Seismology

Seismology studies elastic waves generated by earthquakes, impacts and controlled sources, and uses them to investigate planetary interiors. A seismometer converts ground motion into a recorded seismogram. Arrival times, waveforms and amplitudes then constrain both the source and the material traversed by the waves.

P waves respond to compression and S waves to shear. Their different speeds and their reflections or conversions at interfaces help distinguish layers. Locating an earthquake requires several stations because its origin time and location must be estimated along with the propagation model. A single late arrival can reflect a deeper source, a slower path or a mistaken seismic phase assignment.

A coordinated collection of recording stations with known positions and time bases. Distributed source–receiver paths constrain earthquake locations and internal wave speeds more effectively than one station. Geometry, noise, synchronization and instrument response determine which structures can be resolved; adding stations without crossing ray paths need not remove an inversion trade-off.

A processed record that isolates relative arrival times and amplitudes of converted and reverberated seismic waves by reducing the common source and propagation response. P-to-S conversion and S-to-P conversion at contrasts in seismic velocity can constrain interface depths and sharpness. Interpretation requires a velocity model and checks for crustal multiples, dipping interfaces and seismic anisotropy.

A propagating disturbance in or along the solid Earth, including body P waves, body S waves and surface modes. A water layer supports compressional acoustic motion but not a bulk shear restoring force.

A natural or artificial generator of seismic waves. For an active survey its bandwidth, pulse duration, directivity and energy influence penetration, resolution and acquisition design.

A controlled vibratory seismic source that transmits a known frequency sweep through a coupled ground contact. Repeated sweeps can improve useful signal strength, and Vibroseis correlation compresses the long source signal into a shorter wavelet. Access, ground coupling, finite source force, bandwidth and nonlinear harmonics affect the resulting survey data.

Let a linear Earth response $h$ be recorded using a known source sweep $s$, giving $d=s\ast h+n$. Cross-correlating the record with the sweep yieldsThus a compact sweep autocorrelation $R_{ss}$ produces pulse-compressed arrival wavelets, while unrelated noise has a smaller coherent response. The Fourier transform of the result is $H(\omega )|S(\omega ){|}^2+N(\omega )\overline{S(\omega )}$. Correlation is a matched-filter operation, not an exact inversion of missing source frequencies. Spectral shaping or stabilized deconvolution can further adjust the wavelet within the measured bandwidth.^[ref]

A seismic source producing an impulsive disturbance by a professionally managed subsurface explosion. Its applications include surveys needing strong, broad-band energy, but source coupling, repeatability, ground disturbance, site access and specialist operational requirements influence suitability. Vibroseis instead distributes controlled energy over a known frequency sweep.

An array of compressed-air sources fired in water to generate repeatable broadband pressure pulses for a seismic reflection survey. Source tuning reduces coherent bubble oscillations but does not eliminate seismic source ghosts. Its energy and low-frequency content can provide greater penetration than small high-frequency boomer seismic sources or sparker seismic sources.

A delayed, usually polarity-reversed copy of source radiation reflected at the water free surface. Interference with the direct wave shapes the usable spectrum; source depth controls the delay.

A marine acoustic source using a rapid underwater electrical discharge and its expanding bubble to generate a pulse. Its energy and spectrum can be chosen to investigate shallow sub-bottom structure; bubble behaviour and seismic wavelet shape affect resolution.

An impulsive marine acoustic source in which an electrically driven plate displaces water. The emitted broadband pulse is useful for shallow seismic reflection profiling, with effective penetration and resolution depending on source energy and substrate.

A shallow marine acoustic profiling system transmitting a swept-frequency pulse and using matched processing to resolve interfaces below the seabed. Fine vertical seismic resolution must be balanced against penetration and seismic attenuation.

An acquisition and interpretation method using seismic travel times, commonly first arrivals, to constrain a velocity model. Seismic head waves can constrain faster layers, but missing velocity contrasts, hidden thin layers and low-velocity layers limit simple interpretations.

