Application of a long-range forecasting model to earthquakes in the Japan mainland testing region
© 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: 12 April 2010
Accepted: 10 August 2010
Published: 4 March 2011
The Every Earthquake a Precursor According to Scale (EEPAS) model is a long-range forecasting method which has been previously applied to a number of regions, including Japan. The Collaboratory for the Study of Earthquake Predictability (CSEP) forecasting experiment in Japan provides an opportunity to test the model at lower magnitudes than previously and to compare it with other competing models. The model sums contributions to the rate density from past earthquakes based on predictive scaling relations derived from the precursory scale increase phenomenon. Two features of the earthquake catalogue in the Japan mainland region create difficulties in applying the model, namely magnitude-dependence in the proportion of aftershocks and in the Gutenberg-Richter b-value. To accommodate these features, the model was fitted separately to earthquakes in three different target magnitude classes over the period 2000–2009. There are some substantial unexplained differences in parameters between classes, but the time and magnitude distributions of the individual earthquake contributions are such that the model is suitable for three-month testing at M ≥ 4 and for one-year testing at M ≥ 5. In retrospective analyses, the mean probability gain of the EEPAS model over a spatially smoothed seismicity model increases with magnitude. The same trend is expected in prospective testing. The Proximity to Past Earthquakes (PPE) model has been submitted to the same testing classes as the EEPAS model. Its role is that of a spatially-smoothed reference model, against which the performance of time-varying models can be compared.
Key wordsStatistical seismology earthquake forecasting Japan
The Collaboratory for the Study of Earthquake Predictability (CSEP) earthquake forecasting experiment in Japan presents an opportunity to apply the EEPAS model at a lower magnitude threshold than previously, and to have it formally and independently tested against other competing models. The Ω scaling relation between magnitude and precursor time (Evison and Rhoades, 2004) indicates that it might be possible to use the model either for three-month forecasts at M ≥ 4.0 or for annual forecasts at M ≥ 5.0. The model uses the minor earthquakes to forecast the major events, and requires catalogue completeness at nearly two orders of magnitude below the target magnitude threshold. With the current dense network of seismograph stations covering the Japan mainland region, the magnitude threshold of completeness is likely to be sufficiently low in recent years for fitting the model at M ≥ 4.0. This suggests that it may be possible to enter the EEPAS model into the CSEP tests in the Japan mainland region for both three-month and one-year forecasts, but first it must be adapted to the testing region.
2. Model Description
In Eq. (10), r i is the distance in km between (x, y) and the epicentre (x i , y i ) of the ith precursory earthquake; and a, d and s are constant parameters.
3. Application to the Japan Mainland Region
The mean weights plotted in Fig. 4 are generally lower and more strongly dependent on m than in previous catalogues to which EEPAS has been applied. For example, Rhoades (2009) reported p I values of 0.71 for California and 0.83 for the Kanto region at m = 5.0. This magnitude dependence is problematical for the EEPAS model, which implicitly assumes that p I is independent of magnitude. The same assumption was made in a version of EEPAS allowing explicitly for aftershocks of forecasted events (Rhoades, 2009). Deviation of the data from this assumption will undoubtedly affect the fit and performance of the standard EEPAS model applied here, which is the only version so far adapted for CSEP testing. The situation is complicated further by the fact that the dependence of p I on m is bound to be somewhat time dependent, if only because the maximum magnitude in R will differ in different time periods. The maximum magnitude contributing to Fig. 4 is 7.3; a very large earthquake in R with M > 8 would be expected to have numerous aftershocks with M > 6, which would have the effect of reducing p I for M ≥ 6 to lower values than are shown in Fig. 4.
The EEPAS model is primarily designed to forecast the major earthquakes, rather than their aftershocks. The dependence of b and p I on the magnitude threshold precipitated a decision to fit both the PPE and EEPAS models within restricted target magnitude classes, namely 3.95 < m < 4.45, 4.45 < m < 4.95, and 4.95 < m < 9.05, so that the performance at higher magnitudes is not adversely affected by fitting the model to a much larger number of predominantly smaller events.
3.1 Fitted parameters
The EEPAS model was found to fit all three target magnitude classes better with aftershocks down-weighted than with equal weights. The optimal parameters for both models and all three magnitude classes are listed in Table 1. Note that the PPE and EEPAS models use different Gutenberg-Richter b-values. The PPE model b-value (“b” in Table 1) is needed for magnitude scaling of earthquakes within the target magnitude range (Eq. (9)). It is estimated from the unweighted b-value for magnitudes in the target range. For the EEPAS model, the b-value (“bEEPAS” in Table 1) plays a role in the normalising function (Eq. (6)), that is needed to retain an appropriate relation between the number of “small” precursory earthquakes and the number of “large” predicted events. For a given target magnitude range, it was arbitrarily set taking into account the weighted b-value pertaining to the most influential magnitudes of precursory earthquakes (Fig. 5). The parameter also plays a role in the normalising function. The value of this parameter was arbitrarily adjusted also for each magnitude class so that the number of earthquakes predicted by the EEPAS model approximately matched the actual number over the fitting period. The parameter b M was set to 1, as in several previous applications of the EEPAS model (Rhoades and Evison, 2005, 2006; Rhoades, 2007), and thus a constant difference is assumed between the magnitude of a precursor and the mean magnitude of its contribution to the rate density. The other parameters (three for PPE and eight for EEPAS) were optimised by maximum likelihood, within upper and lower bounds as indicated in Table 1. Such constraints are necessary, inter alia, to limit the distorting effect of aftershocks on the fitted parameters.
