Building Earth Images

Learning objectives

By the end of this lecture:

• Explain the difference between forward modeling ($\mathbf{d} = F\mathbf{m}$) and migration ($\hat{\mathbf{m}} = F^\top\mathbf{d}$)
• Show why even flat reflectors need a layered velocity model to be correctly depth-converted
• Derive the migration corrections $\Delta x = d\sin\theta$ and $\tau = t\cos\theta$ for dipping layers
• Describe how Kirchhoff migration collapses diffractions to their source points
• Diagnose from a migrated image whether the velocity was correct, too slow (frowns), or too fast (smiles)
• Apply the 8-step iterative workflow combining refraction and reflection
• Evaluate a deep-learning surrogate: who provided its training data?

The Hikurangi subduction margin

Pacific plate subducts beneath North Island, New Zealand.
How do we image the plate interface at 5–20 km depth?
Same airgun shots → refractions to OBS (long offset) + reflections to streamer (short offset)

Two windows, one Earth

Refraction imaging (OBS, long offsets)
→ absolute layer velocities $v_1, v_2, \ldots$
→ reliable to ~1–10 km depth

Reflection imaging (streamer, shorter offsets)
→ structural image at all depths
→ velocities are relative, not absolute

Neither method alone gives a quantitatively correct depth image.
Their combination, iterated, does.

The framework: forward and inverse modeling

$$\mathbf{d} = F\,\mathbf{m} \qquad \hat{\mathbf{m}} = F^\top\,\mathbf{d}$$

Both need $v(x,z)$. Migration ($F^\top$) is not the true inverse — it's the adjoint. Its quality depends entirely on the velocity model.

Building the image: four cases

Each case adds one physical complication. Each case shows the same lesson: you need an accurate velocity model.

Case 1 — Flat layer, constant velocity

For a flat reflector at depth $z$, velocity $v$, zero-offset geometry:

$$t = \frac{2z}{v} \quad\Longrightarrow\quad z = \frac{vt}{2}$$

• Normal ray is vertical — the display is correct
• "Migration" = multiply by $v/2$ → exact time-to-depth conversion
• No horizontal shift, no depth error

This is why intro courses skip migration for flat layers. The display is already right.

Case 2 — Multiple flat layers: the velocity matters

Three layers with velocities $v_1 < v_2 < v_3$, interfaces at $z_1 < z_2 < z_3$:

$$t_1 = \frac{2z_1}{v_1}, \quad t_2 = t_1 + \frac{2(z_2-z_1)}{v_2}, \quad t_3 = t_2 + \frac{2(z_3-z_2)}{v_3}$$

To recover depths → need interval velocities $v_1, v_2, v_3$ via Dix:

$$v_n^2 = \frac{V_{{\rm rms},n}^2\,t_n - V_{{\rm rms},n-1}^2\,t_{n-1}}{t_n - t_{n-1}}$$

Problem: Dix integrates downward. Error in $v_1$ propagates into $v_2$, $v_3$.
Solution: Refraction gives absolute $v_1$ → anchors the chain.

Case 2 — The depth image depends on velocity

Same two-way times. Two velocity assumptions. Very different images.
Constant $v = 2.0$ km/s → deepest interface 31% wrong.
Refraction + Dix → all three interfaces exact.

Case 3 — Dipping layers: mispositioning

For a dipping reflector ($\theta$ from horizontal), the normal ray is not vertical.
The instrument records only $t = 2d/v$ — no directional information.
Conventional display: plot event straight down at depth $d$. Two errors result:

$$\Delta x = d\sin\theta \quad (\text{too far downdip})$$

$$\tau = t\cos\theta \quad (\text{corrected time, shallower depth})$$

Hand-migration formulas (using slope $p_0 = \partial t / \partial y$):

$$\Delta x = \frac{v^2 p_0 t}{4}, \qquad \tau = t\sqrt{1 - \frac{v^2 p_0^2}{4}}$$

Both $\to 0$ when $\theta \to 0$ (flat-layer limit).

Case 3 — Flat to dipping: error grows with dip

Migration applies the corrections. Both corrections depend on the velocity — again.

Case 4 — Diffractions: signature of structure

Any geometric discontinuity (fault tip, unconformity edge, salt flank) generates a diffraction.

In zero-offset data, a point scatterer at $(x_0, z_0)$ produces a hyperbola:

$$t(y) = \sqrt{\left(\frac{2z_0}{v}\right)^2 + \left(\frac{2(y-x_0)}{v}\right)^2}$$

An unprocessed section over complex geology is full of overlapping hyperbolas.

Three challenges in a real accretionary wedge:

• Dipping reflectors → mispositioning (Case 3)
• Surface multiples → same $V_{\rm rms}$, hard to remove
• Diffraction hyperbolas → geological structure hidden until migrated

Case 4 — Diffractions in the Cascadia wedge

The diffraction hyperbola at the fault tip is not noise — it is the fault.
Kirchhoff migration collapses it to a point at the fault tip location.

