Elastic and anisotropic FWI

Part 6, Full-Waveform Inversion

Learning objectives

Explain why acoustic FWI systematically mis-fits elastic reflection data at non-zero offsets
Describe the Aki-Richards $R_{\mathrm{PP}}$ and $R_{\mathrm{PS}}$ reflection coefficients as functions of angle
Identify the three principal elastic parameters ( $V_{\mathrm P}$ , $V_{\mathrm S}$ , $\rho$ ) and their cross-talk in elastic FWI
Summarise what anisotropic (VTI/TTI) FWI adds and when it is required

Everything in Sections 6.1-6.3 assumed the acoustic wave equation: one parameter (velocity) per pixel, P-waves only, no shear. Real rocks support shear and real seismic data contains mode-converted and shear-wave arrivals. Running acoustic FWI on elastic data leaves a systematic residual: the converted events are not in the acoustic synthetic at all, and the inversion can only mimic them by inventing structure. Elastic FWI is the physics upgrade: invert for $V\_{\\mathrm P}$ , $V\_{\\mathrm S}$ and $\\rho$ simultaneously, using the elastic wave equation to simulate the full vector wavefield.

1. The missing physics: mode conversion at oblique incidence

At a flat interface between two elastic layers, an incident P-wave produces four arrivals: reflected P, transmitted P, reflected mode-converted S (SV), and transmitted SV. The Zoeppritz equations give the exact amplitudes; Aki and Richards' linearisation is the small-contrast approximation used in production:

R_{\mathrm{PP}}(\theta) \approx \tfrac{1}{2}\!\left(\tfrac{\Delta V_{\mathrm P}}{V_{\mathrm P}} + \tfrac{\Delta\rho}{\rho}\right) + \tfrac{1}{2}\,\tfrac{\Delta V_{\mathrm P}}{V_{\mathrm P}}\,\tan^2\theta - 2\,\tfrac{V_{\mathrm S}^2}{V_{\mathrm P}^2}\,\sin^2\theta\left(2\tfrac{\Delta V_{\mathrm S}}{V_{\mathrm S}} + \tfrac{\Delta\rho}{\rho}\right)

where the velocities and the density are averages across the interface and each contrast is taken relative to them. The PP reflection has a normal-incidence intercept $\\tfrac{1}{2}(\\Delta V\_{\\mathrm P}/V\_{\\mathrm P} + \\Delta\\rho/\\rho)$ and a gradient whose elastic part is the shear-modulus contrast, $2\\Delta V\_{\\mathrm S}/V\_{\\mathrm S} + \\Delta\\rho/\\rho = \\Delta\\mu/\\mu$ with $\\mu = \\rho V\_{\\mathrm S}^2$ . The mode-converted PS coefficient is zero at normal incidence, grows roughly as $\\sin\\theta$ and, for typical contrasts, peaks between about 30° and 45°:

R_{\mathrm{PS}}(\theta) \approx -\frac{\sin\theta}{2\cos\varphi}\Bigl[\Bigl(1 - 2\tfrac{V_{\mathrm S}^2}{V_{\mathrm P}^2}\sin^2\theta + 2\tfrac{V_{\mathrm S}}{V_{\mathrm P}}\cos\theta\cos\varphi\Bigr)\tfrac{\Delta\rho}{\rho} - \Bigl(4\tfrac{V_{\mathrm S}^2}{V_{\mathrm P}^2}\sin^2\theta - 4\tfrac{V_{\mathrm S}}{V_{\mathrm P}}\cos\theta\cos\varphi\Bigr)\tfrac{\Delta V_{\mathrm S}}{V_{\mathrm S}}\Bigr]

with Snell's law $\\sin\\varphi = (V\_{\\mathrm S}/V\_{\\mathrm P})\\sin\\theta$ giving the S reflection angle. To first order $R\_{\\mathrm{PS}}$ depends on $\\Delta V\_{\\mathrm S}$ and $\\Delta\\rho$ only; it carries no information about $\\Delta V\_{\\mathrm P}$ beyond the background ratio $V\_{\\mathrm S}/V\_{\\mathrm P}$ .

2. What an acoustic model can and cannot fit

Figure 6.4 records one such interface with a shot gather and asks the best acoustic model of the interface to explain it. Open the spread with $\\theta$ , the widest reflection angle it records, change the contrasts and the rock above, and watch two things: what is left over in (d), and what the fit does to the P contrast.

