Seismic ray theory

Seismic ray theory approximates high-frequency propagation through smoothly varying media. The travel-time field satisfies the eikonal equation, and its characteristics are paths of stationary travel time according to Fermat's principle. Diffraction and caustics require refinements beyond a simple geometrical-ray approximation.

Table of contents
- Seismic ray parameter in horizontal layering Seismic ray theory
  - Seismic intercept time Seismic ray parameter in horizontal layering

Seismic ray parameter in horizontal layering ( $p = sin i / v = d T / d x$ )

 0  0

In horizontally layered isotropic media, horizontal slowness is conserved along a ray. The incidence angle

i

is measured from vertical. A smooth turning depth satisfies

p v = 1

. The range and travel time are

x = 2 \int p v d z / 1 - p^{2} v^{2}

and

T = 2 \int d z / [v 1 - p^{2} v^{2}]

for a symmetric surface-to-surface turning ray.

Seismic intercept time ( $τ = T - p x$ )

 0  0

A ray's intercept time subtracts its tangent travel-time contribution at the receiver range. For a smooth branch,

d T / d x = p

implies

d τ / d p = - x

. Thus an intercept-time curve encodes range as its negative slope. Separate turning branches around a low-velocity zone must be retained rather than joined by fictitious rays.