Seismic waves: P, S, surface waves; seismograph interpretation and Earth's interior

Intuition Beginner

Earthquakes release energy as seismic waves when rocks break along faults. P-waves (primary) are the fastest, compressing and expanding rock like a pushed slinky. They travel through solids, liquids, and gases at roughly 5 to 8 kilometers per second in the crust. Because they arrive first at a seismograph, they earn the name primary.

S-waves (secondary) arrive next. They shear rock side-to-side, like shaking a rope up and down, and travel only through solids. Liquids cannot support the shear stress these waves need. This single fact — that S-waves vanish beyond a certain angular distance — told us Earth has a liquid outer core.

Surface waves arrive last but cause the most destruction. Love waves shift the ground horizontally. Rayleigh waves roll the surface in an elliptical motion, like ocean swell hitting a beach. Both are confined to the surface and carry larger amplitudes than body waves.

Seismographs record the arrival times of each wave type. When at least three stations measure the time gap between P and S arrivals, scientists triangulate the earthquake's epicenter. The larger the gap, the farther the station is from the source.

The disappearance of S-waves in certain directions was the key evidence Richard Oldham used in 1906 to argue Earth has a dense liquid core. Inge Lehmann later showed in 1936 that within this liquid core lies a solid inner core, detected through faint P-wave arrivals in the S-wave shadow zone.

Visual Beginner

Wave type	Motion	Speed	Travels through	Damage potential
P-wave (body)	Compressional (push-pull)	Fastest (5-8 km/s crust)	Solids, liquids, gases	Moderate
S-wave (body)	Shear (side-to-side)	~3-4.5 km/s	Solids only	Significant
Love wave (surface)	Horizontal shear	Slower than S	Surface only	High
Rayleigh wave (surface)	Elliptical retrograde	Slowest	Surface only	Highest

Worked example Beginner

On October 17, 1989, the Loma Prieta earthquake (magnitude 6.9) struck the Santa Cruz Mountains south of San Francisco. Three seismograph stations recorded the event. Station A, closest to the epicenter, measured a P-S arrival time gap of 8 seconds. Station B measured a gap of 20 seconds. Station C, the most distant, measured 30 seconds.

The P-S gap grows with distance because P-waves travel faster than S-waves. In typical crustal rock, the gap increases by roughly 8 seconds per 50 kilometers of distance. Station A's 8-second gap places it about 50 kilometers away. Station B's 20-second gap means roughly 125 kilometers. Station C's 30-second gap means roughly 190 kilometers.

By drawing circles of the matching radius around each station on a map, the three circles intersect at one point — the epicenter. In practice this triangulation is done digitally, but the principle is unchanged. The Loma Prieta epicenter was located in the Santa Cruz Mountains, consistent with the trace of the San Andreas Fault system.

Check your understanding Beginner

Formal definition Intermediate+

Body waves propagate through the Earth's interior. P-waves (compressional, primary) are longitudinal: particle motion is parallel to the direction of propagation. S-waves (shear, secondary) are transverse: particle motion is perpendicular to propagation.

Surface waves are guided by the Earth's free surface. Love waves are horizontally polarized shear waves trapped in low-velocity surface layers. Rayleigh waves arise from coupling of P and SV motion, producing retrograde elliptical particle motion at the surface.

Wave velocities and elastic moduli

Body wave velocities are governed by the elastic moduli and density of the medium:

$$v_P=\sqrt{\frac{K+\frac{4}{3}\mu }{\rho }}$$ $$v_S=\sqrt{\frac{\mu }{\rho }}$$

where $v_P$ is P-wave velocity, $v_S$ is S-wave velocity, $K$ is the bulk modulus, $\mu$ is the shear modulus, and $\rho$ is density. P-waves propagate through any medium with $K>0$, including liquids where $\mu =0$. S-waves require $\mu >0$ and therefore cannot exist in fluids.

The velocity ratio $v_P/v_S=\sqrt{K/\mu +4/3}$ provides diagnostic information. A decrease in this ratio may signal fluids or partial melt in the subsurface, a principle exploited in volcanic monitoring and earthquake precursor studies.

