Beam / Gaussian-beam migration

Kirchhoff PSDM (§5.4) sums data along an infinitesimal ray path for each image pixel. That is mathematically clean but physically wasteful: a real reflection from a scatterer is coherent across a Fresnel zone around the ray — a lateral neighbourhood whose width is set by frequency and distance. Kirchhoff ignores that width and treats every ray in isolation. Full one-way or two-way wave-equation methods (§5.6, §5.7) respect it rigorously but cost an order of magnitude more. Beam migration sits in between: replace each ray by a Gaussian beam with a controlled transverse width and sum the beams. This captures Fresnel-zone physics at a fraction of the full-wavefield cost.

1. A beam = ray + Gaussian envelope

Given a central ray parameterised by arc length $s$, a Gaussian beam assigns an amplitude

at every point $(s, y)$, where $y$ is the perpendicular distance from the ray. The beam is effectively zero for $|y| \gtrsim W(s)$ and maximal on the ray itself. Its width grows with distance along the ray:

— $W_0$ is the initial width at $s = 0$, and $\lambda = V/f$ is the wavelength. The beam is narrow near the source and wider far away; the rate of spread is proportional to $\lambda$ (longer waves spread faster) and inversely proportional to $W_0$ (narrower starts spread faster).

2. The widget

A single beam is drawn from a source at $x_s = 400\ \text{m}$ heading downward at the user-selected takeoff angle. Background intensity is $A(s, y) = \exp(-y^2/W(s)^2)$; the dashed white line is the central ray; white perpendicular ticks at depths 500, 1000, 1500 m show the beam half-width there. The info strip reports the wavelength, the three beam widths, and the Gauss-optimal $W_0$ for 1500 m target depth.

3. Gauss-optimal W₀

The width formula has a minimum in $W_0$ for a fixed target depth. Differentiating $W(s)^2$ with respect to $W_0$ and setting it to zero gives

That is the narrowest beam you can thread to a target at distance $s$ with wavelength $\lambda$. Production beam-migration codes pick $W_0$ per target depth to minimise aliasing at the image. Play with the $W_0$ slider in the widget at default $f, \theta$: the displayed Gauss-optimal value is around 160–200 m at 1500 m depth for 2500 m/s / 30 Hz. At $W_0 = 50\ \text{m}$ the beam balloons out because the second term dominates; at $W_0 = 350\ \text{m}$ the beam is too fat near the source.

4. How beam migration uses this

Instead of a single Kirchhoff summation along a single ray, beam migration:

• At each surface source (or receiver), shoot a fan of central rays at a discrete set of takeoff angles — typically 5 to 15 angles spanning the dip aperture.
• Assign each ray a Gaussian beam with its computed $W(s)$.
• For each image pixel, identify all beams that pass within a few $W(s)$ of that pixel and weight each by its beam amplitude and local ray attributes.
• Sum the weighted contributions. The result is like a Kirchhoff sum but over beams, not individual rays.

5. What beams fix that Kirchhoff misses

• Aliasing and footprint. Sparse shot spacing can produce aliased “Kirchhoff noise” in the image. The finite width of a beam naturally low-pass filters across space, suppressing alias artefacts.
• Shadow zones. A Kirchhoff first-arrival ray may miss a region where no direct ray reaches. A wider beam (or a few rays at nearby takeoff angles) covers the shadow with low but non-zero amplitude.
• Caustics. Rays crossing at caustics produce infinite Kirchhoff amplitudes. Gaussian beams superpose with the correct phase (they carry a complex-valued amplitude), and the caustic integrates out to a finite result.
• Amplitude fidelity. Beam amplitudes incorporate the ray Jacobian (how much the beam spreads or focuses along its path), so beam migrations preserve amplitudes much better than simple Kirchhoff sums.

6. What beams still miss

• Turning waves. Beam tracing typically uses a single central ray per takeoff angle; waves that turn and come back up require special handling (multi-arrival beam migrations) or full RTM.
• Severe multipathing. In extremely complex media (sub-salt, salt-flank), a beam's Gaussian assumption breaks near the boundary and amplitudes are wrong. RTM is safer.
• Anisotropic complexities. Azimuthally anisotropic beams exist but the math becomes considerably more involved; production anisotropic imaging often jumps straight to anisotropic RTM.

7. The cost/benefit sweet spot

Beam migration costs roughly 2–5× Kirchhoff PSDM but produces images that, for moderate-complexity targets, are within a few percent of RTM quality. That makes it the default choice for many exploration settings where sub-salt RTM is overkill but PSTM or PSDM falls short. Typical use: foothills imaging, shallow-salt margins, deep-water basins where salt is absent but lateral velocity variation is 5–10 % per km.

Where this goes next

§5.6 goes further and propagates a full wavefield downward one depth step at a time — one-way wave-equation migration. Instead of beams or rays, the operator is a downward continuation of the entire recorded wavefield, sidestepping the Gaussian-beam approximation and handling complex media correctly (except for turning waves).

References

• Etgen, J., Gray, S. H., Zhang, Y. (2009). An overview of depth imaging in exploration geophysics. Geophysics, 74, WCA5.
• Yilmaz, Ö. (2001). Seismic Data Analysis (2 vols.). SEG.
• Claerbout, J. F. (1985). Imaging the Earth’s Interior. Blackwell.
• Stolt, R. H., Benson, A. K. (1986). Seismic Migration: Theory and Practice. Geophysical Press.