True-amplitude migration

Section 7.1 fixed the pre-migration processing; this section fixes the migration step itself. Plain Kirchhoff (Section 5.2) is a structural migration: it focuses diffractions and repositions dips correctly, but its output amplitudes are not calibrated. The magnitude of a migrated event depends on the operator's aperture, the obliquity of the contributing rays, and the geometric spreading of the recorded wavefield. A QI-grade pipeline replaces that plain sum with a true-amplitude Kirchhoff sum whose weights cancel those effects.

1. The true-amp Kirchhoff operator

For each output image point $(x\_o, z\_o)$, instead of the plain sum

the true-amp operator is

where, for a flat reflector in a constant velocity $V$ recorded along one line with point sources, the weight that returns the reflection coefficient is

with $L\_s$ and $L\_r$ the lengths of the source and receiver legs to the image point at depth $z\_0$; each factor is 1 at vertical incidence. The obliquity factor $O$, which is $\\cos\\theta$ at the reflection point, comes from the normal derivative in the Kirchhoff integral: a ray that reaches the recording surface obliquely contributes less flux per unit of surface. The spreading factor $S$ undoes the $1/(L\_s + L\_r)$ amplitude loss of the two legs. The square root, the two-dimensional form of Beylkin's determinant, sets how wide a band of midpoints adds in phase. All three are kinematic: they depend on the velocity model and the ray geometry, not on the reflectivity being estimated (Bleistein, 1987; Schleicher, Tygel and Hubral, 1993). Together they turn migrated amplitude into a quantitative reflectivity estimate.

2. The constant-R test

The figure records one flat reflector along a line and migrates the same data four ways, chosen with the Weighting control: the plain sum, the obliquity factor alone, the spreading factor alone, and the full weight. It opens on a test reflector whose $R$ is 0.10 at every angle, migrated with the plain sum. Read the amplitude picked at each angle in (e), fit $\\hat A + \\hat B\\sin^2\\theta$, and ask whether the gradient belongs to the earth or to the migration.

With the plain sum the test reflector comes back 7.4 % too strong at 30°, and the fit returns $\\hat B = +0.036$ where the earth has $B = 0$: a relative gradient $\\hat B/\\hat A$ of +0.36, which fails a 0.2 tolerance. By stationary phase, at the image point the four weightings return

• Plain sum: $R/\\sqrt{\\cos\\theta}$. The far angles brighten, so the fit finds a spurious positive gradient, and a soft brine sand that dims with angle is pulled toward Class III: its fitted gradient falls from +0.040 to +0.018.
• Obliquity alone: $R\\sqrt{\\cos\\theta}$. The far angles dim: on the gas sand the fitted gradient falls from −0.080 to −0.038, and about half of the brightening that marks the gas is gone.
• Spreading alone: $R/\\cos^{3/2}\\theta$. The far angles brighten strongly: the brine sand, Class IV in the earth, comes back with a gradient of −0.038 instead of +0.040 and plots as Class III, a gas response the earth does not have.
• Full weight: $R$. The picks lie on the earth's curve, the test reflector returns $\\hat B = 0.000$, and every reflector keeps its class.

A migration with the wrong weights bends the AVO gradient in either direction: a weight that dims the far angles can mute a real gas sand, and one that brightens them can invent one. The weights are also only as good as the velocity model. With the full weight but a migration velocity 8 % fast, the gas sand comes back with its intercept 7.8 % too strong and its gradient at −0.101 instead of −0.080, because the reflector is imaged too deep and every offset is assigned the wrong angle.

