Essay: Ensemble Smoothed Seismicity Models for the New Italian Probabilistic Seismic Hazard Map

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Abstract

The smoothed seismicity models plan to be introduced into the new Italian Probabilistic Seismic Hazard Maps 2017-2018. In this study we report progress on the use of smoothed seismicity models developed by using fixed and the adaptive smoothing algorithms and present an earthquake rate forecast developed from an ensemble smoothed seismicity model. Recent developments in adaptive smoothing methods and statistical tests for evaluating and comparing rate models prompt us to investigate the appropriateness of adaptive smoothing together with the fixed smoothing seismicity models for the new Italian seismic hazard maps. The gridded-seismicity models are based on both historical and instrumental earthquakes and assume that larger earthquakes occur at or near clusters of previous smaller earthquakes. In general, the approach of using spatially smoothed historical seismicity is different from the one used previously by Working group MSP04 (2004), and Slejko et al. (1998) for Italy, in which source zones were drawn around the seismicity and the tectonic provinces and is the first to be used for the new probabilistic seismic hazard maps for Italy.  We develop two different smoothed seismicity models following the well-known and widely applied fixed (Frankel, 1995) and adaptive smoothing methods (Helmstetter et al., 2007) and compare the resulting models (Moschetti, 2015) by calculating and evaluating the likelihood test. In this framework, the smoothed seismicity models are constructed by using both the historical Catalogue Parametrico dei Terremoti Italiani, CPTI15, (1000-2014) (Rovida et al. 2015) and the instrumental Italian catalog (1981-20165) (Gasperini et al., 2013) and their associated completeness levels to produce a space-time forecast of the future Italian seismicity.  We follow guidance from previous studies to optimize the smoothing seismicity parameters; the correlation smoothing distance (fixed smoothing) and the neighboring number (adaptive smoothing) by comparing model likelihood values, which estimate the likelihood that the observed earthquake epicenters from the recent catalog are derived from the smoothed rate models. We compare likelihood values from all rate models to rank the smoothing methods. We also compare two our models with the best two models of the several Italian Collaboratory for the Study of Earthquake Probability (CSEP) experiment models, to check their relative performances. Finally we create six an ensemble models combining using two different smoothing models (adaptive and fixed), which are weighted equally with different weights through a logic-tree approach to improve the forecast capability (Marzocchi et al., 2012; Taroni et al., 2014) and to estimate the uncertainty associated with the models.

Introduction

Probabilistic seismic hazard analysis (PSHA) quantifies the probability of ground shaking at a specified site that exceeds a specified intensity level (Cornell 1968, SSHAC 1997). It contains two basic ingredientsparts: (1) the earthquake rate and rupture models, for which specifyication of the statistical distribution of earthquakes in time, space and size (magnitude); and (2) the ground motion prediction equations, for which estimateion of the expected ground shaking level (and distribution) at a site for each earthquake rupture (Field et al. 2005). Recently developed eEarthquake rate models which are the inputs to PSHA, developed for PSHA, may include estimates of earthquake recurrence from historical, geological, and paleo-seismological observations, and inferencesas well as the fromestimates from the crustal deformations and seismicity rates (Stirling et al., 2002; Field et al., 2014; Petersen et al., 2014; Adams et al., 2015). In these models we mostly perform past earthquake rates estimating future rates from the spatial smoothing of earthquake locations from catalogs, especially in regions where we have little, limited or incomplete information about the active faults. Modern earthquake rupture forecasts commonly use past earthquake rates to estimate future rates. Particularly in regions where knowledge about active faults is limited and not complete, earthquake rate forecasts are mostly developed from the spatial smoothing of earthquake locations from catalogs (i.e., smoothed seismicity models). In fact the Ssmoothed seismicity models have been commonlybroadly performedapplied over the past two decades in developing earthquake rupture forecasts (Kagan and Jackson, 1994; Frankel, 1995; Wang et al., 2011; Akinci, 2010) and seismic-hazard models (Frankel et al., 1996; Petersen et al., 2014; Adams et al., 2015; Akinci et al., 2004; Akinci et al., 2009).

Early smoothed seismicity models employed spatially uniform smoothing parameters (i.e., fixed smoothing), such that the kernels used to smooth catalog-derived seismicity rates were invariant to spatial variations in seismicity rate. However, some recently developed methods (Stock and Smith, 2002; Helmstetter et al., 2007; Werner et al., 2011) spatially adapt the smoothing parameters to the earthquake rate (i.e., adaptive smoothing). Adaptive smoothing methods have the general effect of concentrating seismicity rates near the clusters of high activity and reducing them in areas of low-background seismicity compared to models derived from fixed smoothing.

