Ground-motion prediction for reference rock conditions

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Reference motion and site effects in SHA

Precise seismic hazard evaluations are essential for site-specific engineering applications, such as the design of critical facilities. Remarkably, the seismic safety of nuclear facilities requires periodic site-specific hazard reassessments that permit to consider the advances of methods and knowledge in engineering seismology. The latter can lead to significant modifications of expected hazard levels, which imposes verifications of facilities and equipment to cope with the new hazard levels. As a result, if the concerned facilities are sensible to the new hazard levels, they are usually shut down or reinforced.
Until today, an absolute prediction of an earthquake occurrence and ground shaking level is not possible in a purely theoretical way due to several complexities of the earth’s structure. Instead, researchers perform regressions on recorded data in the past to predict ground shaking levels of future earthquakes. The latter is only possible under the assumption that earthquake spatial and temporal properties are invariant with time. The equations used in the regressions are given the generic term “ground motion prediction equations” or “GMPEs”. They equally provide a statistical estimate of the expected ground motion level and a standard deviation representing the uncertainty for a given earthquake scenario. Thus, GMPEs do not literally predict future earthquakes but instead evaluate average scenario-specific intensities based on previous observations.
In the context of site-specific SHA, site effects are a compulsory part to consider. However, different approaches are proposed to integrate site effects into SHA studies, which can be very simple or detailed. Despite the latter, the common principle is to assess ground motion on reference site conditions with the help of GMPEs and then integrate site-specific parameters to account for local site responses (Renault, 2014; Rodriguez‐Marek et al., 2014; Ameri et al., 2017) and illustrated in Figure 1.1. Thus, the identification of reference motion is a critical question in such studies. Since site-responses are defined as the effects undergone by ground motion between the underlying seismic bedrock and the surface level, the reference motion is generally meant to be at the bedrock level, where no site effects are yet acquired. It is common to consider that reference ground motion, for a specific site, exists on a near outcropping bedrock, which is not always easy to find. However, the definition of reference motion has always been a sensitive issue since the early 1990s (Steidl et al., 1996).
In the following sections, the main aspects of site-specific SHA are briefly detailed to illustrate the background of the PhD work. Therefore, we will elaborate on the basics of GMPEs related to our work, namely accounting for site effects in GMPEs. Since reference conditions, the main target of the present work, represent “complete absence of site effects”, we propose defining reference ground motion by removing the associated site responses from recordings in a given region. This step is named the “deconvolution approach”, and it requires a proper estimation of site effects for all sites in the dataset. For this aim, we will address the state-of-the-practice to define reference motion, and we will consequently detail the most common methods (and mainly those addressed in this text) for precise site effects estimations.
Figure 1.1: The concept of site-specific seismic hazard assessment: (1) GMPEs are used to estimate the reference motion(2). Then, integration of local site responses (3) is necessary to obtain site-specific hazard estimates (4).

Ground-motion prediction equations (GMPE)

GMPEs are models mainly resulting from regressions on empirical and sometimes simulated data. In either case, the regressions use simplified functional forms to model the recorded data robustly. These models ought to describe the relationship between a desired intensity measure that describes the ground motion and a group of variables related to the seismic scenario and stands for the different contributions (source, path, and site), as in Eq(1.1).
Here, is a function that mainly scales ground motion prediction with magnitude ( ), ℎ is the scaling with source-to-site distance ( ), and is the part accounting for site-effects that can be a function of site-specific information as VS30 or other parameters. This relation is usually established by using multivariate mixed-effects nonlinear regressions, which results in a residual term .

Intensity measures (IMs)

