Key structural parameters affecting earthquake ground motion in 2D and 3D sedimentary structures
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Abstract
Alluvial valleys generate strong effects on earthquake ground motion (EGM). These effects are rarely accounted for even in sitespecific studies because of (a) the cost of the required geophysical surveys to constrain the site model, (b) lack of data for empirical prediction, and (c) poor knowledge of the key controlling parameters. We performed 3D, 2D and 1D simulations for six typical sedimentary valleys of various width and depth, and for a variety of modifications of these 6 “nominal models” to investigate sensitivity of EGM characteristics to impedance contrast, attenuation, velocity gradient and geometry. We calculated amplification factors, and 2D/1D and 3D/2D aggravation factors for 10 EGM characteristics, using a representative set of recorded accelerograms to account for input motion variability. The largest values of the amplification and aggravation factors are found for the Arias intensity and cumulative absolute velocity, the lowest for the rootmeansquare acceleration. The aggravation factors are largest for the vertical component. For each model, at least one EGM characteristic exhibits a significant 2D/1D aggravation factor, while all EGM characteristics exhibit significant 2D/1D aggravation factor on the vertical component. For all investigated sites, there is always an area in the valley for which 1D estimates are not sufficient. 2D estimates are insufficient at several sites. The key structural parameters are the shape ratio and overall geometry of the sedimentbedrock interface, impedance contrast at the sedimentbedrock interface, and attenuation in sediments. The amplification factors may largely exceed the values that are usually considered in GMPEs between soft soils and rock sites.
Keywords
Site acceleration Earthquake ground motion Amplification factor Aggravation factor Numerical modelling1 Introduction
Soft sediments have been known for a long time now as modifying (most often amplifying) earthquake ground motion and increasing damage: this resulted in their accounting in hazard assessment studies and building codes through simplified site characterization, coupled with sitedependent spectral shapes and levels of motions. All the site proxies used for such purposes implicitly refer to the local, 1D, structure of the site, and do not take into account the possibility of variations of the underground structure around the considered site, and the potentially associated effects on the intensity and characteristics of ground motion.
Meanwhile, a large number of cities or critical facilities are located in alluvial valleys or sedimentary basins with pronounced 2D or 3D underground geometry. Many investigations of different kinds have thus been carried out over the last 5 decades about the seismic response of such non1D soil structures. Some addressed mainly the physics of wave propagation in generic valleys or basins, some were dedicated to case studies for specific sites either hit by strong events (such as Caracas, Mexico City, Los Angeles or SeattleVancouver basins, KobéOsaka areas, Kanto plain, for example) or being the focus of dense instrumentation (Mygdonian basin in Greece, alpine valleys such as the Grenoble and Martigny areas in Western European Alps); some other had more applied objectives, from the benchmarking of numerical codes on canonical cases or real sites (see, e.g., Moczo and Irikura 1999; Day et al. 2003; Chaljub et al. 2010; Maufroy et al. 2015, 2016, 2017), to the attempt to quantify the associated amplification effects (e.g., SISMOVALP project, Lacave and Lemeille 2006; Barnaba et al. 2010).
Summarizing the results of all these studies is much beyond the scope of the present article. In short, the lateral variations of thickness in alluvial valleys or basins have been shown to generate peculiar wave propagation phenomena (diffraction of surface waves, possible focusing of body waves, vertical and lateral reverberations) leading to increased wave trapping and interferences, and significant differences (increased duration, mostly overamplification, sometimes deamplification) with respect to the case of horizontally stratified layers (“1D soil columns”). Despite their qualitative prediction by theory for several decades, and their actual observation in real recordings or damage distribution (e.g., in Kobe in 1995), such effects are only very rarely accounted for even in sitespecific studies, because of (a) the cost of the required geophysical surveys to constrain geomechanical parameters of the underground structure not only underneath but also around the target site, (b) the insufficient number of welldocumented observations that prevents any statistical treatment for a purely empirical prediction, and (c) the lack of comprehensive enough parameter study that would allow to identify the key controlling parameters and to quantify their effects.
The last issue has been addressed in a number of recent works, one of the last ones being the European project NERA (see deliverable D11.5 at http://www.orfeuseu.org/organization/projects/NERA/Deliverables/), as partly illustrated by Riga et al. (2016): following the initial paper by ChávezGarcía and Faccioli (2000), it has become usual to characterize these geometrical effects through an “aggravation factor” which quantifies the ratio between 2D (or 3D) and 1D ground motion amplification at a given site (see also ChávezGarcía 2007). This was done, e.g., by Makra et al. (2001, 2005) for the Euroseistest site (Mygdonian basin, Greece), by Paolucci and Morstabilini (2006) for a family of hypothetical basin edges (walls or wedges), by Kumar and Narayan (2008), Narayan and Richharia (2008), Hasal and Iyisan (2012), Gelagoti et al. (2012) and Vessia and Russo (2013) for hypothetical basins in the nonlinear and linear equivalent cases. Such engineeringoriented studies use most often the aggravation factor on the responsespectra amplification factor, and generally end up with values of aggravation factor below 2—except for Makra et al. (2001, 2005) and Hasal and Iyisan (2012) for whom aggravation factor reach values up to 4 in specific cases. However, despite the significance of the associated computational efforts, there is not yet a wide consensus on the amount of such “overamplification” (which may indeed sometimes consist in “underamplification, especially close to the edges, see, e.g., Riga et al. 2016), and the task is so huge and the way is long before such coefficients can be accepted and used in building codes.
 a.
What could be the amount of actual amplification for realistic situations, especially as it may exceed by far what is predicted by GMPEs based on simple, 1D site proxies?
 b.
In which case is the aggravation factor significant, or in other terms, when 2D or 3D site response studies should be required?

