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.

Pattern match with the truth $0.724$

Anomaly amplitude recovered $96.5\,\%$

Misfit per ray (chi-square) $0.171$

Mean resolution-matrix diagonal $0.422$

Invert the delays, then ask what is resolved

Pick noise (ms)

\sigma_t

20

Damping, log scale

\log_{10}\varepsilon

-1

Smoothing, log scale

\log_{10}\eta

-0.5

Stations

True model

Drag a ring in the true Earth and watch the reconstruction follow; drag the green probe cell to see how one cell is smeared.

Model, units and sources

$\hat{\mathbf m}=\left(\mathbf G^{\mathsf T}\mathbf C_d^{-1}\mathbf G+\varepsilon^2\tau\mathbf I+\eta^2\tau\mathbf D^{\mathsf T}\mathbf D\right)^{-1}\mathbf G^{\mathsf T}\mathbf C_d^{-1}\mathbf d$

The block is 40 km square; anomalies change the P speed by a few per cent.

Read the course chapter behind this Lab

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

Travel-Time Tomography: Ray Coverage, Resolution and Damping

BackgroundSlowness 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 mattersDivide 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 misconceptionIf the checkerboard is recovered, every feature of the real image is reliable.

Better mental modelA 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

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.
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.
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.

Travel-Time Tomography: Ray Coverage, Resolution and Damping

Travel-Time Tomography: Ray Coverage, Resolution and Damping

Core mathematical model

Regularized least squares

Resolution matrix

Common difficulties

A good checkerboard is not a guarantee

Run the experiment

Compare station layouts

Run the checkerboard test

Take away the damping

Related experiments and articles

Seismic waves and imaging, from rock boundaries to tomography