Geoscience Reference
In-Depth Information
In the generalized (or exact) and WKBJ (Wenzel, Kramers, Brillouin and
Jeffreys) ray methods (e.g., Helmberger 1968; Chapman 1978), the total response
of a stack of layers is expressed as an infinite sum of the impulse responses of
partial rays. Each partial ray is associated with a different propagation path
through the structure, and together they include all interconversions and rever-
berations. The final synthetic seismogram is obtained by convolving the impulse
response sum with the source and receiver functions. The problem with these
methods comes from the termination of this infinite sum of impulse responses:
rays that would make a significant contribution may inadvertently be excluded
from the calculation. Often only primary reflections are considered, making this
a cheap method.
The classical ray method allows the inclusion of lateral variation in velocity
and of curved interfaces (e.g., Cerveny 1979). In this method, the particle dis-
placement is expressed as an infinite sum, known as a ray series. For seismic body
waves (reflected and refracted P- and S-waves), it is often sufficient to consider
only the first term in the ray series: this is the solution obtained according to
the principles of geometrical optics. This approximation is not valid near crit-
ical points, caustics or shadow zones; corrections must be applied there. Ray
paths to be included in the computation must be specified (in practice, frequently
only primary rays are considered, multiples being neglected). This method is not
as accurate as the generalized ray method but has the significant advantage of
including the effects of lateral variation.
The Maslov integral method (Chapman and Drummond 1982), an asymp-
totic high-frequency extension of ray theory, is useful in that it includes the
signals in shadow zones and at caustics, but has the disadvantage of serious prob-
lems near pseudo-caustics (Kendall and Thompson 1993). However, an exten-
sion of the Maslov method in which a Kirchoff integration is used to reduce the
problems associated with pseudo-caustics (Huang
et al
. 1998)isaconsiderable
improvement.
The most comprehensive method for simple structures, although also the most
costly, is the reflectivity method (Fuchs and Muller 1971;Muller 1985). In this
technique, all the calculations are performed in the frequency domain, and all
reflections and interconversions are included by using a matrix formulation for
the reflection and transmission coefficients. The attenuation
Q
can also be spec-
ified for each layer (Kennett 1975)aswell as
. The computation is
reduced by calculating only arrivals within a specified phase-velocity window,
and by including only the multiples and interconversions for the 'reflection zone',
usually the lower part of the model. The layers above the reflection zone, which
need not be the same beneath source and receiver, are assumed to introduce
only transmission losses; no reflections are calculated for these upper layers. The
source and receiver responses are included in the frequency domain by multi-
plication, and Fourier transformation then gives the synthetic seismogram as a
time series. The disadvantage of this method is the amount of computing time
α
,
β
and
ρ
