---
title: "Whole Earth Imaging: From Travel-Time Observations to the 1-D Earth"
week: 6
lecture: 11
date: "2026-05-04"
topic: "Global body-wave phases, shadow zones, PREM, 1-D Earth structure"
course_lo: ["LO-2", "LO-3", "LO-5", "LO-7"]
learning_outcomes: ["LO-OUT-B", "LO-OUT-C", "LO-OUT-E"]
open_sources:
  - "Dziewonski & Anderson 1981, PREM (open access via ScienceDirect)"
  - "Kennett et al. 1995, AK135 (open access, GJI)"
  - "Lowrie & Fichtner 2020 Ch. 3 (UW Libraries)"
  - "IRIS/EarthScope Global Stacks (open access)"
---

# Whole Earth Imaging: From Travel-Time Observations to the 1-D Earth

:::{seealso}
📊 **Lecture slides** — <a href="https://uw-geophysics-edu.github.io/ess314/slides/lecture_11_slides.html" target="_blank">open in new tab ↗</a>
:::

::::{dropdown} Learning Objectives
:color: primary
:icon: target

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

- **[LO-11.1]** Describe how seismic waves recorded around the globe serve as a probe of Earth's internal structure, and explain why travel-time observations alone can constrain the radial profile of P- and S-wave velocity and the location of major discontinuities.
- **[LO-11.2]** Read a global travel-time diagram, identify the major body-wave phases (P, S, PP, SS, PcP, ScS, PKP, PKIKP, SKS), and explain how each phase's existence or absence constrains a specific property of Earth's interior.
- **[LO-11.3]** Explain the physical evidence for the three canonical whole-Earth discoveries — Oldham's fluid outer core (1906), Gutenberg's depth to the CMB (1914), and Lehmann's solid inner core (1936) — and reproduce the shadow-zone reasoning that anchors each one.

::::

::::{dropdown} Syllabus Alignment
:color: secondary
:icon: list-task

| | |
|---|---|
| **Course LOs addressed** | LO-2, LO-3, LO-5, LO-7 |
| **Learning outcomes practiced** | LO-OUT-B (interpret travel-time curves), LO-OUT-C (explain physical reasoning from shadow zones), LO-OUT-E (evaluate 1-D Earth models) |
| **Prior lectures** | Lectures 3–4 (Snell's law, ray-wavefront duality, slowness integral), Lecture 10 (forward/inverse problem framing) |
| **Next lecture** | Lecture 12 — Seismic Tomography |
| **Lab connection** | Lab 3: students compute AK135 travel times with obspy.taup and compare to observed phase picks |
| **Discussion connection** | Discussion 6: How shadow zones led to discovery — evaluating historical claims from first principles |

::::

## Prerequisites

Students should be comfortable with: Snell's law and refraction at
planar interfaces (Lecture 3), ray-wavefront duality and the concept
of a travel time as a path integral of slowness (Lecture 4), and the
distinction between the forward and inverse problems (introduced in
the module overview).

---

## 1. The framing question: what is inside the Earth, and how would you know?

No one has ever been more than ten kilometres beneath the surface.
The deepest borehole ever drilled — the Kola Superdeep — reached
12.3 kilometres, roughly one five-hundredth of the way to the centre
of the planet. Yet seismologists routinely quote the radius of the
core to within a kilometre, the depth of the core-mantle boundary to
within a few kilometres, and the velocity of sound inside the core to
three significant figures. The tool that makes these numbers
knowable is the global seismogram: a recording of ground motion from
distant earthquakes, measured at stations distributed around the
world.

This lecture is the capstone of the subsurface-imaging module. In
Lectures 7 through 10 the target was the upper crust: a reflection
survey off the Washington coast, a refraction line across a basalt
flow. The physics is the same — ray paths, travel times, impedance
contrasts — but the scale is now global. A ray emerging at an
epicentral distance $\Delta = 100^\circ$ has sampled the entire
mantle. A ray emerging at $\Delta = 170^\circ$ has passed through
the outer core twice and the inner core once. Each travel time is a
line integral along a path that samples a different slice of the
planet. Collect enough of them, invert, and a radial profile of
velocity emerges.

The historical arc of that inversion is the subject of this lecture.

