Anisotropic Seismic Imaging

I confess that anisotropy is not a subject I ever had any passion for, but its relevance for seismic imaging is undisputed and increasing in profile. I attempt to untangle the complex nomenclature below and highlight implications of the various theory without including the mathematics. The relevance of different degrees of anisotropy are summarized for various global geological settings.

Terminology

Let us begin by defining the terms stress and strain. Stress is a physical quantity that expresses the force applied to some material exert on each other, while strain is the measure of the deformation of the material.

The interplay between all the permutations of stress and strain affecting some material can be very complex. According to Hooke’s law for general anisotropic, linear elastic solid, there are eighty one (81) possible ways to relate stress and strain! 

Note that elastic means the deformation is reversible. In reality, a seismic wave propagating through the earth loses some energy as heat, but we can generally consider a rock under the stress of a pressure wave to deform in an elastic manner.

(left) Illustration of typical stresses (arrows) across various surface elements on the boundary of a particle (sphere), in a homogeneous material under uniform (but not isotropic) triaxial stress. (right) Cartesian representation of 3D stress tensors.

Unsurprisingly perhaps, acoustic wave propagation is described by the acoustic wave equation and elastic wave propagation is described by the elastic wave equation.

The acoustic approximation says that shear effects in the data are negligible (i.e. no mode conversions at each interface) and the dominant wave type is a compressional wave, a wave where the particle motion is parallel to the propagation direction.

Correspondingly, the theory of stress-strain relationships and elastic constants described above is relaxed somewhat so that less parameters are needed to describe various anisotropic acoustic media. 

Hang in there, that’s (almost) the end of the mathematics.

OK, so there seem to be an impossibly large number of ways to describe the response of an anisotropic material to stresses. The good news is that the 81 relationships between stress and strain reduce to a mathematical maximum of twenty one (21), and in isotropic medium to only two (2), which is where we have been operating for much of the history of seismic imaging.

Anisotropy in the Layered Earth

It was always recognized of course that the Earth of interest to seismic exploration is layered, and this layering imposes an anisotropic fabric onto the rocks whereby pressure waves travel somewhat faster along the layers in comparison to their speed across layer boundaries.

The mathematical description of this form of anisotropy is known as hexagonal or transversely isotropic, whereby five (5) unique (elastic) stress-strain relationships exist. There was, however, no elegant way to incorporate this into seismic processing and imaging algorithms, even when anisotropy was clearly creating problems.

Almost thirty years ago, Leon Thomsen, then working with Amaco, published what is now the most quoted paper in SEG history (Weak elastic anisotropy).

Thomsen recognized that most anisotropy in rocks is ‘weak’ rather than behaving in strictly textbook fashion, and therefore he was able to provide simple mathematical approximations defined by the dimensionless ‘Thomsen parameters’ epsilon (ε), gamma (g) and delta (d), which together with a knowledge of the P-wave and S-wave velocities made anisotropy tractable for seismic processing and imaging.

For acoustic media, epsilon (ε) quantifies the velocity difference for wave propagation along and perpendicular to the symmetry axis, delta (d) controls the P-wave propagation for angles near the symmetry axis, and essentially captures the relationship between the velocity required to flatten gathers (the NMO velocity) and the zero-offset average velocity as recorded by check shots. As we are only concerned with P-wave propagation gamma (g) can be ignored (set to zero).

Practical Solutions for Acoustic Media (No knowledge of Vs required)

Tariq Alkhalifah and Ilya Tsvankin from Colorado School of Mines published a paper almost ten years after Thomsen’s paper that allowed the most common form of transverse isotropy to be described in processing and imaging with only one additional parameter, eta (h), which is a function of epsilon (ε) and delta (d).

Again, there are several mathematical approximations going on in the background, but this bought anisotropic processing and imaging into the mainstream in the late-1990s.

The anisotropic described is ‘transverse anisotropy with a vertical axis of symmetry’ or VTI. What this refers to is horizontally layered rocks where the azimuth of wave propagation is irrelevant (symmetric), but the wave propagation does depend upon the vertical angle.

In processing, two parameters are sought: The velocity along the axis of symmetry (V0, the vertical velocity in the case of VTI anisotropy) and eta (h).

The number of terms required to describe more complex anisotropy increases as illustrated below.

Comparison of tilted transverse isotropy (TTI) with tilted orthorhombic (t-ORT). Note that the parameter terminology is specific to PGS algorithms but is nevertheless also representative of common seismic imaging industry nomenclature. (upper) TTI: The local stress field is affected by the fracture plane with three (local coordinate) orthogonal symmetry planes defined by angles slopex, slopey, and α. The anisotropic parameters are defined with respect to the bedding planes (X-Y local coordinates) and then and then α defines the rotation between the shooting grid (inline/xline) and the local X-Y coordinates.

