Earthquake Phenomena II

Learning objectives

By the end of today, you will be able to:

• [LO-14.1] Explain why magnitude scales use base-10 logarithms and convert magnitude differences into amplitude / moment / energy ratios.
• [LO-14.2] Distinguish $M_L$, $m_b$, $M_S$, and $M_W$ — phase, period, instrument, and saturation behaviour.
• [LO-14.3] Apply $M_0 = \mu A \bar{s}$ and the Gutenberg-Richter and Omori laws to interpret a catalogue.

Course LOs: LO-1, LO-2, LO-4, LO-7 • Outcomes: LO-OUT-B, C, E, H

1. How do we put one number on an earthquake?

• 11 March 2011, 05:46 UTC: a rupture begins offshore Tōhoku, Japan.
• 14 minutes later, the first long-period waves reach PNSN broadband stations in Washington.
• Within hours, the world is told this was an "$M_W$ 9.1".

What does that one number actually measure, and what does it leave out?

Read more → Lecture 15 §1

2. Why a logarithm?

Three observations from Charles F. Richter (1935):

1. Wave amplitudes from one event decay smoothly with epicentral distance $\Delta$.
2. Different events trace parallel decay curves.
3. Amplitudes span eight orders of magnitude (μm → m).

→ Compress the dynamic range with $\log_{10}$, then read source size as the vertical offset between the curves.

2. (continued) Richter's foundational observation

Source size lives in the curve's offset; path lives in its shape.

Read more → Lecture 15 §2

3. A hierarchy of magnitudes

3a. Richter's local magnitude

$$M_L = \log_{10} A(\Delta) - \log_{10} A_0(\Delta)$$

• $A(\Delta)$ — peak amplitude on a Wood-Anderson seismometer, in mm.
• $A_0(\Delta)$ — empirical distance correction, anchored at $A_0 = 1$ μm at $\Delta = 100$ km.
• One unit of $M_L$ 10× ground motion ~32× radiated energy.

3a. (continued) Reading the Richter nomogram

Read more → Lecture 15 §3a

3b. Going global: $m_b$ and $M_S$

$$m_b = \log_{10}\!\left(\frac{A}{T}\right) + Q(\Delta, h)$$

$$M_S = \log_{10}\!\left(\frac{A}{T}\right) + 1.66 \log_{10}\Delta + 3.30$$

• $m_b$ uses 1-s P-waves at teleseismic distances → fast, but narrow-band.
• $M_S$ uses 20-s Rayleigh waves → broader-band, but only shallow events.
• Both saturate at large source size.

3c. Why magnitudes saturate

For a rupture of duration $\tau \sim L / v_r$:

$$f_c \approx \frac{1}{\tau} \quad \Rightarrow \quad f_c \;\downarrow\; \text{ as } \; L \;\uparrow$$

When $f_c$ falls below the scale's measurement frequency, you stop seeing the source — the seismogram averages a constant amplitude.

→ $m_b$ saturates near $f_c \sim 1$ Hz; $M_S$ saturates near $f_c \sim 0.05$ Hz.

3d. Seismic moment and moment magnitude

$$\boxed{\; M_0 = \mu \, A \, \bar{s} \;} \qquad \boxed{\; M_W = \tfrac{2}{3} \log_{10} M_0 - 6.03 \;}$$

$M_0$ in N·m. • 1 unit of $M_W$ ~32× more moment ~32× more energy. • $M_W$ does not saturate.

3d. Magnitude saturation, visually

Read more → Lecture 15 §3c–3d

4. Forward problem — worked example

A circular rupture, $A = \pi R^2$, on a crustal fault.

Given: $R = 5$ km, $\bar{s} = 0.4$ m, $\mu = 30$ GPa.

$$M_0 = (3{\times}10^{10}) \cdot \pi (5{\times}10^3)^2 \cdot 0.4 \approx 9.4 \times 10^{17} \;\text{N·m}$$

$$M_W = \tfrac{2}{3}\log_{10}(9.4{\times}10^{17}) - 6.03 \approx 5.93$$

→ Realistic for a small subduction event.

Read more → Lecture 15 §4

5. Inverse problem — going from seismograms to $M_0$

Three modern routes:

1. Long-period spectral level — read the flat low-frequency plateau of the displacement spectrum.
2. Centroid moment tensor (CMT) — fit synthetic body and surface waves to long-period data ($T > 100$ s) → full moment tensor.
3. Finite-fault inversion — invert the slip distribution $s(\xi, \tau)$ on a discretised fault.

GCMT catalogue: ~50 routine solutions per month worldwide.

Read more → Lecture 15 §5

6a. The Gutenberg–Richter law

Two empirical laws govern earthquake catalogues — Gutenberg-Richter for size, Omori for time.

$$\log_{10} N(M' > M) = a - bM$$

• $a$ — total seismicity level (regional).
• $b$ — slope; globally $b \approx 1$.

→ Each unit increase in $M$ divides the rate by 10.

