Integration of Site Effects Into Probabilistic Seismic Hazard Assessment (PSHA): a Comparison Between Two Fully Probabilistic Methods on the Euroseistest Site

geosciences

Article Integration of Site Effects into Probabilistic Seismic Hazard Assessment (PSHA): A Comparison between Two Fully Probabilistic Methods on the Euroseistest Site

Claudia Aristizábal *, Pierre-Yves Bard ID ,Céline Beauval and Juan Camilo Gómez Université Grenoble Alpes, Université Savoie Mont Blanc, CNRS, IRD, IFSTTAR, ISTerre, 38000 Grenoble, France; [email protected] (P.-Y.B.); [email protected] (C.B.); [email protected] (J.-C.G.) * Correspondence: [email protected] or [email protected]

Received: 16 April 2018; Accepted: 26 July 2018; Published: 30 July 2018

Abstract: The integration of site effects into Probabilistic Seismic Hazard Assessment (PSHA) is still an open issue within the seismic hazard community. Several approaches have been proposed varying from deterministic to fully probabilistic, through hybrid (probabilistic-deterministic) approaches. The present study compares the hazard curves that have been obtained for a thick, soft non-linear site with two different fully probabilistic, site-specific seismic hazard methods: (1) The analytical approximation of the full convolution method (AM) proposed by Bazzurro and Cornell 2004a,b and (2) what we call the Full Probabilistic Stochastic Method (SM). The AM computes the site-specific s hazard curve on soil, HC(Sa( f )), by convolving for each oscillator frequency the bedrock hazard r curve, HC(Sa( f )), with a simplified representation of the probability distribution of the amplification function, AF( f ), at the considered site The SM hazard curve is built from stochastic time histories on soil or rock corresponding to a representative, long enough synthetic catalog of seismic events. This comparison is performed for the example case of the Euroseistest site near Thessaloniki (Greece). For this purpose, we generate a long synthetic earthquake catalog, we calculate synthetic time histories on rock with the stochastic point source approach, and then scale them using an adhoc frequency-dependent correction factor to fit the specific rock target hazard. We then propagate the rock stochastic time histories, from depth to surface using two different one-dimensional (1D) numerical site response analyses, while using an equivalent-linear (EL) and a non-linear (NL) code to account for code-to-code variability. Lastly, we compute the probability distribution of the non-linear site amplification function, AF( f ), for both site response analyses, and derive the site-specific hazard curve with both AM and SM methods, to account for method-to-method variability. The code-to-code variability (EL and NL) is found to be significant, providing a much larger contribution to the uncertainty in hazard estimates, than the method-to-method variability: AM and SM results are found comparable whenever simultaneously applicable. However, the AM method is also shown to exhibit severe limitations in the case of strong non-linearity, leading to ground motion “saturation”, so that finally the SM method is to be preferred, despite its much higher computational price. Finally, we encourage the use of ground-motion simulations to integrate site effects into PSHA, since models with different levels of complexity can be included (e.g., point source, extended source, 1D, two-dimensional (2D), and three-dimensional (3D) site response analysis, kappa effect, hard rock ... ), and the corresponding variability of the site response can be quantified.

Keywords: site response; epistemic uncertainty; PSHA; non-linear behavior; host-to-target adjustment; stochastic simulation

Geosciences 2018, 8, 285; doi:10.3390/geosciences8080285 www.mdpi.com/journal/geosciences Geosciences 2018, 8, 285 2 of 28

1. Introduction The integration of site effects into Probabilistic Seismic Hazard Assessment (PSHA) is a constant subject of discussion within the seismic hazard community due to its high impact on hazard estimates. Preliminary versions on how to integrate them have been presented in [1–3]. Furthermore, [4–9] have also published similar studies on this subject. A relevant overview with enlightening examples can be found, for instance in [10,11] and [12,13]. However, research on this topic is still needed since the integration of site effects is still treated in a rather crude way in most engineering studies, as the variability associated to the non-linear behavior of soft soils and its ground-motion frequency dependence is generally neglected. For instance, the various approaches that are discussed in [14,15] correspond to hybrid (deterministic–probabilistic) approaches [16], where the (probabilistic) uniform hazard spectrum on standard rock at a given site is simply multiplied by the linear or nonlinear median site-specific amplification function. The main drawback of those different hybrid-based approaches (currently the most common way to include site effects into PSHA) is that once the hazard curve on rock is multiplied with the local non-linear frequency-dependent amplification function, the exceedance probability at the soil surface is no longer the one that is defined initially for the bedrock hazard. At a given frequency, the same soil surface ground motion can be reached with different reference bedrock motion (corresponding to different return periods and/or different earthquake contributions) with different non-linear site responses. Ref. [10,11] concluded from their studies at two different sites (clay and sand) that the hybrid-based method tends to be non-conservative at all frequencies and at all mean return periods with respect to their approximation of the fully probabilistic method (AM) that is considered here. The present research work constitutes a further step along the same direction, trying to overcome the limitation of hybrid-based approaches for strongly non-linear sites by providing a fully probabilistic description of the site-specific hazard curve. The aim of this work is to present a comparison exercise between hazard curves that are obtained with two different, fully probabilistic site-specific seismic hazard approaches: (1) The analytical approximation of the full convolution method (AM) proposed by [10,11] and (2) what we call the Full Probabilistic Stochastic Method (SM). The AM computes the site-specific hazard on soil by convolving the hazard curve at the bedrock level, with a simplified representation of the probability distribution of the site-specific amplification function. The SM hazard curve is directly derived from a set of synthetic time histories at the site surface, combining point-source stochastic time histories for bedrock that is inferred from a long enough virtual earthquake catalog, and non-linear site response. This comparative exercise has been implemented for the Euroseistest site, a multidisciplinary European experimental site for integrated studies in earthquake engineering, engineering seismology, seismology, and soil dynamics [17]. To perform PSHA calculations, the area source model developed in the SHARE project is used [18].

2. Study Area: The Euroseistest Site As stated in [18], the Euroseistest site is a multidisciplinary experimental site for integrated studies in earthquake engineering, engineering seismology, seismology, and soil dynamics. It has been established thanks to European funding in Mygdonia valley, the epicentral area of the 1978, M6.4 earthquake, located about 30 km to the NE of the city of Thessaloniki in northern Greece, Figure1a [ 19–21]. The ground motion instrumentation consists of a three-dimensional (3D) strong motion array, which has been operational for slightly more than two decades. This site was selected as an appropriate site to perform this exercise, because of the availability of extensive geological, geotechnical, and seismological surveys. The velocity model of the Euroseistest basin has been published by several authors [22–25], varying from one-dimensional (1D) up to two-dimensional (2D) and 3D models (Figure1b). These models were used to deﬁne the 1D soil column for the present exercise (Figure2a and Table1). Degradation curves are also available to characterize each soil layer, (Figure2b,c) [ 26], as well as local earthquake recordings to calibrate the model in the linear (weak motion) domain. For the present exercise, in the absence of reliable Geosciences 2018, 8, x FOR PEER REVIEW 3 of 28

(1D) up to two-dimensional (2D) and 3D models (Figure 1b). These models were used to define the 1DGeosciences soil column2018, 8, for 285 the present exercise (Figure 2a and Table 1). Degradation curves are also available3 of 28 to characterize each soil layer, (Figure 2b,c) [26], as well as local earthquake recordings to calibrate the model in the linear (weak motion) domain. For the present exercise, in the absence of reliable measurements along the whole proﬁle, the cohesion was assumed to be zero in all layers (C = 0 measurements along the whole profile, the cohesion was assumed to be zero in all layers (C = 0 kPa). kPa). As non-zero cohesion is generally associated with non-zero plasticity index, and a shift of As non-zero cohesion is generally associated with non-zero plasticity index, and a shift of non-linear non-linear degradation curves towards higher strains, this zero-cohesion assumption is likely to result degradation curves towards higher strains, this zero-cohesion assumption is likely to result in a in a stronger impact of the NL site response, so that the present case is to be considered as an example stronger impact of the NL site response, so that the present case is to be considered as an example case for a highly non-linear, deep soft site. case for a highly non-linear, deep soft site. Several studies performed at the Euroseistest, both instrumental and numerical, have consistently Several studies performed at the Euroseistest, both instrumental and numerical, have shown a fundamental frequency (f 0) around 0.6–0.7 Hz [23,27–30] with an average shear wave velocity consistently shown a fundamental frequency (f0) around 0.6–0.7 Hz [23,27,28,29,30] with an average over the top 30 m (VS30) of 186 m/s. The 1D simulations that are performed in this study are based shear wave velocity over the top 30 m (VS30) of 186 m/s. The 1D simulations that are performed in this on the parameters listed in Table1, which have been shown to be consistent with the observed study are based on the parameters listed in Table 1, which have been shown to be consistent with the instrumental ampliﬁcation functions, AF( f ), at least in the linear domain [23,27–30]. observed instrumental amplification functions, (), at least in the linear domain [23,27,28,29,30].

(a) (b)

Figure 1. (a) Map showing the broader region of occurrence of the 20 June 1978 (M6.5) earthquake Figure 1. (a) Map showing the broader region of occurrence of the 20 June 1978 (M6.5) earthquake sequence. Epicenter and focal mechanism [19] of the mainshock (red) and epicenters of the largest sequence. Epicenter and focal mechanism [19] of the mainshock (red) and epicenters of the events of the sequence (green, yellow) [20] are also depicted. Dotted lines mark the Mygdonian graben largest events of the sequence (green, yellow) [20] are also depicted. Dotted lines mark the andMygdonian surface ruptures graben after and surfacethe 1978 rupturesearthquake after [21] the are 1978shown earthquake as red lines. [21 (b]) areThree-dimensional shown as red lines.(3D) model(b) Three-dimensional of the Mygdonian (3D) model of thebasin Mygdonian geologic basinal geologicalstructure structure[25]. [25(Figures]. (Figures from from http://euroseisdb.civihttp://euroseisdb.civil.auth.gr/geotecl.auth.gr/geotec). ).

Geosciences 2018, 8, 285 4 of 28 Geosciences 2018, 8, x FOR PEER REVIEW 4 of 28

(a)

(b) (c)

Figure 2. (a) One-dimensional (1D) shear wave, Vs, and compressive wave, Vp, velocity profiles Figure 2. (a) One-dimensional (1D) shear wave, Vs, and compressive wave, Vp, velocity proﬁles between TST0 and TST196 stations, located at the middle of the Euroseistest basin, at surface and 196 between TST0 and TST196 stations, located at the middle of the Euroseistest basin, at surface and 196 m depth, respectively. (b) Euroseistest shear modulus and (c) damping ratio degradation curves [27]. m depth, respectively. (b) Euroseistest shear modulus and (c) damping ratio degradation curves [27].

Table 1. Material properties of the Euroseistest soil profile used on this study to perform the different Table 1. Material properties of the Euroseistest soil proﬁle used on this study to perform the different site response calculations. site response calculations. Depth Vs Vp ρ Ko Layer Soil Type Q ϕ (°) Depth Vs Vp ρ 3 Ko Layer Soil Type (m) (m/s) (m/s) (kg/m )Q φ (◦) (adm) (m) (m/s) (m/s) (kg/m3) (adm) 1 Silty Clay Sand 0.0 144 1524 2077 14.4 47 0.26 2 1 SiltySilty Clay Sand* Sand 0.05.5 144177 15241583 20772083 14.417.7 4719 0.26 0.67 2 Silty Sand* 5.5 177 1583 2083 17.7 19 0.67 3 Lean Clay 17.6 264 1741 2097 26.4 19 0.68 3 Lean Clay 17.6 264 1741 2097 26.4 19 0.68 4 4Fat Fat Clay Clay 54.254.2 388388 19521952 21172117 38.838.8 2727 0.54 0.54 5 5Silty Silty Clay 81.281.2 526526 22002200 21512151 52.652.6 4242 0.33 0.33 6 6Weathered Weathered RockRock 131.1131.1 701701 25202520 22152215 70.170.1 69 69 0.07 0.07 7 7Hard Hard Rock 183.0183.0 26002600 45004500 24462446 ------ρ ** Water Water table table at 1at m 1 depth. m depth. Vs: shearVs: shear wave wave velocity. velocity. Vp: Compressive Vp: Compressive wave velocity. wave :velocity. soil density. ρ: soil Q: Anelastic density. attenuation factor. φ: Friction angle. Ko: Coefﬁcient of earth at rest. Q: Anelastic attenuation factor. ϕ: Friction angle. Ko: Coefficient of earth at rest.