---

## 2. The physics: how a depth-dependent velocity profile bends rays

The foundational observation is that seismic waves do not travel in
straight lines inside the Earth. They curve. The reason is that
seismic velocity increases with depth (with two notable exceptions —
the low-velocity zone near 150 km, and the catastrophic drop at the
core-mantle boundary), and Snell's law at each infinitesimal interface
bends the ray back toward the lower-velocity side, which from the
deep interior's perspective is the surface.

Consider two hypothetical Earths. In the first, the P-wave velocity
is constant everywhere — say, 10 km/s. Rays travel in straight
chords from source to receiver, and the travel time is simply chord
length divided by velocity:

```{math}
:label: eq:tt-constant-v
T(\Delta) = \frac{2R\sin(\Delta/2)}{V}.
```

In the second Earth, the velocity increases monotonically with depth.
Rays descending at steeper take-off angles encounter faster material
and bend concave-up, returning to the surface at a greater distance
than a chord of the same length would reach. At large $\Delta$, the
integrated velocity along the longer curved path is larger than the
constant-$V$ reference, and the travel time is shorter. Figure
{numref}`fig-constant-vs-gradient` compares the two.

```{figure} ../assets/figures/fig_11_constant_v_vs_gradient.png
:name: fig-constant-vs-gradient
:alt: Three-panel comparison. Top-left quarter-Earth cross-section shows
  straight blue ray paths (chords) from a source at the top to points
  along a half-circle, for a constant-velocity Earth. Bottom-left shows
  orange concave-up ray paths that turn at progressively greater depths
  for rays reaching larger epicentral distances, for a velocity-
  increasing-with-depth Earth. The right panel plots travel time T
  versus angular distance from 0 to 180 degrees; the solid blue
  constant-V curve is always above the dashed orange gradient curve at
  large distances, and an annotation arrow points to the orange curve
  labelled "rays return earlier at large Delta".
:width: 90%

Ray paths and travel-time curves for two hypothetical Earths. In a
constant-velocity Earth, rays travel in straight chords and the
travel-time curve is the chord length divided by velocity. When
velocity increases with depth, Snell's law bends rays concave-up and
they emerge at greater distance with shorter travel times.
```

The key takeaway: **the shape of the $T(\Delta)$ curve encodes the
depth dependence of velocity**. A seismologist who measures travel
times from many earthquakes recorded at many stations and plots them
on a single $T(\Delta)$ diagram is, without yet writing any
equations, already measuring the interior of the Earth.

---

## 3. The mathematical framework: travel time as an integral of slowness

Along a ray path parametrised by arc length $s$, the travel time is

```{math}
:label: eq:tt-integral
T \;=\; \int_{\mathrm{source}}^{\mathrm{receiver}} \frac{ds}{V(\mathbf{r})}
    \;=\; \int_{\mathrm{source}}^{\mathrm{receiver}} u(\mathbf{r})\, ds,
```

where $u = 1/V$ is the slowness. In a spherically symmetric Earth,
the ray path depends only on radius, and the travel time for a ray
that turns at radius $r_p$ can be written explicitly as

```{math}
:label: eq:tt-sphere
T(p) \;=\; 2\,p\,\Delta(p) + 2 \int_{r_p}^{R}\!\sqrt{\eta^2(r) - p^2}\;\frac{dr}{r},
\qquad
\eta(r) = r/V(r),
```

where $p$ is the ray parameter (the analogue of the horizontal
slowness in a flat Earth) and $\Delta(p)$ is the angular distance
travelled. You will not be asked to derive or apply this integral in
this course; we cite it only so that you know the machinery exists
and is numerically tractable. What matters pedagogically is the
inverse: **given a set of measured travel times $T_i$ at known
distances $\Delta_i$, find the function $V(r)$ that is consistent
with them**. Solving that inverse problem is what gives us the
preliminary reference Earth model (PREM) in Section 7.

---

## 4. The forward problem: predicting shadow zones from a layered Earth

A spherically layered Earth with a fluid outer core (no S-wave
propagation; a sharp velocity decrease for P-waves at the CMB) makes
two predictions that are qualitative enough for a student to check
from first principles.

