Inversions de colonnes de sol 1D lors du séisme de Tohoku, Japon Mars 2011

Rapport de stage MARS-SEPTEMBRE Maimouna MINT BRAHIM

Master de Sciences et technologies de l’Université Pierre et Marie Curie Mention Mathématiques et applications Spécialité Ingénierie mathématique Parcours Mathématiques pour l’entreprise

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Abstract This report describes the six months internship I spent at the french geological survey (BRGM) in Orléans. The BRGM is France’s leading public institution in Earth science applications for the management of surface and subsurface resources and risks. During my internship, I worked within the service of natural risks in the unity of seismic risks. I had three primary goals : understanding the theory of the inverse problem used to quantify site effects via borehole station installed in sedimentary basin and understanding the genetic algorithm used to solve the inverse problem, adding two methods to the genetic algorithm the first method is a niching method used to reduce the effect of genetic drift and therefor prevent from the stagnation of the individuals of a population of the genetic algorithm around a local minimum and the second method is used to search around the best individuals of a population (the individuals are compared based on the value of the objective function) of the genetic algorithm in order to replace the current best individual with the closest best, this second method was developed to reduce the run time of the genetic algorithm. The third goal of my internship was to perform inversions using the genetic algorithm and using data from different Japanese borehole stations hoping to find evidence of nonlinear behavior of soft sediments during strong ground motions.

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Table des matières 1 Inversion of Borehole Soil Structure : Theory 6 1.1 Elastic waves propagation . . . . . . . . . . 6 1.2 Thomson-Haskell propagator matrix method 8 1.3 Standard Genetic Algorithm . . . . . . . . . 11 2 Two New Methods 2.1 Niching Method . . 2.1.1 Principle . . 2.1.2 Validation . 2.2 Search Around The 2.2.1 Principle . . 2.2.2 Validation .

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3 Results of the Genetic Algorithm Inversions 3.1 Before The Inversions . . . . . . . . . . . . . 3.2 Objective Function And S-wave Window Selection . . . . . . . . . . . . . . . . . . . . . 3.3 Genetic Algorithm Inversions . . . . . . . . 3.4 MYGH10 . . . . . . . . . . . . . . . . . . . 3.4.1 Velocity Profile And Database . . . . 3.4.2 Results . . . . . . . . . . . . . . . . . 3.5 IBRH11 . . . . . . . . . . . . . . . . . . . . 3.5.1 Velocity Profile And Database . . . . 3.5.2 Results . . . . . . . . . . . . . . . . . 3.6 IBRH10 . . . . . . . . . . . . . . . . . . . . 3.6.1 Velocity Profile And Database . . . .

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15 15 16 17 29 29 31 36 36 37 37 37 37 39 50 50 52 71 71

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3.6.2 Results . . . . . . . . . . . . . IBRH19 . . . . . . . . . . . . . . . . 3.7.1 Velocity Profile And Database 3.7.2 Results . . . . . . . . . . . . .

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Chapitre 1

Inversion of Borehole Soil Structure : Theory The inverse problem studied here is the one used to quantify site effects via borehole station installed in sedimentary basin whose depth ranges from few meters to hundred of meters. Such inverse problem is a special case of a general class of inverse problems often used in physical sciences and in engineering that, giving the excitation introduced in a system (input) and its related response (output), aims at reconstructing the excitation and/or internal structure using the response. Because of the complex nature of these problems there mathematical formulation leads to complex multimodal optimization functions with multiple minimums and other irregularities and in this scenario, Genetic Algorithms (GA) arise as an interesting alternative to analytical-numerical techniques such as descent methods [2]. In this chapter is presented a summary of the work that was done before me and for more details readers are referred to the first chapter of Florent De Martin’s Phd thesis[3]. In the following, we present in Section 1.1 a formulation of the direct problem, in Section 1.2 we present the Thomson-Haskell propagator matrix method which is used to solve the direct problem of waves propagation in a sedimentary basin and in Section 1.3 we present the standard genetic algorithm that is used to invert model parameters using borehole station data.

1.1

Elastic waves propagation

In this section we present the fundamental equations used in dynamic elasticity to obtain the motion equation of a particle of a linear elastic medium wish is the first step to determine wave propagation equation. To analyze the deformation of a medium, solid or fluid, elastic or inelastic, we define the infinitesimal strain tensor defined in a Cartesian coordinate system (x1 , x2 , x3 ) as ϵij = 1/2(ui,j + uj,i )

(1.1)

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with ϵij the ijth component of strain tensor and ui the ith component of displacement. Where ui,j = ∂ui /∂xj . To analyze the internal forces acting mutually between adjacent particles within a continuum, we use the concepts of traction and stress tensor related by Ti = τij nj

(1.2)

with Ti the ith component of the traction vector acting on the plane of normal n, τij the ijth component of stress tensor and nj the jth component of the unit vector normal to the surface where acts the traction T. To obtain the equation of motion of a general particle, we equate the rate of change of momentum of particles constituting a volume V with surface S to the forces acting on these particles as ∂ ∂t

∫∫∫ V

∂u ρ dV = ∂t

∫∫

∫∫∫

T dS

f+

(1.3)

S

V

where ρ is bulk density of the media. We apply Gauss’s divergence theorem : ∫∫

∫∫

∫∫∫

Ti dS = S

τij nj dS = S

τij,j dV

(1.4)

V

to Equation 1.3 we find for a general volume V that ∫∫∫ V

(ρ¨ ui − fi − τij,j )dV = 0

(1.5)

This integrand must be zero wherever it is continuous, otherwise a volume V could be found that violates Equation 1.5, hence we obtain the equation of motion of a general particle ρ¨ ui = fi + τij,j

(1.6)

The Equation 1.6 is the key equation for wave propagation. If the medium is linear elastic, then the generalization of Hooke’s law can be used to express the stress tensor as a linear combination of all components of the strain tensor as τij = cijkl ϵkl

(1.7)

cijkl = λδij δkl + µ(δij δjl + δil δjk )

(1.8)

with

and λ and µ the Lamé moduli. The stress strain-relation becomes τij = λδij ϵkk + 2µϵij

(1.9)

and the equation of motion presented in Equation 1.6 becomes ρ¨ ui = fi + (λ + µ)uj,ji + µui,jj

(1.10) 7

We keep in mind that the linear elastic stress-strain relation of Equation 1.9 is an idealized behavior ; such equation is not valid when it comes to modeling real life wave propagation problems because it does not take in consideration the dissipation of energy during the propagation of the wave in the medium. The effect of this energy dissipation is quantified in seismology using the damping ratio ξ . We give in the next section a way to modify the stress-strain relation by adding a term for the dissipation of energy.

1.2

Thomson-Haskell propagator matrix method

In order to establish the mathematical formulation of the direct problem of waves propagation in a sedimentary basin two hypothesis were made : - we consider the Earth (in our case, a sedimentary basin) as a stack of welded homogeneous horizontal layers as shown in Figure 1.1. - we consider only the case of a plane wave incident on the stack of homogeneous layers.

Figure 1.1 – Horizontally layered medium used for computation of spectral ratio in inverse problems. Within each layer, parameters are constant. An example of propagation of a vertically incident SH wave is shown. Moreover, for the resolution of the direct problem, we consider the medium as isotropic linear elastic governed by Equation 1.10. The coordinate system associated with the mathematical model is shown in Figure 1.1. The plane (x,z) is associated with the components ux and uz of 8

the coupled P-waves and SV-waves. The y axis corresponds to the uncoupled SH-waves of component uy . The Thomson-Haskell method make use of motion-stress equations in order to relate the motion-stress vector at depth z to the one at depth z0 via the propagator matrix computed using the layers properties between z and z0 (see Figure 1.1 for notation). The motion-stress equation is obtained by combining the equations of motion with the constitutive relation in such a way that only first-order depth derivatives of stress and displacement are needed. To show how the Thomson-Haskell method works we consider the equation of motion for the SH-waves case (with τyy,y = 0) which gives ρ¨ uy (x, z, t) = τyz,z (x, z, t) + τyx,x (x, z, t)

(1.11)

And this equation combined with constitutive relation which becomes for SH waves τyz = µ∂uy /∂z , τyx = µ∂uy /∂x give the system df = Af dz with

(1.12) (

f = f(z) = and

(

A=

uy τyz

)

(1.13) )

0 µ− 1 ω 2 (µp2 − ρ) 0

(1.14)

where ω is the angular frequency, p the ray parameter, t the time and ρ the bulk density and uy (x, z, t) = exp[iw(px − t)] where i2 = −1. Solutions of equations of Equation 1.12 are given by f = vα exp[λα (z − zref )]

(1.15)

where v α and λα (α = 1, 2 for SH-waves) are the eigenvectors and eigenvalues of A. The most general solution f is a linear combination of type f = Fw

(1.16)

where F is a matrix whose columns consist in solutions of type 1.15 and w is a constant weighting vector. The two eigenvalues of A are √

λ1,2 = ±iω 1/β 2 − p2 = ±η

(1.17)

where β is the S-wave velocity and η is the vertical slowness for S waves and the corresponding eigenvectors are (

v

1,2

=

1 ±iωµη

)

(1.18) 9

The general general solution of Equation 1.12 for SH case is f = Fw = (

(1.19) )(

exp[+iωη(z − zref )] exp[−iωη(z − zref )] +iωµη exp[+iωη(z − zref )] −iωµη exp[−iωη(z − zref )]

w1 w2

)

In seismology F is called the layer matrix. The propagator matrix P(z, z0 ) relates the motion-stress vector at depth z to the one at depth z0 as F(z) = P(z, z0 )F(z0 )

(1.20)

and we have F(z2 ) = P(z2 , z1 )F(z1 )

(1.21)

= P(z2 , z1 )P(z1 , z0 )F(z0 ) Hence, for a layered medium as shown in Figure 1.1, the propagator matrix P(z, z0 ) for zk?1 < z < zk can be written as F(z) = P(z, zk−1 )P(zk−1 , zk−2 ) · · · P(z1 , z0 )f(z0 ) = P(z, z0 )f(z0 )(1.22) And knowing that z and z0 are in the homogeneous layer, we have P(z, z0 ) = F(z)F−1 (z0 )

(1.23)

For SH case with F(z) given by Equation 1.19 (

cos[ωη(z − z0 )] P(z, z0 ) −ωµη sin[ωη(z − z0 )]

sin[ωη(z − z0 )] cos[ωη(z − z0 )]

1 ωµη

)

(1.24)

The Thomson-Haskell theory also allows to deconvolve a wave recorded at a point in order to come back to the incident wave at the borehole station (i.e., the wave formed by the source and the path effects). The above formulation was derived for a linear elastic stress-strain relationship without attenuation. The addition of attenuation in the ThomsonHaskell Method can be done by using the viscoelastic Kelvin-Voigt solid which allows the use of damping through a complex shear modulus. A Kelvin-Voigt solid (subjected to a one-dimensional vertically incident shearwave) has a stress-strain relationship of the form (in the SH case)

τyz = µγyz + η

∂γyz ∂t

(1.25)

where τyz is the shear stress,γyz the shear strain, and η the viscosity of the material.

