**3.2. Numerical examples**

In this subsection, we conduct the numerical implementation of the fluid-solid formulation of the introduced method. Three examples are calculated for validity testing, considering both plane wave incidence and SV-wave source excitation. Although the models used in this subsection seem very simple, they are often used as examples for validity testing of some seismogram synthetic method.

We first calculate the case of a plane *P*-wave (Primary or pressure wave) vertically incident onto irregularly layered fluid-solid structures in the two models, as shown in Figure 11. The reason for the selection of the two models is because the synthetic results of seismoacoustic scattering due to the irregular fluid-solid interface are available in [20]. The existing results for the two models calculated by the reflection/transmission matrix method in [20] were validated in detail by the finite difference method [20], which concluded that the reflection/transmission matrix method could give more accurate results. Therefore, we will compare our results with those calculated by the reflection/ transmission matrix method.

**Figure 11.** The two models used to test the fluid-solid formulation: (a) Model 1; (b) Model 2 [20]

In order to make our comparison convincing, not only are the model parameters set to be the same as those used in [20] but also the time function of the input plane wave is selected as

$$f(t) = \begin{cases} \sqrt{a/\pi} \exp(-at^2), -T\_s/2 \le t \le T\_s/2\\ 0, & t \ge T\_s/2 \text{ , } t \le -T\_s/2 \end{cases} \tag{53}$$

where the pulse duration *T*S is set to 4.2 s and *α* is set to 2.1. Eq. (53) is exactly the same as the Eq. (88) in [20] and its shape and spectrum are shown in Figure 12.

**Figure 12.** The time function (a) and its Fourier spectra (b) used for the two models in Figure 11

#### *3.2.1. Model 1 subjected to a plane wave incidence*

144 Wave Processes in Classical and New Solids

**3.2. Numerical examples** 

seismogram synthetic method.

as

Fluid Vf

<sup>f</sup>

=1500 m/s

=1000 kg/m3

Depth (km)

(a)

details of which are suppressed here.

boundary element method have been developed.

0 32 64

Plane P-wave

=6000 m/s

=3460 m/s =2500 kg/m3

Solid Vp

Vs

Distance (km)

water surface 

order of calculating the global matrix propagators in the fluid layer and the solid layer, the

Although the boundary element method is known as the best way to model wave propagation problems in unbounded media, it is still necessary to make some special treatment on the truncation edges of the model in order to avoid the appearance of some unphysical waves. For such purpose, many different techniques [33-40] applicable to the

In this subsection, we conduct the numerical implementation of the fluid-solid formulation of the introduced method. Three examples are calculated for validity testing, considering both plane wave incidence and SV-wave source excitation. Although the models used in this subsection seem very simple, they are often used as examples for validity testing of some

We first calculate the case of a plane *P*-wave (Primary or pressure wave) vertically incident onto irregularly layered fluid-solid structures in the two models, as shown in Figure 11. The reason for the selection of the two models is because the synthetic results of seismoacoustic scattering due to the irregular fluid-solid interface are available in [20]. The existing results for the two models calculated by the reflection/transmission matrix method in [20] were validated in detail by the finite difference method [20], which concluded that the reflection/transmission matrix method could give more accurate results. Therefore, we will compare our results with those calculated by the reflection/ transmission matrix method.

**Figure 11.** The two models used to test the fluid-solid formulation: (a) Model 1; (b) Model 2 [20]

In order to make our comparison convincing, not only are the model parameters set to be the same as those used in [20] but also the time function of the input plane wave is selected

(a) (b)

(b)

Fluid Vf

<sup>f</sup>

=1500 m/s

=1000 kg/m3

Depth (km)

�(�) � � ��/�exp(����)� ���/2 � � � ��/2

 �� � � ��/2 � � � ���/2 , (53)

0 32 64

Plane P-wave

=3460 m/s =2500 kg/m3

Solid Vp=6000 m/s

Vs

Distance (km)

water surface 

Figure 13 shows the displacements of both horizontal and vertical components along the fluid-solid interface of *Model* 1. The one on the left is calculated by the reflection/transmission matrix method in [20]. And the one on the right is calculated by the present method. Inside the irregular part of the model, i.e., the basin structure, the largeamplitude multiple reflections in the water basin are well observed in the synthetic waveforms calculated by both our method and the reflection/transmission matrix method. Both the scattered *P* wave (P) and *Rayleigh* wave (R), which are indicated in the left plot, can be well and clearly observed in our calculated seismograms.

