Computation of amplification factor of earthquake ground motion for a local sedimentary structure
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Abstract
We present methodology of calculating acceleration and corresponding earthquake ground motion characteristics at a site of interest assuming acceleration at a reference site for two basic configurations. In one configuration we assume that the reference ground motion is not affected by the local structure beneath the site of interest. In the other configuration we assume that the reference ground motion is affected by the local structure. Consequently, the two configurations differ from each other by the presence of the reference site within the computational model. For each of the two configurations we assume two wavefield excitations: a vertical planewave incidence and a point doublecouple source. We illustrate the methodology on the example of the Grenoble valley. The extensive investigation of effects of local surface sedimentary structures based on the developed methodology is presented in the accompanying article by Moczo et al. (Bull Earthq Eng, 2018) in this volume.
Keywords
Site acceleration Earthquake ground motion Amplification factor Aggravation factor Numerical modelling1 Introduction
SIGMA was a Research & Development program of EDF (Electricité de France), AREVA, CEA (Commissariat à l’énergie atomique et aux énergies alternative) and ENEL in 2011–2015 (Senfaute et al. 2015; Pecker et al. 2016, 2017). Being focused on characterization of seismic ground motion in France and nearby countries, its main goal was to obtain robust and stable estimates of the seismic hazard based on proper characterization of uncertainties. The Work Package 3 (WP3) of the project aimed in developing methods of predicting whether a specific site needs a special investigation with respect to its site conditions in view of including site effects in the seismic hazard estimation. One of specific items of WP3 was to investigate potential of a few typical, virtual site configurations to undergo specific site effects associated to the 2D or 3D geometry of the underground structure. This was done using 1D, 2D and 3D forward numerical simulations to identify key parameters controlling the site amplification, and to quantitatively characterize their amount, with a special focus on the modifications to “classical” 1D effects by the 2D or 3D geometry.
A set of 7 nominal models for typical, virtual underground configurations has been defined (including a few real sites such as the Grenoble Alpine valley), together with a set of modifications of the nominal models to investigate effects of variations of a few, carefully selected, structural parameters on earthquake ground motion. Forward numerical simulations were performed in the linear domain with the finitedifference (FD) method (Moczo et al. 2014; Chaljub et al. 2010, 2015). In addition to 3D simulations for 3D models assuming a vertical plane wave incidence and/or point doublecouple (DC) source, 2D simulations were performed for selected 2D profiles in the 3D models and several 2D nominal models assuming the vertical plane wave incidence, together with 1D simulations for 1D models for selected receiver positions along 2D profiles and in 2D models.
If no records or insufficient number of records are available for the investigated site it is necessary to account for a potential variability of earthquake ground motion. It is reasonable to use a set of properly selected accelerograms recorded at different locations to represent the ground motion variability. We may assume that the records represent ground motions at a free surface of a halfspace. This assumption corresponds to the typical situation in probabilistic seismic hazard assessment (PSHA): the estimated characteristics of the earthquake ground motion relate to outcropping “standard” rock. The question is then how the presence of a local surface sedimentary structure (LSSS) modifies the ground motion. In a computational (numericalmodelling) approach we can quantify an effect of LSSS by considering two separate computational models—a model of LSSS (spatial domain for numerical modelling) that does not include the reference site, and the model of the reference site.
In this article we present methodology for computational estimate of acceleration and other ground motion characteristics at a site of interest for a set of accelerograms corresponding to the ground motion at the reference site. We derive formulas for the site acceleration if the reference site is not in the model. First we assume that the ground motion is due to a planewave excitation. This assumption is very common but not very realistic. We thus also assume that the ground motion is due to a point DC source. The pointsource assumption has also limitation because it is not valid for larger magnitudes. The pointsource assumption is inadequate if there is a reasonable argument to assume an extended source near the site of interest. Then it is more appropriate to perform a case study. We complete the methodology by defining the frequencydependent and singlevalued amplification factors for five selected earthquake groundmotion characteristics. We illustrate the methodology by a numerical example for the typical Alpine Grenoble valley in France. Finally we present (in Appendices 1 and 2) derivations of formulas for site accelerations in situation when we cannot exclude the effect of LSSS on ground motion at a reference site and both the reference site and LSSS have to be included in one computational model.