Geosciences 2018, 8, 285 5 of 28

3. Methods The methods that are followed in this work correspond to two different, fully probabilistic procedures to account for highly non-linear soil response in PSHA, a Full Probabilistic Stochastic Method (SM) that is developed for the present work, and the analytical approximation of the full convolution method (AM), as proposed by [10,11]. For both approaches, the first step consists in deriving the hazard curve for the specific bedrock properties of the considered site. As shown in Figure2a and Table1, this bedrock is characterized by a high S-wave velocity (2600 m/s), and is therefore much harder than “standard rock” conditions corresponding to VS30 = 800 m/s. Hence, host-to-target adjustments are required [31–34] to adjust the ground motion estimates that are provided by Ground Motion Prediction Equations (GMPEs) for the larger shear wave velocity and also possibly for the different high-frequency attenuation of the reference rock (such adjustments are commonly known as Vs-kappa effect [35]). For the sake of simplicity in the present exercise, the hazard curve has been derived with only one GMPE (Akkar et al., 2014) (AA14) [36], which is satisfactorily representative of the median hazard curve on rock obtained with seven other GMPEs deemed to be relevant for the European area (see [15]). The host-to-target (HTT) adjustments have been performed here on the basis of simple velocity adjustments calibrated on KiKnet data using the Laurendeau et al., 2107 GMPE [37], since the detailed knowledge of the site characteristics does not compensate the very poor information on the “host” characteristics of any GMPE (in particular those from European data), which results in very large uncertainty levels in classical HTT approaches [38]. For the SM approach, the next step consists in generating a synthetic earthquake catalog by sampling the earthquake recurrence model. The area source model that was developed in SHARE [18] is sampled with the Monte Carlo method to generate the earthquake catalog (OpenQuake engine, Stochastic Event Set Calculator tool [39]). The Boore (2003) Stochastic Method [40,41] is then applied to generate synthetic time histories at the bedrock site corresponding to all earthquakes in the catalog. Some specific adjustments and corrections were required to obtain ground motions compatible with the ground-motion model (AA14) used in the classical PSHA calculation. This step, which is not trivial, is described in detail in the hard rock section. The hard rock, correctly scaled time histories are then propagated to the site surface using two different 1D non-linear site response codes. One set of time histories on soil is generated that is based on the SHAKE91 equivalent-linear (EL) code [42–44], while the second set was derived using the NOAH [45] fully non-linear code (NL). Both of the codes were used with the objective of incorporating somehow the code-to-code variability, which has been shown in all recent benchmarking exercises to be very important [46,47]. The final step in the SM method consists in deriving the site-specific hazard curve by simply calculating the annual rate of exceedance from the set of surface synthetic time series. The AM approach, as proposed by [11], requires, for each frequency, f, a statistical description r of the response spectrum amplification function AF( f , Sa) as a function of input ground motion on r rock, Sa. This is done, for each used site response code, with a piecewise linear function describing the r dependence of the median AF( f ) with loading level Sa( f ), together with an appropriate lognormal r distribution, accounting for its variability as a function of input ground motion on rock, Sa. The large number of synthetic time histories at site surface were used to derive the piecewise linear function of r the AM (AF( f ) vs. Sa( f )). Finally, the bedrock hazard curve (including host-to-target adjustments) is then convolved analytically with the simplified amplification function following [11], to obtain an estimate of the site surface hazard curve, to be compared with the SM estimate.

4. Results for Hard Rock Hazard The site-speciﬁc soil hazard estimates at the Euroseistest using the two different fully probabilistic methods are presented below, with step-by-step explanations of the hypotheses that are considered and the obtained results. Geosciences 2018, 8, 285 6 of 28

4.1. RockGeosciences Target Hazard 2018, 8, x FOR PEER REVIEW 6 of 28

The4.1. first Rock step Target consists Hazard in defining the target hazard on a standard rock at the Euroseistest, to be able to generate aThe set first of consistent step consists synthetic in defining time the target histories. hazard Following on a standard the rock methodology at the Euroseistest, that is to described be in [14,15],able the to hazard generate curve a set calculated of consistent using synthetic the AA14 time histories. GMPE was Following selected the as methodology the target hazardthat is curve for EC8 standarddescribed in rock [14,15], (VS30 the= hazard 800 m/s) curve for calculated this exercise. using the Figure AA143 GMPE displays was theselected AA14 as hazardthe target curves at threehazard different curve spectral for EC8 periods standard (PGA, rock (V 0.2S30 =s 800 and m/s) 1.0 for s, this respectively) exercise. Figure using 3 displays 5% spectral the AA14 damping hazard curves at three different spectral periods (PGA, 0.2 s and 1.0 s, respectively) using 5% spectral ratio (ξ = 5%). The hazard is calculated while using the area source model that was developed in the damping ratio (ξ = 5%). The hazard is calculated while using the area source model that was Europeandeveloped SHARE in project the European [18] (xml SHARE input project files available[18] (xml input on the files efher.org available website)on the efher.org and the website) Openquake Engine [39and,48 the]. Openquake Engine [39,48].

1.0000 1.0000 1.0000 (PGA) (0.2s) (1.0s) AA14 (800 m/s) 0.1000 0.1000 0.1000

0.0100 0.0100 0.0100

Tr=475 Tr=475 Tr=475 0.0010 0.0010 0.0010

Tr=4975 Tr=4975 Tr=4975 Annual Rate of Exceedance Annual Rate of Exceedance Annual Rate of Exceedance 0.0001 0.0001 0.0001 0.01 0.10 1.00 0.01 0.10 1.00 0.01 0.10 1.00 Acceleration (g) Acceleration (g) Acceleration (g)

Figure 3.FigureTarget 3. hazardTarget hazard curve curve on standard on standard rock rock (V (VS30S30= = 800800 m/s) atat the the Euroseistest Euroseistest calculated calculated using using Akkar etAkkar al., 2014 et al., Ground 2014 Ground Motion Motion Prediction Prediction Equations Equations (GMPE) (AA14) (AA14) for forthree three different different spectral spectral periods (PGA, 0.2 s, 1.0 s). periods (PGA, 0.2 s, 1.0 s). 4.2. Host-to-Target Adjustments 4.2. Host-to-Target Adjustments As presented in more detail in [21,22], the bedrock beneath the Euroseistest basin (~196 m depth) As presentedhas a shear inwave more velocity detail of in about [21, 222600], them/s, bedrock which is beneathmuch larger the than Euroseistest the validity basin range (~196 of most m depth) has a shearGMPEs. wave It velocityis therefore of aboutmandatory 2600 to m/s, perform which wh isat much is known larger in thanliterature the validityas “host-to-target range of most adjustment” corrections (HTT [31,32,33,34]). As discussed in [14,15], the high level of complexity and GMPEs. It is therefore mandatory to perform what is known in literature as “host-to-target adjustment” uncertainty associated to the current HTT adjustments procedures [16,37,49,50,51], have motivated correctionsus to (HTT use here [31 another,–34]). Assimpler, discussed straightforward in [14,15 wa],y theto account high levelfor rock of to complexity hard rock correction and uncertainty for associatedprobabilistic to the current seismic HTT hazard adjustments purposes. procedures [16,37,49–51], have motivated us to use here another, simpler,This simpler straightforward approach is waybased to on account the work for by rock Laurendeau to hard rocket al., correction2017 (LA17) for [37], probabilistic who seismic hazardproposed purposes. a new rock GMPE that is valid for surface velocities ranging from 500 m/s to 3000 m/s and Thisderived simpler on approach the basis of is the based Japanese on the KiK-net work datase by Laurendeaut [52]. This GMPE et al., presents 2017 (LA17) the advantage [37], who to rely proposed only on the value of rock velocity, without requiring the κ0 values either for the host region or for the a new rocktarget GMPE site, and that therefore is valid avoiding for surface the associated velocities uncertainties. ranging from The adjustments 500 m/s to are 3000 thus m/s performed and derived on the basison the of basis the Japaneseon one single KiK-net proxy for dataset rock, as [52 it ].is Thisusually GMPE done for presents all soil deposits. the advantage It may account to rely for only on the valuethe of variability rock velocity, of some without other mechanical requiring characteristics the κ0 values (such either as κ0, foror the the fundamental host region frequency or for the f0), target site, andbut therefore only through avoiding their potential the associated correlations uncertainties. with VS30. We The consider, adjustments as discussed are in thus [37,38], performed that such on the basis onan one approach single proxyis as reliable for rock, as the as conventional it is usually HTT done approach for all soildue to deposits. the huge Ituncertainties may account and for the possible bias linked with the measurement of the physical parameter κ0. It provides rather robust variability of some other mechanical characteristics (such as κ , or the fundamental frequency f ), but predictions of rock motion whatever the approach used for their0 derivation: deconvolution of surface 0 only throughrecordings their to potential outcropping correlations bedrock with with the VS30 known. We consider,1D profile, as or discussed the correction in [ 37of, 38down-hole], that such an approachrecordings, is as reliable see [37] as for the the conventional corresponding HTT discussion. approach due to the huge uncertainties and possible bias linked withThe LA17 the measurement GMPE is based of on the a combined physical dataset parameter includingκ0. Itsurface provides recordings rather from robust stiff predictions sites of rock motionand hard whatever rock motion the estimated approach through used deconvolut for their derivation:ion of the same deconvolution recordings by the of corresponding surface recordings to outcropping1D linear bedrock site response. with theThe knownsite term 1D in the proﬁle, propos ored the (simple) correction GMPE of equation down-hole is given recordings, by a simple see [37] dependence on rock S-wave velocity value VS30 in the form (). ( /1000), where is tuned for the corresponding discussion. from actual recordings for each spectral oscillator period (). The HTT adjustment factor can thus be Thecomputed, LA17 GMPE as follows is based in Equation on a combined(1): dataset including surface recordings from stiff sites and hard rock motion estimated through deconvolution of the same recordings by the corresponding 1D linear site response. The site term in the proposed (simple) GMPE equation is given by a simple dependence on rock S-wave velocity value VS30 in the form S1 (T). ln(VS30/1000), where S1 is tuned from actual recordings for each spectral oscillator period (T). The HTT adjustment factor can thus be computed, as follows in Equation (1): Geosciences 2018, 8, 285 7 of 28

Hard Rock HC (Vs = 2600 m/s) HTT Adjutment Factors(T) = = S (T).ln (2600/800) (1) Geosciences 2018, 8, x FOR PEER REVIEWStandard Rock HC (Vs = 800 m/s) 1 7 of 28 Since the site terms in LA17, for such hard and standard rock, do not include any non-linearity, ( = 2600 /) () = =(). (2600/800) (1) the resulting HTT adjustment factors are ground-motion( = 800 / independent,) and thus return period (Tr) independent. The correction procedure consists simply in computing the hazard curves for Since the site terms in LA17, for such hard and standard rock, do not include any non-linearity, standardthe rock resulting (VS30 HTT= adjustment 800 m/s) factors using are AA14 ground-motion GMPE, and independent, then correcting and thus itreturn with period the so(Tr) defined, frequency-dependentindependent. The LA17correction HTT procedure factor. Theconsists values simply listed in computing in Table the2 indicate hazard curves that hardfor standard rock motion is systematicallyrock (VS30 = smaller800 m/s) thanusingthe AA14 standard GMPE, and rock then motion correcting at all it ofwith the the spectral so defined, periods, frequency- with larger reductiondependent factors atLA17 short HTT periods. factor. OfThe course, values these listed corrections in Table 2 are,indicate in principle, that hard only rock valid motion for is Japanese rock sites,systematically but on the smaller other handthan mostthe standard of HTT rock methods motion are at implicitlyall of the spectral using, inperiods, practice, with typical larger United reduction factors at short periods. Of course, these corrections are, in principle, only valid for States-California (US-California) velocity rock profiles at least for the host side. There is no obvious Japanese rock sites, but on the other hand most of HTT methods are implicitly using, in practice, reason whytypical such United US States-California profiles would (US-California) be more acceptable velocity rock for Europeprofiles at than least Japanese for the host results. side. There Ultimately, the hard-rockis no obvious hazard reason curve why is thensuch derivedUS profiles by would simply be scalingmore acceptable the AA14 for hazard Europe curvethan Japanese with the HTT adjustmentsresults. factors Ultimately, for each the hard-rock period (Figure hazard 4curve). is then derived by simply scaling the AA14 hazard curve with the HTT adjustments factors for each period (Figure 4). Table 2. Host-to-target adjustments factors derived using LA17 hazard curves from a standard rock Table 2. Host-to-target adjustments factors derived using LA17 hazard curves from a standard rock (VS30 = 800 m/s) to the Euroseistest hard rock (VS30 = 2600 m/s). (VS30 = 800 m/s) to the Euroseistest hard rock (VS30 = 2600 m/s). Spectral Period (s) PGA 0.05 0.1 0.2 0.5 1.0 2.0 Spectral Period (s) PGA 0.05 0.1 0.2 0.5 1.0 2.0 HTT AdjustmentHTT Adjustment Factors 0.47Factors 0.45 0.47 0.45 0.38 0.38 0.510.51 0.70 0.70 0.78 0.780.86 0.86

1.0000 1.0000 1.0000 (PGA) (0.2s) (1.0s) AA14 (800 m/s) 0.1000 0.1000 0.1000 AA14 (2600 m/s)

