Wave Equation PSDM - Petroleum Geo-Services

A Publication of PGS Geophysical Vol. 5 No. 9
October 2005
Wave Equation Pre-Stack Depth Migration at PGS
Introduction
Pre-stack depth migration (PSDM)
of 3D seismic data is a challenging and
complex process. An almost countless
array of PSDM solutions can be found
in the literature. Each solution will
pursue an improvement in a specific
geophysical component of imaging
(e.g. the accuracy of steep dips or the
preservation of “true” amplitudes), or
will attempt to overcome the significant
computational costs that have delayed
the entry of PSDM into mainstream
processing. As computational
resources become faster and cheaper it
becomes more feasible to use PSDM on
larger size 3D data volumes.
Simultaneously, it becomes more
feasible to implement increasingly
sophisticated PSDM solutions, thereby
addressing challenges to successful
seismic imaging from increasingly
complex geology.
The industry market for
commercial PSDM processing has
matured rapidly during the last decade.
PSDM algorithms using Kirchhoff
integral techniques are now routinely
used on large-scale 3D volumes.
Significantly, “wave equation” PSDM
(WEPSDM) algorithms are now being
used on large 3D surveys in the Gulf of
Mexico, and smaller 3D volumes
elsewhere.
WEPSDM algorithms use
wavefield extrapolation techniques to
image pre-stack seismic data in depth.
In contrast to most Kirchhoff
implementations, they can naturally
handle “multi-arrivals” or
“triplications”, where multi-pathing
occurs between specific source and
receiver locations. Complex scattering,
focusing and defocusing effects are
implicitly also accounted for.
WEPSDM does not make the high
frequency assumptions of Kirchhoffbased
PSDM and, therefore, can
produce higher quality images in areas
of complex geology (rapid lateral
variations in velocity). As noted, the
increasingly attractive price
/performance ratios of Linux-based PC
clusters allow the pursuit of more
accurate and more computationally
intensive WEPSDM algorithms.
Discussion is given below to the
selection of different WEPSDM
algorithms for different geological
scenarios.
It is clear that ongoing
developments in WEPSDM technology
will see its application to increasingly
larger projects, in a cost effective
manner. From a technical viewpoint,
we will eventually see wavefield
extrapolators being developed that are
capable of handling wave propagation
at all dips in a medium with all arbitrary
lateral velocity variations.
Wave Equation PSDM
Fundamentals
The many names given to WE
migration methods are often confusing
Continued on next page
Summary
PGS has invested significant
resources in the development of
wave equation pre-stack depth
migration (WEPSDM) routines.
These include Common Azimuth
migration and shot migrations at
increasing levels of sophistication:
Gazdag Phase Shift, Split-Step
Fourier, and Finite Difference
methods. Gazdag Phase Shift and
Split-Step Fourier are highly
efficient extrapolation methods
suitable for mild lateral velocity
variations. Finite Difference
methods are more accurate for
strong velocity changes, and may
be classified into explicit and
implicit approaches. Each
technique presents inherent merits
and challenges with respect to
efficiency and cost. The critical
virtue is that a flexible range of
WEPSDM solutions are available
to be customized to the demands
of any imaging project. In other
words, the solution is matched to
the complexity of the earth model
and the geophysical objectives.
When 3D acquisition is
appropriately parameterized to
enable precise 3D data
regularization, and an accurate 3D
velocity model can be constructed,
WEPSDM will provide the most
sophisticated and accurate
imaging solutions available.

TechLink October 2005 Page 2
Wave Equation PSDM
Fundamentals
Continued from Page 1
because they may refer to the domain in
which the process occurs (e.g.
“frequency-wavenumber”), the manner
in which the data is organized during
the process (e.g. “shot record” or “shotgeophone”),
the methodology used (e.g.
“survey sinking” or “reverse time”) or
the type of extrapolation used (e.g.
“explicit finite difference”).
Irrespective of the many types of
WEPSDM implementations, all
migrations consist of the following two
elements:
• The solution of a differential equation
(the wave equation or its equivalent),
and
• The application of an imaging
condition.