**The S-wave shadow zone.** S-waves cannot propagate through a
fluid. Any S-ray that would have sampled the outer core is
absorbed or converted at the CMB. From a shallow earthquake, the
deepest mantle S-ray turns just above the CMB and emerges at
$\Delta \approx 103^\circ$. Beyond $103^\circ$, no direct S-wave
arrival can exist. The entire far hemisphere is an S-wave shadow.

**The P-wave shadow zone.** P-waves do propagate through the outer
core — but at $8.06$ km/s instead of the $13.7$ km/s just above the
CMB. The sudden velocity drop refracts descending rays strongly
toward the vertical (Snell's law with a low-velocity target layer),
bending them far from where a no-core Earth would send them. The
consequence is a ring, between roughly $103^\circ$ and $143^\circ$,
where neither direct P nor refracted PKP arrives. Beyond $143^\circ$,
PKP emerges from the far side of the core.

```{figure} ../assets/figures/fig_11_shadow_zones.png
:name: fig-shadow-zones
:alt: Two side-by-side schematic Earth cross-sections. Panel (a):
  S-wave shadow. Orange S-rays from a star-shaped source at the top
  of the Earth turn back inside the mantle and reach the surface only
  at angular distances less than 103 degrees. The surface arc from
  103 degrees through the antipode at 180 degrees is shaded in
  vermilion and labelled "S-wave shadow zone". The annotation inside
  the outer core reads "liquid outer core (no S)". Panel (b): P-wave
  shadow. Blue P-rays turn in the mantle and reach the surface within
  103 degrees; light-blue PKP rays refract into the outer core,
  traverse it along curved paths computed from AK135, and emerge at
  the surface beyond the PKP caustic at 143 degrees. Dashed green
  PKIKP rays pass through the outer core and continue through the
  solid inner core, emerging at distances near 155–175 degrees. The
  surface arc from 103 degrees to 143 degrees on each side is shaded
  in vermilion and labelled "P-wave shadow". Angular distance ticks in
  30-degree increments run from 0 to 180 on both sides of each panel.
  Both panels use the colorblind-safe Wong palette.
:width: 100%

Shadow zones are the direct observational signature of the fluid outer
core. The 103° threshold is **geometric** — it is the angular distance
at which a direct mantle ray just grazes the core-mantle boundary. Beyond
103° any direct ray must penetrate the core.

**S-wave shadow** ($\Delta > 103^\circ$): S-wave energy cannot propagate
as a shear wave in the fluid outer core
($\mu = 0 \Rightarrow \beta = \sqrt{\mu/\rho} = 0$), so the S shadow
extends from 103° all the way to the antipode.

**P-wave shadow** ($103^\circ < \Delta < 143^\circ$): P-wave energy
refracts into the core ($\alpha_\text{mantle} \approx 13.7$ km/s,
$\alpha_\text{outer\,core} \approx 8$ km/s; by Snell's law the ray bends
strongly toward the normal) and re-emerges as PKP at or beyond the PKP
caustic near 143°, leaving a gap in the 103°–143° band.

The same kinematic boundary at the CMB combined with one constitutive
fact — fluids carry no shear stress — explains both shadows.

> **Reproducibility.** Ray paths in panel (b) use real obspy.taup AK135
> ray tracing. The precise caustic position and PKP emergence distances
> depend on the radial velocity model; this figure uses AK135
> (Kennett et al. 1995). Source: `assets/scripts/fig_11_shadow_zones.py`.
```

Oldham (1906) observed the S-wave shadow and inferred the fluid
core. Gutenberg (1914) refined the depth estimate to $\approx
2900$ km. Lehmann (1936) found a weak P-wave arrival *inside* the
predicted P-shadow band at distances around $150^\circ$–$160^\circ$
that could not be explained by the fluid-core model alone, and
inferred a solid inner core at $\approx 5150$ km depth. Three
discoveries — each one a simple geometric argument from a missing or
unexpected arrival on a seismogram — built the layered structure of
the planet.

