---
title: "Travel-Time Tomography: Ray Coverage, Resolution and Damping Interactive Lab | Sandbox Physics"
description: "How do many travel times become a map of fast and slow rock, and how far can you trust that map?"
canonical_url: "https://sandboxphysics.com/labs/travel-time-tomography/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/travel-time-tomography/"
last_modified: "2026-10-31"
---

FR36-C · Linear tomography and its resolution matrix

# Travel-Time Tomography: Ray Coverage, Resolution and Damping

**How do many travel times become a map of fast and slow rock, and how far can you trust that map?** Each travel time is a sum of slowness along its ray, so many crossing rays give a linear system for the slowness of every block. Rock that many rays cross from many directions is resolved; elsewhere the image is smeared along the rays. Damping keeps noise from turning into false structure, at the cost of weaker anomalies.

Rays cross a 40 km block that hides a slow and a fast anomaly. The delays are inverted on a 16 by 16 grid. Change the station layout and the damping and see what the image can and cannot resolve.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Model assumptions

Straight rays through a 40 km square at a background P speed of 6 km/s; slowness perturbations are small, so travel-time delays are linear in the model. Forward delays for the smooth anomalies are exact line integrals of Gaussian blobs, computed without the inversion grid; checkerboard and spike tests use the grid itself. The inversion minimizes the weighted data misfit plus damping and first-difference smoothing (Rawlinson and Spakman 2016, Eq. 7) by a direct Cholesky solve; the animation shows conjugate-gradient iterations towards the same answer. The resolution kernel shown is one column of the resolution matrix, which maps a true model to its reconstruction. Ray bending, anisotropy and source mislocation are omitted.

Data: measured delays and the reconstruction’s prediction

- measured, slower ray
- measured, faster ray
- predicted

Where the rays can resolve: resolution-matrix diagonal

- near 1: resolved
- near 0: not resolved

Point-spread function of the probe cell

- probe cell

More readouts and the model record

- **Rays**
- **Cells**
- **Cells no ray crosses**
- **Pick noise**
- **Conjugate-gradient iterations shown**
- **Resolution-matrix diagonal at the probe cell**

The record saves the rays, the measured and noise-free delays, the true model, the reconstruction and the diagonal of the resolution matrix.

[Continue the course: seismic imaging from waves to inversion](https://sandboxphysics.com/stories/seismic-imaging/)

Physics tutorial

## Travel-Time Tomography: Ray Coverage, Resolution and Damping

**Background** Slowness is the inverse of wave speed. A wave that crosses slow rock arrives late; one that crosses fast rock arrives early. The delay of each ray is the sum of the extra slowness along its path.

**Why it matters** Divide the region into cells. Then each delay is a weighted sum $d_i=\sum_j G_{ij}m_j$, with the weights equal to the ray length in each cell. Inverting many such sums is travel-time tomography.

Start with the essentials

- **Focus question**: How do many travel times become a map of fast and slow rock, and how far can you trust that map?
- **One-sentence intuition**: The data fix only combinations of cells that rays sample differently. Regularization picks one model among many that fit, and the resolution matrix shows exactly how each true cell is blurred into the image.

### Core mathematical model

#### Regularized least squares

$$
\Phi(\mathbf m)=(\mathbf G\mathbf m-\mathbf d)^{\mathsf T}\mathbf C_d^{-1}(\mathbf G\mathbf m-\mathbf d)+\varepsilon^2\tau\,\mathbf m^{\mathsf T}\mathbf m+\eta^2\tau\,\mathbf m^{\mathsf T}\mathbf D^{\mathsf T}\mathbf D\mathbf m
$$

Damping $\varepsilon$ pulls the model towards the background; smoothing $\eta$ penalizes differences between neighbouring cells. The mean diagonal of the data term, $\tau$, makes both weights dimensionless.

#### Resolution matrix

$$
\hat{\mathbf m}=\mathbf R\,\mathbf m_{\rm true}+\mathbf G^{-g}\boldsymbol\epsilon,\qquad \mathbf R=\mathbf G^{-g}\mathbf G
$$

With the generalized inverse $\mathbf G^{-g}$, the estimate is the truth filtered by $\mathbf R$ plus mapped noise. A column of $\mathbf R$ is the image of a single spike.

### Common difficulties

#### A good checkerboard is not a guarantee

**Typical misconception** If the checkerboard is recovered, every feature of the real image is reliable.

**Better mental model** A checkerboard tests one pattern at one amplitude. Rawlinson and Spakman (2016) show that it can look good while other structures are smeared; look at point-spread functions as well.

### Run the experiment

1. 01

   #### Compare station layouts

   Start from Stations all around, then choose Two wells: sideways smear.

   **What to observe:** With only near-horizontal rays, both anomalies are smeared sideways and the probe cell’s point-spread function stretches into a horizontal streak.

2. 02

   #### Run the checkerboard test

   Choose Checkerboard test, then switch the stations to Quakes below.

   **What to observe:** The checkerboard comes back well inside the ring and loses its pattern in the corners that rays do not cross.

3. 03

   #### Take away the damping

   Choose No damping: noise takes over and watch the misfit per ray.

   **What to observe:** The misfit falls far below one while the image fills with stripes: the inversion is fitting the noise.

Physical conclusion

Ray coverage decides what can be seen; regularization decides what you choose to show.

## Related experiments and articles

Seismic imaging · Reading path

### Seismic waves and imaging, from rock boundaries to tomography

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **Travel-time tomography and what the resolution matrix shows**](https://sandboxphysics.com/stories/seismic-imaging/travel-time-tomography-and-resolution/)
2. [Article **Reading Rawlinson and Spakman (2016) on resolution tests in seismic tomography**](https://sandboxphysics.com/stories/seismic-imaging/reading-rawlinson-spakman-2016/)

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)