1. Elastic Wavefield Extrapolation in HTI Media
Richard A. Bale and
Gary F. Margrave

2. Outline
Introduction
Elastic wavefield extrapolation in layered HTI media
Extension to laterally heterogeneous media
Conclusions and ongoing work
I’ll follow the outline shown here. After some introductory remarks, I’ll describe our algorithm for elastic wavefield extrapolation in layered HTI media. This will necessarily be brief, but more details are given in the report. I’ll then describe how the algorithm can be extended to laterally heterogeneous media, using PSPI and NSPS type algorithms. I’ll finally draw some conclusions and give an idea where this work is going.

3. Introduction Recursive wavefield extrapolation One-way wave equation
Given u at depth z, generate u at z+Dz
One-way wave equation
Avoids multiple scattered energy
Robust to velocity errors
Elastic anisotropic wave equation
Proper handling of polarizations, mode conversions , shear-wave splitting
Related to Alford rotation
Avoids normal incidence assumption
Extensions to lateral heterogeneity
Making parameters function of x and kx
PSPI, NSPS, AGPS, FFLA
Our algorithm is among those recursive wavefield extrapolation methods, often referred to as “wave-equation” algorithms. Simply stated, given the displacements at depth z we generate a new set of displacements at z+dz.
Wavefield extrapolation can be done either using the full two-way wave equation, or using a one-way wave equation, in which a preferred direction is chosen. The advantage of considering one-way waves is that it avoids generating multiply scattered energy and is therefore less sensitive to velocity errors.
Our algorithm is based on the use of the elastic, anisotropic wave equation. This allows for proper handling of polarizations, mode conversion and shear wave splitting in HTI media. In fact it can be shown that in some sense this extrapolation is a generalization of Alford rotation, in which the assumption of normal incidence is no longer required.
Finally, as with scalar wavefield extrapolation, there are a number of options for extending it to lateral heterogeneity. These essentially consist of replacing a spatially invariant phase shift with one which depends simultaneously on x and kx, and is known as a pseudodifferential operator or a Fourier integral operator. Some of the existing methods can be considered such as PSPI, NSPS, AGPS (adaptive gabor phase shift), or FFLA, which is of course your favourite four letter acronym.

4. Previous Work Etgen (1988) Zhe and Greenhalgh (1997)
Stolt scalar migration, after wavefield decomposition using div and curl operators in wavenumber domain
Dellinger and Etgen (1990) generalized wavefield decomposition to anisotropy using Christoffel eqn.
Zhe and Greenhalgh (1997)
Applied wavefield separation after RTM by a few steps, then split-step scalar migration
Hou and Marfurt (2003)
Extrapolate each component with all velocities, apply separation as part of imaging condition
Let’s look at some previous work.
Etgen described a Stolt algorithm, after first performing wavefield decomposition with div and curl operators in the wavenumber domain. In a later paper Dellinger and Etgen described a wavefield decomposition valid for anisotropy. In a sense, our approach is to combine these two ideas using the anisotropic decomposition and scalar extrapolation. However, we use phase shift rather than Stolt type method for extrapolation.
Zhe and Greenhalgh took a different approach to the wavefield separation step, applying a Reverse Time migration for a few steps, in order to generate the wavefield at several adjacent depths before separating.
Another approach, taken by Hou and Marfurt, is to extrapolate the original displacement wavefield with all velocity fields and apply the wavefield separation within the imaging condition. This is elegant, but has a cost implication, especially for anisotropy, where 3 components would need to be extraplated with 3 velocities.

5. VTI and HTI: decks of Cards
VTI: Vertical
symmetry axis
E.g. Shales f
HTI: Horizontal
symmetry axis,
with azimuthal angle
E.g. Fractured
carbonates
Weak (slow)
direction
Strong (fast)
Let’s just recall what HTI is, using the analogy of a deck of playing cards. I believe this analogy was originally suggested by Leon Thomsen, using business cards, but playing cards are probably more in keeping with a university environment.
First consider VTI anisotropy. This can be though of as a deck of cards lying horizontally on the table. The axis of symmetry is vertical, and wavespeeds depend only the angle relative to this axis.
If we now take the deck of cards and set them on their edges, then we have an HTI medium. The horizontal symmetry axis is the direction the cards face. We need one more parameter to describe HTI than for VTI.
The deck of cards is more compliant in the direction along the axis, giving slower waves, and more rigid parallel to the face, so that we get faster waves. These fast and slow directions lead to shear wave splitting.

