Earthquake Seismology: Sources, Magnitude, Mechanisms, Statistics and Source Theory

This chapter answers the Earthquake Seismology heading of Part B2 of the Geology and Geophysics (GG) paper: “Earthquakes-causes and measurements, magnitude and intensity, focal mechanisms; Earthquake quantification, source characteristics, seismotectonics and seismic hazards; Digital seismographs, Earthquake statistics, quantifying earthquake source from seismological data; Uniqueness Theorem, Representation Theorems of correlation and convolution type, Reciprocity.” It teaches how earthquakes arise and are located, how magnitude differs from intensity and how energy depends on magnitude, the seismic moment and moment magnitude, source size from the corner frequency, how a beach ball is read, how a digital seismograph records ground motion, the Gutenberg–Richter law with recurrence and hazard, and the statements and consequences of the uniqueness, representation and reciprocity theorems. The numericals are epicentral distance from the S−P time, amplitude-ratio magnitude differences, M0 = μAD and Mw, Gutenberg–Richter recurrence, Poisson probability and stress drop.

1. Causes of earthquakes, seismic waves and locating the source

Most earthquakes are tectonic. Plate motion loads a locked fault with elastic strain over decades to centuries; when the shear stress exceeds the fault strength the two sides slip and the stored strain is released as seismic waves. This is H. F. Reid's elastic rebound model, drawn from the 1906 San Francisco earthquake. The point where slip starts is the focus (hypocentre), and the surface point above it is the epicentre. Volcanic activity, reservoir loading, mining and explosions give smaller, often shallow events. By focal depth, earthquakes are shallow (down to about 70 km), intermediate (about 70 to 300 km) or deep (about 300 to 700 km); none occur below about 700 km.

A seismogram shows P waves first (compressional, fastest), then S waves (shear, slower, unable to cross a liquid), then the large, slow surface waves (Love and Rayleigh). In a Poisson solid (λ = μ) Vp/Vs = √3 ≈ 1.73. For a station at distance d in a uniform medium, tP = d/Vp and tS = d/Vs, so d = (tS − tP) · Vp·Vs/(Vp − Vs). With Vp = 6.0 and Vs = 3.0 km/s the multiplier is 6 km per second, so an S−P time of 12 s gives d = 72 km. One station fixes only a distance; circles from three stations meet at the epicentre (triangulation), and the origin time is tP − d/Vp. A Wadati diagram plots tS − tP against tP: the points lie on a line of slope Vp/Vs − 1 that crosses zero at the origin time.

⚠️ S−P gives the distance from the focus
The formula gives the straight-line distance from the focus to the station (hypocentral distance). If the focal depth h is known, the epicentral distance is √(d² − h²). A stem that asks for the epicentral distance and quotes a depth wants this second step.

2. Magnitude, intensity and energy

Magnitude is an instrumental, logarithmic measure of source size: one value per earthquake. Intensity records shaking and damage at a place from observation (Modified Mercalli, I to XII, or EMS-98), so it changes with distance, local soil and construction; lines joining equal intensity are isoseismals. Richter's local magnitude is ML = log10 A − log10 A0(Δ), with A the peak amplitude on a Wood–Anderson seismograph and A0 the amplitude of a reference zero-magnitude shock at that distance (1 mm at 100 km is ML 3). Since magnitude is a logarithm, two events recorded at the same distance on the same instrument differ by ΔM = log10(A1/A2): an amplitude ratio of 10 is one unit and a ratio of 100 is two.

Common magnitude scales
ScaleBased onBehaviour for the largest events
MLPeak amplitude on a Wood–Anderson seismograph, local distancesSaturates
mbP-wave amplitude and period (about 1 s)Saturates near 6
MsRayleigh-wave amplitude at about 20 s periodSaturates near 8
MwSeismic moment M0Does not saturate

Energy. The Gutenberg–Richter relation is log10 E = 1.5 M + 4.8 (E in joules). One magnitude unit therefore multiplies radiated energy by 10^1.5 ≈ 31.6 and amplitude by 10; two units multiply energy by 1000 and amplitude by 100. An M 6.0 event radiates 10^13.8 ≈ 6 × 10¹³ J.