3. Which migration flavours support true-amp?

• Kirchhoff PSTM (Section 5.3): true-amp variant exists, widely available. Apply the weight at output-point construction.
• Kirchhoff PSDM (Section 5.4): true-amp version (Bleistein weighting) is standard for QI-grade processing. Cost is roughly 1.2-1.5× plain PSDM.
• Beam migration (Section 5.5): beams carry amplitude through the ray Jacobian and the Gaussian envelope, and Gaussian-beam migration has a well-established true-amplitude form (Gray and Bleistein, 2009) that makes beams a good choice for QI.
• One-way WE (Section 5.6): amplitudes become reliable with true-amplitude one-way propagators and an inverse-oriented imaging condition; the plain adjoint alone is not amplitude-preserving. Good for QI if the dip limit is respected.
• RTM (Section 5.7): the zero-lag imaging condition is not amplitude-preserving. Amplitude-preserving RTM uses an inverse-scattering imaging condition or deconvolution imaging condition, or post-migration amplitude calibration to wells. Production amplitude-preserving RTM is harder than the amplitude-preserving versions of Kirchhoff or beam migration.
• FWI output (Section 6): FWI outputs a model, not an image. It fits amplitudes as part of its misfit, and its velocity model supplies the background for amplitude-preserving migration and the low-frequency trend for simultaneous inversion.

4. Practical considerations

• Aperture taper. The weighted sum uses a finite aperture; abrupt truncation at the edge creates ringing and small amplitude offsets. Apply a raised-cosine taper over the outermost 10-20 % of the aperture width. The aperture must also hold the band of midpoints that adds in phase, which widens with angle: in the figure a 400 m half-aperture with a hard edge spoils the far angles first and invents a gradient of +0.026 on the test reflector, while 800 m is exact at the default 1500 m depth.
• Angle limits. Beyond about 45° post-critical effects contaminate the sum. Production true-amp flows often mute at the critical angle (for known velocity contrasts) or at 45-60° (conservative).
• Model dependence. Weights depend on the velocity model and raypath geometry. Small velocity errors produce small amplitude errors; large ones can undermine the AVO signal. The velocity model driving a QI migration should be FWI-grade (Section 6).
• Absolute vs relative. True-amp Kirchhoff gives amplitudes proportional to reflectivity; getting the absolute scale right requires a calibration step (well tie) because the source wavelet amplitude is usually not known exactly.

5. How to tell if the migration is true-amp

Apply the constant-R diagnostic, or better, compare against the AVO response modelled from a nearby well. Take a reflector whose AVO is known (modelled from the well's $V\_P$, $V\_S$ and $\\rho$ with Zoeppritz or Shuey), extract its angle gather from the pre-stack migration output and fit $\\hat A + \\hat B\\sin^2\\theta$. The difference from the modelled gradient should be zero within noise. A relative gradient error $|\\hat B/\\hat A - B/A|$ larger than about 0.2 indicates the migration is not true-amp, or that something earlier in the pre-processing is changing amplitudes with offset. Either way, the QI flow cannot be trusted.

Where this goes next

Section 7.3 handles the next amplitude-killing effect: intrinsic attenuation. Earth is lossy; high-frequency energy decays exponentially with travel time. Compensating for this (Q-compensation) restores the spectrum and preserves amplitudes for inversion.

References

• Stolt, R. H., Benson, A. K. (1986). Seismic Migration: Theory and Practice. Geophysical Press.
• Schneider, W. A. (1978). Integral formulation for migration in two and three dimensions. Geophysics, 43, 49.
• Yilmaz, Ö. (2001). Seismic Data Analysis (2 vols.). SEG.
• Etgen, J., Gray, S. H., Zhang, Y. (2009). An overview of depth imaging in exploration geophysics. Geophysics, 74, WCA5.
• Castagna, J. P., Backus, M. M. (1993). Offset-Dependent Reflectivity. SEG.
• Bleistein, N. (1987). On the imaging of reflectors in the earth. Geophysics, 52, 931-942.
• Schleicher, J., Tygel, M., Hubral, P. (1993). 3-D true-amplitude finite-offset migration. Geophysics, 58, 1112-1126.
• Rutherford, S. R., Williams, R. H. (1989). Amplitude-versus-offset variations in gas sands. Geophysics, 54, 680-688.
• Castagna, J. P., Swan, H. W. (1997). Principles of AVO crossplotting. The Leading Edge, 16, 337-342.
• Gray, S. H., Bleistein, N. (2009). True-amplitude Gaussian-beam migration. Geophysics, 74, S11-S23.