Geology Reference
In-Depth Information
velocity-depth relationship and this is used to construct
the wavefront segments passing through each reflection
event to be migrated.
An alternative approach to migration is to assume
that any continuous reflector is composed of a series of
closely-spaced point reflectors, each of which is a source
of diffractions, and that the continuity of any reflection
event results from the constructive and destructive inter-
ference of these individual diffraction events.A set of dif-
fracted arrivals from a single point reflector embedded in
a uniform-velocity medium is shown in Fig. 4.31. The
two-way reflection times to different surface locations
define a hyperbola. If arcs of circles (wavefront segments)
are drawn through each reflection event, they intersect at
the actual point of diffraction (Fig. 4.31). In the case of a
variable velocity above the point reflector the diffraction
event will not be a hyperbola but a curve of similar con-
vex shape. No reflection event on a seismic section can
have a greater convexity than a diffraction event, hence
the latter is referred to as a
curve of maximum convexity
.In
diffraction migration
all dipping reflection events are as-
sumed to be tangential to some curve of maximum con-
vexity. By the use of a wavefront chart appropriate to the
prevailing velocity-depth relationship, wavefront seg-
ments can be drawn through dipping reflection events
on seismic sections and the events migrated back to their
diffraction points (Fig. 4.31). Events so migrated will,
overall, map the prevailing reflector geometry.
All modern approaches to migration use the
seismic
wave equation
which is a partial differential equation de-
scribing the motion of waves within a medium that have
been generated by a wave source. The migration prob-
lem can be considered in terms of wave propagation
through the ground in the following way. For any reflec-
tion event, the form of the seismic wavefield at the
surface can be reconstructed from the travel times of
reflected arrivals to different source-detector locations.
For the purpose of migration it is required to reconstruct
the form of the wavefield within the ground, in the
vicinity of a reflecting interface.This reconstruction can
be achieved by solution of the wave equation, effectively
tracing the propagation of the wave backwards in time.
Propagation of the wavefield of a reflection event half-
way back to its origin time should place the wave on the
reflecting interface, hence, the form of the wavefield at
that time should define the reflector geometry.
Migration using the wave equation is known as
wave
equation migration
(Robinson & Treitel 2000). There are
several approaches to the problem of solving the wave
equation and these give rise to specific types of wave
equation migration such as
finite difference migration
,in
which the wave equation is approximated by a finite dif-
ference equation suitable for solution by computer, and
frequency-domain migration
, in which the wave equation is
solved by means of Fourier transformations, the neces-
sary spatial transformations to achieve migration being
enacted in the frequency domain and recovered by an
inverse Fourier transformation.
Migration by computer can also be carried out by
Source-detector
Actual
reflection
point
Locus of all
reflection points
with equal
travel times
Display position
on seismic section
Fig. 4.29
For a given reflection time, the reflection point may be
anywhere on the arc of a circle centred on the source-detector
position. On a non-migrated seismic section the point is mapped
to be immediately below the source-detector.
Reflector
surface
Record
surface
Fig. 4.30
A planar-dipping reflector
surface and its associated record surface
derived from a non-migrated seismic
section.
α
t
α
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