Significant improvements of the space-time ETAS model for forecasting of accurate baseline seismicity
- Yosihiko Ogata^{1}Email author
https://doi.org/10.5047/eps.2010.09.001
© TThe 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: 4 June 2010
Accepted: 7 September 2010
Published: 4 March 2011
Abstract
The space-time version of the epidemic type aftershock sequence (ETAS) model is based on the empirical laws for aftershocks, and constructed with a certain space-time function for earthquake clustering. For more accurate seismic prediction, we modify it to deal with not only anisotropic clustering but also regionally distinct characteristics of seismicity. The former needs a quasi-real-time cluster analysis that identifies the aftershock centroids and correlation coefficient of a cluster distribution. The latter needs the space-time ETAS model with location dependent parameters. Together with the Gutenberg-Richter’s magnitude-frequency law with location-dependent b-values, the elaborated model is applied for short-term, intermediate-term and long-term forecasting of baseline seismic activity.
Key words
Anisotropic clusters Bayesian method b-values, Delaunay tessellation location dependent parameters probability forecasting1. Introduction
Seismicity patterns vary substantially from place to place, showing various clustering features, though some of the fundamental physical processes leading to earthquakes may be common to all events. Kanamori (1981) postulates that fault zone heterogeneity and complexity are responsible for the observed variations. Such complex features have been tackled in terms of stochastic point-process models for earthquake occurrence. The stochastic models have to be accurate enough in the sense that they are spatio-temporally well adapted to and predict various local patterns of normal activity. The epidemic type aftershock sequence (ETAS) model and its space-time extension have been introduced for such a purpose (Ogata, 1985, 1988, 1993, 1998).
However, their postulate is that the parameter values are assumed to be the same throughout the whole region and time span considered. We learn by experience that the difference of parameter values of the model at different subre-gions becomes more significant as the catalog size increases by lowering the magnitude threshold or as the area of the investigation becomes larger. For example, the p-value of the aftershock decay varies from place to place (Utsu, 1969), besides the background seismicity that obviously depends on the location. If the space-time ETAS model is fitted to such a dataset, the parameter estimates on average are obtained for the seismicity on the whole area, but they lead to biased seismicity prediction in the subregions where the seismicity pattern is significantly different from the one estimated for the whole area (see Ogata, 1988, for example).
Therefore, the best fitted case among the candidates of the space-time ETAS models in Ogata (1998) was extended to the hierarchical version of the model (the hierarchical space-time ETAS model, HIST-ETAS model in short) in which the parameters depend on the location of the earthquakes (Ogata et al., 2003; Ogata, 2004). The software package of the computing programs is in preparation for publishing (Ogata et al., 2010).
Using the present HIST-ETAS model together with Gutenberg-Richter’s magnitude frequency (Gutenberg and Richter, 1944) with the location dependent b-values, we are able to forecast the baseline seismic activity more accurately than ever, and thus we take a part in the Earthquake Forecast Testing Experiment in Japan (EFTEJ) for a short-term, intermediate-term and long-term future in and around Japan (http://www.eic.eri.u-tokyo.ac.jp/ZISINyosoku/wiki.en/wiki.cgi). This manuscript describes a sequence of procedures of pre-treatment (recompiling) of the space-time data, parameter estimation of the HIST-ETAS model as well as estimation of the location dependent b-values to undertake the short-, intermediate- and long-term forecasting.
2. Location Dependent Space-Time ETAS Model
As will be specifically described in Section 5, each of the parameters μ(x, y), K (x, y), α(x, y), p(x, y) and q(x, y) is represented by a piecewise function whose value at any location (x,y) is interpolated by the three values (the coefficients) at the locations of the nearest three earthquakes (Delaunay triangle vertices) on the planed tessellated by epicenters. The coefficients of the parameter functions are simultaneously estimated by maximizing a penalized log-likelihood function that determines the optimum trade-off between the goodness of fit to the data and uniformity constraints of the functions (i.e., facets of each piecewise linear function being as flat as possible). Here, such optimum trade-off is objectively attained by minimizing the Akaike Bayesian Information Criterion (ABIC; Akaike, 1980; see Section 4) that actually evaluates the expected predictive error of Bayesian models based on the data used for the estimation (e.g., Ogata, 2004).
