**Motivation**

Seismic modelling proves very helpful
to support the interpretation
of tunnel seismic data and
to validate reflection imaging.
Modelling
is particularly important in tunnel seismology because poor spatial
resolution may often leave room for ambiguities and tunnelling noise
often compromises data quality.
Wave scattering
at or close to the excavation damaged zone around the
tunnel is an important target for seismic modelling,
in particular tunnel surface
waves that circulate around the tunnel.
A technique that is suitable for tunnel
seismic modelling
must be able to treat correctly arbitrarily complex geologies.
In particular seismic interactions at heterogeneities from empty
microcracks to water-filled breccia zones as well as the free surface
of the tunnel must be modelled correctly for arbitrary measurement
configurations.

**Method**

What method should we choose? Ray-tracing techniques are valid for
high frequencies only and may not be able to deal with the required
complexity of models, at least not with an affordable computational effort.
Finite element methods represent a possible choice, in particular if the
modelling space can be divided into areas of distinctively different scales
of heterogeneity. We prefer
finite-difference (FD) methods
because they
are easier to use and their computation times are nearly independent
of the model heterogeneity. FD schemes discretise the wave equations on a grid.
Finite differences over neighboring gridpoints replace spatial derivatives.
Among the different published algorithms
we prefer the diagonal differencing algorithm because it
is stable even at locations of extreme
elastic contrasts as they are of particular importance in tunnel seismology.
In a recent publication (Kneib, G., Leykam, A., 2004,
Finite Difference Modelling
for Tunnel Seismology, Near Surface Geophysics, 2, 71-93.),
we present
an extension of the diagonal
differencing algorithm
in three ways: (1) We apply FD operators which
are high-order accurate in space and time,
(2) we automatically switch to 2^{nd} order spatial operators
at strong heterogeneities, and
(3) we implement effective absorbing boundaries.
This allows 3D simulations of realistic models on affordable PC-hardware.

**Example**

The example below illustrates wave scattering ahead of a tunnel boring machine.
The 3-D elastic model of 6m X 6m X 2m
includes the high-velocity steel body of the
tunnelling machine and continues with the scratching tools
which reach into the loose sandy ground ahead.
Around and ahead of the machine we observe an excavation-damaged zone
where velocities and density gradually decrease towards the tunnel.
The undisturbed ground has a moderate vertical gradient.

The source that is integrated in the cutting wheel of a tunnel
boring machine applies a directional force in the forward (Z) direction,
emitting P-Waves towards the viewer and mostly S-waves to the sides.
The ten snapshots below are taken every msec in a plane immediately ahead of the
cutting wheel. They show orthogonal components X, Y, Z, the S-wave (curl),
and the P-wave (divergence).

Apart from the direct P- and S-waves we observe strong scattering
and wave conversions.
Shear-wave scattering is dominant because the source emits S-waves
into the plane shown and because dominant S-wavelengths are about 20 cm
that is the approximate dimension of the tools.
Near-source S-wave scattering
results in several wavefronts immediately following each other.

Last updated November 1, 2005. © 2005, Tunnelseis. All rights reserved.