Lecture: Earthquake Source Dynamics

Lecture: Earthquake Source Dynamics#

Learning objectives#

  • Directivity: Explain how rupture velocity \(V_r\) deforms observed STFs and biases inferred corner frequency with azimuth.

  • Stress drop from spectra: Connect corner frequency \(f_c\) to source radius \(r\) and show why the Brune \(k\)-factor and \(V_r\) contaminate \(\Delta\sigma\).

  • Radiated energy & apparent stress: Define \(E_R\) and \(\sigma_a = \mu E_R / M_0\) as dynamic measures of the energy budget.

  • Energy budget: Write the balance \(\Delta E_\text{strain} = E_R + E_G + E_H\) and identify where fracture energy \(G\) enters.

  • Modern context: Recognize that much scatter in catalog stress drops reflects inversion assumptions and STF complexity rather than physical variability.

📖 Shearer reference: Sections 9.7 (source spectra) and 10.3 (radiated energy)

1. The observable: source time function and source spectrum#

The source time function (STF) \(\dot{M}(t)\) is the moment-rate function recovered after removing path and site effects from observed waveforms. Its Fourier transform is the source spectrum:

\[ \tilde{\dot{M}}(f) = \int_{-\infty}^{\infty} \dot{M}(t)\, e^{-2\pi i f t}\, dt. \]

The \(\omega\)-square (Brune) model predicts that the displacement spectrum \(|\tilde{u}(f)|\) is flat below the corner frequency \(f_c\) and falls as \(f^{-2}\) above it. The long-period plateau gives \(M_0\); the corner frequency \(f_c\) encodes source size.

Note

The mapping STF duration → source size is only valid under the simplest rupture models. Real STFs are more complex. The companion notebook lets you explore what “complexity” does to \(f_c\) estimates.

2. Rupture velocity and directivity#

For a unilateral rupture propagating at velocity \(V_r\), observers in different directions see different STF durations. A minimal Doppler-like relation is:

\[ \boxed{T_\text{obs}(\theta) \approx T_0\!\left(1 - \frac{V_r}{c}\cos\theta\right), \qquad f_{c,\text{obs}}(\theta) \sim \frac{1}{T_\text{obs}(\theta)}} \]

where \(c\) is the relevant wave speed (\(\beta\) for S-wave intuition) and \(\theta\) is the angle from the rupture propagation direction.

\(V_r\)

\(\theta = 0°\) (forward)

\(\theta = 180°\) (backward)

\(V_r \to 0\)

\(T_\text{obs} \approx T_0\), no bias

same

\(V_r = 0.6\,c\)

\(T_\text{obs} = 0.4\,T_0\) (compressed)

\(T_\text{obs} = 1.6\,T_0\) (stretched)

Warning

If you ignore directivity, a higher \(f_c\) observed in the forward direction will be misinterpreted as a smaller source radius and higher stress drop. This is one of the most common sources of systematic error in spectral stress-drop studies [Ji et al., 2022].

The directivity signature is azimuthal variation of \(f_c\): a single event shows systematically higher \(f_c\) in the forward direction and lower \(f_c\) in the backward direction, while the long-period plateau (\(M_0\)) remains the same.

3. Stress drop from corner frequency#

For a circular rupture the Brune-style scaling relates \(M_0\), source radius \(r\), and stress drop \(\Delta\sigma\):

\[ \Delta\sigma \approx \frac{7}{16}\,\frac{M_0}{r^3}, \qquad r \approx k\,\frac{\beta}{f_c}. \]

Because \(\Delta\sigma \propto f_c^3\), a 20% overestimate of \(f_c\) inflates \(\Delta\sigma\) by \((1.20)^3 \approx 1.7\times\).

Note

Kilgore and Beeler [2025] showed using laboratory experiments that frictional slip unsteadiness introduces apparent scatter in spectral stress-drop even when the true on-fault stress drop is controlled. Lin et al. [2025] demonstrated numerically that for rate-and-state fault simulations the Brune \(k\) factor is not constant — it depends on the dynamic rupture trajectory.

Key message: Spectral stress drop is powerful but is systematically sensitive to the choice of \(k\), \(V_r\), and the assumed spectral model. Ji et al. [2022] showed that incorporating radiated energy \(E_R\) constrains this degeneracy.