:::{admonition} Key Equation — shadow-zone prediction
:class: important

For a spherical Earth with mantle P-velocity increasing monotonically
from $V_P \approx 8$ km/s at the Moho to $V_P \approx 13.7$ km/s
just above the CMB, and an abrupt drop to $V_P \approx 8.06$ km/s in
the outer core, Snell's law predicts:

$$
\Delta_{S\text{-shadow start}} \approx 103^\circ,
\qquad
103^\circ \lesssim \Delta_{P\text{-shadow}} \lesssim 143^\circ.
$$

These are numbers that students can derive themselves given the
velocity contrast and the ray-parameter conservation rule $p =
r\sin i / V$. Lab 6 walks through this calculation.
:::

:::{admonition} Reasoning sketch — why $p = r\sin i / V$ is the right invariant
:class: tip

In a flat-layered Earth, Snell's law conserves $\sin i / V$ at every
interface. In a *spherically symmetric* Earth the geometry has changed:
"horizontal" is no longer a single direction. What is preserved instead
is the **Bullen ray parameter**

$$
p \;=\; \frac{r \sin i(r)}{V(r)} \;=\; \text{constant along a ray},
$$

where $r$ is radius and $i(r)$ is the angle between the ray and the
local radial direction. Two facts make this work, and you should be
able to argue both:

1. **Spherical symmetry.** Velocity depends only on $r$; rotating the
   source–receiver geometry around Earth's centre does not change the
   physics. Noether-style: a continuous symmetry implies a conserved
   quantity. Here the conserved quantity is the angular momentum-like
   product $r\sin i / V$.
2. **Turning condition.** A ray turns where $i = 90°$, i.e. where
   $r/V(r) = p$. Plotting $r/V(r)$ for AK135 makes the shadow-zone
   numbers visible: every $p$ that maps onto the *mantle* part of the
   curve corresponds to a ray that emerges at $\Delta < 103°$; the
   first $p$ small enough to reach the CMB defines the 103° edge; rays
   that hit the velocity drop at the CMB refract steeply into the core
   and re-emerge as PKP beyond 143°.

The reasoning chain is: **symmetry → conserved quantity → turning
depth → emergence distance**. Once internalised, every global travel-time
diagnostic in this lecture and Lab 6 is a consequence.
:::

---

## 5. The inverse problem: from global seismograms to a 1-D Earth

Travel-time curves are assembled empirically. Global networks such
as the IRIS Data Management Center and GEOSCOPE archive seismograms
from thousands of earthquakes at hundreds of stations. When many
seismograms are aligned by origin time and plotted against
source-receiver distance, coherent arrival curves emerge for each
body-wave phase. Figure {numref}`fig-tt-curves` shows a modern
computed version, using the AK135 reference Earth model (Kennett et
al. 1995) as the interior structure and the `obspy.taup`
ray-tracing package to calculate travel times.

```{figure} ../assets/figures/fig_11_traveltime_curves.png
:name: fig-tt-curves
:alt: Global travel-time diagram plotting travel time in minutes on
  the vertical axis from 0 to 45 versus epicentral distance in degrees
  on the horizontal axis from 0 to 180. Curves for the phases P (solid
  blue), S (solid orange), PP (dotted blue), SS (dotted orange), PcP
  (dashed light blue), ScS (dashed red-orange), PKP (green), PKIKP
  (magenta) and SKS (dashed black) are plotted. A pale orange vertical
  band from 103 to 143 degrees is labelled "P shadow" and shows P and
  PP terminating at the left edge while PKP and PKIKP emerge at the
  right edge. Phase labels sit at the right ends of each curve.
:width: 95%

Global travel-time curves for the AK135 reference Earth model,
computed for a 10-km-deep source using `obspy.taup`. Each curve is
the forward prediction of a different phase. The shaded band marks
the P-wave shadow zone: P and PP terminate at its left edge, and the
core-refracted phases PKP and PKIKP emerge at its right edge. SKS
appears near $80^\circ$ where S-wave energy can convert to P at the
CMB and back.
```

Reading this diagram is a fundamental skill. A student who can
identify which travel-time branch corresponds to which phase, and
who understands what part of Earth each phase has sampled, can
interpret almost any global seismogram. **Phase identification is
also the foundation on which every higher-order seismology skill is
built.** Lab 6 puts this skill into practice using live IRIS data.

