Earthquake Focal Mechanisms and Faults

Learning Objectives

By the end of this lecture, you should be able to:

• LO-OUT-A — Explain the four-quadrant compression / dilatation P-wave radiation pattern.
• LO-OUT-C — Read a beach ball; identify fault style and approximate strike, dip, rake.
• LO-OUT-D — Interpret the fault-plane / auxiliary-plane ambiguity and how to resolve it.
• LO-OUT-E — Connect focal mechanisms to plate-boundary types and PNW tectonics.

Aligned course objectives: LO-1, LO-2, LO-3, LO-4.

A Puget Sound morning, 28 February 2001

• M 6.8 Nisqually earthquake ruptured at 53 km depth beneath Anderson Island.
• The State Capitol was damaged; ≈250,000 people felt strong shaking.
• The earthquake was deep and extensional — within the descending Juan de Fuca slab.

How is any of that known, when no one was at the source?

→ The focal mechanism: a compact summary of source geometry, inferred from the first wiggles to arrive on regional seismograms.

A focal mechanism is...

• The orientation of the fault plane that ruptured.
• The slip direction of the hanging wall on it.
• The pattern of P-wave radiation that geometry produces.

Every PNSN earthquake larger than ≈M3 has one.
The Global CMT project has produced >65,000 of them since 1976.

Focal mechanisms are the bridge between the seismogram and plate tectonics.

The push–pull pattern

A slipping fault loads its surroundings with a four-quadrant stress pattern:

• Two quadrants are pushed outward → P-wave begins as compression (first motion up).
• Two quadrants are pulled inward → P-wave begins as dilatation (first motion down).
• The boundary between quadrants is the fault plane and an auxiliary plane perpendicular to it.

Right-lateral strike-slip fault: NE & SW quadrants record upward first motions; NW & SE record downward. Adapted from Lowrie & Fichtner (2020); regenerated open-licensed.

Read more → Lecture 16 §2b

The double-couple equivalence

A point shear dislocation on a fault is mathematically equivalent to a pair of perpendicular force couples — a double couple.

$$M_{ij} = M_0 \, (n_i d_j + n_j d_i)$$

• $\mathbf{n}$ = fault normal · $\hat{\mathbf{d}}$ = slip direction · $M_0 = \mu A |\mathbf{u}|$
• The form is symmetric under $\mathbf{n} \leftrightarrow \hat{\mathbf{d}}$.
• Far-field seismic radiation depends only on $M_{ij}$.

→ Two perpendicular planes radiate identically. Fault-plane / auxiliary-plane ambiguity is built into the physics.

Read more → Lecture 16 §2a

The P-wave radiation pattern

For a vertical strike-slip fault striking north, the radiation in the horizontal plane reduces to:

$$F^P(\phi) = \sin(2\phi)$$

• Positive (compression) for $0° < \phi < 90°$ and $180° < \phi < 270°$.
• Negative (dilatation) for $90° < \phi < 180°$ and $270° < \phi < 360°$.
• Vanishes on the nodal planes $\phi = 0°, 90°, 180°, 270°$.

The general form (Aki & Richards 2002, eq. 4.84) depends on $(\phi_s, \delta, \lambda, i, \phi)$.

Read more → Lecture 16 §3c

The focal sphere

The lower hemisphere of an imaginary sphere around the source is projected stereographically to a disk — the beach ball. Filled = compression; open = dilatation.

Read more → Lecture 16 §2d

Strike, dip, and rake

• Strike $\phi_s$ — azimuth of fault trace, 0–360° from north.
• Dip $\delta$ — angle of fault plane below horizontal, 0–90°.
• Rake $\lambda$ — slip direction in the fault plane from strike, $-180°$ to $180°$.

Read more → Lecture 16 §3d

Rake convention

Rake encodes the style of slip on the fault plane. Memorise the four end-members: 0° = left-lateral; 90° = thrust; 180° = right-lateral; $-90°$ = normal.

Three end-member fault styles

Fault-style cheat sheet

Beach-ball signatures:

• Strike-slip → four quadrants meeting at centre.
• Normal → white inner section, dark crescents.
• Thrust → dark inner section, white crescents.

Forward problem

Given $(\phi_s, \delta, \lambda)$ and source–station geometry, predict the polarity at every station.

