Alford rotation

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The ubiquity of shear-wave splitting in the sedimentary crust was first documented [1] at the "Amoco Anisotrophy Session" [sic!] in 1986. An efficient and effective way to separate the fast and slow shear modes was also presented; this is now called Alford rotation [2] [3].

Background

In anisotropic media, two different shear body-waves propagate, each polarized in one of the two orthogonal "principal polarizations" ("S1" and "S2"), which are properties of the medium and the direction of propagation, not properties of the source (cf., e.g. [4]) . These propagate with different velocities, hence they progressively "split" as they travel.

A shear wave sourced at the surface will not, in general, be aligned with either of these principal polarizations, and so will excite both modes, in trigonometric proportion. Upon reflection and return to a surface receiver, two arrivals (with a short delay between them) will be recorded, instead of the one expected in isotropic media. For example, in a 2D "SH" survey, the source is polarized crossline, and the receivers are also polarized crossline. When the split shear waves arrive at the receiver, both will be recorded, and they will also possess an inline component of motion (which of course is not recorded by the crossline receivers). These two arrivals will superpose upon one another; this will cause difficulties in interpretation.

In two-dimensional 2c x 2C acquisition (not to be confused with "4C acquisition", with three vector geophones and one hydrophone), at each source point both crossline and inline sources are activated (sequentially), and the signals from both are recorded on receivers also oriented both crossline and inline. This makes a 2 x 2 matrix of traces, indexed by source and receiver locations. In general, each element of the matrix contains arrivals from both S1 and S2 modes, superposing.

A 2C x 2C matrix of images, as acquired, each calculated by conventional imaging techniques

From each of these elements, an image may be calculated (using any imaging method of choice); this results in a 2 x 2 matrix of images, such as shown at the right [5]. The "traditional" SH image is at the lower right; a corresponding SV image (seldom acquired) is at the upper left. In isotropic theory, the off-diagonal images (with source and receiver polarizations mis-matched) would be null, but here they are just as energetic as the diagonal images. And, none of the images is readily interpretable.




A 2C x 2C matrix of images, but rotated by the magic "Alford angle", producing nul off-diagonal images

Alford (2C x 2C) rotation

Alford rotation consists of the matrix (tensor) rotation of this matrix of images, about the vertical axis, by successive amounts of etc. With each rotation angle, energy moves between the different elements of the matrix. Commonly, one can find in this way a magic "Alford angle", in which the off-diagonal images are null (or almost null), as in the figure at the right.

This rotation has moved all the S1 energy to one of the diagonal elements, and all of the S2 energy to the other. (One may determine which is the fast (S1) mode by cross-correlating the images.) A further rotation by reverses the positions of these two pure-mode images. In effect, this rotation has replaced the acquisition source and receiver orientations with orientations along the principal axes of azimuthal anisotropy (for vertical propagation), so that the rotated sources only excite one shear mode, or the other, not both together. Experience has shown that it is common to empirically discover a single Alford angle which accomplishes this, at all recorded times. In this respect, the image behaves as a set of normal-incidence traces (noise-reduced), even though most of the traces which enter the image come from non-zero source-receiver offsets.

The Alford angle is the angle between the line of acquisition and one of the principal axes of azimuthal anisotropy. In the simplest interpretation, the fast (S1) polarization is aligned with a single set of parallel vertical cracks, as shown in the figure (although more complicated interpretations may be more realistic). The travel-time dependence of the correlation is a measure of the distribution of the azimuthal anisotropy (i.e. seismic density of the inferred cracks) with depth.

It may happen that no single rotation angle nulls the off-diagonal images for all times and lateral positions. The simplest interpretation of this situation is that the azimuthal anisotropy (i.e. the inferred cracks) vary in orientation, laterally and/or vertically. Vertical variation requires a layer-stripping procedure [6] to determine the Alford angle at each depth.

1C x 2C rotation

In 2D acquisition with only 1 source orientation, but 2 horizontal receiver components, a less robust technique is available[4]. One simply rotates the 2C vector of horizontal image components by successive angles, as above. The magic angle is that angle for which the two rotated images most resemble each other (with a correlation lag). This criterion replaces the Alford criterion of null off-diagonal images. It is less robust, but cheaper, since the source effort is less. This technique works with either crossline or inline source polarization.

WAZ C-wave rotation

In a C-wave survey, the P-wave incident upon a reflecting boundary excites a reflected shear wave, with an inline polarization. In isotropic media, this would propagate upwards with SV polarization, but in azimuthally anisotropic media, it progressively splits, as described above; this may be called a split C-wave. If the acquisition is narrow azimuth, the 1C x 2C rotation technique mentioned above suffices to separate the 2C recording into its "C1" and "C2" modes. However, if the acquisition is wide azimuth, then a Common Conversion Point gather has a multiplicity of initial polarizations. This permits a generalization of Alford rotation to multiple polarizations of source [7] An elegant reformulation of this solution is termed tensor migration. [8].

References

  1. Willis, H., G, Rethford, and E. Bielanski, 1986. Azimuthal anisotropy: The occurrence and effect on shear wave data quality: Soc. Expl. Geoph. Conv. Expnd. Absts. 56, 479-480.
  2. Alford, R. M., 1986. Shear data in the presence of azimuthal anisotropy-Dilley, Texas: Soc. Expl. Geoph. Conv. Expnd. Absts. 56, 476-479.
  3. Thomsen, L. 1988. Reflection seismology over azimuthally anisotropic media, Geophysics, 53(3), P. 304-313.
  4. 4.0 4.1 Thomsen, L., 2014. Seismic Anisotropy in Exploration and Exploitation, the SEG/EAGE Distinguished Instructor Short Course #5 Lecture Notes, 2nd Edition, Soc. Expl. Geoph., Tulsa.
  5. Beaudoin G.J., T.A. Chaimov, W.W. Haggard, M.C. Mueller, L.A. Thomsen, 1998. The Use of Multi-component Seismology in CBM Exploration, in Second International Mining Technology Symposium: Groundwater Hazard Control/Coal Bed Methane Development and Application Techniques, Xian, China (October 15-17, 1996).
  6. Thomsen, L, I. Tsvankin, and M. Mueller, 1999. Coarse-Layer Stripping of Azimuthal Anisotropy from Reflection Shear-wave data, Geophysics, 64(4), 1126-1138.
  7. Gaiser, J., 1997. 3-D converted shear wave rotation with layer stripping, U. S. Patent 5,610,875.
  8. Dellinger, J., R. Clarke, and L. Thomsen, 2002: Alford rotation after tensor migration, Soc. Expl. Geoph. Conv. Expnd. Absts., 72, 982-985.