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1.3

Standard Genetic Algorithm

Genetic algorithms (GAs) are a class of evolutionary methods that use techniques inspired by evolutionary biology (e.g. inheritance, mutation, selection and crossover) to find exact or approximate solutions to optimization problems. GAs are best used when the objective function is : discontinuous, highly nonlinear, stochastic or has unreliable or undefined derivatives because they do not make any assumption about the underlying fitness landscape (the shape of the fitness function or objective function). In this section we give a brief presentation of a genetic algorithm that captures most of the essential components of every genetic algorithm (inheritance, mutation, selection and crossover) so it is commonly called the standard genetic algorithm (or the simple genetic algorithm). This algorithm is used in the two next chapters to invert the parameters of the soil structure of a borehole station. This inversion code was developed in 2010 by Florent De Martin for his Phd Thesis and it is written in FORTRAN 90. The algorithm is started with a set of solutions represented in binary as strings of 0s and 1s called population. The initial population of solutions is generated by selecting the best individuals ( = solutions) based on there fitness (i.e. the value of the objective function) found after running a Monte Carlo search that uses the Monte Carlo method to explore uniformly the search space. In each generation, the fitness of every individual in the population is evaluated. The individuals that are selected to form the new individuals (offspring) are selected according to their fitness - the more suitable they are the more chances they have to reproduce. This is repeated until some condition (for example number of populations or improvement of the best solution) is satisfied. The flowchart of the genetic algorithm used in this study is shown in Figure 1.2. The process of discretization of a 1-D soil column is shown in Figure 1.3.

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Figure 1.2 – Flowchart of genetic algorithm inversion preceded by a Monte Carlo search.

Figure 1.3 – Discretization of the search space for the genetic algorithm optimization. In the left-hand side graphic, the search space of the S-wave velocity is shown in solid line and an example of soil column within the search space is shown in dashed line. The discretization process is emphasized on the right-hand side. An example of individual is shown at the bottom.

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Figure 1.4 – Example of a tournament pool. The individuals who participate to the tournament are randomly selected in the population P. The winner of the tournament (i.e., the individual with the best fitness function) is injected in the population of reproduction.

Figure 1.5 – Example of a crossover and a mutation. In the population of reproduction, the first two parents are representated by a gray and a black binary. The children are generated by a uniform crossover from these parents. A mutation is the change of a bit from 0 to 1 or vice-versa. Prior to the genetic algorithm optimization, a Monte Carlo search is done in order to explore the search space. Then, the best individuals found during the Monte Carlo search are used to initialize the first population of the GA optimization. During each successive generation of the GA, a proportion of the existing population is selected based on there fitness to breed a new generation. And the rest of the production population are generated randomly to keep the diversity of the GA population and prevent from convergence towards local minimums. Popular selection methods include roulette wheel selection and tournament selection and we can see in Figure 1.4 an example of tournament selection. The next step is to generate a second generation population of solutions from those selected through genetic operators : crossover (also called recombination) and/or mutation. An example of crossover and mutation is shown in Figure 1.5. This generational process is repeated until a termination condition has been 13

reached. Examples of termination conditions : a solution is found that satisfies minimum criteria for the objective function, a fixed number of generations (i.e., populations) is reached, etc.

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Chapitre 2

Two New Methods One of the main goals of my internship is to increase the number of inverted parameters with accuracy in a single inversion using the genetic algorithm but the inversion of more parameters without adding more information (enlarging the S-Wave window time for example) could lead the problem to be ill-conditioned which could lead the genetic algorithm to be trapped in local minimums. And that’s when methods like niching methods come to use. In fact recent studies have shown that a niching method when added to a genetic algorithm prevent the stagnation of the individuals of the genetic algorithms around a local minimum and therefor increases the chances of finding the global minimum if it exists. Along with a niching method another method was added to the genetic algorithm in the purpose of accelerating the convergence process and the performance of the genetic algorithm. In the following we present in Section 2.1 the niching method and the results of it’s validation tests and in Section 2.2 the second method and the results of it’s validation tests.

2.1

Niching Method

Niching methods have been developed to reduce the effect of genetic drift resulting from the selection operator in the standard genetic algorithm. They maintain population diversity and permit the genetic algorithm to search many optima in parallel and therefor prevent the genetic algorithm from being trapped in local minimums of the search space. Nishing genetic algorithms are inspired by the mechanics of natural ecosystems. In nature, animals compete to survive by hunting, feeding, grazing, breeding, etc., and different species evolve to fill each role. A niche can be viewed as a subspace in the environment that can support different types of life. A species is defined as a group of individuals with similar biological features. For each niche, the physical resources are finite and must be shared among the population of that niche or fought for and won by the strongest individuals.

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Figure 2.1 – Flowchart of genetic algorithm with niching method. The most famous niching methods are : crowding, sharing and clearing. We chose the clearing method due to it’s compatibility with elitist selection that is used in the genetic algorithm. In the following we give in Section 2.1.1 the principles of the clearing method and an algorithm of the method and in Section 2.1.2 we give the results of the validation test.

2.1.1

Principle

The clearing procedure is applied after evaluating the fitness of individuals and before applying the selection operator (2.1). It consists of : initially we construct n niches around the n best individuals of the genetic algorithm population each niche is constructed by calculating the distance between all the individuals of the GA population and the n best individuals and if the distance is less than the radius of the niche we increase the residue of that individual to avoid it’s selection by the selection operator. With this step, we make sure that in all the next populations we have n niches and every niche only contains one individual that is one of the n best individuals of the population. To calculate the distance between 2 individuals of the same GA population we define a distance that uses binary genotypes. For example, in the case where Nparameters = 1 the distance between an individual A = (a1 a2 a3 a4 a5 ) and an individual B = (b1 b2 b3 b4 b5 ) in binary represen∑ ∑ tation with 5 bits is d = | i ai ∗ 25−i − i bi ∗ 25−i | with ai and bi ∈ {0, 1}.

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for All the individuals of the current population do if An individual I is close enough to B then Increase the residue of I; end end Algorithm 1: Algorithm of the clearing method.

f(x,y)

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Figure 2.2 – Plot of 2-D Rastrigin function represented in [0,4]x[0,4].

2.1.2

Validation

To test the new genetic algorithm we used the 2-D rastrigin function defined by f (x) = 2A +

2 ∑

x2i − A cos(2πxi )

(2.1)

i=1

where A = 10. This function has one global minimum (Figure 2.2) in (0, 0) where f (0, 0) = 0 and several local minimums. Figure 2.3 shows the results of sixteen independent inversions performed to find the global minimum of the Rastrigin function. Eight of the inversions were performed with the GA without niching and eight with the GA with the niching method. We can see that both GA without niching and GA with niching found the global minimum in (0,0) with most of the inversions and both programs were not trapped in local minimums. At this point we can not see which version of the program is better because of the simplicity of 17

without niching

With Niching

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Figure 2.3 – left : results of eight independent inversions with the genetic algorithm without niching. Right : results of eight independent inversions with genetic algorithm + niching. the 2-D optimization function. In order to see if the version of the program with the niching method gives better results than the original version without niching we made some tests on a more complex inverse problem. We inverted 1-D soil profiles using synthetic data that was calculated with the Thomson Haskell propagator matrix and with the use of initial velocity profile of the Japaneses Kik-net station : IBRH11 (see Chapter 3 for more details on this station). The inversions that we performed use a time-domain objective function E(x) defined by : ∫

E(x) = S−wave window

T HT (x) − T HS (x)2 18

(2.2)

Where T HT stands for targeted time history and T HS stands for simulated time history and x ∈ {solutions}. The objective function is computed on filtered time history data. In our case, we use time history band-pass filtered in the frequency range [0.1 - 10.0] Hz. The S-wave portion used as objective function is determined by examination of the particle motion on acceleration time history in the [transverse - up down] plane. To test the new genetic algorithm with the niching method, we first invert 6 parameters (the shear-wave velocity of each layer and a constant damping along the entire soil column) and after that we kept on increasing the number of inverted parameters ( maximum number of inverted parameters : 12) to see the difference between the two programs. For every test (with a different number of inverted parameters) we performed sixteen different inversions, eight with the original version of the program and eight with the new version of the program, to compute a mean and standard deviation of the inverted parameters. Each inversion runs over 2500 populations. A Monte-Carlo search is performed on the first 500 populations composed by 2048 individuals (one individual represents the velocity and damping soil structure from free surface to downhole). Then, the genetic algorithm inversion runs over 2000 populations composed by 1024 individuals. For every inversion, elitism is activated so that the 10 best soil columns are always present from one population to another. Selection is performed by tournament and the probability of reproduction is 85%. The probability of mutation is 0.1%. Figure 2.4, Figure 2.5 and Figure 2.6 show the results of inversions performed with 6 inverted parameters : the shear-wave velocity of each layer and a constant damping along the entire soil column. Figure 2.7, Figure 2.8 and Figure 2.9 show the results of inversions performed with 7 inverted parameters : the shear-wave velocity of each layer, the damping of the last layer and a constant damping along the four other layers of the soil column. Figure 2.10, Figure 2.11 and Figure 2.12 show the results of inversions performed with 8 inverted parameters : the shear-wave velocity of each layer, the damping of the last two layers and a constant damping along the three other layers of the soil column. Figure 2.13, Figure 2.14 and Figure 2.15 show the results of inversions performed with 9 inverted parameters : the shear-wave velocity of each layer, the damping of the last three layers and a constant damping along the two other layers of the soil column. Figure 2.16, Figure 2.17 and Figure 2.18 show the results of inversions performed with 10 inverted parameters : the shear-wave velocity and the damping of each layer of the soil column. To invert up to 12 (results : Figure 2.19, Figure 2.20 and Figure 2.21) parameters we divided the last layer of the soil column to two layers and we

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Acceleration (gal)

Station IBRH11 50 40 30 20 10 0 -10 -20 -30 -40 -50

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Figure 2.4 – Top panel : comparison between observed acceleration time history (black) and inverted time histories (blue) using the GA with Niching. Bottom panel : comparison between observed acceleration time history (black) and inverted time histories (grey) using the GA without Niching.

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Figure 2.5 – Shear wave velocity and damping profile. Search space (black), Mean from eight inversions and standard deviation (grey) calculated with GA without Niching and Mean Mean from eight inversions and standard deviation (blue) calculated with GA with Niching.

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100 Global minimum residual

with niching without niching 10 1 0.1 0.01 0.001 0

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1000 1500 Population

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Figure 2.6 – Convergence curves of eight independent inversions, GA without Niching (grey) and GA with Niching (blue).

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Figure 2.7 – Top panel : comparison between observed acceleration time history (black) and inverted time histories (blue) using the GA with Niching. Bottom panel : comparison between observed acceleration time history (black) and inverted time histories (grey) using the GA without Niching.

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Figure 2.8 – Shear wave velocity and damping profile. Search space (black), Mean from eight inversions and standard deviation (grey) calculated with GA without Niching and Mean Mean from eight inversions and standard deviation (blue) calculated with GA with Niching.

