8.1 Elastic Rebound Theory
Reid's Discovery (1910)
Following the devastating 1906 San Francisco earthquake (Mw 7.9), Harry Fielding Reid analyzed geodetic survey data collected before and after the rupture along the San Andreas Fault. He observed that markers on opposite sides of the fault had been gradually displaced over decades prior to the earthquake, accumulating elastic strain in the surrounding rock. At the moment of failure, the fault slipped suddenly, and the strained rock “rebounded” elastically to an approximately unstrained configuration.
This observation led Reid to propose the elastic rebound theory, which remains the foundational model for understanding shallow earthquakes. The theory posits that tectonic forces slowly deform rocks on either side of a locked fault, storing elastic strain energy. When the accumulated shear stress exceeds the frictional strength of the fault, sudden slip occurs, releasing the stored energy as seismic waves and heat.
The 1906 rupture extended approximately 470 km along the San Andreas Fault, with maximum surface offsets reaching 6–7 m near Olema, north of San Francisco. Triangulation surveys showed that points as far as 30 km from the fault had been displaced by the gradual interseismic strain accumulation.
The Earthquake Cycle
Modern geodesy (GPS, InSAR) has confirmed and refined Reid's model into a four-phase earthquake cycle. Each phase is characterized by distinct deformation patterns observable at Earth's surface:
1. Interseismic Phase (decades to millennia)
The fault is locked from the surface down to the locking depth (typically 10–25 km), while the deeper fault and surrounding asthenosphere creep steadily. Elastic strain accumulates in the upper crust at a rate governed by the far-field plate velocity. GPS stations near the fault record a smooth, arctangent-shaped velocity profile across the fault.
2. Preseismic Phase (days to years?)
Some faults exhibit accelerating creep, foreshock sequences, or anomalous strain transients before large earthquakes. However, reliable precursors remain elusive. The 2011 Tōhoku earthquake (Mw 9.1) was preceded by a M 7.3 foreshock two days earlier, but most large earthquakes lack identifiable preseismic signals. This phase remains the most uncertain and controversial aspect of the cycle.
3. Coseismic Phase (seconds to minutes)
Sudden fault slip releases accumulated strain energy. Rupture propagates at 2–3 km/s (sub-Rayleigh) or occasionally at supershear velocities (> 3.5 km/s). The coseismic displacement field, measurable by GPS and InSAR, provides constraints on the slip distribution through inverse modeling using elastic dislocation theory.
4. Postseismic Phase (months to decades)
After the earthquake, transient deformation continues through three mechanisms: (1) afterslip on and around the rupture zone, (2) viscoelastic relaxation of the lower crust and upper mantle, and (3) poroelastic rebound as pore fluids re-equilibrate. These processes are distinguishable by their spatial patterns and time constants. Afterslip dominates in the first months; viscoelastic relaxation operates over decades.
Interseismic Strain Accumulation
During the interseismic period, the locked fault concentrates deformation in a zone whose width scales with the locking depth. The strain rate at the fault trace is highest and decreases with distance. For an infinitely long, vertical strike-slip fault, the interseismic velocity field is well described by the screw dislocation model of Savage & Burford (1973):
Interseismic Velocity Profile
\[ v(x) = \frac{v_{\text{plate}}}{\pi} \arctan\!\left(\frac{x}{d}\right) \]
where v(x) is the fault-parallel velocity at perpendicular distance x from the fault, vplate is the full plate velocity across the fault, and d is the locking depth. The strain rate at the fault trace is \( \dot{\varepsilon} = v_{\text{plate}} / (\pi \, d) \).
Interseismic Strain Rate
\[ \dot{\varepsilon} = \frac{v_{\text{plate}}}{2 \, W} \]
where W is the effective locking width (related to the locking depth and fault geometry). For the San Andreas Fault with vplate ≈ 35 mm/yr and d ≈ 15 km, the strain rate at the fault is approximately 0.7 μstrain/yr, consistent with strainmeter observations.
Coseismic Stress Drop
The stress drop during an earthquake — the difference between the initial shear stress on the fault and the final (residual) stress after slip — is a fundamental source parameter. Despite the enormous range of earthquake magnitudes (spanning 17 orders of magnitude in moment), stress drops are remarkably constant:
Stress Drop Relation
\[ \Delta\sigma = C \, \frac{\mu \, \bar{D}}{L} \]
where Δσ is the static stress drop, μ is the shear modulus (~30 GPa for crustal rocks), D̄ is the average slip, L is the fault length, and C is a geometric constant of order unity depending on fault shape. Typical values range from 1–10 MPa for tectonic earthquakes, with a median near 3 MPa.
The near-constancy of stress drop is a key observation in earthquake physics. It implies self-similarity: small and large earthquakes are mechanically similar processes, differing only in the area that ruptures, not in the intensity of stress release. This self-similarity has important implications for seismic hazard assessment and the scaling of ground motion with magnitude.