⚠️ Amplitude ratio is not energy ratio
A difference of 2 magnitude units means 100 times the amplitude but 1000 times the energy. Intensity has no such arithmetic: intensity VIII is not twice intensity IV. Saturation means a scale stops rising for very large events because the short-period waves it uses stop growing, not that the earthquake stops growing.

3. Quantifying the source: seismic moment, moment magnitude and spectra

A tectonic earthquake is modelled as shear slip on a fault. Its size is the seismic moment M0 = μ A D, with μ the rigidity (about 3 × 10¹⁰ Pa in the crust), A the rupture area and D the average slip; the unit is N·m (1 N·m = 10⁷ dyne·cm). The moment magnitude is Mw = (2/3)(log10 M0 − 9.1) with M0 in N·m (older books use 9.05; the difference is 0.03, and a stem states the constant). Mw does not saturate, and one magnitude unit means M0 larger by 10^1.5 ≈ 31.6. Example: a rupture 40 km long and 15 km wide, D = 2.0 m, μ = 3.0 × 10¹⁰ Pa has A = 6.0 × 10⁸ m², M0 = 3.6 × 10¹⁹ N·m, log10 M0 = 19.56 and Mw = (2/3)(19.56 − 9.1) = 6.97.

Source spectrum. The far-field displacement spectrum of body waves is flat at low frequency, with a plateau proportional to M0, and falls as f⁻² above the corner frequency fc (Brune's ω² model); larger events have lower fc. In Brune's circular-crack model the source radius is r = 2.34β/(2π fc) ≈ 0.37 β/fc (β the S-wave speed), and the static stress drop is Δσ = 7 M0/(16 r³). Example: β = 3.5 km/s and fc = 0.40 Hz give r ≈ 3.2 km; with M0 = 5 × 10¹⁷ N·m, Δσ ≈ 6 MPa, typical of earthquakes.

  • First-motion polarities at many stations give the fault-plane solution: orientation and sense of slip, but no size.
  • Waveform (moment-tensor) inversion fits the observed seismograms with synthetics built from Green's functions. Seismograms are linear in the six independent moment-tensor components, so the mechanism and M0 come out together.
  • Spectral analysis of P or S waves, once path and instrument effects are removed, gives M0 (plateau), fc, r and Δσ.
  • Finite-fault inversion maps slip over the fault plane, from which A and D follow directly.
⚠️ M0 is not energy
M0 = μAD measures the permanent offset across the fault, not the radiated energy. Two earthquakes with equal M0 can radiate different energy if their stress drops differ. The relation log10 E = 1.5 M + 4.8 is an average empirical trend, not an identity.

4. Focal mechanisms (fault-plane solutions)

Each station records whether the first P motion is compression (ground moves away from the focus, an upward first motion on a vertical seismometer) or dilatation (towards the focus, downward). Plotted on a lower-hemisphere projection of the focal sphere, compressions and dilatations fall in four quadrants separated by two perpendicular great circles, the nodal planes: one is the fault plane, the other the auxiliary plane. The diagram is shaded black for compression and white for dilatation (the “beach ball”). The P axis lies in the middle of the dilatational quadrants and the T axis in the middle of the compressional quadrants, each 45° from both nodal planes.

Faulting style and its beach ball (idealised for dip-slip planes dipping at 45°)
StyleRake (Aki–Richards)P axisT axisCentre of the ball
Strike-slip0° or 180°HorizontalHorizontalNodal planes cross: four quadrants meet
Normal−90°VerticalHorizontalWhite (dilatational)
Thrust (reverse)+90°HorizontalVerticalBlack (compressional)

A double couple has no preferred plane, so first motions alone cannot say which nodal plane is the fault; surface rupture, the line of aftershocks, the tectonic setting or rupture directivity decides. In moment-tensor terms a pure double couple has zero trace and eigenvalues +M0, 0 and −M0. An explosion or implosion adds an isotropic part (equal eigenvalues), and a compensated linear vector dipole (CLVD) part appears in some volcanic and deep events.