Time histories of earthquakes are not directly used in GMPE developments. Instead, they are treated and used to estimate IMs, which are parameters that best describe the ground motion impact on buildings. In other words, they are often used to quantify ground motion severity. The common IMs are:
• Peak-based, which are measures of absolute maximum values of ground motion’s time history characteristics. A large portion of GMPEs is derived from horizontal peak ground accelerations (PGA), while fewer ones are derived for peak ground velocities and displacements (PGV and PGD, respectively). PGA is known for its close relation to dynamic forces, for which structures are exposed. Also, it can be considered close to the response of very stiff structures. PGV and PGD could be more related to lower frequency responses (higher proper periods) and have several uses. For example, in addition to PGA, they were used to constrain the elastic response spectra for design (Newmark and Hall, 1982). Also, PGV/PGA ratios were examined in some studies for their correlation with inelastic structural deformations (Zhu et al., 1988; Sucuoğlu et Hussein SHIBLE al., 1998) or relation to nonlinear soil behavior (Chandra et al., 2016; Guéguen, 2016).
• Duration-based, which results from the integration of functions related to ground motion time histories over the entire duration. Some of these measures are the cumulative absolute velocity (CAV), Arias Intensity (IA), root-mean-square acceleration, and characteristic intensity (IC). Since these duration-based IMs are not exploited within this text, further related details are omitted.
• Frequency-response-based, which are computed from the response of single degree of freedom oscillators (SDOF). The structure response is commonly approximated by that of an SDOF, characterized by a specific resonance frequency. Since the natural frequency can vary depending on the structure characteristics, a frequency-dependent acceleration response spectrum is usually derived. The most common is compute pseudo-spectral acceleration response (PSA) for a 5% damped oscillator. Figure 1.2 simplifies the concept of a response spectrum given an input earthquake signal (borrowed from Kramer, 1996). The response spectrum is principally constructed by calculating maximum oscillator response at several resonance frequencies, and they are combined in the form of a complete spectrum.
The vast majority of GMPEs are derived for PGA and PSA, as detailed in the reports of Douglas (2020). This is mainly due to the fact that most earthquake-resistant designs are based on PGA and PSA (Bommer and Alarcon, 2006). In the nuclear industry particularly, the prediction of response spectral accelerations over a specific frequency band grabs more attention than PGA. These facilities are sensitive to high-frequency motions between 10 Hz and 20 Hz, which can be higher than a PGA value. The latter leads to limited use of PGA in such precise hazard analysis.
Figure 1.2: The figure describes how the response acceleration spectrum is obtained from recorded signals as input motion for SDOF systems. The maximal acceleration obtained at different natural frequencies (periods) of the SDOF oscillator constitutes the response spectrum (figure adopted from Kramer, 1996).

How GMPEs are derived

A vital step before deriving a GMPE is to construct a ground motion database with enough metadata and well-defined IMs. First, a network of seismic stations should have been installed and recording for several years, covering as many earthquakes as possible. Figure 1.3 illustrates the concept of strong-motion database construction. Each earthquake “ ” recorded is usually characterized by a series of source-related parameters, e.g., epicenter, depth, magnitudes, etc. On the other side, each recording station “ ” is usually characterized by site-specific parameters such as localization, installation type, and a basic idea about the sub-surface soil structure. Seismic signals recorded at each station are treated and used to construct strong-motion databases used for GMPE regressions Before Brillinger and Preisler (1985), regressions on recorded data were performed by the fixed effects models as described in Eq(1.2). In these models, is simply the error of the regression model for earthquake “ ” at site “ ”, which is normally distributed. Also, they are called fixed effects models because their main concern is the regression coefficients and no particular analysis is performed for residuals .
Brillinger and Preisler (1985) first presented an algorithm to include random effects, in addition to fixed effects. Random effects are proposed to account for dependencies in data that we do not consider in the functional form used in regressions. The essence of the method lay in the decomposition of residuals ( ) into between-event and within-event residuals to better understand event-related terms. Then, this modeling type was revised by Abrahamson and Youngs (1992) and delivered a more stable algorithm that was afterward used in most of GMPE derivations in the last few decades. Recently, Bates et al. (2015) published an R-package dedicated to nonlinear mixed-effects regressions (nlmer) that greatly simplified the regression task for GMPE developers. The importance of residuals decomposition in GMPEs will be highlighted afterward.
Figure 1.3: An illustration of a strong-motion dataset construction, which is aimed to develop GMPEs, using each event at each site .
First, GMPEs started in the late 1960s with elementary functional forms (as reported in Douglas, 2020). The limited amount and quality of recordings available were not enough to observe detailed aspects of ground motion considered in hazard analyses nowadays. After several devastating earthquakes in different parts of the world, seismic hazard analyses received increased attention. Thus, deployment of seismic networks and construction of strong-motion databases massively started in the regions exposed to seismic hazard such as in Japan (KiK-net, K-net, F-net, etc.), United States (NGA East-1 and-2), and Europe (RESORCE then ESM datasets). The worldwide actions lead over three decades to enlarge the databases and include vast amounts of records that allowed addressing additional complexities in GMPEs.
In a GMPE functional form, dependence on the source ( ) is usually described by the magnitude . Additional source details and complexities are considered if they are available and well-constrained such as faulting style, hanging-wall effects, depth to rupture surface, and many others. However, in this text, we will avoid more detailing about source aspects and limit the study to magnitude dependence.
Despite the uncertainties related to the definition of (local magnitudes, moment magnitudes, etc.), a clear dependence of on is always found and described in different ways in GMPE derivations. When datasets are limited to low-to-moderate seismicity context, a linear and simple form is usually considered (e.g., Berge-Thierry et al., 2003). Then, the linear dependence is extended to a quadratic polynomial in moderate seismicity contexts (e.g., Bindi et al., 2014). Afterward, a hinge magnitude ( ℎ) started to be introduced when recordings of large events appeared in databases (Rodriguez-Marek et al., 2011; Kotha et al., 2016), beyond which scaling returns to a linear form. Other studies with a wide range of magnitudes preferred to include a piece-wise linear relation with multiple frequency-dependent magnitude hinges ( ℎ1 and ℎ2) as in Boore et al. (2014) and Kotha et al. (2018).
The path terms ( ℎ ) are inspired by physical models that are used for crustal wave propagation. First, anelastic attenuation is usually included by an exponential decay with distance while geometrical spreading is inversely proportional to . Distance decay in GMPE derivations was found to be magnitude dependent, as Cotton et al. (2008) first suggested, and it was considered in various GMPE derivations afterward. Another aspect of distance scaling is the near-source saturation effects of ground motion explored in several studies (e.g., Atkinson et al., 2016). Near-source effects are usually introduced in ℎ in the form of effective depths at short distances.
Early functional forms of GMPEs were not prepared to account for site effects in an term. Full ergodic assumptions were assumed by default for application purposes. Advances in the last decades showed significant importance to take site-specific ground-motion variations. As each site has its geological structure, ground motion characteristics are expected to vary differently from one site to another. First, site effects were considered similar for sites of the same category (i.e., soil or rock conditions, as in (Berge-Thierry et al., 2003; Zhao et al., 2006, 2016). Alternatively, site effects are also considered through site proxies, and mainly by the time-averaged S-wave velocity in the first 30 meters below the soil surface, i.e., the VS30. However, accounting for site effects using VS30 underwent different forms, either directly using the value itself or indirectly using classified sites into VS30-based soil categories.