The parameters of the PPE model and several parameters of the EEPAS model do not vary appreciably across the three magnitude classes. In all cases, a M attained its minimum bound of 1.0; this rather low value may be an effect of the model compromising between forecasting major earthquakes and their aftershocks. This effect appears to be confirmed by the fitted values of μ (<0.1), which represents the proportion of unpredicted earthquakes in the EEPAS model. Such low values can be compared to the relatively high proportion (>0.5) of aftershocks in each magnitude class, and indicate that the EEPAS model has adjusted itself to successfully forecast many of the aftershocks. The values of the parameters a T and σ a differ across the three magnitude classes for unknown reasons, but perhaps reflecting differing adjustments of the scales of precursor time and area to accommodate the aftershocks, or latent complexities in the scaling of precursor time and area which are not evident from Fig. 1. As an example of such complexities, it seems possible that time and area scaling could be affected locally by the average rate of earthquake occurrence (Rhoades, 2009). The values for the highest magnitude class are nearest to those found in previous studies of the Japan region.
The fitted values of σ m and σ T are generally lower than in previous applications of the model. A consequence of the low values of σ m is that the precursory earthquakes contributing to the rate density for target earthquakes of a given magnitude fall into a rather narrow magnitude range. For example, at a target magnitude of m = 4.0, more than 99.9 percent of the contributions to the rate density come from earthquakes in the magnitude range 2.45 < m < 3.45. Evidently, with these fitted parameters, the chosen minimum magnitude threshold of m0 = 2.15 is more than adequate to ensure that all precursory earthquakes relevant to the target range are included in the analysis. This threshold could be raised to 2.45 with hardly any loss of information.
3.2 Time and magnitude distributions of contributions to rate density
Let us consider the time distribution of the contribution from an earthquake of m 2.5, which is at the lower end of the precursor magnitude range that could make an effective contribution to the forecast in the lowest targeted magnitude range. Using Eq. (3) and the parameters a T and b T from Table 1, the logarithm of the time in days, starting from the time of its occurrence, is normally distributed with mean 3.07 and standard deviation 0.15. The cumulative distribution is less than 10−13 at 3 months after its occurrence. Therefore no earthquakes occurring during a three-month forecasting period would have any appreciable influence on the rate density before the end of the period. The same is true for contributions to the middle targeted magnitude range but not for those to the highest magnitude range, for which the parameter a T is lower.
3.3 Goodness of fit and expected forecasting performance
The information rates of the PPE and EEPAS models for each magnitude class are given in Table 2. It is the difference Ieepas — Ippe that is of most interest. This difference is seen to be 0.24, 0.42 and 1.02 for the three magnitude classes in increasing order. The geometric mean probability gain per earthquake is calculated as exp(Ieepas — Ippe). This statistic is given in parenthesis in Table 2, and is seen to increase from 1.27 for the 3.95 < m < 4.45 magnitude class to 2.77 for the 4.95 < m < 9.05 magnitude class. The last value is towards the high end of probability gains found in previous applications of the EEPAS model. Although probability gains estimated from fitting are not necessarily a good guide to future performance, these values indicate that the model is likely to perform betterathigher magnitudes thanatlower magnitudes.
The effect on the information rate of fitting the model separately to different magnitude classes is substantial. For example, the EEPAS parameters fitted to the 3.95 < m < 4.45 class give an information rate of only 1.43 when applied to the 4.95 < m < 9.05 class. This is less than that of the PPE model, and corresponds to a probability gain of 0.74. Conversely, when the EEPAS parameters fitted to the 4.95 < m < 9.05 class give an information rate of only 1.80 when applied the 3.95 < m < 4.45 class. This is again less than that of the PPE model, and corresponds to a probability gain of 0.61.
3.4 Implementation for the testing center
In order to implement the EEPAS and PPE models for the Japan testing center, the continuous rate densities of these models are integrated over the latitude, longitude, magnitude and time limits of each cell to compute the expected number of earthquakes. The integration is performed by averaging the rate densities over an inner grid of 34 points regularly spaced in time, magnitude, latitude and longitude within each cell. This discretization into cells is bound to affect the performance of the models to some extent, but the cell sizes are small enough so that the effects on performance should be quite small.
The EEPAS and PPE models have been submitted to the Japan Testing Center for testing in the Japan mainland region in the three-months and one-year forecast classes. The role of the PPE model here is that of a spatially-smoothed reference model, against which the performance of time-varying models such as EEPAS can be compared.
The EEPAS model has never before been applied with a magnitude threshold as low as 4.0, which is the minimum magnitude for the three-month class. The adaptation of the model to such a low target magnitude has brought into focus some issues which cannot be resolved without further research. In particular, it has shown that the model needs further development to rigorously handle magnitude-dependence in the proportion of aftershocks and in the b-value. Moreover, the unexplained differences in parameters of the EEPAS time and location distributions between target magnitude classes require further investigation. Also, the possibility of adapting the EEPAS model for testing in the wider Japan region can be considered. Further research into these matters is likely to result in one or more new models being submitted for testing in the future.
The EEPAS model forecasts could be applied in linear combination with one or more models from the one-day class to generate new one-day forecasting models (Rhoades and Gerstenberger, 2009). The performance of such hybrid models would help to clarify the relative information value of the precursory scale increase phenomenon with that of short-term clustering for forecasting over a range of target magnitudes. The analysis performed here suggests that the relative value of the precursory scale increase phenomenon should increase as the target magnitude is increased.
This research was supported by the New Zealand Foundation for Research, Science and Technology under contract C050804 and by GNS Science Capability Funding. A. Christophersen, M. Stirling, W. Smith, M. Werner, E. Papadimitriou and editor K. Nanjo provided helpful reviews of the manuscript.
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