:::{admonition} Phase naming convention
:class: note

A single upper-case letter denotes a leg of the ray path. The
conventions are: **P** = P-wave in the mantle; **S** = S-wave in the
mantle; **K** = P-wave in the outer core; **I** = P-wave in the inner
core; **J** = S-wave in the inner core. A lower-case **c** denotes
reflection at the CMB, a lower-case **i** denotes reflection at the
ICB. Thus `PcP` is a P-wave that descends through the mantle,
reflects off the CMB, and returns through the mantle; `PKIKP` passes
through mantle, outer core, inner core, outer core, and back through
the mantle.
:::

```{figure} ../assets/figures/fig_11_phase_nomenclature.png
:name: fig-phase-key
:alt: Left: cross-section of Earth with a source at the top and
  labelled ray paths for P (solid blue arcing through mantle to the
  right), S (dashed orange arcing through mantle to the left), PP
  (light blue, two concave arcs meeting at a surface reflection
  midway), PcP (pink, V-shape reflecting off the CMB), PKP (green,
  four-leg path refracting through the outer core), and PKIKP (dark
  purple, dashed, passing through mantle, outer core, inner core,
  outer core, and back through mantle). Tick marks in 30-degree
  increments from 0 to 180 on both sides. Right: phase-nomenclature
  key listing P, S, K, I, J, c, i with their meanings, and example
  readings for PcP, PKIKP, and SKS.
:width: 98%

The major body-wave phases and the naming convention that encodes
their ray paths. Each segment of a phase name describes one leg of
the journey through Earth's interior.
```

---

## 6. A worked example: measuring the depth to the CMB

Suppose we know the P-wave travel time for a direct mantle arrival
at $\Delta = 90^\circ$ is $T_P \approx 12.8$ min = 768 s, and the
PcP reflection time at the same distance is $T_{PcP} \approx 13.4$
min = 804 s (both from AK135). The differential time
$T_{PcP} - T_P = 36$ s corresponds to the extra path travelled by
PcP, which descends all the way to the CMB and back before returning
to the same receiver. Order-of-magnitude estimation says that extra
path is approximately $2(d_{\mathrm{CMB}} - d_{\mathrm{turning\ depth\ of\ P}})$
divided by the average mantle P-velocity of $\sim$ 11 km/s, so

$$
d_{\mathrm{CMB}} - d_{P\text{-turn}} \;\sim\; \tfrac{1}{2} \cdot 11 \;
\mathrm{km/s} \cdot 36\;\mathrm{s} \approx 200\;\mathrm{km}.
$$

Combined with the estimate that the direct P-ray at $90^\circ$ turns
near $\sim 2700$ km depth, this places the CMB at roughly $2900$ km.
That is the Gutenberg discontinuity, and the crude estimate is
within a few percent of the modern value of $2891$ km.

---

## 7. The answer: the Preliminary Reference Earth Model (PREM)

Three quarters of a century of global travel-time measurements,
normal-mode observations, and mass-and-inertia constraints were
inverted jointly by Dziewonski and Anderson (1981) to produce the
Preliminary Reference Earth Model (PREM). PREM reports $V_P$, $V_S$,
density $\rho$, attenuation $Q$, and transverse anisotropy as
functions of radius. It remains the standard 1-D reference 45 years
after publication, and every 3-D tomographic model published since
is reported as a *deviation* from PREM or a close relative such as
AK135.

```{figure} ../assets/figures/fig_11_prem_profile.png
:name: fig-prem
:alt: Vertical profile plot with depth increasing downward from 0 at
  the surface to 6371 km at Earth's centre, and velocity or density on
  the horizontal axis from 0 to 14.5 km/s or g per cubic cm. Blue
  solid curve shows P-wave velocity Vp rising stepwise from 5.8 km/s
  at the surface through jumps at the Moho (24 km), 410 km, and 660
  km, reaching 13.7 km/s just above the core-mantle boundary at 2891
  km, dropping abruptly to 8.1 km/s across the CMB, then increasing
  through the outer core to about 10.3 km/s at the inner-core boundary
  at 5150 km where it jumps to about 11.0 km/s in the inner core.
  Orange solid curve shows S-wave velocity Vs, rising similarly
  through the mantle from 3.2 to 7.3 km/s, then dropping to zero in
  the outer core (a gap in the line) and reappearing at about 3.5
  km/s in the solid inner core. Dashed green curve shows density,
  increasing from about 2.6 at the surface to about 13.1 g per cubic
  cm at Earth's centre. Horizontal guide lines mark the Moho, LAB,
  410, 660, CMB, and ICB; layer labels on the left identify crust and
  lithosphere, mantle, liquid outer core (with no Vs), and solid
  inner core.
:width: 70%

The Preliminary Reference Earth Model (PREM; Dziewonski and Anderson
1981). The three curves are $V_P$ (blue), $V_S$ (orange), and density
(green dashed). Major discontinuities are annotated. The gap in the
$V_S$ curve across the outer core is the seismological signature of a
liquid layer: shear waves cannot propagate. The jump in $V_P$ at the
inner-core boundary, together with the reappearance of $V_S$ inside
the inner core, indicates solid iron.
```

