8.3 Focal Mechanisms
The Double-Couple Source
Shear faulting on a planar surface within the Earth produces a characteristic radiation pattern that can be represented by a double couple — two equal and opposite force couples acting at the source point. This equivalent body-force system produces the same far-field seismic radiation as the actual slip on the fault, without requiring any net force or net torque (consistent with conservation of momentum).
The double-couple model was established through the work of Maruyama (1963) and Burridge & Knopoff (1964), who showed that a shear dislocation on a fault is mathematically equivalent to a system of body forces with no net resultant. This equivalence is fundamental: it means that seismograms alone cannot distinguish between slip on a fault plane and slip on its conjugate (auxiliary) plane — the so-called fault-plane ambiguity.
The P-wave radiation pattern from a double couple has four lobes: two compressional (outward first motion) and two dilatational (inward first motion), separated by two orthogonal nodal planes. One nodal plane is the actual fault plane; the other is the auxiliary plane. Distinguishing between them requires additional information such as mapped surface rupture, aftershock distributions, or directivity effects.
The Seismic Moment Tensor
The most general point-source representation of an earthquake is the moment tensor Mij, a symmetric 3×3 matrix that fully describes the equivalent forces at the source. For a shear dislocation on a planar fault:
Moment Tensor Components
\[ M_{ij} = \mu \, A \left( s_i \, n_j + s_j \, n_i \right) \]
where μ is the shear modulus, A is the fault area, s is the unit slip vector (direction of hanging-wall motion), and n is the unit normal to the fault plane. The tensor is symmetric (Mij = Mji), so it has at most 6 independent components. For a pure double couple, the trace is zero (Mkk = 0) and the determinant is zero.
Scalar Seismic Moment
\[ M_0 = \mu \, A \, \bar{D} \]
where M0 is the scalar seismic moment (units: N·m), μ ≈ 30 GPa (crust) to 70 GPa (upper mantle), A is the rupture area, and D̄ is the average slip. The scalar moment is the most physically meaningful measure of earthquake size, directly proportional to the total work done against friction. It spans from ~106 N·m for microearthquakes to ~1023 N·m for the 1960 Chile earthquake.
Beach Ball Diagrams
The focal mechanism or “beach ball” is a stereographic projection of the P-wave first-motion pattern onto the lower-hemisphere focal sphere. Compressional first motions (upward P-wave onset) are plotted as filled (dark) quadrants, and dilatational first motions (downward onset) as open (white) quadrants. The boundaries between quadrants are the two nodal planes.
P-Wave First Motion Polarity
At each seismic station, the first arriving P-wave is either compressional (ground motion initially away from the source, upward on a vertical seismogram for a surface station) or dilatational (ground motion initially toward the source, downward on a vertical seismogram). By plotting the polarity observed at each station on the focal sphere according to its takeoff angle and azimuth from the source, the four-lobed radiation pattern emerges and the two nodal planes can be fit.
Historically, focal mechanisms were determined from hand-picked P-wave first motions at many stations. Modern practice uses full waveform inversion of broadband seismograms to determine the complete moment tensor, which is more robust and provides additional information such as the non-double-couple component.
Recognizing Fault Types from Beach Balls
Normal Fault
The T-axis (tension, or minimum compressive stress) is approximately vertical. On the beach ball, the center is white (dilatational) and the sides are filled. The filled quadrants appear as two “wings” on the left and right. This pattern reflects extensional tectonics, as seen along mid-ocean ridges and continental rifts.
Example: Normal faulting earthquakes along the East African Rift, the Basin and Range Province, and along the axes of mid-ocean ridges.
Thrust (Reverse) Fault
The P-axis (pressure, or maximum compressive stress) is approximately vertical. The beach ball shows a filled center with white (dilatational) quadrants on the sides. This pattern arises from horizontal compression, characteristic of subduction zones and collisional plate boundaries.
Example: Megathrust earthquakes at subduction zones (2004 Sumatra Mw 9.1, 2011 Tōhoku Mw 9.1), and crustal thrust events in collisional belts (Himalayas, Zagros, Andes).
Strike-Slip Fault
Both P and T axes are approximately horizontal. The beach ball shows alternating filled and white quadrants in an X-pattern, with the null (B) axis vertical. Right-lateral and left-lateral mechanisms are distinguished by the orientation of the filled quadrants relative to the mapped fault trace.
Example: San Andreas Fault earthquakes (right-lateral), North Anatolian Fault (right-lateral), Alpine Fault of New Zealand (right-lateral), Dead Sea Transform (left-lateral).
