Earthquake Phenomena I

Learning Objectives

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

• [LO-1] Identify P, S, and surface waves on a seismogram and explain why they arrive in that order
• [LO-2] Convert an $S$-minus-$P$ time into a hypocentral distance using a known velocity model
• [LO-3] Frame earthquake location as a forward / inverse problem in $(x_0, y_0, z_0, t_0)$
• [LO-5] Explain the geometric origin of location uncertainty
• [LO-7] Critique an AI-picked phase or relocated catalog

1. The framing question

The Pacific Northwest sits above the Cascadia subduction zone.

• Last megathrust event: $M_w \sim 9$, on 26 January 1700
• Hundreds of smaller earthquakes per month, recorded by PNSN
• Every earthquake is hidden underground — the focus is never directly observed during rupture
• Yet, from surface records, we routinely infer where, when, how big, and what kind

This lecture: where and when.

Read more → Lecture 14 §1

What the source geometry looks like

Focus (hypocenter): point where rupture initiates
Epicenter: vertical projection of focus to surface
Focal depth $h$: vertical distance between them

2. Three pieces of physics combine

1. Two body-wave modes. P and S waves leave the source together; $V_P/V_S = \sqrt{3}$ in a Poisson solid

2. Spherical wavefronts. Geometric spreading reduces amplitudes, but arrival times are governed by the integral of slowness along the ray

3. The free surface. Converts body-wave energy into surface waves and produces the depth phases (pP) used for teleseismic depth determination

Read more → Lecture 14 §2

Wave propagation in time

The three wavefronts spread at distinct speeds → arrivals always in P → S → surface order at every station.

3. The seismogram, anatomized

The interval $T_S - T_P$ is the diagnostic measurement for distance.

Read more → Lecture 14 §3

The S-minus-P relation

Subtract the P arrival time from the S arrival time at one station:

$$T_S - T_P \;=\; D \left( \frac{1}{V_S} - \frac{1}{V_P} \right)$$

Solving for hypocentral distance:

$$\boxed{\;D \;=\; \frac{V_P\, V_S}{V_P - V_S}\,(T_S - T_P)\;}$$

For average crust ($V_P = 6.0$, $V_S = 3.46$ km/s):

$$D \;\approx\; 8.2 \times (T_S - T_P)$$

— the textbook rule of eight.

S-P time as a function of distance

Slower velocity contrast → steeper slope → small velocity-model errors → large distance errors.

Single-station back-azimuth

$$\mathrm{AZI} \;=\; \arctan(A_E / A_N)$$

Vertical-component polarity resolves the $180°$ ambiguity.

3d. Triangulation: the multi-station epicenter

• 3 stations → epicenter (3 unknowns: $x_0, y_0, t_0$)
• 4 stations → epicenter + depth (4 unknowns)

3e. Resolving focal depth

• Local distance: right triangle, $h = \sqrt{D^2 - \Delta^2}$
• Teleseismic distance: depth phase, $t_{pP} - t_P$ → focal depth

Read more → Lecture 14 §3e

4. The forward problem

Given a candidate hypocenter $\mathbf{m} = (x_0, y_0, z_0, t_0)$, predict the P arrival time at every station:

$$T_P^{(i)\,\mathrm{pred}} \;=\; t_0 \;+\; \frac{1}{V_P}\,\sqrt{(x_i - x_0)^2 + (y_i - y_0)^2 + (z_i - z_0)^2}$$

Two key properties:

• Linear in $t_0$ — origin time enters as an additive constant
• Non-linear in $(x_0, y_0, z_0)$ — distance enters through a square root

This decomposition is what Geiger's 1912 algorithm exploits.

Read more → Lecture 14 §4

5. The inverse problem

Define the residual at observation $i$:

$$r_i(\mathbf{m}) \;=\; d_i^{\,\mathrm{obs}} - G_i(\mathbf{m})$$

Minimize the misfit:

$$\Phi_2(\mathbf{m}) = \sum_i \left( \frac{r_i}{\sigma_i} \right)^{\!2} \quad\text{(L$_2$, Gaussian errors)}$$

$$\Phi_1(\mathbf{m}) = \sum_i \left| \frac{r_i}{\sigma_i} \right| \quad\text{(L$_1$, robust to outliers)}$$

Iterative: linearize about $\mathbf{m}_k$, take a least-squares step, repeat.

Read more → Lecture 14 §5

Why location uncertainty is geometric

• Clustered network → ellipse points radially away
• Distant stations only → depth and $t_0$ trade off

Relative location: HypoDD

When two earthquakes are close together, the difference of their arrival times depends only on the difference of their coordinates — velocity-model errors cancel.

• {cite:t}Waldhauser2000 — double-difference algorithm
• Routinely achieves tens of metres relative precision
• Resolves fault-plane structures invisible in absolute catalogs
• {cite:t}Hauksson2012 (SoCal), {cite:t}Shelly2016 (Long Valley), {cite:t}Ross2019 (San Jacinto)

6. Worked example — a Puget Lowland event

A station at $\Delta = 50$ km records $T_P = 14.2$ s, $T_S = 21.1$ s, with $A_N = 0.74$, $A_E = 0.32$, $A_Z = +0.92$.