In this study, we apply both fixed and adaptive smoothed seismicity methods to develop a long-term earthquake rate model for independent events. The smoothed seismicity is an alternative to the approach that was used previously for the hazard calculation in Italy, in which area source zones were drawn around seismic or tectonic provinces. Special zones allow for local variability in seismicity characteristics within a zone (for example, hypocentral depth changes, changes in b-value, changes in maximum magnitude Mmax, and uniform seismicity characteristics). These models are combined to account for the suite of potential earthquakes that can affect a site. However, one advantage of the smoothed-seismicity methods is to avoid choosing zone boundaries that are sometimes poorly controlled by data and drawn by subjectively merging geological and seismological informationprobability. The delimitation and parametric characterization of small zones can be responsible for introducing uncertainties into the hazard evaluation. In its purest form, the smoothed-seismicity method simply assumes that patterns of historical earthquakes predict future activity, but it can easily be supplemented by tectonic – or geodetic – based zones or other model elements if there is reason to suspect that seismicity catalogs are insufficient. Even though in the approach by Frankel (1995) no seismicity source zones are needed, some model parameters can be taken as homogeneous throughout regional sub-zones. For example in this study, the study region is subdivided into seven broad zones mainly on the basis of the different catalog characteristics/completeness of the region.

In this study the smoothed seismicity models are constructed based on the spatial distribution of both the historical (Catalogue Parametrico dei Terremoti Italiani, CPTI15) and the instrumental Italian catalogues and their associated completeness levels produced for the Mappa di Pericolosita’ Sismica (MPS16) to attain a space-time forecast of the future Italian seismicity. The fixed smoothing model follows the method of Frankel (1995) and uses spatially uniform smoothing parameters (i.e., fixed smoothing), such that the kernels used to smooth catalog-derived seismicity rates are invariant to spatial variations in seismicity rate. However, the adaptive smoothing method (Helmstetter et al., 2007) defines a unique smoothing distance for each earthquake epicenter from the distance to the n-th nearest neighbor. We examined the ability of both models to predict the spatial distribution of seismicity from the recent part of the earthquake catalog using standard likelihood calculations. We followed the si
milar likelihood testing methods that have been spurred by the Regional Earthquake Likelihood Models (RELM) (Schorlemmer et al., 2007) and the Collaboratory for the Study of Earthquake Predictability (CSEP) (Zechar, 2010) testing centers for evaluation and comparison of earthquake forecast models (Schorlemmer et al., 2007; Zechar, et al., 2010). The aim of this effort is to develop and provide models for the seismicity rates using the most recently practiced methodologies along with updated databases (seismic catalogs) to be performed towards to new probabilistic seismic hazard maps for Italy.

Earthquake Catalogs and Completeness

The seismicity based source models benefit the smoothed seismicity rates calculated on a spatial grid platform from the two catalogs:

1) The parametric catalog of Italian earthquakes (Catalogue Parametrico dei Terremoti Italiani, CPTI15) that contains larger earthquakes up to magnitude 7.3 since 1000 up to year of 2014;

2) The instrumental catalog of Italian seismicity that contains the small earthquakes down to Mw 1.0, with a maximum of Mw 6.53, over the past 365 years (1981-201675).

The shallow background seismicity rates are calculated on a spatial grid using the historical CPTI15 catalog (Rovida et al. 2015), consisting of 442718 records (M≥>4.0) in the time window from 1000 AD to 2014. Since PSHA assumes a Poissonian process of earthquakes, where seismic events are considered temporally independent, the CPTI15 catalog is declustered using Gardner and Knopoff (1974) method to remove large fluctuations of seismicity rates in space and time due to aftershock sequences, and we select total of 28971594 main shocks for the final computation with depth ≤30 kKm. The completeness periods of the CPTI15 catalog were identified by the Working Group 2015 (http;//emidius.mi.ingv.it/CPTI15-DBMI15/data/CPTI15_descrizione.pdf), based on both historical and the statistical analyses. The completeness magnitude threshold was defined over 157 magnitude bins, and represents the centers of each magnitude class with a width of 0.232, starting from a minimum magnitude of Mcw 3.734.0 (as the central value of the first magnitude class) to a maximum magnitude of Mcw 7.41410 for different periods of time using both the historical and statistical approach (Albarello et al., 2001). For example, the lower magnitude limit 4.650 in the complete catalog represents the center of the 4.5353.96 to 4.76518 class. Table 1a and b shows the completeness of the catalog in the five six zones in terms of the variability at the beginning of this completeness time over seventeentwelve magnitude ranges from the historical and the statistical approaches, respectively. Figure 1a shows these seven six zones, which are indicated by #zone numberdifferent colors for different historical completeness time intervals and magnitude thresholds, as given in Table 1a. These zones have been used also to estimate b values through the Gutenberg-Richter (GR) relation (Gutenberg and Richter, 1949), for the fixed smoothed seismicity model (see following section). Only earthquakes with focal depths less than 30 km are considered in this study.

