Geology Reference
In-Depth Information
9
Seismic Rays
Abstract
In this second chapter on seismology, I introduce seismic ray theory
starting from the Eikonal equation. The classic concepts about travel-
time curves are discussed, as well as the seismic phase's nomenclature
at regional and global scale.
parameters vary smoothly from a grid cell to its
neighboring cells. In this instance, at each grid
center we can write the wave equations in the
form:
9.1
The Eikonal Equation
Seismic rays are the continuum mechanics
analogous of the usual light rays of geometrical
optics. They have been used since the dawn of
seismological science in the interpretation of
earthquake data and still furnish the simplest
computational approach to a wide class of prob-
lems, including the localization of earthquake
foci, the determination of focal mechanisms, and
seismic tomography. However, we shall see that
the range of applicability of this approach is lim-
ited to the propagation of high-frequency waves.
Furthermore, seismic rays do not adequately
describe non-geometrical phenomena such as
diffraction. The starting point of seismic ray
theory is the so-called
eikonal equation
,which
determines the relation between the geometry
of a wavefront and the velocity fields '
D
'(
r
)
and “
D
“(
r
). We know that these fields depend
in turn from the mechanical parameters of the
that a non-homogeneous but isotropic material
can be approximated by a regular grid of small
homogeneous regions and that the mechanical
@
2
@t
2
D
0
1
'
2
.r/
2
r
(9.1)
@
2
1
“
2
.r/
@t
2
D
0
2
r
(9.2)
These equations can be solved locally to deter-
mine the mode of propagation of seismic waves
within specific grid cells. Let ®
D
®(
r
,
t
)anyof
the field components (
r
,
t
)or
i
(
r
,
t
). The form
of Eqs. (
9.1
)and(
9.2
) suggests a monochromatic
solution with amplitude and phase that depend
from the position.
Therefore, we make the ansatz:
®.
r
;t/
D
A.r/e
i!.tT.r//
(9.3)
where
T
D
T
(
r
) is a phase factor and
A
D
A
(
r
)
is the local wave amplitude. Taking the gradient
and then the divergence of (
9.3
)gives:
