Seismic amplifications from offshore to shore

Applied Ocean Research 53 (2015) 200–207

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Seismic amplifications from offshore to shore ˜ a, R. Ávila-Carrera a , N. Flores-Guzmán b , E. Olivera-Villasenor a,c,∗ a A. Rodríguez-Castellanos , J.E. Rodríguez-Sánchez a

Instituto Mexicano del Petróleo, Eje Central Lázaro Cárdenas 152, Gustavo A. Madero, México, DF, Mexico Centro de Investigación en Matemáticas, Jalisco s/n, Mineral de Valenciana, Guanajuato, Guanajuato, Mexico Instituto Politécnico Nacional, Unidad Profesional Zacatenco, SEPI-ESIME, Ingeniería de Sistemas, Av. Instituto Politécnico Nacional, Gustavo A. Madero, México, DF, Mexico b c

a r t i c l e

i n f o

Article history: Received 16 January 2015 Received in revised form 1 July 2015 Accepted 17 August 2015 Available online 1 October 2015 Keywords: Offshore structures Seismic amplifications Oil industry Earthquake Elastic waves Boundary Element Method

a b s t r a c t The objective of this study is to determine numerical estimations of seismic amplifications of waves traveling from offshore to shore considering the effect of sea floor configurations. According to the Boundary Element Method, boundary elements were used to irradiate waves and density force can be determined for each element. From this hypothesis, Huygens’ Principle is implemented since diffracted waves are constructed at the boundary from which they are radiated and this is equivalent to Somigliana’s theorem. Application of boundary conditions leads to determine a system of integral equations of Fredholm type of second kind, which is solved by the Gaussian method. Various numerical models were analyzed, a first one was used to validate the proposed formulation and some other models were used to show various ideal sea floor configurations to estimate seismic amplifications. Once the formulation was validated, basic slope configurations were studied for estimating spectra of seismic amplifications for various sea floor materials. In general terms, compressional waves (P-waves) can produce seismic amplifications of the incident wave in the order of 2–5. On the other hand, distortional waves (S-waves) can produce amplifications up to 5.5 times the incident wave. A relevant finding is that the highest seismic amplifications due to an offshore earthquake are always located near the shore-line and not offshore despite the seafloor configuration. © 2015 Elsevier Ltd. All rights reserved.

1. Introduction The majority of all natural earthquakes have epicenters in offshore areas [1]. Seaquakes are characterized by the propagation of vertical earthquake motions on the sea floor as a compressional wave and are reported to cause damage to ships and their effect on floating structures is a matter of great concern (Takamura et al. [2]). Trevorrow et al. [3] developed measurements of ambient seismic noise using ocean-bottom seismometers. They obtained the vertical components of seabed acceleration, in shallow waters, only, and then extrapolations of the measured pressures and seabed motions to deeper water conditions were made. Ocean-bottom seismometers were also used to quantify gravity-wave-coupled seabed motion, and to determine the propagation velocity and spectral characteristics of micro seismic noise [4]. Moreover, underwater sensor stations were used for continuous registration of bottom movements in the North Sea. The energy of the sea waves

∗ Corresponding author at: Instituto Mexicano del Petróleo, Eje Central Lázaro Cárdenas 152, Gustavo A. Madero, México, DF, Mexico. Tel.: +52 5591758145. E-mail address: [email protected] (A. Rodríguez-Castellanos). 0141-1187/$ – see front matter © 2015 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.apor.2015.08.003

mainly induces the bottom motion and microseisms [5]. Other important contributions to data acquisition in seabed for seismic applications can be consulted in [6–9]. Marine structures are generally vulnerable to strong seismic motions. However, the attention given to the seismic response of marine structures under strong seismic wave has been limited [10]. Using the Finite Element Method (FEM), Jianhong [10] evaluated the seismic amplifications in breakwater structures and found that the amplification of the horizontal seismic response is stronger than the vertical seismic one. FEM was also employed to study the propagation of tsunamis generated by earthquakes and the impact of the water waves against the coast of a circular island was calculated [11]. Moreover, the seismic performance of the submarine pipeline and the water/pipeline interactions during the seismic events was studied by Zeinoddini et al. [12] and Yan and Cheng [13]. FEM has been also used to model submerged floating tunnels under seismic propagation [14]. On the other hand, Boundary Element Method (BEM) was applied to study the earthquake-induced hydrodynamic pressures on rigid axisymmetric offshore structures of arbitrary shape [15]. A boundary integral equation was derived assuming that the seabed is a semi-infinite homogeneous elastic solid in order to analyze the seaquake induced hydrodynamic pressure acting on the