0.0100 0.0100 0.0100

Tr=475 Tr=475 Tr=475 0.0010 0.0010 0.0010

Tr=4975 Tr=4975 Tr=4975 Annual Rate of Exceedance of Rate Annual Exceedance of Rate Annual Exceedance of Rate Annual 0.0001 0.0001 0.0001 0.01 0.10 1.00 0.01 0.10 1.00 0.01 0.10 1.00 Acceleration (g) Acceleration (g) Acceleration (g) Figure 4.FigureHazard 4. Hazard curves curves at Euroseistest, at Euroseistest for, forthree three different spectral spectral periods periods (PGA, (PGA, 0.2 s, 1.0 0.2 s); s, blue: 1.0s); blue: standard rock (VS30 = 800 m/s); green: hard rock (2600 m/s), calculated from the hazard curves on standard rock (VS30 = 800 m/s); green: hard rock (2600 m/s), calculated from the hazard curves on standard rock scaled with LA17 host-to-target (HTT) adjustment factors. standard rock scaled with LA17 host-to-target (HTT) adjustment factors. One might wonder why scaling AA14 curve instead of using directly LA17 in the calculations. OneThe might main reason wonder was why that AA14 scaling (VS30 AA14 = 800 m/s) curve provides instead a good of using approximation directly to LA17 the median in the hazard calculations. estimate at the Euroseistest on standard rock from seven explored GMPES [15], while the estimates The main reason was that AA14 (VS30 = 800 m/s) provides a good approximation to the median hazard with LA17 (VS30 = 800 m/s) is located outside the uncertainty range of the same selected representative estimate at the Euroseistest on standard rock from seven explored GMPES [15], while the estimates GMPEs. This is probably due to the very simple functional form that is used in LA17, and to the fact with LA17that(V it S30was= elaborated 800 m/s) isonly located from outsideJapanesethe data. uncertainty Its main interest range ofhowever the same is to selected provide representativea rock GMPEs.adjustment This is probably factor calibrated due toon theactual very data simple from a large functional number formof rock that and ishard used rock in sites, LA17, without and to the fact thatany it wasassumptions elaborated regarding only fromnon-me Japaneseasured parameters, data. Its mainsuch as interest κ0. Once however the approach is to provide that is a rock adjustmentproposed factor here calibrated and initially on based actual in dataLA17 fromis tested a large on other number data sets, of rockthe corresponding and hard rock results sites, can without be used to provide alternative scaling factors between hard rock and standard rock. Note, however, any assumptions regarding non-measured parameters, such as κ0. Once the approach that is proposed that this modulation approach through another kind of GMPE is exactly similar to what has been here and initially based in LA17 is tested on other data sets, the corresponding results can be used proposed in [16], where stochastic simulations are used for the VS-κ corrections without being used to provideas GMPEs. alternative scaling factors between hard rock and standard rock. Note, however, that this modulation approach through another kind of GMPE is exactly similar to what has been proposed 4.3. Synthetic Earthquake Catalog in [16], where stochastic simulations are used for the VS-κ corrections without being used as GMPEs. Once defined the target hazard on hard rock, we proceed to generate a long enough seismic 4.3. Syntheticcatalog Earthquake to build synthetic Catalog hazard curves both on rock and soil at the selected Euroseistest site. The Once deﬁned the target hazard on hard rock, we proceed to generate a long enough seismic catalog to build synthetic hazard curves both on rock and soil at the selected Euroseistest site. The earthquake catalog is obtained by sampling the area source model of SHARE. To handle a reasonable Geosciences 2018, 8, 285 8 of 28 amount of events, only the sources that are contributing to the hazard must be sampled. To identify which areaGeosciences sources 2018, 8, contribute x FOR PEER REVIEW to the hazard at the Euroseistest site, the individual contributions8 of 28 from the crustal sources in the neighborhood of the site were calculated. The individual hazard curves correspondingearthquake to catalog each is source obtained were by sampling compared the area to the source total model hazard of SHARE. curve To using handle all a sources,reasonable showing that theamount area source of events, enclosing only the sources the Euroseistest that are contributing site (GRAS390) to the hazard fully must controls be sampled. the hazard To identify at all of the which area sources contribute to the hazard at the Euroseistest site, the individual contributions from considered periods, so that we can neglect the other sources in our calculations. the crustal sources in the neighborhood of the site were calculated. The individual hazard curves Thecorresponding stochastic to earthquake each source were catalog compared was to generated the total hazard by samplingcurve using theall sources, magnitude-frequency showing distributionthat the (truncated area source enclosing Gutenberg-Richter the Euroseistest model) site (GRAS390) of the area fully sourcecontrols zone,the hazard and atearthquakes all of the are distributedconsidered homogeneously periods, so that within we can theneglect area the source other sources zone. in For our thiscalculations. purpose, we use the Stochastic Event Set CalculatorThe stochastic tool earthquake from the Openquakecatalog was gene Enginerated [ 39by,48 sampling]. The generated the magnitude-frequency catalog must be long enoughdistribution to estimate (truncated correctly Gutenberg-Richter the hazard for themodel) return of the period area source of interest, zone, 5000and earthquakes years (equivalent are to a distributed homogeneously within the area source zone. For this purpose, we use the Stochastic Event probability of exceedance of 1.0% in 50 years). The length of the catalog is obviously a key issue when Set Calculator tool from the Openquake Engine [39,48]. The generated catalog must be long enough calculatingto estimate seismic correctly hazard the with hazard the Montefor the Carloreturn technique. period of interest, 5000 years (equivalent to a Toprobability identify which of exceedance minimum of 1.0% length in 50 isyears). required, The length a sensitivity of the catalog analysis is obviously is led consisting a key issue inwhen comparing the differentcalculating magnitude-frequency seismic hazard with the distribution Monte Carlo technique. curves and hazard curves that were obtained with different catalogTo identify lengths. which Figure minimum5 displays length theis required, magnitude-frequency a sensitivity analysis distribution is led consisting curves built in using increasingcomparing catalog the lengths different (500, magnitude-frequency 5000, 25,000, 35,500, distribution 50,000 curves years), and when hazard compared curves that to were the original obtained with different catalog lengths. Figure 5 displays the magnitude-frequency distribution SHAREcurves model built (black using solid increasing line). catalog For short lengths catalog (500, 5000, lengths 25,000, (here 35,500, 500 50,000 years), years), the recurrence when compared model is not correctlyto the sampled original in SHARE the upper model magnitude (black solid range. line). TheFor short catalog catalog length lengths is too (here short 500 with years), respect the to the recurrencerecurrence times model of large is not events. correctly When sampled considering in the upper longer magnitude catalog range. lengths The catalog (from length 5000 yearsis too on), the full magnitudeshort with range respect is to sampled. the recurrence However, times theof large correct events. annual When rates considering for the longer upper catalog magnitude lengths range are obtained(from with 5000 a reasonableyears on), the error full magnitude (~5%) only range for is very sampled. long However, earthquake the correct catalog annual lengths rates (larger for the or equal upper magnitude range are obtained with a reasonable error (~5%) only for very long earthquake to 50,000 years). Most important is that the magnitude range contributing to the hazard at the return catalog lengths (larger or equal to 50,000 years). Most important is that the magnitude range periodscontributing of interest to are the correctly hazard at sampled.the return periods of interest are correctly sampled.

(a) (b) (c)

(d) (e) (f)

FigureFigure 5. (a– e5.) ( Truncateda–e) Truncated Gutenberg-Richter Gutenberg-Richter recurrence models models derived derived from fromfive different ﬁve different stochastic stochastic catalogcatalog lengths lengths (500, (500, 5000, 5000, 25,000, 25,000, 35,500, 35,500, andand 50, 50,000000 years), years), compared compared with withthe original the original recurrence recurrence model of the source zone GRAS390 in the SHARE model. (f) The 5 recurrence curves obtained are superimposed to the original recurrence model. Geosciences 2018, 8, x FOR PEER REVIEW 9 of 28

model of the source zone GRAS390 in the SHARE model. (f) The 5 recurrence curves obtained are Geosciences 2018, 8, 285 9 of 28 superimposed to the original recurrence model.

FigureFigure 66aa displaydisplay thethe stochasticstochastic hazardshazards curvescurves onon rockrock obtainedobtained fromfrom thethe ﬁvefive catalogscatalogs withwith increasingincreasing lengths, lengths, as as well well as as the the hazard hazard curve curve calc calculatedulated with with the the classical classical PSHA PSHA calculation calculation (using (using AA14AA14 ground-motion ground-motion model). model). Figure Figure 66bb illustrates the ratio between the PGAs relying on the Monte CarloCarlo method method and and the the PGAs PGAs relying relying on on the the classical classical approach, approach, interpolated interpolated for for different different annual annual rates. rates. BasedBased on on these these results, results, we we decide decide to to use use a a 50,000 50,000 years years earthquake earthquake catalog catalog to to estimate estimate the the hazard, hazard, −4 sincesince the the difference difference between PGAs is lower than 5% for allall ofof thethe annualannual ratesrates downdown toto 1010−4 (or(or return return periodsperiods up up to to 10,000 10,000 years). years). The The final ﬁnal selected selected catalo catalogg of of 5000 5000 years years consists consists of of ~21,800 ~21,800 events events ranging ranging from the minimum considered magnitude Mw = 4.5 to the maximum magnitude Mw = 7.8, with a from the minimum considered magnitude Mw = 4.5 to the maximum magnitude Mw = 7.8, with a distancedistance range range (Joyner (Joyner and and Boore) Boore) from from 0 to 150 km and a depth range from 0 to 30 km.

1.0000 1.1 Classical 500 yrs 5,000 yrs 1.0 0.1000 25,000 yrs 37,500 yrs 50,000 yrs 0.9

0.0100 0.8

Tr=475 0.7 500 yrs 0.0010 5,000 yrs

Annual Rate of Exceedance of Rate Annual 0.6 25,000 yrs Tr=4975 37,500 yrs PGA (Stochastic)PGA(Classical) / 50,000 yrs 0.0001 0.5 0.01 0.10 1.00 1.0000 0.1000 0.0100 0.0010 0.0001 Acceleration (g) Annual Rate of Exceedance (a) (b)

FigureFigure 6. 6. (a(a) )Hazard Hazard curves curves built built from from five ﬁve earthquake earthquake ca catalogstalogs with with increasing increasing lengths lengths (from (from 500 500 to 50,000to 50,000 years). years). (b) (bRatio) Ratio between between the the acceleration acceleration obtained obtained from from the the Monte Monte Carlo Carlo method method and and the the expectedexpected acceleration, acceleration, interpolated interpolated for for given given annual annual rates. rates.

4.4. Synthetic Time Histories on Rock 4.4. Synthetic Time Histories on Rock Now, we can proceed to generate synthetic time histories on hard rock. We propose to define Now, we can proceed to generate synthetic time histories on hard rock. We propose to define the input motions combining a simple approach to generate synthetic time histories with the AA14 the input motions combining a simple approach to generate synthetic time histories with the AA14 GMPE. An arbitrary seismological model is used to generate the waveforms that are then scaled to GMPE. An arbitrary seismological model is used to generate the waveforms that are then scaled to match the target accelerations, using a continuous function. We consider that it is reasonable: (1) to match the target accelerations, using a continuous function. We consider that it is reasonable: (1) to define the input motions using the European AA14 GMPE and (2) to generate synthetic time histories define the input motions using the European AA14 GMPE and (2) to generate synthetic time histories that are compatible to the selected GMPE on the basis of a simple, point source seismological model that are compatible to the selected GMPE on the basis of a simple, point source seismological model with plausible but arbitrary source and path parameters, providing waveforms that are later scaled with plausible but arbitrary source and path parameters, providing waveforms that are later scaled using a continuous function to match the target accelerations at different spectral periods. using a continuous function to match the target accelerations at different spectral periods. We use the well-known stochastic method that was proposed by D. Boore [40] for the generation We use the well-known stochastic method that was proposed by D. Boore [40] for the generation of ground-motion time histories, and its corresponding Fortran code SMSIM [41], which is freely of ground-motion time histories, and its corresponding Fortran code SMSIM [41], which is freely provided by the author on his website. Any other suitable method to generate synthetic time histories provided by the author on his website. Any other suitable method to generate synthetic time histories could be used. This step is one of the most time consuming of the entire process, since the length of could be used. This step is one of the most time consuming of the entire process, since the length of the earthquake catalog implies a very large number (~21,800) of events and thus the generation of the earthquake catalog implies a very large number (~21,800) of events and thus the generation of numerous time histories at the site. The main input parameters are [40]: the source characteristics numerous time histories at the site. The main input parameters are [40]: the source characteristics (seismic moment M0 and stress drop Δσ or corner frequency fc), the path terms with geometrical (seismic moment M and stress drop ∆σ or corner frequency fc), the path terms with geometrical spreading G(R), anelastic0 attenuation Q(f), and duration terms Tgm(f) related to hypocentral distance spreading G(R), anelastic attenuation Q(f), and duration terms Tgm(f) related to hypocentral distance Dhyp, the frequency , and the site terms including the crustal amplification factor CAF(f) derived Dhyp, the frequency f , and the site terms including the crustal amplification factor CAF(f) derived from from the site velocity profile VS(z), together with the site specific shallow attenuation factor the site velocity profile VS(z), together with the site specific shallow attenuation factor characterized characterized by the κ0 value. When considering the published, locally available data at the by the κ0 value. When considering the published, locally available data at the Euroseistest site, the Boore stochastic method has been applied with the crustal velocity structure detailed in [26], a κ0 value Geosciences 2018, 8, 285 10 of 28

Geosciences 2018, 8, x FOR PEER REVIEW 10 of 28 of 0.024 s [53], and a lognormal distribution of the stress drop with a median value of 30 bars and standardEuroseistest deviation site, of the 0.68 Boore (natural stochastic log). Themethod ground-motion has been applied duration with (Tgm)the crustal used velocity here corresponds structure to detailed in [26], a κ0 value of 0.024 s [53], and a lognormal distribution of the stress drop with a median the sum of the source duration, which is related to the inverse of the corner frequency (1/fc) and a value of 30 bars and standard deviation of 0.68 (natural log). The ground-motion duration (Tgm) path-dependent duration (0.05 R follows [54] model). used here corresponds to the sum of the source duration, which is related to the inverse of the corner Wefrequency calculate (1/fc) the and hazard a path-dependent curves on duration rock based (0.05 on R fo thellows synthetic [54] model). time histories generated using SMSIM (FigureWe calculate7, PGA, the 0.2 hazard s and 1.0curves s). The on rock annual based rates on of the the synthetic generated time time histories histories generated are significantly using lowerSMSIM than the(Figure rates 7, of PGA, the target0.2 s hard-rockand 1.0 s). hazardThe annual curve rates relying of the on generated AA14 GMPE time histories (green curve are on Figuresignificantly7). Such a lower discrepancy than the isrates related of the to target the lackhard-rock of consistency hazard curve between relying theon AA14 theoretical GMPE (green stochastic approachcurve(source, on Figure path, 7). Such and a site discrepancy term) and is related the empirical to the lack AA14 of consistency GMPE. A between correction the shouldtheoretical thus be appliedstochastic to the approach results of (source, the stochastic path, and simulation, site term) and which the empirical can be doneAA14 inGMPE. several A correction possible should ways. The first waythus consistsbe applied in to tuning the results the SMSIMof the stochastic ground-motion simulation, model, whichby can performing be done in several a sensitivity possible analysis ways. on The first way consists in tuning the SMSIM ground-motion model, by performing a sensitivity the input parameters (geometrical spreading term, crustal attenuation, ∆σ, κo) until a hazard curve analysis on the input parameters (geometrical spreading term, crustal attenuation, Δσ, κo) until a closehazard to the targetcurve close hazard to the curve target is hazard obtained. curve We is triedobtained. this We approach tried this on approach a trial-and-error on a trial-and-error basis, but we foundbasis, it to but be verywe found time-consuming it to be very time-consuming and rather inefficient and rather in obtaining inefficient timein obtaining histories time simultaneously histories compatiblesimultaneously at all spectral compatible periods. at all Actually,spectral periods. such an Actually, approach such would an approach require would a full require inversion a full of the “free”inversion parameters of the mentioned “free” parameters above,as mentioned performed abov recently,e, as performed for instance recently, by [ 55for]. instance Hence, weby [55]. opt for a muchHence, faster, we easier, opt for and a much entirely faster, pragmatic easier, and scaling entirely technique, pragmatic asscaling described techniqu below.e, as described below.

