True-amplitude migration

§7.1 fixed the pre-migration processing; this section fixes the migration step itself. Plain Kirchhoff (§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 forward wavefield. A QI-grade pipeline replaces that plain sum with a true-amplitude Kirchhoff sum that includes the cancelling weights.

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 the weight is approximately

with $\theta$ the incidence angle on the reflector at the image point, $L$ the half-raypath length, and $v$ the local velocity. The $\cos\theta$ is the obliquity factor (reflections are stronger at perpendicular incidence and weaker at grazing); the $1/L$ cancels the forward-modelled geometric spreading. Both factors are purely geometric — no subsurface property enters. They transform migrated amplitude into a quantitative reflectivity estimate.

2. The constant-R test

Set up an imagined reflector with angle-independent reflectivity $R = 0.10$ — a flat, non-anomalous interface that has the same reflectivity at every angle. Migrate it two ways:

• True-amp Kirchhoff (teal): applies $W \propto \cos\theta/L$. The migrated angle gather is flat at $\hat{A} = R, \hat{B} = 0$. Exactly what we want — reflectivity recovered without angle-dependent bias.
• Structural Kirchhoff (yellow): no weight. Migrated amplitude decays with angle as $\cos^{2n}\theta$ for some bias exponent $n > 0$ (the widget lets you dial $n$ with the "geometric bias strength" slider). An interpreter fitting $\hat{A} + \hat{B}\sin^2\theta$ to the structural output finds a spurious negative gradient — this non-anomalous reflector appears to have a Class II or Class III AVO anomaly that does not exist in the earth.

The widget reports both recovered (Â, B̂). True-amp: (R, 0). Structural: depending on the bias, B̂ can reach −0.05 to −0.15 — a false AVO anomaly caused entirely by the migration.

3. Which migration flavours support true-amp?

• Kirchhoff PSTM (§5.3): true-amp variant exists, widely available. Apply the weight at output-point construction.
• Kirchhoff PSDM (§5.4): true-amp version (Bleistein weighting) is standard for QI-grade processing. Cost is roughly 1.2–1.5× plain PSDM.
• Beam migration (§5.5): beams carry amplitude naturally via the ray Jacobian and Gaussian envelope, so Gaussian-beam migration is amplitude-preserving by construction. A strong reason to prefer beams over plain Kirchhoff for QI.
• One-way WE (§5.6): amplitudes are well-behaved by construction as long as the adjoint operator is used (not a substitute transpose). Good for QI if the dip limit is respected.
• RTM (§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 (§6): by construction amplitude-preserving because the forward operator is the full wave equation. FWI velocity/impedance models feed directly into 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.
• 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 (§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: find a reflector that is known or strongly expected to be angle-independent (a brine-saturated sand of uniform thickness, say). Extract its angle gather from the pre-stack migration output. The gradient should be zero within noise. A negative gradient of −0.05 or more indicates the migration is not true-amp, or (alternatively) that something earlier in the pre-processing is killing amplitudes. Either way, the QI flow cannot be trusted.

Where this goes next

§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.