R. Ávila-Carrera et al. / Applied Ocean Research 53 (2015) 200–207

floating structure [2]. Pressure profile through water depth and seismic amplifications in irregular bathymetries using BEM were reported by Rodríguez-Castellanos et al. [16] and Martínez-Calzada et al. [17]. This paper applies the Boundary Element Method (BEM) to calculate the seismic amplifications due to the incidence of P- and S-waves on the seabed. Wave amplifications due to the configuration of the sea bottom are highlighted. Our formulation can be considered as a numerical implementation of Huygens’ Principle in which the diffracted waves are constructed at the boundary from which they are radiated. Thus, mathematically it is fully equivalent to the classical Somigliana’s representation theorem. Our results are compared with those previously published. Several seabed configurations and materials are modeled to exemplify the seismic amplification. In the following paragraphs a brief explanation of the BEM applied to sea bottom subjected to seismic motions is given.

sea floor. Regarding sea floor configuration it can be modeled as various slope-plateau steps from offshore to shore. Considering this sea floor model, a two dimensional (2D) coordinate system is chosen to develop the formulation proposed in this paper. To apply our integral formulation we need to define the boundary conditions and the regions in which the problem is divided, then Fig. 2 establishes. Consider the movement of an elastic, homogeneous and isotropic solid of volume E , delimited by the boundary of density i (x, ω)  E (∂1 E ∪ ∂2 E). Introducing fictitious sources   on  E , the total fields of displacements





tjE (x, ω)

uEj (x, ω)

and tractions

can be written, in frequency domain, as Banerjee and

Butterfield [18]:

 uEj (x, ω)

=



E

GijE (x, ␰, ω)i (␰, ω)dE E

+

2. Formulation of the problem

˝E

This paper provides a numerical solution to estimate seismic amplifications of waves propagating from offshore to shore considering sea floor configuration effects. Fig. 1 shows various types of offshore structures for the oil industry, it can be observed that the type range vary from self-supported to floating, which is dictated by water depth. Seismic response of these structures is a function of the stiffness system that links the structure to the

201

tjE (x, ω) =

GijE (x, ␰, ω)bi (␰, ω)d˝E + uoj (x, ω),



c1 i (x, ω)ıij +



E

(1)

TijE (x, ␰, ω)i (␰, ω)dE E

+ ˝E

TijE (x, ␰, ω)bi (␰, ω)d˝E + tjo (x, ω)

where GijE (x, ␰) and TijE (x, ␰) are the Green’s functions for displacements and tractions, respectively, which can be found in [16]. The

Fig. 1. Concept solutions for the offshore oil industry and slope-plateau sea floor model used for the seismic response study. A two dimensional problem is considered.

Fig. 2. (a) Boundary conditions; (b) boundary mesh.

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suffices i, j = 1, 3, which are related to the two dimensional reference E E system (see Fig. 2a). uoj (x, ω) and tjo (x, ω) are free terms depending on the type of elastic waves impinging on the body (soil), for this study is the occurrence of P- and S-waves. x, ␰ represent the receiver and source points, respectively (x = {x1 , x3 } , ␰ = {x1 , x3 }) and ω is the circular frequency. bi (␰, ω) are the body forces. The jump term is represented by c(c1 or c2 ), c = 0.5 if x tends to the boundary  ( E or  F ) “from inside” the region, c = −0.5 if x tends  “from outside” the region, or c = 0 if x is not at  . For this case, c1 = 0.5 for ∂1 E and ∂2 E; and c2 = −0.5 for ∂1 F and ∂2 F. ıij is the Kronecker’s delta. If the body is made by a fluid (˝F and  F = ∂1 F ∪ ∂2 F), the following functions represent the displacement and pressure fields:



uFn (x, ω) =

c2 (x, ω) +



+

F ω2

1

 pF (x, ω)

1

F ω2

˝F

∂GF (x, ␰, ω) F b (␰, ω)d˝F ,  ∂n

(2)



GF (x, ␰, ω)

=

F

∂GF (x, ␰, ω) (␰, ω)d F  ∂n

F

(␰, ω)d F 

+ ˝F

GF (x, ␰, ω)

(2) H0 (ωr/c F ),

= 0,

(2) H0

(3)

E

∀ x ∈ ∂2 E

N+K 

(xp , ω)

= −p (x, ω),



∀ x ∈ ∂1 E, ∂1 F ∀ x ∈ ∂1 E, ∂1 F

(6)