At the starting setting, contrasts of +15, +10 and +8 % in $V\_{\\mathrm P}$ , $V\_{\\mathrm S}$ and $\\rho$ recorded out to 30°, the best acoustic model leaves 21 % of the gather's energy unexplained, and most of that is the converted PS event: a separate, later arrival with its own moveout, which no acoustic model of the interface can make. The fit also goes wrong where it seems to succeed. It matches the PP amplitudes by putting the P contrast at −11 % where the truth is +15 %, because the only way an acoustic model can make the PP amplitude fall with angle is a lower $V\_{\\mathrm P}$ : the elastic part of the gradient, the shear-modulus contrast, leaks into $V\_{\\mathrm P}$ . Hold $\\mu$ fixed (exercise 3) and the fit recovers $V\_{\\mathrm P}$ while the converted event stays.

The consequence for FWI: a trace recorded at 2 km offset from a flat reflector at 2 km depth samples a reflection angle near 27° (straight rays; refraction in a velocity gradient raises it toward 30°). At such angles the converted wave is not small. With the figure's default contrasts $|R\_{\\mathrm{PS}}|$ is 73 % of $|R\_{\\mathrm{PP}}|$ at 30°, and when the P-impedance contrast is weak beside the shear contrast the converted wave can outgrow PP. An acoustic synthetic has no PS, so whatever the earth produces is a systematic residual. The gradient tries to reduce it by changing $V\_{\\mathrm P}$ , but to first order the converted wave has nothing to do with $V\_{\\mathrm P}$ : the inversion fits a little of it by inventing structure, and the model is distorted in the process. The PP reflection adds a second bias, the one the figure measures: its shear-modulus term leaks into $V\_{\\mathrm P}$ . The symptom: false structures along the illumination direction, correlated with the source-receiver azimuth.

3. Elastic FWI: three parameters per pixel

Elastic FWI solves the elastic wave equation forward and adjoint (at 3-4× the cost of acoustic) and inverts for three parameters per grid cell instead of one: $V\_{\\mathrm P}(x,z), V\_{\\mathrm S}(x,z), \\rho(x,z)$ . The gradient now has three components per pixel; the Hessian is $3N \\times 3N$ and its condition number is worse than the acoustic case because the three parameters trade off in the data:

$V\_{\\mathrm P}$ and $\\rho$ trade off in the intercept of $R\_{\\mathrm{PP}}$ (they appear as the sum $\\Delta V\_{\\mathrm P}/V\_{\\mathrm P} + \\Delta\\rho/\\rho$ ). Only the change of amplitude with angle separates them, which is why Figure 6.4 quotes no fitted $\\Delta V\_{\\mathrm P}/V\_{\\mathrm P}$ below 15°.
$V\_{\\mathrm S}$ and $\\rho$ trade off in the PS coefficient in a similar way.
$V\_{\\mathrm P}$ and $V\_{\\mathrm S}$ both affect the AVO gradient, but for most rocks the shear-modulus term dominates.

To mitigate parameter cross-talk, elastic FWI uses re-parameterisations that decorrelate the gradients:

Impedance: $I\_{\\mathrm P} = V\_{\\mathrm P}\\rho$ , $I\_{\\mathrm S} = V\_{\\mathrm S}\\rho$ , constrains the AVO intercept/gradient separately.
$V\_{\\mathrm P}/V\_{\\mathrm S}$ ratio and $V\_{\\mathrm P}$ : $V\_{\\mathrm P}/V\_{\\mathrm S}$ is sensitive to fluid content and lithology and is often more stable than the individual velocities.
Poisson’s ratio and $V\_{\\mathrm P}$ : similar motivation; Poisson is well-constrained from AVO.

4. Anisotropic FWI: adding Thomsen parameters

Real sedimentary rocks, especially shales, exhibit VTI anisotropy: the vertical velocity differs from the horizontal. The velocity surface is rotationally symmetric about the vertical axis. It is described by two axial velocities and Thomsen's anisotropy parameters, and it is an ellipse only when $\\epsilon = \\delta$ :

$V\_{\\mathrm P0}$ : P-wave velocity along the symmetry (vertical) axis.
$V\_{\\mathrm S0}$ : S-wave velocity along the symmetry axis.
$\\epsilon$ : P-wave anisotropy (typically 0.05-0.2 for shales). Controls the difference between horizontal and vertical P velocity.
$\\delta$ : the near-vertical P-wave anisotropy. It sets the short-spread (small-angle) P-wave NMO velocity, distinct from $\\epsilon$ , the horizontal-velocity term.