#### *3.2.2. Model 2 subjected to a plane wave incidence*

Figure 14 shows the displacements of both horizontal and vertical components at the fluidsolid interface of *Model* 2. This model is a little bit more complicated than *Model* 1. The flat fluid layer on both sides of *Model* 2 will cause multiple reflections of the incident plane *P*wave easily. The multiple reflections will be interfered by the waves scattered by the irregular fluid-solid interface, which leads to the complexity of the wave field. In this case, the 'observation points' are all located at the fluid layer bottom. The one on the left is calculated by the reflection/transmission matrix method in [20]. And the one on the right is calculated by the present method. Again, the synthetic seismograms calculated by both methods agree with each other very well. The periodic multiple reflections at the flat portion of the fluid layer are clearly observed and interfered by the scattered waves from the inside of the irregular part of the model, which are modeled very well by both methods.

An Efficient Approach for Seismoacoustic Scattering Simulation Based on Boundary Element Method 147

��� � ���� � ���(�)�(�)�(�), (54)

U x

The above two testing calculations show the validity of the present method and the correctness of our computational program, respectively. However, the two numerical examples are only plane wave incidence cases. To validate the present method further, we show one more example using an SV-wave (Shear wave polarized in the vertical plane)

where the source function *f*(*t*) is a Ricker wavelet with central frequency 0.25 Hz. The model used for the calculation is similar to *Model* 1 but with a width of 80 km in total, and the basin-like fluid-solid interface having a width of 60 km and thickness of 2.1 km. The material properties of the model is little different from *Model* 1, details of which can refer to [23]. The SV-wave source is located at (distance, depth) = (0.1 km, 4.1 km). As for this case, the results calculated by the discrete wavenumber method are available in [23] so that we

**Figure 15.** Time responses along the basin-like fluid-solid interface: the one on the left calculated in [23]

0

5

1 0

1 5

2 0

) s ( e T i m

2 5

3 0

3 5

4 0

0

5

1 0

1 5

2 0

 e ( s ) T i m

U z

2 5

3 0

3 5

4 0


0 D i s t a n c e ( k m )

4 0

Figure 15 shows the displacements of both horizontal and vertical components along the basin-like fluid-solid interface due to the SV-wave source presented in Eq. (54). The one on the left is calculated by the discrete wavenumber method in [23]. And the one on the right is calculated by the present method. It can be clearly seen that the large-amplitude multiple

with a time offset of 8 seconds, and the one on the right calculated by the present method.

*3.2.3. Model 1 subjected to an SV-wave point source* 

excitation source as [23]

can make a full waveform comparison.

**Figure 13.** Time responses along the fluid-solid interface of *Model* 1: the one on the left calculated in [20], and the one on the right calculated by the present method

**Figure 14.** The same as Figure 13, but for *Model* 2

#### *3.2.3. Model 1 subjected to an SV-wave point source*

146 Wave Processes in Classical and New Solids

**Figure 13.** Time responses along the fluid-solid interface of *Model* 1: the one on the left calculated in

0

0

) Time (s

)s( Time

U x

U x

1 2

1 2

0

) Time (s

0

)s( Time

U z

0

1 2

0

1 2

3 2 D i s t a n c e ( k m )

6 4

3 2 D i s t a n c e ( k m )