4. Radiated energy and apparent stress#

Radiated energy \(E_R\) is the total seismic energy carried away in far-field waves and proportional to moment. To compare \(E_R\) with \(\Delta\sigma\), we define the apparent stress:

\[ \boxed{\sigma_a \equiv \mu\,\frac{E_R}{M_0}.} \]

\(\sigma_a\) is a dynamic measure: it is sensitive to high-frequency radiation and to energy partitioning at the fault. Two earthquakes with the same \(M_0\) but different rupture styles can have very different \(\sigma_a\).

Note

Observations consistently show \(\sigma_a \ll \overline{\Delta\sigma}\) for natural earthquakes, implying most released strain energy goes to heat and fracture rather than radiation. Laboratory stick-slip events with controlled friction can approach higher radiation efficiencies [Kilgore and Beeler, 2025].

5. Energy budget#

A minimal energy balance for a seismic rupture is:

\[ \boxed{\Delta E_\text{strain} = E_R + E_G + E_H} \]

where \(E_G\) is fracture energy (breakdown work at the rupture front) and \(E_H\) is frictional heat (far-field dissipation). The radiation efficiency \(\eta_R = E_R / (E_R + E_G)\) quantifies how much of the non-heat energy is radiated.

In the slip-weakening framework, fracture energy per unit area is:

\[ G = \int_0^{D_c} \!\left[\tau(\delta) - \tau_r\right] d\delta, \]

where \(D_c\) is the weakening distance, \(\tau(\delta)\) is the shear strength as slip accumulates, and \(\tau_r\) is the residual (dynamic) frictional stress. Cocco et al. [2023] provides a comprehensive review of how \(G\) is measured and how it scales with event size.

Warning

“Missing energy” in the budget — the gap between \(\Delta E_\text{strain}\) and \(E_R\) — is often large and poorly constrained. It reflects our incomplete knowledge of \(D_c\) and the friction law operating during rapid, large-amplitude slip.

6. STF complexity and its observational consequences#

Natural STFs are rarely simple Brune pulses. Sub-event complexity (multiple asperity ruptures, stopping phases, healing fronts) modifies the spectrum above \(f_c\):

  • Adding a second sub-event separated by \(\Delta t\) creates spectral notches at \(f = n / \Delta t\).

  • The apparent \(f_c\) estimated from a single-corner fit may be biased high or low.

  • Neely et al. [2024] showed that STF complexity systematically biases stress-drop estimates and can generate apparent magnitude trends that are in fact complexity trends.

Implication: reported trends of increasing \(\Delta\sigma\) with magnitude in some catalogs may partly reflect the greater complexity of large events rather than physical variation in on-fault conditions.

7. Research Relevance#

The three core observables — corner frequency / directivity, stress drop, and radiated energy — are each the focus of recent work, which shows they are not equally direct windows into rupture physics.

7.1 Directivity and what STFs/spectra actually measure#

Rupture velocity \(V_r\) causes directivity: forward stations see shorter apparent durations and higher \(f_c\); backward stations see the opposite. Recent work shows source complexity produces the same signatures:

  • Observed: Neely et al. [2024] show that real STFs are more complex than a single Brune pulse, and this complexity degrades agreement between time-domain and frequency-domain stress-drop methods.

  • Simulated: Lin et al. [2025] simulate elongated, unilateral, and ring-like ruptures and find that inferred \(f_c\) depends strongly on source geometry and station coverage — not just \(V_r\).

  • Laboratory: Kilgore and Beeler [2025] provide a control case with sources too small for sustained directivity, isolating the spectral signature of local frictional dynamics and energy partition.

Warning

A short STF pulse is not automatically a fast rupture — source complexity and geometry produce the same observational signature as high \(V_r\).

7.2 Stress drop: useful, but model dependent#

The standard route

\[ M_0 , f_c \to r \approx k\,\frac{\beta}{f_c} \to \Delta\sigma \sim \frac{M_0}{r^3} \]

yields an inference, not a direct observable.

  • Observed: Ji et al. [2022] show \(\Delta\sigma\) estimates differ by factors of several across source spectral models. Neely et al. [2024] show STF complexity further degrades agreement between time- and frequency-domain methods.