Six realistic site geometries have been selected (including two sites with extensive validation against data—Euroseistest and Grenoble), for which 1D, 2D and sometimes 3D computations have been performed and systematically compared for a large number of receivers (5251 in total) located either within the valleys of basins, or on the bedrock edges.

Several earthquake ground motion characteristics have been selected because of their engineering relevance for such a comparison: for each of them (scalar or vector) both the amplification levels (with respect to the reference) and the aggravation factors have been computed as detailed in the companion article by Kristek et al. (2018). The characteristictocharacteristic have also been analysed to identify a limited set of quasiindependent characteristic to be analysed in more detail.

In addition to the six “nominal models” with a given geometry and a given set of mechanical parameters (velocities, density, attenuation), sensitivity studies have been performed to investigate the sensitivity to impedance contrast, attenuation, velocity gradient and geometry), in view of identifying the key parameters to be known/measured for a new site.

A specific attention has been paid also to the rock sites located on each side of the valleys in order to assess the amount of variability of the rock ground motion due to the feedback from the valley or basin.
2 Overview of sites and computations
We perform extensive numerical simulations for two sets of structural models: the first set of “nominal” models corresponds to real sites representing a meaningful scatter of local surface sedimentary and bedrock conditions, whereas the second set consists of modifications to the nominal models for investigating sensitivity of several “EGM” (Earthquake Ground Motion) characteristics to important structural features. The set of nominal models consists of a few typical surface sedimentary structures: Site 1 (Mygdonian basin, near Thessaloniki, Greece)—a shallow sedimentfilled basin, Site 2 (Grenoble valley, France)—a typical deep Alpine sedimentfilled valley, Site 4—a small shallow sedimentfilled valley, Site 5—a midsize sedimentfilled valley, Site 6—a relatively small shallow sedimentfilled valley, and Site 7—a shallow relatively large sedimentfilled valley. The set of modified models includes variations in velocity and attenuation in sediments, velocity in bedrock, geometry of the border slope, velocity and thickness of sediments, and 3D meander extension. Forward numerical simulations were performed in the linear domain with the finitedifference (FD) method (Moczo et al. 2014; Kristek and Moczo 2014; Chaljub et al. 2010, 2015). 3D simulations were performed for Sites 1, 2 and 6, assuming a vertical plane wave incidence, and also for several doublecouple (DC) point sources located at different positions. 2D simulations were performed for selected 2D profiles in the 3D models (Sites 1 and 2) and also all other 2D nominal models and their variants (Sites 4–7) assuming the vertical plane wave incidence. 1D simulations have been performed with 1D models corresponding to all the selected receiver positions along all the 2D profiles.
2.1 Sites
2.1.1 3D models
2.1.2 2D models
Site 4 is the smallest of the investigated sedimentary structures, consisting of a shallow valley (thickness 120 m, width 920 m) with velocity gradient in sediments (Online Resource 4) and local fundamental resonant frequencies above 2 Hz. Site 5 is a midsize, deep sedimentfilled valley (thickness 581 m, width 3.5 km) with relatively strong velocity gradient in sediments (Online Resource 5), relatively large contrast at the sedimentbedrock interface and local fundamental resonant frequencies below 1 Hz, the minimum being around 0.5 Hz. Site 6 has an intermediate size (thickness 161 m, width 2.2 km), between sizes of sites 4 and 5, and presents a thin lowvelocity layer at the surface and a large velocity contrast at the sedimentbedrock interface; the corresponding local fundamental resonant frequencies are around 1 Hz. There are 2 alternative models: one with a thin, soft layer (\( V_{S} \,\, = \,\,230\,\,{\text{m/s}} \), 5 m thickness) overlying homogeneous sediments with a 600 m/s Swave velocity, one with the same top thin, soft layer, overlying sediments with a gradient in the P and Swave velocities (Online Resource 6). Site 7 is the relatively large, shallow valley (thickness 510 m, width 6.2 km) with three layers of sediments, two of them with strong gradients (Online Resource 7). There is a large velocity contrast at the sedimentbedrock interface. The fundamental resonant frequencies are below 1 Hz, the minimum being approximately 0.5 Hz.
2.2 Sensitivity studies

In the model of the Grenoble valley a highvelocity layer at the free surface was alternatively assumed, corresponding to the realistic situation where recent, fluviatile gravel layers overtop older, softer, lacustrine clayey deposits.

Alternative attenuation parameters were also considered for the Grenoble valley models (\( Q_{S} \) derived from 1D nonlinear simulations to investigate the impact of larger attenuation throughout the very thick deposits on the 3D and 2D responses) and Sites 5–7 (different \( Q_{S} \,\,  \,\,V_{S} \) scaling, from no attenuation at all to \( Q_{S} \,\, = \,\,{{V_{S} } \mathord{\left/ {\vphantom {{V_{S} } {20}}} \right. \kern0pt} {20}} \) and \( Q_{S} \,\, = \,\,{{V_{S} } \mathord{\left/ {\vphantom {{V_{S} } {40}}} \right. \kern0pt} {40}} \)).

The effect of the detail of valley edge geometry was investigated by considering alternative sloping angles for the sedimentbedrock interface for various 2D models: Grenoble profile P1, Sites 5 and 6.

The impact of the sediment/bedrock impedance contrast was investigated by considering alternative bedrock velocities for models of Sites 5–7, and also by alternative velocities in sediments for site S6h.

In two modifications of model S6h simultaneous variations in the velocity and thickness of sediments were considered: the velocity in the homogeneous sediments is increased by 40% in one modification and decreased by 40% in the other while the local sediment thickness is increased/decreased, respectively, in order to keep the local 1D fundamental resonant frequency unchanged. The sediment Pwave velocity is however kept unchanged, as they correspond to high Poisson ratio, water saturated sediments.