6 4

U z

[20], and the one on the right calculated by the present method

**Figure 14.** The same as Figure 13, but for *Model* 2

The above two testing calculations show the validity of the present method and the correctness of our computational program, respectively. However, the two numerical examples are only plane wave incidence cases. To validate the present method further, we show one more example using an SV-wave (Shear wave polarized in the vertical plane) excitation source as [23]

$$M\_{\chi x} = -M\_{z\chi} = -Mf(t)\delta(\mathfrak{x})\delta(\mathfrak{z}),\tag{54}$$

where the source function *f*(*t*) is a Ricker wavelet with central frequency 0.25 Hz. The model used for the calculation is similar to *Model* 1 but with a width of 80 km in total, and the basin-like fluid-solid interface having a width of 60 km and thickness of 2.1 km. The material properties of the model is little different from *Model* 1, details of which can refer to [23]. The SV-wave source is located at (distance, depth) = (0.1 km, 4.1 km). As for this case, the results calculated by the discrete wavenumber method are available in [23] so that we can make a full waveform comparison.

**Figure 15.** Time responses along the basin-like fluid-solid interface: the one on the left calculated in [23] with a time offset of 8 seconds, and the one on the right calculated by the present method.

Figure 15 shows the displacements of both horizontal and vertical components along the basin-like fluid-solid interface due to the SV-wave source presented in Eq. (54). The one on the left is calculated by the discrete wavenumber method in [23]. And the one on the right is calculated by the present method. It can be clearly seen that the large-amplitude multiple reflections in the fluid-solid basin part are well observed in the synthetic waveforms calculated by both the present method and the discrete wavenumber method. Furthermore, the first scattered *P*-waves, the first scattered *Rayleigh* waves, and the secondary scattered *Rayleigh* waves, which are separately indicated as *P*, *R* and *R*2 in the plot on the left, can be well and clearly observed in our calculated seismograms. Considering the time offset used in [23], the very good agreement between the results by the two methods further confirms the validity of the present method and the correctness of the fluid-solid formulation.

An Efficient Approach for Seismoacoustic Scattering Simulation Based on Boundary Element Method 149

Figure 17 shows the time responses of the horizontal and vertical motions along the irregular solid interface due to a vertically incident plane *P* wave for the models with and without a fluid layer, respectively. It can be seen from Figure 17 that the direct waves appearing on the vertical component keep the same amplitude along the solid interface, regardless of the existence of the uppermost fluid layer. Outside the irregular part of the interface, later arrivals on the vertical component are mainly due to the wave reflection at the upper part of the irregular interface but they seem to contain some contribution of the diffracted waves since their amplitude changes very slowly. Judging from their particle motions and apparent velocities, we can say that they appear to become *Rayleigh* waves soon after the departure from the edges of the irregular part of the interface. Comparison between the results for the models with and without the fluid layer can be clearly seen in Figure 17. For the model shown in Figure 16(a), although the later arrivals on the vertical component inside the irregular part of the interface seem complicated, the multiple later arrivals outside the irregular part of the interface due to the multiple reflections caused by the fluid layer are clearly observed. In the case of the absence of the fluid layer, i.e. the model shown in Figure 16(b), there is only one later arrival on the vertical component, even the previous complicated later arrivals appearing inside the irregular part of the interface in the case of the presence of the fluid layer completely

**Figure 17.** Time responses along the irregular solid interface in the models shown in Figure 16 due to a

(a) (b)

0

T i m e (s)

U x

1 2

0

T i m e (s)

0

(b)

1 2

3 2 D i s t a n c e ( k m )

6 4

U z

vertically incident plane *P* wave: (a) with a fluid layer; (b) without a fluid layer.

0

(a)

1 2

3 2 D i s t a n c e ( k m )

6 4

*4.1.1. Plane wave incidence* 

0

)s( T i m e

U x

1 2

0

)s( T i m e

disappear, which appears to be very interesting results.

U z

Up to now, we have validated the fluid-solid formulation by three examples, which results in the conclusion that the introduced approach can accurately cover the seismoacoustic scattering due to an irregular fluid-solid interface. In the next Section, we will show how to use the introduced approach to simulate the effects of a fluid layer on the synthetic seismograms and the water reverberation by three preliminary examples, respectively.