• Distance: $D = 8.2 \times 6.9 \approx 56$ km
• Back-azimuth: $\mathrm{AZI} = \arctan(0.32/0.74) \approx 23°$
• Depth: $h = \sqrt{56^2 - 50^2} \approx 25$ km

A 25 km focal depth is consistent with a deep intra-slab event in the subducting Juan de Fuca plate — the same regime as the 2001 $M_w$ 6.8 Nisqually earthquake.

Did anyone in this room feel Nisqually? Stories from across the Puget Lowland — things falling off shelves at home, bricks tumbling off the State Capitol — are exactly the ground-motion data this lecture’s methods turned into a hypocenter the same morning.

Read more → Lecture 14 §6

7. Course connections

• Lecture 12 (Tomography): same forward/inverse framework, different unknown
• Lecture 15 (next): takes location as known, asks how big — magnitude, $M_0$
• Lectures 18, 23: gravity and magnetic inverse problems — the same non-uniqueness reappears
• Week 5 lab: phase picking and location with ObsPy and PNSN data

Read more → Lecture 14 §11

8. Research horizon — ML phase picking

• PhaseNet {cite:p}Zhu2019PhaseNet: U-Net trained on 600,000 NCEDC waveforms; ~96% precision on P
• EQTransformer {cite:p}Mousavi2020EQT: hierarchical attention; hundreds of microearthquakes detected with one-third of typical networks
• PhaseNO {cite:p}Sun2023PhaseNO: multi-station Fourier neural operator
• Cascadia ML catalog: re-trained EQTransformer on 20 years of PNSN data
• Not a replacement for the physics — a fast front-end that supplies the $(T_P, T_S)$ that the inverse problem consumes

8. Research horizon — Distributed Acoustic Sensing

• A single fibre-optic cable, interrogated by laser pulses, becomes a dense seismic array of thousands of channels
• Submarine fibres off Cascadia {cite:p}Wilcock2025: detect offshore earthquakes invisible to onshore networks
• Crucial for early warning of offshore megathrust ruptures
• Active research area at UW (Denolle group): semi-supervised picking on DAS strain-rate data {cite:p}Zhu2023DAS

Read more → Lecture 14 §8

7. Connecting to Cascadia — ShakeAlert

• Operational across Washington and Oregon since 2021
• Ingests data from ~1500 PNSN seismic stations + ~760 GNSS sensors
• Real-time location and magnitude → seconds-to-minutes of warning
• Cascadia M9: tens of seconds of warning in Seattle
• Nisqually-style intra-slab event: ~10 s of warning typical
• GFAST {cite:p}Crowell2024GFAST: geodetic algorithm avoids magnitude saturation at $M_w$ 7

Read more → Lecture 14 §7

9. AI Literacy — when to trust an ML phase pick

ML pickers achieve ~95% precision on data that look like their training data.

• Recall drops 30–40% across regions {cite:p}Munchmeyer2022
• Even worse on ocean-bottom, borehole, DAS, or mining data
• Three habits:
  1. Know the training distribution
  2. Verify a sample by eye
  3. Carry the velocity-model assumption forward — a pick is not a location

Read more → Lecture 14 §9

10. Concept check

1. If the velocity model used $V_P = 5.5$ instead of $V_P = 6.0$ km/s (same $V_P/V_S$), how would the calculated $D$ change?

2. For an event $\Delta = 5$ km from the closest station, do you trust the single-station $D$ more, or the multi-station triangulated epicenter?

3. With only teleseismic stations, which of $(x_0, y_0, z_0, t_0)$ is best constrained, and which is most degenerate?

Read more → Lecture 14 §10

Beyond Earth — locating marsquakes

NASA's InSight lander (2018–2022) carried a single three-component seismometer (SEIS) to Elysium Planitia.

• One station, no triangulation possible.
• Distance came from $T_S - T_P$ — exactly equation $D = \dfrac{V_P V_S}{V_P - V_S}(T_S - T_P)$, with a Mars velocity model.
• Back-azimuth came from P-wave polarization — exactly section 3c.
• Depth was nearly unconstrained — a planetary-scale version of the depth–origin-time trade-off.

The same physics that locates a Puget Sound earthquake located the InSight S1222a marsquake ($M_w \sim 4.7$, May 2022).

If you're newer to Python — what to focus on this week

You don't need to write a phase picker from scratch. The lab uses ObsPy, a high-level Python library where one line gets you a seismogram:

from obspy.clients.fdsn import Client
st = Client("IRIS").get_waveforms("UW", "SEP", "*", "BHZ",
                                  t1, t1 + 600)

What matters for this lecture:

• Read the S-minus-P time off a plot — by eye, not by code
• Apply $D = 8.2\,(T_S - T_P)$ — on paper, with a calculator
• Reason about which parameters are well-constrained and which are not

The physics is what we are testing. The Python is the medium.