1.0000 1.0000 1.0000 (PGA) (0.2s) (1.0s) AA14 (800 m/s) 0.1000 0.1000 0.1000 AA14 (2600 m/s) SMSIM(2600 m/s)

0.0100 0.0100 0.0100

Tr=475 Tr=475 Tr=475 0.0010 0.0010 0.0010

Tr=4975 Tr=4975 Tr=4975 Annual Rate of Exceedance of Rate Annual Exceedance of Rate Annual Exceedance of Rate Annual 0.0001 0.0001 0.0001 0.01 0.10 1.00 0.01 0.10 1.00 0.01 0.10 1.00 Acceleration (g) Acceleration (g) Acceleration (g) Figure 7. Hazard curves at the Euroseistest site for three different spectral periods (PGA, 0.2 s, 1.0 s), Figure 7. Hazard curves at the Euroseistest site for three different spectral periods (PGA, 0.2 s, 1.0 s), applying AA14 GMPE (blue for standard rock, green for hard rock—with HTT adjustments) or built applying AA14 GMPE (blue for standard rock, green for hard rock—with HTT adjustments) or built from stochastic time histories on hard rock generated using SMSIM code (Vs = 2600 m/s, κ0 = 0.024 s, fromΔσ stochastic = 30 ± 0.68 time bars, histories CAF(f) onfrom hard [26]). rock generated using SMSIM code (Vs = 2600 m/s, κ0 = 0.024 s, ∆σ = 30 ± 0.68 bars, CAF(f) from [26]). First we calculated the median accelerations predicted by AA14 (VS30 = 800 m/s), for the magnitude set Mw = [4.7, 5.3, 5.9, 6.5, 7.1, 7.3] and distances Dhyp = [1, 5, 10, 20, 50, 75, 100, 150] km at First we calculated the median accelerations predicted by AA14 (VS30 = 800 m/s), for the a series of spectral periods T = [0.01, 0.05, 0.2, 0.2, 0.5, 1.0, 2.0, 3.0, 4.0] s. These AA14 predictions were magnitude set M = [4.7, 5.3, 5.9, 6.5, 7.1, 7.3] and distances D = [1, 5, 10, 20, 50, 75, 100, 150] km at corrected whilew using the site term of the LA17 GMPE to derivehyp the expected motion for a hard rock a series of spectral periods T = [0.01, 0.05, 0.2, 0.2, 0.5, 1.0, 2.0, 3.0, 4.0] s. These AA14 predictions were with Vs = 2600 m/s (see HTT, Section 4.2). Then, for each event (Mw, Dhyp), we calculate the ratios corrected while using the site term of the LA17 GMPE to derive the expected motion for a hard rock between the synthetic spectral acceleration and the corrected AA14 acceleration (VS30 = 2600 m/s). The with Vsratios = 2600that were m/s (seeobtained HTT, for Section different 4.2 spectral). Then, periods for each are event displayed (Mw ,Din hypFigure), we 8a. calculate The error the bar ratios betweenrepresents the synthetic the median spectral ± one accelerationstandard deviation and the calculated corrected over AA14 all the acceleration ratios available (VS30 for= a 2600given m/s). The ratiosmagnitude. that were The variability obtained forcomes different from the spectral incons periodsistency between are displayed the distance in Figure term8 a.in Thethe AA14 error bar representsGMPE theand median its equivalent± one standardcontribution deviation in the calculatedSMSIM approach over all (i.e., the ratiosthe combination available forof the a given magnitude.geometrical The spreading variability term comes and the from crustal/site the inconsistency attenuation terms). between the distance term in the AA14 GMPE andThe its horizontal equivalent black contribution lines in inFigure the SMSIM 8a represent approach the (i.e.,average the combinationcorrection factors of the when geometrical all magnitudes and distances are considered and the associated standard deviation (dashed lines). These spreading term and the crustal/site attenuation terms). values are displayed in Figure 8b as a function of the spectral period, showing that the largest Thecorrection horizontal corresponds black to lines rather in long Figure periods8a (aroun representd 2.0 thes), suggesting average that correction part of this factors discrepancy when all magnitudescould be and due distances to the fact are that considered only point and sources the associatedwere considered. standard Indeed, deviation the correction (dashed factors lines). in These valuesFigure are displayed 8a increase in with Figure magnitude,8b as a function which suggests of the spectralthat the use period, of a code showing considering that the extended largest source correction correspondsshould be to preferred. rather long Figure periods 8b also (around displays 2.0the s),values suggesting of the median that partcorrection of this factor discrepancy + one and couldtwo be due to the fact that only point sources were considered. Indeed, the correction factors in Figure8a increase with magnitude, which suggests that the use of a code considering extended source should be Geosciences 2018, 8, 285 11 of 28 preferred. Figure8b also displays the values of the median correction factor + one and two standard deviations, and their interpolation from period to period to derive a continuous, period-dependent average correctionGeosciences 2018 factor,, 8, x FOR required PEER REVIEW to scale the generated stochastic time histories to match11 of 28 the target hazard at all spectral periods. standard deviations, and their interpolation from period to period to derive a continuous, period- The nextdependent step consistedaverage correction in scaling factor, the required ~21,800 to scale stochastic the generated time stochastic histories time by histories applying to match the continuous correctionthe factors target (Figurehazard at8 allb) spectral derived periods. in the response spectra domain to each one of the time histories in the FourierThe domain, next step and consisted posteriorly in scaling retrieving the ~21,800 the newstochastic scaled time time histories histories by applying by inverse the Fourier transform,continuous as shown correction in Figure factors8c. After (Figure the 8b) scaling derived processin the response is performed, spectra domain we calculateto each one the of the new hazard time histories in the Fourier domain, and posteriorly retrieving the new scaled time histories by curves basedinverse on Fourier the scaled transform, accelerograms. as shown in Figure Figure 8c. After9a shows the scaling the process newly is scaled performed, hazard we calculate curve, this time being closerthe new to the hazard target curves hazard based curveon the scaled (AA14 accelero 2600grams. m/s) Figure than 9a the shows original the newly one, scaled but hazard still with lower annual rates.curve, We this applied time being the closer same to correctionthe target hazard procedure curve (AA14 using 2600 the m/s) median than the + original one standard one, but deviation (CF+1σ) andstill with median lower + annual two standardrates. We applied deviations the same (CF+2 correctionσ) correction procedure factors.using the median The resulting + one hazard standard deviation (CF+1σ) and median + two standard deviations (CF+2σ) correction factors. The curves areresulting displayed hazard in curves Figure are9b,c. displayed in Figure 9b,c.

(a)

10.0

Sa=0.01 Sa=0.05 Sa=0.1 Sa=0.2 1.0 Sa=0.5 Sa=1.0 Sa=2.0 CF=-0.17X2+0.69X+1.61μCF(T)=-0.17(T)2+0.69(T)+1.61 Sa=3.0 CF+μCF(T)+1< =-0.28X2+1.09σ=-0.28(T)2+1.09(T)+2.51 Sa=4.0 CF+2μCF(T)+2< =-0.39X2+1.48σ=-039(T)2+1.48(T)+3.41 Correction Factor (CF) 0.1 0.01 0.10 1.00 Period (s)

(b) (c)

Figure 8. (a) Correction factors or ratios between synthetic accelerations and accelerations predicted Figure 8. a (by) AA14 Correction (adjusted factors at VS30 = or 2600 ratios m/s), between for a series synthetic of magnitudes accelerations at different spectral and accelerations periods: median predicted by AA14 (adjusted± one standard at VS30 deviation= 2600 m/s),of all available for a series ratios of for magnitudes a given magnitude at different (Mw), μ spectralis the median periods: of all median ± one standardmedian deviation correction of allfactors available when considering ratios for aall given magnitudes, magnitude and σ (M thew ),correspondingµ is the median standard of all median correctiondeviation. factors ( whenb) Variation considering of the correction all magnitudes, factor with spectral and σperiodthe correspondingestimated over all magnitudes standard deviation. and distances: black solid line: interpolated frequency-dependent median correction factor; dashed (b) Variationline: of median the correction + one standard factor deviation; with dot- spectraldashed: period median estimated + two standard over deviations. all magnitudes (c) Example and of distances: black solid line: interpolated frequency-dependent median correction factor; dashed line: median +

one standard deviation; dot-dashed: median + two standard deviations. (c) Example of an Mw = 7.9 accelerogram generated using SMSIM (red) and then scaled by the interpolated median + two standard deviations correction factor (black). Geosciences 2018, 8, x FOR PEER REVIEW 12 of 28

an Mw = 7.9 accelerogram generated using SMSIM (red) and then scaled by the interpolated median + Geosciences 2018, 8, 285 12 of 28 two standard deviations correction factor (black).

(a)

(b)

(c)

Figure 9. InIn black, black, hazard hazard curves curves for for three three spectral spectral pe periodsriods (PGA, (PGA, 0.2 0.2 s, s, 1.0 1.0 s) built from SMSIM stochastic time histories scaled by the: ( a) Median correction factor (CF), (b) Median + one standard deviation correction factor (CF+1 σσ),), ( (cc)) Median + two two standard standard deviation deviation correction correction factor factor (CF+2 σσ).). Also superimposed the hazard curves obtained with AA14 at VS30 = 800 m/s (blue), hazard curves Also superimposed the hazard curves obtained with AA14 at VS30 = 800 m/s (blue), hazard curves obtained with AA14 corrected at VS30 = 2600 m/s (green), and hazard curves derived from SMSIM obtained with AA14 corrected at VS30 = 2600 m/s (green), and hazard curves derived from SMSIM original time histories (red).

The ratios between the target AA14 accelerations (VS30 = 2600 m/s) and the scaled stochastic The ratios between the target AA14 accelerations (V = 2600 m/s) and the scaled stochastic accelerations, interpolated for a series of annual exceedanceS30 rates, are shown in Figure 10. The best accelerations, interpolated for a series of annual exceedance rates, are shown in Figure 10. The best agreement is obtained by applying the (CF+2σ) scaling factor; the resulting ratio is close to 1.0 for agreement is obtained by applying the (CF+2σ) scaling factor; the resulting ratio is close to 1.0 for most most annual rates of exceedance (and especially the smallest) at the three displayed spectral periods. annual rates of exceedance (and especially the smallest) at the three displayed spectral periods. It is It is important to mention that the (CF+2σ) correction factor is a pragmatic non-physical parameter important to mention that the (CF+2σ) correction factor is a pragmatic non-physical parameter that we that we used to scale the accelerograms in order to fit a certain target hazard level. used to scale the accelerograms in order to ﬁt a certain target hazard level.

Geosciences 2018, 8, 285 13 of 28 Geosciences 2018, 8, x FOR PEER REVIEW 13 of 28

Geosciences 2018, 8, x FOR PEER REVIEW 13 of 28

Figure 10. AA14 (2600 m/s)/SMSIM hazard curve ratios for three different spectral periods (PGA, left; Figure 10. AA14 (2600 m/s)/SMSIM hazard curve ratios for three different spectral periods (PGA, left; 0.2 s, middle; 1.0 s, right), showing the fit of the hazard curve built from scaled stochastic time histories 0.2Figure s, middle; 10. AA14 1.0 s,(2600 right), m/s)/SMSIM showing the hazard fit of thecurve hazard ratios curve for three built different from scaled spectral stochastic periods time (PGA, histories left; to the target hazard curve, using three different correction factors CF, CF+1σ, and CF+2σ. The best fit to0.2 the s, middle; target hazard 1.0 s, right), curve, showing using three the differentfit of the h correctionazard curve factors built CF, from CF+1 scaledσ, and stochastic CF+2σ. time The besthistories fit is is obtained using the scaling factor CF+2σ at all spectral periods. obtainedto the target using hazard the scaling curve, factorusing CF+2threeσ differentat all spectral correction periods. factors CF, CF+1σ, and CF+2σ. The best fit is obtained using the scaling factor CF+2σ at all spectral periods. Figure 11 displays the synthetic mean spectral acceleration values before and after the scaling Figure 11 displays the synthetic mean spectral acceleration values before and after the scaling process and is compared to Akkar et al., 2013 [36] scaled while using the HTT factors (Figure 11). This processFigure and is11 compareddisplays the to Akkarsynthetic et al., mean 2013 spectral [36] scaled acceleration while using values the HTT before factors and (Figureafter the 11 scaling). This plot is shown for three different spectral periods (PGA, 0.2 s, 1.0 s) and for events of Mw = [4.7, 6.5, plotprocess is shown and is for compared three different to Akkar spectral et al., periods2013 [36] (PGA, scaled 0.2 while s, 1.0 using s) and the for HTT events factors of M w(Figure= [4.7, 11). 6.5, This 7.5]. 7.5]. Events in the catalog were generated for different hypocentral distances (Dhyp) and a scaling Eventsplot is shown in the catalogfor three were different generated spectral for differentperiods (PGA, hypocentral 0.2 s, 1.0 distances s) and for (D events) and of a M scalingw = [4.7, factor 6.5, factor at +2σ was applied to generate compatible hazard curves as explainedhyp in the previous step. As at7.5]. +2 Eventsσ was appliedin the catalog to generate were compatiblegenerated for hazard different curves hypocentral as explained distances in the (D previoushyp) and step.a scaling As mentioned before, this correction is only a pragmatic way of scaling accelerograms and it does not mentionedfactor at +2σ before, was applied this correction to generate is only compatible a pragmatic hazard way curves of scaling as explained accelerograms in the previous and it does step. not As rely on a physics-based scaling strategy. Nonetheless, it does provide a very satisfactory fit between relymentioned on a physics-based before, this correction scaling strategy. is only Nonetheless, a pragmatic itway does of providescaling aaccelerograms very satisfactory and fitit betweendoes not the stochastic hazard curves and the target hazard curves at all spectral periods, and it upholds the therely stochasticon a physics-based hazard curves scaling and strategy. the target Nonetheless, hazard curves it does at provide all spectral a very periods, satisfactory and fit it upholdsbetween original variability of the accelerograms. Future applications should improve this step to find a thethe originalstochastic variability hazard curves of the and accelerograms. the target hazard Future curves applications at all spectral should periods, improve and this it step upholds to find the a physics-based optimal fitting procedure (such as higher, possibly magnitude-dependent, stress physics-basedoriginal variability optimal of fittingthe accelerograms. procedure (such Future as higher, applications possibly should magnitude-dependent, improve this step stress to drops,find a drops, frequency-dependent geometrical spreading, and crustal quality factor). frequency-dependentphysics-based optimal geometrical fitting procedure spreading, (such and as crustal higher, quality possibly factor). magnitude-dependent, stress drops, frequency-dependent geometrical spreading, and crustal quality factor).