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with niching without niching 10 1 0.1 0.01 0.001 0

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Figure 2.9 – Convergence curves of eight independent inversions, GA without Niching (grey) and GA with Niching (blue).

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Figure 2.10 – Top panel : comparison between observed acceleration time history (black) and inverted time histories (blue) using the GA with Niching. Bottom panel : comparison between observed acceleration time history (black) and inverted time histories (grey) using the GA without Niching.

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Figure 2.11 – Shear wave velocity and damping profile. Search space (black), Mean from eight inversions and standard deviation (grey) calculated with GA without Niching and Mean Mean from eight inversions and standard deviation (blue) calculated with GA with Niching.

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with niching without niching 10 1 0.1 0.01 0.001 0

500

1000 1500 Population

2000

2500

Figure 2.12 – Convergence curves of eight independent inversions, GA without Niching (grey) and GA with Niching (blue).

Acceleration (gal)

Station IBRH11 50 40 30 20 10 0 -10 -20 -30 -40 -50

Target S-wave window

Acceleration (gal)

18

19

20

21

50 40 30 20 10 0 -10 -20 -30 -40 -50

22

23

24

Target time history with niching

25 26 Time (s)

27

28

29

Target S-wave window

18

19

20

21

22

23

24

30

31

32

33

32

33

Target time history without niching

25

26

27

28

29

30

31

Time (s)

Figure 2.13 – Top panel : comparison between observed acceleration time history (black) and inverted time histories (blue) using the GA with Niching. Bottom panel : comparison between observed acceleration time history (black) and inverted time histories (grey) using the GA without Niching.

24

IBRH11 0

0

Depth (m)

Depth (m)

Search space Target profile Std dev. with niching Std dev. without niching

50

100

50

100 0

500

1000 1500 2000 S-wave velocity (m/s)

2500

3000

-1

0

1

2 3 ξ0 (%)

4

5

6

Figure 2.14 – Shear wave velocity and damping profile. Search space (black), Mean from eight inversions and standard deviation (grey) calculated with GA without Niching and Mean Mean from eight inversions and standard deviation (blue) calculated with GA with Niching.

100 Global minimum residual

with niching without niching 10 1 0.1 0.01 0.001 0

500

1000 1500 Population

2000

2500

Figure 2.15 – Convergence curves of eight independent inversions, GA without Niching (grey) and GA with Niching (blue).

25

Acceleration (gal)

Station IBRH11 50 40 30 20 10 0 -10 -20 -30 -40 -50

Target S-wave window

Acceleration (gal)

18

19

20

21

22

50 40 30 20 10 0 -10 -20 -30 -40 -50

23

24

Target time history with niching

25 26 Time (s)

27

28

29

Target S-wave window

18

19

20

21

22

23

24

30

31

32

33

32

33

Target time history without niching

25

26

27

28

29

30

31

Time (s)

Figure 2.16 – Top panel : comparison between observed acceleration time history (black) and inverted time histories (blue) using the GA with Niching. Bottom panel : comparison between observed acceleration time history (black) and inverted time histories (grey) using the GA without Niching.

IBRH11 0

0

Depth (m)

Depth (m)

Search space Target profile Std dev. with niching Std dev. without niching

50

100

50

100 0

500

1000

1500

2000

2500

3000

-1

0

1

2

3

4

5

6

ξ0 (%)

S-wave velocity (m/s)

Figure 2.17 – Shear wave velocity and damping profile. Search space (black), Mean from eight inversions and standard deviation (grey) calculated with GA without Niching and Mean Mean from eight inversions and standard deviation (blue) calculated with GA with Niching.

26

100 Global minimum residual

with niching without niching 10 1 0.1 0.01 0.001 0

500

1000 1500 Population

2000

2500

Figure 2.18 – Convergence curves of eight independent inversions, GA without Niching (grey) and GA with Niching (blue).

Acceleration (gal)

Station IBRH11 50 40 30 20 10 0 -10 -20 -30 -40 -50

Target S-wave window

Acceleration (gal)

18

19

20

21

50 40 30 20 10 0 -10 -20 -30 -40 -50

22

23

24

Target time history with niching

25 26 Time (s)

27

28

29

Target S-wave window

18

19

20

21

22

23

24

30

31

32

33

32

33

Target time history without niching

25

26

27

28

29

30

31

Time (s)

Figure 2.19 – Top panel : comparison between observed acceleration time history (black) and inverted time histories (blue) using the GA with Niching. Bottom panel : comparison between observed acceleration time history (black) and inverted time histories (grey) using the GA without Niching.

27

IBRH11 0

0

Depth (m)

Depth (m)

Search space Target profile Std dev. with niching Std dev. without niching

50

100

50

100 0

500

1000 1500 2000 S-wave velocity (m/s)

2500

3000

-1

0

1

2 3 ξ0 (%)

4

5

6

Figure 2.20 – Shear wave velocity and damping profile. Search space (black), Mean from eight inversions and standard deviation (grey) calculated with GA without Niching and Mean Mean from eight inversions and standard deviation (blue) calculated with GA with Niching.

100 Global minimum residual

with niching without niching 10 1 0.1 0.01 0.001 0

500

1000 1500 Population

2000

2500

Figure 2.21 – Convergence curves of eight independent inversions, GA without Niching (grey) and GA with Niching (blue).

28

inverted the shear-wave velocity and the damping of each layer (total number of layers :6) of the new soil column. Knowing that niching methods prevent the genetic algorithm from being trapped in local minimums, we expected the tests to show that the new version of the algorithm with the niching method find better solutions spatially when the number of parameters increases and the problem becomes more and more ill-conditioned. But the results of the tests do not show what was expected, and we can see that for most cases both versions of the program find the same results which could be explain by the fact that in the standard genetic algorithm, when the individuals are selected for the reproduction only some of this individuals are from a selected from the GA population and the rest are generated randomly which prevent from the stagnation of the individuals around a local minimum and increases the chances of finding the global minimum.

2.2

Search Around The Best Individuals

The "search around the best individuals" method is a method that is used to search around the best individuals of a GA population to replace them with close individuals with better objective function value. In the following we give in Section 2.2.1 the principles of the clearing method and an algorithm of the method and in Section 2.2.2 we give the results of the validation test.

2.2.1

Principle

To explain how the method works we suppose that we are running an inversion in 2-D space. If B = (a1 a2 a3 a4 a5 a6 a7 a8 b1 b2 b3 b4 b5 b6 b7 b8 ) represented in binary with 8 bits is one of the n saved best individuals of the GA population where an represent the first parameter and bn represent the second parameter with an and bn ∈ {0, 1}. The method calculate two ∑ ∑ integers i an j with i = 5n=1 an ∗ 25−n and j = 8n=1 bn ∗ 27−n and uses i and j to generate 3N bP arametre − 1 = 32 − 1 = 8 individuals, each one of this individuals is determined by a vector with two components x and y such as x ∈ {i, i − 1, i + 1} and y ∈ {j, j − 1, j + 1} (Figure 2.22). After this step, the objective function is calculated for all this 8 individuals and compared with the value of the objective function of B and if an individual A of the new 8 individuals have a lower value of the objective function than B we replace B with A and restart the search process for the new B. The search stops for B when none of the eight close individuals of the best B have a lower objective function value. And this search process is done for all the best individuals of the GA population.

29

7

6 (i-1,j+1)

Parametre 2

5

(i,j+1)

(i-1,j)

4

(i+1,j+1)

(i,j)

(i+1,j)

B (i-1,j-1)

3

(i,j-1)

(i+1,j-1)

2

1

0 0

1

2

3 4 Parametre 1

5

6

7

Figure 2.22 – Search around one of the best individual B.

for I = 1 : N bOf AllBestIndividuals do Best = False; while Best = False do NbOfBestIndividuals = 0; for J=1 :3N bP arameters − 1 do if Residue(J) 0 then Replace I with the best of the saved Js; else Best = True; end end end Algorithm 2: Algorithm of the "search around the best individuals" method.

30

2.2.2

Validation

To test the new method we used it to find the global minimum of the 2-D Rastrigin function in the subspace [0,4]x[0,4]. With Niching Population 15

Niching + SAB Population 11

Population 6

Population 4

4

4

4

4

3

3

3

3

2

2

2

2

1

1

1

1

0

0 0

1

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Population 15

1

2

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Population 15

1

2

3

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4

4

4

3

3

3

3

2

2

2

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Population 15

1

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1

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Population 15

4

4

4

3

3

3

3

2

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2

1

1

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0 1

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0 0

1

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2

3

4

1

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3

4

0

4

4

3

3

3

3

2

2

2

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1

1

1

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0 2

3

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0 0

1

2

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3

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4

1

1

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4

0

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0

4

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4

0

3

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0

2 Population 6

4

0

1

Population 15

0 0

1

2

3

4

0

1

2

Figure 2.23 – left : results of eight independent inversions with the genetic algorithm + niching. Right : results of eight independent inversions with genetic algorithm + niching + search around best. Figure 2.23 shows the results of sixteen independent inversions performed to find the global minimum of the Rastrigin function. Eight of the inversions were performed with the GA with niching and eight with the GA with the niching method and the "search around best individuals" method. For this inversions we had two termination conditions : the program stops if the minimum residue (= 10−10 in our case) is reached or if the maximum number of population (= 15 in our case) is reached. And we can see that in the case of the GA + Niching + "search around best" five among eight inversions reached the minimum residue while, in the case of the GA + Ni31

ching, only two inversions among eight have reached the minimum residue which means that the genetic algorithm with the niching method and the "search around best individuals" method converges faster than the GA with niching and without niching (see the results of the same test with the genetic algorithm without the new methods 2.3). To test the new method on a more complex problem we did the same test with synthetic data that we did in Section 2.1.2 with the niching method. We made two tests, in one we inverted 8 parameters and in the other we inverted 9 parameters and for both test we performed eight different inversions using the GA + Niching + "search around best individuals" to compute a mean and standard deviation of the inverted parameters. Each inversion runs over 2500 populations. A Monte-Carlo search is performed on the first 500 populations composed by 2048 individuals (one individual represents the velocity and damping soil structure from free surface to downhole). Then, the genetic algorithm with niching runs over 2000 populations composed by 1024 individuals and the "search around best individuals " method is activated at 1000th (500 Monte Carlo + 500 GA ) population and after that every 10 populations For every inversion, elitism is activated so that the 10 best soil columns are always present from one population to another. Selection is performed by tournament and the probability of reproduction is 85%. The probability of mutation is 0.1%.

Acceleration (gal)

Station IBRH11 50 40 30 20 10 0 -10 -20 -30 -40 -50

Target S-wave window

18

19

20

21

22

23

24

Target time history 8 independent inversions

25

26

27

28

29

30

31

32

Time (s)

Figure 2.24 – Target synthetic time history (black) and inverted time histories (gray) from eight independent inversions. 32

33

IBRH11 0

0

Depth (m)

Depth (m)

Search space Target profile Std dev. Niching+SAB-1000 Niching Std dev. without niching Std dev.