⚠️ P and T are not the stress axes
P and T are kinematic axes that bisect the nodal planes. They coincide with the principal stress axes only for a newly formed fracture; a reactivated fault slips under a stress field whose axes can differ from them.

5. Digital seismographs

A seismometer is a damped mass on a spring in a frame fixed to the ground. At periods much shorter than its natural period T0 the mass stays nearly still while the frame moves past it, so the relative motion records ground motion. Damping near 0.7 of critical gives a flat response. A moving-coil transducer gives an output proportional to ground velocity above the natural frequency, while a force-balance (feedback) sensor holds the mass nearly still and records acceleration, flat down to very low frequencies. Broadband sensors (flat velocity response from about 0.01 to 50 Hz or more) have replaced separate short-period (T0 about 1 s) and long-period instruments.

  • Analogue-to-digital converter. An N-bit converter has 2^N levels and a dynamic range of 20 log10(2^N) ≈ 6.02 N dB, so 24 bits give about 144 dB (real systems reach less because of noise).
  • Sampling. A rate fs gives the Nyquist frequency fN = fs/2. Energy above fN aliases to a lower frequency: with fs = 40 samples/s, a 25 Hz signal (fN = 20 Hz) appears at 40 − 25 = 15 Hz. An anti-alias filter must remove it before sampling.
  • Timing. GPS clocks give absolute time to better than a millisecond, which is essential because location depends on arrival times.
  • Instrument response. The recorded counts equal ground motion convolved with the sensor, filter and converter response (poles and zeros plus a sensitivity in counts per m/s). Magnitudes, spectra and moments need this response removed first.
ℹ️ Near the source, range matters
Near a large earthquake a sensitive broadband sensor saturates (clips). A strong-motion accelerometer, which trades sensitivity for range, records the shaking that engineers need.

6. Earthquake statistics, seismotectonics and seismic hazard

The Gutenberg–Richter law is log10 N = a − bM, with N the number of events of magnitude ≥ M in a given area and time. The b value (about 1 globally) is the slope: with b = 1 each unit of magnitude has ten times fewer events. The a value measures overall activity and depends on the area and period. The mean recurrence interval of events ≥ M is T = 1/N. Example: log10 N = 4.0 − 1.0 M per year gives N(≥6) = 10⁻² per year and T = 100 years; M ≥ 7 has T = 1000 years. For a catalogue complete above Mmin, Aki's maximum-likelihood estimate is b = log10 e/(M̄ − Mmin) = 0.434/(M̄ − Mmin). Energy grows 31.6 times per unit while numbers fall 10 times, so most seismic energy comes from the largest events.

Occurrence in time is modelled as a Poisson process: the chance of at least one event in t years is P = 1 − exp(−t/T). For T = 475 years and t = 50 years, P = 0.10, the usual design level of “10% in 50 years”. Aftershock rate follows Omori's law n(t) = K/(c + t)^p with p near 1, and Båth's law puts the largest aftershock about 1.2 magnitude units below the mainshock.

Seismotectonic settings
SettingTypical mechanismDepth and size
Subduction zone (Wadati–Benioff zone)Thrust on the megathrust; normal and thrust events inside the slabShallow to about 700 km; the largest events (Chile 1960, about Mw 9.5)
Mid-ocean ridgeNormal faultingShallow; small to moderate
Oceanic transform faultStrike-slipShallow; moderate to large
Continental collision (Himalaya)ThrustShallow; great earthquakes of about Mw 8
Stable continental interior (Latur 1993, Bhuj 2001)Reverse faulting on old structuresShallow; rare but damaging

Seismic hazard is the probability of exceeding a level of ground shaking at a site; risk combines hazard with exposure and vulnerability. Probabilistic seismic hazard analysis (PSHA) defines source zones, gives each a Gutenberg–Richter recurrence relation with a maximum magnitude, adds a ground-motion prediction equation (shaking against magnitude and distance) and sums over all sources to give the annual rate of exceeding each level, the hazard curve. A deterministic analysis instead takes one scenario earthquake. India's code map (IS 1893) groups the country into zones II to V; zone V, the most severe, includes parts of the Himalayan arc, the north-east and Kutch.