=



j (xp , ω)

F

=

(9)

 + F

E

N+K   ∂GF (xp , ␰r , ω)

(xp , ω)



ni

∂n



rF

+ c2 ıpr

p = 1, . . ., N

(15)

Eq. (11) can be expressed as:



N+M

+

N+K 

(xp , ω)

TijE (xp , ␰q , ω) qE

+ c1 ıij ıpq



 ni

GF (xp , ␰r , ω) rF

r=1

=

E −tno (xp , ω),

p = 1, . . ., N

(16)

And Eq. (12) as:



N+M

j (xp , ω)

= −t oE (xp , ω),

ni − c2 (x, ω) (10)



 c1 j (x, ω)ıij +

GijE (xp , ␰q , ω) qE

TijE (xp , ␰q , ω) qE

+ c1 ıij ıpq



 (ıij − ni nj )

q=1

∂GF (x, ␰, ω) E (␰, ω)dF = −uon (x, ω) ∂n



−

E −tio (x, ω)

 



N+M

(7)

(8)

TijE (x, ␰, ω)j (␰, ω)dE

GijE (x, ␰, ω)j (␰, ω)dE

1 F ω2



(14)

q=1

∀x ∈ ∂1 E, ∂1 F



−

p = 1, . . ., M

And Eq. (10) as:

j (xp , ω)



E



(5)

GF (x, ␰, ω) (␰, ω)dF = 0

E

(13)

q=1 E −t oi (xp , ω),

E



c1 j (x, ω)ıij +

p = 1, . . ., K

TijE (xp , ␰q , ω) qE + c1 ıij ıpq

= −uon (xp , ω),

ni is the outward vector to the boundary. According to the boundary conditions, Eqs. (3)–(7) and taking into account Eqs. (1) and (2), and neglecting the body forces, we can write:

F

GF (xp , ␰q , ω) qF = 0,

r=1

Shear stress is zero in solid–water interface: tjE (x, ω)(ıij − ni nj ) = 0,

E

N+M

j (xp , ω)

(4)

Stresses in the solid are balanced with water pressure: F

E

The variables uon (x, ω), tno (x, ω) and t o (x, ω) are the incident displacement and traction fields expressed in terms of the normal (n) and tangential ( ) directions to the boundary. Eqs. (8)–(12) represent the system of integral equations of Fredholm type of second kind and zero order to be solved. The unknowns (x, ω) and (x, ω) can be obtained once a discretization process is applied to the domain, see Fig. 2b. Region F is between borders ∂1 F (N elements) and ∂2 F (K elements). While, Region E is formed by ∂1 E (N elements) and ∂2 E (M elements). Thus, discrete form of Eq. (8) can be expressed as in Eq. (13).

q=1

In the seabed: Continuity of normal displacement:

tiE (x, ω)ni

(12)

Eq. (9) can be written as:

∀ x ∈ ∂2 F

uEi (x, ω)ni = uFn (x, ω),

(ıij − ni nj )

= −t oE (x, ω)

At the free soil surface: tiE (x, ω)

E

TijE (x, ␰, ω)j (␰, ω)dE

q=1

by = 4i is the Hankel function of the second kind and zero order, r is the distance between x and  and cF is the fluid velocity. The superscript “F” denotes fluid, i is the imaginary unit and c2 was defined previously. The boundary conditions of the problem studied are (see Fig. 2a): At the free water surface the pressure is zero, i.e.: pF (x, ω) = 0,



 c1 j (x, ω)ıij +

GF (x, ␰, ω)bF (␰, ω)d˝F , 

where (x, ω) is the force density in the fluid, F is the fluid density, GF (x, ␰, ω) is the Green function for the fluid and is given F ω2



TijE (x, ␰, ω)j (␰, ω)dE

GF (x, ␰, ω) (␰, ω)dF = −tnoE (x, ω)

ni

p = 1, . . ., N

(17)

Eqs. (13)–(17) represent the system of integral equations of Fredholm type of second kind and zero order to be solved by the Gaussian elimination method. The following section shows the validation of the previous formulae. 3. Validation

(11)

Wong [19] and Kawase [20] reported the seismic amplifications for the case of a semicircular canyon (see detail in Fig. 3d).

R. Ávila-Carrera et al. / Applied Ocean Research 53 (2015) 200–207

203

Fig. 3. Seismic amplification for a semicircular topography. Circles represent results by Wong [19] and squares by Kawase [20]. Plots for the proposed formulae are with a dash-dot-dot line (vacuum–solid interface) and with a solid line (air–solid interface).