In terms of both accuracy and cost,
the extrapolator used is the most
defining issue. The most accurate
extrapolator is that used by reverse time
migration, which is a “two way”
solution. Two way extrapolation can
handle multiples, is able to image high
dip and turning waves, provides better
(relative) amplitudes preservation and
better accommodates numerical noise
and imaging artefacts. Unfortunately,
two way extrapolators are about an
order of magnitude slower than one
way extrapolators, so all current
commercial WEPSDM algorithms use
one way (downward) extrapolators.
One way extrapolators cannot handle
multiples (they must be removed prior
to imaging), are restricted to dips less
than 90°, and will accommodate lateral
velocity variations according to their
complexity (and therefore cost). In
order of increasing complexity,
examples include Gazdag phase shift,
Split-Step Fourier, phase shift plus
interpolation, and (implicit or explicit)
finite difference.
The two most
common one way
extrapolation
implementations are
“common shot” and
“survey sinking”
implementations.
Common shot domain
techniques extrapolate
the shot and receiver
w a v e f i e l d s
independently, and
obtain an image for
each shot by comparing
the two wavefields at
each spatial location
and depth. Plane wave
techniques (e.g.
“delayed shot”) are a cost-saving The PGS Wave Equation
implementation of common shot PSDM Portfolio
techniques that migrate several shots PGS currently offers five
simultaneously. Survey sinking WEPSDM solutions contained in three
techniques (such as common or narrow routines in its Cube Manager
azimuth WEPSDM) simultaneously
extrapolate both the shot and receiver
wavefields, effectively simulating the
wavefield as if it was recorded at each
extrapolated depth (thus the “survey
sinking” name). The output image at
each spatial location is obtained when
the extrapolated shot and receiver are
collocated (i.e. zero offset and time).
Common shot implementations are the
most accurate WEPSDM solutions, and
form the majority of the PGS
WEPSDM portfolio.
Numerous wavefield extrapolation
techniques are available within the PGS
WEPSDM portfolio. The choice of
which to use is driven by: 1. The
complexity of the velocity model,
notably the lateral variability, and 2.
The computational speed requirements
of the particular project. The benefits
of a large portfolio of WEPSDM
solutions are flexibility and efficiency,
as illustrated in Figure 1.
TM
Figure 1: The variety of wavefield extrapolators available
within the PGS WEPSDM portfolio. Gazdag phase shift
migration is exact for no lateral velocity variation, and is
the fastest available option (much faster than Kirchhoff).
Finite difference migration provides the most accurate
solution in complex velocity media (including VTI
anisotropy), but is also the most computationally expensive.
PGS does provide the ability to select the most efficient
extrapolation solution at each depth step, constrained by
the lateral variability in the velocity model at the output
depth step.
processing system:
1. A fast Common Azimuth WEPSDM
that is particularly suitable for typical
marine seismic geometries, and for
early iterations in depth imaging
when the velocity model is being
refined. Lateral velocity variations
are elegantly handled, as is the phase
correction required at each depth step.
2. A unique, multi-algorithm Adaptive
WE PSDM that efficiently images
geology where the overburden has
relatively mild velocity variation.
The strategy involves the automated
selection of the most appropriate
algorithm based on the measured
velocity complexity at each depth
step (refer also to Figure 1). Thus,
Gazdag Phase Shift and Split-Step
Fourier methods are used where mild
velocity variation allows. An
inline/cross-line splitting Implicit
Finite Difference solution, with phase

Page 3 A Publication of PGS Geophysical
correction, is applied in the case of
more complex velocity variation. Our
adaptive approach is also available in
shot and plane wave modes, and
incorporates WKBJ corrections in the
wavefield extrapolation for correct
amplitude preservation. The
application may also be run entirely in
single-algorithm mode for each of the
three respective algorithms.
3. An Explicit Finite Difference
WEPSDM. This is the PGS flagship
WEPSDM implementation, and uses
an innovative “constrained” operator
design for downward extrapolation
that ensures stability, efficiency and
high dip accuracy. The
implementation handles isotropic and
VTI anisotropic velocity fields, steep
dips, and is available in both full shot
and plane wave modes. Efficiency
measures inherent in the
implementation include the ability to
handle rectangular bins and the
adaptive design of variable step sizes.