Comments on site effects in GMPEs

Since we are interested in the site-specific ground motion predictions in this text, it is important to recall some of the main aspects of site conditions considered in GMPEs. Site effects can be accounted for by two main approaches: a generic and site-specific one. Generic methods provide average estimates of site effects and accept lower accuracy than site-specific methods, intending to provide a quick and simple practice for engineering applications. These methods mainly rely on simple site proxies such as VS30, which are used either directly (e.g., Rodriguez-Marek et al., 2011; Bindi et al., 2017) or indirectly by classifying soil conditions into different categories (e.g., Zhao et al., 2016). However, a site-specific method delivers accurate site responses resulting from site-specific ground response analyses, which require additional costs and efforts.
In the context of SHA for critical facilities, an increased level of attention is generally recommended for site effects. Recommendations for nuclear facilities usually consider categorization or parametrization of site-effects as an exaggerated simplification. Instead, a detailed site-by-site to account for amplification is strongly desired. Some of the reasons standing behind these recommendations are mentioned hereafter:
• VS30 is usually obtained by geophysical surveys investigating soil properties profiles up to at least 30 m depths. Since the surveys are usually expensive in time and money, a low portion of well-characterized sites is present in the strong-motion databases. Instead, geological and topographical correlations with VS30 are performed (based on existing measured profiles) to complete this missing site information. However, these correlations inject additional uncertainties and bias in the amplification estimations.
• The proportion of measured VS30 in sites is relatively lower than inferred sites, as in several databases.
• Many sites can have the same VS30 (or same site category, usually based on VS30) but can reveal very different site responses. VS30 describes the first 30 meters of soil profiles, while deeper soil layers can play an essential role in site effects. Thus, there is a non-unique site-effect for the same VS30. In another way, VS30 parameterizations lead to an average response without considering local resonance effects.
• A detailed site-response assessment may require investigations for lateral and vertical spatial variations of soil properties at every single site. A few hundred meters between two stations can hold significant changes in site responses.
• Since the work of Boore and Joyner (1997), an additional site proxy to VS30 started to be used, which is the high-frequency attenuation term, . This term was initially introduced by Anderson and Hough, (1984) to describe the unexplained high-frequency attenuation observed on Fourier spectra. However, clear physical bases of such parameterization remained absent until today, with correspondingly high levels of uncertainties in its measurements (Ktenidou et al., 2015).

