1. Richard A. Bale and Gary F. Margrave
Elastic wave-equation migration for laterally varying isotropic and HTI media
Richard A. Bale and
Gary F. Margrave
Good afternoon ladies and gentlemen. This afternoon, I’m going to describe some efforts I’ve been making with my Ph.D. advisor Gary Margrave to apply wave-equation migration to elastic data. The method is applicable to HTI media, but I’ll also show an interesting isotropic example.

2. Outline Introduction Theory Examples Conclusions
Elastic wavefield extrapolation
Extension to laterally heterogeneous media
Migration imaging condition
Examples
Elastic HTI modeled data
Marmousi 2: elastic OBC modeled data
Conclusions

3. Introduction Drawbacks to scalar extrapolation for elastic migration:
Neglects mode conversions
Fails to keep track of polarization changes
Difficult to fully account for anisotropic effects, in particular shear wave splitting (birefringence) for HTI media
The earth is not acoustic but elastic.
When we do migration, by wavefield extrapolation or other means, we usually make an acoustic approximation, and use a scalar wave equation. This is true for conventional P-wave seismic, and usually true for converted wave seismic. In that case a separate scalar wave equation for the S-waves is used. Generally the scalar approach has been very successful.
There are circumstances where scalar extrapolation is expected to fail. For example if mode conversions are strong, or if polarization changes are rapid. An example where both these are true is when we encounter shear wave splitting due to HTI anisotropy.

4. VTI and HTI: decks of Cards
VTI: Vertical
symmetry axis
E.g. Shales f
HTI: Horizontal
symmetry axis
E.g. Fractured
carbonates
Weak (slow)
direction
Strong (fast) x y
Leon Thomsen uses the analogy of business cards to understand tranverse isotropy. If the deck is lying with faces on the table, we have VTI. If we sit it on edge, we have HTI. HTI has an additional parameter which is the direction of the symmetry axis. Shear wave propagating vertically split into fast and slow polarized waves, aligned along the cards or fractures and perpendicular to them. We have to be more careful when considering non-vertical propagation.
Before reviewing HTI anisotropy, we first remind ourselves of a more well known type VTI anisotropy. For VTI the wave speed depends on the angle relative to vertical, but not on azimuth. In other words, the symmetry axis is vertical. A deck of cards lying face down on the table is a good conceptual model.
For HTI, the cards are on edge, and the symmetry axis (normal to the card face) is horizontal. Fractured carbonates are often considered to be examples of HTI. With regard to shear waves, HTI is a birefringent medium, in which the polarization is decomposed into so called fast and slow directions.

5. Variation of Polarization with Slowness: HTI
The variation of the two polarizations with propagation direction in an HTI medium is illustrated here. At vertical incidence the S1 and S2 modes (red and green respectively) are oriented at + and – 45 degrees to the inline direction. However at high angles, the directions are quite different and would not be properly oriented by a 45 degree rotation in the horizontal plane. Also, the terms fast and slow become ambiguous as we move away from vertical, since the two slowness sheets intersect.

6. Introduction Standard processing of birefringent shear waves:
Assumes vertical incidence waves
Neglects variation of shear wave polarization with propagation angle
Neglects changes in velocity, (and time delay) with propagation angle
Often neglect variations of symmetry axis with depth
Various methods are in common use dealing with “shear wave splitting”. Typically, they are based on simple assumptions, which are accurate for waves propagating vertically in the medium, but become increasingly inaccurate as propagation angles increase.
The key idea of this paper is to incorporate the correction for shear wave splitting within the migration extrapolation, instead of treating them as two separate problems.

7. Outline Introduction Theory Examples Conclusions
Elastic wavefield extrapolation
Extension to laterally heterogeneous media
Migration imaging condition
Examples
Elastic HTI modeled data
Marmousi 2: elastic OBC modeled data
Conclusions
Now let’s take a look a the theory used for elastic wave-equation migration.

8. V(z) Extrapolation p : horizontal slowness zn : nth depth level
v : wave-mode vector in k-w domain ( k = pw )
Ln : diagonal matrix of eigenvalues (vert. slowness)
Extrapolation across a homogeneous layer is given by the diagonal phase-shift equation using the mode vector v. This is nothing more than the scalar wave extrapolation applied 3 times for P, S1 and S2 wavemodes. However, v is not continuous.

9. V(z) Extrapolation recomposition extrapolation decomposition
p : horizontal slowness
zn : nth depth level
v : wave-mode vector in k-w domain ( k = pw )
Ln : diagonal matrix of vertical slowness (P, S1, S2)
b : displacement-stress vector in k-w domain
Dn : eigenvector matrix (from polarizations)
Transformation to b allows the continuity condition to be satisfied.
Hence, extrapolation through a single depth step in general consists of three distinct steps: decomposition from b to v using the inverse of D, phase shifting to move downwards through a layer and recomposition to b at the next depth using D.
Now let’s consider the extension to laterally varying media.