Seismic refraction and Snell's law

At a boundary between layers of different velocity, seismic waves refract according to Snell's law:

$$\frac{\sin i}{v_1}=\frac{\sin r}{v_2}$$

where $i$ is the angle of incidence, $r$ is the angle of refraction, and $v_1$, $v_2$ are the layer velocities. When $v_2>v_1$, a critical angle $i_c={\sin }^{-1}(v_1/v_2)$ exists beyond which energy is totally internally reflected. At this critical angle, a head wave travels along the boundary, returning energy to the surface — the basis of seismic refraction surveys.

Shadow zones and Earth's layered structure

Seismic shadow zones are angular distance ranges where particular wave types are absent or severely diminished:

• P-wave shadow zone: 104 to 140 degrees from the epicenter. The liquid outer core sharply reduces P-wave velocity at the core-mantle boundary, refracting P-waves inward and away from their original path.
• S-wave shadow zone: Beyond 104 degrees. S-waves cannot traverse the liquid outer core at all.

These shadow zones were the primary evidence for Earth's internal layering. The Mohorovicic discontinuity (Moho, 1909) separates crust from mantle at depths of 5-70 km, marked by a sharp velocity increase. Oldham (1906) identified the core through travel-time anomalies. Gutenberg (1914) placed the core-mantle boundary at approximately 2,900 km. Lehmann (1936) detected the solid inner core at approximately 5,150 km depth through weak P-wave arrivals within the shadow zone.

Travel-time curves and epicenter location

A travel-time curve plots seismic phase arrival time as a function of epicentral distance. The S-P time interval increases systematically with distance, providing a direct measure of epicentral distance from a single station. Locating the epicenter requires at least three stations: each station's S-P interval defines a distance, and the intersection of three distance circles yields the epicenter. Modern practice uses least-squares inversion of arrivals from many stations and phases.

Magnitude scales

The Richter scale (local magnitude $M_L$, 1935) measures wave amplitude on a Wood-Anderson seismograph and saturates above about magnitude 7.0. The moment magnitude $M_w$, based on seismic moment $M_0=\mu AD$, does not saturate:

$$M_w=\frac{2}{3}{\log }_{10}(M_0)-10.7$$

where $M_0$ is in dyne-centimeters. Each whole-number increase in $M_w$ represents roughly 32 times more energy release.

Focal mechanisms

Focal mechanisms (beach ball diagrams) represent the radiation pattern of seismic waves from a double-couple source — motion on a planar fault. The four-lobed pattern of compressional and dilatational first motions yields two possible fault-plane orientations (the actual fault and an auxiliary plane), determined from the polarity of first arrivals at stations distributed around the earthquake.

Key result: seismic mapping of Earth's layered interior Intermediate+

The discovery of Earth's internal layering — crust, mantle, liquid outer core, solid inner core — was accomplished by analyzing how seismic waves propagate through the planet. The key steps in this chain of evidence are:

1. The Moho (1909). Mohorovicic observed two distinct P-wave arrivals from a Croatian earthquake. The faster set traveled through a deeper, higher-velocity layer (the mantle). The velocity discontinuity at 5-70 km depth marks the crust-mantle boundary.

2. The core (Oldham 1906, Gutenberg 1914). P-wave travel times beyond 104 degrees were anomalously delayed, and direct S-waves were absent. A deep liquid layer with lower P-wave velocity explained both observations. Gutenberg placed the core-mantle boundary at approximately 2,900 km.

3. The inner core (Lehmann 1936). Within the P-wave shadow zone (104-140 degrees), weak arrivals were observed that a uniform liquid core could not produce. A solid inner core with higher velocity refracts P-waves back into the shadow zone. The inner core boundary lies at approximately 5,150 km.

These inferences require integrating travel-time measurements with Snell's law and the velocity–modulus relationships. The result is a self-consistent model in which density and velocity increase with depth, punctuated by sharp discontinuities at the Moho, the core-mantle boundary, and the inner core boundary. This model has been confirmed by independent constraints from normal modes, gravity, and geomagnetism, and is encoded in reference models such as PREM (Preliminary Reference Earth Model, Dziewonski and Anderson 1981).

Exercises Intermediate+

Advanced results Master

Seismic tomography

Seismic tomography constructs three-dimensional images of Earth's interior from travel-time residuals — the differences between observed and predicted travel times. Local velocity anomalies reveal thermal and compositional heterogeneity.

