Ground Motion After Seismic Event:Scattering off a Small Mountain

Ground Motion After Seismic Event:
Scattering off a Small Mountain

This tutorial studies the evolution of the ground motion after a seismic event in a two-dimensional isotropic linear elastic half space with a small mountain on its top. The seismic event is due to a vertical load applied at a point on the top of the half space to the right of the mountain. As the result of the applied load, various types of elastic waves propagate into the bulk and along the top surface of the linear elastic half space. In particular, these include longitudinal, shear, head (von Schmidt) waves as well as Rayleigh waves.

The mountain acts as a scatterer, which results in the new envelope of backscattering waves of the same nature.

Model Definition

The half space y < 0 with a small mountain on its top is occupied by a linear elastic material with the properties given in Table 1.

Table 1: Material Properties.
ρ	cp	cs
2200 kg/m3	3.2 km/s	1.8475 km/s

The mountain has the hight of 200 m. Its shape is given by the Gaussian bell

located 5 km to the left of the vertical line y = 0. This setup repeats the one used in Ref. 1.

The half space is modeled as a 40 km wide and 20 km high rectangle with the center of its top side being at the coordinate system origin (0, 0).

The reflections of the waves that reach the outer boundary are suppressed by adding the Absorbing Layers to the left, right, and bottom sides of the rectangle. The layers thickness is 2 km. In addition, a Low-Reflecting Boundary condition is imposed on the outer boundaries.

The top of the half space is free except for the load at the coordinate system origin. Its distribution in time is given by a Ricker wavelet shown in Figure 1.

Figure 1: Source distribution in time.

The free top boundary is of particular interest, because it gives rise to Rayleigh waves that propagate in a shallow region under the surface with the speed lower than that of the shear wave. One of the classical estimates of the Rayleigh wave speed, vR, is

where ν is the Poisson’s ratio. This is therefore important to use the Rayleigh wave speed while building the mesh in order to resolve the Rayleigh waves properly. That is, the minimum wavelength used to define the maximum mesh element size will be

Results and Discussion

Figure 2 shows the velocity magnitude profiles in the physical domain computed at 3, 6, 9, and 12 s. In the upper-left picture, one can see the propagation of the longitudinal (faster) and the shear (slower) waves in the medium at t = 3 s. They reach the mountain, which results in the backscattering of the wave. This is seen in the upper-right picture at t = 6 s. The Rayleigh wave also becomes distinguishable in the shallow region below the top of the ground, as it travels slower than the shear wave. One can also see the head (von Schmidt) waves that has a conical shape and propagates at the critical angle β to the ground top, such that sinβ = cs/cp. In the lower-left (t = 9 s) and lower-right (t = 12 s) pictures, the initial and the scattered waves leave the computational domain.

Figure 2: Velocity magnitude profiles at 4 different time steps.

The second principal invariant of the stress tensor contains information on the pressure and shear waves traveling through the solid. Pressure waves present a positive value of the invariant while shear waves present a negative value. As this invariant has units of squared stress, we will take the square root of the absolute value of the invariant and multiply it with the sign of the invariant to create Figure 3.

Figure 3: Signed squared root of the second principal invariant at 4 different time steps.

Notes About the COMSOL Implementation

The same notes as in the Isotropic-Anisotropic Sample: Elastic Wave Propagation model apply here.

Reference

1. D. Appelö and N. Anders Petersson, “A Stable Finite Difference Method for the Elastic Wave Equation on Complex Geometries with Free Surfaces,” Commun. Comput. Phys., vol. 5, pp. 84–107, 2009.

Acoustics_Module/Elastic_Waves/ground_motion_seismic_event

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

Click .

Click

In the tree, select .

Click

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file ground_motion_seismic_event_parameters.txt.

Create the source space and time functions given by a Gaussian and a Ricker wavelet, respectively.

Analytic 1 (an1)

In the toolbar, click

and choose .

In the window for , type G_space in the text field.

Locate the section. In the text field, type 1/sqrt(pi*dS)*exp(-x^2/dS).

Locate the section. In the text field, type m.

In the text field, type 1.

Analytic 2 (an2)

In the toolbar, click

and choose .

In the window for , type G_time in the text field.

Locate the section. In the text field, type 1e9*(2*(pi*f0*(t - t0))^2 - 1)*exp(-(pi*f0*(t - t0))^2).

In the text field, type t.

Locate the section. In the text field, type s.

In the text field, type Pa.

Locate the section. In the table, enter the following settings:


Argument	Lower limit	Upper limit
t	0	3

Click

The signal should look like the one in Figure 1.

Results

Impulse Frequency Content

In the window for , type Impulse Frequency Content in the text field.

Click to expand the section. In the text area, type FFT of G_time (Pa).

Locate the section. Select the check box.

In the associated text field, type FFT of the signal (Pa).

Function 1

In the window, expand the node, then click .

In the window for , locate the section.

From the list, choose .

In the toolbar, click

Select the check box.

In the text field, type 10.

In the toolbar, click

The Fourier transformation of the signal should look like this. The plot indicates that the mesh should resolve frequency content up to 4 to 5 Hz.

Geometry 1

Rectangle 1 (r1)

In the toolbar, click

In the window for , locate the section.

In the text field, type 40[km].

In the text field, type 20[km].

Locate the section. In the text field, type -20[km].

In the text field, type -20[km].

Surround the computational domain by layers from the left, right, and bottom. They will be used to set up the absorbing layers (sponge layers) to mimic the wave propagation in open domain.

Click to expand the section. Select the check box.

Select the check box.

In the table, enter the following settings:


Layer name	Thickness (m)
Layer 1	2[km]

Create a 0.2 km high and 1 km wide mountain on the top of the rectangle.

Parametric Curve 1 (pc1)

In the toolbar, click

and choose .

In the window for , locate the section.

In the text field, type 2[km].

Locate the section. In the text field, type s.

In the text field, type 0.2*(exp(-((s - 1[km])/0.3[km])^2) - exp(-(1[km]/0.3[km])^2))[km].

Here, the subtraction ensures that the mountain base lies on the line y = 0.

Locate the section. In the text field, type -6[km].

Click

Rectangle 2 (r2)

In the toolbar, click

In the window for , locate the section.

In the text field, type 2.4[km].

In the text field, type 0.7[km].

Locate the section. In the text field, type -6.2[km].

Point 1 (pt1)

In the toolbar, click

In the window for , click

Partition Objects 1 (par1)

In the toolbar, click

and choose .

Select the object only.

In the window for , locate the section.

Find the subsection. Select the

toggle button.

Select the object only.

Click

Delete Entities 1 (del1)

In the window, right-click and choose .

In the window for , locate the section.

From the list, choose .

On the object , select Domain 1 only.

Click

Form Union (fin)

In the toolbar, click

Form Composite Domains 1 (cmd1)

In the toolbar, click

and choose .

On the object , select Domains 4 and 5 only.

In the window for , click

Definitions

Absorbing Layer 1 (ab1)

In the toolbar, click

Select Domains 1–3, 5, and 6 only.

Materials

Ground Material

In the window, under right-click and choose .

In the window for , type Ground Material in the text field.

Elastic Waves, Time Explicit (elte)

Elastic Waves, Time Explicit Model 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

Create an isotropic material with the desired properties.

Materials

Ground Material (mat1)

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Pressure-wave speed	cp	3.2[km/s]	m/s	Pressure-wave and shear-wave speeds
Shear-wave speed	cs	1.8475[km/s]	m/s	Pressure-wave and shear-wave speeds
Density	rho	2200[kg/m^3]	kg/m³	Basic