7. Uniqueness, representation and reciprocity theorems

Seismic displacement u in an elastic body obeys ρ ∂²uᵢ/∂t² = ∂ⱼ(cᵢⱼₖₗ ∂ₗuₖ) + fᵢ, with body force f. The Green's function Gᵢⱼ(x, t; ξ, τ) is the i-component of displacement at (x, t) due to a unit impulsive force in direction j at (ξ, τ); in a time-invariant medium it depends on t − τ. The four theorems below are statements to know, with their consequences.

Theorems of elastodynamics and what follows
TheoremStatementConsequence
UniquenessFor a given medium, body force and initial displacement and velocity, with displacement or traction given on the boundary for all later times, the displacement field is unique.Any solution, however found, is the solution, so equivalent force systems can stand in for the real source. It does not make the inverse problem unique.
Representation, convolution typeDisplacement at any point is a volume integral of the force convolved in time with G, plus a surface integral of boundary displacement and traction with G.A seismogram is source time function ∗ Green's function ∗ instrument. A dislocation acts as a double couple, uₙ = M_pq(t) ∗ Gₙp,q, linear in M_pq: the basis of moment-tensor inversion and deconvolution.
Representation, correlation typeThe same relation with one wavefield time-reversed, so that products become cross-correlations; it needs a lossless (time-reversal-invariant) medium.Cross-correlating recordings at two receivers retrieves the Green's function between them (seismic interferometry, ambient-noise methods); also time-reversal focusing.
ReciprocityGᵢⱼ(x, t; ξ, 0) = Gⱼᵢ(ξ, t; x, 0): source and receiver positions swap together with the force and component directions, with the same boundary conditions.The field of many sources at one receiver can be computed from one calculation run from the receiver; it checks numerical Green's functions and underlies adjoint methods.
⚠️ Reciprocity swaps directions too
A force in direction j at A recorded in component i at B equals a force in direction i at B recorded in component j at A. Swapping positions alone is correct only when i = j, for instance vertical force and vertical component. Uniqueness likewise concerns the forward problem: different sources can radiate fields that no station network can tell apart.

Key takeaways

  • Tectonic earthquakes are elastic rebound on faults. Epicentral distance from the S−P time is d = Δt · Vp·Vs/(Vp − Vs), which is the distance from the focus; subtract depth in quadrature, √(d² − h²), for the epicentre. Poisson solid: Vp/Vs = √3.
  • Magnitude is one logarithmic value per event; intensity is local, observed and varies with distance and site. ΔM = log10(A1/A2); log10 E = 1.5 M + 4.8 gives 31.6 times the energy and 10 times the amplitude per unit.
  • M0 = μAD in N·m and Mw = (2/3)(log10 M0 − 9.1) do not saturate. The spectrum has a plateau ∝ M0 and falls as f⁻² above fc; in Brune's model r ≈ 0.37 β/fc and Δσ = 7M0/(16r³).
  • In a beach ball black is compression and white dilatation; the P axis is in the white and the T axis in the black quadrants. Thrust has a black centre, normal faulting a white centre, strike-slip four quadrants meeting. Two nodal planes cannot be told apart by first motions alone.
  • A digital seismograph is sensor, converter, clock and response. Nyquist fN = fs/2, and energy above it aliases; N bits give about 6.02 N dB. Gutenberg–Richter log10 N = a − bM gives T = 1/N, Aki's b = 0.434/(M̄ − Mmin) and Poisson P = 1 − exp(−t/T).
  • Uniqueness fixes the forward solution; the convolution-type representation gives seismogram = source ∗ Green's function ∗ instrument; the correlation type needs a lossless medium and gives interferometry; reciprocity Gᵢⱼ(x;ξ) = Gⱼᵢ(ξ;x) swaps directions as well as positions.

