Beam / Gaussian-beam migration

Part 5, Imaging (migration)

Learning objectives

Describe a Gaussian beam as a central ray with a transverse amplitude envelope of finite width
Derive the beam-width formula and the start width that keeps a beam narrowest at a target
Explain how beam migration gives Kirchhoff’s zero-width rays a Fresnel-scale width, and what that buys in robustness
Identify the geophysical cases where beams are the right compromise between Kirchhoff and full wavefield

Kirchhoff PSDM (Section 5.4) sums data along the diffraction curve of each image point, with traveltimes and amplitudes computed along zero-width rays, usually one arrival per point. A real reflection is coherent across a Fresnel zone around the ray, a lateral neighbourhood whose width is set by frequency and distance, and zero-width rays ignore it; where rays fold and cross, their amplitudes even become infinite. Wave-equation methods (Sections 5.6, 5.7) respect that width rigorously but cost far more. Beam migration sits in between: it gives each ray a controlled, frequency-dependent Gaussian width and sums the beams.

1. A beam is a ray with a Gaussian width

Given a central ray parameterised by arc length $s$ , a 2D Gaussian beam assigns an amplitude

A(s, n) = \sqrt{W_0/W(s)}\,\exp(-n^2 / W(s)^2)

at every point a distance $n$ from the ray. The beam is effectively zero for $|n| \\gtrsim W(s)$ , and its peak falls as it widens, because it carries the same energy across a wider cross-section. Its half-width grows with distance along the ray:

W(s) = \sqrt{W_0^2 + \left(s\lambda / (\pi W_0)\right)^2}

$W\_0$ is the half-width at $s = 0$ , and $\\lambda = V/f$ is the wavelength. The rate of spread is proportional to $\\lambda$ (longer waves spread faster) and inversely proportional to $W\_0$ (narrower starts spread faster). The beam stays close to its start width out to the Rayleigh range $z\_R = \\pi W\_0^2/\\lambda$ , where it has widened by a factor of $\\sqrt 2$ .

2. The figure

Start with one beam: drag the start width and find the one that keeps the beam narrowest at the target depth, then change the frequency and the takeoff angle to see what sets that best start. Then send a fan of beams through a slow lens and compare what the beams and the zero-width rays deliver along the target line.

At the default start of 150 m the beam has a Rayleigh range of only 848 m against the 1732 m slant path to the target, so it widens to 341 m there, 13 % wider than the 303 m that the best start, 214 m, would give. Panel (c) is the whole trade-off: narrow starts diffract and balloon, wide starts are fat from the beginning, and the minimum between them is the best start. In (e), where the lens folds the rays into caustics at 1439 and 1561 m, ray theory's amplitude is infinite, while the sum of 21 beams peaks at 5.2 times the lens-free ray amplitude, and that peak moves by under 2 % whether the beams start 100 or 400 m wide. The lens is broad, 550 m to its 1/e radius, against beams a few hundred metres wide, so their paraxial sum converges: set the target to 900 m, above the caustic, and the sum of 41 beams follows ray theory to within 7 % at 30 Hz for starts of 100 to 300 m. Too few beams break the sum in a different way: each beam spreads about $\\lambda/(\\pi W\_0)$ radians about its ray, and when the fan's step is wider than that, as with 3 beams, the sum strays far from ray theory even without the lens.

3. The best start width

For a target a slant distance $s$ away, $W(s)$ has a minimum in $W\_0$ . Differentiating $W(s)^2$ with respect to $W\_0$ and setting it to zero gives

W_{0,\mathrm{opt}} = \sqrt{s\lambda/\pi} \quad\Longrightarrow\quad W_{\min}(s) = \sqrt{2 s \lambda / \pi}

That is the narrowest beam you can thread to a target at distance $s$ : at the best start the Rayleigh range equals $s$ , and the beam arrives $\\sqrt 2$ times as wide as it started. The Fresnel radius at that distance is $\\sqrt{\\lambda s/2}$ , so the narrowest beam is $\\sqrt{4/\\pi} \\approx 1.13$ Fresnel radii at any distance: the best beam carries the Fresnel-zone physics that a zero-width ray ignores. For 2500 m/s and 30 Hz the best start is 199 m straight down to a 1500 m target and 214 m at a 30° takeoff, because the slant path $s = z/\\cos\\theta$ lengthens as the ray tilts away from vertical; it also grows as the target deepens. Production codes choose $W\_0$ so the beam stays narrow over the whole target depth range, with beam centres spaced about one $W\_0$ apart; the best start here is the single-target version of that choice.

4. How beam migration uses this

Instead of summing along one diffraction curve per image point, beam migration:

Divides the surface into beam centres spaced about $W\_0$ apart.
At each centre, windows the recorded data and slant-stacks them into local plane waves, one per ray parameter $p$ (typically tens of values spanning the dip aperture).
Shoots one central ray per $p$ and gives it a Gaussian beam, traced with the paraxial (dynamic) ray equations so that its width and amplitude follow the velocity model.
Images each point by summing the contributions of every beam that passes within a few $W$ of it. The fan in (d) stands for this step, with its beams leaving one point.

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 and multipathing. Rays crossing at caustics produce infinite ray-theory amplitudes, and a single-arrival traveltime table keeps only one of the arrivals where rays cross. Every beam that reaches an image point contributes, and a beam's amplitude stays finite because its paraxial $Q$ is complex and never zero, so the sum is finite and multipathed energy is imaged, as (e) shows.
Turning waves. Beams follow rays, and rays turn naturally where velocity increases with depth, so turned and overturned energy is imaged; that limitation belongs to one-way methods (Section 5.6), not to beams.
Amplitude fidelity. Each beam carries the amplitude of dynamic ray tracing (how much its ray tube spreads or focuses along the path). Where the beams are narrow against the velocity variation and close enough in angle to overlap, their sum reproduces ray-theory amplitudes, and at a caustic, where ray theory is infinite, it stays finite. Where the beams are as wide as the velocity variation, the paraxial expansion around each ray fails and the sum depends on the start width, which is one reason the start is chosen for the target (Hill, 1990).

6. What beams still miss

Sharp velocity contrasts. The Gaussian beam is a paraxial approximation around its central ray; where velocity changes strongly within one beam width (salt flanks, karst), the beam's shape and amplitude are wrong and RTM is safer.
Head waves and diffraction around sharp bodies. Ray-based beams do not carry energy that has no ray, such as head waves along a salt top or diffractions around a sharp salt edge; wave-equation methods do.
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 about as much as Kirchhoff PSDM and far less than RTM, and for moderate-complexity targets its images approach RTM's. That makes it the default choice for many exploration settings where sub-salt RTM is overkill but PSTM or Kirchhoff 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.

The one sentence to remember

Beam migration gives Kirchhoff's zero-width rays a Gaussian width $W(s) = \sqrt{W_0^2 + (s\lambda/(\pi W_0))^2}$ , about one Fresnel zone at the best start, so it images multipathed energy and stays finite at caustics at roughly Kirchhoff cost: the standard compromise between Kirchhoff PSDM and RTM.

Where this goes next

Section 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 strong lateral velocity variation without rays, within the dip limits of its one-way operator (and without 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.
Hill, N. R. (1990). Gaussian beam migration. Geophysics, 55, 1416–1428.
Hill, N. R. (2001). Prestack Gaussian-beam depth migration. Geophysics, 66, 1240–1250.
Červený, V. (2001). Seismic Ray Theory. Cambridge University Press.