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## Apparent Source time functions

Another way to characterize source parameters is to analyze the moment rate function 𝑀̇(𝑡) (also called source time functions), which can provide information about integrated source parameters like source duration or stress drop. By studying apparent moment rate functions (that is moment rate functions observed at different stations around fault ruptures), one can also get information about the rupture propagation such as the average rupture velocity. This approach was conducted for example by (Chounet et al. 2017) who concluded that the distributions of stress drop and rupture velocity are not independent, but anti-correlated. Besides, (Archuleta and Ji 2016) also looked at the apparent moment rate functions and observed that their Fourier amplitude spectra must be characterized not by a single but two corner frequencies to match observations of PGA and PGV values.

### Link between frictional and kinematic rupture parameters

With dynamic simulations, seismologists have analyzed the relationship between the stresses and strengths along the fault, and the slip and the rupture velocity during the earthquake. (Madariaga 1983; Pulido and Dalguer 2009; Mena, Dalguer, and Mai 2012; Madariaga and Ruiz 2016) dynamically simulated ground motion and showed that geometrical obstacles or barriers (that is area of high fault strength) cause abrupt changes in rupture velocity. Besides, (Bouchon 1997) observed that the regions of the fault that break with a high stress drop are also the regions where slip is large. He also related the change of the rupture speed to the strength excess: a high strength excess value implies that either the fault strength was high or the shear stress was low at that specific area of the fault prior to the earthquake. Thus, the spatial distribution of the strength excess over the fault plane appears to be inversely correlated to the local rupture velocity. Rupture propagates slowly where the relative fault strength is high and accelerates over low strength regions. In addition, dynamic simulations can help representing more realistic ruptures in kinematic simulations: (Mai et al. 2017) conducted a suite of dynamic earthquake simulations for various rough-fault realizations and suggested to take into account fault roughness in kinematic models behavior by retaining the moment tensor orientations but neglecting their off-fault positions.

#### Correlation between kinematic rupture parameters

Last but not least, simulating dynamic source models and collecting the resulting source parameters in a catalogue is a way to study statistical relationships between various rupture parameters (e.g. Song and Somerville 2010). (Schmedes, Archuleta, and Lavallee 2010) analyzed a database of dynamic strike-slip rupture models computed using different models of initial conditions of stresses and strength properties, and concluded that slip strongly correlates with rise time. Whereas they reported that the correlation pattern between the rupture velocity and the slip is unclear at least under certain conditions of dynamic rupture models. (Bizzarri 2012) considered a wide catalogue of dynamic 3D models, and observed that the peak slip rate and the rupture speed are positively correlated, which is supported by theory (Ida 1973). (Oglesby and Day 2002) used 3D dynamic models with variable assumptions on strength and stress heterogeneity and concluded that rupture velocity, rise time, and slip are associated with the fault strength and stress drop, as well as each other, however the connections between these quantities are not simple. On the other hand, (Trugman and Dunham 2014) presented a 2D pseudo-dynamic model that emulates earthquake source parameters on rough faults where final slip, local rupture speed and peak slip velocity are anti-correlated with the observed-fault roughness. Thus, the correlation patterns between the different kinematic rupture parameters remain unclear. This is mainly because dynamic simulations are sensitive to the assumptions on the input friction laws, which are poorly constrained. (Song 2015) showed that the source parameter correlation structures can be significantly affected by the input fracture energy distribution (Figure 1-15).

A compilation of references to papers that studied the correlation between source parameters at different scales, between kinematic and dynamic characteristics, is presented in Table 1-2.

**From source rupture towards ground motion**

These local and large-scale rupture features largely affect the ground motion. The waves transport the complexity of the rupture process through the Earth structure to the ground surface in terms of displacement, velocity or acceleration ground motion. Each source parameter induces its own signature on the ground motion, in various frequency ranges. For instance, ground motion is strongly affected by the fault dimension for a given moment magnitude (Aagaard, Hall, and Heaton 2001), the hypocenter position (Aagaard, Hall, and Heaton 2001; Somerville et al. 1997; Ripperger, Mai, and Ampuero 2008), rupture speed and rise time especially in the fault vicinity (Aagaard, Hall, and Heaton 2001). Recent near-fault ground motion simulation studies (e.g. Moschetti et al. 2017) all show that earthquake ground motions and its variability are highly sensitive to the choice of slip distribution, rupture speed, slip velocities and hypocenter locations. It shows that there is a real need for further characterization of the kinematic source parameters for probabilistic seismic hazard analyses.

