Seismology

Seismology studies elastic waves in a planet and uses their propagation to infer its interior and the sources of ground motion.

A toroidal normal mode of the Earth is a free oscillation with tangential, divergence-free displacement. It is the spherical counterpart of the transverse shear motion in a Love wave. A bounded spherical Earth has discrete angular orders and eigenfrequencies, so the arbitrarily low-frequency continuum of a flat infinite layer is not a literal statement about global nonrigid oscillations.

A seismic phase labels a propagation type and path, such as direct P, direct S, a reflection or a converted wave. Misidentifying the phase assigns the wrong travel-time function even if its arrival is picked precisely.

A low-velocity zone has a wave-speed decrease with depth over some interval. Surface rays do not turn within its negative-gradient interior; rays penetrating it can return only after encountering sufficient speed recovery below. This can produce shadow ranges, multiple travel-time branches and poor turning-depth coverage. A wave-specific low-velocity zone is not identical to a mechanical layer boundary.

A seismic shadow range is an interval of receiver distances missing a specified family of geometrical rays. A velocity decrease can separate shallow-turning and deep-turning arrivals; diffraction or other wave families can still deliver energy there.

Arrival-time location estimates a hypocenter and origin time by comparing picked phase arrivals with predicted travel times. Linearization gives residual changes $\delta t_i=\delta t_0+\nabla T_i\cdot \delta \mathbf{x}$. Poorly distinct derivative columns cause tradeoffs, while incorrect wave speeds or phase labels cause systematic bias even with many stations.

For surface stations at similar distances, the travel-time derivatives with respect to depth can be small or nearly equal. Their arrival-time shifts are then difficult to distinguish from a common change in earthquake origin time. A deeper source can be paired with an earlier origin. Nearby stations, broad distance coverage, and depth-sensitive phases help separate the two parameters.

If $r=v_A/v_B$ and both phases share path length, arrival time of A is affine in the interphase delay. With unbiased independent clock errors, the intercept estimates earthquake origin time and slope estimates $1/(r-1)$. A common clock bias is indistinguishable from an origin-time shift; unrestricted station offsets also destroy speed-ratio identifiability. A statistical regression must state its clock-error assumptions.

In a homogeneous medium, two phases along the same geometric path have delay $\Delta t=KR$, where $K=1/v_B-1/v_A$. The squared delay is a quadratic in station position. Its vertex locates the epicentre, while the ratio of intercept to quadratic coefficient gives squared depth. These spatial parameters do not depend on a particular assigned speed ratio if both speeds are inferred from the same data. Local clock offsets cancel, although uncertainties in the velocity model do not.

An earthquake origin time is the estimated onset time of rupture. Predicted arrivals add this time to a propagation travel time; an uncertain origin time therefore couples to uncertain source depth in some station geometries.

Differences between arrival times of nearby earthquakes at a common station largely cancel common propagation errors. Comparing observed and predicted differences constrains relative source separation and origin-time differences. Closely similar waveforms permit precise differential picks by cross-correlation. Absolute position still needs an accurate model or external anchor.

Seismic anisotropy is direction dependence of elastic-wave propagation. Aligned crystals or layering can produce it; using it to infer deformation requires a model connecting fabric to flow.

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 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.

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.

Seismic surface-wave dispersion is the frequency dependence of Rayleigh wave or Love wave propagation caused by depth-dependent elastic properties, mode structure and possibly intrinsic anelasticity. Phase velocity is $\omega /k$; group velocity is $d\omega /dk$. Different periods and branches have different depth sensitivities, enabling surface-wave tomography.

For surface phase speed $c$ on a spherical Earth, the conserved angular ray parameter is $p=R_E/c$. The local horizontal slowness at radius $r$ is $p/r$, giving squared radial shear wavenumber $q_r^2={\omega }^2[\beta (r{)}^{-2}-p^2/r^2]$ in ray theory. The displayed inequality makes it negative. A jump in shear speed can turn a propagating wave above the interface into an evanescent wave below it. This is a local condition, not proof of evanescence at every greater depth.

A seismogram records ground motion through an instrument response. Comparing it with a synthetic seismogram requires consistent component orientation, sampling, timing, filtering and instrument calibration.

Repeated reflection and conversion between a layer surface and its lower interface produces additional delayed phases. Their travel times and amplitude depend on layer thickness, wave speeds, impedance contrasts, attenuation and source spectrum.

In a chemically homogeneous adiabatic spherical region, $d\rho /dr=-\rho g/\Phi$, coupled to $dm/dr=4\pi r^2\rho$ and $g=Gm/r^2$. Seismic velocities supply $\Phi$; density normalization and jumps require further data and assumptions.

The adiabatic bulk modulus divided by density, $\Phi =K_S/\rho ={\alpha }^2-4{\beta }^2/3$ for an isotropic solid. It has units of squared speed and is a key input to the Adams-Williamson density relation.

A seismometer compares ground motion with the motion of an inertial test mass. A mechanical seismograph records their relative displacement. Its frequency response depends on the suspension and on which relative or absolute velocity enters the damping law.