
Whitepaper: 3D Wave Equation Shot Record Migration
Wave Imaging's shot record migration employs the
Phase Shift Plus Interpolation (PSPI) algorithm to downward continue both the
data collected at the surface and simultaneously a model of the source wave
field. These two fields are correlated at every point in the subsurface to
form an image.
Algorithm specifications and highlights:
 Migration from topography  this can imply land topography,
OBC/nodes, or 3D VSP.
 Expanding aperture option with increasing depth
 Makes one large depth step through constant velocity layers
 Increase depth step with increasing velocity
 Interpolation before applying imaging condition to avoid aliasing
 Shot consolidation improves efficiency without discarding data
 Patentpending 3D angle decomposition to generate full volume
incidence angle gathers and azimuth angle gathers.
 Can currently handle VTI and TTI media
 Can easily migrate PS converted waves by downward continuing source
and receiver fields with different velocity functions.
 Illumination compensation for irregular geometry and lateral
velocity variation.
Algorithm Details:
 PSPI  It has long been know that the PSPI algorithm gives
good images in the presence of virtually all complex velocity models
encountered in exploration geophysics. However there has been no truly
satisfying theory of the interpolation part of the method to back up this
observation. In a
2009 EAGE abstract, we put the
interpolation part of the PSPI method of migration on firm theoretical ground.
It turns out that this theoretical footing is critical to a perhaps more
important application: angle decomposition.
 Angle Stacks 
Wave Imaging filed a patent in 2009 on a method to measure
the 3D propagation direction of a wavefield in a oneway WEM algorithm, and
then to use the propagation direction vectors for the source and receiver
wavefields in a shot record WEM to determine incidence angle, azimuth angle,
and dip angle. In this fashion, we efficiently and directly form true angle
gathers. We incidence angle gathers for velocity analysis and AVA, and
azimuth angle gathers for fracture analysis.
Wave Imaging's method is vastly more efficient than many other methods in
the literature. Running a WEM with angle decomposition produces a full volume
of angle gathers. For instance, the user might desire incidence angles from 0
to 48 degrees, sampled at 4 degrees, or 12 angle volumes.
 Wide Aperture  An objection that has been raised in the past to
wave equation shot record migration is the limited aperture of the computation
that can miss steep dip events at depth. Wave Imaging's answer to this is to expand the
aperture of the migration with depth and yet lose almost no computational
efficiency. This is possible for two reasons: velocity increases with depth,
causing the spatial wavelength of a particular frequency in the data to
increase, and allowing a coarser grid to support wave propagation.
Second, deep reflections are known to lose frequency content, which also
enables coarser grid spacing at depth. In practice, we can increase the
migration aperture 35 times at 30,000 ft depth.
 Antialiasing 
Suppose that sin(wt) from the source wave field is correlated with
cos(wt) from the surface detected wave field. In the frequency domain
correlation is multiplication so sin(wt) cos(wt) = 0.5 sin(2wt). The
frequency has doubled from, wt, to, 2wt. Physically it means that the grid
that supports the image should be twice as finely spaced as the detectors on
the surface that measured the signal. Wave Imaging's PSPI shot record depth migration
algorithm has an option for using FFT interpolation or a more efficient but
less accurate interpolation to resample the source and receiver fields before
correlation.
 From topography  In 1985 Joe Higginbotham and coauthors from
Texaco published an article in Geophysics the first example of
tiltedcoordinate WEM.
The authors showed that the method could also be used to image from a notflat
datum. One method for doing this was described using a zero velocity above the
datum  the same method later named zerovelocity migration by Western
Geophysical. This method of imaging from topography is employed by Wave
Imaging to
allow the shot record migration algorithm to image from topography  the
algorithm works in mountainous and hilly regions and also with 3D VSP data or
OBC/node data.