Apollo lunar seismology uses ground-motion recordings from instruments placed on the Moon during the Apollo missions. Natural moonquakes and impacts provide sources whose waves sample the lunar interior. The station positions, source locations and seismic velocity model jointly determine the paths expected for each arrival.

Weak deep-interior phases are difficult to identify amid long scattered wave trains. Combining records from repeated deep moonquakes and applying array-processing methods can enhance arrivals reflected or converted near the core. Such reanalysis supplied evidence for a solid inner and fluid outer lunar core; its layer sizes remain an inference from a propagation model.^[ref]

A seismic event originating within the Moon. Its arrival times and waveforms constrain lunar structure when recorded by a suitable seismic network. Lunar impacts are another useful seismic source but have a different origin; their known locations can help calibrate an otherwise sparse network.

The propagation speed of a seismic disturbance, depending on elastic properties, density, composition, temperature and microstructure.

The propagation speed of a shear elastic wave. In an ideal isotropic elastic medium it is $V_s=\sqrt{\mu /\rho }$, where $\mu$ is shear modulus and $\rho$ mass density. Seismic anisotropy, frequency-dependent relaxation and melt can complicate the inference from an observed shear-wave velocity to temperature or composition.

The ratio of compressional to shear elastic wave speeds. Composition, cracks, temperature and fluid saturation all affect it.

The spatial and temporal occurrence of earthquakes, including their depth and magnitude distributions.

An inclined distribution of earthquake sources associated with a descending slab. Shallow interface earthquakes and deeper intraslab earthquakes need not share the same failure mechanism. The pattern helps trace subduction, subject to the limits of source location and the distribution of seismically active material. Named for Kiyoo Wadati and Hugo Benioff.

A localized seismic waveform used to describe a source or reflected pulse. Its phase and sign determine how an impedance-driven seismic reflection polarity appears in a recorded trace.

A description of seismic wave speed as a function of position and, if needed, direction. It converts travel times into geometric constraints and is distinct from a direct depth measurement.

The return of an elastic wave from a contrast in material properties. At normal incidence in an isotropic medium, the conventional stress-amplitude reflection coefficient is $(Z_2-Z_1)/(Z_2+Z_1)$ for the relevant P wave or shear-wave impedance.

The ratio of reflected to incident wave amplitude under a stated physical-variable and polarity convention. For normal incidence between isotropic layers with impedances $Z_a,Z_b$, the stress-amplitude convention gives $R_{\sigma }=(Z_b-Z_a)/(Z_a+Z_b)$, while particle velocity gives $R_v=-R_{\sigma }$. The reflected energy fraction is $R_{\sigma }^2$ for a single lossless propagating mode. A receiver trace requires the relevant seismic wavelet, travel times and transmission factors as well as this local coefficient.

The inclination of a reflecting interface to horizontal. It influences seismic travel times, offset sampling and seismic migration; a reflector is not necessarily a fault plane.

The sign of a reflected seismic seismic wavelet relative to an incident or reference wave under a stated recording convention. A negative normal-incidence stress-amplitude reflection coefficient reverses that polarity.

A controlled-source survey recording reflected elastic waves to infer subsurface interfaces and their properties. Travel times constrain geometry through a velocity model; amplitudes also depend on source, propagation and impedance contrasts.

The number of source–receiver traces contributing to a midpoint bin or image location under a stated acquisition and processing scheme. Greater useful fold can improve the signal-to-noise ratio, but does not compensate for missing spatial bandwidth or an incorrect velocity model.

Processing that moves recorded reflection energy to its inferred subsurface positions using a velocity model. It corrects apparent positions of dipping reflectors and collapses suitable diffraction events; insufficient spatial sampling and incorrect velocities impair the result.

The ability to distinguish vertically adjacent reflectors. A quarter-wavelength thickness is a useful wavelet-tuning scale, rather than a universal exact limit; bandwidth, seismic wavelet shape and noise also matter.

Adjustment for the change in reflected travel time with source–receiver separation. A horizontal-reflector approximation gives $t(x{)}^2=t_0^2+x^2/V_{\mathrm{NMO}}^2$ under appropriate assumptions; a velocity model is required.