Figure 11. Cont.

Geosciences 2018, 8, 285 14 of 28 Geosciences 2018, 8, x FOR PEER REVIEW 14 of 28

Figure 11. Mean spectral acceleration decay with respect to distance (km) predicted by AA14∙HTT ± Figure 11. Mean spectral acceleration decay with respect to distance (km) predicted by AA14·HTT ± 2σ (corrected for hard rock using HTT factors) for three different spectral periods (PGA, 0.2 s, 1.0 s), 2σ (corrected for hard rock using HTT factors) for three different spectral periods (PGA, 0.2 s, 1.0 s), superimposed to the original SMSIM accelerations for hard rock (blue dots) and the scaled accelerations superimposed to the original SMSIM accelerations for hard rock (blue dots) and the scaled accelerations after applying a correction factor of CF+2σ (red dots) for events of Mw = [4.7, 6.5, 7.5]. after applying a correction factor of CF+2σ (red dots) for events of Mw = [4.7, 6.5, 7.5]. 5. Results for Soft Soil Hazard 5. Results for Soft Soil Hazard

5.1. Synthetic Synthetic Time Histories on Soil Based on the (CF+2σσ)) scaled scaled synthetic synthetic time time histories histories on on hard rock, we can now generate the corresponding timetime historieshistories onon soilsoil byby performing performing a a 1D 1D equivalent-linear equivalent-linear (EL, (EL, SHAKE91 SHAKE91 [42 [42,43,44])–44]) and 1Dand non-linear 1D non-linear (NL, (NL, NOAH NOAH [45]) [45]) site responsesite response analyses. analyses. The soil profile profile (Table 1 1)) andand thethe degradationdegradation curvescurves forfor allall layerslayers (Figure(Figure2 b,c)2b,c) were were used used to to perform the 1D site response calculations. SHAKE91 SHAKE91 requires requires only degradation curves, while NOAH requires the maximummaximum stresses,stresses, relatedrelated to to the the cohesion, cohesion, friction friction angle angleϕ φ, and, and the the coefficient coefficient of of earth earth at at rest K0. For the present exercise, the cohesion was assumed to be zero in all layers since no sufficient rest K0. For the present exercise, the cohesion was assumed to be zero in all layers since no sufficient information waswas availableavailable beingbeing relatedrelated to to this this parameter, parameter, especially especially at at large large depths. depths. It isIt importantis important to tohighlight highlight that that the the (CF+2 (CF+2σ) scaledσ) scaled synthetic synthetic time time histories histories on hard on hard rock arerock representative are representative of the of AA14 the AA14 (VS30 = 2600 m/s) ground motion predictions for outcropping conditions, and this should be (VS30 = 2600 m/s) ground motion predictions for outcropping conditions, and this should be specified specified inside the site response input files in order to appropriately remove the free field surface inside the site response input files in order to appropriately remove the free field surface effect (about effect (about a factor of ~2.0). a factor of ~2.0).

5.1.1. Focus Focus on Nonlinear Ground Motion Saturation An interesting way to analyze the surface syntheti syntheticscs is to compare the spectral acceleration on soil, with respect to the corresponding input ground motion on rock. Figure 12a12a shows the results at three spectral periods (PGA, 0.2 0.2 s and 1.0 s) for the SHAKE91 computations and Figure 12b12b for the NOAH computations. In both both cases, cases, the the soil soil presents presents significant signiﬁcant non nonlinearlinear behavior, eveneven atat weak/intermediateweak/intermediate motion motion levels (0.01(0.01 g–0.1g–0.1 g), g), and and de-ampliﬁcation de-amplification is observedis observed on soilon withsoil with respect respect to the to rock the motion rock motion at ground at groundmotion levelsmotion around levels (0.1around g–0.3 (0.1 g). g–0.3 Moreover, g). Moreover a saturation, a saturation level of thelevel acceleration of the acceleration is always isreached always reachedin both soilin both cases soil (SHAKE91 cases (SHAKE91 and NOAH) and andNOAH) at all and spectral at all periods. spectral The periods. point The where point the where saturation the saturationlimit is reached limit is depends reached stronglydepends onstrongly the spectral on the periodspectral (at period least (at for least this example),for this example), and less and on less the onpropagation the propagation code that code is usedthat is for used the for site the response site response analysis. analysis. The saturation The saturation limit appears limit appears faster at faster high atfrequency high frequency than at than low frequencyat low frequency for both for codes. both codes. These resultsThese results also indicate also indicate a larger a larger variability variability of the ofdata the for data a given for a rockgiven input rock levelinput when level usingwhen SHAKE91using SHAKE91 when comparedwhen compared to the NOAHto the NOAH results. results. Using UsingSHAKE91, SHAKE91, the dispersion the dispersion of calculated of calculated accelerations accelerations increases increases with the with spectral the spectral period (thisperiod is not(this the is notcase the using case NOAH). using NOAH).

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(a)

(b)

FigureFigure 12. 12. SpectralSpectral acceleration acceleration on on soil soil Vs. Vs. Spectral Spectral acceleration acceleration on on ro rock,ck, for for three three different different spectral spectral periodsperiods (PGA, 0.20.2 ss and and 1.0 1.0 s); s); (a )( Equivalent-lineara) Equivalent-linear (EL) (EL) results results (SHAKE91); (SHAKE91); (b) Non-linear (b) Non-linear (NL) results (NL) results(NOAH). (NOAH).

5.1.2. Ground Motion Variability 5.1.2. Ground Motion Variability It is also of interest to better comprehend what happens with the ground motion variability after It is also of interest to better comprehend what happens with the ground motion variability after performing the 1D site response analysis with the two selected codes. For this purpose, we compared performing the 1D site response analysis with the two selected codes. For this purpose, we compared the probability density function (PDF) of the ground motion acceleration (PGA) on hard rock at depth the probability density function (PDF) of the ground motion acceleration (PGA) on hard rock at depth with respect to the PDF on soil, for different magnitude and distance ranges (Figure 13). with respect to the PDF on soil, for different magnitude and distance ranges (Figure 13). A systematic amplification of the median ground motion at soil surface (SHAKE91 and NOAH) A systematic amplification of the median ground motion at soil surface (SHAKE91 and NOAH) with respect to hard rock at depth is observed for small magnitudes M = [4.7–5.1] and all distances with respect to hard rock at depth is observed for small magnitudes M = [4.7–5.1] and all distances (Figure 13a). Non-linearity of the soil is not observed, indicating mostly linear behavior of the soil. (Figure 13a). Non-linearity of the soil is not observed, indicating mostly linear behavior of the soil. Similarly, Figure 13b shows the ground motion variability at the intermediate magnitude range Mw Similarly, Figure 13b shows the ground motion variability at the intermediate magnitude range = [6.1–6.5] at different distances (Dhyp) ranges. Amplification of the ground motion with respect to Mw = [6.1–6.5] at different distances (Dhyp) ranges. Amplification of the ground motion with respect hard rock at depth is again observed. However, this time a clear saturation of the ground motion to hard rock at depth is again observed. However, this time a clear saturation of the ground motion occurs at all distances due to the nonlinear effect of the soil (PDF truncated on the strong motion side, occurs at all distances due to the nonlinear effect of the soil (PDF truncated on the strong motion side, with more pronounced effect at short distances). This saturation is stronger using NOAH rather than with more pronounced effect at short distances). This saturation is stronger using NOAH rather than SHAKE91, since NOAH is a fully nonlinear code. However, part of the difference can be also explained SHAKE91, since NOAH is a fully nonlinear code. However, part of the difference can be also explained by intrinsic code differences and different degradation curves definition. At short distances, a smaller by intrinsic code differences and different degradation curves definition. At short distances, a smaller amplification and even deamplification on soil with respect to rock is observed. amplification and even deamplification on soil with respect to rock is observed. Finally, for large magnitudes Mw = [7.1–7.9], a strong saturation of ground motion can be Finally, for large magnitudes Mw = [7.1–7.9], a strong saturation of ground motion can be observed observed (Figure 13c), especially when using NOAH. It is also interesting to notice that the variability (Figure 13c), especially when using NOAH. It is also interesting to notice that the variability of the of the soil ground motion, σS(), is smaller than the variability of the hard rock ground motion, soil ground motion, σSs ( f ), is smaller than the variability of the hard rock ground motion, σSr ( f ).[2] σS(). [2] also discusseda this phenomenon, which has also been observed from real recordingsa on also discussed this phenomenon, which has also been observed from real recordings on rock and soil rock and soil conditions (see [56,57,58,59]). conditions (see [56–59]).

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Hard Rock Soil (SHAKE) Soil (NOAH)

FigureFigure 13. Probability13. Probability density density function function (PDF)(PDF) of the the ground ground motion motion acceleration acceleration (PGA) (PGA) on hard on hardrock rock (blue)(blue) and and on soilon soil after after performing performing 1D 1D sitesite responseresponse analysis analysis using using SHAKE9 SHAKE911 (red) (red) and NOAH and NOAH (green) (green) for different ranges of magnitudes (columns) and distances (rows). for different ranges of magnitudes (columns) and distances (rows). 5.2. Full Probabilistic Stochastic Method (SM) 5.2. Full Probabilistic Stochastic Method (SM) What we call here the full probabilistic stochastic method is nothing else than the hazard curve builtWhat from we callstochastic here the time full histories probabilistic (~21,800 stochastic accelerograms), method corresponding is nothing else to the than strong-motion the hazard curve builthistory from stochastic at the site timeduring histories 50,000 years. (~21,800 accelerograms), corresponding to the strong-motion history at the siteTo during build the 50,000 stochastic years. hazard curves, we calculate the annual rate of exceedance, λ(x≥X), of a seriesTo build of ground the stochastic motion levels hazard (X), curves, for a given we calculate spectral period the annual (T), by rate counting of exceedance, the numberλ (ofx ≥ X), occurrences of ground-motions (N) exceeding a certain ground motion level x, as follows: of a series of ground motion levels (X), for a given spectral period (T), by counting the number of N (x ≥ X) occurrences of ground-motions (N)λ exceeding(x≥X) = a certain ground motion level x, as follows: (2) Catalogue time length N (x ≥ X) λT(x ≥ X) = (2) 5.2.1. SM Results for Hard Rock Catalogue time length () 5.2.1. SMThe Results stochastic for Hard hazard Rock curve on hard rock, , is thus built from the annual rate of exceedance of the scaled synthetic time histories set, as expressed in Equation (2), for three different spectralThe stochastic periods hazard(PGA, 0.2 curve s and on 1.0 hard s), Figure rock, 14HC (black).( f ), is thus built from the annual rate of exceedance of the scaled synthetic time histories set, as expressed in Equation (2), for three different spectral periods5.2.2. (PGA, SM Results 0.2 s andfor Soft 1.0 Soil s), Figure Using SHAKE9114 (black). Similarly, we calculate the hazard curves on soil at the site for different spectral periods, while 5.2.2.using SM Resultsthe synthetic for Soft time Soil histories Using on SHAKE91 soil that was obtained with the SHAKE91 equivalent-linear site response analysis code. The results displayed in Figure 14 exhibit two characteristic features when Similarly, we calculate the hazard curves on soil at the site for different spectral periods, while compared to their rock equivalent. For all spectral periods, the soil hazard curves exhibit a much using the synthetic time histories on soil that was obtained with the SHAKE91 equivalent-linear site stronger convexity than for the rock hazard curves: this is related to the non-linear behavior of soil, response analysis code. The results displayed in Figure 14 exhibit two characteristic features when compared to their rock equivalent. For all spectral periods, the soil hazard curves exhibit a much stronger convexity than for the rock hazard curves: this is related to the non-linear behavior of soil, Geosciences 2018, 8, 285 17 of 28

Geosciences 2018, 8, x FOR PEER REVIEW 17 of 28 which saturates the ground motion for the upper acceleration range at long return periods. This is also true forwhich the saturates largest spectral the ground period motion considered for the upper here acce (1.0leration s), because range it at corresponds long return periods. to a shorter This periodis also true for the largest spectral period considered here (1.0 s), because it corresponds to a shorter than the site fundamental period (around ~1.5 s), and therefore it is affected by the non-linear behavior period than the site fundamental period (around ~1.5 s), and therefore it is affected by the non-linear over the whole soil column. behavior over the whole soil column. 5.2.3. SM Results for Soft Soil Using NOAH 5.2.3. SM Results for Soft Soil Using NOAH Lastly,Lastly, we calculate we calculate the hazard the hazard curves curves on soil on atsoil the at site the basedsite based on the on synthetics the synthetics that werethat were obtained with NOAHobtained (seewith Section NOAH 5.1 (see). ResultsSection 5.1). displayed Results in displayed Figure 14 in, exhibitFigure 14, the exhibit same overallthe same features overall than the SHAKE91features than results: the SHAKE91 a pronounced results: convexity a pronounced related convexity with therelated existence with the of existence an upper of bound an upper for the surfacebound ground for motionthe surface due ground to the non-linearmotion due behavior. to the non-linear As the saturation behavior. isAs better the saturation accounted is forbetter in fully NL codesaccounted than inforequivalent-linear in fully NL codes approaches,than in equivale thent-linear hazard curveapproaches, that was the predictedhazard curve using that NOAH was is saturatedpredicted at acceleration using NOAH levels is saturated lower thanat acceleration in the SHAKE91 levels lower case. than in the SHAKE91 case.