Eventually, 3D meander extensions were considered for models S6h and S6g. Thus 3D and 2D models for Sites 1, 2 and 6 make it possible to quantify the differences between 2D and 3D site responses.
2.3 Numerical simulations and computations of ground motion characteristics
3 Overview of calculated EGM aggravation factors
For each component of motion the following EGM scalar characteristics are evaluated: \( \overline{AF} \left\{ {pga} \right\} \)—average amplification factor for \( pga \), \( \overline{AF} \left\{ {pgv} \right\} \)—average amplification factor for \( pgv \), \( \overline{{F_{A} }} \)—shortperiod amplification factor, \( \overline{{F_{V} }} \)—longperiod amplification factor, \( \overline{{F_{0} }} \)—amplification factor centred about the minimum fundamental resonant frequency \( f_{00} \), \( \overline{{F_{L} }} \)—amplification factor centred about the local fundamental resonant frequency \( f_{0} \), \( \overline{AF} \left\{ {a_{rms} } \right\} \)—average amplification factor for rootmeansquare acceleration, \( \overline{AF} \left\{ {SI} \right\} \)—average amplification factor for spectrum intensity, \( \overline{AF} \left\{ {CAV} \right\} \)—average amplification factor for cumulative absolute velocity, and \( \overline{AF} \left\{ {I_{A} } \right\} \)—average amplification factor for Arias intensity. For a given position at a site it is reasonable to define a 3D/2D aggravation factor as ratio of values of an EGM characteristic obtained from 3D and 2D numerical simulations. Analogously, a 2D/1D aggravation factor compares values of an EGM characteristic obtained from 2D and 1D simulations. We refer to Eq. (62) of the accompanying article by Kristek et al. (2018).
 a.
\( \overline{AF} \left\{ {pga} \right\} \) is correlated with \( \overline{{F_{A} }} \). \( \overline{AF} \left\{ {pga} \right\} \) is correlated with other quantities more than \( \overline{{F_{A} }} \) is. Thus, \( \overline{AF} \left\{ {pga} \right\} \) will be excluded in the following.
 b.
\( \overline{AF} \left\{ {SI} \right\} \) is correlated with \( \overline{AF} \left\{ {pgv} \right\} \) and also with \( \overline{{F_{V} }} \), which basically means that SI is an intermediate frequency characteristic. As \( \overline{AF} \left\{ {pgv} \right\} \) and \( \overline{{F_{V} }} \) are more common EGM characteristics, \( \overline{AF} \left\{ {SI} \right\} \) will be excluded in the following.
 c.
\( \overline{AF} \left\{ {I_{A} } \right\} \) is correlated with \( \overline{AF} \left\{ {CAV} \right\} \). \( \overline{AF} \left\{ {I_{A} } \right\} \) is correlated with other quantities more than \( \overline{AF} \left\{ {CAV} \right\} \) is. Thus, \( \overline{AF} \left\{ {I_{A} } \right\} \) will be excluded to the benefit of \( \overline{AF} \left\{ {CAV} \right\} \) as both characteristics take into account in some way both the motion amplitude and its duration.
 d.
\( \overline{AF} \left\{ {a_{rms} } \right\} \) is correlated with \( \overline{{F_{A} }} \) mainly for values of aggravation factor larger than 1.25. \( \overline{{F_{A} }} \) is kept due to correlation with \( \overline{AF} \left\{ {pga} \right\} \). Thus, \( \overline{AF} \left\{ {a_{rms} } \right\} \) will be excluded.
 e.
\( \overline{{F_{0} }} \) is correlated with \( \overline{{F_{L} }} \). \( \overline{{F_{0} }} \) is more artificial (less founded) quantity since it is not related to the considered receiver, but the whole valley. Thus, \( \overline{{F_{0} }} \) will be excluded.
 f.
\( \overline{AF} \left\{ {pgv} \right\} \) is correlated with \( \overline{{F_{V} }} \) and \( \overline{{F_{L} }} \). \( \overline{{F_{V} }} \) and \( \overline{{F_{L} }} \) are less correlated. Thus, \( \overline{AF} \left\{ {pgv} \right\} \) will be excluded.