The application of the methodology is presented in the accompanying article by Moczo et al. (2018).
2 Computation of amplification factors of earthquake ground motion
2.1 Site acceleration
2.1.1 Reference site is not in the model, planewave excitation
For this configuration, shown in Fig. 1, we assume ground motion due to a vertical incidence of a plane wave.
Pseudoimpulse input signal
Denote the Fourier spectrum of the input signal by \({\mathcal{F}}p\left( f \right)\).
Matrix of the timedomain pseudoimpulse responses at a site
Matrix of the Fourier transfer functions
Acceleration at the free surface of a halfspace
Assume acceleration \(\vec{a}_{{ i}} (t)\) at site HAL, that is, at the free surface of a homogeneous halfspace. This means, that \(\tfrac{1}{2} \vec{a}_{{ i}} (t)\) is the acceleration of the vertically incident plane wave.
Note, however, that in the numerical simulations we cannot use exactly \(\tfrac{1}{2} \vec{a}_{{ i}} (t)\) for a convolution in the local structure. This is because the numerically evaluated transfer function includes effects of grid dispersion (although optionally small, corresponding to a chosen spatial discretization). Consequently, if we replaced the local structure by a homogeneous medium (getting thus the model of a homogeneous halfspace), we would not get exactly \(\vec{a}_{{ i}} (t)\) at the free surface for \(\tfrac{1}{2} \vec{a}_{{ i}} (t)\) in the incident wave. Therefore we simulate propagation of the plane wave in the grid and obtain \(\tfrac{1}{2} \vec{a}_{{ i}} (t)\) affected by the grid dispersion. We then apply such numerically obtained \(\tfrac{1}{2} \vec{a}_{{ i}} (t)\) in the convolution.
Site acceleration
2.1.2 Reference site is not in the model, excitation by a doublecouple point source
In the configuration shown in Fig. 2 we assume that a reference acceleration \(\vec{a}_{{ i}} (t); i \in \left\{ {1, \ldots ,n} \right\},\) at the free surface of a homogeneous halfspace is caused by some DC point source at a specified hypocentre position. We want to know the corresponding acceleration \(\vec{s}_{i} \,(t)\) at a site of interest SIT.
First we have to find the DC mechanism. Because an arbitrary DC point source can be expressed as a linear combination of 6 independent elementary sources (canonically oriented dipoles and couples) we have to numerically simulate surface motion separately for each of the 6 elementary sources at the hypocentre position. From the 3 components of the reference acceleration and 6 × 3 components of the accelerations caused by the 6 elementary sources we have to find 6 coefficients of the linear combination of the elementary sources that gives the sought DC mechanism producing \(\vec{s}_{i} \,(t)\) at SIT. Let us note why we have to determine specific mechanism if we know the source of each of the accelerograms considered for representing scatter in the wavefield excitation. We want to assume, e.g., 3 hypocentre positions for the Grenoble valley (the case investigated in the companion paper). We want to assume for each chosen pointsource hypocentre all reference accelerograms. Therefore we have to find specific mechanism for each of the accelerograms separately. Such a procedure also allows to compare the amplification factors for point sources and incident plane waves.
If we simulate surface motions for each of 6 elementary sources also in the model with LSSS, we can use coefficients of linear combination for \(\vec{a}_{i} \left( t \right)\) for evaluating \(\vec{s}_{i} \,(t)\) corresponding to the ith DC source.
In the following we present the detailed theory.
Elementary sources e
Particle velocity at halfspace due to an elementary source
Let \({\mathbf{A}}_{elem}^{\text{HAL}}\) denote the matrix of the corresponding acceleration seismograms.
Particle velocity at a site due to an elementary source
Let \({\mathbf{A}}_{elem}^{\text{SIT}}\) Denote the matrix of the corresponding acceleration seismograms.
Acceleration at the free surface of a halfspace
Site acceleration
In fact, given our numericalmodelling method, the velocitystress FD scheme, we primarily obtain \({\mathcal{F}} {\mathbf{S}}_{elem}^{\text{HAL}}\) and \({\mathcal{F}} {\mathbf{S}}_{elem}^{\text{SIT}}\), and therefore we make use of relation (21) for determining MESH.
2.2 Amplification factors
The amplification factors are looked for on several ground motion characteristics (singlevalued or frequencydependent vector values as specified in Sect. 2.2.2) that are not related linearly with their values for the input motion (unlike a Fourier spectral ratio). It is thus needed to consider several realistic input accelerograms, in order to get robust estimates on the corresponding average amplification factors (and their signaltosignal variability).
2.2.1 Selection of reference accelerograms