Global tomography uses the worldwide network to image mantle structure at resolutions of hundreds to thousands of kilometers. It has revealed two large low-shear-velocity provinces (LLSVPs) beneath Africa and the Pacific, regions where waves travel more slowly, consistent with hotter or compositionally distinct material. Subducted slabs appear as high-velocity anomalies; some penetrate through the 660-km discontinuity to the core-mantle boundary.

Regional tomography uses dense local arrays for higher resolution. Body-wave tomography exploits crossing ray paths from many earthquake-station pairs. Surface-wave tomography exploits frequency-dependent dispersion to resolve structure at different depths.

Ambient noise tomography uses continuous seismic noise — ocean waves, atmospheric pressure, human activity — rather than earthquake signals. Cross-correlating noise at station pairs extracts the empirical Green's function between them, enabling imaging in aseismic regions.

Normal modes and free oscillations

Large earthquakes excite Earth's normal modes — standing waves causing the entire planet to vibrate at discrete frequencies. Spheroidal modes involve radial displacement; toroidal modes involve tangential displacement. The fundamental spheroidal mode ${}_0{S}_0$ has a period of about 20.5 minutes.

Normal mode frequencies constrain density and elastic structure globally, complementing body-wave travel times that sample only along ray paths. Mode splitting — the separation of degenerate modes into components with slightly different frequencies — reveals Earth's rotation, ellipticity, and lateral heterogeneity.

The 1960 Chilean earthquake (magnitude 9.5, the largest instrumentally recorded) provided the first clear observation of Earth's free oscillations, confirming theoretical predictions from decades earlier.

Seismic anisotropy and mantle flow

Seismic anisotropy — the directional dependence of wave velocity — probes mantle deformation. In the upper mantle, anisotropy arises from lattice-preferred orientation of olivine crystals deformed by convective flow. Fast shear-wave polarization directions align with mantle flow in many settings.

Shear-wave splitting measurements detect S-wave birefringence through anisotropic regions. Beneath mid-ocean ridges, fast directions align with divergent flow. Beneath subduction zones, they typically parallel plate motion, indicating shear between plate and asthenosphere. Anisotropy in the D" layer at the base of the mantle may reflect post-perovskite crystal alignment in the deforming thermal boundary layer above the core.

Receiver function analysis

Receiver functions isolate P-to-S converted waves generated when teleseismic P-waves hit velocity discontinuities beneath a station. The timing and amplitude of converted phases constrain the depth and impedance contrast of the Moho, the lithosphere-asthenosphere boundary, and the 410-km and 660-km discontinuities.

H-k stacking exploits the moveout of converted phases as a function of ray parameter to simultaneously determine crustal thickness $H$ and $v_P/v_S$ ratio $k$. This has been applied globally to map crustal properties.

Earthquake source theory: the double-couple model

The double-couple model represents the earthquake source as two opposing force couples acting on a fault plane, producing a four-lobed radiation pattern. The moment tensor, a symmetric second-rank tensor with zero trace for a pure double-couple, generalizes this.

Deviations from the double-couple — the compensated linear vector dipole (CLVD) component — may indicate complex faulting, volumetric sources (magma injection), or landslides. The isotropic component reveals explosive sources, making moment tensor analysis central to nuclear test discrimination under the Comprehensive Nuclear-Test-Ban Treaty (CTBT).

The Gutenberg-Richter law and earthquake statistics

The frequency-magnitude relation ${\log }_{10}N=a-bM$ describes earthquake populations, with $b\approx 1.0$ for tectonic events. Each unit decrease in magnitude yields approximately 10 times more events. This power law is characteristic of self-organized critical systems.

The characteristic earthquake model proposes that individual faults preferentially produce events of a specific size, deviating from Gutenberg-Richter statistics. The time-predictable model holds that recurrence time scales with prior slip. Neither has been conclusively validated, and the debate has significant implications for hazard assessment.

Paleoseismology and the seismic cycle

Paleoseismology extends the earthquake record beyond instrumental and historical periods through geological evidence — fault trenching, displaced sedimentary layers, tsunami deposits, and liquefaction features dated by radiocarbon, luminescence, or cosmogenic methods.

The Cascadia subduction zone was recognized as a seismic hazard through paleoseismic evidence: marsh subsidence, drowned forests, tsunami sand sheets, and Japanese records of an orphan tsunami in January 1700. These converged on a magnitude 9.0 earthquake on January 26, 1700, confirming a recurrence interval of approximately 300-600 years.