**Impact of seismic rupture on surface ground motion**

Understanding and mitigating earthquake risk depends critically on predicting the intensity of strong ground motion, including estimates of the aleatory variability, which remains a scientific challenge. Here, the term ground motion variability refers to the variability due to source effects only, that is the variability one would expect for repeating events of the same magnitude on a given fault, recorded at the same station. The fault rupture process that generates seismic waves is complex and incompletely understood. In this chapter, we aim to better quantify the link between the rupture properties and the Peak Ground Acceleration (PGA) and the Peak Ground Velocity (PGV), which are two commonly used measures of the ground motion intensity. Note that these two quantities are controlled by different frequency ranges. As explained in section 1.1.4.1, the frequency content of ground motion is, at the first order, controlled by the corner frequency fc. The ground displacement amplitude spectrum decays with a slope of -2 for frequencies larger than fc. Hence, the velocity spectrum increases with the frequencies up to fc and then decreases as f-1, and the acceleration spectrum is flat above fc. This implies that PGA is essentially controlled by frequencies larger than fc, while the PGV is expected to be driven by lower frequencies (note that reported values of fc for various moment magnitudes are displayed on Figure 1-13). A more detailed analysis of the frequency range that mostly controls PGA and PGV values is proposed in chapter 2.

**PGA controlled by the large-scale source parameters: role of average stress drop**

The stress drop Δτ has become a key parameter to measure the strength of the observed high-frequency ground motion. Many authors referred to stress drop Δτ and its variability driving the high-frequency ground motion: (Lavallée and Archuleta 2005) noticed that the variability of the PGA, which represents a measure of the high-frequency ground motion, is not that different from the variability of the Δτ. (Cotton, Archuleta, and Causse 2013) identified that the PGA variability should translate directly into earthquake Δτ variability as described by equation (1-13), derived under the assumption of the classical omega squared model and the random vibration theory: 𝝈𝐥𝐧(Δ𝝉)=𝟔𝟓𝝈𝐥𝐧(𝑷𝑮𝑨).

**Mechanism of PGA generation in kinematic source models**

Earthquake ruptures generate seismic waves that travel from the source to the surface and cause ground motions over a wide range of frequencies. One approach to describe the source process is the so-called kinematic approach, which consists in a priori prescribing the displacement discontinuity across the fault surface. The local slip-rate function needs to be specified (e.g. Liu, Archuleta, and Hartzell 2006; Tinti et al. 2005) to describe the space-time evolution of slip along the fault by means of kinematic parameters. We use the pseudo-dynamic source model developed by (Song, Dalguer, and Mai 2014) for a rectangular fault plane. In this model, kinematic source parameters are calibrated using a suite of spontaneous heterogeneous dynamic rupture simulations. The rupture starts from the hypocenter and expands over the fault plane with a rupture speed 𝑉𝑟. Each point on the fault slips as it is reached by the rupture front and is characterized by the final slip value (𝐷) and the peak slip velocity (𝑝𝑠𝑣) or the slip duration, also called the rise time (𝑇𝑟𝑖𝑠𝑒). In order to characterize the spatial variability of the kinematic source parameters (𝑉𝑟, 𝐷 and 𝑝𝑠𝑣) over the fault area, two statistical properties are considered. First, the 1-point statistics is defined for a given fault point by the mean value (𝜇) and the standard deviation (𝜎) of the considered source parameter, considering a normal distribution. Second, the 2-point statistics is defined by the correlation lengths (𝑎𝑥 and 𝑎𝑧, representing the characteristic length of heterogeneities along-strike and along-dip, respectively) and the spatial cross-correlation, defined by the correlation coefficient (𝜌) between any pair of kinematic parameters at a given point, and by a correlation function. We use a Von Karman autocorrelation function (Mai and Beroza 2002). Note that our statistical model is stationary, which implies that the statistics of any parameter is constant over the fault plane.

**Earthquake source parameterization**

We generate rupture models equivalent to a moment magnitude M = 7 (Figure 2-2). The rupture length L = 70 km and width W = 14 km are derived from the 𝑀−𝐿 scaling relationship by (Papazachos et al. 2004). Note that the relationship provided by (Thingbaijam, Mai, and Goda 2017) result in close values of the rupture dimension (L = 67 km and W = 19 km). The mean value of the slip 𝜇𝐷 is then defined by: 𝜇𝐷=𝑀0𝐺 𝐿 𝑊, where 𝑀0 is the seismic moment and 𝐺 is the shear modulus. We make sure that the maximum slip does not exceed the ceiling defined by (McGarr and Fletcher 2003) as a function of magnitude (500 cm in our case). For a strike slip rupture with L>>W, the average stress drop Δ𝜏 is expressed as: Δ𝜏=𝐶∗𝐺∗𝜇𝐷𝑊.