They considered an elastic solid medium (with Poisson’s ratio of 0.33) in contact with a vacuum medium showing seismic amplificax tions in the solid surface between −2 ≤ a1 ≤ 2, for a dimensionless ωa = 2. Where: ω is the angular frequency, ˇ = Sfrequency of  = ˇ

4. Study cases Figs. 4 and 5 show seismic amplifications produced by compressional and distortional waves, P- and S-waves respectively, due to their impact on the seafloor. Material parameters used for the simulations correspond to material 5 in Table 2 (Borejko [22]). From Fig. 4, amplifications show a broad range for analyzed locations (A, B, C or D) and a high correlation with frequency. In any case the seismic amplifications shown in Fig. 4 (for = 0◦ ) reach a value of 4 times or less than the incident wave. For the case shown in Fig. 4a, where the location is located onshore, horizontal displacement amplifications of 3.6 with frequency  = 0.40 are observed. For Fig. 4e, the maximum amplification value is 3.5 for the vertical component at a frequency  = 1.5. Location B shows a peak at frequency  = 2.3 reaching an amplification of 3.8 for horizontal displacements, see Fig. 4b. Fig. 4g from location C shows several peaks that reach amplifications in the order of 3.9 in the frequency range  = 0.5–1.5. Location D shows an average of horizontal and vertical displacement amplifications of 2 due to S waves, see Fig. 5d and due to P waves, see Fig. 4h. Thus, it can be deduced that for a far location from the shore either on the slope or on a plateau, the maximum seismic amplification theoretical value that can be reached is 2 for P- or S-waves, as expected. All displacements are normalized with respect the incident wave (see, Appendix). Fig. 5b, c, e and g (for = 10◦ ) show amplifications greater than 4.0 times the incident wave. The higher amplification value is 5.5 for  = 1.75 and corresponds to S-waves, (see point B in Fig. 5b). Impact of P-waves generates an amplification close to 5.0 times the vertical component, (see Fig. 5g). As in the previous case, location D shows oscillations with an amplification factor close to 2.0 for

wave velocity for the region E and a = the radio of the semicircular canyon. The incidence of seismic waves considered by Wong [19] and Kawase [20] are P-and S-waves with incident angles = 0◦ and

= 30◦ , for each one. In our formulation, it is possible to consider that the height of acoustic medium is approaching to infinity (Ha → ∞) and such properties are of the air (see Table 1). Moreover, this method can deal with a solid–vacuum interface, in such way; we solve exactly the same problem solved by Wong [19] and Kawase [20]. The horizontal displacements obtained for the acoustic (air or vacuum) medium in contact with a solid one are displayed in Fig. 3. Circles represent results by Wong and squares by Kawase. Results for a vacuum–solid interface are plotted with a dash-dot-dot line and for air with a solid line. In general, it is possible to appreciate good matches between the results found with the current formulation and those at the references mentioned, for both P- and S-wave incidences. The main differences obtained are for incident case of x S-waves with = 30◦ for a1 ≤ −1 (see Fig. 3d). For all the other cases, good agreement can be seen. Moreover, for the incidence of S-waves to  = 30◦ it can be seen a strong amplification of seisx mic response at a1 = −1 reaching a horizontal displacement 3 times larger than = 0◦ (see Fig. 3b and d). All displacements are normalized with respect the incident wave (see, Appendix). The following sections examine amplifications for displacements in the sea bottom due to seismic events that impact at the seabed.

Table 1 Elastic properties for elastic and acoustic media used for verification.

Bedford and Drumheller [21] Elastic medium Wong [19] and Kawase [20]

P-wave velocity (˛) (m/s)

S-wave velocity (ˇ) (m/s)

Density () (kg/m3 )

Observations

330 1998

– 1000

1.29 2500

Only for verification purposes Only for verification purposes

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R. Ávila-Carrera et al. / Applied Ocean Research 53 (2015) 200–207

Fig. 4. Seismic amplifications in a slope-plateau model. P- and S-waves normal incidence ( = 0◦ ).

Fig. 5. Seismic amplifications in a slope-plateau model. Oblique incidence ( = 10◦ ) of P- and S-waves.