6a. Global frequency–magnitude statistics

Read more → Lecture 15 §6a

6b. Aftershocks and Omori's law

$$n(t) = \frac{K}{(t + c)^p}, \qquad p \approx 1$$

• Aftershock rate decays roughly as $1/t$.
• Largest aftershock typically about one magnitude below the mainshock (Båth's law).
• ~5% of "aftershocks" turn out to be larger than the prior shock — relabelling them retrospectively as foreshocks.

6b. Northridge 1994 — Omori in action

The same pattern reappears in every aftershock sequence — see §6b of the notes for the 2020 Alaska Peninsula example with two superposed Omori decays.

Read more → Lecture 15 §6b

7. Connecting to Cascadia and the Pacific Northwest

• Cascadia subduction zone — capable of $M_W \approx 9$ ruptures (last one 26 January 1700 CE).
• Frankel et al. 2018 — physics-based ground-motion simulations for $M_W$ 9 Cascadia (BSSA 108: 2347–2369).
• A factor-of-four difference in source moment ($M_W$ 8.4 → $M_W$ 9.0) corresponds to ~64× more energy and 2 minutes of strong shaking instead of 30 seconds.
• PNSN + ShakeAlert + GFAST (2024) put real-time, non-saturating geodetic magnitude into your phone.

Did anyone in this room — or your family — feel the 2001 Nisqually $M_W$ 6.8 event? That intra-slab shock was small enough that all four magnitude scales agreed; the next Cascadia megathrust will not be.

Read more → Lecture 15 §7

8. Putting it all together — energy, frequency, size

Read more → Lecture 15 §8

8. (continued) Cumulative moment release, 1900–2024

Most of the global moment budget is released by a handful of $M \geq 9$ events. The $b$-value tells you the rate; $M_W$ tells you the budget.

9. Research horizon — post-2020 highlights

• Real-time GNSS magnitude. Goldberg et al. Seismica 2024 — PGD → $M_W$ in seconds for tsunami warning (doi:10.26443/seismica.v3i1.1129).
• GFAST in ShakeAlert (2024). UW-developed geodetic algorithm now in operational alerts (Crowell, PNSN 2024) — finally a non-saturating estimator for Cascadia.
• ML phase pickers + magnitude. Mousavi & Beroza ARES 2023 (open access); PhaseNO multi-station extension (Sun et al. 2023). Magnitude of completeness drops by ~1 unit.
• ETAS forecasting. Hardebeck et al. ARES 2024 — cascading aftershocks in real time; deployed for the 2023 Türkiye doublet (Mai et al. 2023).
• DAS magnitude. Yin et al. GJI 2023 — fibre-optic cables now produce $M_W$ within 0.2 unit of catalogue, including offshore Cascadia (Wilcock 2025).

Read more → Lecture 15 §9

9. (continued) Magnitude beyond Earth — InSight on Mars

• InSight (2018–2022): the first single-station planetary seismic catalogue.
• Magnitudes reported on $M_W^{\rm Ma}$ — uses Mars's crustal rigidity ($\mu \approx 25$ GPa) and a Martian distance correction (Böhm et al. 2022).
• S1222a (4 May 2022): $M_W^{\rm Ma} \approx 4.7$ — largest non-impact marsquake observed, the first to excite detectable surface waves (Kawamura et al. 2023).
• Same equation: $M_0 = \mu A_f \bar{s}$. Different planet.

For the planetary-science track: the moment-magnitude framework you just learned is the same machinery that placed Mars on the seismicity hierarchy.

10. AI literacy — magnitude as a reasoning partner

Prompt 1. "Explain magnitude saturation to a 10th-grader." — Critique the LLM's analogy.

Prompt 2. "An earthquake has $M_W$ 7.0, fault area 200 km², rigidity 30 GPa. What is the average slip?" — Verify each algebra step; LLMs occasionally substitute $M_W$ for $M_0$ in $\bar{s} = M_0 / (\mu A_f)$.

Prompt 3. "Why does $M_S$ saturate but $M_W$ doesn't?" — Compare the LLM's answer to the corner-frequency argument from §3c.

→ Submit the better of your answer or the LLM's, with a short rubric-based critique.

Read more → Lecture 15 §10

A note on Lab 4 — coding anxiety welcome here

• Lab 4 uses ObsPy to download a PNSN waveform and compute $M_L$ — the physics is what's being assessed, not your Python fluency.
• ObsPy abstracts the heavy lifting (SEED parsing, instrument response, taper/filter). You write ~20 lines of code.
• Office hours and the #labs Slack channel are explicitly for "I'm stuck on a Python error" questions — those are not stupid, they are part of every working seismologist's day.

If you have never run a Jupyter notebook before, the lab handout walks through every step. The grading rubric is on the physics, not the syntax.

11. Concept check

1. Two earthquakes have the same $M_W$ but one occurred at 10 km depth and the other at 200 km depth. Why might their reported $M_S$ values differ — and which scale would you trust?

2. A regional catalogue gives $b = 0.7$. What does that tell you, physically, compared to a region with $b = 1.0$?

3. Two ruptures release the same $M_0$. One has area $A_1 = 100$ km² and the other $A_2 = 400$ km². What is the ratio of their average slip values? Of their stress drops (assume same fault shape)?