Figure 14. Hazard curves calculated using the Full Probabilistic Stochastic Method (SM), for three Figure 14. Hazard curves calculated using the Full Probabilistic Stochastic Method (SM), for three different spectral periods (PGA, 0.2 s, 1.0 s). Time histories on: hard rock (VS30 = 2600 m/s, black), soft differentsoil spectralobtained periodswith SHAKE91 (PGA, 0.2site s, response 1.0 s). Time code histories(red), soft on: soil hard obtained rock (VwithS30 NOAH= 2600 site m/s, response black), soft soil obtainedcode (green). with SHAKE91 site response code (red), soft soil obtained with NOAH site response code (green). 5.3. Analytical Approximation of the Full Convolution Method (AM) 5.3. AnalyticalIn order Approximation to calculate ofa fully the Full probabilistic Convolution hazard Method curve (AM) on soil at the surface (or at any desired Indepth), order [10,11] to calculate proposed a fully three probabilistic different approaches hazard to curve calculate on soil hazard at the on surface soil, consisting (or at any in the desired depth),convolution [10,11] proposed of the site-specific three different hazard approachescurve at the tobedrock calculate level hazardwith simple on soil, to more consisting complex in the descriptions of the site amplification function. These approaches provide more precise surface convolution of the site-specific hazard curve at the bedrock level with simple to more complex ground-motion hazard estimates than those found by means of standard GMPEs for generic soil descriptions of the site amplification function. These approaches provide more precise surface conditions. These three different methods are: (1) Full Convolution of the reference rock hazard curve ground-motion hazard estimates than those found by means of standard GMPEs for generic soil with the NL amplification function, AF(, ), and its variability. (2) PSHA with a Site-Specific GMPE conditions.(i.e., with These the generic three different site term methodsof the original are: GM (1) FullPE being Convolution replaced by of a the site reference-specific term, rock S, hazard derived curve r with thefrom NL the amplification site-specific amplification function, AF functions,( f , Sa), and (, its variability.), and the (2)original PSHA aleatory with a variability Site-Specific being GMPE (i.e., withalso modified the generic to siteaccount term for of the the actual original variability GMPE beingin the replacedsite-specific by aamplification site-specific function). term, S, derived(3) Analytical estimate of the soil hazard convolving the rprobabilistic rock hazard curve with a statistical from the site-specific amplification functions, AF( f , Sa), and the original aleatory variability being also modifieddescription to account of the for non-linear the actual site variability response and in theits variability. site-specific amplification function). (3) Analytical estimate ofInthe the soil present hazard work, convolving we decided the probabilisticto use the th rockird approach, hazard curve which with is the a statistical simplest descriptionone to implement. The “Analytical Estimate of the Soil Hazard” method (AM) provides a simple, closed- of the non-linear site response and its variability. form solution that appropriately modifies the hazard results at the rock level to obtain the hazard Incurve the presenton soil. work,Under we several decided assumptions, to use the thirdas disc approach,ussed in whichtheir paper, is the simplest[10,11] stated one tothat implement. the The “Analytical Estimate of the Soil Hazard” method (AM) provides a simple, closed-form solution acceleration on soil, (), can be predicted from the acceleration on rock, (), as follows: that appropriately modifies the hazard results at the rock level to obtain the hazard curve on soil. ()= + ( +1) (()) (3) Under several assumptions, as discussed in their paper, [10,11 ] stated that the acceleration on soil, s The amplification function is: r Sa( f ), can be predicted from the acceleration on rock, Sa( f ), as follows: () ln(Ss ( f )) = C + (C + 1) ln(Sr ( f )) (3) a (0)= 1 a (4) ()

The ampliﬁcation function is: Ss ( f ) ( ) = a AF f r (4) Sa( f ) Geosciences 2018, 8, 285 18 of 28

Equation 4 simply corresponds to a log-linear dependence of the ampliﬁcation factor AF( f ) in r r Equation (4) and the loading level Sa( f ) in the form ln(AF( f )) = C0 + C1 ln(Sa( f )), with a variability described by a standard deviation σln(AF( f )), as in Figure 15. r They also assumed that the rock hazard curve HC(Sa( f )) may be locally described with a power law dependenceGeosciences 2018, on 8, x the FOR spectral PEER REVIEW acceleration, as follows: 18 of 28

Equation 4 simply correspondsHC to a( Slog-linearr ( f )) = dependenceH ·Sr −k1 of the amplification factor () in (5) a 0 a Equation (4) and the loading level () in the form (()) = + 1 (()) , with a σln(( Byvariability convolving described the rockby a standard hazard curvedeviation with the lognormal)), as in Figure distribution 15. of the site amplification They also assumed that the rock hazard curve (()) may be locally described with a power factor and its linear dependence on the loading level, they could then derive a closed-form expression law dependence on the spectral acceleration, as follows: for the hazard curve at soil site, as follows: ( ()) = ∙ (5) 2 2 ( 1 k1 σ ) By convolving the rock hazards curve with the lognormalr 2 distribution2 of the site amplification HC(S ( f )) = HC(S ( f ))·e (C1+1) (6) factor and its linear dependence on thea loading level, theya could then derive a closed-form expression for the hazard curve at soil site, as follows: where σ corresponds to the standard deviation of the amplification function in log-normal scale, r σln(AF( f )), and the spectral acceleration on rock, S ( f ), is related to the spectral acceleration on a () (6) s (())=( ())∙ r soil, Sa( f ), through the corresponding median amplification curve AF( f , Sa). In the case where the uncertaintywhere σ can corresponds be associated to the to standard the rock deviation hazardcurve of the (e.g.,amplification when a logicfunction tree in is log-normal used with scale, different GMPEs),( the( resulting)), and the uncertainty spectral acceleration in the soil on siterock, hazard (), curveis related may to the be written,spectral acceleration as proposed on bysoil, [11 ]: (), through the corresponding median amplification curve (, ). In the case where the q uncertainty can be associated to the rock hazard2 curve (e.g., 2when a logic 2tree is used with different s ( f ) ≈ (C + ) · σ r ( f ) + σ ( f ) GMPEs), the resulting uncertaintyσlnSa in the soil1 site1 hazardlnSa curve maylnAF be written, as proposed by [11]: (7)

()≈ (1 + 1) ⋅σ () +σ () The first step in the proposed procedure consists in calculating the amplification functions,(7)AF ( f ), in the response spectra domain for all input ground motions and for each site response approach The first step in the proposed procedure consists in calculating the amplification functions, (SHAKE91 and NOAH), and then proceed to fit piecewise linear regressions to obtain the values (), in the response spectra domain for all input ground motions and for each site response of the coefficients C and C , together with the associated variability, σln(AF( f )). The results are approach (SHAKE910 and1 NOAH), and then proceed to fit piecewise linear regressions to obtain the illustrated in Figure 15 for the two site-response codes and for the three considered spectral periods. It values of the coefficients C0 and C1, together with the associated variability, σ(()). The results is interestingare illustrated to notice in Figure that non-linear15 for the two behavior site-response is observed codes evenand fo atr relativelythe three considered large spectral spectral periods (~1.0periods. s)—for largeIt is interesting loading to levels—and notice that non-linear also at short behavior periods is observed (PGA andeven 0.2at relatively s) from ratherlarge spectral low input accelerationperiods levels(~1.0 s)—for (blue large regression loading lines levels—and in Figure also 15 ata,b). short periods (PGA and 0.2 s) from rather low input acceleration levels (blue regression lines in Figure 15a,b).

Figure 15. Cont.

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(a) (b)

r FigureFigure 15. 15.Trilinear Trilinear piecewise piecewise regressionregression models of of ()AF( f on) on ()Sa( ffor) for three three different different spectral spectral periodsperiods (PGA, (PGA, 0.2 0.2 s, 1.0s, 1.0 s): s): (a )(a Figures) Figures built built usingusing the the accelerogr accelerogramsams propagated propagated with with EL-SHAKE91. EL-SHAKE91. (b) Regression(b) Regression on on the the accelerograms accelerograms propagated propagated wi withth NL-NOAH. NL-NOAH. The The regression regression coefficients coefﬁcients are are indicatedindicated inside inside each each plot plot (for (for each each linear linear segment), segment), see see Table Table 3.3.

The explanation for the former (long period non-linear behavior) can be related to the low fundamentalThe explanation frequency for theof the former site (~[0.6–0.7] (long period Hz): everything non-linear beyond behavior) that canfrequency be related is primarily to the low fundamentalcontrolled frequencyby the site response, of the site and (~[0.6–0.7] the large velocity Hz): everything contrast at ~200 beyond m depth that is frequency likely to generate is primarily controlledsignificant by thestrains site over response, the whole and soil the column. large velocity The latter contrast observation, at ~200 however, m depth seems is more likely likely to generate to significantbe related strains to the over dependence the whole of amplification soil column. factor The latter on the observation, response spectra however, on the seemsactual frequency more likely to be relatedcontents to theof the dependence input waveform, of amplification leading to factorthe magnitude on the response dependence spectra of the on response the actual spectral frequency contentsamplification, of the input as reported waveform, in [58], leading than to a to true the NL magnitude behavior at dependence very low strains, of the which response cannot spectralbe identified in the corresponding Fourier domain amplifications [60]. amplification, as reported in [58], than to a true NL behavior at very low strains, which cannot be The regression coefficients are listed in Table 3 together with their standard deviations. The identifiedhazard in curves the corresponding on soil can now Fourier be estimated domain accordin amplificationsg to Equation [60]. (6). It is worth discussing the limitationsThe regression of this coefficients simplified are analytical listed in Tablemethod3 together (AM), as with already their standardacknowledged deviations. by [11]. The The hazard curvesfunctional on soil canform now provided be estimated in Equation according (6) presents to Equation numerical (6). limitations It is worth when discussing the argument the limitationsof the of thisexponential simplified is too analytical large, i.e., method when C (AM),1 is close as toalready −1.0, and/or acknowledged when the variability by [11 ].of Thethe amplification functional form providedfactor in (σ Equation) is large. The (6) presentsformer situation numerical does limitationsoccur at Euroseistest when the for argument high acceleration of the exponential levels (greenis too large,lines i.e., in when Figure C1 15a,b),is close which to − 1.0,corresponds and/or whento the thesaturation variability of surface of the ground amplification motion factordisplayed (σ) isin large. The formerFigure 12. situation does occur at Euroseistest for high acceleration levels (green lines in Figure 15a,b), I which correspondsThe hazard tocurves the saturationobtained are of presented surface groundin Figure motion 16a (SHAKE91) displayed Figure inFigure 16b (NOAH). 12. n most cases of this particular example (all except in Figure 16b, i.e., for the NL code NOAH at PGA level), The hazard curves obtained are presented in Figure 16a (SHAKE91) Figure 16b (NOAH). In most the nonlinearity is so important for both site response models, that the AM approach is not even cases of this particular example (all except in Figure 16b, i.e., for the NL code NOAH at PGA level), the capable to provide hazard curves results for the most common 475 years return period: C1 values are nonlinearitygetting to is soclose important to −1.0, where for both Equation site response (6) diverges. models, This that limitation the AM approachis actually isone not of even the capablemain to providedrawbacks hazard curvesof the AM results approach, for the which most commoncannot be 475 used years in the return case period:of strong C 1non-linearity.values are getting Also, toas close to −1.0,acknowledged where Equation by [11], (6) the diverges. AM should This be limitation used with iscaution actually when one the of correction the main factor drawbacks in Equation of the AM approach,(6) takes which on cannotvalues that be used are greater in the case than of 10 strong (which non-linearity. may also result Also, from as acknowledged a large variability by [11on], the the AM shouldamplification be used with factor caution for different when waveforms). the correction factor in Equation (6) takes on values that are greater than 10 (which may also result from a large variability on the amplification factor for different waveforms). Table 3. Intercept (C0), slope (C1), and standard deviation (σ) of the piecewise-linear regression models (PLM) for () vs. (). Table 3. Intercept (C0), slope (C1), and standard deviation (σ) of the piecewise-linear regression models (PLM) for AF( f ) vs. Sr ( fPiecewise-Linear). SHAKE91 NOAH Spectral Period (s) a Models C0 C1 σ C0 C1 σ Piecewise-Linear SHAKE91 NOAH Spectral Period (s) PLM 1 (Blue) 0.7487 −0.1089 0.15 0.5585 −0.1833 0.11 Models C C σ C C σ PGA PLM 2 (Red) −0.12020 −10.2906 0.19 −00.2539 1 −0.3649 0.08 − − PLMPLM 3 1(Green) (Blue) 0.7487−1.3990 0.1089−1.21400.15 0.19 0.5585−1.6390 0.1833−0.85940.11 0.08 PGA PLM 2 (Red) −0.1202 −0.2906 0.19 −0.2539 −0.3649 0.08 PLMPLM 31 (Green)(Blue) −1.39900.2855 −1.2140−0.14750.19 0.19 −1.63901.0730− 0.8594−0.10200.08 0.12 (0.2) PLMPLM 2 1 (Blue)(Red) 0.2855−0.5356 −0.1475−0.40430.19 0.18 1.0730−0.0906− 0.1020−0.47890.12 0.13 (0.2) PLMPLM 3 2(Green) (Red) −−0.53561.1360 −0.4043−1.19700.18 0.27 −0.0906−0.9453− 0.4789−1.11720.13 0.35 PLM 3 (Green) −1.1360 −1.1970 0.27 −0.9453 −1.1172 0.35 PLM 1 (Blue) 0.8846 −0.0561 0.27 1.7030 −0.0170 0.11 (1.0) PLM 2 (Red) 0.4191 −0.1438 0.14 0.7180 −0.3298 0.22 PLM 3 (Green) 0.9606 −1.2900 0.08 2.2100 −1.4930 0.26 Geosciences 2018, 8, x FOR PEER REVIEW 20 of 28