Oblique Mechanisms
Many earthquakes have oblique mechanisms combining components of strike-slip with either normal or thrust faulting. The San Andreas system, for example, includes transpressional segments (strike-slip + thrust) in the Big Bend region and transtensional segments (strike-slip + normal) in the Salton Trough. Oblique mechanisms appear as rotated or asymmetric beach balls.
Global CMT Catalog
The Global Centroid-Moment-Tensor (CMT) project (originally the Harvard CMT project, established by Adam Dziewonski in 1976; now maintained by Göran Ekström at Columbia University) routinely determines full moment tensors for all earthquakes with Mw ≥ 5.0 worldwide. The catalog contains over 60,000 solutions dating back to 1976 and is updated monthly.
Centroid vs. Hypocenter
The hypocenter is the point where rupture nucleates, typically determined from arrival times of the first P-waves. The centroid is the average location and time of moment release, determined by fitting long-period waveforms. For large earthquakes, the centroid can be tens to hundreds of kilometers from the hypocenter. For the 2004 Sumatra earthquake, the hypocenter was near the southern end of the rupture, while the centroid was located ~500 km to the north, near the center of the ~1300-km-long rupture.
CMT Inversion Method
The CMT algorithm inverts long-period (T > 40 s) body and surface waveforms recorded at global seismic stations. It solves simultaneously for the six moment-tensor components, the centroid location (latitude, longitude, depth), and the centroid time. The solution uses a 3-D Earth model for synthetic seismograms, ensuring accurate path corrections. Uncertainty estimates are provided for all parameters.
Non-Double-Couple Components
While most tectonic earthquakes are well described by pure double-couple sources, the full moment tensor can contain non-double-couple (non-DC) components. Any symmetric moment tensor can be decomposed into three parts:
- Isotropic component: Volume change (explosion or implosion). Zero for tectonic earthquakes; nonzero for nuclear explosions, volcanic events, and mine collapses.
- Double-couple (DC): Pure shear faulting on a plane. The dominant component for most tectonic earthquakes.
- Compensated Linear Vector Dipole (CLVD): A non-shear deviatoric component with no volume change. Can arise from simultaneous slip on non-planar fault geometries, triggered slip on multiple sub-faults, or opening-mode (tensile) cracks. Significant CLVD components are observed for deep-focus earthquakes, volcanic events, and geothermal/induced seismicity.
| Source Type | Isotropic | DC (%) | CLVD (%) | Example |
|---|---|---|---|---|
| Tectonic (shallow) | ~0 | 85–100 | 0–15 | San Andreas, megathrust events |
| Deep-focus (> 300 km) | ~0 | 60–90 | 10–40 | 1994 Bolivia Mw 8.2 |
| Volcanic | Variable | 30–80 | 20–70 | Caldera collapse events |
| Nuclear explosion | Dominant | 0–30 | 0–20 | Underground nuclear tests |
| Induced / geothermal | Small | 50–90 | 10–50 | Wastewater injection, geothermal |
Regional Stress from Focal Mechanism Inversions
While individual focal mechanisms constrain the orientation of the P and T axes (which approximate but do not exactly equal the maximum and minimum principal stress directions), the regional stress tensor can be determined by inverting a population of focal mechanisms. The method, developed by Gephart & Forsyth (1984) and Michael (1984), finds the stress tensor that best explains the observed diversity of fault-plane orientations and slip directions in a region.
The inversion determines four parameters: the orientations of the three principal stress axes (σ1 ≥ σ2 ≥ σ3) and the stress ratio R = (σ1 − σ2) / (σ1 − σ3). Absolute stress magnitudes cannot be determined from focal mechanisms alone. The assumption underlying the method is that all faults in the region experience the same uniform stress field — a reasonable approximation for regions smaller than ~100 km.
Applications include mapping the transition from compressive to extensional stress regimes across orogens, identifying stress rotations near major faults (e.g., low shear stress on the San Andreas), and monitoring temporal stress changes associated with volcanic unrest or reservoir impoundment. The World Stress Map project compiles stress orientations from focal mechanisms, borehole breakouts, and hydraulic fracturing data worldwide.
Key Takeaways
- Shear faulting produces a double-couple radiation pattern with four P-wave polarity lobes
- The moment tensor Mij = μA(sinj + sjni) fully characterizes the source
- Scalar seismic moment M0 = μAD̄ is the most physically meaningful size measure
- Beach balls project the P-wave first-motion pattern onto the focal sphere: filled = compressional
- Normal faults: T-axis vertical (white center); thrust faults: P-axis vertical (filled center); strike-slip: X-pattern
- The Global CMT catalog provides moment tensors for all M ≥ 5 earthquakes since 1976
- Non-double-couple components (CLVD, isotropic) arise from complex or non-shear sources
- Focal mechanism inversions constrain the regional stress tensor orientation and shape ratio