The instrumental catalog 1981-201567 (actually it starts on 1/1/1981 and ends on 30/4/2017) is based on the Gasperini et al (2013) catalog, updated for the new seismic Italian hazard map. After declustering (method by Gardner and Knopoff 1974), from this catalog we select 2560412 shallow earthquakes (depth < 30 Km) inside the CSEP Italy zone, with magnitude M≥3.0 (Figure 1b2). Completeness magnitude Mcw 3.0 is the same suggested by the authors of the catalog (Gasperini et al, 2013).

Constructing Smoothed Seismicity Models

The gridded seismicity sources assume earthquake rate models with generating cumulative earthquake counts by assuming the magnitude-frequency distribution with the truncated form of the GR relation (Gutenberg and Richter, 1949),

Log10 N (MMc) = a – bMcM, (2)

where N(Mc) is the number of events with magnitude equal to or greater than McM, and the b-value is the slope of the distribution that describes the relative frequency of small and large earthquakes within the completeness range MminMcM≤Mmax. The Weichert maximum-likelihood method (Weichert, 1980) was used to obtain the 10a values, with the completeness magnitude thresholds over different periods of time as given by the MSP16 Working Group (2016) (see Table 1a and b), so that the rare large and the smaller earthquakes with short recording times were obtained more accurately. Based on the assumption that the seismicity rate is constant with time, the method included each earthquake that was counted as of equal weight in the rate calculations. Thus, the seismicity rates during time periods that have more countable earthquakes (for example; time periods with lower completeness thresholds) will be affected by the completeness period much more strongly than the seismicity rate at other times (Felzer 2008). Cumulative numbers of events are sampled on a 0.1-by-0.1 degree grid by counting the number of earthquakes with Mmin≥4.5Mc. , However which is the minimum magnitude threshold stabilized by in the MPS16 Project, is M4.5 in each cell of the grid across the Italian territory and surrounding region for the final earthquake rate model to be used in the probabilistic seismic hazard calculations. In order to calculate the probabilistic ground motions from our smoothed seismicity models, however, we convert from cumulative recurrence to incremental recurrence using established relations (Herrmann 1977) which also requires a b-value. Therefore, the resulting “agrid” gives the annual rate of earthquakes in each grid cell as an incremental 10a in the Gutenberg-Richter notation, between plus or minus 0.05 magnitude units or a 0.1 bin-width centered on M= 0 in each grid cell.

We use two different approaches to estimate the b-value for the two catalogs. For the historical catalog CPTI15 the b-value is assumed to be variable throughout the sixeven regions (Figure 1a,b2) and calculated using the Weichert (1980) formulation for for both catalog from different magnitude-time completeness interval (Table 2). We start our computation from the third bin of magnitude (M 4.535) because the completeness of the first two bin is not entirely reliable (as suggested by an internal report of the project).

To estimate the maximum magnitude for the historical catalog, we use a classical approach, where maximum possible magnitude is equal to the maximum observed magnitude plus a constant, that can be computed using statistical consideration, or using the standard error associated with the maximum observed magnitude (Kijko 2004). In our case, the maximum observed magnitude is M7.3 (Val di Noto earthquake 1693) and the associated standard error is 0.2; so we chose a maximum magnitude equal to 7.5. We use the same maximum magnitude for the whole Italian territory.