The scatter matrices and values of the correlation coefficients for the 2D or 3D amplification factors (shown in Kristek et al. 2015) are largely consistent with those obtained for the aggravation factors. Consequently, we further focused our investigations and results on the four EGM characteristics that exhibit (in average, all components considered) the lowest correlations, i.e. those corresponding to \( \overline{{F_{A} }} \), \( \overline{{F_{V} }} \), \( \overline{{F_{L} }} \) and \( \overline{AF} \left\{ {CAV} \right\} \). The two first are short and intermediateperiod characteristics which are more and more often used in earthquake engineering for common constructions (Borcherdt 1994, 2002; Paolucci and Smerzini 2017). The third one focuses on the behaviour around the site fundamental frequency \( f_{0} \), a characteristic that is more and more often proposed for site classification (Castellaro et al. 2008; Luzi et al. 2011; Cadet et al. 2012; Pitilakis et al. 2012, 2013). The fourth one is used as an index related to potential damage in nuclear engineering and to soil liquefaction, and was found by Campbell and Bozorgnia (2010) as providing the NGA GMPEs with the lowest aleatory variability values, i.e., as the “best predictable” ground motion characteristic.
4 Example results: aggravation factors for cumulative absolute velocity
Analogous figures for \( \overline{{F_{A} }} \), \( \overline{{F_{V} }} \) and \( \overline{{F_{L} }} \) (shown in the Online material 11–13) reveal similar features except that mean and median values of the aggravation factors on the horizontal components are smaller than those for \( \overline{AF} \left\{ {CAV} \right\} \).
Figure 11b shows 3D/2D aggravation factors for \( \overline{AF} \left\{ {CAV} \right\} \)—again separately for the sediment sites and rock sites. 3D effects are significant in the Grenoble valley (Site 2)—profiles P1–P4, in blue. 3D effects are present also in the Mygdonian basin but they are not dominant. Unlike the 2D/1D aggravation factors, the vertical component is not significantly different from the horizontal components: the impact of the StoP conversions and edgegenerated Rayleigh waves is included in both 2D and 3D simulations, while it is not in 1D simulations.
Analogous figures for \( \overline{{F_{A} }} \), \( \overline{{F_{V} }} \) and \( \overline{{F_{L} }} \) (not shown here) reveal similar features.
Figures 12 and 13 detail the values of the aggravation factors along profiles P1 and P4 in the Grenoble valley, Site 2, respectively (see Fig. 2 for the location of 2D profiles). Figure 12 shows that the 3D effects along profile P1 are not negligible (i.e., aggravation factors larger than 1.25) despite the intuitive impression that the profile might be suitable for 2D modelling. This is clear not only from the values of the aggravation factor but also from the fact that the aggravationfactor curve does not simply reflect the geometry of the profile itself. The 2D/1D aggravation factor on the vertical component is again significantly larger than those on the horizontal components. Such difference is not seen in the 3D/2D aggravation factors. Figure 13 confirms that 3D effects are significant along profile P4—as we could intuitively expect from the sedimentbedrock interface geometry in that part of the valley—see the position of profile P4 at the intersection of three valley branches in Fig. 2.
5 Major results from the sensitivity analysis
As indicated in Fig. 6, several “variants” were considered for each site to further investigate the impact of several geomechanical parameters on the amplification and aggravation factors. Rather than detailing the results for each site, we group below the sensitivity results according to different subsurface parameters, starting with velocity in bedrock and sediments, then addressing the effect of sediment attenuation, and finally considering the effects of subsurface geometry (coupled thickness and velocity variations, edge slopes and 3D effects in meandertype quasicylindrical valleys). In each case, we display results only for one or two EGM characteristics from the four preselected ones.
We performed simulation only for one alternative attenuation model in 3D—for the Grenoble valley. Therefore we do not have sufficient sensitivity results for 3D. However, that sole simulation provides results similar to those seen based on 2D simulations: larger attenuation leads to a smaller aggravation factor (Kristek et al. 2015).
6 Conclusions