  • Simulated: Lin et al. [2025] find true fault stress drops of 1.5–5 MPa yet seismologically inferred values spanning 0.01–100 MPa; second-moment methods outperform simple spectral fitting.

  • Laboratory: Kilgore and Beeler [2025] show that even with controlled geometry the measured spectral stress drop depends on how the source populates the spectrum (\(f^{-1}\) vs \(f^{-2}\) behavior).

Key message: Spectral stress drop is a model-dependent summary statistic, not a uniquely determined physical property.

7.3 Radiated energy and the energy budget#

Apparent stress \(\sigma_a = \mu E_R / M_0\) is a less model-dependent anchor than \(f_c\)-based stress drop:

  • Observed: Ji et al. [2022] show that different spectral models imply broadly similar radiation efficiencies even when their \(\Delta\sigma\) estimates differ — \(E_R\) stabilizes interpretation.

  • Simulated: Lin et al. [2025] compute on-fault stress-drop averages directly, enabling comparison of what the fault did versus what far-field waveforms imply.

  • Laboratory: Kilgore and Beeler [2025] infer that ≥95% of released energy does not escape to the far field; high-frequency spectral richness reflects energy partitioning, not just source size.

Cocco et al. [2023] frame the full balance \(\Delta W = E_R + E_{FZ}\) and partition the fault-zone term into rupture-propagation, on-fault, and off-fault contributions. In the linear slip-weakening framework the energy dissipated at the rupture front is:

\[ G = \tfrac{1}{2}(\tau_p - \tau_r)\,D_c. \]

Key message: \(E_R\) and \(\sigma_a\) bridge spectra to the energy budget; missing energy partitions into on-fault dissipation, off-fault damage, and fracture energy.

7.4 Synthesis#

Perspective

Paper

Key result

Observed

Neely et al. [2024]

STF complexity biases \(f_c\) and \(\Delta\sigma\); creates apparent magnitude trends

Simulated

Lin et al. [2025]

Most catalog \(\Delta\sigma\) scatter is consistent with geometry/complexity artifacts

Laboratory

Kilgore and Beeler [2025]

Earthquake-like spectra emerge when ≥95% of energy stays near the source

Energy budget

Cocco et al. [2023]

Non-radiated energy partitions into on-fault dissipation, off-fault damage, and fracture energy

Check your understanding#

  1. A network observes \(f_c = 0.5\) Hz in the forward direction and \(f_c = 0.2\) Hz in the backward direction for the same event. Estimate \(V_r / c\) assuming unilateral rupture.

  2. \(f_c\) is overestimated by 30% due to directivity. By what factor is \(\Delta\sigma\) overestimated?

  3. Two \(M_w\, 6\) earthquakes share the same \(M_0\) but one has \(\sigma_a\) three times larger. What must differ?

  4. In the energy budget, if \(E_G\) increases while \(M_0\) is held fixed, what happens to \(\eta_R\)?

  5. Explain in one sentence why catalog scatter in \(\Delta\sigma\) does not necessarily imply physical variability in on-fault stress release.

What we deliberately did not cover#

  • Full waveform inversion for dynamic rupture parameters

  • Rate-and-state friction derivations and thermal pressurization models of \(D_c\) (the instability mechanism and slip-weakening framework are introduced in Module 8)

  • Off-fault damage effects on the energy budget

  • Finite-fault slip models and their relationship to spectral \(f_c\)

Looking ahead#

The companion notebook (Lab 7d) lets you build synthetic STFs, apply the toy-directivity warp, estimate \(f_c\) from spectra, compute a radiated-energy proxy, and explore the energy-budget sandbox — all with the equations developed here.

In Module 8 we turn from observing source dynamics to explaining them: why does slip become unstable, what controls the nucleation zone size, and what determines whether rupture propagates or arrests.

Reading#

📖 Shearer: Sections 9.5, 9.6

  • Cocco et al. [2023] — fracture energy and breakdown work (comprehensive review)

  • Ji et al. [2022] — reconciling spectral stress-drop variability via \(E_R\)

  • Kilgore and Beeler [2025] — laboratory stress drop and spectral estimates

  • Lin et al. [2025] — dynamic simulation stress-drop estimates on rate-and-state faults

  • Neely et al. [2024] — STF complexity and stress-drop bias

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