Adaptively Smoothed Seismicity Earthquake Forecasts for Italy

We present a model for estimating the probabilities of future earthquakes of magnitudes m>4.95 in Italy. The model, a slightly modified version of the one proposed for California by Helmstetter et al. (2007) and Werner et al. (2010), approximates seismicity by a spatially heterogeneous, temporally homogeneous Poisson point process. The temporal, spatial and magnitude dimensions are entirely decoupled. Magnitudes are independently and identically distributed according to a tapered Gutenberg-Richter magnitude distribution. We estimated the spatial distribution of future seismicity by smoothing the locations of past earthquakes listed in two Italian catalogs: a short instrumental catalog and a longer instrumental and historical catalog. The bandwidth of the adaptive spatial kernel is estimated by optimizing the predictive power of the kernel estimate of the spatial earthquake density in retrospective forecasts. When available and trustworthy, we used small earthquakes m>2.95 to illuminate active fault structures and likely future epicenters. By calibrating the model on two catalogs of different duration to create two forecasts, we intend to quantify the loss (or gain) of predictability incurred when only a short but recent data record is available. Both forecasts, scaled to five and ten years, were submitted to the Italian prospective forecasting experiment of the global Collaboratory for the Study of Earthquake Predictability (CSEP). An earlier forecast from the model was submitted by Helmstetter et al. (2007) to the Regional Earthquake Likelihood Model (RELM) experiment in California, and, with over half of the five-year experiment over, the forecast performs better than its competitors.

which we estimated future earthquake potential in Italy. Section 3 describes the model and its calibration 87 on the two data sets. We present the earthquake forecasts in section 4 before concluding in section 5.   to the kth nearest neighbor. To estimate the optimal number of neighbors to include in the smoothing, 149 we divided the catalog into two non-overlapping sets: a learning catalog and a testing catalog. In section 150 3.3, we determine the optimal number of neighbors by calculating the spatial density of seismicity from 151 the learning catalog and evaluating its predictive power on the testing catalog. The spatial density 152 was scaled to the number of expected earthquakes by using the mean number of observed earthquakes 153 (section 3.6). Finally, to obtain a rate-space-magnitude forecast, we multiplied the scaled spatial density   As in these prior studies, we set the input parameters to r f act = 8, x k = 0.5, p1 = 0.95, τmin = 1 day and 162 τmax = 5 days. We varied x mef f according to the different learning catalogs we used. As the interaction  We estimated the density of spontaneous seismicity in each 0.1 by 0.1 degree cell by smoothing the 170 location of each earthquake i with an isotropic adaptive power-law kernel K d i ( r): where di is the adaptive smoothing distance and C(di) is a normalizing factor, so that the integral of 172 K d i ( r) over an infinite area equals 1. 173 We measured the smoothing distance di associated with an earthquake i as the horizontal distance 174 between event i and its kth closest neighbor. The number of neighbors, k, is an adjustable parameter, 175 estimated by optimizing the forecasts (see section 3.3). We imposed di > 0.5 km to account for location 176 uncertainty. The kernel bandwith di thus decreases if the density of seismicity is high at the location ri 177 of the earthquake i, so that we have higher resolution (smaller di) where the density is higher.

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The density at any point r was estimated by where N l is the total number of earthquakes in the learning catalog. However, the forecasts are given as 180 an expected number of events in each cell of 0.1 • . We therefore integrated equation (2) over each cell 181 and summed over all contributing earthquakes to obtain the seismicity rate of each cell. the total expected number of events, we optimized the normalized spatial density estimate in each cell 188 (ix, iy) using where Nt is the number of observed target events. The expected number of events for the model µ * thus 190 equals the observed number Nt.

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The log-likelihood of the model is given by where n is the number of events that occurred in cell (ix, iy). To adhere to the rules of the CSEP-Italy 193 predictability experiment, we assumed that the probability p of observing n events in cell (ix, iy) given 194 a forecast of µ * (ix, iy) in that cell is given by the Poisson distribution We built a large set of background models µ * by varying (i) the starting times, end times and magnitude 196 thresholds of the learning and testing catalogs, and (ii) the catalog (either the MIC or the CPTI catalog). 197 We evaluated the performance of each model by calculating its probability gain per target earthquake 198 relative to a model with a uniform spatial density: where L0 is the log-likelihood of the spatially homogeneous model. test whether including small earthquakes results in greater predictability of future m ≥ 4.95 earthquakes. 206 We also changed the target periods to test the robustness of the results.

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In Figure 1, we show the probability gains per earthquake against the magnitude threshold of the two

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Whenever target earthquakes occur in previously active regions, the optimal amount of smoothing is 226 small (k = 1), and the gains tend to be higher (see, e.g., model 13 in Table 1 Table 1 obtained from the MIC and model 14 ! in Table 2 obtained from the CPTI catalog), we had to 233 decide which magnitude threshold to apply to the learning catalog, which smoothing parameter to use, 234 and whether to use all existing data up until the end of the two catalogs. We decided to use all available 235 data in each catalog for the final density estimate so that the forecasts could benefit from as much 236 data as possible. Moreover, despite the observed variability in gains against the magnitude threshold of 237 the input catalog (discussed above), Figure 1 shows that, on average, there seems to be an advantage 238 in including small earthquakes for estimating the predictive spatial density (see also the discussion in section 5). Therefore, to calculate the spatial densities for the final forecasts, we used the lowest reliable  Table 1 and model 14 ! in Table 2, respectively. We discuss future improvements of the 246 spatial optimization method in section 5.