PREM is an inverse solution in the cleanest possible sense: an
enormous number of observations (travel times of many phases at many
distances, periods of free-oscillation eigenmodes, the mass and
moment of inertia of the planet) are combined with a
parametrisation of $V_P(r)$, $V_S(r)$, $\rho(r)$ through several
hundred parameters, and optimisation finds the profile that best
fits all data jointly. The solution is *non-unique*; ten different
radial profiles that differ by small oscillations can fit the same
travel-time data to within their error. PREM was chosen because it
is smooth and physically reasonable, not because it is the only
profile consistent with the data. Lecture 12 will return to this
point.

:::{admonition} Reasoning prompt — why density needs separate data
:class: tip

Travel-time data of body waves constrain $V_P(r)$ and $V_S(r)$ but
**not** $\rho(r)$ directly: a P-wave travel time depends on
$V_P = \sqrt{(\kappa + \tfrac{4}{3}\mu)/\rho}$, so a $\rho$ change can
always be absorbed by a corresponding change in $\kappa$ or $\mu$.
Density enters PREM through three independent constraints that body-wave
times alone cannot supply:

1. **Total mass** $M_\oplus = 5.972\times 10^{24}$ kg (from Newtonian
   gravity + the Moon's orbit) sets $\int_0^R \rho(r)\,4\pi r^2 dr$.
2. **Moment of inertia** $I/M_\oplus R^2 \approx 0.3307$ (from precession
   and lunar laser ranging) sets the *radial weighting* of $\rho(r)$ —
   a denser core lowers $I/M R^2$.
3. **Free-oscillation eigenfrequencies** depend on the elastic moduli
   *and* on $\rho$ through the equations of motion, breaking the
   $\rho$–modulus trade-off at specific depths.

Reasoning question to take to office hours: if we only had body-wave
travel times (no mass, no moment, no normal modes), could we still
prove the inner core is solid? Argue from the role $V_S$ plays in the
PKIKP raypath, then explain what we *could not* determine.
:::

:::{admonition} Reasoning prompt — what the shadow zone tells us, and what it doesn't
:class: tip

The shadow zone proves the outer core is liquid (no $V_S$ at any depth
in the 103°–180° band) and bounds the core radius (the geometric 103°
edge fixes where the CMB sits). It does **not** by itself prove:

- The core is iron-rich (density argument from mass + moment).
- The inner core is solid (Lehmann's PKIKP energy in the gap;
  J-wave detections by Tkalčić & Pham 2018).
- The inner core rotates differentially (decadal PKIKP residuals).

Distinguishing **what a single observation constrains** from **what
is layered on top of it through additional data** is the central
reasoning skill of whole-Earth seismology.
:::

---

## 8. Connecting to Cascadia and the Pacific Northwest

The travel-time curves and the PREM profile describe a spherically
symmetric reference Earth — a useful zeroth-order model, but not
literally the planet we live on. Three-dimensional deviations from
PREM reveal the Juan de Fuca slab subducting beneath our feet, the
mantle wedge that feeds the Cascade arc, and the plume tail beneath
Yellowstone. Every time a Cascadia seismologist measures a teleseism
arriving at a PNSN station, the first-arrival time is compared to the
AK135 prediction; the residual, positive or negative by a few
seconds, is a direct measurement of 3-D mantle structure beneath the
station. This is the material of Lecture 12.