R. Ávila-Carrera et al. / Applied Ocean Research 53 (2015) 200–207 Table 2 Parameters used for the analyses. Material

S-wave velocity (ˇ) (m/s)

Density () (kg/m3 )

Poisson’s ratio ()

Reference

1

3000

2100

0.25

Huerta-Lopez et al. [23]

2 3 4 5

400 190 90 1645

1700 1400 1300 2400

0.35 0.40 0.45 0.38

Borejko [22]

the horizontal component of S-waves and the vertical component of P-waves. Locations on the seabed further away from the slopes reach a constant amplification theoretical value of 2.0. Figs. 6 and 7 show seismic amplifications calculated in locations 1, 2 and 3 for a basic seabed configuration as in Fig. 6d and Fig. 7h. Elastic medium and fluid properties are the same as used in the solutions given in Figs. 4 and 5. Fig. 6 shows displacement spectra for the horizontal and vertical components. The medium is excited by P-waves with an incidence angle = 0◦ . Location 1 is on the shore line, location 2 is at the end of the slope on the seabed and location 3 is on the seabed at a horizontal distance 1.5 Ha from location 1. For the horizontal component of displacement, location 1 shows amplifications close to 3 times the incident wave and is the location that shows higher amplitudes, see top line of Fig. 6a–d. As 

205

increases, amplification factor at location 1 reduces this is, 2.2 for  = 15◦ , 1.7 for  = 30◦ and 1.3 for  = 45◦ . Regarding amplifications for the vertical component of the displacement, they reach values of 4 for  = 2.9, 3.6 for  = 1.65, 3.2 for  = 1.1 and 3.0 for  = 1.15 for location 1, see top line of Fig. 6e–h, respectively. As in the previous case, amplifications decrease as  increases. Fig. 7 shows the case of an incident S-wave for the same medium and configuration as described in Fig. 6, where  is increased. It can be seen relevant seismic amplifications at location 1, reaching values for the horizontal component of 4.2, 3.7, 3.1, and 2.8, see top lines of Fig. 7a–d, respectively, as previously the higher amplifications are present for  = 0◦ . For the vertical displacement, the maximum amplification at location 1 is always lower to 2.7 for  = 0◦ and 1.8 for  = 15◦ ; for the cases  = 30◦ and  = 45◦ , the seismic amplifications at any location are negligible, see Fig. 7g and h. On the other hand, relevant parameters involved in the study of seismic amplifications in seabed configurations, as previously described, are the medium elastic properties in which the seismic waves propagate. Some authors have characterized the dynamic properties of medium by considering the propagation velocities, which are also related to Poisson’s ratio. Huerta-Lopez et al. [24] studied propagation velocities for seabed applications as shown in Table 2 for various seabed materials. It is worth noticing that material 1 has a shear propagation velocity ˇ = 3000 m/s while material

Fig. 6. Seismic amplifications for a basic seabed model. Normal incidence ( = 0◦ ) of P-waves.

Fig. 7. Seismic amplifications for a basic seabed model. Normal incidence ( = 0◦ ) of S-waves.

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R. Ávila-Carrera et al. / Applied Ocean Research 53 (2015) 200–207

Fig. 8. Seismic amplifications for a basic seabed model and various seabed materials. Normal incidence ( = 0◦ ) of P- and S-waves.

4 has ˇ = 90 m/s. Fig. 8 shows seismic amplifications for materials 1 to 4 according to Table 2, for locations 1, 2 and 3 as described in Figs. 6 and 7 and for  = 0◦ . From Fig. 8, it can be observed that for all the cases analyzed the higher seismic amplification is at location 1 for P- and S-waves. For P-wave incidence, seismic amplifications in the horizontal component are in the order of 3 for material 1 and for materials 2, 3 and 4 seismic amplifications are in the order of 2. Also for P-wave incidence, seismic amplifications in the vertical component reach values of 3.8 for material 1 and 3.2 for material 4. On the other hand, for S-wave incidence, seismic amplification in the horizontal component is 4.0 (Fig. 8i), 4.3 (Fig. 8j), 4.35 (Fig. 8k) and 4.30 (Fig. 8l). Additionally, the maximum displacement in the vertical component is not greater than 2.8 for location 1 and not greater than 1.5 for locations 2 and 3.