1. Compute the moment tensor $M_{ij}$ from the three angles.
2. Ray-trace to find the take-off direction $(i, \phi)$ for each station.
3. Evaluate $F^P(i, \phi; \phi_s, \delta, \lambda)$ — the sign predicts polarity.

→ Implemented in labs/lab_06_focal_mechanisms.ipynb using ObsPy.

Read more → Lecture 16 §4

Inverse problem

Find $(\phi_s, \delta, \lambda)$ that best separate up-first-motion stations from down-first-motion stations on the focal sphere.

• FPFIT, HASH — first-motion grid search.
• Global CMT, W-phase — full-waveform inversion of body and surface waves.
• ML methods (Mousavi et al. 2020; Zhu et al. 2022) — neural-network polarity pickers and mechanism estimators.

A small, classical inverse problem — but with one fundamental ambiguity.

Read more → Lecture 16 §5

The fault-plane ambiguity

Both nodal planes radiate identically. To pick the fault, you need independent information:

• Aftershock cloud — usually outlines the actual rupture plane.
• Surface rupture — geological mapping resolves it directly.
• Geodesy — InSAR / GPS displacement fields fit one plane better.
• Tectonic context — slab geometry, regional stress, etc.

This is a research problem in itself. A focal mechanism alone is consistent with two different earthquakes.

Plate boundaries — three signatures

Each plate-boundary type carries its own focal-mechanism signature. Subduction zones host multiple styles — a key reason Cascadia is so geophysically rich.

Worked example: 2001 Nisqually

$$M_w = 6.8, \quad h = 53\,\mathrm{km}, \quad \phi_s = 357°, \quad \delta = 83°, \quad \lambda = -104°$$

• Style? $\lambda = -104°$ → normal. Hanging wall dropped.
• Why so deep? The hypocentre is in the descending Juan de Fuca slab.
• Physical interpretation: downdip extension within the slab under its own weight.
• Which nodal plane is the fault? Steep N–S plane (the shallow plane is geometrically inconsistent with slab orientation).

Read more → Lecture 16 §6

Cascadia: one margin, four mechanisms

Megathrust · intraslab normal · crustal thrust · oceanic transform — within ~600 km.

Read more → Lecture 16 §7

Hazard implications

→ Each style requires different building-code provisions and different warning protocols.

AI literacy: AI Epistemics

Activity (15 min, groups of 3):

1. Design first — sketch your own workflow for an M5 event in central Puget Lowland.
2. Query — ask Claude / ChatGPT / Gemini the same question.
3. Evaluate the AI response on physics correctness, assumption transparency, operational feasibility, failure-mode flagging.
4. Compare — where did your design beat the AI? Where did it lose? Find one thing AI was confidently wrong about.

→ Goal: treat AI outputs as hypotheses to test, not as answers to accept.

Concept check (1/3)

A vertical-component seismometer records a downward first motion for the P-wave from a nearby earthquake. What does that tell you, geometrically?

::: speaker-notes
The station sits in a dilatational quadrant of the source's radiation pattern. The fault and auxiliary planes pass between this station and any station that recorded an upward first motion.
:::

Concept check (2/3)

A focal mechanism shows two nodal planes: one striking 005°, dipping 12°E; the other striking 185°, dipping 78°W. Aftershocks lie in a thin layer dipping ≈12°E.

Which is the fault? What style?

::: speaker-notes
The shallow east-dipping plane is the fault — aftershocks resolve the ambiguity. With dip 12° and a thrust signature implied by the question, this is consistent with a subduction megathrust.
:::

Concept check (3/3)

The Global CMT solution for the 2011 M9 Tōhoku earthquake has $\phi_s = 203°$, $\delta = 10°$, $\lambda = 88°$.

What style of faulting? What plate-boundary type?

::: speaker-notes
Rake $\lambda = 88°$ ∈ (45°, 135°) → thrust. Dip 10° → very shallow. This is the textbook subduction megathrust signature.
:::

Further reading

• Aki & Richards (2002) Quantitative Seismology, Ch. 4.
• Lowrie & Fichtner (2020) Ch. 3, §3.5. (UW Libraries)
• McCrory et al. (2012) Juan de Fuca slab geometry, JGR. doi:10.1029/2012JB009407
• Mousavi et al. (2020) Earthquake Transformer, Nat. Commun. doi:10.1038/s41467-020-17591-w
• Global CMT globalcmt.org · PNSN pnsn.org

→ Lab 6 next week: first-motion analysis with ObsPy on PNSN data.