Wavefronts, rays, and Huygens

Every piece of acquisition physics rests on one picture: a source releases energy that travels outward as an expanding wavefront. In uniform isotropic rock of speed $V$ the wavefront is a sphere (a circle in 2D) of radius $Vt$ at time $t$. Real rock is not uniform: compaction makes it faster with depth, and even the simplest model of that, a speed that grows linearly with depth, $V(z) = V\_0 + kz$, changes the shape of everything.

The three equivalent views

• Wavefronts, the set of points the wave has reached at time $t$: surfaces of equal travel time (isochrons).
• Rays, the paths along which the energy travels. In isotropic rock a ray crosses every wavefront at a right angle, and its direction obeys Snell’s law: $\\sin\\theta/V(z)$ stays constant along the ray, where $\\theta$ is the angle from the vertical. Rays are the right tool when the wavelength is short compared with the distance over which the speed changes.
• Huygens’ construction: every point on the current wavefront acts as the source of a secondary wavelet, of radius $V\\,\\Delta t$ at the local speed after a step $\\Delta t$. The forward envelope of the wavelets is the next wavefront; the backward envelope is not a wave (Kirchhoff’s obliquity factor removes it).

In the figure a shot fires at the surface of rock with $V\_0 = 1800\\ \\mathrm{m/s}$ and a gradient $k = 0.6\\ \\mathrm{s}^{-1}$. Lower the gradient to zero, then raise it again, and watch how far down and how far along the surface the front in (a) has gone.

What the gradient does

After 800 ms the front is 1848 m deep but reaches only 1454 m along the surface; uniform rock at $V\_0$ would put it 1440 m out in every direction. The front stretches downward because its deepest part runs through the fastest rock, and for this model the stretch is exact: the depth reached is $e^{kt/2}$ times the reach along the surface. Every slanted ray bends toward the slower rock above, turns, and comes back up: these are diving waves. They reach far offsets sooner than a straight ray at $V\_0$ could, so the first breaks in (b) curve away from the line $x/V\_0$: at 3 km the front arrives 63 ms early. Plate (c) shows the cause, the speed growing from 1800 m/s at the surface.

The three views stay consistent however curved they become. Every ray in (a) crosses the fronts at 90.0°, and a Huygens step of 150 ms with wavelets at the local speed lands within 20 m of the true front. That gap is the construction’s first-order error: it shrinks as $\\Delta t^2$, so halving the step cuts it about four times. Like any marching scheme, the construction is only as good as its step.

Why all three matter for acquisition

You will design a survey by choosing offsets and azimuths so that rays reflected from your target reach your receivers, and you will check illumination by tracing rays from the sources down to the target and back up to the receivers. The first breaks you pick for refraction statics are the diving and refracted waves of plate (b), and their curvature measures the near-surface gradient. You will reason about diffractions from the edges of salt, faults and reservoirs with Huygens: each point along the edge acts as a secondary source, which places the diffracted front; how strong that front is needs Fresnel’s and Kirchhoff’s refinement of the same idea.

References

• Sheriff, R. E., Geldart, L. P. (1995). Exploration Seismology (2nd ed.). Cambridge University Press.
• Slotnick, M. M. (1959). Lessons in Seismic Computing. SEG.
• Aki, K., Richards, P. G. (2002). Quantitative Seismology (2nd ed.). University Science Books.
• Yilmaz, Ö. (2001). Seismic Data Analysis: Processing, Inversion, and Interpretation of Seismic Data (2 vols.). SEG Investigations in Geophysics 10.