Earthquake Seismology, Earth's Interior and Heat Flow

Earth's interior discontinuities and core phases

Seismic velocities change in steps at a few named discontinuities, and their names form the core vocabulary of this topic. The Mohorovičić discontinuity (Moho) separates crust from mantle, and both P and S velocities increase downward across it. The Gutenberg discontinuity, the core–mantle boundary at about 2900 km depth, is the largest contrast: the S-wave velocity drops to zero because the outer core is liquid, and the P-wave velocity falls from about 13.7 km/s to about 8 km/s. This drop is a decrease, opposite in sign to the increases at the Moho and at the inner core boundary.

The density jump at the same boundary is abrupt because it is compositional. Silicate mantle rock lies on an iron–nickel alloy core, so the chemical contrast produces a step of several g/cm³. Temperature and pressure change density only gradually, through thermal expansion and compression, and cannot produce a sudden step.

The Lehmann discontinuity marks the top of the solid inner core, where velocities increase again on entering the solid iron core.

Within the upper mantle, olivine transforms at about 410 km depth to the modified spinel structure of wadsleyite, which raises velocity and density and produces the 410 km discontinuity. Plagioclase is stable only in the shallow crust-to-uppermost-mantle range, so plagioclase-related reactions do not occur at 400 km. Near 660 km most silicates transform to the perovskite structure, which separates the upper and lower mantle.

The P-wave velocity increases with depth through the mantle, from about 10.8 km/s near 660 km to about 13.7 km/s just above the core–mantle boundary. The lower mantle is therefore the layer in which V_P first exceeds 13 km/s, whereas the liquid outer core transmits P waves at about 8 to 10 km/s and the solid inner core at about 11 km/s.

Phase names encode the ray path. Capital P and S are compressional and shear legs in the mantle, K is a P leg in the liquid outer core, I is a P leg in the solid inner core, and J is an S leg in the inner core. Lowercase letters mark reflections: c at the core–mantle boundary, m at the Moho, i at the inner core boundary, and a lowercase p placed before a phase marks a leg that leaves the source upward and reflects at the free surface.

• PKIKP — a P wave that crosses both the outer and inner core without changing mode, with I as its inner-core P leg. e.g., a P arrival that crosses the Earth's centre as P throughout is PKIKP, not PKJKP.
• PKJKP — a P wave that converts to an S wave on crossing the solid inner core, so J is the shear leg. e.g., a core phase whose inner-core leg carries shear is PKJKP.
• SKKS — an S wave in the mantle that passes through the outer core as P twice and returns to the mantle as S, never entering the inner core. e.g., its K legs show that the outer core transmits P waves but not S waves.
• PKP — a P wave refracted through the outer core and emerging as P, with no inner core leg. e.g., a path that turns in the outer core without touching the inner core is PKP.
• PKiKP — a P wave reflected at the inner core boundary that never enters the inner core. e.g., its path ends at the boundary rather than continuing through it.
• PmP — a P wave reflected at the Moho and returned to the surface. e.g., a reflection from the base of the crust is PmP, whereas Pn is a head wave refracted along the top of the upper mantle, not a reflection.

The geomagnetic field is produced by the geodynamo, in which convection of the electrically conducting liquid iron–nickel alloy in the outer core sustains electric currents that regenerate the magnetic field. The solid inner core cannot convect, and the lithosphere and asthenosphere lie outside the core, so neither is the seat of the geodynamo.

In a convecting layer the temperature follows an adiabat, whose magnitude is |dT/dr|_S = \alpha g T / c_p, where \alpha is thermal expansivity, g is gravity, T is absolute temperature and c_p is specific heat. The gradient grows with T/c_p, so a layer that is hotter and has a lower specific heat has a steeper adiabat. The outer core, being hotter than the lower mantle, shows such an increase.

Some items in the same question bank test surface landforms, which are easily confused with deep-structure terms. Karst topography develops by dissolution of soluble carbonate rocks such as limestone, and oxbow lakes form when a river meander is cut off in a fluvial setting. Fjords are glacially eroded valleys rather than aeolian features, and ventifacts are stones faceted by wind-blown sand rather than by glaciers.

Radiogenic heat production decay

Radiogenic heat is released by the decay of U, Th and K, and in continental crust it is concentrated near the surface. If the production falls off exponentially, A(z) = A_0 e^{-z/h_r}, where A_0 is the surface value and h_r is the characteristic depth, often taken as about 10 km in cratonic crust. The mean over a layer of thickness h is \bar{A} = \dfrac{A_0 h_r}{h}\left(1 - e^{-h/h_r}\right). For A_0 = 3\ \muW m^{-3}, h_r = 10 km and h = 30 km, the mean is about 0.95 \muW m^{-3}, far below the surface value.

For the granitic expression Q_r = \alpha C_U + \beta C_{Th} + \gamma C_K, the coefficients are ordered by the energy each element releases per unit of its concentration. The uranium decay series releases the most energy per unit mass, thorium is intermediate, and potassium gives the smallest contribution because only a tiny fraction of natural potassium is the radioactive ^{40}K. The ordering is therefore \alpha > \beta > \gamma, so per unit concentration K contributes far less heat than U or Th.

When heat-producing elements are uniformly distributed through a layer of thickness D, the surface heat flow equals the reduced heat flow plus the layer's contribution, q_s = q_r + AD, so A = (q_s - q_r)/D. Convert mW m^{-2} and km to W m^{-2} and m before dividing. For example, a 20 mW m^{-2} excess over a 10 km layer gives A = 2\ \muW m^{-3}.