50

100

50

100 0

500

1000 1500 2000 S-wave velocity (m/s)

2500

3000

-1

0

1

2 3 ξ0 (%)

4

5

6

Figure 2.25 – Shear wave velocity and damping profile. Search space (black) and Mean from eight inversions and standard deviation, GA+Niching+"search around best individuals" (grey), GA+Niching (blue) and Standard GA (red).

Global minimum residual

100 Niching+SAB-1000 Niching without niching

10 1 0.1 0.01 0.001 0

500

1000 1500 Population

2000

2500

Figure 2.26 – Convergence curves of eight independent inversions, GA+Niching+"search around best individuals" (grey), GA+Niching (blue) and Standard GA (red).

33

Acceleration (gal)

Station IBRH11 50 40 30 20 10 0 -10 -20 -30 -40 -50

Target S-wave window

18

19

20

21

22

23

Target time history 8 independent inversions

24

25

26

27

28

29

30

31

32

Time (s)

Figure 2.27 – Target synthetic time history (black) and inverted time histories (gray) from eight independent inversions.

IBRH11 0

0

Depth (m)

Depth (m)

Search space Target profile Std dev. Niching+SAB-1000 Niching Std dev. without niching Std dev.

50

100

50

100 0

500

1000 1500 2000 S-wave velocity (m/s)

2500

3000

-1

0

1

2 3 ξ0 (%)

4

5

6

Figure 2.28 – Shear wave velocity and damping profile. Search space (black) and Mean from eight inversions and standard deviation, GA+Niching+"search around best individuals" (grey), GA+Niching (blue) and Standard GA (red).

34

33

Global minimum residual

100 Niching+SAB-1000 Niching without niching

10 1 0.1 0.01 0.001 0

500

1000 1500 Population

2000

2500

Figure 2.29 – Convergence curves of eight independent inversions, GA+Niching+"search around best individuals" (grey), GA+Niching (blue) and Standard GA (red). Figure 2.24, Figure 2.25 and Figure 2.26 show the results of inversions performed with 8 inverted parameters : the shear-wave velocity of each layer, the damping of the last two layers and a constant damping along the three other layers of the soil column. Figure 2.27, Figure 2.28 and Figure 2.29 show the results of inversions performed with 9 inverted parameters : the shear-wave velocity of each layer, the damping of the last three layers and a constant damping along the two other layers of the soil column. We can see that even after increasing the number of inverted parameters we still can not say if the "search around best" have improved the performance of the genetic algorithm.

35

Chapitre 3

Results of the Genetic Algorithm Inversions A very important part of the work that I did during my internship was performing inversions with the genetic algorithm using initial velocity profile data of several Japanese KiK-net stations that were chosen based on there geological placements and soil proprieties hoping to find proofs of non-linear soil behavior during strong ground motions. The inversions were performed using data from four different KiK-net stations : MYGH10, IBRH11, IBRH10 and IBRH19. And for three of the stations (MYGH10, IBRH11 and IBRH19), we performed inversions using data recorded during several weak ground motions in order to obtain a better initial soil column than the soil column given by the KiK-net station and after that we performed inversions using data recorded during strong ground motions and using the new initial soil column. As for the fourth station (IBRH10) we were not able to find good results with the inversions and we didn’t have enough time to understand the reason for that. In Section 3.1 we present and explain the steps that we fallow to process the type of data that we use in the inversions. In the other sections we give, for all four KiK-net stations, the proprieties of the initial soil column and a list of all the earthquakes that was used for the inversions and finally we present the results of the inversions.

3.1

Before The Inversions

Before lunching an inversion, the data that is provided by the Kik-net station is not yet ready to be used by the genetic algorithm. When the data has been downloaded we lunch a program that uses the KiK-net files and give gnuplot format files wish is very helpful for plotting the seismograms and gives an idea to when the S-wave starts and when it ends. The second step is to filter the KiK-net data using ten different frequencies. The third step is to estimate the S-wave window time using the acceleration particle motion in the transverse - up down plane. 36

3.2

Objective Function And S-wave Window Selection

The inversions that we performed use a time-domain objective function E(x) defined by : ∫

E(x) = S−wave window

T HO (x) − T HS (x)2

(3.1)

Where T HO stands for observed time history and T HS stands for simulated time history and x ∈ {solutions}.

3.3

Genetic Algorithm Inversions

For each earthquake that was used for the inversions, we perform several independent genetic algorithm inversions in order to compute a mean and standard deviation of the inverted parameters. Each genetic algorithm inversion runs overs 2500 populations. A Monte-Carlo search is performed on the first 500 populations composed by 2048 individuals (one individual represents the velocity and damping soil structure from free surface to downhole). Then, the genetic algorithm inversion runs over 2000 populations composed by 1024 individuals. Consequently, for an earthquake, a total of 3,072,000 soil columns are compared with the observation. Such an inversion runs for about 13 hours. For a single inversion, elitism is activated so that the 10 best soil columns are always present from one population to another. Selection is performed by tournament and the probability of reproduction is 80%. The probability of mutation is 0.1%.

3.4

MYGH10

In this section we present the results of the inversions performed using the data from the KiK-net station MYGH10 and using the genetic algorithm before adding the niching method or the "search around best individuals" method.

3.4.1

Velocity Profile And Database

The initial velocity profile of MYGH10 provided by KiK-net website is shown in Fig.3.1. The shear-wave velocity Vs range from 110 m/s at the free surface to 770 m/s at the downhole (205 m). The initial soil column used for inversions at MYGH10 is presented in Table 3.1. Ground motions used for inversions are listed in Table 3.2.

37

MYGH10 0

Vs Vp

Depth (m)

50

100

150

200 0

500 1000 1500 Wave velocity (m/s)

2000

Figure 3.1 – Initial velocity profile of MYGH10 provided by KiK-net.

N1 1 2 3 4 5

Thikness (m) 1 2 31 80 91

Vs (m/s) 110 250 390 590 770

Density (kg/m3 ) 1750 1800 1850 1950 2030

Damping factor 1.0 1.0 1.0 1.0 1.0

Table 3.1 – Soil column at MYGH10. S-wave factors are used to determine the search space of genetic algorithm inversions of weak motions data.

38

N˚ EQ1 EQ2 EQ3 EQ4 EQ5 EQ6 EQ7

Date YY.MM.DD/HH.MM 2011.06.04/12.37 2011.02.21/20.34 2007.04.12/22.50 2007.04.05/21.39 2005.05.07/00.27 2004.05.09/01.30 2001.12.23/01.41

Lon. (˚) 141.265 141.026 141.147 141.149 141.307 141.065 140.970

Lat. (˚) 38.272 37.982 38.199 38.202 38.125 38.195 38.078

Mag. Mw 4.4 3.9 4.5 4.5 4 4.1 4.5

Table 3.2 – Earthquakes used for inversions.

3.4.2

Results

In this sections we give the results of the inversions that we performed using the data of the earthquakes listed in Table 3.2. We can see that for all eight earthquakes the simulated time history is closer to the targeted time history compared to the time history from Kiknet profile spatially in the S-wave time window. And based on that we know that for all seven earthquakes the S-wave velocity and damping profile resulting from the inversions give a better estimation of the real S-wave velocity and damping profile than the Kik-net profile.

39

EQ 2011.06.04

Acceleration (gal)

30

Observation (transverse) 8 independent inversions

Target S-wave window 1

20

Target S-wave window 2

10 0 -10 -20 24

25

26

27

28

29

30

31

32

33

Time (s) EQ 2011.06.04

Acceleration (gal)

30

Observation (transverse) 8 independent inversions KiK-net velocity profile Target S-wave window 2

Target S-wave window 1

20 10 0 -10 -20 24

25

26

27

28

29

30

31

32

33

Time (s)

Figure 3.2 – For EQ1. Top panel : comparison between observed acceleration time history (black) and inverted time histories (grey). Bottom panel : comparison between observed acceleration time history (black) and inverted time histories (grey) and time history from KiK-net profile.

EQ 2011.06.04 0

0

KiK-net profile Mean Std dev. Search space

50

Depth (m)

Depth (m)

50

100

150

100

150

200

200 0

500 1000 S-wave velocity (m/s)

-1

0

1

2 3 ξ0 (%)

4

5

6

Figure 3.3 – For EQ1 : Shear wave velocity and damping profile. Search space (black), KiK-net (red) and Mean from the eight inversions and standard deviation (grey).

40

EQ 2011.06.04 Global minimum residual

45 8 independent inversions 40 35 30 25 20 0

500

1000 1500 Population

2000

2500

Figure 3.4 – For EQ1 : Convergence curves of eight independent inversions.

Acceleration (gal)

EQ 2011.02.21 Observation (transverse) 8 independent inversions

10 Target S-wave window 1

Target S-wave window 2 0

23

24

25

26

27

28

29

30

Time (s)

Acceleration (gal)

EQ 2011.02.21 Observation (transverse) 8 independent inversions KiK-net velocity profile

10 Target S-wave window 1

Target S-wave window 2 0

23

24

25

26

27

28

29

Time (s)

Figure 3.5 – For EQ2. Top panel : comparison between observed acceleration time history (black) and inverted time histories (grey). Bottom panel : comparison between observed acceleration time history (black) and inverted time histories (grey) and time history from KiK-net profile.

41

30

EQ 2011.02.21 0

0

KiK-net profile Mean Std dev. Search space

50

Depth (m)

Depth (m)

50

100

150

100

150

200

200 0

500 1000 S-wave velocity (m/s)

-1

0

1

2 3 ξ0 (%)

4

5

6

Figure 3.6 – For EQ2 : Shear wave velocity and damping profile. Search space (black), KiK-net (red) and Mean from the eight inversions and standard deviation (grey).

EQ 2011.02.21 Global minimum residual

5 8 independent inversions

4.5 4 3.5 3 2.5 2 1.5 0

500

1000 1500 Population

2000

2500

Figure 3.7 – For EQ2 : Convergence curves of eight independent inversions.

42

EQ 2007.04.12

Acceleration (gal)

30

Observation (transverse) 8 independent inversions

Target S-wave window 1

20

window 2

10 0 -10 -20 18

19

20

21

22

23

24

25

26

27

28

Time (s) EQ 2007.04.12

Acceleration (gal)

30

Observation (transverse) 8 independent inversions KiK-net velocity profile window 2

Target S-wave window 1

20 10 0 -10 -20 18

19

20

21

22

23

24

25

26

27

28

Time (s)

Figure 3.8 – For EQ3. Top panel : comparison between observed acceleration time history (black) and inverted time histories (grey). Bottom panel : comparison between observed acceleration time history (black) and inverted time histories (grey) and time history from KiK-net profile.

EQ 2007.04.12 0

0

KiK-net profile Mean Std dev. Search space

50

Depth (m)

Depth (m)

50

100

150

100

150

200

200 0

500 1000 S-wave velocity (m/s)

-1

0

1

2 3 ξ0 (%)

4

5

6

Figure 3.9 – For EQ3 : Shear wave velocity and damping profile. Search space (black), KiK-net (red) and Mean from the eight inversions and standard deviation (grey).