We performed 3D simulations for 3 3D local surface sedimentary structures (one of them being an artificial/virtual 3D extension of a 2D structure), 2D simulations for 12 2D crosssections (7 of them being selected 2D profiles in the 3D structures), and 1D simulations for local 1D models in the 2D models. Using pseudoimpulse responses from direct finitedifference simulations and a set of specified 11 reference accelerograms we investigated the amplification and aggravation factors in the specified local “nominal” sedimentary structures in terms of average (over the 11 reference accelerograms) amplification and aggravation factors for several EGM characteristics. In addition, the effects of uncertainty in the bedrock velocity, velocity in sediments, attenuation in sediments, interface geometry and simultaneous variations in velocity and thickness of sediments was investigated through sensitivity studies by varying the mechanical or geometrical parameters around their nominal values.
The values of amplification and aggravation factors depend on the considered EGM characteristic: largest values are found for the Arias intensity \( I_{A} \), ahead of the cumulative absolute velocity \( CAV \), lowest values are found for root mean square acceleration. An analysis of ground motion on the rocky valley edge sites indicates the existence of some feedback effects which slightly modify the “freefield” rock motion; however, the associated ground modifications remain most often smaller than 20–25%, even for the most sensitive EGM characteristics. We have therefore considered as “significant” all the valley effects resulting in aggravation factor larger than 1.25.
For all the considered sites, there is at least one surface point for which at least one EGM characteristic exhibits a significant 2D/1D aggravation factor (i.e., larger than 1.25). In particular, all EGM characteristics exhibit significant 2D/1D aggravation factor on the vertical component.
The 2D/1D aggravation actors are component dependent: they are found systematically the largest for the vertical component, and the smallest for the inplane component. For the antiplane case, there is no componenttocomponent exchange of energy, while for the inplane case, the generation of Rayleigh waves on valley edges, together with the SV to P and (to a less extent) P to SV conversions, result in a partitioning of energy on the two components, leading to larger aggravation factors on vertical component, and lower ones on the inplane component. Such a large sensitivity of amplification of vertical ground motion to the 2D (or 3D) underground structure should be kept in mind for the design of structures or equipment which should be designed for vertical motion (such as bridges or overhead cranes), especially as a large majority of the structural engineering community considers there is no amplification for vertical motion on soft soils (see for instance the recommended design spectra in EC8 regulations). The latter is in our opinion a direct consequence of the traditional 1D modelling approach considering only vertically incident plane waves: for the vertical component it should be supplemented with the consideration of obliquely incident S waves at least to account for the already existing StoP conversions for either obliquely incident waves or sloping sedimentbedrock interfaces.
For the considered cases (three sites), the 3D/2D aggravation factors are smaller than the corresponding 2D/1D values. They are the most pronounced mainly in the Grenoble site (Site 2) because of the Y shape which cannot be approximated by a 2D profile (profile P4). However, even for such a complex 3D site, the classical, vertically impinging plane wave assumption, is found to provide rather robust and reliable estimates of amplification factors. The planewave excitations should not, however, replace a DC point source (or extended sources) if such sources are identified to represent a possible or likely excitation for a given 3D site.
The main partial conclusion is that 1D estimates of EGM characteristics are not sufficient at any of the investigated sites.
This series of computations also led to several semiquantitative/semiqualitative findings concerning the sensitivity of aggravation factors to several structural parameters, and to identify the key impact of some of them.