The surface heat flow contributed by a layer is q = \rho H h, where \rho is density in kg m^{-3}, H is heat production per unit mass in W kg^{-1} and h is thickness in m, so the product is in W m^{-2}. e.g., with \rho = 2700 kg m^{-3}, H = 1.2\times10^{-9} W kg^{-1} and h = 5 km, the flow is q = 16.2 mW m^{-2}.

Earthquake source depth and magnitude

Focal depth is found from the delay between the direct P wave and pP, the phase that leaves the source upward, reflects at the free surface and then follows the downward path. The delay grows with source depth, so the difference t_{pP} - t_P gives focal depth, whereas PP is a surface reflection far from the source and does not give depth this way.

The seismic moment is M_0 = \mu A D, where \mu is rigidity, A is the fault area and D is the average slip. The moment magnitude is M_w = \frac{2}{3}\left(\log_{10} M_0 - 9.1\right) with M_0 in N m, which is equivalent to the common form \frac{2}{3}\log_{10}M_0 - 6.07. e.g., a 10 km by 10 km patch slipping 1 m in rock with \mu = 3\times10^{10} Pa has M_0 = 3\times10^{18} N m and M_w of about 6.25.

Explosions are shallow, isotropic compressional sources. Their first P motion is compressional at every azimuth, and they radiate weaker S and surface waves than earthquakes of similar size. At equal surface-wave magnitude M_s, an explosion has a larger body-wave magnitude m_b than an earthquake, so the two populations do not share the same m_b–M_s relation, and this difference is the basis of the discrimination.

Peak ground acceleration at a site depends on the distance from the source, through geometric spreading and anelastic attenuation, on rupture directivity, which focuses energy forward along the rupture, and on local site conditions, such as soft soils that amplify shaking. The origin time of the earthquake does not change the amplitude recorded at a station.

Wadati diagram and Vp/Vs ratio

A Wadati diagram plots the S–P interval t_S - t_P against the P travel time t_P for several stations recording one local earthquake. Since both waves cover the same distance, t_S - t_P = (V_P/V_S - 1)\,t_P, so the slope is V_P/V_S - 1, and the line crosses the t_P axis at the origin time. For a Poisson solid, \lambda = \mu, which gives V_P/V_S = \sqrt{3} and a slope of about 0.73.

The same geometry gives the distance from the S–P time: \Delta t = D\left(1/V_S - 1/V_P\right), so D = \dfrac{\Delta t\, V_P V_S}{V_P - V_S}. For example, with V_P = 6 km/s, V_P/V_S = 1.5 and \Delta t = 2 s, the distance is D = 24 km. Using the same relation, a 1 s S–P difference with V_P = 3 km/s and V_P/V_S = 2.0 corresponds to a distance of 3.0 km.

Haskell fault rupture model

In the Haskell model a rupture spreads along the fault at speed V_r, and the radiated waves are compressed in the direction of rupture and stretched behind it. The apparent duration at a station at angle \theta from the rupture direction is T = \frac{L}{V_r}\left(1 - \frac{V_r}{V_s}\cos\theta\right), where L is the fault length and V_s is the speed of the radiated wave. Stations ahead of a unilateral rupture therefore see a shortened pulse, while stations behind see a lengthened one.

For a station straight ahead of the rupture, \cos\theta = 1, and when P waves are used the apparent time reduces to L\left(1/V_r - 1/V_p\right). For example, a 40 km fault rupturing at 2 km/s with V_p = 8 km/s has a true rupture time of 20 s and an apparent time of 40(0.5 - 0.125) = 15 s.

Bulk Earth composition

The bulk Earth is dominated by iron, oxygen, silicon and magnesium, which together make up about 90% of its mass, while Al, Ca and Na are minor constituents. Magnesium is the major element that completes this group, held in olivine and pyroxene in the mantle.

Iron meteorites are fragments of differentiated planetesimal cores and consist mainly of an iron–nickel alloy. Their composition therefore matches that of Earth's metallic core, not any silicate reservoir such as the primitive mantle, the depleted mantle or the bulk silicate Earth.

• bulk silicate Earth — the whole Earth minus its metallic core, so it is silicate only. e.g., removing the Fe–Ni core from the bulk Earth leaves the bulk silicate Earth.
• primitive mantle — the silicate mantle left after core formation and before melt extraction has depleted it. e.g., the primitive mantle is the reference composition for trace-element budgets.

Heat flow from thermal gradient

Conductive heat flow follows Fourier's law, q = k\,\dfrac{dT}{dz}, where k is thermal conductivity in W m^{-1} K^{-1} and dT/dz is the geothermal gradient. A gradient given in °C km^{-1} must be converted to K m^{-1} by a factor of 0.001 before use. For example, a gradient of 25 °C km^{-1} in rock with k = 2.5 W m^{-1} K^{-1} gives q = 62.5 mW m^{-2}.

In a conductive lithosphere the temperature gradient is set by the heat flux entering from below, so a larger basal heat flow makes the geotherm steeper and raises temperature at every depth. The new profile is therefore the steepest of the candidate curves, which departs most from the original at depth.

Geophysical method depth resolution

Depth resolution is the smallest depth interval over which a geophysical method can distinguish two structures. It is controlled by the wavelength of the probing field and by how strongly that field is attenuated or smeared with depth. Seismic waves have wavelengths of kilometres to tens of kilometres at mantle depths and pass through the entire mantle and core, so travel-time tomography images velocity structure on the scale of 1000 km with the finest resolution among the common methods. This is why seismology resolves the core-mantle boundary and mantle discontinuities. Gravity and magnetic anomalies are integrals over volume. They decay with distance from the source and are non-unique, so their resolution at great depth is coarse. Electrical and electromagnetic methods are limited by the attenuation of fields in conducting material, which restricts both penetration and resolution. When a question asks for the highest resolution of deep Earth structure, choose the seismic method.