43

EQ 2007.04.12 Global minimum residual

72 8 independent inversions

71 70 69 68 67 66 65 64 0

500

1000 1500 Population

2000

2500

Figure 3.10 – For EQ3 : Convergence curves of eight independent inversions.

Acceleration (gal)

EQ 2007.04.05 Target S-wave window 1

30 20 10 0 -10 -20 -30 18

19

20

21

22

Observation (transverse) 8 independent inversions

Target S-wave window 2

23

24

25

26

27

28

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30

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30

Time (s)

Acceleration (gal)

EQ 2007.04.05 Target S-wave window 1

30 20 10 0 -10 -20 -30 18

19

20

21

22

Observation (transverse) 8 independent inversions KiK-net velocity profile

Target S-wave window 2

23

24

25

26

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28

Time (s)

Figure 3.11 – For EQ4. Top panel : comparison between observed acceleration time history (black) and inverted time histories (grey). Bottom panel : comparison between observed acceleration time history (black) and inverted time histories (grey) and time history from KiK-net profile.

44

EQ 2007.04.05 0

0

KiK-net profile Mean Std dev. Search space

50

Depth (m)

Depth (m)

50

100

150

100

150

200

200 0

500 1000 S-wave velocity (m/s)

-1

0

1

2 3 ξ0 (%)

4

5

6

Figure 3.12 – For EQ4 : Shear wave velocity and damping profile. Search space (black), KiK-net (red) and Mean from the eight inversions and standard deviation (grey).

EQ 2007.04.05 Global minimum residual

220 8 independent inversions

210 200 190 180 170 160 150 0

500

1000 1500 Population

2000

2500

Figure 3.13 – For EQ4 : Convergence curves of eight independent inversions.

45

EQ 2005.05.07 Observation (transverse) 8 independent inversions

Acceleration (gal)

Target S-wave window 5 0 -5 23

24

25

26

27

28

29

30

Time (s) EQ 2005.05.07 Observation (transverse) 8 independent inversions KiK-net velocity profile

Acceleration (gal)

Target S-wave window 5 0 -5 23

24

25

26

27

28

29

30

Time (s)

Figure 3.14 – For EQ5. Top panel : comparison between observed acceleration time history (black) and inverted time histories (grey). Bottom panel : comparison between observed acceleration time history (black) and inverted time histories (grey) and time history from KiK-net profile.

EQ 2005.05.07 0

0

KiK-net profile Mean Std dev. Search space

50

Depth (m)

Depth (m)

50

100

150

100

150

200

200 0

500 1000 S-wave velocity (m/s)

-1

0

1

2 3 ξ0 (%)

4

5

6

Figure 3.15 – For EQ5 : Shear wave velocity and damping profile. Search space (black), KiK-net (red) and Mean from the eight inversions and standard deviation (grey).

46

EQ 2005.05.07 Global minimum residual

8.5 8 independent inversions

8 7.5 7 6.5 6 5.5 5 0

500

1000 1500 Population

2000

2500

Figure 3.16 – For EQ5 : Convergence curves of eight independent inversions.

EQ 2004.05.09

Acceleration (gal)

S-wave window 1

S-wave window 2

S-wave window 3

Observation (transverse) 8 independent inversions

0

14

15

16

17

18

19

20

21

22

23

Time (s) EQ 2004.05.09

Acceleration (gal)

S-wave window 1

S-wave window 2

S-wave window 3

Observation (transverse) 8 independent inversions KiK-net velocity profile

0

14

15

16

17

18

19

20

21

22

Time (s)

Figure 3.17 – For EQ6. Top panel : comparison between observed acceleration time history (black) and inverted time histories (grey). Bottom panel : comparison between observed acceleration time history (black) and inverted time histories (grey) and time history from KiK-net profile.

47

23

EQ 2004.05.09 0

0

KiK-net profile Mean Std dev. Search space

50

Depth (m)

Depth (m)

50

100

150

100

150

200

200 0

500 1000 S-wave velocity (m/s)

-1

0

1

2 3 ξ0 (%)

4

5

6

Figure 3.18 – For EQ6 : Shear wave velocity and damping profile. Search space (black), KiK-net (red) and Mean from the eight inversions and standard deviation (grey).

Global minimum residual

EQ 2004.05.09 10.5 10 9.5 9 8.5 8 7.5 7 6.5 6

8 independent inversions

0

500

1000 1500 Population

2000

2500

Figure 3.19 – For EQ6 : Convergence curves of eight independent inversions.

48

Acceleration (gal)

EQ 2001.12.23 15 10 5 0 -5 -10 -15

Target S-wave window 1

24

25

Observation (transverse) 8 independent inversions

Target S-wave window 2

26

27

28

29

30

Time (s)

Acceleration (gal)

EQ 2001.12.23 15 10 5 0 -5 -10 -15

Target S-wave window 1

24

25

Observation (transverse) 8 independent inversions KiK-net velocity profile

Target S-wave window 2

26

27

28

29

30

Time (s)

Figure 3.20 – For EQ7. Top panel : comparison between observed acceleration time history (black) and inverted time histories (grey). Bottom panel : comparison between observed acceleration time history (black) and inverted time histories (grey) and time history from KiK-net profile.

EQ 2001.12.23 0

0

KiK-net profile Mean Std dev. Search space

50

Depth (m)

Depth (m)

50

100

150

100

150

200

200 0

500 1000 S-wave velocity (m/s)

-1

0

1

2 3 ξ0 (%)

4

5

6

Figure 3.21 – For EQ7 : Shear wave velocity and damping profile. Search space (black), KiK-net (red) and Mean from the eight inversions and standard deviation (grey).

49

EQ 2001.12.23 100 Global minimum residual

8 independent inversions

10 0

500

1000 1500 Population

2000

2500

Figure 3.22 – For EQ7 : Convergence curves of eight independent inversions. N1 1 2 3 4 5

Thikness (m) 2 8 10 10 71

Vs (m/s) 130 180 240 450 2100

Density (kg/m3 ) 1750 1750 1800 1900 2400

Damping factor 1.0 1.0 1.0 1.0 1.0

Table 3.3 – Soil column at IBRH11. S-wave factors are used to determine the search space of genetic algorithm inversions of weak motions data.

3.5

IBRH11

In this section we present the results of the inversions performed using the data from the KiK-net station IBRH11 and using the genetic algorithm with the niching method.

3.5.1

Velocity Profile And Database

The initial velocity profile of IBRH11 provided by KiK-net website is shown in Fig.3.23. The shear-wave velocity Vs range from 130 m/s at the free surface to 2100 m/s at the downhole (101 m). The initial soil column used for inversions at IBRH11 is presented in Table 3.3. Ground motions used for inversions are listed in Table 3.4.

50

IBRH11

Depth (m)

0

Vs Vp

50

100 0

1000 2000 3000 4000 5000 Wave velocity (m/s)

Figure 3.23 – Initial velocity profile of IBRH11 provided by KiK-net.

N˚ EQ1 EQ2 EQ3 EQ4 EQ5 EQ6 EQ7 EQ8 EQ9 EQ10 EQ11 EQ12

Date YY.MM.DD/HH.MM 2011.04.16/11.19 2011.07.15/21.01 2012.03.01/07.32 2008.03.08/01.55 2011.04.19/23.10 2002.06.14/11.42 2011.03.24/08.56 2000.08.15/03.55 2008.06.05/20.37 2001.11.08/02.14 2008.11.21/19.10 2006.10.15/19.58

Lon. (˚) 139.945 140.083 140.625 140.612 139.807 139.977 140.042 140.037 140.102 140.089 140.478 140.080

Lat. (˚) 36.340 36.163 36.438 36.452 36.168 36.215 36.177 36.206 36.138 36.140 36.248 35.987

Table 3.4 – Earthquakes used for inversions.

51

Mag. Mw 5.9 5.4 5.3 5.2 5 4.9 4.8 3.9 3.6 3.6 3.5 3.5

Acceleration (gal)

EQ 2000.08.15 60 50 40 30 20 10 0 -10 -20 -30 -40 -50

Observation (transverse) 8 independent inversions

Target S-wave window

18

19

20

21

22

23

24

25

26

27

28

Time (s)

Acceleration (gal)

EQ 2000.08.15 60 50 40 30 20 10 0 -10 -20 -30 -40 -50

Observation (transverse) 8 independent inversions KiK-net velocity profile

Target S-wave window

18

19

20

21

22

23

24

25

26

27

Time (s)

Figure 3.24 – For EQ8. Top panel : comparison between observed acceleration time history (black) and inverted time histories (grey). Bottom panel : comparison between observed acceleration time history (black) and inverted time histories (grey) and time history from KiK-net profile.

3.5.2

Results

In this sections we give the results of the inversions that we performed using the data of the earthquakes listed in Table 3.4. We start with giving the results for the weak motions : EQ8, EQ9, EQ10, EQ11 and EQ12. For the weak motions we inverted seven parameters : the shear-wave velocity of each layer, the damping of the last layer and a constant damping along the four other layers of the soil column. And we have shown in Chapter 2 Section 2.1 with tests using synthetic data that with only 0.5 S S-wave window we can invert robustly up to ten parameters using all the versions of the program (without niching, with niching and with niching + "search around best"). We can see that for all eight earthquakes the simulated time history is closer to the targeted time history compared to the time history from Kiknet profile spatially in the S-wave time window. We plotted in Figure 3.39, for all five weak motions the shear wave velocity and damping profile and we chose EQ8, EQ9, EQ11 and EQ12 to calculate a mean of the shear wave velocity profiles to use for the inversions of strong ground motions data. We did not choose EQ10 because we can see in Figure 3.39 that for the last layer the S-wave velocity is very low compared to EQ8, EQ9, EQ11 and EQ12. 52

28

EQ 2000.08.15 0

KiK-net profile Mean Std dev. Search space

Depth (m)

Depth (m)

0

50

100

50

100 0

500

1000 1500 2000 S-wave velocity (m/s)

2500

3000

-1

0

1

2 3 ξ0 (%)

4

5

6

Figure 3.25 – For EQ8 : Shear wave velocity and damping profile. Search space (black), KiK-net (red) and Mean from the eight inversions and standard deviation (grey).

Global minimum residual

EQ 2000.08.15 270 260 250 240 230 220 210 200 190 180 170 160

8 independent inversions

0

500

1000 1500 Population

2000

2500

Figure 3.26 – For EQ8 : Convergence curves of eight independent inversions.

53

EQ 2001.11.08 10

Observation (transverse) 8 independent inversions

Acceleration (gal)

Target S-wave window

0

-10 13

14

15

16

17

18

19

20

21

22

23

Time (s) EQ 2001.11.08 10

Observation (transverse) 8 independent inversions KiK-net velocity profile

Acceleration (gal)

Target S-wave window

0

-10 13

14

15

16

17

18

19

20

21

22

23

Time (s)

Figure 3.27 – For EQ9. Top panel : comparison between observed acceleration time history (black) and inverted time histories (grey). Bottom panel : comparison between observed acceleration time history (black) and inverted time histories (grey) and time history from KiK-net profile.