3. Data Processing for Anisotropic Clusters
According to the format required by the EFTEJ, we use the hypocenter catalog of the Japan Meteorological Agency (JMA) for the period 1926-2008 as the original source. Furthermore, we combine the catalog with the Utsu catalog (Utsu, 1982, 1985) for the period 1886–1925, whose magnitudes are consistent with the JMA catalog. Actually, the detection rate of smaller earthquakes is low in early period. Nevertheless, we utilize such large earthquakes as the history in the ETAS model in the precursory period because they are possibly influential to the seismicity in the target period. The accuracy of the hypocenter depth of the JMA catalogue is not satisfactory especially in offshore regions, so that we ignore the depth axis and consider only longitude and latitude for the location of an earthquake restricting ourselves to shallow events down to 100 km depth. Also, we should be sensitive to and avoid the constrained epicenters in such a way that they are subsequently located at the same place or on lattice coordinates because these cause odd or biased estimates of the space-time ETAS models.
As requested by the EFTEJ, we consider two target periods with different threshold magnitudes for the long- and short-term forecasts, taking the evolution of detection capability of earthquakes by the seismic network of the JMA. The former one is 1926–2008 with threshold magnitude M 5.0, and the latter is 2000–2008 with threshold magnitude M 4.0. These are regarded as almost completely detected throughout the respective target period and the Japan area except for the north-end off-shore and southern end of Izu-Ogasawara (Izu-Bornin) Islands in early years. We use a moderate number of large earthquakes (M 6 or larger) in the precursory period to the target period of the analysis, as the history of the ETAS model. Then, based on this earthquake data, we form the Delaunay tessellation that is necessary to apply the location dependent space-time ETAS model as specified in Section 5.
4. Optimization and Selection of Bayesian Models
5. Hierarchical Modelling on Tessellated Spatial Region
5.1 Delaunay interpolation functions
Therefore, our alternative proposal for the present case is as follows. Consider the Delaunay triangulation (e.g., Green and Sibson, 1978); that is to say, the whole rectangular region A is tessellated by triangles with the vertex locations of earthquakes and some additional points {(x_{ i },y_{ i }),i = 1,…, N + n}, where N is the number of earthquakes and n is the number of the additional points on the rectangular boundary including the corners. Here, for successfully fulfilling a Delaunay tessellation, we sometimes need very small perturbation of epicenters to avoid lattice structure or duplicated locations in a local domain. Figure 2(b) shows such a tessellation based on the epicenters of the present dataset (Fig. 2(a)) and the additional points on the boundaries.
Then, define the piecewise linear function ϕ(x, y) on the tessellated region such that its value at any location (x, y) in each triangle is linearly interpolated by the three values at the vertices. Specifically, consider a Delaunay triangle and the coordinates of its vertices (x_{ i }, y_{ i }), i = 1, 2, 3. Then, for the values ϕ_{ i } = ϕ(x_{ i }, y_{ i }), i = 1, 2, 3, the function value at any location inside the triangle is given as follows:
5.2 Spatial ETAS with all parameters constant
Estimates of the models applied to the M ≥ 4 data.
Model unit | μ events/day/deg^{2} | K events/day/deg^{2} | c days | α 1/mag | p — | d deg^{2} | q — | AIC, ABIC — |
---|---|---|---|---|---|---|---|---|
Space-Time ETAS0iso | 1.88E-04 | 2.19E-04 | 1.58E-03 | 0.808 | 0.865 | 3.32E-04 | 1.368 | 49528.5 |
Space-Time ETAS0aniso | 1.90E-04 | 2.14E-04 | 1.59E-03 | 0.823 | 0.865 | 3.18E-04 | 1.367 | 49407.1 |
Space-Time ETASiso | 7.77E-05 | 9.63E-05 | 1.24E-03 | 1.197 | 0.853 | 2.32E-04 | 1.415 | 47972.0 |
Space-Time ETASaniso | 7.93E-05 | 9.44E-05 | 1.24E-03 | 1.204 | 0.853 | 2.21E-04 | 1.414 | 47821.1 |
μK -HIST-ETAS0 weights | 1.54E-02 | 2.34E-06 | 1.06E-02 | 1.680 | 1.150 | 7.57E-05 | 1.660 | — |
0.429 | 0.134 | — | — | — | — | — | 39982.2 | |
μK-HIST-ETAS weights | 1.52E-02 | 2.35E-06 | 1.10E-02 | 1.430 | 1.160 | 1.06E-04 | 1.700 | — |
0.381 | 0.138 | — | — | — | — | — | 39503.4 | |
HIST-ETAS0 weights | 1.49E-02 | 1.24E-05 | 8.82E-03 | 1.470 | 1.140 | 1.57E-04 | 1.580 | — |
0.571 | 0.901 | — | 8.0 | 24.4 | — | 1440 | 38340.7 | |
HIST-ETAS weights | 1.76E-02 | 2.11E-06 | 1.10E-02 | 1.440 | 1.160 | 9.16E-05 | 1.690 | — |
0.445 | 0.239 | — | 17.1 | 19.6 | — | 1790 | 37903.5 |
The estimates of the models applied to the M ≥ 5 data. The same caption as for Table 1.
