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SWLGS Luncheon Topics

Updated January 20, 2004

January 14, 2003
11:30 am
Lafayette Petroleum Club

The Importance of Turning Rays and Anisotropy in Prestack Time Migration
Paul Fowler, Everett Mobley, WesternGeco

Abstract Summary: 

Anisotropy is the rule, not the exception, in most sedimentary sequences around the world. The importance of including anisotropy and turning waves in depth imaging is generally recognized; however, in time imaging these effects are often handled with approximations that unnecessarily limit the migrated result. In this paper, we show that Kirchhoff pre-stack time migration can, with little additional cost, image steep or overturned events provided turning wave energy and anisotropy are taken into account.

Pre-stack vs Post-stack Migration

Post-stack migration algorithms deal mainly with rays traveling at moderate angles from vertical. Rays traveling at large angles are required only to image overturned reflectors. This is not the case with wide offset, pre-stack data. Even for moderately dipping events, a ray from either source or detector may turn. The intrinsic anisotropy in layered sedimentary sequences may result in horizontal velocities 2-15% higher than vertical velocities. To image reflections from dipping events recorded with today's wide offset acquisitions requires both faithful handling of vertical velocity gradients and attention to anisotropy.

Kirchhoff Time vs Depth Migration

Kirchhoff time migration differs from Kirchhoff depth migration in both the choice of vertical axis and traveltime approximations. Simple implementations use RMS velocities and double square root traveltimes. Most programs today implement at least a high-order moveout correction or ray-tracing to take into account curved raypaths. In general, any traveltime method consistent with the assumption that rays stay within a vertical plane qualifies as time migration. Arbitrary vertical velocity variation, including turning waves, and anisotropy with a vertical symmetry axis are permitted. Strictly speaking, this assumption implies laterally invariant velocities; however, modest spatial variation can be tolerated by assuming the traveltime between two points is governed by laterally invariant velocities.

Since images are formed in vertical traveltime rather than depth, anisotropy can be accomodated with a two parameter approximate VTI model expressed in time rather than a three parameter model in depth.

Traveltimes for KpreSTM may be computed on the fly, or precomputed and saved as two-dimensional tables, greatly simplifying storage requirements and data distribution. Velocities may be computed using 1-D update methods. Anisotropic, steep dip imaging requires a more critical velocity analysis than that for stacking, but provides a better initial velocity model for subsequent depth migration.

Recursive Traveltimes by Fermat Minimization

Our traveltime method is a first arrival scheme based on the Fermat principle. Traveltimes are computed for each output bin, using 1-D interval velocities defined at output locations. Anisotropy is defined by short spread NMO interval velocity, VNMO, and interval ?, or equivalently, VNMO and interval horizontal velocity Vh

The use of Fermat's principle to compute traveltimes has been suggested previously, for example, by Moser [1] to compute synthetic seismograms. As a method for computing KpreSTM traveltimes, Fermat's principle offers several advantages over ray-tracing. For example, the lateral invariance of the velocity field is exploited; the method is less susceptible to shadow zones and areas of poor ray density than ray tracing; the suppression of spurious events at layer boundaries, such as headwaves, is accomodated, and the method runs in bounded time, regardless of the complexity of the velocity model.

The algorithm we employ recursively extrapolates traveltime from a source located at the origin downward from layer to layer and outward within a layer. Fermat's principle is applied to determine the traveltime from the source to a point at pseudo-depth ti, distance xj by finding the minimum traveltime from points ti-1, xk, k = j, using a set of test ray segments computed from velocities in layer i.

Turning waves are accomodated by performing a second upward extrapolation pass, using the first pass traveltimes as initial data.

Anisotropy is accomodated using a two parameter VTI group velocity approximation obtained by Fowler [2], based on quasi-acoustic approximations developed by Alkhalifah [3].


Inclusion of turning rays can significantly improve KpreSTM images in areas where lateral velocity variation is modest. VTI traveltimes require a second input field, interval ? or Vh. ? can be determined from long offset moveout, from migration scans of dipping events, from well information, or VSP data. While the effort to determine ? is not insignificant, a regional estimate is often sufficient for time imaging. The traveltime method we employ does not make high-order moveout approximations. The KpreSTM migration velocity field which yields the best image is a superior initial model for depth migration model building. Correct imaging of steeply dipping events requires both accurate handling of ray bending beyond 90 degrees and the use of anisotropic velocities.


  1. Moser, T.J., Shortest path calculation of seismic rays: Geophysics, 56, 1991, 59-67.
  2. Fowler, P., Two- and three-parameter approximations for wave propogation in vertical transverse isotropic media: to be published, 2002.
  3. Alkhalifah, T., Acoustic approximations for processing in transversely isotropic media: Geophysics, 63, 623-631.

Biographical Sketch: 

Robert Vauthrin has the title of area geophysicist for WesternGeco and has worked for Western Geophysical his entire career. Although he has been in Houston for the last 21 years since he left college, he has supported the South American offices and made trips to Rio and Buenos Aires. His list of publications include Shallow Waterflow Detection Using Prestack Inverstion, Surface Multiple Attention: A 3D Example, Impact of Field Parameters on Surface Multiple Attenuation, A 3D Example, Using Legacy Seismic Data in an Intergrated Time-Lapse Study, Lena Field, Gulf of Mexico, and Improvements of 4-D Legacy Data Quality and Interpretability Through Reprocessing Lena Field. Robert received the 2000 Best Paper Award in Geophysics for Using Legacy Seismic Data in an Integrated Time Lapse Study: Lena Field, Gulf of Mexico. Robert graduated from Saint Mary's University in San Antonio, Texas with a degree in Mathematics. He has over 21 years experience processing geophysical data. He was born in Kansas, attended high school in Alabama and New Jersey and attended college in San Antonio due to his father working for the government and accepting several transfers.