5. Convergence study A convergence study was carried out varying the number of boundary elements used for the mesh. In general, the boundaries of each region are discretized into linear elements whose size depends on the shortest wavelength (six boundary segments per S-wavelength) and within each element Gaussian integration of 3 points is used. As mentioned, the size of the elements has dependence on the frequency, then, the number of boundary elements is great for high frequencies. This criterion is commonly used in wave propagation phenomenon and guarantees accuracy in the computations. For the models used in Figs. 4 and 5, M = 82, N = 409 and K = 380 were considered, while for Figs. 6–8, M = 100, N = 125 and K = 115; all of them for a frequency of  = 3.0 (the highest studied

Table 3 Convergence check. Case

Boundary elements per S-wavelenght

1

Boundary elements

Vertical displacement amplitude at location 1 (see Fig. 6) ( = 3.0 and  = 0)

Relative error er × 100 (%)

2

K = 28 M = 138 N = 128

3.2504584

–

4

K = 56 M = 276 N = 256

3.5924585

9.52

6

K = 82 M = 409 N = 380

3.7761023

4.86

12

K = 168 M = 828 N = 768

3.7895124

0.35

2

3

4

R. Ávila-Carrera et al. / Applied Ocean Research 53 (2015) 200–207

frequency). In the following, we vary the number of boundary elements used for the model of Fig. 6 and observe the response at the location 1 (at  = 3.0 and  = 0◦ ). To this end, the discretization is done using a criterion of 2, 4, 6 and 12 boundary segments per S-wavelength, as mentioned above. The following table shows the response (vertical displacement due to the normal P-wave incidence) and the relative error reached. Table 3 shows the convergence check. Checking for convergence requires that results obtained from at least two models using different number of elements provide an acceptable error. The number of boundary elements used for the analyses (third column) depends on the number of boundary elements considered per each S-wavelength (second column). For these circumstances, the vertical displacements obtained exhibit several relative errors. The use of 6 and 12 boundary elements per each S-wavelength provided a relative error of 0.35% which can be considered as acceptable for engineering purposes thus, for the calculations presented in this paper, 6 boundary elements per S-wavelength were considered. The relative error was calculated using: er = |Drm − Dpm |/ Drm . Where Drm is the displacement obtained from the model with a higher ratio of boundary elements per S-wavelenght, while Dpm stands for displacement of the previous mesh. For example, for case 4 Drm = 3.7895124 and Dpm = 3.7761023 thus, er = 0.35%. 6. Conclusions The present paper uses the Boundary Element Method (BEM) to calculate the seismic amplifications due to the incidence of P- and Swaves on the seabed. Wave amplifications due to the configuration of the sea bottom are highlighted. Several seabed configurations and materials are modeled to exemplify the seismic amplification. It has been found that the compressional waves (P-waves) produce seismic amplifications that can vary from 2 to 5 times the amplitude of the incident wave. On the other hand, the distortional waves (S-waves) can produce seismic amplifications up to 5.5 times the incident wave. The relevant finding is that the highest seismic amplifications due to an offshore earthquake are always located near the shoreline and not offshore, despite the seafloor configuration. Moreover, the seabed configurations have huge importance in seismic amplifications and their effect depend on the slope gradient. Finally, the results obtained suggest that the medium elastic properties have a strong influence in the amplification of seismic waves. Appendix. Normalization with the incident wave amplitude (horizontal and vertical displacements) For the case of plane strain conditions, the amplitude of incident waves can be expressed in terms of two displacement potentials  and ␺ (for P- and S-waves, respectively) as follows (see Manolis and Beskos [24]): E

uoj (x, ω) = ∇ ˚ + ∇ × ,

(A.1)

oE

where uj (x, ω) is the amplitude of the incident wave (j = 1, 3, regarding the two dimensional reference system (see Fig. 2a)) and  is the Laplacian operator. According to this equation, the incident displacement field can be expressed using the gradient and rotational of the displacement potentials. For the case of the incident P-wave field, we have: E

uoj (x, ω) = ∇ ˚ =

∂˚ ∂˚ + , with ˚ = ˚0 e−ikP x e−iωt . ∂x1 ∂x3

(A.2)

207

For this case, note that ∇ ×  = 0 must be enforced. ˚0 = i/kP ; kP = ω ˛ = P-wave ˛ = P-wave-number; ω is the angular frequency, √ velocity for the region E, i = imaginary unit = −1, and t = time. In the following, the time dependent term e−iωt will be omitted. For the incident S-wave field, we have: E

uoj (x, ω) = ∇ ×  =

∂ ∂ − , with  = 0 e−ikS x , ∂x1 ∂x3

(A.3)

where  0 = i/kS and x1 = x3 = 0. Note that ∇  = 0; kS = ω/ˇ and is also equal to the S-wave-number and ˇ = the S-wave velocity for the region E. Thus, the normalized displacements for P- and S-wave incidences could be defined as:

E E E uj (x, ω)/uoj (x, ω) ⇒ uEj (∗)/uoj (∗)

(A.4)

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