Wave Imaging Technology Inc.
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
  • Patent-pending 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 one-way 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 3-5 times at 30,000 ft depth.

  • Anti-aliasing - 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 co-authors from Texaco published an article in Geophysics the first example of tilted-coordinate WEM. The authors showed that the method could also be used to image from a not-flat datum. One method for doing this was described using a zero velocity above the datum - the same method later named zero-velocity 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.
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