Of significance to the extraction of
reservoir parameters, this solution
includes an imaging condition that
extracts data reflectivity.
Naturally, the accuracy of the 3D
velocity model used for WEPSDM is
critical, and represents a critical
challenge to any imaging success. PGS
has the ability to generate (reflection)
angle gathers for velocity model
updates, and the ability to perform
velocity scanning for velocity model
updates, particularly in sub-salt areas.
Figure 2 illustrates an explicit
finite difference WEPSDM application
to the well-known SIGSBEE2A
synthetic model.
Figure 3 illustrates the
improvements possible by replacing
Kirchhoff PSDM with explicit finite
difference WEPSDM, as applied to a
Continued on next page
Figure 2: The explicit finite difference PSDM enables very accurate imaging below salt
structures, as illustrated with this SIGSBEE2A example. Subtle faults and sub-salt structures
are evident, and image focus is excellent in all spatial locations. Thus, WEPSDM is proven to
have significant value in locations of complex velocity variation.
Figure 3: Kirchhoff PSDM (left) vs. Explicit Finite Difference WEPSDM (right) in the Gulf of
Mexico. Note the benefits of WEPSDM for imaging sedimentary features on the flanks of the
salt and below the salt.
Figure 4: Isotropic (left) vs. Anisotropic (right) Explicit Finite Difference PSDM comparison
of images from the Marmousi model dataset. Incorporation of VTI anisotropy parameters
allows superior focusing of events, notably in the upper half of the section.

TechLink October 2005 Page 4
The PGS Wave Equation
PSDM Portfolio
Continued from Page 3
real dataset is from the Gulf of Mexico.
Significant improvements were
achieved in terms of image resolution
and clarity.
Finally, Figure 4 illustrates the
benefits of VTI anisotropic capability
when the explicit finite difference
WEPSDM is applied to the well-known
Marmousi synthetic model.
Optimum Acquisition
Considerations for WEPSDM
All “wave equation” processes
assume that the input dataset is
uniformly sampled. As any 3D
streamer acquisition survey will
inherently involve variable vessel
trajectories, streamer feathering, and
variable sea state, pre-stack 3D data is
by nature irregularly sampled. Aside
from lower computational cost,
Kirchhoff migration methods have been
the historical method of choice because
they are far more robust in their ability
to accommodate irregular data. The
solution to this dilemma is effective
data “regularization”. 3D
regularization is a significant area of
C O N T A C T
PGS Geophysical
London
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Oslo
Tel: 47-67-52-6400
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© 2005 Petroleum Geo-Services. All Rights Reserved
ongoing research within the industry
and academia, and has a history as
challenging as that of the development
of imaging methods themselves.
Whilst outside the scope of this
discussion, it can be briefly observed
that regularization methods have a full
spectrum of technical complexity vs.
cost efficiency. Compromises in the
integrity of regularization processing
will translate to compromises in the
integrity of the final imaging result. In
the most sophisticated scenario, Fourier
regularization is not theoretically
limited to plane events or a limited
number of dips, does not use a
geophysical model to describe the input
data, and provided that the input data is
not unacceptably aliased, can be
considered as an “exact” process.
Therein lays an assumption with
implications for the ideal 3D
acquisition platform. High Density 3D
(HD3D) acquisition presents the
acquisition platform that minimizes
data aliasing in any direction, and can
be customized to satisfy specific
assumptions about cross-line aperture
or target illumination, as demanded by
other high-end processing algorithms.
When allowed to work successfully,
Fourier regularization will use a
parametric inversion to output the data
to a perfectly uniform data grid, as
demanded by the user. Properly
sampled (HD3D) data will present the
best platform for regularization, which
in turn will present the optimum
platform for the successful application
of WEPSDM.
Figure 5: Dense spatial sampling of the reflected wavefield is provided by HD3D acquisition
to avoid aliasing and ensure the successful application of regularization and WEPSDM
technology. The “dark” regions on the wavefront shown (from a single shot in a complex
model, as produced by 3D ray tracing) are “multi-arrivals”. A specific receiver location at
the surface will receive three discrete arrivals from the same wavefront. This observation
reinforces the value of dense sampling for accurate reconstruction of the target during
(WEPSDM) imaging.
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