= Seismic travel-time fold caustic
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,
$$
\frac{dX}{dp}=0,\qquad \frac{d^2X}{dp^2}\ne0.
$$
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 $dT/dp=p\,dX/dp$ 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.
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