10. V(z) Extrapolation decomposition phase shift continuity recomposition
The extrapolation of the vector wavefield by one depth step is illustrated here. The decomposition and recomposition steps allow us to use a diagonal operator applied to the vector of wave-modes, for the actual phase shift, but to map back to a vector which satisfies continuity across the interface. In this way we handle the changes of polarization which may occur from one step to the next.
recomposition

11. V(x,z) Extrapolation Operator
Lateral Dependence
Fourier Transform:
Lateral variation introduces an x dependence within the phase shift operator, just as it does for scalar extrapolation. However, in this case we must also allow for the composition and decomposition operators to depend on x. The equation which results is mathematically a pseudodifferential operator and may be written in a number of ways. Here we show a generalised PSPI form of the equation.
Extrapolation:

12. PSPI Elastic Extrapolationz
z+Dz
D(x1),
L(x1)
D(x2),
L(x2)
D(x3),
L(x3)
The PSPI elastic extrapolator for an HTI medium is illustrated schematically with this diagram. The coloured boxes represent a spatially variable elastic model, due either to velocity variation, variation in the HTI symmetry axis, or both. The wavefield at depth z is extrapolated by each in turn, with a windowing or interpolation operation to spatially localize the output wavefield at x+dz. vP
vS1
vS2
f :

13. Spatial Interpolation: PSPI
Standard PSPI
Extrapolate with N reference velocities
Interpolate based on actual velocity at each spatial position
Isotropic elastic case: dependence on VP and VS is (almost) separable
 cost  NVp + NVs OK
HTI elastic case: non-separable dependence on 6 parameters
 cost  (NVpNVs)(NeNdNg)Nf BAD!
We now consider two different approaches to spatial interpolation of the elastic wavefield extrapolation.
The first is referred to as Standard PSPI. Basically the idea is to apply the usual PSPI algorithm separately to extrapolation of the P-wave and S-wave modes. In this approach the extrapolation is performed for N reference velocities and the result at each x position is computed by interpolating between appropriate reference wavefields. For the isotropic case, this leads to a cost proportional to N, and is perhaps the most efficient approach. However, for the HTI case, the number of reference wavefields required becomes problematic, since generally combinations of 6 parameters are needed..

14. Spatial Interpolation: PSPAW
“Phase shift plus adaptive windowing”
Windows (“molecules”) constructed from elementary small windows (“atoms”)
c.f. Scalar adaptive method (Grossman et al., 2002)
Compute phase slowness for P, S1, S2 modes as a function of lateral position and phase angle
For each molecule, atom acceptance based on:
Maximum phase error over slownesses
Maximum variation of HTI symmetry axis
Begin new molecule if either criteria are violated
 Cost  # Windows (usually OK)
The second approach, which we have recently started calling “Phase shift plus adatpive windowing”, is based on a similar method used for the scalar wave equation by Jeff Grossman. Here we apply spatial windowing in an adaptive fashion. First we define a minimal window size, referred to as an “atom”. Adjacent atoms are combined to form “molecules”, provided that the resulting phase error and variation of symmetry axis are not too large.
The cost is typically higher than standard PSPI, for isotropic media, but is much more cost effective for HTI media. The cost depends also on the degree of lateral heterogeneity.

15. Adaptive Windowing:
A simple example illustrating the acceptance criteria for adaptive windowing is shown here. The model consists of two adjacent blocks, the left side being HTI, and the right side isotropic. Phase slowness is computed as a function of lateral position (CMP number in this case) and phase angle. The upper surface is the P-wave phase slowness, and the lower surfaces are the S1 and S2 phase slownesses. These of course coincide in the isotropic half of the model.
Phase slowness for HTI to Isotropic transition model

16. Imaging Condition P-wave S1-wave S2-wave
Forward extrapolated source wavefield:
Backward extrapolated receiver wavefield:
etc. …
P-P Image
One last detail of the algorithm is needed. The imaging condition is applied to the wave modes at each depth step. The basic ideas is to use the forward extrapolated source wavefield to deconvolve the backward extrapolated receiver wavefield, for each combination of modes needed.
Depending on the desired output images there could be up to 9 different imaging conditions, corresponding to 3 different source wavefields and 3 different receiver wavefields.
P-S1 Image

17. Outline Introduction Theory Examples Conclusions
Elastic wavefield extrapolation
Extension to laterally heterogeneous media
Migration imaging condition
Examples
Elastic HTI modeled data
Marmousi 2: elastic OBC modeled data
Conclusions

18. Isotropic Model
We first test the imaging of the elastic migration using a model with 4 isotropic layers, but with a normal fault in the second layer. This figure shows the P-wave model.

19. HTI Model Iso: “S1”=SH=90°, “S2”=SV=0° HTI: S1=-45°, S2=45°
inline
Symmetry axis
Now we modify our model to introduce HTI anisotropy into the faulted layer. Otherwise the model is unchanged. The symmetry axis of the anisotropy is horizontal with an azimuth of 45 degrees from the line direction. Because the output images will include both isotropic and anisotropic parts of the model, it is necessary to define a convention on labelling the SV and SH parts of the image. The convention used is that SH is regarded as S1 and SV is regarded as S2 where the model is isotropic. In other words, the isotropic parts are treated as degenerate HTI media with symmetry axes aligned with the line direction. It is important to realize this is a convention. One could equally well equate SH to S2 and SV to S1 for isotropic parts of the model.
NOTE: In following images, we (arbitrarily) assign SH mode to S1, and SV to S2, for isotropic layers.