**PGA computation in the far-field approximation**

The ground displacement for a homogeneous elastic medium in the far-field approximation 𝑢𝐹𝐹(𝑡) is proportional to the source time function, also called the moment rate function 𝑀̇(𝑡). We first use Equation (2-4) to compute the displacement in the frequency domain: 𝑢𝐹𝐹(𝑋,𝑓)=14𝜋𝜌0𝑉𝑠31X RP 𝑀̇(𝑓) 𝑒−2𝜋𝑖𝑓𝑋𝑉𝑠 𝑒−𝜋𝑓𝑋𝑉𝑠𝑄𝑠 (2-4).

where 𝑋 is the distance to the rupture, assumed equal to 100 km, 𝜌0 is the rock density (𝜌 = 2.7 g/cm3), 𝑉𝑠 is the shear wave speed (𝑉𝑠 = 3.58 km/s), 𝑅𝑃 is the average radiation pattern of the shear waves (according to (Boore and Boatwright 1984), we assume RP = 0.63), 𝑄𝑠 is the attenuation factor (we choose 𝑄𝑠= 220 (e.g. Heacock, Research, and Mines 1977). The attenuation of ground motion is represented by the geometrical attenuation 1/𝑋 and by the anelastic attenuation 𝑒𝑥𝑝(−𝜋𝑓𝑋𝑉𝑠 𝑄𝑠). We then use inverse Fourier transform to obtain ground motion in the time domain and compute PGA as the maximum absolute value of the displacement second derivative. Note that a quarter-period-cosine taper is applied to the first second of the acceleration to remove the strong phase due to the sharp increase of rupture velocity at the rupture nucleation. Theoretical studies show that the rupture velocity increases smoothly during nucleation (e.g. Latour et al., 2011) and such initiation phases are not observed on real seismograms.

**Mechanism of PGA generation for homogeneous ruptures**

We start by investigating homogeneous ruptures, in order to identify the mechanisms of the far-field PGA generation in a simple rupture case. The slip, the rupture speed and the rise time are then constant along the rupture (𝜎=0), except at the fault boundaries due to the applied tapering. The parameters used for simulations are summarized in Table 2-2 (simulation A). We use the concept of isochrones to extract the part of the rupture that produces the PGA (Spudich and Frazer 1984). Isochrones are all the points on the fault that radiate elastic waves such that the waves arrive at a given station at the same time. In the case of the far-field approximation (Equation (2-4), the isochrone at the PGA time is simply the rupture front at the PGA time (Figure 2-3 a, b, c and d). (Spudich and Frazer 1984) demonstrated that ground acceleration is proportional to the variations of isochrones velocity. In the far-field approximation, ground motion is then proportional to the variations of rupture velocity. Thus, for homogeneous ruptures, ground acceleration is essentially dominated by four peaks (Figure 2-3 e-1) corresponding to the times where the rupture reaches the four fault boundaries. For the chosen rupture nucleation position and fault boundary tapering function, the stopping phase generated by the rupture arrest at the fault top is responsible for the PGA. Since the tapering function determines the sharpness of the rupture stopping, it highly controls the PGA value. Thus, increasing the tapering length from 20% to 30% (a factor of 1.5), while holding the mean values of the parameters (D, Vr, Trise, psv) unchanged tends to decrease the PGA by a factor of 1.4 (Figure 2-3 e-2). In the following we explore the impact of various kinematic parameters on the PGA.

By decreasing the length of the rupture L while preserving the magnitude, we increase the slip D and therefore the stress drop Δ𝜏. Considering a decrease of L from 71 km to 55 km (that is by a factor of 1.29), D and Δ𝜏 increase by a factor of 1.29 (note that the mean value of Trise is unchanged, hence psv also increases by a factor of 1.29). According to Equation (2-2), PGA should then increase by a factor of ~1.22. Though the PGA changes by a slightly higher factor of 1.30, similar to the slip and the stress drop increase (Figure 2-3 e-3). Equation (2-2) is derived assuming a simple Brune’s source model and random phases for the source spectrum (Brune 1970). PGA is then estimated using the random vibration theory and depends only on the corner frequency (that is, the overall rupture duration). Our source model also matches a Brune’s source model (Figure 2-3 f). However PGA is not driven by the corner frequency but is controlled locally by the phase generated by the rupture stopping at the fault top, which is proportional to the local slip and hence to the stress drop.