PLM 1 (Blue) 0.8846 −0.0561 0.27 1.7030 −0.0170 0.11 (1.0) PLM 2 (Red) 0.4191 −0.1438 0.14 0.7180 −0.3298 0.22 Geosciences 2018, 8, 285 20 of 28 PLM 3 (Green) 0.9606 −1.2900 0.08 2.2100 −1.4930 0.26

(a)

(b)

Figure 16. Hazard curves on soil estimated with the analytical approximation method (AM), for three Figure 16. Hazard curves on soil estimated with the analytical approximation method (AM), for three different spectral periods (PGA, 0.2 s and 1.0 s). (a) Hazard curve on rock and soil derived from differentSHAKE91 spectral accelerograms. periods (PGA, (b) hazard 0.2 s curve and 1.0on soil s). derived (a) Hazard from NOAH curve onaccelerograms. rock and soil The derived blue, red from SHAKE91and green accelerograms. points correspond (b) hazard to the curvedifferent on regressi soil derivedon sections from used NOAH to mo accelerograms.dify the original The hazard blue, red and greencurve points(see Figure correspond 15a). Hazard to the curves different on hard regression rock (black). sections used to modify the original hazard curve (see Figure 15a). Hazard curves on hard rock (black). 5.4. Robustness of the () Prediction 5.4. RobustnessAnother of theissue AF that( f ) Predictionmay be discussed that is related to the accuracy and performance of the model, is the required amount of accelerograms to robustly predict the median of the amplification Another issue that may be discussed that is related to the accuracy and performance of the model, function (). Bazzurro, P. [10] proposed that “the number of records, n, needed to keep the is the required amount of accelerograms to robustly predict the median of the ampliﬁcation function standard error of the median of (()) at any spectral acceleration level within a specified ( ) ln AFfractional. Bazzurro, accuracy, P. [ 10ζ, ]is proposed approximately that (Strictly, “the number Equation of (9) records, is validn for, needed the median to keep () the standardat a errorlognormal of the median spectral-acceleration of ln(AF( f )) levelat corresponding any spectral to acceleration the sample median levelwithin of the spectral-acceleration a speciﬁed fractional accuracy,valuesζ ,used is approximately to produce the piece-wise (Strictly, Equation regression. (9) The is standard valid for error the of median the medianln AF of( ()f ) at a lognormalgrows spectral-accelerationat values that are levelaway corresponding from the average. to theSo, to sample maximize median accuracy, of the the spectral-acceleration record intensities should values be used to producechosen the in the piece-wise appropriate regression. () Theneighborhood standard error(see [10]).) of the given median by” Equation of ln(AF (8):) grows at values that are away from the average. So, to maximize accuracy,σ() the record intensities should be chosen in the r = (8) appropriate ln Sa( f ) neighborhood (see [10]).) givenζ by” Equation (8):

The standard deviation of the median of () is in both2 cases (SHAKE91 and NOAH) and at σlnAF( f ) all spectral periods significantly lower thann = 0.3 (Figure 15a,b and Table 3, most values between 0.01 (8) and 0.2). The criterion defined by [10] in page 2099ζ is therefore fulfilled: “Given that the residual standard deviation of the quadratic model in () is 0.3 or less across the entire frequency range, The standard deviation of the median of ln(AF) is in both cases (SHAKE91 and NOAH) and at all the median value of () associated with a given () on rock can be estimated within ±10% (i.e., spectralζ = 0.1) periods by using significantly not more than lower 10 thanrecords”. 0.3(Figure 15a,b and Table3 , most values between 0.01 and 0.2). The criterion defined by [10] in page 2099 is therefore fulfilled: “Given that the residual standard r deviation of the quadratic model in ln Sa( f ) is 0.3 or less across the entire frequency range, the median r value of AF( f ) associated with a given Sa( f ) on rock can be estimated within ±10% (i.e., ζ = 0.1) by using not more than 10 records”. To test this assertion, we randomly selected from 2 to 200 accelerograms (1000 times each) in our r r set of stochastic time histories and for different Sa( f ) bins, Sa( f )bins = [0.001:0.001:0.01; 0.01:0.01:0.1; 0.1:0.1:2]. Posteriorly, we calculated efficiency curves showing the percentage of random samples where the number of accelerograms, n, predicted a median ln AF( f ) within a 10% error margin (with Geosciences 2018, 8, x FOR PEER REVIEW 21 of 28

To test this assertion, we randomly selected from 2 to 200 accelerograms (1000 times each) in our Geosciences 2018, 8, 285 21 of 28 set of stochastic time histories and for different () bins, ()bins = [0.001:0.001:0.01; 0.01:0.01:0.1; 0.1:0.1:2]. Posteriorly, we calculated efficiency curves showing the percentage of random samples () whererespect the to median numberln of AF accelerograms,( f ), estimated from, predicted the full set a median of stochastic time series;within Figure a 10% 17 error and Tablemargin4). () (withOne can respect observe to median that the number of, estimated accelerograms from thatthe isfull required set of stochastic is larger using time SHAKE91series; Figure than 17 using and TableNOAH. 4). ThisOne can is due observe to the that larger the variabilitynumber of ofaccelerograms SHAKE91 data that withis required respect is to larger NOAH using data. SHAKE91 Hence, thanmore using accelerograms NOAH. This are is needed due to for the a larger robust variability estimation. of FromSHAKE91 this particular data with example,respect to we NOAH determine data. Hence,that an more amount accelerograms of 60 accelerograms are needed per for acceleration a robust estimation. bin Sr ( f ) are From required this particular to robustly example, predict thewe a determine that an amount of 60 accelerograms per acceleration bin ()r are required to robustly median of the ampliﬁcation function ln(AF). The different centers of the Sa( f ) bins that were used are () () predictdisplayed the in median Figures of 18 the and amplification 19. function . The different centers of the bins that were used are displayed in Figures 18 and 19. samples fulfilling criteria Percentage (%) of random

(a) samples fulfilling criteria samples fulfilling criteria Percentage (%) of random Percentage

(b)

Figure 17. (a) Efficiency curves: the median of () is estimated from accelerograms randomly Figure 17. (a) Efﬁciency curves: the median of AF( f ) is estimated from n accelerograms randomly selected in our set of stochastic time histories, this is repeated 1000 times, and the percentage of samples selected in our set of stochastic time histories, this is repeated 1000 times, and the percentage of samples where median () is within a 10% error with respect to the true median value (estimated over the where median AF( f ) is within a 10% error with respect to the true median value (estimated over the full set of time histories) is determined. varies between 2 and 200. (a) SHAKE case; (b) NOAH case. full set of time histories) is determined. n varies between 2 and 200. (a) SHAKE case; (b) NOAH case.

Table 4. Minimum number of records, n, required to predict a median value of () within ±10% Table 4. Minimum number of records, n, required to predict a median value of AF( f ) within ±10% of of the true value for three different spectral periods (PGA, 0.2 s and 1.0 s). the true value for three different spectral periods (PGA, 0.2 s and 1.0 s). Code n, (PGA) n, (0.2 s) n, (1.0 s) SHAKE91Code n,30 (PGA) n 60, (0.2 s) 50n, (1.0 s) SHAKE91NOAH 8 30 50 60 30 50 NOAH 8 50 30 We consider that the differences between [10,11] work and this study, might be associated to the followingWe consider reasons: that (1) The the differencesauthors in [10,11] between calculated [10,11] work the “true” and this reference study, mightmedian be () associated based to theon onlyfollowing 78 ground reasons: motion (1) The records authors alon ing [ 10all,11 of] calculatedthe acceleration the “true” levels, reference (), while median in lnour(AF case,) based we r calculatedon only 78 the ground “true” motion reference records median, along all() of the, accelerationwith more than levels, ~21,800Sa( f ), whileground in motion our case, time we histories.calculated Authors the “true” in reference[10,11] might median, haveln (beenAF) ,calculating with more thanstatistics ~21,800 on a ground sample motion too small. time (2) histories. Since Authorsboth studies in [10 are,11 site-specific] might have studies, been calculating it is very statisticslikely that on the a sample minimum too small.number (2) of Since records both to studies estimate are thesite-speciﬁc median studies,() it with is very a given likely accuracy that the is minimum expected number to be different. of records However, to estimate we the found median that,ln in( AFour) withcase, athe given expression accuracy in is expectedEquation to(8) be is different. insufficient However, and under we found conservative, that, in our and case, such the differences expression cannotin Equation be explained (8) is insufﬁcient by the selected and under Mw conservative,and Dhyp range and of suchthe accelerograms, differences cannot since be in explained their study, by the the authorsselected considered Mw and Dhyp recordsrange with of the magnitudes accelerograms, and distances since in their that study,were between the authors Mw = considered [5.0–7.4] and records Dhyp with= [0.1–142] magnitudes km, while and we distances explore that the weremagnitudes between and Mw distances= [5.0–7.4] between and D hypMw= = [0.1–142][4.7–7.9] and km, D whilehyp = (1– we 150).explore the magnitudes and distances between Mw = [4.7–7.9] and Dhyp = (1–150).

Geosciences 2018, 8, 285 22 of 28 Geosciences 2018, 8, x FOR PEER REVIEW 22 of 28 Geosciences 2018, 8, x FOR PEER REVIEW 22 of 28

r WithWith respect respect to to the the robustness robustness of of the the standard standard deviation deviation of ofAF AF(,( f , S a),), nothing nothing was was said said by by the the With respect to the robustness of the standard deviation of AF(, ), nothing was said by the authors.authors. We We decided decided to to explore explore the the behavior behavior of of the the standard standard deviation deviation while while considering considering 30, 30, 50, 50, and and authors. We decided to explore the behavior of the standard deviation while considering 30,r 50, and thenthen whole whole set set of of accelerograms. accelerograms. Figures Figures 18 18 and and 19 19 display display the the ampliﬁcation amplification factor,factor,AF AF(,( f , S a)),, (top) (top) then whole set of accelerograms. Figures 18 and 19 display the amplification rfactor, AF(, ), (top) andand its its standard standard deviation deviation (bottom), (bottom), versusversus thethe inputinput hazardhazard onon hardhard rock,rock,S a(()f ),, using using SHAKE91 SHAKE91 and its standard deviation (bottom), versus the input hazard on hard rock, (), using SHAKE91 andand NOAH NOAH results, results, respectively. respectively. The The median median standard standard deviation deviation as as a a function function of of the the input input hazard hazard on on and NOAHr results, respectively. The median standard deviation as a function of the input hazard on hardhard rock, rock,S a(f()), does, does not not vary vary signiﬁcantly significantly when when increasing increasing the the number number of of accelerograms, accelerograms, for for both both hard rock, (), does not vary significantly when increasing the number of accelerograms, for both SHAKE91SHAKE91 and and NOAH NOAH results. results. The The apparent apparent larger larger variation variation at highat high input input acceleration acceleration levels levels is due is due to SHAKE91 and NOAH results. The apparent larger variation at high input acceleration levels is due theto the lack lack of data of data on suchon such acceleration acceleration levels. levels. to the lack of data on such acceleration levels.

(a) (b) (c) (a) (b) (c) Figure 18. Amplification factor, (), (top) and standard deviation (bottom), versus input hazard Figure 18. AF( f ) Figure 18.Ampliﬁcation Amplification factor, factor, (), (top), (top) and and standard standard deviation deviation (bottom), (bottom), versus versus input input hazard hazard on hard rock, r (), derived using accelerograms from the 1D SHAKE91 propagation (randomly onon hard hard rock, rock,S a(f()), derived, derived using usingn accelerograms accelerograms from from the the 1D 1D SHAKE91 SHAKE91 propagation propagation (randomly (randomly sampled): (a) =30 accelerograms; (b) =60 accelerograms, (c) = accelerograms. The sampled): (a) n =30= 30 accelerograms; (b) n =60= 60 accelerograms, (c) n == all accelerograms. The sampled): (a) accelerograms; (b) accelerograms, (c) accelerograms. The different colors correspond to the r () bins used to calculate the standard deviation. differentdifferent colors colors correspond correspond to to the theS a(f()) bins bins used used to to calculate calculate the the standard standard deviation. deviation.

(a) (b) (c) (a) (b) (c) Figure 19. Amplification factor, (), (top) and standard deviation (bottom), versus input hazard Figure 19. Amplification factor, (), (top) and standard deviation (bottom), versus input hazard Figureon hard 19. rock,Ampliﬁcation (), derived factor, AFusing( f ) ,n (top) accelerograms and standard from deviation the 1D (bottom), NOAH versuspropagation input hazard(randomly on on hard rock, (), derived using n accelerograms from the 1D NOAH propagation (randomly hardsampled): rock, S r((af) ), derived = 30 accelerograms; using n accelerograms (b) = from 60 theaccelerograms, 1D NOAH propagation (c) = All (randomly accelerograms. sampled): The a = 3 = 6 sampled): (a) 0 accelerograms; (b) 0 accelerograms, (c) = All accelerograms. The (adifferent) n = 30 colorsaccelerograms; correspond ( bto) nthe= 60()accelerograms, bins used to calculate (c) n = All the accelerograms. standard deviation. The different colors different colors correspondr to the () bins used to calculate the standard deviation. correspond to the Sa( f ) bins used to calculate the standard deviation.

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6.6. Discussion Discussion TheThe direct direct comparison comparison of of both both approaches approaches in in Figure Figure 2020 shows shows that that the the AM AM and and SM SM hazard hazard curves curves presentpresent a a good good fit fit at at all all available available return return periods periods fo forr the the three three studied studied spectral spectral periods. periods. Nevertheless, Nevertheless, despitedespite this good fit,fit, thethe above-mentioned above-mentioned numerical numerical limitation limitation of theof the AM AM method method prevents prevents this study this studyfrom performingfrom performing such comparisonsuch comparison even ateven the at most the commonmost common 475 years 475 years return return period, period, which which raises raisesquestions questions about theabout effectiveness the effectiveness of the AM of the approach AM approach for the incorporation for the incorporation of site effects of site in site-specific effects in site-specificPSHA. It is PSHA. important It is toimportant emphasize to emphasize that this limitation that this limitation will only affectwill only sites affect with sites strong with nonlinear strong nonlinearand/or high and/or seismic high demandsseismic demands such as thesuch Euroseistest as the Euroseistest site. The site. vertical The vertical asymptotes asymptotes of the hazardof the hazardcurves curves on soil on in Figuresoil in Figure20 correspond, 20 correspond, for each for spectral each spectral period, period, to the slope to the (green) slope (green) in Figure in 12Figure, and 12,it thereforeand it therefore corresponds corresponds to a non-linear to a non-linear saturation satura effecttion ofeffect the groundof the ground motion motion at the at surface the surface under understrong strong shaking, shaking, as it can as beit can better be better appreciated appreciated in Figure in Figure 21. 21.