20. HTI Data P-P Image (PSPAW)
First, here is the P-wave image for reference.

21. HTI Data P-S1 Image (PSPAW)
Here is the P-S1 image after migration with the true HTI model. In addition to the imaging of the faulted layer, note that the bottom reflector is absent from this section. Recall that the S1 image here represents the response with polarization of 45 degrees for the HTI layer, but 90 degrees for the isotropic layers. Since our source generates no SH wave, this is zero as expected.
Migrated with true HTI model

22. HTI Data P-S1 Image (PSPAW)
Incorrect Imaging of isotropic interface
(no SH mode should exist)
If, instead we migrated with the isotropic model based on the fastest velocity of the HTI layer, then the imaging of the faulted layer is less well defined. Also, there is an incorrect image of the isotropic interface.
Migrated with isotropic model

23. HTI Model P-S2 Image (PSPAW)
Here is the PS2 image, migrated with the true model. The bottom reflector corresponds to the SV mode, which is present, so it’s good to see the reflector imaged.
Migrated with true HTI model

24. HTI Data P-S2 Image (PSPAW)
Focussing degrades
Once again, the migration with the isotropic model results in a loss of focussing for the anisotropic layer, and for the reflector below. This is a general point : failure to account for anisotropy results in loss of resolution for deeper structures.
Migrated with isotropic model

25. The Marmousi-2 Elastic OBC Model
From Martin, Marfurt and Larsen, “Marmousi-2: an updated model for the investigation of AVO in structurally complex areas”, SEG 2002
Original (acoustic)
Marmousi model
The last example is the elastic Marmousi model, which is isotropic but highly heterogeneous. The elastic Marmousi dataset was generated by the Allied Geophysical Laboratory at the University of Houston. It is based upon the standard acoustic Marmousi model but with several modifications and extensions. First of all it has been extended laterally as shown here to a total line length of 17 km. The extensions are less structurally complex than the central section, but include interesting stratitgraphic features and hydrocarbon accumulations. Secondly it has been submerged under 500 m of water. Finally it is an elastic model with density and shear velocity determined from Castagna equations. The modelled data are very rich including both OBC and towed streamer data. We have confined attention to the X and Z components of the obc dataset.
We will look at results from two parts of the model, first from the traditional Marmousi area in the middle, and secondly from a less structural shallow area to the left hand side.

26. Marmousi-2 Mid-section: P-Impedance
Water Layer
Here we see the central section, roughly equivalent to the original Marmousi area. The secion shows the acoustic impedance.

27. Marmousi -2 Mid-Section: PP Image (PSPI)
Diffraction
Noise
Water layer
multiples
We applied our elastic migration algorithm to the X and Z component input data, and computed both PP and PS images. Here we see the PP image. There are a number of unwanted features resulting from the water layer, in particular strong water layer multiples and some diffraction noise arising from the fault intersecting the ocean bottom. Nevertheless, we feel it is quite an encouraging result.

28. Marmousi-2 Mid-section: S-Impedance
Water Layer: IS=0
Here is the shear wave impedance section …

29. Marmousi -2 Mid-Section: PS Image (PSPI)
Water layer
Multiple?
And the PS image. This is less compelling than the PP image, and represents work in progress. Nevertheless, the main structural features are present, if imperfectly imaged.

30. Marmousi-2 Shallow: IP Water Wet Sand Gas Charged Sand
The corresponding P-wave velocity profile clearly shows the gas as a low velocity feature.
Gas Charged
Sand

31. Marmousi -2 Shallow: PP Image
The PS image of this area shows very clear delineation of the sand channels. The relative lack of amplitude enhancement for the gas would correctly confirm this as a P-wave bright spot. Note in general the much higher resolution of the PS image compared with PP, as expected in the absence of attenuation.

32. Marmousi -2 Shallow: PS Image
Here is the corrsponding PP image, which shows a strong response to the gas.

33. Marmousi-2 Shallow: IS Water Wet Sand Gas Charged Sand
By contrast the Shear velocity shows a slightly higher, much less anomalous, velocity in the gas sand.
Gas Charged
Sand

34. Conclusions
Developed elastic wave-equation migration applicable to HTI anisotropy
AVO response compares well to Zoeppritz for flat reflector under isotropic layer
Two PSPI-type algorithms for spatial variations
“Standard” PSPI for isotropic cases
PSPAW for HTI
HTI migration focuses S1 and S2 images - isotropic migration fails to
Marmousi tests demonstrate:
Multiples and aliased noise are problematic
Imaging in structural area: PP better than PS
Shallow resolution of PS better than PP
Fluid lithology discrimination
Conclusions are as shown here.

35. 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
Allied Geophysical Laboratory, University of Houston for permission to use the Marmousi II data
Prof. Robert Wiley
Gary Martin (GX Technology)
Kevin Hall, CREWES