6. Outline
Introduction
Elastic wavefield extrapolation in layered HTI media
Extension to laterally heterogeneous media
Conclusions and ongoing work
Now let’s have a more detailed look at the method, and some results.

7. Elastic Wave-equation Migration
b = displacement-traction vector
b continuous
at z, z+Dz
vM = magnitude of mode M
vM discontinuous
at z, z+Dz
Extrapolation is the key element in wave-equation migration. We start with a shot and set of receivers at the surface. The shot wavefield is forward extrapolated downwards, whereas the wavefield recorded by the receivers is backwards extraplated downwards, one depth step at a time.
The basic wavefield defined at each depth is a vector of 3 displacements and 3 tractions, quantities which satisfy continuity across any horizontal interface. To propagate we have to decompose this vector into a wave-mode vector, v, consisting of a P-wave and two S-wave modes. However v is discontinuous, and so at least in principle, the wavefield should be recomposed at the next depth before progressing onwards.
To obtain P, S1 and S2 images at each depth, we have some kind of imaging conditions. These are outside the scope of this talk, but discussed in a paper in the report. Here we focus just on the wavefield extrapolation.
+ some kind of imaging conditions

8. A Generic Extrapolation Step
Decomposition of displacement-traction into P and S waves (SV, SH or S1, S2), at depth z
Scalar extrapolation using 3 different dispersion relations
Recomposition to displacement-traction at depth z+Dz
Let’s now consider a generic extrapolation step. It consists of decomposition into P and S waves, separate extrapolation, then recomposition. Strictly speaking the decomposition and recomposition are only needed if there is a change in the medium.

9. Extrapolations Compared
Theoretical model
Decomposition
Extrapolation
Acoustic, isotropic
None
Single scalar w.e.: operator derived analytically (square-root)
Elastic, isotropic
Helmholtz decomposition 1 P-wave  s (x-free) 2 S-waves  s (.-free)
P, SV and SH (=SV) scalar w.e.s: operator derived analytically
Elastic, anisotropic
De/Re-composition matrix from eigenvectors of Christoffel equation
P, S1 and S2 scalar w.e.s: operator from eigenvalues of Christoffel equation
It is instructive to compare extrapolations for different cases.
In the acoustic case, we have only a scalar wavefield, and no decomposition is required. The operator is derived analytically, since the vertical slowness is given by a simple square root.
In the elastic isotropic case, the decomposition is determined by the Helmholtz decomposition, in which the P-wave is curl free and the two S-waves are divergence free. In this case we still have analytically derived operator with different velocities for the P and S-waves.
Finally the elastic anisotropic case requires analysis of the eigenvectors and eigenvalues of the Christoffel equation. Whether the extrapolation is analytic depends on the symmetry of anisotropy. Fortunately for most useful cases, including HTI, there exists analytic solutions.

10. Slowness Curves: Isotropic
This figure shows the vertical slowness as a function of horizontal slowness for the isotropic case. All 6 solutions are present but the two shear wave solutions are coincident. The dotted lines are the imaginary parts of the vertical slowness, corresponding to evanescent behaviour.

11. Slowness Curves: HTI (45°)
This shows the same for a symmetry axis lying at 45 degrees to the x-axis. In this case we have 3 different values of sx for the onset of evanescent waves.

12. Isotropic Extrapolator: Impulse (on Z)
Here is the case for extrapolation by 200 m with P-wave velocity 3000m/s and S-wave velocity 1500m/s. The input is an impulse on the vertical component. This shows the P and SV responses before recomposition.

13. Isotropic Extrapolator: Impulse (on Z), Decomposed to P-SV
Here is the case for extrapolation by 200 m with P-wave velocity 3000m/s and S-wave velocity 1500m/s. The input is an impulse on the vertical component. This shows the P and SV responses before recomposition.

14. Isotropic Extrapolator: Impulse (on Z) Response, P-SV
Here is the case for extrapolation by 200 m with P-wave velocity 3000m/s and S-wave velocity 1500m/s. The input is an impulse on the vertical component. This shows the P and SV responses before recomposition.

15. Isotropic Extrapolator: Impulse (on Z) Response, X-Z
Here we see the X and Z displacement responses after recomposition. For non-zero incidence both P and S waves are present on the horizontal component, but they both vanish at zero offset.