EQ 2001.11.08 0

KiK-net profile Mean Std dev. Search space

Depth (m)

Depth (m)

0

50

100

50

100 0

500

1000 1500 2000 S-wave velocity (m/s)

2500

3000

-1

0

1

2 3 ξ0 (%)

4

5

6

Figure 3.28 – For EQ9 : Shear wave velocity and damping profile. Search space (black), KiK-net (red) and Mean from the eight inversions and standard deviation (grey).

54

EQ 2001.11.08 1 Global minimum residual

8 independent inversions

0.1

0.01 0

500

1000 1500 Population

2000

2500

Figure 3.29 – For EQ9 : Convergence curves of eight independent inversions.

EQ 2006.10.15

Acceleration (gal)

10

Observation (transverse) 4 independent inversions Target S-wave window 1 Target S-wave window 2

0

13

14

15

16

17

18

19

20

21

22

23

Time (s) EQ 2006.10.15

Acceleration (gal)

10

Observation (transverse) 4 independent inversions KiK-net velocity profile

Target S-wave window 1

Target S-wave window 2 0

13

14

15

16

17

18

19

20

21

22

Time (s)

Figure 3.30 – For EQ10. Top panel : comparison between observed acceleration time history (black) and inverted time histories (grey). Bottom panel : comparison between observed acceleration time history (black) and inverted time histories (grey) and time history from KiK-net profile.

55

23

EQ 2006.10.15 0

KiK-net profile Mean Std dev. Search space

Depth (m)

Depth (m)

0

50

100

50

100 0

500

1000 1500 2000 S-wave velocity (m/s)

2500

3000

-1

0

1

2 3 ξ0 (%)

4

5

6

Figure 3.31 – For EQ10 : Shear wave velocity and damping profile. Search space (black), KiK-net (red) and Mean from the eight inversions and standard deviation (grey).

Global minimum residual

EQ 2006.10.15 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3

4 independent inversions

0

500

1000 1500 Population

2000

2500

Figure 3.32 – For EQ10 : Convergence curves of eight independent inversions.

56

EQ 2008.06.05 10

Observation (transverse) 8 independent inversions

Acceleration (gal)

Target S-wave window

0

-10 13

14

15

16

17

18

19

20

21

22

23

Time (s) EQ 2008.06.05 10

Observation (transverse) 8 independent inversions KiK-net velocity profile

Acceleration (gal)

Target S-wave window

0

-10 13

14

15

16

17

18

19

20

21

22

23

Time (s)

Figure 3.33 – For EQ11. Top panel : comparison between observed acceleration time history (black) and inverted time histories (grey). Bottom panel : comparison between observed acceleration time history (black) and inverted time histories (grey) and time history from KiK-net profile.

EQ 2008.06.05 0

KiK-net profile Mean Std dev. Search space

Depth (m)

Depth (m)

0

50

100

50

100 0

500

1000 1500 2000 S-wave velocity (m/s)

2500

3000

-1

0

1

2 3 ξ0 (%)

4

5

6

Figure 3.34 – For EQ11 : Shear wave velocity and damping profile. Search space (black), KiK-net (red) and Mean from the eight inversions and standard deviation (grey).

57

EQ 2008.06.05 5.5 Global minimum residual

8 independent inversions 5 4.5 4 3.5 3 2.5 2 0

500

1000 1500 Population

2000

2500

Figure 3.35 – For EQ11 : Convergence curves of eight independent inversions.

EQ 2008.11.21 10

Observation (transverse) 8 independent inversions

Acceleration (gal)

Target S-wave window

0

-10 13

14

15

16

17

18

19

20

21

22

23

Time (s) EQ 2008.11.21 10

Observation (transverse) 8 independent inversions KiK-net velocity profile

Acceleration (gal)

Target S-wave window

0

-10 13

14

15

16

17

18

19

20

21

22

Time (s)

Figure 3.36 – For EQ12. Top panel : comparison between observed acceleration time history (black) and inverted time histories (grey). Bottom panel : comparison between observed acceleration time history (black) and inverted time histories (grey) and time history from KiK-net profile.

58

23

EQ 2008.11.21 0

KiK-net profile Mean Std dev. Search space

Depth (m)

Depth (m)

0

50

100

50

100 0

500

1000 1500 2000 S-wave velocity (m/s)

2500

3000

-1

0

1

2 3 ξ0 (%)

4

5

6

Figure 3.37 – For EQ12 : Shear wave velocity and damping profile. Search space (black), KiK-net (red) and Mean from the eight inversions and standard deviation (grey).

EQ 2008.11.21 100 Global minimum residual

8 independent inversions

10

1 0

500

1000 1500 Population

2000

2500

Figure 3.38 – For EQ12 : Convergence curves of eight independent inversions.

59

IBRH11 0

KiK-net profile EQ8 EQ10 EQ12 EQ9 EQ11

Depth (m)

Depth (m)

0

50

50

100

100 0

500

1000 1500 2000 S-wave velocity (m/s)

2500

-1

0

1

2 3 ξ0 (%)

4

5

6

Figure 3.39 – Shear wave velocity and damping profile. Search space (black), KiK-net (red) and Mean from the five weak motions.

In the following we give the results for the inversions of strong motions data : EQ1, EQ2, EQ3, EQ4, EQ5, EQ6 and EQ7. In Figure 3.40, Figure 3.43, Figure 3.46, Figure 3.49, Figure 3.52, Figure 3.55 and in Figure 3.58 we can see that for the seven earthquakes : EQ1, EQ2, EQ3, EQ4, EQ5, EQ6 and EQ7 the simulated time histories fit very good the observations specially in the S-wave time window. In Figure 3.47 an Figure 3.59 we can see that for EQ3 and EQ7 in the bedrock (last layer) the mean S-wave velocity of the eight inversions has the same value than the mean of S-wave velocity of the weak motions, but in the other layers we can see a decrease of the S-wave velocity and an increase of the damping which are signs of nonlinear behavior.

Acceleration (gal)

EQ 2011.04.16 250 200 150 100 50 0 -50 -100 -150

Target S-wave window

21

22

23

24

25

Observation (transverse) 4 independent inversions

26

27

28

29

Time (s)

Figure 3.40 – For EQ1 : comparison between observed acceleration time history (black) and inverted time histories (grey).

60

30

EQ 2011.04.16 0

Mean weak motions KiK-net profile Mean Std dev. Search space

Depth (m)

Depth (m)

0

50

100

50

100 0

500

1000 1500 2000 S-wave velocity (m/s)

2500

0

10

20 30 ξ0 (%)

40

50

Figure 3.41 – For EQ1 : Shear wave velocity and damping profile. Search space (black), KiK-net (red), Mean from weak motions (blue), Mean from the eight inversions (black points) and standard deviation (grey).

EQ 2011.04.16 4500 Global minimum residual

4 independent inversions 4000 3500 3000 2500 2000 1500 1000 0

500

1000 1500 Population

2000

2500

Figure 3.42 – For EQ1 : Convergence curves of eight independent inversions.

61

EQ 2011.07.15

Acceleration (gal)

150

Target S-wave window

Observation (transverse) 4 independent inversions

100 50 0 -50 -100 21

22

23

24

25

26

27

28

29

30

Time (s)

Figure 3.43 – For EQ2 : comparison between observed acceleration time history (black) and inverted time histories (grey).

EQ 2011.07.15 0

Mean weak motions KiK-net profile Mean Std dev. Search space

Depth (m)

Depth (m)

0

50

100

50

100 0

500

1000 1500 2000 S-wave velocity (m/s)

2500

0

10

20 30 ξ0 (%)

40

50

Figure 3.44 – For EQ2 : Shear wave velocity and damping profile. Search space (black), KiK-net (red), Mean from weak motions (blue), Mean from the eight inversions (black points) and standard deviation (grey).

62

EQ 2011.07.15 1400 Global minimum residual

4 independent inversions 1200 1000 800 600 400 200 0

500

1000 1500 Population

2000

2500

Figure 3.45 – For EQ2 : Convergence curves of eight independent inversions.

Acceleration (gal)

EQ 2012.03.01 200 150 100 50 0 -50 -100 -150

Target S-wave window 1

21

22

23

Observation (transverse) 4 independent inversions

Target S-wave window 2

24

25

26

27

28

29

Time (s)

Figure 3.46 – For EQ3 : comparison between observed acceleration time history (black) and inverted time histories (grey).

63

30

EQ 2012.03.01 0

Mean weak motions KiK-net profile Mean Std dev. Search space

Depth (m)

Depth (m)

0

50

100

50

100 0

500

1000 1500 2000 S-wave velocity (m/s)

2500

0

10

20 30 ξ0 (%)

40

50

Figure 3.47 – For EQ3 : Shear wave velocity and damping profile. Search space (black), KiK-net (red), Mean from weak motions (blue), Mean from the eight inversions (black points) and standard deviation (grey).

EQ 2012.03.01 10000 Global minimum residual

4 independent inversions

1000 0

500

1000 1500 Population

2000

2500

Figure 3.48 – For EQ3 : Convergence curves of eight independent inversions.

64

EQ 2008.03.08

Acceleration (gal)

150

Target S-wave window

Observation (transverse) 4 independent inversions

100 50 0 -50 -100 -150 21

22

23

24

25

26

27

28

29

30

Time (s)

Figure 3.49 – For EQ4 : comparison between observed acceleration time history (black) and inverted time histories (grey).

EQ 2008.03.08 0

Mean weak motions KiK-net profile Mean Std dev. Search space

Depth (m)

Depth (m)

0

50

100

50

100 0

500

1000 1500 2000 S-wave velocity (m/s)

2500

0

10

20 30 ξ0 (%)

40

50

Figure 3.50 – For EQ4 : Shear wave velocity and damping profile. Search space (black), KiK-net (red), Mean from weak motions (blue), Mean from the eight inversions (black points) and standard deviation (grey).

65

EQ 2008.03.08 1600 4 independent inversions Global minimum residual

1400 1200 1000 800 600 400 200 0

500

1000 1500 Population

2000

2500

Figure 3.51 – For EQ4 : Convergence curves of eight independent inversions.

Acceleration (gal)

EQ 2011.04.19 100 80 60 40 20 0 -20 -40 -60 -80

Observation (transverse) 4 independent inversions

Target S-wave window

20

21

22

23

24

25

26

27

28

29

Time (s)

Figure 3.52 – For EQ5 : comparison between observed acceleration time history (black) and inverted time histories (grey).

66

30

EQ 2011.04.19 0

Mean weak motions KiK-net profile Mean Std dev. Search space

Depth (m)

Depth (m)

0

50

100

50

100 0

500

1000 1500 2000 S-wave velocity (m/s)

2500

0

10

20 30 ξ0 (%)

40

50

Figure 3.53 – For EQ5 : Shear wave velocity and damping profile. Search space (black), KiK-net (red), Mean from weak motions (blue), Mean from the eight inversions (black points) and standard deviation (grey).