This VTI theory can also easily be adapted to areas affected by pronounced vertical fracturing where the term used is ‘transverse anisotropy with a horizontal axis of symmetry’ or HTI. What this refers to is vertically affected rocks where the vertical angle of wave propagation is irrelevant (not completely realistic), but the wave propagation does depend upon the azimuth. HTI anisotropy has most famously been applied to onshore prospects with thick, highly fractured chalk layers.

And then of course things start to get ugly because as the appreciation of anisotropy and its affects upon seismic imaging grew, people wanted increasingly accurate implementations. In addition to being layered, the earth has structure—the layers are ‘tilted’, so Tilted Transverse Isotropy (TTI) became in demand—VTI anisotropy applied to tilted bedding planes.

Unfortunately, TTI requires the explicit knowledge of epsilon (ε) and delta (d), in addition to the bedding plane dip with respect to the inline and cross-line grid axes . Combined with knowledge of V0, this means TTI anisotropy models have five (5) parameters to solve.

Table of parameters required for the various types of anisotropic imaging.

The principle of scientific deduction says that when we have several unknown variables we usually try to solve one at a time. In the context of TTI anisotropic velocity model building this means an iterative process is required, taking the processing geophysicist through several parameter estimation phases.

In addition to being layered, the earth tends to have an azimuthal component of an isotropy too, typically associated with oriented stresses of fracturing. This scenario, referred to as orthorhombic isotropy (usually abbreviated as ‘ORT’ as in the figure above), seven (7) parameters must be known, including a new parameter describing the horizontal direction of ‘fastest’ velocity. Then comes tilted orthorhombic (t-ORT) which also acknowledges the structural dip of the affected layers--also illustrated in the figure above and the figure below.

An orthorhombic model produced by parallel vertical cracks embedded in a medium composed of thin horizontal layers; a scenario associated with ORT or t-ORT anisotropy. An expanding 3D seismic wavefront is also superimposed within the model to illustrate how velocity varies in different directions (see also below).

3D Seismic Acquisition Caveats when Implementing High Order Anisotropic Imaging

The imaging operators for VTI, TTI, ORT and t-ORT isotropy are actually fairly straightforward to implement—once you know the anisotropic model parameters—and therein lies the bottleneck.

In practice, t-ORT model building really requires three available survey azimuths for initial model building (i.e. overlapping surveys, preferably at least one of them multi-vessel wide-azimuth)

Implementing t-ORT Anisotropy in Seismic Migration

For mathematical convenience we first assume that the vectors normal to the ORT symmetry planes (refer to the figure with annotated nomenclature above) are defined on a local coordinate system (Z, X, Y), as schematically illustrated in the following figure. In this case, and under the acoustic approximation, the wave propagation requires six independent parameters (V₀ , ε₁ , δ₁ , ε₂ , δ₂ , δ₃).

Snapshot of a wavefield modeled in an axis-aligned ORT medium. The three panels show a vertical X-Z slice (top left), a vertical Y-Z slice (top right), and a horizontal X-Y slice (bottom) through the source location. The pressure source is initiated in the center of the grid.

The t-ORT model parameters ε₁ and δ₁ are solved with respect to the Y-Z symmetry plane, ε₂ and δ₂ are solved with respect to the X-Z symmetry plane, δ₃ is solved with respect to the X-Y symmetry plane, and it is generally assumed that ε₁ > ε₂ as ε₁ is the ‘fast’ direction.

In t-ORT media three rotation angles are necessary to transform the elastic tensor from a local system to a global one. This is due to the fact that the physical properties are not symmetric in the local X-Y plane in t-ORT media.

Therefore, two angles (azimuth, q, and tilt, f) are used to define the vertical axis at each grid point, while a third angle, a, is introduced to rotate the elastic tensor in the local x-y plane to describe the orientation of the fracture systems.

Consequently, the description of wave propagation in t-ORT media requires nine (9) parameters (V₀, ε₁, δ₁, ε₂, δ₂, δ₃, q, f, a).

Pragmatic Solutions in Complex Media

While a t-ORT medium is generally described by nine independent parameters, for most common geological settings that number can be reduced.

One clear example is the case of fracturing combined with thin layering in the matrix in relatively low relief structures. If the fracture directions coincide with the orientation of the processing grid, then six parameters are needed (ORT).

If the fractures have variable spatial orientation, then seven parameters are necessary (one extra angle, a).

In the Gulf of Mexico, salt diapirs tend to induce radial faulting, in which case the orientation of a will conform to the local dip and azimuth of reflectors, resulting in a structurally conformable T-ORT system.

In a rift setting such as the NW Shelf of Australia, dips are generally benign, and VTI anisotropic models often suffice. The use of ORT and T-ORT anisotropic imaging is increasing nevertheless. The complexity of the anisotropic model is bounded by our ability to accurately estimate parameters using surface seismic and/or well data. The outcome on migrated images will be improvements in imaging focusing and spatial structure, albeit often modest.

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