---

## 9. Research Horizon

Modern whole-Earth seismology is moving beyond the 1-D reference
picture in several directions.

- **Inner-core differential rotation.** Cross-comparisons of PKIKP
  travel times from repeating earthquake sources recorded over
  decades (Song and Richards 1996; Tkalčić and Pham 2018) have
  identified small but resolvable inner-core rotation rates relative
  to the mantle. The most recent analysis of multi-decade PKIKP
  doublets reports *annual-scale* variability in both the rotation
  rate and the near-surface structure of the inner core
  ({cite:t}`Vidale2025InnerCore`), reframing the long-running
  "rotation vs. surface change" debate as a coupled, time-varying
  process rather than a steady differential rotation.

- **Comparative planetology.** The NASA InSight mission placed a
  broadband seismometer on Mars in 2018 and recorded more than 1300
  marsquakes through 2022. SKS-like converted phases have been used
  to image the Martian core, establishing that it is liquid and
  larger than previously thought (Stähler et al. 2021,
  https://doi.org/10.1126/science.abi7730). Apollo-era seismometers
  on the Moon similarly imaged a small lunar core (Weber et al.
  2011, https://doi.org/10.1126/science.1199375).

- **CMB and inner-core boundary texture.** Ultra-low velocity zones
  (ULVZs) and D″ phase transitions detected with ScS precursors and
  PKKP-diffracted phases are now being mapped globally, with
  implications for mantle plume anchoring (Cottaar and Lekić 2016).

- **AI-assisted global tomography.** Machine-learning phase-picking
  (e.g., PhaseNet, Zhu and Beroza 2019,
  https://doi.org/10.1093/gji/ggy423; EQTransformer, Mousavi et al.
  2020, https://doi.org/10.1038/s41467-020-17591-w) has increased
  global travel-time catalogs by one to two orders of magnitude over
  the past five years, enabling higher-resolution 1-D and 3-D models.
  The methodology is discussed in Lecture 12.

---

## 10. AI Literacy

:::{admonition} AI Literacy — Shadow Zone Epistemics (LO-7)
:class: tip

**Prompt Lab.** Ask a chat assistant: *"Why does the S-wave shadow
zone extend to 180° but the P-wave shadow zone only goes from 103°
to 143°?"* Then evaluate the response against Figure
{numref}`fig-shadow-zones`.

Common failure modes to watch for:

1. **Attributing both shadows to absorption** rather than distinguishing
   the two distinct mechanisms (fluid constitutive property for S;
   refraction geometry for P).
2. **Confusing 103° geometry with the width of the P-shadow**: 103° is
   set by the ray that grazes the CMB; the outer edge at 143° is set
   by the PKP caustic, a separate piece of physics.
3. **Hallucinating velocity numbers**: correct values are
   $\alpha_\text{mantle} \approx 13.7$ km/s and
   $\alpha_\text{outer\,core} \approx 8$ km/s just across the CMB;
   $\beta_\text{outer\,core} = 0$.

**The epistemic skill.** Compare what the model says against the figure
and against `obspy.taup.TauPyModel('ak135')`. Report numerical discrepancies
explicitly and treat AI-generated shadow-zone explanations as *drafts to
cross-check*, not authoritative answers.
:::

:::{admonition} AI Literacy — Phase-Name Epistemics (LO-7)
:class: tip

**Prompt to try.** *"List the seismic phases that pass through
Earth's inner core and give their typical travel-time ranges for a
teleseism at 150 degrees epicentral distance."*

**What to check.** A modern language model will readily produce a
confident-sounding list: PKIKP, PKJKP, PKiKP, and perhaps others.
Your task is to verify, independently, each claim the model makes.
For each phase the model names, answer three questions: (i) is the
phase name syntactically valid under the P/S/K/I/J/c/i convention?
(ii) does the phase actually exist — i.e., has it been observed on
real seismograms? (iii) is the travel-time range the model gives
consistent with what `obspy.taup.TauPyModel` (ak135 or iasp91)
returns for 150 degrees? Phase names like `PKJKP` and `PKiKP` are
controversial — some have been claimed in the literature but never
unambiguously confirmed. A model that reports them without flagging
the controversy is hallucinating authority.