Reduction of seismic amplitude through intrinsic energy dissipation and scattering, distinguished from geometrical spreading.

Change in wave amplitude as the same propagating energy is distributed across changing wavefront area. It is distinct from intrinsic material absorption even though both can reduce observed amplitude with distance.

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.

For two phases travelling the same path length $R$ at constant wave speeds $v_A>v_B$, let $r=v_A/v_B>1$, $t_A=t_0+R/v_A$ and $\Delta t=t_B-t_A$. ThenThe intercept of this geometric relation is earthquake origin time $t_0$ and its slope is $1/(r-1)$.

A common station clock offset $c_i$ added to both phase arrivals cancels from $\Delta t$, but remains in $t_A$. Ordinary linear regression can then estimate the relation under zero-mean offsets uncorrelated with the true delays. A common systematic clock bias instead shifts the intercept, and unrestricted fixed station offsets destroy identifiability of the speed ratio.

Phase-specific arrival-picking errors are a different issue. If independent errors ${\epsilon }_A,{\epsilon }_B$ affect the two picks, the predictor error is ${\epsilon }_B-{\epsilon }_A$ and the response error contains ${\epsilon }_A$. Their covariance is $-\mathrm{var}({\epsilon }_A)$, so even individually unbiased picks do not produce an ordinary regression with error-free predictor and independent response noise. A justified errors-in-variables model or a joint arrival-time likelihood must account for that shared measurement error. More stations alone do not remove this structural bias.

In a homogeneous medium with surface stations along coordinate $x$, an hypocenter at horizontal position $x_0$ and depth $z_0>0$ has straight-ray distance $R=\sqrt{(x-x_0{)}^2+z_0^2}$. For two phases along that path, set $K=1/v_B-1/v_A>0$. Their arrival delay satisfieswith $A=K^2$, $B=-2K^2{x}_0$ and $C=K^2(x_0^2+z_0^2)$. Completing the square yieldsThe ratio of intercept to curvature gives squared depth only after centering at the fitted vertex: $(\Delta t{)}^2=A(x-x_0{)}^2+A{z}_0^2$. The raw uncentered ratio $C/A$ includes $x_0^2$ and generally overestimates squared depth.

The fitted curvature also determines $K=\sqrt{A}$, but it does not separately identify both wave speeds. Shared clock offsets cancel in the interphase delay, while a wrong homogeneous velocity model or phase-picking errors still affect the result. Squaring noisy delays changes their error probability distribution; practical inference should model that transformation or fit the unsquared delay relation directly.

The delay between corresponding S wave and P wave arrivals. For a common uniform path of length $r$, $\Delta t=r(1/V_s-1/V_p)$, cancelling earthquake origin time. Matched phase identification and an appropriate velocity model are needed to infer distance.

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 seismic ray is a trajectory describing high-frequency seismic energy propagation in a medium whose properties vary slowly compared with the wavelength. In an isotropic medium it is normal to local travel-time wavefronts, and its travel time is $T=\int ds/v$ along the path.

Changes in wave speed bend rays, while interfaces can reflect or refract them. The approximation is useful for interpreting arrival times, but diffraction, interference and caustics require more than a single-ray description. In anisotropic media, energy propagation and the wavefront-normal direction need not coincide.

The time for a specified seismic phase to propagate along its ray. Arrival time additionally includes earthquake origin time; those quantities must not be interchanged on an absolute-clock seismogram.

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.

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.

A seismic phase reflected more than once before recording. Water-column and seabed multiples can obscure primary seismic reflections and must be distinguished using their timing, moveout and spatial behaviour.

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.

A sensor of ground motion, commonly giving an electrical response proportional to particle velocity over a specified frequency band. Three-component geophones record different motion directions and can help distinguish P- and S-wave responses. Ground coupling and orientation are essential for interpreting amplitudes and seismic reflection polarity.

An underwater acoustic pressure sensor. A hydrophone streamer records fluid P waves; observing ground S waves directly instead requires coupled solid-motion sensors.