16. Isotropic Extrapolator, X-Z: FK Amplitude Spectrum
Propagating P and S
Propagating S,
Evanescent P
Here is the FK amplitude spectrum of the recomposed X and Z responses. Three regions are obvious especially on the vertical displacement. In the center are the wavenumbers for which both P and S are propagating modes. Flanking these are wavenumbers for which P-waves have become evanescent, but S-waves are propagating. Then outside of these are regions where both modes are evanescent.
Evanescent P and S

17. Isotropic Extrapolator: Downward and Upward, X-Z
Here we see the X and Z displacement responses after recomposition. For non-zero incidence both P and S waves are present on the horizontal component, but they both vanish at zero offset.

18. Elastic Extrapolation Example
To illustrate the application of the wavefield extrapolation, I have generated some synthetic test data using anisotropic pseudospectral modelling. The geometry of the model data is shown here. The source is a vertical displacement source at 1000 m depth. The response is recorded at 3 different depths indicated by A b and C. To test the extrapolation I take the data at A and extrapolate it downwards to B, then compare with the modelled data at that depth. In this case I haven’t included C. Note that there are some effects encountered due to different apertures at A and B.

19. Pseudospectral Modelled Wavefield at A (HTI - 45°)
Here is the displacement wavefield measured at A. The medium is homogeneous HTI, with a symmetry axis at 45 degrees to the X direction. There are some low amplitude artifacts from an imperfect absorbing boundary, but the key events are the 3 direct arrivals: P S1 and S2..

20. After Wavefield Decomposition
After application of the wavefield decomposition the three arrivals are separated out. Recall that this is based upon the eigenvectors of the Christoffel equation, and cannot be achieved by conventional wavefield separation algorithms.

21. After Extrapolation and Recomposition at B
And are then recombined to get the displacements at the new depth.

22. Pseudospectral Modelled Wavefield at B (HTI - 45°)
Here is the modelled wavefield at depth B. The extrapolated wavefield compares well except for artifacts due to the differences in aperture.

23. Elastic Extrapolation Example
To illustrate the application of the wavefield extrapolation, I have generated some synthetic test data using anisotropic pseudospectral modelling. The geometry of the model data is shown here. The source is a vertical displacement source at 1000 m depth. The response is recorded at 3 different depths indicated by A b and C. To test the extrapolation I take the data at A and extrapolate it downwards to B, then compare with the modelled data at that depth. In this case I haven’t included C. Note that there are some effects encountered due to different apertures at A and B.

24. Wavefield from C, after Extrapolation and Recomposition at B
And are then recombined to get the displacements at the new depth.

25. Pseudospectral Modelled Wavefield at B (HTI - 45°)
Here is the modelled wavefield at depth B. The extrapolated wavefield compares well except for artifacts due to the differences in aperture.

26. Outline
Introduction
Elastic wavefield extrapolation in layered HTI media
Extension to laterally heterogeneous media
Conclusions and ongoing work
I’ll now describe two methods for extending the algorithm to laterally heterogeneous media.

27. PSPI Elastic Extrapolationz
z+Dz
Cij(x1)
Cij(x2)
Cij(x3)
The next couple of pictures are adapted from Margrave and Geiger in last years CREWES report.
The first approach is an elastic version of the PSPI algorithm. In it’s limiting form, each output point at the new depth is produced by applying a Fourier operator to the input slice, where the parameters of the operator (both slownesses and polarizations) are determined by the elastic properties at the output location. Here this is symbolized by the colour of the lines connecting input to output. This could be considered as a “pull” type operation.
A spatially varying elastic tensor is used, which could be constructed from knowledge of P velocities S1 and S2 velocities, and the orientation of the symmetry axis. Two other anisotropy parameters are also required but not illustrated here. So, for example the medium might be isotropic to the left hand side, and then become azimuthally anisotropic as we move to the right, but with variation in the orientation of the axis.
In practice of course the operator is not evaluated for every output point, but is interpolated from results obtained with reference models. Here we depart from the standard PSPI approach where reference velocities are selected by sampling velocities between a minimum and maximum value. With up to 6 different parameters involved, this would be prohibitive, since each combination of parameter choices gives a new operator. Instead we sample the medium spatially, calculating a new operator every so many CMP positions. This approach could be made adaptive, to handle sudden changes in medium. vP
vS1
vS2
f :

28. NSPS Elastic Extrapolationz
z+Dz
Cij(x1)
Cij(x2)
Cij(x3)
An alternative algorithm, is an elastic version of nonstationary phase shift. In NSPS, the operator is defined for each input location and used to generate contributions to the complete output slice. If PSPI is thought of as a pull operator, then NSPS is a push operator. Once again, in practice the input positions are not single points, but are windowed portions of the input slice, with the model parameters used based on the centre point of the window. vP
vS1
vS2
f :