**The epistemic skill.** Never treat an AI-generated list of named
scientific objects as a primary source. Cross-check against a
first-principles tool (here, `obspy.taup`) and a peer-reviewed
review paper. Report discrepancies as discrepancies, not as minor
details to paper over.
:::

---

## 11. Concept Checks

1. **[LO-11.1]** If the Earth had a uniform P-wave velocity of
   $V_P = 10$ km/s everywhere from surface to centre, what would the
   travel time be for a ray emerging at $\Delta = 180^\circ$?
   Compare to the actual PKIKP time (about 20 minutes) and explain
   the sign of the difference.

2. **[LO-11.2]** A seismogram records a clear first arrival at
   $T = 7$ min for a known earthquake. You also identify an S-like
   arrival at $T = 13$ min. Using Figure {numref}`fig-tt-curves`,
   estimate the epicentral distance. What phases would you expect
   to see next, and at what times?

3. **[LO-11.3]** Lehmann's 1936 discovery of the inner core rested
   on observing P-wave energy *inside* the predicted P-wave shadow
   zone. Explain, in terms of Snell's law, why a solid inner core
   with $V_P > V_P$(outer core) would produce such arrivals. What
   would the PKIKP travel-time curve look like if the inner core did
   not exist?

4. **[LO-11.3 — reasoning chain]** Suppose a future probe of an
   exoplanet records body-wave travel times that show **no** S-wave
   shadow zone at any distance, but a P-wave shadow at
   $\sim 110^\circ$–$140^\circ$. What can you infer about that
   planet's interior, and what can you *not* infer? Build the
   argument step by step from the shadow-zone reasoning of Section 4.

5. **[LO-11.2 — quantitative reasoning]** Two stations record the
   same earthquake at $\Delta = 60^\circ$ and $\Delta = 95^\circ$.
   Sketch how the Bullen ray parameter $p = r\sin i / V$ differs
   between the two rays, predict which ray turns deeper, and explain
   why the deeper ray has the *smaller* $p$. Use this to argue why
   the slope of the global $T(\Delta)$ curve is monotonically
   decreasing with distance up to $\sim 100^\circ$.

---

## 12. Connections

- **Previous lectures.** This lecture reuses Snell's law (Lecture 3)
  and the concept of the forward/inverse problem (Lecture 4).

- **Companion lab.** Lab 6 asks you to use `obspy` to download three
  real teleseisms recorded at IRIS-network stations, to pick the
  arrival times of the major phases, and to compare to AK135
  predictions. The AI Literacy component of Lab 6 critically
  evaluates ML-based phase pickers against manual picks.

- **Next lecture.** Lecture 12 takes the residuals — the
  observed-minus-predicted travel times — and uses them as data to
  invert for 3-D structure. The forward problem is the same one
  that built PREM; the twist is that the model $\mathbf{m}$ now
  represents cell-by-cell velocity perturbations rather than a
  smooth radial profile.

---

## Further Reading

**Open-access and freely available:**

- Kennett, B.L.N., Engdahl, E.R., Buland, R., 1995. *Constraints on
  seismic velocities in the Earth from travel times.* Geophys. J.
  Int. 122(1), 108-124.
  https://doi.org/10.1111/j.1365-246X.1995.tb03540.x

- Stein, S. and Wysession, M., 2003. *An Introduction to Seismology,
  Earthquakes, and Earth Structure.* Blackwell. (UW Libraries
  electronic access.) Chapters 3.3-3.5 cover whole-Earth travel
  times and phase identification.

- IRIS / EarthScope Global Stacks and Seismic Phases explorer:
  https://ds.iris.edu/spud/eventplot

- Tkalčić, H. and Pham, T.-S., 2018. *Shear properties of Earth's
  inner core constrained by a detection of J waves in global
  correlation wavefield.* Science 362(6412), 329-332.
  https://doi.org/10.1126/science.aau7649

**Primary textbook reference (required for this course):**

- Lowrie, W. and Fichtner, A., 2020. *Fundamentals of Geophysics*,
  3rd ed., Cambridge University Press. Chapter 3.5-3.6. (Free via
  UW Libraries.)

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```