Elastic Dislocation Models
The coseismic displacement field at Earth's surface can be computed analytically using dislocation theory in an elastic half-space. The foundational solutions were derived by Okada (1985), who provided closed-form expressions for surface displacements, strains, and tilts due to a rectangular fault with uniform slip embedded in a homogeneous elastic half-space.
Okada's solutions form the basis for geodetic earthquake source inversions: observed GPS displacements or InSAR interferograms are fit by summing contributions from many sub-faults, each with adjustable slip, to recover the spatially variable slip distribution on the rupture plane. These finite-fault models are routinely produced for significant earthquakes (M > 6) within hours of occurrence.
Key Assumptions of Okada's Model
- Homogeneous, isotropic, elastic half-space (no layering or lateral heterogeneity)
- Planar, rectangular fault geometry with uniform or piecewise-uniform slip
- Static equilibrium (no wave propagation effects)
- Small-strain approximation (valid for slip/distance ratios < ~0.01)
Seismic Coupling & Slip Deficit
Not all plate-boundary slip occurs seismically. The seismic coupling coefficient (χ) is defined as the fraction of total relative plate motion accommodated by earthquakes:
\( \chi = \frac{\sum M_0}{\mu \, A \, v_{\text{plate}} \, T} \)
where the numerator sums seismic moments over time interval T, and the denominator represents the moment expected if all slip were seismic. Values range from χ ≈ 0 (aseismic creeping faults like the Hayward Fault) to χ ≈ 1 (fully locked subduction megathrusts like Cascadia and the Nankai Trough).
| Fault / Boundary | Coupling χ | Slip Rate (mm/yr) | Behavior |
|---|---|---|---|
| Chile subduction zone | ~0.9 | 65–80 | Strongly locked |
| Cascadia megathrust | ~1.0 | 35–40 | Fully locked |
| San Andreas (Parkfield) | ~0.5 | 34 | Partially creeping |
| Hayward Fault (central) | ~0.1 | 9 | Mostly creeping |
| Mariana subduction | ~0.0 | 40–70 | Aseismic |
Coulomb Stress Transfer & Triggering
When an earthquake ruptures, the resulting stress redistribution can promote or inhibit failure on nearby faults. This interaction is quantified by the change in Coulomb failure function (CFF):
Coulomb Failure Function Change
\[ \Delta \text{CFF} = \Delta\tau + \mu' \, \Delta\sigma_n \]
where Δτ is the change in shear stress resolved in the slip direction (positive promotes failure), Δσn is the change in normal stress (positive = unclamping), and μ′ is the effective coefficient of friction (typically 0.2–0.6, accounting for pore pressure effects). Failure is promoted when ΔCFF > 0.
Stress Shadows
Regions where ΔCFF < 0 experience reduced seismicity rates after a mainshock. These “stress shadows” can suppress earthquake activity for decades. The 1906 San Francisco earthquake created a stress shadow that reduced seismicity rates along the central San Andreas for most of the 20th century.
Stress Bright Spots
Regions where ΔCFF > 0 (even by as little as 0.01 MPa) show enhanced seismicity. The 1992 Landers earthquake (Mw 7.3) triggered the 1999 Hector Mine earthquake (Mw 7.1) on a nearby fault loaded by the Coulomb stress transfer. Aftershock distributions consistently align with positive ΔCFF lobes.
Earthquake Recurrence Models
Two fundamentally different models describe how earthquakes recur on individual faults:
Characteristic Earthquake Model
Proposed by Schwartz & Coppersmith (1984). A given fault segment repeatedly produces earthquakes of approximately the same size at quasi-regular intervals. The recurrence time is estimated from:
\( T_{\text{recur}} = \frac{\bar{D}}{v_{\text{slip}}} \)
where D̄ is the characteristic slip and vslip is the long-term fault slip rate. For the Wasatch Fault: D̄ ≈ 2 m, v ≈ 1.5 mm/yr ⇒ T ≈ 1300 yr.
Gutenberg–Richter Model
Earthquakes on a fault follow a power-law frequency–size distribution at all magnitudes. There is no preferred or “characteristic” size. Slip occurs in a self-similar cascade of events from small to large. This model is well supported for regional seismicity but debated for individual fault segments.
Recent paleoseismic data suggest that many faults exhibit behavior intermediate between these two end-members, with some tendency toward quasi-periodic large events superimposed on a background GR distribution.
Key Takeaways
- Elastic rebound theory (Reid, 1910) explains earthquakes as sudden release of gradually accumulated elastic strain
- The earthquake cycle comprises interseismic, preseismic, coseismic, and postseismic phases
- Interseismic velocity fields follow an arctangent profile governed by the fault locking depth
- Coseismic stress drops are remarkably constant at 1–10 MPa across all earthquake sizes (self-similarity)
- Okada (1985) elastic half-space solutions form the basis of modern geodetic source inversions
- Seismic coupling varies from 0 (aseismic) to 1 (fully locked) and controls hazard potential
- Coulomb stress transfer (ΔCFF) explains aftershock patterns, stress shadows, and earthquake triggering