29. Heterogeneous HTI Model
f = 0°
f = 45°
400m 0m
1280m
2560m
Reference Models:
f = 0°
A simple model illustrates the different behaviour of the elastic PSPI and NSPS approaches for a single extrapolation step of 200m. The model consists of two azimuthally anisotropic media, which have identical velocities, but differ in the axis of symmetry by 45 degrees. I will refer to this as the half-and-half model. We then generate the impulse response for both PSPI (click) and NSPS (click) operators in this model.
For purposes of comparison we have also generated results for 2 reference models which are homogeneous slabs having a constant symmetry axis, one with 0 and one with 45 degrees. I’ll refer to these as the left and right reference models.
f = 45°

30. Homogeneous HTI, f =0° V(z) Elastic Extrapolation
The first result shown is from the homogeneous reference model with 0 degree axis of symmetry. The input is three bandlimited spikes on the vertical component. Since the XZ plane is a plane of symmetry here, there is only a response for the P and S2 components.
In all the examples, the results are shown before recomposition to X Y and Z displacements, for the sake of clarity.

31. Heterogeneous HTI, PSPI Elastic Extrapolation
We now see the result of using PSPI for the half-and-half model. The output is comparable with the 0 degree reference model on the left hand side (flip back). On the right hand side, it matches the 45 degree reference model (flip forward). The transition occurs over only 8 CMPS, and therefore appears almost discontinuous. As anticipated the spatial variations are driven by the output.

32. Homogeneous HTI, f =45° V(z) Elastic Extrapolation
Here again is the reference result for the 45 degree axis. The rotation of the symmetry axis has the effect of transferring energy onto both S1 and S2 components, since each of these planes differs from XZ by 45 degrees.

33. Heterogeneous HTI, NSPS Elastic Extrapolation
Compare this with the NSPS result in the half-and-half model. NSPS applies the windowing operation before extrapolation, so that the impulses which lie in the left half are fully extrapolated using a 0 degree symmetry axis, whereas those in the right are extrapolated using a 45 degree axis. The center pulse lies within the overlap of a left model window and a right model window, and is a mixture.

34. Homogeneous HTI, f =0° V(z) Elastic Extrapolation
Here is the comparison with the left hand reference model again.

35. PSPI downward 400m PSPI upward 400m
We now look at some examples of reverse extrapolation, applied to the previous PSPI result. First we attempt to reverse extrapolate using the same algorithm. This does a reasonable job, though there is residual energy on both X and Y components, and the Z-component impulses are somewhat asymmetric.

36. PSPI downward 400m NSPS upward 400m
Now consider the result of using NSPS reverse extrapolation after PSPI forward extrapolation. There is less energy on the Y component, and the impulses are much more symmetric. We are led to the conjecture, that just as in the scalar case, PSPI and NSPS elastic extrapolators are natural adjoints of each other.

37. Outline
Introduction
Elastic wavefield extrapolation in layered HTI media
Extension to laterally heterogeneous media
Conclusions and ongoing work
Now it is time to draw some conclusions and point the way forward.

38. Conclusions
Anisotropic elastic extrapolators are constructed from eigen-solutions of the Christoffel matrix
Based on continuity of displacement-stress vector
Extrapolation includes decomposition / recomposition + scalar extrapolation
Decomposition operator derived from eigenvectors (polarizations)
Scalar extrapolation operator derived from eigenvalues (vertical slownesses)
JUST READ IT.

39. Conclusions Different evanescent cutoffs for each mode
Cause of residual terms after inverse extrapolation
V(z) algorithm tested with data from pseudospectral elastic modeling
Extension to lateral heterogeneity via elastic versions of PSPI or NSPS algorithms
Elastic NSPS natural adjoint (pseudo-inverse) of Elastic PSPI and vice versa
JUST READ IT.

40. Ongoing Work Elastic extrapolation with interface- propagators
Equivalent to decomposition/recomposition for V(z) medium
Accuracy of PSPI/NSPS form?
Shot record migration: paper in report (Bale & Margrave)
V(z) version coded so far: tested on model with density anomaly
Imaging of P-P, P-S1, P-S2 etc.
Will extend with PSPI or NSPS elastic extrapolators
JUST READ IT.

41. Acknowledgements Sponsors of CREWES
Sponsors of POTSI: Pseudodifferential Operator Theory and Seismic Imaging
MITACS: Mathematics of Information Technology and Complex Systems
PIMS: Pacific Institute of the Mathematical Sciences
NSERC: Natural Sciences and Engineering Research Council of Canada
Ed Krebes, Jeff Grossman, Hugh Geiger
JUST READ IT.