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GMPE residuals

Despite accounting for different contributions in GMPEs, ground motion representation remains very simple compared to the complexities of the physical processes involved in ground motion generation and propagation. Thus, the departure of data from the GMPE average predicted is considered to be in a random manner. Ground-motion residuals are defined as the difference between observations and predictions to capture the unexplained dispersions from this average as in Eq(1.3).
Ground-motion residuals are generally assumed to follow a normal distribution with zero mean and a total standard deviation ( ). This allows seismologists to express ground-motion predictions into an explained (fixed-effects) and an unexplained ( ) component that represents the related variability. In the context of hazard analyses, it is customary to distinguish between epistemic uncertainty, which results from incomplete knowledge of physical processes, and aleatory uncertainty due to the random nature of the processes. However, it is sometimes challenging to separate epistemic from aleatory uncertainties regarding the models used in regressions, where explainable processes not accounted for in a GMPE can appear as random variations. To simplify the separation, one can consider as epistemic all the variabilities that can be reduced by improved knowledge and data. At the same time, all the rest falls by default into aleatory variability.
Random effects are introduced in regressions to extract more information from residuals, in addition to fixed effects, permitting the identification of ≪repeated residuals≫ for a specific occurrence. In particular, mixed-effects (fixed and random) regressions were initially introduced by Brillinger and Preisler (1985) and revised afterward by Abrahamson and Youngs (1992) to extract event-related residuals treated as random effects. These residuals were called between-event residuals ( ) to capture what we do not account for in regressions. Based on this, what remains from residuals after deduction of is called within-event residuals ( = − ). Each of the latter variability components is considered to follow zero-mean normal distributions with standard deviations for between-event terms and for within-event terms.
The variability associated with GMPEs (i.e. ) appear to be important since it describes the uncertainty on the median predictions, and they can have a powerful influence on final hazard estimates (especially those of PSHA). In fact, median and standard deviations of GMPEs are obtained using a broad range of earthquakes and sites, and they are applied to analyze the hazard at a single source-site combination. This practice was always applied under the ergodic assumption (Anderson and Brune, 1999). However, the increased amount of well-recorded earthquakes allowed researchers to relax such an assumption, even if partially. It has been shown in several studies (e.g., Lin et al. 2010; Chen and Tsai 2002; Atkinson 2006; Morikawa et al. 2008; Anderson and Uchiyama 2010) that removing the ergodic assumption leads to lower variability of ground-motion. However, removing the ergodic assumption requires the definition of the site- and path-specific ground-motion models, which need enough data to constrain them.
Under a full ergodic assumption, all the variability observed is automatically classified as aleatory variability. Thus, the key to reducing the aleatory is to identify repeatable effects at single sites and events that can be transferred from aleatory variability into epistemic uncertainty. For this aim, researchers started to break down ground-motion residuals into several components to improve seismic hazard analyses (e.g., Walling 2009; Atik et al., 2010). The first necessary step was to decompose residuals into smaller variability parts and attribute to each one a specific acronym. The within-event residuals of GMPEs include systematic site- or path-specific effects that can be identified. Similarly, between-event residuals contain systematic source-specific effects. Removing these systematic effects from ground-motion residuals is considered as the key to remove ergodic assumption, reduce the aleatory and improve hazard estimates. At the end, all repeatable effects that can be identified can thus be transferred into epistemic uncertainties, and all that remains unexplained will be thus considered as a random process.
The level to which can be decomposed depends on the assumption and the level of the event- and site-specific information available. Though the present ways to further decompose and into smaller variability components can vary from one application to another. These decompositions mainly break down the ergodic hypotheses about events, regions, path, and site levels. With the help of mixed-effects regressions, between and within-event components can be decomposed as follows:
• Between-event ( ): Earthquakes of similar parameters can produce different ground motions (e.g., Joyner and Boore, 1981) probably because of distinct physical characteristics that could not be accounted for in GMPEs. For a well-recorded event, such unaccounted effects can be viewed as event-specific non-ergodic effects with random effects and highlighted by . If more information is available about earthquake seismogenic zones, GMPE developers might be interested in location-to-location variabilities 2 ~ (0, ) of earthquakes. The latter component is a location-specific one that follows that identifies repeatable effects more accurately than .
• Within-event (): Excluding the source, two main explainable components can be generally identified: site-to-site and path-to-path components. First, site-to-site variability represents the systemic deviation of the observed amplification from the predicted one (the result of site-proxies) and is generally denoted 2 ~ (0, 2 ). Second, path-to-path variabilities denoted 2 ~ (0, 2 ), represent the average shift of event- and site-corrected ground-motion from the median site-specific model prediction. In other words, 2 signifies how many specific characteristics of the travel path can lead to systematic deviations from the median predictions. All the remaining unaccounted effects are shifted to a residual term 0,es~ (0, 0) accounting for record-to-record variabilities. In some studies, as Rodriguez-Marek et al.
(2011), 0 is also denoted as (“ss” standing for single station). It is worth mentioning that it is common to merge the path-to-path component into the residual term 0,es whenever it is not addressed.
Since we are working in the context of site-specific hazard evaluation in this text, we will need to tackle later on the within-event residual components of ground motion. More precisely, we will use the site-to-site residuals due to their close relation to site-specific amplification factors.