1.0000 1.0000 1.0000 (PGA) (0.2s) (1.0s) ROCK FSM-NOAH 0.1000 0.1000 ASM-SHAKEFN 0.1000 AM-NOAH AAM-SHAKE

0.0100 0.0100 0.0100

Tr=475 yrs Tr=475 Tr=475

0.0010 0.0010 0.0010 Annual Rate Exceedance of Tr=4975 yrs Annual Rate Exceedance of Annual Rate of Exccedance of Annual Rate Tr=4975 Tr=4975

0.0001 0.0001 0.0001 0.01 0.10 1.00 0.01 0.10 1.00 0.01 0.10 1.00 Sa (g) Sa (g) Sa (g) Figure 20. Hazard curves on hard rock (blue) and soil using the equivalent-linear SHAKE91 Figure 20. Hazard curves on hard rock (blue) and soil using the equivalent-linear SHAKE91 simulations simulations(red) and nonlinear (red) and NOAH nonlinear simulations NOAH simulations (green) for both(green) AM for (points) both AM and (points) SM (solid and line) SM methods,(solid line) at methods,three different at three spectral different periods spectr (PGA,al periods 0.2 s and(PGA, 1.0 0.2 s). s and 1.0 s).

These results also show that the code-to-code variability introduces much more uncertainty on These results also show that the code-to-code variability introduces much more uncertainty on the hazard estimates than the method-to-method (AM, SM) variability. The use of SHAKE91 leads to the hazard estimates than the method-to-method (AM, SM) variability. The use of SHAKE91 leads to larger PGA values for return periods that are larger than 100 years, while the use of NOAH leads to larger PGA values for return periods that are larger than 100 years, while the use of NOAH leads to larger spectral accelerations for all of the return periods at intermediate and long spectral periods (0.2 larger spectral accelerations for all of the return periods at intermediate and long spectral periods (0.2 s s and 1.0 s). It is interesting to notice that nonlinear behavior is observed, even at relatively large and 1.0 s). It is interesting to notice that nonlinear behavior is observed, even at relatively large spectral spectral periods (1.0 s) and low return periods. However, the deamplification behavior with respect periods (1.0 s) and low return periods. However, the deamplification behavior with respect to rock to rock occurs only at short periods (PGA, 0.2 s), while at longer period (1.0 s), amplification can be occurs only at short periods (PGA, 0.2 s), while at longer period (1.0 s), amplification can be observed observed for all explored return periods. This is simply because the (1.0 s) period is close to the for all explored return periods. This is simply because the (1.0 s) period is close to the fundamental fundamental period of the site (~1.5 s), where linear amplification is strong (close to a factor of 10). Hence,period the of thenonlinearity site (~1.5 s),due where to the linear soil shear amplification modulus is degradation strong (close and to alarger factor damping of 10). Hence, (induced the undernonlinearity strong duestrain to thelevels) soil is shear not modulusstrong enough degradation to generate and larger deamplification damping (induced with respect under to strong the groundstrain levels) motion is on not rock. strong enough to generate deamplification with respect to the ground motion on rock.Concerning the ground motion variability analysis, the standard deviation of the ground motion Concerning the ground motion variability analysis, the standard deviation of the ground motion on soil, σS(), remains in both cases (SHAKE91 and NOAH) of the same order of magnitude at low σ s ( f ) groundon soil, motionSa , levels remains (Figure in both 13a), cases and (SHAKE91 it is larger and for NOAH) SHAKE91 of thethan same for orderNOAH of at magnitude strong ground at low motionground levels motion (Figure levels 13c). (Figure This 13 cana), andbe partially it is larger explained for SHAKE91 by the larger than for variability NOAH atof strongamplification ground ()motion using levels SHAKE (Figure than 13c). NOAH This can (Figure be partially 15, Table explained 3, σ() by). theAdditionally, larger variability our numerical of amplification results AF( f ) σ(AF) are in qualitativeusing SHAKE agreement than NOAH with observations (Figure 15, Tablefrom 3real, recordings). Additionally, on soil and our rock numerical [57,58], where results are in qualitative agreement with observations from real recordings on soil and rock [57,58], where the standard deviation on soil, σS(), is smaller than the standard deviation on rock, σS(), for all σ s ( f ) σ r ( f ) groundthe standard motion deviation levels (see on soil,FigureSa 13),. isCurrently, smaller than this thephenomenon standard deviation is presumed on rock, to beS adue, forto . nonlinearityall ground motionof the soil levels response (see Figure during 13 severe) Currently, shaking this[11], phenomenon and also the effect is presumed of a lower to bedamping due to atnonlinearity low strain levels of the on soil rock response that makes during the severe ground shaking motion [11 more], and sensitive also the effectto the of input a lower motion. damping at low strain levels on rock that makes the ground motion more sensitive to the input motion.

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Figure 21. Equivalent-linear (SHAKE91, red) and nonlinear (NOAH, green) predicted accelerations Figure 21. comparedEquivalent-linear with data recorded (SHAKE91, at Euroseistest red) (yellow), and nonlinear for three (NOAH, different green)spectral predicted periods (PGA, accelerations 0.2 s comparedand 1.0 with s). data recorded at Euroseistest (yellow), for three different spectral periods (PGA, 0.2 s and 1.0 s). Furthermore, we compared the predictions with current available data at the site. Figure 21 plots Furthermore,the spectral acceleration we compared on soil, the as predictions a function withof the current corresponding available input data spectral at the site.acceleration Figure 21on plots rock, for both nonlinear codes tested, together with recordings from the Euroseistest database [17]. the spectral acceleration on soil, as a function of the corresponding input spectral acceleration on rock, The same amount of observed earthquakes recordings (28 records) are used at the three different for both nonlinear codes tested, together with recordings from the Euroseistest database [17]. The spectral periods, but accelerations on rock below 0.001g are not displayed. same amountFrom ofvisual observed inspection, earthquakes the few recordingsobservations (28 that records) are available are used at atlow/intermediate the three different ground spectral periods,motion but levels accelerations are roughly on rockwithin below the range 0.001g pred areicted not by displayed. the two models. We can thus expect the Fromactual hazard visual curve inspection, of the site the to fewbe comprise observationsd between that the are two available proposed athazard low/intermediate models. SHAKE91 ground motionmight levels better are predict roughly site within surface the motion range for predicted low input by ground the two motion models. levels We can[S( thus) < 0.01 expect g], while the actual hazardNOAH curve predictions of the site might to be be comprised better for intermed betweeniate the input two proposedground motion hazard levels models. [0.01 g SHAKE91≤ S() ≤ 0.1 might betterg]. predict However, site many surface more motion recordings for low would input be groundnecessary motion to derive levels sound [Sr (conclusions.f ) < 0.01 g], For while higher NOAH a input acceleration levels [S() > 0.1 g], nothing can be said, as they are no recordings. As somer of the predictions might be better for intermediate input ground motion levels [0.01 g ≤ Sa( f ) ≤ 0.1 g]. accelerations recorded are located outside the scatter range, it seems that the simulations somewhat However, many more recordings would be necessary to derive sound conclusions. For higher input underestimate the ractual variability. acceleration levels [Sa( f ) > 0.1 g], nothing can be said, as they are no recordings. As some of the accelerations7. Conclusions recorded are located outside the scatter range, it seems that the simulations somewhat underestimate the actual variability. A comparison between two fully probabilistic methods to integrate site effects into PSHA has 7. Conclusionsbeen presented along this work. We found that the analytical approximation of the full convolution method, AM, is computationally less expensive, but it breaks down in the case of strong non-linearity Adue comparison to a numerical between limitation two (as fully for the probabilistic Euroseistest methods site). On tothe integrate other hand, site the effects full probabilistic into PSHA has beenstochastic presented method, along thisSM, work.requires We much found heavier that thecomp analyticalutations, approximationbut it provides results of the over full convolutionthe full method,acceleration AM, is computationallyrange, even when strong less expensive, non-linearity but occurs. it breaks Both down of the in methods the case provide of strong comparable non-linearity due toannual a numerical exceedance limitation rates over (as their for common the Euroseistest validity range. site). On the other hand, the full probabilistic To build the hazard curves on soil, two different propagation codes were used to calculate the stochastic method, SM, requires much heavier computations, but it provides results over the full time histories on soil at the surface, EL-SHAKE91 [42,43,44] and NL-NOAH [45]. In the Euroseistest acceleration range, even when strong non-linearity occurs. Both of the methods provide comparable case study, the code-to-code variability is found to control the uncertainty on the hazard estimates, annualrather exceedance than the method-to-method rates over their common variability validity (AM or range.SM). We found that the soil variability that was Toobtained build as the a result hazard after curves performing on soil, the two propagation different is propagation smaller than codesthe input were rock used variability. to calculate This the timeresult histories is in on qualitative soil at the agreement surface, with EL-SHAKE91 observations [42 from–44 ]real and recordings NL-NOAH on soil [45]. and In rock the Euroseistest[57,58]. case study,Currently, the this code-to-code phenomenon variability is presumed is foundto be due to to control nonlinearity the uncertainty of the soil response on the hazardduring severe estimates, rathershaking than the[11]. method-to-methodWe also found that the variability expression (AMdescribing or SM). the minimum We found required that the amount soil variability of records that was obtainedproposed asby a[11] result falls afterrather performing short in our thecase, propagation being insufficient is smaller and under than conservative, the input rock especially variability. This resultfor capturing is in qualitative the whole agreementvariability of with the site observations response. For from this real particular recordings example, on soil 8 to and 60 records rock [57 ,58]. per acceleration bin on rock were needed to predict the amplification within a 10% error margin (with Currently, this phenomenon is presumed to be due to nonlinearity of the soil response during severe respect to the true amplification based on the full stochastic sample), for both propagation codes and shaking [11]. We also found that the expression describing the minimum required amount of records at the three studies spectral periods (PGA, 0.2 s and 1.0 s). proposed by [11] falls rather short in our case, being insufficient and under conservative, especially for capturing the whole variability of the site response. For this particular example, 8 to 60 records per acceleration bin on rock were needed to predict the amplification within a 10% error margin (with respect to the true amplification based on the full stochastic sample), for both propagation codes and at the three studies spectral periods (PGA, 0.2 s and 1.0 s). Geosciences 2018, 8, 285 25 of 28

By comparing the two models with real in-situ data, we observed that the few available recordings at the Euroseistest are within the acceleration range that was predicted by both site response models (EL and NL). However, more data might be needed to validate this observation. We are aware that the selection of the synthetic time histories and the scaling procedure that we followed here might raise several questions that are related to the ‘true ground motion’ at the Euroseistest. However, the selection of the input time histories on rock is simply an intermediate step that, in our opinion, should not impact significantly the conclusions reached here. The accelerogram selection and generation process would impact the site nonlinear response in case of a deterministic study with one or only a few times series. We believe that the large number of computations performed here allows for considering that the results are statistically significant and quite robust with respect to the actual phase contents of the input rock time series. It must nevertheless be emphasized that the selection of accelerograms compatible to a given GMPE is still an open issue, which needs to be better addressed in the future. Recent works, such as those proposed by [55,60], allow for a more physical tuning of the source and crustal propagation parameters, while not warranting however to get the actual regional site and source parameters, neither avoiding the potential trade-off between some of them [38]. We are also aware that the nonlinear parameters that are assumed for the soil column (in particular the zero-cohesion) results in over-emphasizing the impact of the non-linearity of site response on the hazard curve, and thus limiting the validity domain of the AM approach. The present results, especially in terms of AM/SM comparison, are thus to be considered as an ‘extreme’ example for highly non-linear, thick soft site. It is likely that the validity domain of the AM approach would be much larger for shallow, high-plasticity clayey deposits. Finally, we encourage the use of ground-motion simulations calibrated with real data to integrate site effects into PSHA, since models with different levels of complexity can be included (e.g., point source, extended source, 1D, 2D, and 3D site response analysis, kappa, hard rock ... ), and the corresponding variability of the site response can be quantified. In most practical cases, this is presently not possible with real data because of their scarcity at high acceleration levels.

Author Contributions: C.A., P.-Y.B. and C.B. conceived and designed the experiments; C.A. and J.-C.G. performed the experiments; C.A. and P.-Y.B. analyzed the data; C.A., J.-C.G., contributed/materials/analysis tools; C.A., P.-Y.B. and C.B. wrote the paper. Funding: This work has been supported by a grant from Labex OSUG@2020 (Investissements d’avenir—ANR10 LABX56—http://www.osug.fr/labex-osug-2020/) and by the European STREST project (Harmonized approach to stress tests for critical infrastructures against natural hazards—http://www.strest-eu.org). Acknowledgments: Special thanks are due to Benjamin Edwards, Adrián Rodriguez-Marek, Valerio Poggi and an anonymous reviewer for fruitful discussions. Also, special thanks to Fabian Bonilla for facilitating the nonlinear code NOAH and additional technical assistance on its usage, and to Fabrice Hollender for providing valuable geotechnical and site-specific data to build the nonlinear models. All ground-motion records used in this study were generated using Boore 2003 Stochastic Method and his open source code SMSIM (http: //www.daveboore.com/). Hazard calculations on rock have been performed using the Openquake engine (http://www.globalquakemodel.org/), as well as the SHARE area source model (http://www.share-eu.org/). The post-processing toolkits developed by the GEM Foundation have been used. Special thanks to Geosciences (Switzerland) for the invitation to participate to this special issue. Conflicts of Interest: The authors declare no conflict of interest.

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