Past exam of the physics of the Earth as a Planet course of the University of Cambridge / 2014 / Paper 1 / 3

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.^[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.

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 usingThe 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 byThe $x$–$p$ 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.^[ref]

Oceanic thermal subsidence. Newly formed oceanic lithosphere is hot and buoyant. Cooling increases its column density, causing subsidence by isostasy as it moves away from a ridge. The half-space cooling model has profile isWith volumetric thermal expansion coefficient $\alpha$, the excess column mass relative to hot mantle isCompensation with seawater density ${\rho }_w$ givesThus young-to-intermediate-age seafloor generally deepens approximately as the square root of age. At old ages, finite thermal-plate thickness and basal heat supply flatten this trend; indefinite half-space deepening is inappropriate. Sediments, variable crust, volcanic swells and dynamic topography also perturb bathymetry. Ocean-floor depth and heat flow jointly constrain cooling models.^[ref]

Effective elastic thickness. This is the thickness of a homogeneous elastic plate with the observed effective flexural rigidity, not necessarily the crustal or thermal lithospheric thickness:Infer it by modelling deflection around known loads, such as seamounts or trenches, or fitting gravity-topography admittance and gravity-topography coherence across wavelengths. A simple Fourier balance is $(D{k}^4+\Delta \rho g)w_k=q_k$ for a signed transverse load. The transition from buoyancy-dominated to bending-dominated behaviour constrains $D$; then assumed Young's modulus $E$ and Poisson's ratio $\nu$ give $T_e$.

Good inference includes load geometry and density, buried interfaces, sediment and internal loading, boundary conditions and data resolution. These can trade off with rigidity. Plastic yielding, faulting, mechanical layering and creep make $T_e$ an effective, timescale- and stress-dependent quantity. Older, colder lithosphere is often stronger, but age at loading and subsequent relaxation matter; a single universal age-to-$T_e$ conversion is unjustified.