Early warning systems

Earthquake early warning exploits the velocity difference between P-waves ($\sim$6 km/s) and the more damaging S-waves and surface waves ($\sim$3.5 km/s). Detection of the P-wave at a nearby station enables magnitude and location estimation, then prediction of shaking intensity and arrival time at distant sites. Warning times range from seconds to tens of seconds.

Japan's system, operational since 2007, automatically slows trains, shuts off industrial processes, and issues public alerts. The US ShakeAlert system became operational on the west coast in 2021.

Induced seismicity

Human activities can trigger earthquakes by altering subsurface stress or pore pressure. Wastewater injection induced the 2016 magnitude 5.8 Pawnee, Oklahoma earthquake by reducing effective normal stress on pre-stressed faults through elevated pore fluid pressure. Hydraulic fracturing typically produces events below magnitude 3.0. Reservoir impoundment, geothermal production, and mining have also induced seismicity.

Nuclear test monitoring and the CTBT

The International Monitoring System (IMS) under the CTBT uses seismic, hydroacoustic, infrasound, and radionuclide stations to detect nuclear explosions. Seismic discrimination criteria include: compressional first motions in all directions (no four-lobed pattern), shallow source depth, excess high-frequency P-wave energy, and reduced S-wave energy relative to earthquakes of comparable magnitude.

The D" layer and deep mantle discontinuities

The D" layer, extending roughly 200-300 km above the core-mantle boundary, exhibits strong lateral heterogeneity, seismic anisotropy, and a velocity discontinuity at its top. The post-perovskite phase transition, occurring at lowermost-mantle pressures, explains many D" observations.

Ultra-low velocity zones (ULVZs) within the D" layer show velocity reductions of 10-30 percent, potentially representing partial melt, iron-enriched material, or both. These zones may be the root of some mantle plumes.

Connections Master

Connections to plate tectonics

Seismic velocity structure directly maps the lithosphere-asthenosphere system that underlies plate tectonics. The low-velocity zone at 100-250 km depth corresponds to the asthenosphere. Seismic anisotropy reveals mantle flow direction, confirming that plates are dragged by convective circulation. Wadati-Benioff zones — planes of deep seismicity descending to 700 km — trace subducting lithosphere through the mantle, providing the three-dimensional geometry of convergence.

Connections to volcanic hazard assessment

Volcanic earthquakes produce distinct seismic signatures. Volcanic tremor (sustained, monochromatic shaking) and harmonic tremor often precede eruptions. Earthquake swarms beneath a volcano indicate magma migration at depth. Monitoring these signals is a primary tool for eruption forecasting. S-wave attenuation tomography beneath volcanoes can reveal the geometry and melt fraction of magma chambers, informing assessments of eruption potential.

Connections to mineral and energy exploration

Reflection seismology — using artificial sources to image subsurface structure at meter-scale resolution — applies the same physics of wave propagation and reflection used to probe Earth's deep interior. Hydrocarbon exploration relies on reflection surveys to map sedimentary basins, identify structural traps, and characterize reservoirs. Refraction surveys determine layer velocities that constrain rock type and fluid content.

Connections to planetary science

Seismology on other bodies provides unique constraints on internal structure. Apollo seismometers on the Moon revealed a small core and thick lithosphere. NASA's InSight lander (2018) deployed a seismometer on Mars, detecting marsquakes that constrained the size and density of the Martian core. These comparisons show how planetary size, composition, and thermal history control internal layering.

Connections to structural engineering

Seismic wave characteristics directly inform earthquake-resistant design. Surface waves dominate ground motion at periods of 1-10 seconds, the range most damaging to mid-rise buildings. Site amplification — the enhancement of shaking by soft sedimentary basins — depends on the impedance contrast between surficial deposits and bedrock, a direct consequence of the velocity equations for $v_P$ and $v_S$.

Connections to geodesy

GPS and InSAR measurements of co-seismic displacement complement seismic data by providing independent constraints on fault slip and earthquake magnitude. Combined seismic and geodetic inversions yield the most complete source models. Interseismic GPS strain rates map the buildup of elastic strain that will be released in future earthquakes.