EQ 2011.04.19 1000 Global minimum residual

8 independent inversions

100

10

1 0

500

1000 1500 Population

2000

2500

Figure 3.54 – For EQ5 : Convergence curves of eight independent inversions.

67

Acceleration (gal)

EQ 2002.06.14 200 150 100 50 0 -50 -100 -150 -200

Target S-wave window 1

18

19

20

S-wave window 2

21

22

23

24

Observation (transverse) 4 independent inversions

25

26

27

28

Time (s)

Figure 3.55 – For EQ6 : comparison between observed acceleration time history (black) and inverted time histories (grey).

EQ 2002.06.14 0

Mean weak motions KiK-net profile Mean Std dev. Search space

Depth (m)

Depth (m)

0

50

100

50

100 0

500

1000 1500 2000 S-wave velocity (m/s)

2500

0

10

20 30 ξ0 (%)

40

50

Figure 3.56 – For EQ6 : Shear wave velocity and damping profile. Search space (black), KiK-net (red), Mean from weak motions (blue), Mean from the eight inversions (black points) and standard deviation (grey).

68

EQ 2002.06.14 4500 8 independent inversions

Global minimum residual

4000 3500 3000 2500 2000 1500 1000 500 0

500

1000 1500 Population

2000

2500

Figure 3.57 – For EQ6 : Convergence curves of eight independent inversions.

EQ 2011.03.24

Acceleration (gal)

200

Target S-wave window 1 Target S-wave window 2

100

Observation (transverse) 8 independent inversions

0 -100 -200 18

19

20

21

22

23

24

25

26

27

Time (s)

Figure 3.58 – For EQ7 : comparison between observed acceleration time history (black) and inverted time histories (grey).

69

28

EQ 2011.03.24 0

Mean weak motions KiK-net profile Mean Std dev. Search space

Depth (m)

Depth (m)

0

50

100

50

100 0

500

1000 1500 2000 S-wave velocity (m/s)

2500

0

10

20 30 ξ0 (%)

40

50

Figure 3.59 – For EQ7 : Shear wave velocity and damping profile. Search space (black), KiK-net (red), Mean from weak motions (blue), Mean from the eight inversions (black points) and standard deviation (grey).

EQ 2011.03.24 16000 Global minimum residual

8 independent inversions 14000 12000 10000 8000 6000 4000 2000 0

500

1000 1500 Population

2000

2500

Figure 3.60 – For EQ7 : Convergence curves of eight independent inversions.

70

N˚ 1 2 3 4 5

Thikness (m) 20 170 220 108 382

Vs (m/s) 110 380 530 850 2350

Density (kg/m3 ) 1750 1800 1850 1900 2800

Damping factor 1.0 1.0 1.0 1.0 1.0

Table 3.5 – Soil column at IBRH10. S-wave factors are used to determine the search space of genetic algorithm inversions of weak motions data. As for EQ1, EQ2, EQ4, EQ5 and EQ6 we can see in Figure 3.41, Figure 3.44, Figure 3.50, Figure 3.53 and Figure 3.56 that we have an increase of the S-wave velocity in the bedrock which could also be a proof of nonlinear behavior but could also be purely numeric. So, to understand if we have found proof of nonlinear behavior, we tried to invert data of the same earthquakes but with different stations : IBRH10 and IBRH19 to see if we will have the same results.

3.6

IBRH10

In this section we present the results of the inversions performed using the data from the KiK-net station IBRH10 and using the genetic algorithm with the niching method.

3.6.1

Velocity Profile And Database

The initial velocity profile of IBRH10 provided by KiK-net website is shown in Fig.3.61. The shear-wave velocity Vs range from 110 m/s at the free surface to 2800 m/s at the downhole (900 m). The initial soil column used for inversions at IBRH10 is presented in Table 3.5. Ground motions used for inversions are listed in Table 3.2.

3.6.2

Results

After performing inversions on several earthquakes, we found that for ground motions with very high accelerations the inversions were enabled to reproduce the targeted time history and we did not have enough time to understand this results. In the following we give the results of inversions of EQ1 and EQ6 of Table 3.4. In both cases we performed eight independent inversions using the same objective function in Section 3.2 and the GA with niching.

71

IBRH10 0

Vs Vp

100 200

Depth (m)

300 400 500 600 700 800 900 0

1000 2000 3000 4000 5000 Wave velocity (m/s)

Figure 3.61 – Initial velocity profile of IBRH10 provided by KiK-net.

Acceleration (gal)

EQ 2002/06/14 100 80 60 40 20 0 -20 -40 -60 -80

Target S-wave window

18

19

20

21

22

Observation (transverse) 4 independent inversions

23

24

25

26

27

Time (s)

Figure 3.62 – For EQ6 : comparison between observed acceleration time history (black) and inverted time histories (grey).

72

28

EQ 2011/04/16

Acceleration (gal)

150

Target S-wave window

Observation (transverse) 4 independent inversions

100 50 0 -50 -100 23

24

25

26

27

28

29

30

31

32

Time (s)

Figure 3.63 – For EQ1 : comparison between observed acceleration time history (black) and inverted time histories (grey). We can see that for EQ1 (Figure 3.63) the inversions were not able to reproduce the targeted time history which was not the case for EQ6 ( Figure 3.62).

3.7

IBRH19

In this section we present the results of the inversions performed using the data from the KiK-net station IBRH10 and using the genetic algorithm with the niching method.

3.7.1

Velocity Profile And Database

The initial velocity profile of IBRH19 provided by KiK-net website is shown in Fig.3.64. The shear-wave velocity Vs range from 110 m/s at the free surface to 2800 m/s at the downhole (900 m). The initial soil column used for inversions at IBRH19 is presented in Table 3.6. Ground motions used for inversions are listed in Table 3.7. In Table 3.7, EQ1, EQ2, EQ3, and EQ4 are same ones in Table 3.4. And EQ5 in Table 3.7 is EQ7 in Table 3.4.

N˚ 1 2 3 4

Thikness (m) 2 34 56 118

Vs (m/s) 100 1200 2100 2800

Density (kg/m3 ) 1750 2000 2400 2900

Damping factor 1.0 1.0 1.0 1.0

Table 3.6 – Soil column at IBRH19. S-wave factors are used to determine the search space of genetic algorithm inversions of weak motions data. 73

33

IBRH19 0

Vs Vp

Depth (m)

50

100

150

200 0

1500 3000 4500 Wave velocity (m/s)

6000

Figure 3.64 – Initial velocity profile of IBRH19 provided by KiK-net.

N˚ EQ1 EQ2 EQ3 EQ4 EQ5 EQ6 EQ7 EQ8 EQ9 EQ10 EQ11

Date YY.MM.DD/HH.MM 2011.04.16/11.19 2011.07.15/21.01 2012.03.01/07.32 2008.03.08/01.55 2011.04.19/23.10 2011.03.24/08.56 2011.08.26/21.26 2008.03.20/01.36 2012.04.18/07.13 2011.04.27/02.17 2008.06.05/20.37

Lon. (˚) 139.945 140.083 140.625 140.612 139.807 140.042 139.868 139.885 139.912 139.955 140.102

Lat. (˚) 36.340 36.163 36.438 36.452 36.168 36.177 36.098 36.087 36.060 36.200 36.138

Table 3.7 – Earthquakes used for inversions.

74

Mag. Mw 5.9 5.4 5.3 5.2 5 4.8 3.8 3.8 3.6 3.6 3.6

3.7.2

Results

In this section we give the results of the inversions that we performed using the data of the earthquakes listed in Table 3.7. We start with giving the results for the weak motions : EQ6, EQ7, EQ8, EQ9 and EQ10. For the weak motions we inverted seven parameters : the shear-wave velocity of each layer, the damping of the last layer and a constant damping along the four other layers of the soil column.

Acceleration (gal)

EQ 2011/08/26 Observation (transverse) 4 independent inversions

Target S-wave window

10 5 0 -5 18

19

20

21

22

23

24

25

26

27

28

Time (s)

Acceleration (gal)

EQ 2011/08/26 Observation (transverse) 4 independent inversions KiK-net velocity profile

Target S-wave window

10 5 0 -5 18

19

20

21

22

23

24

25

26

27

Time (s)

Figure 3.65 – For EQ6. Top panel : comparison between observed acceleration time history (black) and inverted time histories (grey). Bottom panel : comparison between observed acceleration time history (black) and inverted time histories (grey) and time history from KiK-net profile.

75

28

EQ 2011/08/26 0

0

KiK-net profile Mean Std dev. Search space

50

Depth (m)

Depth (m)

50

100

100

150

150

200

200 0

500 1000 1500 2000 2500 3000 3500 4000 S-wave velocity (m/s)

-1

0

1

2 3 ξ0 (%)

4

5

6

Figure 3.66 – For EQ6 : Shear wave velocity and damping profile. Search space (black), KiK-net (red) and Mean from the eight inversions and standard deviation (grey).

EQ 2011/08/26 10 Global minimum residual

4 independent inversions

1 0

500

1000 1500 Population

2000

2500

Figure 3.67 – For EQ6 : Convergence curves of eight independent inversions.

76

EQ 2008/03/20

Acceleration (gal)

10

Target S-wave window 1

Target S-wave window 2

Observation (transverse) 4 independent inversions

5 0 -5 -10 18

19

20

21

22

23

24

25

26

27

28

Time (s) EQ 2008/03/20

Acceleration (gal)

10

Target S-wave window 1

Target S-wave window 2

Observation (transverse) 4 independent inversions KiK-net velocity profile

5 0 -5 -10 18

19

20

21

22

23

24

25

26

27

28

Time (s)

Figure 3.68 – For EQ7. Top panel : comparison between observed acceleration time history (black) and inverted time histories (grey). Bottom panel : comparison between observed acceleration time history (black) and inverted time histories (grey) and time history from KiK-net profile.

EQ 2008/03/20 0

0

KiK-net profile Mean Std dev. Search space

50

Depth (m)

Depth (m)

50

100

100

150

150

200

200 0

500 1000 1500 2000 2500 3000 3500 4000 S-wave velocity (m/s)

-1

0

1

2 3 ξ0 (%)

4

5

6

Figure 3.69 – For EQ7 : Shear wave velocity and damping profile. Search space (black), KiK-net (red) and Mean from the eight inversions and standard deviation (grey).

77

EQ 2008/03/20 3.6 4 independent inversions

Global minimum residual

3.4 3.2 3 2.8 2.6 2.4 2.2 2 1.8 0

500

1000 1500 Population

2000

2500

Figure 3.70 – For EQ7 : Convergence curves of eight independent inversions.

Acceleration (gal)

EQ 2012/04/18 4 3 2 1 0 -1 -2 -3

Target S-wave window

13

14

15

16

17

Observation (transverse) 4 independent inversions

18

19

20

21

22

23

Time (s)

Acceleration (gal)

EQ 2012/04/18 4 3 2 1 0 -1 -2 -3

Target S-wave window

13

14

15

16

17

Observation (transverse) 4 independent inversions KiK-net velocity profile

18

19

20

21

22

Time (s)

Figure 3.71 – For EQ8. Top panel : comparison between observed acceleration time history (black) and inverted time histories (grey). Bottom panel : comparison between observed acceleration time history (black) and inverted time histories (grey) and time history from KiK-net profile.