motion recorded on rock or stiff soil sites,

motion recorded in the near source area,

magnitude range,

very good signaltonoise ratio over a wide frequency band with sufficiently low highpass frequency,

wide distribution of peak frequencies of the peak acceleration response spectrum within the considered frequency range,

sufficient number of records for a meaningful statistical analysis.
Specific choice and a table of selected accelerograms will be given in Sect. 3 presenting a numerical example.
2.2.2 Frequencydependent amplification factors
Let s_{ξ,i} be the ξth component of a site acceleration corresponding to the reference ith accelerogram \(\vec{a}_{\,i}\). Then we can define the following frequencydependent and singlevalued amplification factors between the site and reference accelerations.
Amplification factor
Average amplification factor
Standard deviation of the amplification factor
2.2.3 Singlevalued amplification factors
In addition to the frequencydependent response spectra, a few singlevalued ground motion intensity measures are often used to characterize the level of ground motion: amplification factors can therefore be defined also for such parameters. Here we consider some of them, such as the average amplifications factors in some absolute or siterelated frequency ranges.
Shortperiod and longperiod average amplification factors \(\overline{{F_{{A_{\xi } }} }}\) and \(\overline{{F_{{V_{\xi } }} }}\)
The central short period (0.1 s) was deliberately considered lower than in Borcherdt (1994) (0.3 s), considering the importance of highfrequency equipment in the nuclear industry, and the fact that in the response spectra domain, as shown by Bora et al. (2016), the “highfrequency” values are controlled not only by the Fourier content at the same frequency, but by the whole spectral contents below that frequency.
Average amplification factor for [0.75, 3.0] f _{ 0 } and [0.75, 3.0] f _{ 00 } \(\overline{{F_{{L_{\xi } }} }}\) and \(\overline{{F_{{0_{\xi } }} }}.\)
Such definitions are consistent with those the short and midperiod amplification factors (2octave bandwidth), and are centred on 1.5f_{0} and 1.5f_{00}, respectively, instead of f_{0} and f_{00}, to account for the possibility of slight frequency shifts due to 2D or 3D effects (Bard and Bouchon 1985), and for the absence of symmetry around resonance frequencies, as amplification is larger at f_{0} and beyond, than below f_{0}.
Average amplification factor for a singlevalued earthquake ground motion characteristic
Appendix 3 defines characteristics of earthquake ground motion calculated in this study.
3 Numerical examples
Results of extensive numerical simulations and analysis for a set of local surface sedimentary structures is presented in the accompanying article by Moczo et al. (2018). Here we restrict to illustration of the described methodology on the example of the Grenoble valley. The Grenoble valley is a typical deep sedimentfilled Alpine valley. Two aspects make it important: (1) Grenoble urban area with significant population, modern industry and research facilities. (2) Such “alpine valley” configuration occurs in different other areas within the European Alps, and in other mountainous areas with embanked valleys filled with young, postglacial lacustrine sediments.
The numerical simulations of seismic motion are performed using the Fortran95 computer codes FDSim3D and FDSim2D (Kristek and Moczo 2014). The computational algorithm is based on the (2,4) velocitystress staggeredgrid FD explicit heterogeneous scheme on the Cartesian discontinuous spatial grid. Here, (2,4) means the 2ndorder accuracy in time and 4thorder accuracy in space. In the FD method both medium and wavefield are represented by values in the discrete space–time grid. An explicit scheme for updating a particle velocity at a spatial position is obtained by a discrete approximation of the equation of motion and linear stress–strain relation formulated in the particlevelocity vector and stress tensor. The method was concisely described in the SIGMA deliverable D397 (Kristek et al. 2013). The basic references are the book by Moczo et al. (2014), and articles Etemadsaeed et al. (2016) and Kristek et al. (2017).
3.1 Model of the Grenoble valley, France
Mechanical parameters – Grenoble valley (Site 2)
Unit  Position  V_{ P }1  V_{ P } 2  V_{ S } (m/s)  ρ1  ρ2  Q s  Q p  