Practice questions (17)

Attempt each one before opening the answer. Every explanation names the tempting wrong option as well as the right one, because that is where marks are lost.

  1. For a Poisson solid (Lamé constants λ = μ), what is the ratio of the P-wave speed to the S-wave speed?

    1. About 1.41 (√2), because the P-wave speed is the S-wave speed times the square root of two in every elastic solid
    2. About 1.73 (√3), because Vp² = (λ + 2μ)/ρ = 3μ/ρ while Vs² = μ/ρ when λ = μ
    3. About 2.00, because P waves travel twice as fast as S waves in every crustal rock of any composition
    4. About 3.00, because Vp² is three times Vs² and the speeds themselves are therefore in the ratio three
    Show answer

    Answer: B — About 1.73 (√3), because Vp² = (λ + 2μ)/ρ = 3μ/ρ while Vs² = μ/ρ when λ = μ

    With λ = μ, Vp² = (λ + 2μ)/ρ = 3μ/ρ and Vs² = μ/ρ, so Vp/Vs = √3 ≈ 1.73. The ratio 3.00 is the tempting error: it is the ratio of the squares, not of the speeds. √2 holds only for λ = 0, and the ratio is not fixed at 2 for rocks.

  2. One earthquake is felt at Modified Mercalli intensity VII in a town 15 km from the epicentre and at intensity IV in a town 150 km away. Which statement is correct?

    1. The earthquake has one magnitude, while intensity is a local, observed measure that falls with distance and depends on the site
    2. The earthquake has two magnitudes, because magnitude is the logarithm of the shaking amplitude that was felt in each of the two towns
    3. Intensity is the logarithm of the radiated energy, so the nearer town received a thousand times more energy than the farther town
    4. The nearer town must have recorded the larger seismic moment, because seismic moment is measured where the damage is greatest
    Show answer

    Answer: A — The earthquake has one magnitude, while intensity is a local, observed measure that falls with distance and depends on the site

    Magnitude and seismic moment describe the source and are single numbers for the event. Intensity is assigned from observed effects at each place, so it differs between towns with distance, soil and construction. Treating intensity as a logarithm of energy is the tempting error: the scale is an ordinal description of effects and has no energy arithmetic.

  3. A digital seismograph samples at 50 samples per second. Its anti-alias filter is faulty and passes a 32 Hz component of ground motion. At what frequency does this component appear in the digitised record?

    1. At 32 Hz, because digitisation never changes the frequency of a component that lies below the sampling frequency
    2. At 25 Hz, the Nyquist frequency, where all of the energy above the Nyquist frequency is piled up in the record
    3. At 18 Hz, because 50 − 32 folds the component back about the Nyquist frequency into the band below 25 Hz
    4. At 7 Hz, because the excess of the component over the Nyquist frequency, 32 − 25, is what is recorded
    Show answer

    Answer: C — At 18 Hz, because 50 − 32 folds the component back about the Nyquist frequency into the band below 25 Hz

    The Nyquist frequency is 50/2 = 25 Hz. A component at f above it aliases to |f − k·fs|, here 50 − 32 = 18 Hz, the mirror image of 32 Hz about 25 Hz. The record then shows an 18 Hz signal that was never in the ground. The 7 Hz answer treats the excess over the Nyquist frequency as the alias, which would be the mirror distance and not the frequency.

  4. Which description fits the lower-hemisphere beach ball of a pure thrust (reverse) earthquake on a fault plane dipping at 45°, shaded black for compressional first motion?

    1. A white (dilatational) central region, because the P axis is vertical and plots at the centre of the diagram
    2. Four equal quadrants that meet at the centre, because both nodal planes of a thrust are vertical
    3. A completely black ball, because every station sees an upward first motion whenever the hanging wall of a thrust moves up
    4. A black (compressional) central region, because the T axis is vertical and so plots at the centre
    Show answer

    Answer: D — A black (compressional) central region, because the T axis is vertical and so plots at the centre

    In thrust faulting the least compressive (tension) axis is vertical and the P axis is horizontal. The T axis lies in a compressional quadrant, and its vertical direction plots at the centre of a lower-hemisphere diagram, so the centre is black. The white centre belongs to normal faulting, where P is vertical, and the four meeting quadrants belong to strike-slip faulting. A double couple always has dilatational quadrants as well, so the ball is never wholly black.