As expected the aggravation factor is found to increase with the shape ratio (maximum thickness over valley width). Nevertheless, even relatively shallow valleys (shape ratios as low as 0.05) can lead to significant aggravation factors in case of relatively wide, shallow sloping edge (Site 1, Euroseistest, see also Chaljub et al. 2015; Maufroy et al. 2015, 2016, 2017). It is thus important to estimate not only the overall shape ratio, but also the overall geometry of the sedimentbedrock interface, in particular the sloping angles of the sedimentbedrock interface on the valley edges;

The impedance contrast at the sedimentbedrock interface impacts not only the amplification factor, as expected, but also the aggravation factor: for a given geometry, the larger the impedance contrast, the larger the aggravation factor. This is interpreted as related to the more efficient generation and trapping of surface waves in 2D and 3D models.

The attenuation in sediments similarly impacts both the amplification and aggravation factors: larger attenuation results in smaller amplification and aggravation factors, especially for the vertical component.

Finally, it is found that, because of these geometrical effects, the amplification factors may largely exceed the values that are usually considered in GMPEs between soft soils and rock sites: this should be kept in mind when dealing with the design of critical, nuclearlike facilities.
Notes
Acknowledgements
This work was supported in part by project SIGMA (EDF, AREVA, CEA and ENEL) and also by the Slovak Research and Development Agency under the contract APVV150560 (Project IDEFFECTS). Part of the calculations were performed in the Computing Centre of the Slovak Academy of Sciences using the supercomputing infrastructure acquired in project ITMS 26230120002 and 26210120002 (Slovak infrastructure for highperformance computing) supported by the Research and Development Operational Programme funded by the ERDF.
Supplementary material
References
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