78

23

EQ 2012/04/18 0

0

KiK-net profile Mean Std dev. Search space

50

Depth (m)

Depth (m)

50

100

100

150

150

200

200 0

500 1000 1500 2000 2500 3000 3500 4000 S-wave velocity (m/s)

-1

0

1

2 3 ξ0 (%)

4

5

6

Figure 3.72 – For EQ8 : Shear wave velocity and damping profile. Search space (black), KiK-net (red) and Mean from the eight inversions and standard deviation (grey).

EQ 2012/04/18 1.6 Global minimum residual

4 independent inversions 1.55 1.5 1.45 1.4 1.35 1.3 0

500

1000 1500 Population

2000

2500

Figure 3.73 – For EQ8 : Convergence curves of eight independent inversions.

79

Acceleration (gal)

EQ 2011/04/27 4 3 2 1

Observation (transverse) 4 independent inversions

Target S-wave window

0 -1 -2 -3 13

14

15

16

17

18

19

20

21

22

23

Time (s)

Acceleration (gal)

EQ 2011/04/27 4 3 2 1 0 -1 -2 -3

Observation (transverse) 4 independent inversions KiK-net velocity profile

Target S-wave window

13

14

15

16

17

18

19

20

21

22

23

Time (s)

Figure 3.74 – For EQ9. Top panel : comparison between observed acceleration time history (black) and inverted time histories (grey). Bottom panel : comparison between observed acceleration time history (black) and inverted time histories (grey) and time history from KiK-net profile.

EQ 2011/04/27 0

0

KiK-net profile Mean Std dev. Search space

50

Depth (m)

Depth (m)

50

100

100

150

150

200

200 0

500 1000 1500 2000 2500 3000 3500 4000 S-wave velocity (m/s)

-1

0

1

2 3 ξ0 (%)

4

5

6

Figure 3.75 – For EQ9 : Shear wave velocity and damping profile. Search space (black), KiK-net (red) and Mean from the eight inversions and standard deviation (grey).

80

EQ 2011/04/27 0.55 4 independent inversions Global minimum residual

0.5 0.45 0.4 0.35 0.3 0.25 0.2 0

500

1000 1500 Population

2000

2500

Figure 3.76 – For EQ9 : Convergence curves of eight independent inversions.

EQ 2008/06/05

Acceleration (gal)

3

Observation (transverse) 4 independent inversions

Target S-wave window

2 1 0 -1 -2 18

19

20

21

22

23

24

25

26

27

28

Time (s) EQ 2008/06/05

Acceleration (gal)

3

Observation (transverse) 4 independent inversions KiK-net velocity profile

Target S-wave window

2 1 0 -1 -2 18

19

20

21

22

23

24

25

26

27

Time (s)

Figure 3.77 – For EQ10. Top panel : comparison between observed acceleration time history (black) and inverted time histories (grey). Bottom panel : comparison between observed acceleration time history (black) and inverted time histories (grey) and time history from KiK-net profile.

81

28

EQ 2008/06/05 0

0

KiK-net profile Mean Std dev. Search space

50

Depth (m)

Depth (m)

50

100

100

150

150

200

200 0

500 1000 1500 2000 2500 3000 3500 4000 S-wave velocity (m/s)

-1

0

1

2 3 ξ0 (%)

4

5

6

Figure 3.78 – For EQ10 : Shear wave velocity and damping profile. Search space (black), KiK-net (red) and Mean from the eight inversions and standard deviation (grey).

Global minimum residual

EQ 2008/06/05 0.6 0.58 0.56 0.54 0.52 0.5 0.48 0.46 0.44 0.42 0.4 0.38

4 independent inversions

0

500

1000 1500 Population

2000

2500

Figure 3.79 – For EQ10 : Convergence curves of eight independent inversions.

82

IBRH19 0

50

Depth (m)

50

Depth (m)

0

KiK-net profile EQ8 EQ9 EQ10 EQ11 EQ12

100

100

150

150

200

200 0

500 1000 1500 2000 2500 3000 3500 4000 S-wave velocity (m/s)

-1

0

1

2 3 ξ0 (%)

4

5

6

Figure 3.80 – Shear wave velocity and damping profile. Search space (black), KiK-net (red) and Mean from the five weak motions. We can see that for all five weak motions the simulated time history is closer to the targeted time history compared to the time history from Kiknet profile spatially in the S-wave time window. We plotted in Figure 3.80, for all five weak motions the shear wave velocity and damping profile and we used this results to calculate a mean of the shear wave velocity profiles to use for the inversions of strong ground motions data. In the following we give the results for the inversions of strong motions data : EQ1, EQ2, EQ3, EQ4 and EQ5.

Acceleration (gal)

EQ 2011/04/16 80 60 40 20 0 -20 -40

Target S-wave window

21

22

23

24

25

Observation (transverse) 4 independent inversions

26

27

28

29

Time (s)

Figure 3.81 – For EQ1 : comparison between observed acceleration time history (black) and inverted time histories (grey). 83

30

EQ 2011/04/16 0

0

KiK-net profile Mean Mean weak motions Std dev. weak motions Search space

50

Depth (m)

Depth (m)

50

100

100

150

150

200

200 0

500

1000 1500 2000 S-wave velocity (m/s)

2500

3000

0

10

20

30

40

50

ξ0 (%)

Figure 3.82 – For EQ1 : Shear wave velocity and damping profile. Search space (black), KiK-net (red), Mean from weak motions (blue), Mean from the eight inversions (black points) and standard deviation (grey).

EQ 2011/04/16 1000 Global minimum residual

4 independent inversions

100 0

500

1000 1500 Population

2000

2500

Figure 3.83 – For EQ1 : Convergence curves of eight independent inversions.

84

EQ 2011/07/15

Acceleration (gal)

40

Target S-wave window

30

Observation (transverse) 4 independent inversions

20 10 0 -10 -20 19

20

21

22

23

24

25

26

27

28

29

Time (s)

Figure 3.84 – For EQ2 : comparison between observed acceleration time history (black) and inverted time histories (grey).

EQ 2011/07/15 0

50

Depth (m)

50

Depth (m)

0

KiK-net profile Mean Mean weak motions Std dev. weak motions Search space

100

100

150

150

200

200 0

500

1000 1500 2000 S-wave velocity (m/s)

2500

3000

0

10

20

30

40

50

ξ0 (%)

Figure 3.85 – For EQ2 : Shear wave velocity and damping profile. Search space (black), KiK-net (red), Mean from weak motions (blue), Mean from the eight inversions (black points) and standard deviation (grey).

85

EQ 2011/07/15 1000 Global minimum residual

4 independent inversions

100

10 0

500

1000 1500 Population

2000

2500

Figure 3.86 – For EQ2 : Convergence curves of eight independent inversions.

Acceleration (gal)

EQ 2012/03/01 100 80 60 40 20 0 -20 -40 -60 -80

Target S-wave window

21

22

23

24

Observation (transverse) 4 independent inversions

25

26

27

28

29

Time (s)

Figure 3.87 – For EQ3 : comparison between observed acceleration time history (black) and inverted time histories (grey).

86

30

EQ 2012/03/01 0

50

Depth (m)

50

Depth (m)

0

KiK-net profile Mean Mean weak motions Std dev. weak motions Search space

100

100

150

150

200

200 0

500

1000 1500 2000 S-wave velocity (m/s)

2500

3000

0

10

20

30

40

50

ξ0 (%)

Figure 3.88 – For EQ3 : Shear wave velocity and damping profile. Search space (black), KiK-net (red), Mean from weak motions (blue), Mean from the eight inversions (black points) and standard deviation (grey).

EQ 2012/03/01 10000 Global minimum residual

4 independent inversions

1000

100 0

500

1000 1500 Population

2000

2500

Figure 3.89 – For EQ3 : Convergence curves of eight independent inversions.

87

EQ 2008/03/08 30

Observation (transverse) 4 independent inversions

Acceleration (gal)

Target S-wave window 20 10 0 -10 -20 21

22

23

24

25

26

27

28

29

30

Time (s)

Figure 3.90 – For EQ4 : comparison between observed acceleration time history (black) and inverted time histories (grey).

EQ 2008/03/08 0

50

Depth (m)

50

Depth (m)

0

KiK-net profile Mean Mean weak motions Std dev. weak motions Search space

100

100

150

150

200

200 0

500

1000 1500 2000 S-wave velocity (m/s)

2500

3000

0

10

20

30

40

50

ξ0 (%)

Figure 3.91 – For EQ4 : Shear wave velocity and damping profile. Search space (black), KiK-net (red), Mean from weak motions (blue), Mean from the eight inversions (black points) and standard deviation (grey).

88

EQ 2008/03/08 100 Global minimum residual

4 independent inversions

10 0

500

1000 1500 Population

2000

2500

Figure 3.92 – For EQ4 : Convergence curves of eight independent inversions.

EQ 2011/03/24

Acceleration (gal)

60

Observation (transverse) 4 independent inversions

Target S-wave window

40 20 0 -20 -40 18

19

20

21

22

23

24

25

26

27

Time (s)

Figure 3.93 – For EQ5 : comparison between observed acceleration time history (black) and inverted time histories (grey).

89

28

EQ 2011/03/24 0

0

KiK-net profile Mean Mean weak motions Std dev. weak motions Search space

50

Depth (m)

Depth (m)

50

100

100

150

150

200

200 0

500 1000 1500 2000 2500 3000 3500 4000 S-wave velocity (m/s)

0

10

20

30

40

50

ξ0 (%)

Figure 3.94 – For EQ5 : Shear wave velocity and damping profile. Search space (black), KiK-net (red), Mean from weak motions (blue), Mean from the eight inversions (black points) and standard deviation (grey).

EQ 2011/03/24 1000 Global minimum residual

4 independent inversions

100

10 0

500

1000 1500 Population

2000

2500

Figure 3.95 – For EQ5 : Convergence curves of eight independent inversions.

90

In Figure 3.81, Figure 3.84, Figure 3.87, Figure 3.90 and in Figure 3.93 we can see that for the seven earthquakes : EQ1, EQ2, EQ3, EQ4 and EQ5 the simulated time histories fit very good the observations specially in the S-wave time window. For all five strong motions we can see in Figure 3.82, Figure 3.85, Figure 3.88, Figure 3.91 and Figure 3.94 that we have an increase of the S-wave velocity in the bedrock which could also be a proof of nonlinear behavior but could also be purely numeric. And also for this station we did not have enough time to understand if the results that we found are a proof of nonlinear behavior of the bedrock or not.

91

Bibliographie [1] Inverse Problems and Genetic Algorithms. Springer, 2006. [2] Wang Q. J. Using genetic algorithms to solve inverse problems. International journal of modelling & simulation, 1999. [3] Florent De Martin. Influence of the nonlinear behavior of soft soils on strong ground motions. PhD thesis, 2010.

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