*2.2.3. Deep basin model*

Consider the model configuration consisting of a deep basin structure, as shown in Figure 6. The basin has a trapezoidal shape with 1 km depth and 10 km width at the surface. The shear wave velocity of the basin-filling sediments and the half-space are 2.5 km/s and 1 km/s, respectively, and the ratio of mass density is set to 1:1.

An Efficient Approach for Seismoacoustic Scattering Simulation Based on Boundary Element Method 137

basin model as in Figure 7(a) but for a Ricker wavelet of characteristic frequency *fc* = 0.5 Hz (2 s). Very similar phenomena to that shown in Figure 7(a) can be seen, but the amplitude of the Love wave is larger in this case. The X-shaped pattern of the surface-wave arrivals is more clearly shown for high-frequency incident waves, but the maximum amplitude of the Love wave decreases because energy splits into the fundamental mode and the other higher mode. The two seismograms were calculated first by the discrete wavenumber method in [29]. And nice agreement between our calculated and those in [29] can be

Finally, a two-layer model with an irregular interface is considered to have a point source,

( ) 1 cos / 2 / 2,for 3 ,

where *h*(1) = 2.0 km, *h* = 0.5 km, and *d* = 0.5 km. The material parameters are 1 km/s and 1

The synthetic seismograms for the two-layer model are shown in Figure 9. It can be seen from Figure 9 that all the seismic phases can be divided into two groups of hyperbolas, namely, a group of hyperbolas with a center point of *x* = *xs* and another group of hyperbolas with a center point of *x* = 0. The latter ones are due to the reflections by the flat free surface and the flat part of the interface. The former ones, however, are due to the diffractions by the irregular part of the interface. This model was first calculated by the global generalized reflection/transmission matrices method in [8]. Our result shows a nice agreement with that

*zx h x d dh d x d*

, for

 

*h x d*

0, for 3

*x d*

(28)

(1)

g/cm3 for the top layer, and 1.5 km/s and 1 g/cm3 for the bottom layer, respectively.

**Figure 8.** A two-layer model with an irregular interface: the source is at (2 km, 1 km)

observed.

in [8].

*2.2.4. Two-layer model with irregular interface* 

as shown in Figure 8. The irregular interface is described by

**Figure 6.** Deep basin topography and the surrounding homogeneous half-space

**Figure 7.** Time responses along the free surface of the basin in Figure 6 due to a vertically incident *SH*wave: (a) characteristic frequency *fc* is 0.25 Hz (4 sec), (b) *fc* is 0.5 Hz (2 sec)

The time responses at the surface of the basin by a vertically incident *SH* wave are plotted in Figure 7(a). The characteristic frequency *fc* is 0.25 Hz (4 s). This figure shows the horizontally propagating waves generated by the edges of the basin. The amplitude of Love waves (horizontally polarized shear surface waves) is smaller than that of the direct wave. Although these Love waves make the total duration longer, the time interval between the direct wave arrival and the Love wave arrival is less than 10 sec near the edges. Each arrival of reflected Love waves is well separated. Figure 7(b) shows the time responses for the same basin model as in Figure 7(a) but for a Ricker wavelet of characteristic frequency *fc* = 0.5 Hz (2 s). Very similar phenomena to that shown in Figure 7(a) can be seen, but the amplitude of the Love wave is larger in this case. The X-shaped pattern of the surface-wave arrivals is more clearly shown for high-frequency incident waves, but the maximum amplitude of the Love wave decreases because energy splits into the fundamental mode and the other higher mode. The two seismograms were calculated first by the discrete wavenumber method in [29]. And nice agreement between our calculated and those in [29] can be observed.

#### *2.2.4. Two-layer model with irregular interface*

136 Wave Processes in Classical and New Solids

Consider the model configuration consisting of a deep basin structure, as shown in Figure 6. The basin has a trapezoidal shape with 1 km depth and 10 km width at the surface. The shear wave velocity of the basin-filling sediments and the half-space are 2.5 km/s and 1

0

5

10

15

20

TIME (s)

25

30

35

40

**Figure 7.** Time responses along the free surface of the basin in Figure 6 due to a vertically incident *SH-*

(a) (b)


OFFSET (km)

The time responses at the surface of the basin by a vertically incident *SH* wave are plotted in Figure 7(a). The characteristic frequency *fc* is 0.25 Hz (4 s). This figure shows the horizontally propagating waves generated by the edges of the basin. The amplitude of Love waves (horizontally polarized shear surface waves) is smaller than that of the direct wave. Although these Love waves make the total duration longer, the time interval between the direct wave arrival and the Love wave arrival is less than 10 sec near the edges. Each arrival of reflected Love waves is well separated. Figure 7(b) shows the time responses for the same

wave: (a) characteristic frequency *fc* is 0.25 Hz (4 sec), (b) *fc* is 0.5 Hz (2 sec)


OFFSET (km)

km/s, respectively, and the ratio of mass density is set to 1:1.