Connections to the rock cycle and deep Earth geochemistry

Seismic velocities constrain rock type and physical conditions at depth. The Moho separates the felsic-to-intermediate crust from the ultramafic mantle. The 410-km and 660-km discontinuities correspond to phase transitions in olivine (olivine to wadsleyite at 410 km, ringwoodite to bridgmanite plus ferropericlase at 660 km). These mineral physics interpretations link seismic observations to the composition and thermal state of the mantle, connecting seismology to the deep rock cycle and geochemistry.

Historical and philosophical context Master

Oldham and the discovery of the core (1906)

Richard Dixon Oldham, working with data from Indian earthquakes, published his landmark 1906 paper identifying three seismic wave types (P, S, and surface) and using travel times to infer a dense core. His critical observation was that P-wave travel times increased anomalously beyond about 100 degrees angular distance, consistent with a deep region of lower velocity. He also noted the absence of S-waves at large distances, inferring a fluid interior. This work transformed seismology into a quantitative probe of planetary structure.

Mohorovicic and the crust-mantle boundary (1909)

Andrija Mohorovicic, a Croatian seismologist, observed two distinct P-wave arrivals from the 1909 Kulpa Valley earthquake. The faster arrival required a deeper layer with higher velocity — the mantle. The velocity discontinuity at the crust-mantle boundary, varying from about 5 km beneath oceans to 70 km beneath mountain roots, remains the most fundamental boundary in crustal seismology and is the target of the never-completed Mohole Project to drill through the oceanic crust.

Lehmann and the inner core (1936)

Inge Lehmann analyzed P-wave arrivals within the shadow zone that a uniform liquid core could not explain. Her 1936 paper, titled simply "P'," proposed a solid inner core with higher velocity that refracts energy back into the shadow zone. The interpretation was initially controversial but was confirmed by subsequent studies. Lehmann continued her research into her nineties and is recognized as one of the founders of modern seismology.

Gutenberg and the core-mantle boundary (1914)

Beno Gutenberg used travel-time data from many earthquakes to place the core-mantle boundary at approximately 2,900 km depth. Modern studies have refined this value only slightly, confirming the accuracy of his analytical work with limited data. Gutenberg later collaborated with Charles Richter on magnitude scales and the study of global seismicity.

The development of global seismic networks

John Milne, working in Japan, developed the first practical seismograph in the 1890s and established the first global network. The Worldwide Standardized Seismograph Network (WWSSN), deployed by the United States in the 1960s to monitor nuclear testing, enabled the plate tectonic revolution by providing precise earthquake locations mapping plate boundaries.

Modern networks include the IRIS Global Seismographic Network (GSN, over 150 broadband stations) and dense regional arrays like the USArray Transportable Array. These generate the data underpinning contemporary understanding of Earth's interior.

Philosophical dimensions: inference from observation

The seismic mapping of Earth's interior exemplifies a powerful mode of scientific reasoning: inferring properties of an inaccessible object from wave behavior. The chain of inference — from travel times to velocity structure to composition and physical state — integrates seismology with mineral physics, thermodynamics, and experimental petrology. The resulting model, encoded in PREM and its successors, is one of the most thoroughly tested in all of geophysics, constrained independently by body waves, surface waves, normal modes, gravity, and geomagnetism.

Bibliography Master

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2. Stein, S. & Wysession, M. (2003). An Introduction to Seismology, Earthquakes, and Earth Structure. Blackwell. Ch. 1-3: Seismic waves and Earth structure.

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4. Lay, T. et al. (2005). "The great Sumatra-Andaman earthquake of 2004." Science, 308, 1127-1133.

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6. Lehmann, I. (1936). "P'." Publications du Bureau Central Seismologique International, Serie A, Travaux Scientifiques, 14, 87-115.

7. Gutenberg, B. (1914). "Uber Erdbebenwellen. VIIA." Nachrichten von der Koniglichen Gesellschaft der Wissenschaften zu Gottingen, Mathematisch-Physikalische Klasse, 1914, 166-218.

8. Richter, C. F. (1935). "An instrumental earthquake magnitude scale." Bulletin of the Seismological Society of America, 25, 1-32.

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10. Dziewonski, A. M. & Anderson, D. L. (1981). "Preliminary reference Earth model." Physics of the Earth and Planetary Interiors, 25, 297-356.