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.

A seismic ray is a trajectory describing high-frequency seismic energy propagation in a medium whose properties vary slowly compared with the wavelength. In an isotropic medium it is normal to local travel-time wavefronts, and its travel time is $T=\int ds/v$ along the path.

Changes in wave speed bend rays, while interfaces can reflect or refract them. The approximation is useful for interpreting arrival times, but diffraction, interference and caustics require more than a single-ray description. In anisotropic media, energy propagation and the wavefront-normal direction need not coincide.

The time for a specified seismic phase to propagate along its ray. Arrival time additionally includes earthquake origin time; those quantities must not be interchanged on an absolute-clock seismogram.

A ray family can fold in the mapping from ray parameter or turning depth to surface range, so that three rays arrive at the same range with different travel times. A sufficiently strong, thin high-positive-gradient zone can produce this effect. The first-arrival curve is the lower envelope of the branches; other branches represent later arrivals. Gradient contrast and thickness determine whether folds exist.

A seismic travel-time fold caustic occurs where two branches of a seismic ray family merge in the mapping from seismic ray parameter $p$ to surface range $X$. At an ordinary fold,A local Taylor expansion gives $X-X_c\simeq C(p-p_c{)}^2$. On one side of $X_c$ there are two nearby rays; at the fold they merge, and beyond it that pair no longer exists. This is the local geometry underlying a boundary of seismic travel-time triplication.

In a horizontally layered model the travel times also merge, since $dT/dp=pdX/dp$ for the ray family. Geometrical spreading then makes a simple ray-amplitude estimate singular because the same ray bundle is compressed into a small range interval. Actual seismic waves have finite wavelength; their interference requires a wave approximation near the fold instead of interpreting the geometrical singularity as infinite measured motion.

For $v(z)=v_s+\beta z$ with $\beta >0$, a surface-to-surface diving ray is circular. Its turning speed is $v_t=1/p$, and its range and time are $X=2\sqrt{v_t^2-v_s^2}/\beta$ and $T=2\mathrm{arcosh}(v_t/v_s)/\beta$. Differentiation gives $dT/dX=p$, the seismic ray parameter. In the limit $\beta \to 0$ at fixed range, $T\to X/v_s$.

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 $pv=1$. The range and travel time are $x=2\int pvdz/\sqrt{1-p^2{v}^2}$ and $T=2\int dz/[v\sqrt{1-p^2{v}^2}]$ for a symmetric surface-to-surface turning ray.

A ray's intercept time subtracts its tangent travel-time contribution at the receiver range. For a smooth branch, $dT/dx=p$ implies $d\tau /dp=-x$. 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.