  5. A fault-plane solution from P first motions gives two perpendicular nodal planes. Which additional observation can identify which of them is the actual fault plane?

    1. Adding more stations with clear polarities, since the polarity pattern of a double couple differs between the two planes
    2. The alignment of well-located aftershocks, or of mapped surface rupture, along one of the two planes
    3. Finding which plane has the larger seismic moment, since the auxiliary plane carries none of the moment
    4. Comparing the amplitudes of the two P-wave lobes, since the fault plane is always the plane with the smaller lobe
    Show answer

    Answer: B — The alignment of well-located aftershocks, or of mapped surface rupture, along one of the two planes

    A double couple built from slip on one plane radiates exactly the same far field as slip on the auxiliary plane, so no number of first-motion polarities can separate them. Independent geometry can: the line of aftershocks, surface rupture, tectonic setting or rupture directivity. The other suggestions all assume an asymmetry between the planes that a double couple does not have.

  6. In a region log10 N = a − bM holds with b = 1.0, where N is the number of events of magnitude ≥ M. How do the numbers of events with M ≥ 5 and with M ≥ 7 compare?

    1. There are about 10 times as many events with M ≥ 5 as with M ≥ 7, one factor of ten for the whole range
    2. There are about 31.6 times as many, which is the factor by which radiated energy changes for each magnitude unit
    3. There are about 100 times as many events with M ≥ 5, as each of the two magnitude units gives a factor of ten
    4. There are about 1000 times as many, the factor by which radiated energy changes over two magnitude units
    Show answer

    Answer: C — There are about 100 times as many events with M ≥ 5, as each of the two magnitude units gives a factor of ten

    N(≥5)/N(≥7) = 10^(b·2) = 10² = 100 for b = 1. The factors 31.6 and 1000 are energy ratios for one and two magnitude units, and they do not enter the count ratio. Ten is the ratio for a single unit only.

  7. A ground-motion level at a site has a mean return period of 475 years, and its exceedances follow a Poisson process. The probability of at least one exceedance in a 475-year period is closest to which value?

    1. 63%, because P = 1 − exp(−t/T) with t = T gives 1 − e⁻¹
    2. 100%, because a level that is exceeded once per 475 years on average is certain to be exceeded in 475 years
    3. 50%, because half of the events must come sooner than the mean return period and half later than it
    4. 10%, because this is the usual design probability of exceedance attached to a return period of 475 years
    Show answer

    Answer: A — 63%, because P = 1 − exp(−t/T) with t = T gives 1 − e⁻¹

    For a Poisson process P = 1 − exp(−t/T), so with t = T, P = 1 − 1/e = 0.632. The 10% figure belongs to a 50-year window (1 − exp(−50/475) ≈ 0.10), not a 475-year one. A mean return period is not a guarantee, and the median waiting time T ln 2 ≈ 329 years is shorter than the mean, so 50% is also wrong.

  8. Which statement of the elastodynamic reciprocity theorem for the Green's function Gᵢⱼ(x, t; ξ, 0) is correct, with the same boundary conditions in both configurations?