Kirchhoff migration: the adjoint pair

Forward $F$: scatterer $(x_0, z_0)$ $\to$ hyperbola in data. Writes energy along the curve.
Migration $F^\top$: sums data along hyperbola $\to$ focused point. Reads energy along the same curve.

Kirchhoff in pseudocode

for every (ix, iz) in the model:
    for every midpoint y in the data:
        t = sqrt( (2·z[iz]/v)² + (2·(x[ix]−y)/v)² )   # same hyperbola!
        if forward:   data[t, y]    += model[iz, ix]    # F  : spreads
        else:         model[iz, ix] += data[t, y]       # Fᵀ : collapses

Same loop. Same geometry. Opposite direction of the copy operation.
This is what "migration is the adjoint of forward modeling" means concretely.

The velocity–image duality

$$m(x,z) = \sum_{y} w \cdot d\!\left(y,\;\sqrt{\left(\tfrac{2z}{v}\right)^2 + \left(\tfrac{2(x-y)}{v}\right)^2}\right)$$

The summation hyperbola depends on $v$. Wrong $v$ → wrong hyperbola → residual energy.

The image is the velocity diagnostic.

Frowns and smiles

Frowns → migration hyperbola too narrow → velocity too slow.
Smiles → migration hyperbola too wide → velocity too fast.
No borehole needed to read this diagnostic.

Why each method needs the other

Refraction anchors the velocity. Reflection reveals the structure. Migration fuses both.

The 8-step iterative workflow

Loop 6 → 7 → 8 until gathers are flat and diffractions focused.

The feedback that makes iteration work

Refraction → v_shallow (absolute)
    ↓
Dix → v_deep (relative, anchored)
    ↓
Migration (Step 6) → image
    ↓
Frowns? → increase v    ←───────┐
Smiles? → decrease v    ←───────┤
Flat?   → DONE          ←───────┘
              ↑
       Velocity update (Step 8)

The velocity–image duality converts image quality into velocity corrections — without any external reference.

Deep learning as an accelerator

Training data for all three networks was produced by physics-based operators.

DL accelerates the chain. It does not replace the physics upstream of its training data.

Ask this of any surrogate

1. What physics-based operator does this network replace?
2. Who produced the training data, and what physical knowledge was required?
3. What is the training distribution — would you trust this network outside it?

Same questions apply to regression formulas, empirical curves, and neural networks.
The depth of the physics required to answer them is the depth of this course.

Concept check

A zero-offset section shows a diffraction hyperbola with apex at
$(x = 3.0\text{ km},\ t_0 = 1.2\text{ s})$.

After migration with $v = 2.0$ km/s → crisp point image.
After migration with $v = 2.5$ km/s → upward-curving arc.

1. Which velocity is more correct, and how can you tell?
2. What depth does the correct image imply?
3. What would $v = 1.5$ km/s produce?

Discuss with your neighbor. Write one sentence on your index card.

Why Hikurangi matters

• Capable of $M_w > 8.5$ megathrust + trans-Pacific tsunami
• Plate interface depth and coupling → building codes, evacuation zones, shakemaps
• Northern margin: unusually shallow interface (1–2 km below seafloor near trench)
  → highest tsunami hazard → only known from seismic imaging

Published campaigns (Wallace et al. 2009; Barker et al. 2018) applied exactly the 8-step workflow:
OBS first arrivals → shallow $v$ → reflection migration → focused plate-interface image.

GNS Science programme: https://www.gns.cri.nz/research-projects/hikurangi-subduction-margin/

Lab 4 — Design → Simulate → Image

The lab notebook provides:

• A multi-layer synthetic forward model (wave equation simulation)
• A Kirchhoff migration routine (~30 lines of NumPy)
• A set of three migration velocity options

Students will:

1. Run the forward model to generate a synthetic zero-offset section
2. Migrate with correct $v$, $0.80\times v$, and $1.25\times v$
3. Report which image is correct and how they could tell from the image alone
4. Add a 4th migration velocity from refraction-only input — does it improve or degrade the image?

Tying it together

• Forward model $\mathbf{d} = F\mathbf{m}$: given Earth, predict data
• Migration $\hat{\mathbf{m}} = F^\top\mathbf{d}$: given data, estimate Earth (requires $v$)
• Four cases of increasing complexity all demand accurate $v(x,z)$:
  flat → multi-layer → dipping → diffractions
• Refraction + reflection + migration = one iterative loop to estimate $v$ and build the image
• Velocity–image duality: the image itself diagnoses whether $v$ is right
• Deep learning accelerates steps; does not replace the physics upstream of its training data

Further reading

• Claerbout (2010). Basic Earth Imaging, Ch. 3–5. Open: http://sepwww.stanford.edu/sep/prof/bei11.2010.pdf
• Lowrie & Fichtner (2020). Fundamentals of Geophysics, Ch. 3 (UW Libraries)
• Zelt & Barton (1998). Refraction tomography. JGR 103, 7187
• Mardan & Fabien-Ouellet (2024). First-break picking U-Net. Near Surface Geophysics
• Li et al. (2024). Self-supervised denoising. Geophysics
• Yang & Ma (2019). Velocity model building. Geophysical Journal International