004, and 26 aftershocks with M ≥ 5.0 occurred by the end of 2004. In the Ntest, the MGR model expected 5.62 target earthquakes in 2004 whereas 28 earthquakes were observed: the Mid Niigata prefecture earthquake, its aftershocks, and one earthquake in western Hokkaido on December 14, 2004. In addition, the loglikelihood was worse for 2004 than that for other forecast years. Table 10 shows that when target earthquakes were restricted to mainshocks, the total expected number E[n] and total observed number in the Ntest were similar to each other, and the loglikelihood was near the same level as the other forecast years. The definition of target earthquakes is essential for evaluation of models. It is significant that the models in this study were intended to forecast mainly mainshocks, which differs from the framework of CSEP for Japan.
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.
8. Summary
We proposed earthquake forecast models based on the GR law or the modified GR law and compared their performance by retrospective forecast. The results are as follows:

1.
The MGR model, using both the modified GR and GR laws, was better than the Vbv model, using only the GR law.

2.
The Cbv model, based on a spatially constant bvalue, was better than the Vbv model, based on regionally variable bvalues for shortterm (one year) forecasts.

3.
The difference between the MGR and Cbv models was not clear for shortterm forecasts.

4.
The MGR and the Vbv models, using regionally variable bvalues, tended to become better than the Cbv model for longterm (three years or longer) forecasts.
On the basis of our results, we propose the use of the MGR model for CSEP for Japan.
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Acknowledgments
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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Hirose, F., Maeda, K. Earthquake forecast models for inland Japan based on the GR law and the modified GR law. Earth Planet Sp 63, 8 (2011). https://doi.org/10.5047/eps.2010.10.002
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DOI: https://doi.org/10.5047/eps.2010.10.002
Key words
 Earthquake forecast
 GR law
 modified GR law
 bvalue
 modified Omori formula
 minimum limit of expected seismicity
 retrospective forecasts
 CSEP for Japan