    1. Gᵢⱼ(x, t; ξ, 0) = Gᵢⱼ(ξ, t; x, 0): the positions of source and receiver swap while the force and component directions are left unchanged
    2. Gᵢⱼ(x, t; ξ, 0) = −Gⱼᵢ(ξ, t; x, 0): the waveform changes sign when source and receiver exchange roles
    3. Gᵢⱼ(x, t; ξ, 0) = Gⱼᵢ(ξ, −t; x, 0): the roles of source and receiver swap together with a reversal of the time axis
    4. Gᵢⱼ(x, t; ξ, 0) = Gⱼᵢ(ξ, t; x, 0): source and receiver swap positions along with force and component directions
    Show answer

    Answer: D — Gᵢⱼ(x, t; ξ, 0) = Gⱼᵢ(ξ, t; x, 0): source and receiver swap positions along with force and component directions

    Reciprocity states Gᵢⱼ(x, t; ξ, 0) = Gⱼᵢ(ξ, t; x, 0): a force in direction j at ξ recorded in component i at x equals a force in direction i at x recorded in component j at ξ. Swapping positions alone, the tempting version, is true only when i = j. There is no sign change and no time reversal in the statement.

  9. Which statement correctly distinguishes the correlation-type from the convolution-type representation theorem of elastodynamics?

    1. The convolution type requires a lossless medium, whereas the correlation type remains valid in a medium with attenuation and loss
    2. The correlation type is exact only for plane waves in a homogeneous half-space, so it cannot be applied to recorded ambient noise
    3. The correlation type uses one time-reversed wavefield and needs a lossless medium; it gives Green's functions from cross-correlations
    4. The convolution type holds only for P waves in homogeneous media, whereas the correlation type holds for all wave types and structures alike
    Show answer

    Answer: C — The correlation type uses one time-reversed wavefield and needs a lossless medium; it gives Green's functions from cross-correlations

    The convolution-type theorem holds in time-invariant media, with or without attenuation, and gives seismogram = source ∗ Green's function. The correlation-type theorem has one wavefield time-reversed, which is allowed only when the medium is lossless (time-reversal invariant); it underlies seismic interferometry, in which the cross-correlation of recordings at two receivers yields the Green's function between them. The statement that reverses the lossless requirement is the tempting trap.

  10. Which of the following statements about magnitude are correct?

    1. Moment magnitude is derived from the seismic moment M0 = μAD and does not saturate for the largest earthquakes
    2. Two events recorded on identical instruments at the same distance with a peak-amplitude ratio of 100 differ by two magnitude units
    3. A difference of two magnitude units corresponds to a 100-fold difference in the seismic energy radiated by the two earthquakes
    4. Modified Mercalli intensity is a single number that characterises the whole earthquake, as magnitude does, whichever place it is observed at
    Show answer

    Answer: A — Moment magnitude is derived from the seismic moment M0 = μAD and does not saturate for the largest earthquakes; B — Two events recorded on identical instruments at the same distance with a peak-amplitude ratio of 100 differ by two magnitude units

    Mw comes from M0 and does not saturate, and ΔM = log10(A1/A2) = log10(100) = 2 for the same instrument and distance. Energy changes by 10^(1.5 × 2) = 1000 for two units, not 100. Intensity is local and observed, so one earthquake has many intensity values but one magnitude.

  11. Which of the following statements about seismotectonic settings are correct?

    1. Earthquakes along the axes of mid-ocean ridges are shallow and show mainly normal faulting
    2. Oceanic transform faults produce mainly strike-slip earthquakes
    3. Subduction zones host the deepest earthquakes, which extend to about 700 km
    4. Deep-focus earthquakes between 300 and 700 km depth are common beneath mid-ocean ridge axes
    Show answer

    Answer: A — Earthquakes along the axes of mid-ocean ridges are shallow and show mainly normal faulting; B — Oceanic transform faults produce mainly strike-slip earthquakes; C — Subduction zones host the deepest earthquakes, which extend to about 700 km

    Ridge axes are extensional and shallow (normal faulting), transforms slide horizontally (strike-slip), and the Wadati–Benioff zone of a subducting slab carries earthquakes to about 700 km. Beneath a ridge the lithosphere is thin and hot, so no earthquakes occur at 300 to 700 km there; deep events need cold, sinking slabs.