Model unit | μ events/day/deg^{2} | K events/day/deg^{2} | c days | α 1/mag | p — | d deg^{2} | q — | AIC, ABIC — |
---|---|---|---|---|---|---|---|---|
Space-Time ETAS0iso | 1.26E-05 | 1.49E-04 | 4.66E-03 | 1.079 | 0.891 | 5.90E-03 | 1.713 | 82643.0 |
Space-Time ETAS0aniso | 1.27E-05 | 1.47E-04 | 4.66E-03 | 1.083 | 0.890 | 5.66E-03 | 1.706 | 82592.8 |
Space-Time ETASiso | 7.97E-06 | 8.79E-05 | 4.48E-03 | 1.257 | 0.891 | 4.88E-03 | 1.763 | 81893.7 |
Space-Time ETASaniso | 8.04E-06 | 8.68E-05 | 4.48E-03 | 1.263 | 0.891 | 4.67E-03 | 1.756 | 81838.1 |
μK-HIST-ETAS0 weights | 9.47E-04 | 2.62E-05 | 2.46E-02 | 1.310 | 1.090 | 3.00E-03 | 1.830 | — |
0.439 | 0.184 | — | — | — | — | — | 80655.7 | |
μK-HIST-ETAS weights | 1.50E-03 | 1.59E-05 | 2.46E-02 | 1.340 | 1.100 | 2.85E-03 | 1.890 | — |
0.448 | 0.158 | — | — | — | — | — | 78095.5 | |
HIST-ETAS0 weights | 9.47E-04 | 2.62E-05 | 2.46E-02 | 1.310 | 1.090 | 3.00E-03 | 1.830 | — |
0.439 | 0.184 | — | 5.84 | 28.3 | — | 93900 | 80391.9 | |
HIST-ETAS weights | 1.33E-03 | 2.59E-05 | 9.45E-03 | 0.940 | 1.060 | 3.51E-03 | 1.910 | — |
0.461 | 0.241 | — | 5.84 | 28.3 | — | 93900 | 77552.7 |
5.3 ETAS: Spatially varying μ and K
The obtained MLEs under the constant parameter μ for the background seismicity cause the highly biased MLEs for the baseline estimates and in (5) as well as c and d. Without appropriately unbiased initial guess of the baseline parameters, it is not easy to stably obtain the converging solution of the five location-dependent parameters in (5) due to the search in very high dimensional coefficient space. Therefore, before applying the model (2) with (5), we use the MLEs of the space-time ETAS model for the initial guess of the baseline parameters of a special version of the model (2) in which we assume that only the background rates and aftershock productivity rate are location dependent; namely, other functions ϕ_{ k }(x, y), k = 3, 4, 5, in (5) are fixed to be zero. Hereafter we call this restricted model as μK-HIST-ETAS model. In order to estimate ϕ_{ k }(x,y) with each of k = 1, 2, we use more than twice as many coefficients as the number of the earthquake data.
