One-way wave-equation migration

Part 5 — Imaging (migration)

Learning objectives

State the one-way dispersion relation k_z = √((ω/V)² − k_x²) and its propagation condition
Derive the phase-shift downward-continuation operator exp(i k_z Δz)
Compare the exact circle with 15° parabolic and 45° Padé(1,1) approximations and their dip limits
Describe the cross-correlation imaging condition for extracting the image from shot and receiver wavefields

Ray-based methods (§5.4 Kirchhoff, §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 — "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

\tilde{U}(k_x, z, \omega) = \tilde{U}(k_x, 0, \omega)\,\exp(i k_z z)

where the vertical wavenumber is given by the dispersion relation

k_z = \sqrt{(\omega/V)^2 - k_x^2}

This is the phase-shift or Gazdag operator: propagate each wavenumber/frequency pair independently by multiplying by a pure phase factor. It is exact in constant V and implemented in O(n_x n_t log) per depth step via the FFT.

2. What "one-way" means

The full wave equation admits both downgoing $(+k_z)$ and upgoing $(-k_z)$ solutions. Migration needs the downgoing source wavefield and the upgoing receiver wavefield — but not both simultaneously. 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. Explicit finite-difference schemes cannot afford the FFT and approximate the square root by a Taylor or Padé expansion:

15° (parabolic): $k_z \approx k \cdot (1 - \tfrac{1}{2}(k_x/k)^2)$ — a parabola tangent to the circle at the origin. Accurate for small $k_x/k$ (small dip).
45° (Padé 1,1): $k_z \approx k \cdot (1 - (k_x/k)^2 / (1 + \tfrac{1}{2}(k_x/k)^2))$ — a rational approximant tracking the circle further before diverging.
65°, 80°, 88° — higher-order Padé or split-step approximants, each with a higher max-dip limit at the cost of wider stencils or more split-step passes.

The teal quarter-circle is the exact dispersion. The yellow dashed parabola (15°) hugs the circle for small angles but visibly underestimates $k_z$ past ~25°. The red dotted curve (45° Padé) tracks the circle further. The info strip reports the max dip each scheme can resolve within 1 % phase error — about 27° for 15° and 45° for 45° Padé, which is where the scheme names come from. The 5 % tolerance is also reported for reference; at 5 % the schemes extend to 43° and 57° respectively, but 1 % is the convention in the migration literature.

Change the frequency or velocity. The circle scales: $k = 2\pi f/V$ . Higher frequency or lower velocity shrinks the wavelength, but the shape (and therefore the relative error of each approximation) is unchanged. That is why the dip limit of each scheme is frequency-independent: it is a geometric property of how a polynomial fits a circle, not a physical property of the earth.

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:

I(x, z) = \sum_\omega U_s^*(x, z, \omega)\,U_r(x, z, \omega)

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 inside each depth slab correctly. No travel-time table needed. Amplitude-faithful when paired with proper obliquity weighting. O(n_x n_z n_t n_f) overall cost with FFT-based phase-shift, dominated by 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 3–15× PSDM Kirchhoff depending on the approximation order and whether split-step is used. Still significantly cheaper than RTM.

**The one sentence to remember**

One-way WE migration downward-continues the wavefield by multiplying by exp(i·k_z·Δz) with k_z from the dispersion circle, and reads out the image by cross-correlating source and receiver wavefields at each depth — the dip limit of the operator is whatever dip its polynomial approximant of the circle can still follow faithfully.

Where this goes next

§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.