Z_{1}  Z_{2}  
(m)  (m/s)  (kg/m^{3})  
Layer 1  0  24  2200 + 1.2z  320 + 28\(\sqrt z\)  2140 + 0.125z  V_{ S }/10  max (V_{ P }/20, 2Q_{ s })  
Layer 2  24  70  1450 + 1.2z  54.6\(\sqrt z\)  
Layer 3  Variable  300 + 19\(\sqrt z\)  
Bedrock  ∞  5600  3200  2720  ∞  ∞ 
3.2 Selected accelerograms
Parameters of 11 selected accelerograms
RESORCE waveform ID and station name  Site class (EC8)  Earthquake (Name, date, magnitude)  Distance: Epicentral E or RJB R  Component  pga (cm/s^{2})  F_{ peak } (Hz)  Source 

00188—Naso (NAS)  A  BasoTireno, Italy, 15/04/1978 23:33, Mw = 6.1  E18, R16  H1  150  4.2  ITACA 
H2  129  6.7  
V  80  6.7  
6756—Flagbjarnarholt  A  South Iceland, 17/06/2000 15:40, Mw = 6.5  E20, R15  H1  315  2.5  ESMD 
H2  339  10.0  
V  271  13.0  
6802—Thjorsartun  A  South Iceland, 21/06/2000 00:51, Mw = 6.4  E3, R3  H1  669  10.0  ESMD 
H2  544  2.0  
V  331  5.6  
15205—HveragerdiChurch  A  Mt. Hengill, Iceland, 24/08/1997 03:04, Mw = 4.9  E6  H1  168  7.7  ESMD 
H2  67  4.2  
V  42  8.0  
15537—Thjorarbru  A  South Iceland, 17/06/2000 15:42, mb = 5.7  E10  H1  209  3.3  ESMD 
H2  231  3.3  
V  47  7.7  
15560—Thjorarbru  A  South Iceland, 17/06/2000 17:40, Mw = 5.0  E10, R5  H1  176  5.9  ESMD 
H2  281  3.6  
V  124  8.0  
14683—Borgo CerretoTorre  A  UmbriaMarche, 14/10/1997 15:23, Mw = 5.6  E9, R5  H1  333  4.6  ITACA 
H2  329  3.3  
V  157  5.0  
16352—SelfossCity Hall  A  Olfus, Iceland, 29/05/2008 15:45, Mw = 6.1  E5, R3  H1  523  1.1  ESMD 
H2  324  1.3  
V  246  6.7  
15905—Zarrat  A  Firuzabad, Iran, 20/06/1994 09:09, Mw = 5.9  E16, R11  H1  301  4.2  ESMD 
H2  253  4.6  
V  102  7.7  
16996—L’AquilaV. AternoIl Moro—AQM  A  L’Aquila Aftershock, 07/04/2009 21:34, Mw = 4.6  E2, R2  H1  247  8.3  ITACA 
H2  130  50  
V  82  9.5  
17116—MonterealeMTR  A  L’Aquila Aftershock, 09/04/2009 19:38, Mw = 5.3  E10  H1  108  10  ITACA 
H2  90  20  
V  67  25 
3.3 Computed acceleration and ground motion characteristics
4 Conclusions
We developed methodology of calculating acceleration time history and corresponding characteristics of earthquake ground motion at a site of interest assuming acceleration at a reference site for two basic configurations: a reference site is a part of the model with a local surface sedimentary structure, a reference site is not part of the model with a local surface sedimentary structure. For each of the two configurations we assumed two wavefield excitations: a vertical planewave incidence and a point DC source. We illustrated the methodology of computation on the example of the Grenoble valley.
Notes
Acknowledgements
This work was supported in part by project SIGMA (EDF, AREVA, CEA and ENEL) and by the Slovak Research and Development Agency under the contracts APVV150560 (Project IDEFFECTS).
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