= Solution
\b[<Elastic rebound>.] Tectonic motion loads a fault whose shallow portion is locked. Rock on either side deforms and stores elastic <strain> <energy> while slip is inhibited. When failure permits rapid slip, part of the stored <energy> becomes seismic radiation and part is dissipated in <friction> and damage. The adjacent rock partly relaxes, while a permanent displacement remains across the fault. Geodetic <strain> accumulation and coseismic offsets support this loading–release picture.<Reid's Elastic Rebound Theory by United States Geological Survey>[Explanation of accumulated elastic strain, release and permanent fault offset]{ref}
The model is useful for brittle crustal faulting but is not a prediction of precise recurrence, rupture size or frictional evolution. Aseismic creep can release deformation without a large sudden event. Viscoelastic flow, afterslip, fluid-pressure changes and interacting fault segments complicate a simple locked-fault cycle. Deep-focus <earthquakes> and non-shear source components require additional physics; not every <earthquake> source is adequately described by a shallow brittle double couple. Rebound also need not release all the pre-existing <stress>.
\Image[/past-exam-of-the-physics-of-the-earth-as-a-planet-course-of-the-university-of-cambridge/2014/paper-1-pep-elastic-rebound.png]
{title=Original displacement-profile schematic of locked loading and permanent slip after relaxation}
\b[A thin negative-gradient <velocity> zone.] In horizontal layering, use the <seismic ray parameter> $p=\sin i/v$, the conserved horizontal <slowness>. A ray turns when $pv=1$. Downward passage through decreasing <speed> makes $pv$ smaller, so a surface ray cannot turn in the negative-gradient interior. Rays either turn above the zone or penetrate it and turn below where <speed> has recovered sufficiently. This produces missing turning depths and a <seismic shadow range from a low-velocity zone> in the ideal ray model.
Retain all branches using
$$
x(p)=2\int_0^{z_t}\frac{pv(z)}{\sqrt{1-p^2v(z)^2}}\,dz,
\qquad
T(p)=2\int_0^{z_t}\frac{dz}{v(z)\sqrt{1-p^2v(z)^2}}.
$$
The travel-time slope is $dT/dx=p$. Just below the critical value that permits passage beyond the upper <speed> maximum, the ray makes a finite extra excursion through the low-velocity region. Range can jump, and the deep branch can fold back before resuming its normal trend, giving multiple arrivals outside the shadow. A first-arrival curve alone hides some of these branches. Define the <seismic intercept time> by
$$
\boxed{\tau(p)=T(p)-px(p),\qquad\frac{d\tau}{dp}=-x(p).}
$$
The $x$–$p$ representation displays the gap and fold; the $\tau$–$p$ representation has separate shallow and deep branches and a branch jump at the critical ray parameter for the piecewise-linear cusp model. Do not join them by invented turning rays. <Diffraction> and finite-frequency <energy> soften a purely geometrical shadow.
The figures use an explicit illustrative profile, not measured <Earth> parameters: <speed> rises from 1 to 2 over depths 0 to 1, falls to 1.8 by depth 1.2, then rises with its original <gradient>. The <asthenosphere> beneath oceanic <lithosphere> is an important setting for a <low-velocity zone>. A major P-wave <speed> reduction also occurs at the boundary between the <mantle> and the <outer core of the Earth>; <S waves> do not propagate through the liquid <outer core of the Earth>.<Essentials of Geophysics: Seismology by R. D. van der Hilst>[Chapter 4, printed pages 169–170, low-velocity-zone rays, shadow range and Earth examples]{ref}
\Image[/past-exam-of-the-physics-of-the-earth-as-a-planet-course-of-the-university-of-cambridge/2014/paper-1-pep-low-velocity-rays.png]
{title=Analytically traced illustrative velocity reversal: rays, travel times, range–slowness and intercept time}
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