For the instrumental catalog, we use the tapered version of the Gutenberg-Richter law (Kagan and Jackson 2000). We chose to use a different distribution for this catalog to explore the epistemic uncertainty relative to the choice of the type of the Gutenberg-Richter law that must be used (truncated or tapered). For this distribution we need to estimate the b-value and also the corner magnitude (that is a “soft” version of the maximum magnitude): due to the difficulty of estimation of the corner magnitude with a catalog of few years, like the Italian instrumental one, we also use some information of the historical catalog. Then we merge this information to estimate the parameters (Kagan and Schoenberg 2001). In this estimation, the total rate is the observed number of events with magnitude M≥ 4.5. In Appendix A we give more detail of this computation.

The rate i in the each cell ith is calculated following equation;

(3)

where , , and are the smoothing kernel, smoothing distance
, Mc value and event rate respectively, for the jth earthquake from a total of N events, and rij is the distance between the jth event and ith cell.

We then smooth seismicity rates using an isotropic, 2-D Gaussian function as the smoothing kernel:

(4)

The kernel is normalized by the coefficient, C (σj) that is proportional to σj -2. In equation (4), σj is the smoothing distance associated with an earthquake j in the spatial domain as the horizontal distance between event j and its n-th closest neighbor. So that the epicentral distance as “neighbor distance” corresponds to a value N as the “neighbor number”. Adaptive kernels assume that σj vary as a function of the density at location of each earthquake. So that σj smoothing distance decreases when the density of the seismicity is high at the location ri of the earthquake j, providing higher resolution where the density of seismicity is higher. The adaptively smoothed seismicity models make use of spatially varying smoothing distances. In fact the fixed and adaptive smoothed seismicity models differ only in the application of the smoothing distance applied to each earthquake (Moschetti, 2015). Fixed smoothed seismicity models use a single smoothing distance for all earthquakes while the adaptive smoothed seismicity models use unique smoothing distances to each earthquake. Since the adaptive smoothed models vary smoothing distance with the local seismicity density, the resulting smoothing distances are relatively smaller in regions of high seismicity than regions with sparse seismicity. In this study we implemented a maximum smoothing distance of 200 km following Moschetti, (2015) where we have to impose a physical constraint on the smoothing in regions of sparse seismicity in parts of Italy.

Smoothing parameters are based on earthquake location of earthquakes uncertainties and spatial trends seen both in historical and instrumental seismicity. One problem with the smoothing method is apparent in Italy where seismicity that occurs in narrow linear zones is over smoothed into nearby aseismic regions and become important to select optimal and proper smoothing parameters to be used for the earthquake rate forecasting in the study region.

In this study we use both the fixed and adaptive smoothing seismicity parameters in the application of the smoothing distance. For the fixed smoothed seismicity we tested the set of smoothing distances (5,10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95,.. and 100 km) for producing optimized fixed smoothed models, using historical seismicity from the CPTI15 catalog for Mcw≥4.5 events with the completeness magnitude thresholds over different periods of time (see Table 1a and b), so that the rare large and the smaller earthquakes with short recording times were obtained more accurately. However we selected neighbor numbers 1-10 and computed intermediate neighbor distances within this neighbor-number range for identifying optimal adaptive smoothed models. Then we follow the methodology by implementing an adaptive smoothing scheme that provides some smoothing but generally keeps the modeled hazard seismicity rates closer to the original seismicity using respective correlation smoothing distances for directions parallel and normal to dominant seismicity trends.

To construct the adaptive seismicity models, we employ the method of Helmstetter et al. (2007), which determines the smoothing distance for each earthquake from the distance to neighboring earthquake epicenters. We refer to the smoothing models that result from the use of smoothing distances to the n-th nearest neighbor as N-neighbor-number models. We perform compare the use of the instrumental catalog, in which completeness magnitude is smaller Mw 3.0, compare to that of the historical catalog (Mw 4.5), which so that contains more events in the catalog (N=1430 M≥3.0 between 1981-2000 for the instrumental compare to N=675 M≥4.5 between 1895-2000 for the historical catalog) and may be better identifyies the active fault structures and the spatial distribution of the local seismicity that may be locations of the larger events especially in the case of the adaptive smoothing approach. In the following we give optimized smoothing parameters (correlation smoothing distance and the N-neighbor number) obtained through the L-test likelihood testing follows the general CSEP testing methodology (Werner et al., 2011). Finally, for the recurrence time of the seismic activity, we assumed a time-independent (Poisson) model in which the occurrence of an earthquake does not change the probability of occurrence for following events. For a Poisson process, the probability P of occurrence of one or more events, in a time period T of interest, is given by Equation (5):

(5)

where ni is the rate of earthquakes and is the inverse of the average recurrence time.

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