The penalized log-likelihood defines a trade-off between the goodness of fit to the data and the uniformity of each function, namely, the facets of the piecewise linear function being as flat as possible. A smaller weight leads to a higher regional variability of the ϕ-functions. The optimal weights together with the maximizing baseline parameters ( , c, α, p, d, q) are obtained by a Bayesian principle of maximizing the integrated posterior function (see Appendix). Here note that the baseline parameters are automatically determined by the zero sum constraint of the corresponding ϕ-function. This overall maximization can be eventually attained by repeating alternate procedures of the separated maximizations with respected to the parameters (coefficients) and hyper-parameters (weights) described as follows.
First of all, we use the obtained MLEs of the space-time ETAS model for the initial baseline parameter and set ϕ_{1} (x, y) = ϕ_{2}(x,y) = 0 for the initial coefficients. Then, we implement the maximization of the penalized log-likelihood (3) with respect to the coefficients of the ϕ-functions (see Appendix). For the maximization, we adopt a linear search procedure in conjunction with the incomplete Cholesky conjugate gradient (ICCG) method for 2(N + n) dimensional coefficient vectors by using a suitable approximate Hessian matrix (see Appendix), where N is the number of earthquakes and n is the number of the additional points on the rectangular boundary including the corners (see Fig. 2(b)). This makes the convergence very rapid regardless of the high dimensionality of θ if the Gaussian approximations for the posterior function are adequate.
Having attained such convergences for given hyper-parameters τ = (w_{1}, w_{2}, c, α, p, d, q), we eventually need to perform the maximization of Λ (τ) defined in (4) with respect to τ by a direct search such as the simplex method in the 7 dimensional space. Such double optimizations are repeated in turn until the latter maximization converges. The whole optimization procedure usually converges when initial vector values for τ are set in such a way that the penalty is effective enough; otherwise, it may take very many steps to reach the solution. After all, assuming unimodality of the posterior function, one can get the optimal maximum posterior solution for the maximum likelihood estimate .
5.4 ETAS: Spatial variation in 5 parameters
It is also useful to examine whether or not the characteristic parameters, particularly and are significantly uniform (i.e., spatially invariant). For this we can calculate the Akaike Bayesian Information Criterion (ABIC; see Appendix) as a byproduct of the above simplex optimization. A model with a smaller ABIC value indicates a better fit. For example, we can compare the ABIC values of the HIST-ETAS model for the optimal weights with the one for ( , 10^{8}) to examine whether q-value is location dependent or not.
6. Modeling the Spatially Varying b-Values
We further consider that the b-value of the Gutenberg-Richter’s magnitude frequency law is location dependent. Historically, based on the moment method, Utsu (1965) proposed the estimator for the observation of magnitude sequence {M_{ i }, i = 1,…, N} where M_{c} is the lowest bound of the magnitudes above which almost all the earthquakes are detected. This is modified by Utsu (1970) to replace M_{c} by M_{c} − 0.05 for the unbiased estimate of the b-values in case when the given magnitudes are rounded into values with 0.1 unit, and hereafter we follow this modification for the JMA catalog.
Aki (1965) showed that the Utsu’s b-estimator is nothing but the maximum likelihood estimate (MLE) that maximizes the likelihood function , M_{ i } > M_{c} and β = bln 10. Wiemer and Wyss (1997) uses the MLE in ZMAP software to obtain the location dependent b-values using data from moving disk whose radius is adjusted to include the same number of earthquakes. However there remain the issues of optimal selection of the number of earthquakes in the disk and evaluation of significance of the b-value changes.
The estimates for magnitude frequency.
Magntude threshold | Weight | ABIC |
---|---|---|
4.0 | 4.3 | 5804.9 |
5.0 | 5.5 | 4368.8 |
7. Implications of Tables and Figures
We can compare the AIC and ABIC values among the MLE based models and among the Bayesian models, respectively, although we cannot directly compare the AIC value with ABIC values here because we did not adjust the difference in the normalization factors between AIC and ABIC in the considered models. By the entropy concept from which both AIC and ABIC (Akaike, 1974, 1978, 1980) are derived, we can expect a better forecast among the MLE-based models or among the Bayesian models with a smaller AIC or ABIC, respectively, under the assumption that the stochastic structure of future seismicity will not change from the past as the baseline seismicity.
