One-way wave-equation migration

Ray-based methods (Section 5.4 Kirchhoff, Section 5.5 beams) treat each image pixel independently by integrating the data along a ray or beam path. Wave-equation methods are different: they propagate the entire wavefield from the surface downward, depth step by depth step, and read out the image at each depth via an imaging condition. One-way wave-equation migration, also called downward continuation, is the simplest such method and still a workhorse for moderate-complexity targets.

1. The downward-continuation propagator

Start with the recorded surface wavefield $U(x, z=0, t)$. Fourier-transform in time and in $x$ to get $\\tilde{U}(k\_x, z=0, \\omega)$. For a constant-velocity layer the wavefield at depth $z$ is

where the vertical wavenumber is given by the dispersion relation

This is the phase-shift or Gazdag operator: propagate each wavenumber/frequency pair independently by multiplying by a pure phase factor. It is exact when velocity varies only with depth, and each depth step costs $O(n\_\\omega\\, n\_x \\log n\_x)$ with the FFT.

2. What "one-way" means

The full wave equation admits both downgoing $(+k\_z)$ and upgoing $(-k\_z)$ solutions. Migration needs only the downgoing part of the source wavefield and only the upgoing part of the receiver wavefield. A one-way method keeps only one sign of $k\_z$ at a time, dropping the other. That costs you "turning waves" (waves that reflect back upward within the velocity model) but buys stability: no boundary reflections from the computational grid, no aliasing of one direction into the other.

3. The dispersion circle and its approximations

In the $(k\_x, k\_z)$ plane, the dispersion relation is a quarter-circle of radius $k = \\omega/V$. The exact phase-shift operator is that circle. Implicit finite-difference schemes in $(x, \\omega)$, which let velocity vary laterally at every grid point, replace the square root with a Taylor or Padé expansion. With $u = k\_x/k = \\sin\\theta$, where $\\theta$ is the propagation angle from vertical, these are the steps of the continued fraction $R\_{n+1} = 1 - u^2/(1 + R\_n)$, $R\_0 = 1$:

• 15° (parabolic): $k\_z \\approx k \\cdot (1 - \\tfrac{1}{2}(k\_x/k)^2)$, a parabola tangent to the circle at its apex $(k\_x, k\_z) = (0, k)$. Accurate for small $k\_x/k$ (small dip).
• 45° (Padé 1,1): $k\_z \\approx k \\cdot (1 - \\tfrac{3}{4}(k\_x/k)^2) / (1 - \\tfrac{1}{4}(k\_x/k)^2)$, a rational approximant tracking the circle further before diverging.
• 60°: $k\_z \\approx k\\,\\bigl(1 - u^2/(1 + R\_2)\\bigr)$, the third step of the same continued fraction, with $R\_2$ the 45° form.
• 65°, 80° and 90°: further continued-fraction steps or optimised coefficient sets (Lee and Suh, 1985), each with a higher dip limit at the cost of more finite-difference passes per depth step.

Every one of these approximants is at least $\\sqrt{1 - u^2}$: each lies outside the circle, overestimates $k\_z$, and so under-migrates steep dips. The names 15°, 45° and 60° are Claerbout’s historical labels, not measured limits.

4. Measuring the dip limit

In the figure, drag the probe angle $\\theta$ and read how far each operator’s $k\_z$ sits from the circle, first as a relative error and then as a dip error, then compare each operator’s migration impulse response with phase shift’s.

At the default \\theta = 30^\\circ the 15° equation makes $k\_z$ 1.04 % too large, just past a 1 % tolerance that it meets up to 29.7°; the 45° equation meets it up to 44.9° and the 60° equation up to 54.5°. Measure the error instead as a dip error, the true angle less the direction the operator actually sends energy (the normal to its curve), and the 1° limits fall to 19.2°, 37.3° and 50.0°. A dip limit means nothing until you say which error and which tolerance it uses. Plate (c) shows what the bend does to an image: the phase-shift response to one data spike is a semicircle, while the 15° response is an ellipse that falls inside it on steep rays, so at 60° it keeps only 2 % of the smile’s amplitude where phase shift draws it. Past $k\_x = k$ the approximants still give a real $k\_z$, so they propagate energy the exact operator damps, and it appears as faint arcs beneath the 45° and 60° responses.

The axes of (a) are in units of $k = 2\\pi f/V$, so changing the frequency or velocity leaves every curve where it is: the relative error of each approximation depends only on the angle. That is why the dip limit of each scheme is frequency-independent: it is a geometric property of how a rational function fits a circle, not a physical property of the earth. Sampling sets a second limit that does depend on frequency. A one-way wavefield sampled every $\\Delta x$ aliases propagation angles above $\\sin\\theta = V/(2 f \\Delta x)$, the dashed Nyquist line in (a): with 25 m traces and $V$ = 2500 m/s it is absent at 30 Hz and falls to 38.7° at 80 Hz.

5. The imaging condition

Downward continuation gives you the source wavefield $U\_s(x, z, \\omega)$ (propagated downward from a known source wavelet) and the receiver wavefield $U\_r(x, z, \\omega)$ (propagated downward from the recorded data). The image at depth $z$ is their zero-lag cross-correlation:

or equivalently the zero-time-lag of their cross-correlation in time. Intuitively: wherever the downgoing source wave and the upgoing receiver wave are coincident in time at a scatterer location, energy accumulates. Elsewhere it averages to zero. That is the wave-equation analog of the Kirchhoff diffraction-sum: a correlation instead of a summation, operating on wavefields rather than on individual samples.

6. Strengths and limits

• Strengths. Handles lateral velocity variation approximately, through split-step, PSPI or finite-difference corrections in each depth step. No travel-time table needed. Kinematically accurate within its dip limit; true-amplitude variants need modified one-way propagators. Overall cost is $O(n\_z\\, n\_\\omega\\, n\_x \\log n\_x)$ for FFT-based phase shift, dominated by the depth-step extrapolations.
• Limits. Cannot handle turning waves (that is precisely what "one-way" excludes). Max dip determined by the approximation order. Strong lateral velocity gradients inside a depth slab are approximated by split-step or PSPI schemes; sub-salt usually demands RTM.
• Cost. Typically 3 to 15 times the cost of Kirchhoff PSDM, depending on the approximation order and whether split-step is used. Still significantly cheaper than RTM.

Where this goes next

Section 5.7 introduces reverse-time migration (RTM), which drops the one-way assumption entirely. RTM propagates source and receiver wavefields with the full two-way wave equation, handling turning waves, prismatic reflections, and strong lateral velocity contrasts. It is the gold standard for sub-salt imaging and the most expensive migration routinely used in production.

References

• Gazdag, J. (1978). Wave equation migration with the phase-shift method. Geophysics, 43, 1342.
• Stolt, R. H. (1978). Migration by Fourier transform. Geophysics, 43, 23.
• Claerbout, J. F. (1985). Imaging the Earth’s Interior. Blackwell.
• 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.