Seismic travel-time triplication

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.