Thus, Tables 1 and 2 imply several consequences of the present fitting of the models. First, we can say that the fit of the models to the data from the target period associated with the occurrence history of large earthquakes in precursory period will forecast better than those applied to the data during the target period only. Second, the models that take the anisotropic clusters into consideration will forecast better than the models with isotropic clusters only using the original JMA hypocenter data. Third, the five parameter HIST-ETAS models will forecast better than the μK-HIST-ETAS models. Eventually, we expect the best forecasting performance by the 5 parameter HIST-ETAS models that take account of the anisotropic clustering and effect of the history in the precursory periods. Finally, the p < 1 estimate for the uniform background rate μ in space become p > 1 by the location dependent μ estimate. The reason of the p < 1 estimate is that as a compensation of the spatially uniform back ground rate, the time evolution with heavier tailed aftershock decay is easier for the spatial seismicity to concentrate in the active regions.
8. Forecasting
8.1 Short-term forecast
For the short-term forecast, we first reprocess the JMA data in real time as described in Section 3. Namely, during a certain time span (say, one hour) immediately after a large earthquake, the cluster analysis is automatically implemented while during the same period, we can only to make a real time forecast using the generic (null hypothesis model) procedure with the original JMA epicenter coordinates and the identity matrix for isotropic clustering.
8.2 Intermediate-term forecast
8.3 Long-term forecast
During the period [S, T] for a sufficiently large time span T − S, λ(t, x, yǀH_{ S }) is essentially equal to the background seismicity rate μ(x, y) for any location and time. Therefore, the intermediate-term probability above should take a very similar value for the case where we use the background seismicity rate μ(x, y) in place of λ(t, x, yǀH_{ S }) in the above-stated procedure (i)–(iv). Thus, we adopt this as the probability of the long-term forecast of each space-magnitude voxel per unit time.
Relevantly, Ogata (2008) argues that the background rate appears better long-term forecasting for large earthquakes (M ≥ 6.7,15 years period) than the ordinary average occurrence intensity in space, by the retrospective prediction performance. This is mainly because such large earthquakes mostly occurred at the complementary regions of high K-values (e.g., Ogata, 2004) that substantially contribute to the total intensity λ(t,x,yǀH_{ S }).
9. Concluding Remarks
We applied the hierarchical space-time ETAS (HIST-ETAS) model to the short-, intermediate- and long-term forecast of baseline seismicity in and around Japan. Each parameter of the space-time ETAS model is described by a two dimensional piecewise function whose value at a location is interpolated by the three values at the location of the nearest three earthquakes (Delaunay triangle vertices) on the tessellated plane. Such modeling by using Delaunay tessellation is suited for the observation on highly clustered points with accurate locations, and therefore we can expect locally unbiased probability evaluation there. We are particularly concerned with the spatial estimates of the first two parameters of the space-time ETAS model: namely, μ-values of the background seismicity and aftershock productivity K-values. The former is useful for the long-term prediction of the large earthquakes, and the latter for the short-term aftershock probability forecast immediately after a large earthquake.
For the joint probability of space-time-magnitude forecast, we have assumed that the sequences of magnitudes are independent from history of the occurrence times while the reverse relation is highly dependent as described by the ETAS model. Furthermore, we have adopted the exponential distribution (Gutenberg-Richter law) for the magnitude frequency. However, I believe these postulates are not always the case. Indeed, the magnitude sequence of the global large earthquakes is not at all independent between them but possesses a long-range autocorrelations (Ogata and Abe, 1989). Furthermore, Ogata (1989) considered a model for magnitude sequence where the b-value varies in time based on both history of magnitudes and occurrence times of earthquakes. Furthermore, we know that magnitude frequency in a local area is not necessarily exponentially distributed as we see in many swarm activity. These anomalies may provide some hints for a better prediction of large earthquakes than the present models for baseline seismicity.
Declarations
Acknowledgments
I am very grateful to Koichi Katsura and Jiancang Zhuang for their technical assistances. Comments by Annie Chu, Rick Schoenberg and the anonymous referee were useful clarications. We have used hypocenter data provided by the JMA. This study is partly supported by the Japan Society for the Promotion of Science under Grant-in-Aid for Scientifc Research no. 20240027, and by the 2010 projects of the Institute of Statistical Mathematics and the Research Organization of Information and Systems at the Transdisciplinary Research Integration Center, Inter-University Research Institute Corporation.
Authors’ Affiliations
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