  12. Which of the following statements about the theorems of elastodynamics are correct?

    1. For a given body force, initial conditions and boundary data (displacement or traction), the displacement field in the body is unique
    2. By reciprocity, a force in direction j at A recorded in component i at B equals a force in direction i at B recorded in component j at A
    3. The uniqueness theorem proves that a given set of seismograms can have been produced by only one source, however the stations are laid out
    4. The convolution-type representation theorem holds only if the medium is lossless, with no attenuation of any kind
    Show answer

    Answer: A — For a given body force, initial conditions and boundary data (displacement or traction), the displacement field in the body is unique; B — By reciprocity, a force in direction j at A recorded in component i at B equals a force in direction i at B recorded in component j at A

    Uniqueness and reciprocity are stated correctly. Uniqueness concerns the forward problem (given the sources, the field is unique); it says nothing about recovering the source from the field, where different sources can give indistinguishable records. The convolution-type theorem holds also with attenuation; the lossless requirement belongs to the correlation type.

  13. A station records the S arrival 10.0 s after the P arrival. Take uniform speeds Vp = 6.0 km/s and Vs = 3.5 km/s, and a focal depth of 36 km. The epicentral distance of the station, in km to one decimal place, is ____.

    Numerical answer — type the value.

    Show answer

    Answer: 75.9

    Focal distance d = Δt · Vp·Vs/(Vp − Vs) = 10 × (6.0 × 3.5)/2.5 = 10 × 8.4 = 84 km. The epicentral distance removes the depth: √(84² − 36²) = √(7056 − 1296) = √5760 = 75.9 km. Quoting 84 km forgets the depth step.

  14. A rupture is 20 km long and 10 km wide, with average slip 1.5 m and rigidity 3.0 × 10¹⁰ Pa. Using Mw = (2/3)(log10 M0 − 9.1) with M0 in N·m, the moment magnitude to one decimal place is ____.

    Numerical answer — type the value.

    Show answer

    Answer: 6.6

    A = 20 000 × 10 000 = 2.0 × 10⁸ m². M0 = μAD = 3.0 × 10¹⁰ × 2.0 × 10⁸ × 1.5 = 9.0 × 10¹⁸ N·m, so log10 M0 = 18.954 and Mw = (2/3)(18.954 − 9.1) = 6.57, that is 6.6. Entering the lengths in km without converting to metres is the usual slip.

  15. In a region the annual number of earthquakes of magnitude ≥ M follows log10 N = 4.5 − 0.9 M. The mean recurrence interval of earthquakes with M ≥ 6.5, in years to the nearest whole year, is ____.

    Numerical answer — type the value.

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    Answer: 22

    log10 N = 4.5 − 0.9 × 6.5 = 4.5 − 5.85 = −1.35, so N = 10^−1.35 = 0.0447 per year. The recurrence interval is T = 1/N = 10^1.35 = 22.4 years, which is 22 years to the nearest year. Using N itself (0.045) as the interval is the usual slip.

  16. Two earthquakes are recorded at the same distance on identical Wood–Anderson seismographs. The peak trace amplitude of the larger event is 250 times that of the smaller. The difference in local magnitude ML, to two decimal places, is ____.

    Numerical answer — type the value.

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    Answer: 2.40

    With the same instrument and distance the distance term cancels, so ΔML = log10(A1/A2) = log10 250 = 2.398, that is 2.40. Using the energy relation (a factor 10^1.5 per unit) instead of the amplitude ratio would give a different, wrong number.

  17. For an earthquake the S-wave speed is β = 3.5 km/s, the corner frequency is fc = 0.20 Hz and the seismic moment is M0 = 5.0 × 10¹⁸ N·m. Use r = 0.37 β/fc for the source radius and Δσ = 7 M0/(16 r³) for the stress drop. The stress drop in MPa, to the nearest whole number, is ____.

    Numerical answer — type the value.

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    Answer: 8

    r = 0.37 × 3500/0.20 = 6475 m. Then Δσ = 7 × 5.0 × 10¹⁸/(16 × 6475³) = 3.5 × 10¹⁹/(16 × 2.715 × 10¹¹) = 8.06 × 10⁶ Pa, which is 8 MPa. Leaving r in km while M0 is in N·m gives a result off by a factor 10⁹.