Ground-motion prediction for reference rock conditions

After displaying the primary tool in SHA studies, the GMPEs, we address in this section the first key component in site-specific hazard estimates that is ground-motion prediction at reference conditions. Ground-motion is said to be at reference conditions if it has not yet gained site effects while approaching the surface. At the bedrock level below a site, site effects are generally minimal. We usually consider that sites are at reference conditions if their properties are reasonably close to seismic bedrock properties, i.e., hard-rock material with VS exceeding 1.5 km/s. Therefore, researchers have always addressed hard-rock sites to identify recorded reference motion.
GMPEs are derived on datasets with a specific range of site conditions, commonly called “host sites”. As shown in Figure 1.4, existing strong-motion databases are most often representative of stiff-soil to soft-rock site conditions (i.e., VS30 300-800 m/s). It is clear that reference sites are not well constrained (even not at all sometimes) in most GMPEs, and hard-rock sites lie on the periphery, or even outside, of validity domains. Thus, ground-motion prediction at such “target sites” is not straightforward.
The proposed practice to overcome the reference motion issue has been first to derive ground motion on available host sites, then adjust the GMPEs to get a ground motion estimate under reference hard-rock conditions. They often refer to this kind of approach as “Host-to-target-adjustments” (HTTA). Several projects of safety reassessment of critical facilities around the world have considered HTTA. For instance, seismic hazard for Swiss nuclear power plants was first estimated within the PEGASOS and PRP projects and then refined by considering the HTTA approaches (Biro and Renault, 2012). Another application, which used these adjustments, is the “Thyspunt” Nuclear Siting Project (Rodriguez-Marek et al., 2014) and the Hanford site in the U.S. (Coppersmith et al., 2014).
Figure 1.4: Site distributions in VS30 for American NGA (a and b) and European ESM datasets (c), indicating measured and inferred values. Figures are adapted from Bozorgnia et al., (2014) and Lanzano et al., (2019).

The essence of Host-to-Target-Adjustments (HTTA)

The principle of such adjustments is to consider possible differences between host and target sites using physics-based models. The latter requires sufficient knowledge of the underlying physics in addition to a well-defined methodology (or strategy) of application. Since the source and path effects are weakly constrained, they are sometimes neglected, and the focus is on adjustments accounting for site-effect differences. The site differences considered in these corrections are the impedance contrasts based on deep VS-profiles (denoted by VS-correction) and the high-frequency attenuation differences (denoted as -correction). Figure 1.5 illustrates the principle of correction between host and target sites. After defining the adjustments, they are applied in the Fourier domain with IRVT and RVT conversions from and to the response spectral domain.
Figure 1.5: the essence of the Host-to-target adjustments. GMPEs derived on host sites are applied for target sites after physics-based VS- corrections.
• VS-correction is applied in a way to account for differences in impedance contrasts. This is done mainly by modeling site amplification by the QWL approach (initially proposed by Joyner and Boore, 1981). The QWL amplification is computed from impedance ratios (i.e., proportional to VS, surface/VS, bedrock) and results in a simple monotonic function with frequencies. The method uses very deep VS-profiles that go to several kilometers of depth (initially proposed in Boore and Joyner, 1997, then updated in Boore, 2003). It is based on measured deep VS-profiles (Figure 1.6). However, these deep VS-profiles are not always available for target sites or even host sites, making the estimation difficult. Hence, Boore and Joyner (1997) used the VS-profiles through interpolations, based on VS30 only, to obtain a deep generic profile that allows amplification estimation. However, several attempts to improve this practice were made by defining region-specific reference rock profiles (Poggi et al., 2011, 2013). Finally, the ratios between generic-rock and hard-rock amplifications (with the QWL-estimations mentioned above) constitute the VS-correction.
Figure 1.6: a) S-wave velocity profiles commonly adopted for generic rock sites in VS-correction of HTTA. b) The amplification approximation by the quarter-wavelength method (in grey) based on the generic VS-profile. The amplification is compared to other empirical and theoretical estimates. Both figures are adopted from Boore (2003).
• In addition to impedance effects, high-frequency attenuation effects are accounted for with the -correction factors. The “ ” term was initially introduced by Anderson and Hough (1984) to describe the physically unexplained high-frequency slope. The model, which is usually used to parametrize these high-frequency effects of the Fourier amplitude spectrum of an earthquake, is exp(− ). This attenuation model is usually added to QWL amplification to describe both impedance and attenuation effects (Figure 1.7). Based on this, if the attenuation features on target sites are lower than host sites (i.e. < ℎ ), then -correction amplifies the high-frequency content of target sites. This is considered the case for hard-rock sites (usually target sites), which often have values much lower than soil to soft-rock sites. As a consequence of attenuation correction, hard-rock motion presents an amplification compared to standard-rock sites.