A VTI FWI inverts for $V\_{\\mathrm P0}, V\_{\\mathrm S0}, \\epsilon, \\delta$ (and $\\rho$ ) at every pixel, five parameters per grid cell. The numerical cost climbs to 5-7× acoustic. The payoff: in shaley overburdens (most of the North Sea, Gulf of Mexico deep water, Middle Eastern carbonates with shale drapes), anisotropic effects contribute 5-20 % of the velocity field. Isotropic FWI in shaley media pushes the apparent velocity toward the NMO value, distorting depths by 5-10 %. Anisotropic FWI fixes this.

TTI (tilted transverse isotropy) adds a tilt $\\nu$ and an azimuth $\\psi$ of the symmetry axis, seven parameters per pixel. Required for dipping shales and for thrust-belt settings where the shale fabric is rotated. Orthorhombic anisotropy (nine-ish parameters) is needed when the rock has two orthogonal anisotropy directions (fractured reservoirs, horizontally transversely isotropic + VTI combined). FWI at these complexities is on the cutting edge of production.

5. When acoustic FWI is good enough

Short offsets only. If all reflection angles are below about 10°, $R\_{\\mathrm{PS}}$ is small and acoustic FWI is usually adequate: Figure 6.4 leaves under 5 % of the energy at 10° with its default contrasts.
Fluid-filled reservoir rocks. Within a reservoir where the formation behaves close to fluid-like (high $V\_{\\mathrm P}/V\_{\\mathrm S}$ ), mode conversion is weaker, though not absent.
Small shear contrasts. These are two conditions. Acoustic inversion recovers $V\_{\\mathrm P}$ when the shear-modulus contrast $\\Delta\\mu/\\mu = \\Delta\\rho/\\rho + 2\\Delta V\_{\\mathrm S}/V\_{\\mathrm S}$ is near zero (exercises 2 and 3 of Figure 6.4), and the converted wave stays small only when $\\Delta V\_{\\mathrm S}/V\_{\\mathrm S}$ and $\\Delta\\rho/\\rho$ are both small. Small is not zero: with +5 % in each, Figure 6.4 fits a P contrast of +4 % at 30° against a true +15 %.

6. When elastic/anisotropic is mandatory

Shaley overburden above a reservoir target. Anisotropy is 10-20 %, must be included.
Multi-component (OBN, OBC) data. Hydrophone and three-component geophones record both P and S; ignoring shear is throwing away half the data.
Fractured reservoirs. Fracture orientation imprints on azimuthal anisotropy. Orthorhombic FWI extracts that.
AVO and rock-physics inversion. If the end product is a fluid/lithology estimate, you need the full elastic parameter set.

The one sentence to remember

An acoustic model cannot make the converted PS event at all, and the shear part of the PP gradient leaks into the $V_{\mathrm P}$ it fits (Figure 6.4); elastic and anisotropic FWI add the missing parameters, costing 3-7× more per iteration but returning physically meaningful $V_{\mathrm P}$ , $V_{\mathrm S}$ , $\rho$ (and Thomsen parameters) instead of a scrambled $V_{\mathrm P}$ .

Where this goes next

Section 6.5 closes Part 6 with the QC question: given a final FWI model, how do you know it is right? The answer is a synthetic-versus-recorded comparison: propagate through the final model, compare with the data, and accept only when the match is tight across frequency, offset and azimuth.

References

Virieux, J., Operto, S. (2009). An overview of full-waveform inversion in exploration geophysics. Geophysics, 74, WCC1.
Tarantola, A. (1984). Inversion of seismic reflection data in the acoustic approximation. Geophysics, 49, 1259.
Pratt, R. G. (1999). Seismic waveform inversion in the frequency domain, Part 1. Geophysics, 64, 888.
Etgen, J., Gray, S. H., Zhang, Y. (2009). An overview of depth imaging in exploration geophysics. Geophysics, 74, WCA5.