1. Seismic wave simulation in complex
structures to study site amplications
Fabien Coutel and Peter Mora
Abstract
A new technique is used to simulate seismic waves in 2D media
with complex structures. It consists of a numerical solution to the
wave equation on a curved grid whose lines follow interfaces in the
2D medium. Previous work using a centered convolutional nitedierence operator to compute the spatial derivatives in the curved
grid did not yield accurate results for surface wave phenomena at the
free surface. A pseudospectral method using Chebychev polynomials
enables precise spatial derivatives to be computed within the medium
at and below the free surface and hence, overcomes the diculties of
using a centered nite-dierence operator. Tests were conducted using topography and subsurface models adjacent Brisbane airport and
a volcano in the Reunion Island to test the ability of the approach
to handle realistic cases. Ultimately, this approach will be extended
to the 3D case and simulations will be conducted to determine site
amplications of seismic waves in the Brisbane region.
Method
The elastic wave equation transformed in generalized coordinates is solved using a collocation
scheme which can be considered as a generalization of the nite-dierence method. Both the
spatial derivatives of the eld variables and the metrics of the transformation are calculated
using a Chebyshev pseudospectral method. An advantage of the Chebychev method is that
the derivatives can be precisely computed everywhere in the model including at the boundaries. Hence, wave phenomena at the free surface are precisely simulated and absorbing side
boundaries can be easily implemented. Furthermore, Chebyshev's polynomials are denser in
the vicinity of the boundaries which results in a very precise solution of the wave equation
at the free surface, up to the Nyquist wave number of the grid. Absorbing boundary conditions are applied using an highly eective method based on one-dimensional characteristics
perpendicular to the boundaries. More details about the implementation of the method and
preliminary results are presented in Komatitsch et al. 1996 and Carcione 1994.
Propagation of seismic waves through a heterogeneous
medium
To test the ability of the approach to model waves in a heterogeneous model with strong free
surface topography, we used a part of the topography of the Piton de la Fournaise (volcano in
1
QUAKES{1 : Coutel & Mora
Simulation of site eects
the Reunion Island) together with an articial geological model. While the subsurface model
was articial1 1, these simulations allowed us to test the ability of our method to simulate
surface waves and multiple reections in a model with realistic complexity. In these tests, only
the free surface followed the curved grid and not interfaces in the subsurface model. Rather,
the slowness model was interpolated onto the curved grid. This overcomes the problem of
embedding a structured grid into an arbitrary model. The source is an explosion situated
close to the surface. This generates Rayleigh waves which propagate along the bumpy free
surface as seen in Figure 1.1.
:
Figure 1.1: Snapshots of a simulation of seismic waves through a multi-layer medium (including the
surface topography of the Piton de la Fournaise volcano) at dierent times. Colours represent the
vertical component of particle velocity in the medium with red for upwards and blue for downwards.
Propagation of seismic waves produced by an earthquake
at Brisbane Airport
We obtained the topography of Brisbane (see Figure 1.2) for a 10km 10km region including
the City Centre and Brisbane Airport. Figure 1.3 shows three snapshots of the propagation
of the waves caused by a shallow earthquake in the vicinity of the Airport. The geological
model has been deduced from several dierent seismic refraction surveys in this area. As the
Airport is built on sediments (top right layers in Figure 1.3), it induces amplications of the
seismic signal. Reverberations in this layer would further amplify certain frequencies.
1:1
A subsurface model was unavailable.
2
bane topography
Simulation of site eects
0 100
Figure 1.2: Bris-
Topo(m)
QUAKES{1 : Coutel & Mora
Figure 1.3: Snapshots from a simulation of a shallow earthquake in the region of Brisbane Airport
at dierent times. Colours represent the vertical component of particle velocity in the medium with
red for upwards and blue for downwards.
Perspectives
When fully developed for the 3D case, the pseudospectral technique will allow site eects
in Brisbane to be modelled. This will provide a detailed picture of site amplications as a
function of frequency and position in Brisbane. Results will be compared with observational
studies (Smyth, 1994). The observational approach used in this study was based on analysis
of ambient seismic noise and is known to yield unreliable amplication factors although the
frequency of the fundamental mode of amplication is well predicted. Hence, comparison with
3
QUAKES{1 : Coutel & Mora
Simulation of site eects
the observational results will be used to validate the more powerful simulation based approach
and to construct a more precise model of the subsurface layers (sediments). These results
may also provide insights on how to interpret amplication factors provided by observational
studies.
References
1. Carcione, J.M., 1994, The wave equation in generalized coordinates, Geophysics , 59, 1911-1919.
2. Komatitsch, D., Coutel, F., and Mora, P., 1996, Tensorial formulation of the wave-equation for
modeling curved interfaces, in: QUAKES Tech. Rep. # 1 , (The University of Queensland).
3. Smyth, M., 1994, Site response characteristics using spectral ration analysis of microtremors,
B.Sc.App. Honours thesis, Department of Earth Sciences, The Univeristy of Queensland.
4
5