Table of contents :

General Introduction
1.1 Ground-motion prediction equations (GMPE)
1.1.1 Intensity measures (IMs)
1.1.2 How GMPEs are derived
1.1.3 Comments on site effects in GMPEs
1.1.4 GMPE residuals
1.2 Ground-motion prediction for reference rock conditions
1.2.1 The essence of Host-to-Target-Adjustments (HTTA)
1.2.2 Limitations of HTTA
1.2.3 Recent alternatives and advances concerning scaling ratios:
1.3 Site response estimations
1.3.1 Physical background
1.3.2 One-dimensional theoretical estimates
1.3.3 Empirical estimates
1.4 Scope of the work
2.1 Introduction
2.2 Data and observations
2.2.1 Observations
2.2.2 Measurements of VS perturbations and Geostatistical model validation
2.2.3 The geostatistical model proposed
2.3 KiK-net data, empirical estimations, and site signature
2.4 Tuning the model before application:
2.4.1 Quarter wavelength approximation
2.4.2 Sensitivity analyses of key parameters
2.5 Discussions:
2.5.1 An additional constraint to select randomized profiles
2.5.2 Statistical impact for selected sites
2.6 Conclusions
3.1 Introduction
3.2 About Generalized Inversion Techniques
3.2.1 Overview
3.2.2 Methodologies and inversion schemes involved in the benchmark
3.3 The datasets considered
3.4 Generalized inversions on synthetic and real data
3.4.1 Sanity check using a synthetic dataset
3.4.2 The two phases and the reference conditions
3.4.3 Results for the sparse regional dataset, the French Alps
3.4.4 Results for a dense regional dataset, the Central Italy dataset
3.5 Uncertainties associated with GIT results
3.5.1 Strategy for inter-method uncertainties characterization
3.5.2 Quantification of uncertainties
3.5.3 Possible origins of variability: regional Variations
3.6 Exploring the impact of the dataset size
3.7 Summary and conclusions
3.8 Perspectives
4.1 Introduction
4.2 KiK-net dataset adopted
4.2.1 Automated Onset detection
4.2.2 Selection of Ground Motion Recordings
4.2.3 Attenuation regionalization of Japan
4.3 Site response estimations
4.3.1 Empirical
4.3.2 Theoretical
4.3.3 Scales of the estimated functions
4.4 Spectral inversion techniques application
4.4.1 Highlights on the inversion method
4.4.2 Non-parametric attenuation:
4.5 Source and site separation with GIT
4.5.1 Reference site choice
4.5.2 Source results
4.5.3 Site results and some case studies
4.6 Discussions
4.6.1 Revising the 1D classification with absolute terms
4.6.2 Mean comparisons of relative and absolute site responses
4.6.3 Site response estimations using GIT
4.7 Conclusions
5.1 Introduction
5.2 KiK-net dataset explored
5.3 Deconvolution approach
5.4 GMPE functional form and first results
5.4.1 Fd scaling
5.4.2 Fm scaling
5.4.3 Fs scaling
5.4.4 Residuals check
5.5 Results
5.5.1 Model scaling
5.5.2 Variability analysis between SURF1D GMPEs
5.5.3 Variability analysis on SURFALL GMPEs
5.6 Discussions
5.6.1 Results in terms of rock to hard rock scaling
5.6.2 Comparison of amplification factors from recent GMPEs
5.7 Conclusions and perspectives
5.8 Acknowledgments
5.9 Supplements
References

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