**Figure 6.** Deep basin topography and the surrounding homogeneous half-space

*2.2.3. Deep basin model* 

0

5

10

15

20

TIM E (s)

25

30

35

40

Finally, a two-layer model with an irregular interface is considered to have a point source, as shown in Figure 8. The irregular interface is described by

$$\mathbf{z}(\mathbf{x}) = h^{(1)} + \begin{cases} h, & \text{for } \left| \mathbf{x} \right| \le d \\ \left\{ 1 + \cos \left[ \pi \left( \left| \mathbf{x} \right| - d \right) / 2d \right] \right\} h / 2, \text{for } d \le \left| \mathbf{x} \right| \le 3d, \\ \mathbf{0}, & \text{for } \left| \mathbf{x} \right| \ge 3d \end{cases} \tag{28}$$

where *h*(1) = 2.0 km, *h* = 0.5 km, and *d* = 0.5 km. The material parameters are 1 km/s and 1 g/cm3 for the top layer, and 1.5 km/s and 1 g/cm3 for the bottom layer, respectively.

The synthetic seismograms for the two-layer model are shown in Figure 9. It can be seen from Figure 9 that all the seismic phases can be divided into two groups of hyperbolas, namely, a group of hyperbolas with a center point of *x* = *xs* and another group of hyperbolas with a center point of *x* = 0. The latter ones are due to the reflections by the flat free surface and the flat part of the interface. The former ones, however, are due to the diffractions by the irregular part of the interface. This model was first calculated by the global generalized reflection/transmission matrices method in [8]. Our result shows a nice agreement with that in [8].

**Figure 8.** A two-layer model with an irregular interface: the source is at (2 km, 1 km)

An Efficient Approach for Seismoacoustic Scattering Simulation Based on Boundary Element Method 139

���� + (�+�)�� � � + ���� = ��, (29)

Consider the boundary element problem in a region, where the elastic wave equation in

in which **u** is the seismic displacement response in frequency domain, and **f** is the seismic source. Suppose the seismic source distribution consists of a simple point source at a position vector **s**. With the aid of the free-space Green's function, Eq. (29) can thus be transformed into the following boundary integral equation for the displacement *uj*(**r**) on the

∆(�)��(�) <sup>+</sup> � �� � (��)��� (�� ��) ��(��) <sup>=</sup> � �� � (��)��� (�� ��) ��(��) + �����(�� �)

and discretization of interfaces (*<sup>i</sup>*) and (*<sup>i</sup>*+1) each into *N* elements (for

are the coefficient matrices obtained by integrating the fundamental

�(�)�(�) = �(�)�(�) + ����, (31)

and **t**(*i*)

with subscripts (*i*) or (*i*+1) are of the dimension

and **t**(*i*)

(�) � + ����, (32)

are of the dimension

are vectors

where *j* and *k* can be 1, 2. **r** and **r** are the position vectors of "field point" and "source point" on the boundary , respectively. The coefficient (**r**) generally depends on the local geometry at **r**, *tj*(**r**) are the traction components, and *Ujk*(**r**, **r**) and *Tjk*(**r**, **r**) are the fundamental solutions for displacements and tractions, respectively. The source exciting direction can be controlled arbitrarily by changing one of two components *fj*. Application of

containing element displacements and tractions at both interfaces (*<sup>i</sup>*) and (*<sup>i</sup>*+1), respectively.

4*N*1. In the multilayered medium (as shown in Figure 10), considering the continuity conditions of displacements and tractions at inner interfaces, Eq. (31) needs to be rewritten

displacements and tractions at (*<sup>i</sup>*) and (*<sup>i</sup>*+1). Similarly, the subscripts (*i*) and (*i*+1)

For the same purpose as in Section 2, the upward direction is defined as the positive direction for tractions so that they have unique values at each interface [18]. Eq. (32) thus

(�) �(���)

(�) � � �(�) (�)

with subscripts (*i*) and (*i*+1) corresponds separately to the element

indicate the integration over elements at (*<sup>i</sup>*) and (*<sup>i</sup>*+1), respectively.

�(���)

are of the dimension 4*N*4*N*, and **u**(*i*)

(�) �=��(�)

*3.1.1. Formulations in solid layers* 

frequency domain is given by

boundary of the region [30]

Eq. (30) in domain (*i*)

and **G**(*i*)

and **t**(*i*)

Obviously, the matrices **H**(*i*)

and **G**(*i*)

and **G**(*i*)

��(�)

(�) �(���)

(�) � � �(�)

and **G**(*i*)

(�)

�(���)

where **H**(*i*)

into

where **u**(*i*)

in **H**(*i*)

4*N*2*N*.

becomes

Obviously, **H**(*i*)

simplicity, we use constant element) give rise to

solutions *Tjk* and *Ujk* over elements at both interfaces (*<sup>i</sup>*) and (*<sup>i</sup>*+1), and **u**(*i*)

**Figure 9.** Synthetic seismograms for the model in Figure 8. The characteristic frequency is 1.0 Hz.