**Mechanism of PGA generation for heterogeneous ruptures**

Figure 2-4a and 4b show two different realizations of heterogeneous ruptures with the same statistical properties of the source parameters (Table 2-2 – simulation B). These source models are associated with two different mechanisms of PGA generation. The PGA on Figure 2-4a is induced by the rupture stopping at the top fault boundary, as observed for a homogeneous rupture. The PGA is however higher (0.041 g instead of 0.0147 g) because the rupture speed is heterogeneous and gives rise to a stronger rupture speed drop in this case. Furthermore, the PGA on Figure 2-4b is not controlled by the same process because the rupture speed at the top fault above the nucleation is lower. It is generated by the large rupture speed patch located at the right edge of the rupture, resulting in an abrupt change of rupture velocity. Thus, the position of the high rupture speed patches with respect to the rupture nucleation and their interactions with the fault boundaries play a fundamental role in the PGA generation. Figure 2-5 shows the distribution of the PGA time among the various rupture realizations. We consider only rupture models with Vr = 3 km/s. Interestingly, the probability density is maximum at about 3.5 s, which is the average time needed for the rupture to reach the top fault. This implies that the main mechanism and the main fault area implied in the PGA generation remain the same as for homogeneous ruptures.

**Computation of near-fault PGA and PGV**

We synthesize near-fault ground motions in a 1D layered medium (Appendix 2-1) for stations located at rupture distances 𝑅𝑟𝑢𝑝 of 5 km (station S1 and S2), 25 km (station S3 and S4) and 70 km (station S5) (Figure 2-2), using the representation theorem: 𝑈𝑠𝑡𝑎𝑡𝑖𝑜𝑛 𝑖(𝑡)≈ Σ𝐺𝑖(𝑥,𝑦,𝑡)∗𝐹𝑀 (𝑥,y,𝑡)𝑆 (2-5).

where * is the convolution operator. The summation over space integrates the contributions from the finite rectangular fault plane, discretized into a 2-D grid of subfaults. 𝐹𝑀 (𝑥,𝑦,𝑡) is the slip rate function at position (𝑥,𝑦) computed using the source model defined above, while 𝐺𝑖(𝑥,𝑦,𝑡) represents the Green’s functions calculated using the discrete wavenumber technique in the frequency range 0–5 𝐻𝑧 (Bouchon 1981; Cotton and Coutant 1997).

Finally, the PGA and the PGV are computed using an orientation-independent measure proposed by (Boore, Watson-Lamprey, and Abrahamson 2006) (GMRoTD50). This measure comprises a rotation of the two orthogonal components from 1 to 90, and evaluates the peak ground motion from the geometric mean of the rotated time series.

Appendix 2-2 illustrates a realization of kinematic rupture from case 5 (Table 2-2) with the calculated accelerations for both the EW and the NS components (black and red solid lines, respectively). The far-field acceleration computed according to Equation (2-4) is also shown.

**Table of contents :**

**PART 1: SEISMIC SOURCE RUPTURE AND CONSEQUENT GROUND MOTION **

**1 STATE OF ART **

1.1 OVERVIEW OF SOURCE RUPTURE AND NEAR-FAULT GROUND MOTION

1.1.1 GENERAL INTRODUCTION ABOUT EARTHQUAKES

1.1.2 FUNDAMENTAL EQUATIONS FOR EARTHQUAKE GROUND MOTION

1.1.3 EARTHQUAKE GROUND MOTION MODELING

1.1.4 THE SOURCE RUPTURE PROCESS

1.1.5 CONSTRAINING SOURCE PARAMETERS FROM OBSERVATIONS

1.1.6 FROM SOURCE RUPTURE TOWARDS GROUND MOTION

1.2 IMPACT OF SEISMIC RUPTURE ON SURFACE GROUND MOTION

1.2.1 PGA CONTROLLED BY THE LARGE-SCALE SOURCE PARAMETERS: ROLE OF AVERAGE STRESS DROP

1.2.2 PGA CONTROLLED BY THE LOCAL-SCALE HETEROGENEITIES

**2 THE SOURCE PARAMETERS CONTROLLING THE HIGH-FREQUENCY GROUND MOTION **

2.1 ABSTRACT

2.2 INTRODUCTION

2.3 MECHANISM OF PGA GENERATION IN KINEMATIC SOURCE MODELS

2.3.1 EARTHQUAKE SOURCE MODEL

2.3.2 EARTHQUAKE SOURCE PARAMETERIZATION

2.3.3 PGA COMPUTATION IN THE FAR-FIELD APPROXIMATION

2.3.4 MECHANISM OF PGA GENERATION FOR HOMOGENEOUS RUPTURES

2.3.5 MECHANISM OF PGA GENERATION FOR HETEROGENEOUS RUPTURES

2.4 SENSITIVITY OF PEAK GROUND MOTIONS TO SOURCE PARAMETERS

2.4.1 COMPUTATION OF NEAR-FAULT PGA AND PGV

2.4.2 COMPUTATION OF THE PGA AND PGV SENSITIVITY

2.4.3 RESULTS: SENSITIVITY OF PGA AND PGV

2.4.4 EFFECT OF THE NUCLEATION POSITION

2.5 PEAK GROUND MOTIONS VARIABILITY

2.6 CONCLUSION

2.7 APPENDIX

**3 SPATIAL VARIABILITY OF THE DIRECTIVITY PULSE PERIODS OBSERVED DURING AN EARTHQUAKE **

3.1 ABSTRACT

3.2 INTRODUCTION

3.3 RELATIONSHIP BETWEEN PULSE PERIOD, RUPTURE PARAMETERS, AND STATION POSITION BASED ON ANALYSIS OF SYNTHETIC VELOCITY TIME SERIES

3.3.1 SIMULATION OF VELOCITY TIME SERIES

3.3.2 SIMPLE RELATION BETWEEN PULSE PERIOD, RUPTURE PARAMETERS AND STATION POSITION

3.4 COMPARISON BETWEEN PREDICTED PULSE PERIOD (EQUATION (2-1)) AND REAL OBSERVATIONS (NGA-WEST2 DATABASE)

3.4.2 RESULTS

3.4.3 DISCUSSION

3.5 CONCLUSIONS

3.6 DATA AND RESOURCES

3.7 ACKNOWLEDGMENTS

3.8 APPENDIX

**PART 2: LEBANON CASE STUDY **

**4 STATE OF ART **

4.1 OVERVIEW OF THE DEAD SEA FAULT IN THE LEVANT REGION

4.2 OVERVIEW OF THE SEISMICITY WITHIN LEBANON

4.3 SEISMIC RISK IN LEBANON

4.4 OVERVIEW OF THE GEOLOGY IN LEBANON

**5 TOMOGRAPHY OF LEBANON USING SEISMIC AMBIENT NOISE **

5.1 ABSTRACT

5.2 INTRODUCTION

5.3 GEOLOGICAL BACKGROUND

5.4 AMBIENT NOISE CROSS-CORRELATION AND 3D TOMOGRAPHY OF LEBANON

5.4.1 STATIONS DISTRIBUTION AND PERIOD OF RECORDINGS

5.4.2 AMBIENT NOISE DATA PROCESSING AND CROSS-CORRELATION

5.4.3 RAYLEIGH WAVES GROUP VELOCITY MEASUREMENTS

5.4.4 3D SHEAR WAVES VELOCITY INVERSION AND RELATED UNCERTAINTY

5.5 COMPARISON OF INVERTED VS MODEL TO PREVIOUS RESEARCH OUTCOMES

5.5.1 SPATIAL VARIATION OF VS AT DIFFERENT DEPTHS

5.5.2 FIRST ORDER ESTIMATION OF THE MOHO DEPTH?

5.5.3 VERTICAL CROSS-SECTIONS OF VS

5.6 CONCLUSION

5.7 APPENDIX

**6 CASE STUDY: SIMULATION OF NEAR-FAULT GROUND-MOTION FOR RUPTURE SCENARIOS ON THE YAMMOUNEH FAULT (LEBANON) **

6.1 ABSTRACT

6.2 INTRODUCTION

6.3 TECTONIC SETTING AND SEISMIC HAZARD

6.4 FAULT RUPTURE SEGMENT AND TARGET STATIONS

6.5 GROUND-MOTION SIMULATION METHODOLOGY

6.5.1 LOW-FREQUENCY GROUND MOTION (F ≤ 1 HZ)

6.5.2 BROAD-BAND GROUND MOTION (~0.1-10 HZ)

6.6 SIMULATION RESULTS

6.6.1 PEAK GROUND ACCELERATION AT DIFFERENT STATIONS

6.6.2 RESPONSE SPECTRA

6.7 CONCLUSION

CONCLUSIONS AND PERSPECTIVES

**BIBLIOGRAPHY**