Example 02: Radial Anisotropy Checkerboard

This example reproduces SurfATTPP/examples/02_checkerboard_radial_ani. It uses both Rayleigh- and Love-wave phase velocities to test recovery of isotropic velocity structure and radial anisotropy.

Model parameterization

SurfATT represents radial anisotropy using two polarized S-wave velocities:

The effective velocity and radial-anisotropy strength are

The inverse conversion used by SurfATT is

The value of $\zeta$ provides a direct interpretation of the anisotropy:

• $\zeta = 1$: $V_{SH}=V_{SV}$, so the S-wave model is isotropic.
• $\zeta > 1$: $V_{SH}>V_{SV}$, so horizontally polarized S waves are faster.
• $\zeta < 1$: $V_{SH}<V_{SV}$, so vertically polarized S waves are faster.

Internally, the radial-anisotropy inversion updates $V_{SV}$ and

The HDF5 model files expose all four user-facing fields—vsv, vsh, vs, and zeta—so results can be plotted using either velocity components or the more intuitive $(V_s,\zeta)$ parameterization.

What is included

Parameter setup

The default configuration uses:

• radial-anisotropy inversion (model_para_type: 2)
• Rayleigh- and Love-wave phase-velocity data
• model depth range of 0-8 km
• longitude, latitude, and depth intervals of 0.02° / 0.02° / 0.5 km
• up to 40 LBFGS iterations

Setting model_para_type: 2 activates the $V_{SV}$–$\gamma$ radial-anisotropy inversion and requires both Rayleigh and Love data.

Run the example

The example has two dependent stages. First, SURFATT_cb_fwd creates the target checkerboard and synthetic observations. Then, SURFATT_tomo starts from the unperturbed model and inverts those observations.

For a complete automatic run, execute the supplied script from the example directory:

bash run_this_example.sh

The following sections unpack the script step by step.

1. Prepare the forward inputs

Create the output directory and copy the source-receiver geometry to the filenames configured in input_params.yml:

mkdir -p OUTPUT_FILES
cp src_rec_file_ph.csv OUTPUT_FILES/src_rec_file_forward_RL_PH.csv
cp src_rec_file_ph.csv OUTPUT_FILES/src_rec_file_forward_LV_PH.csv

The default configuration contains

wave_type: [True, True]   # Rayleigh and Love
vel_type:  [True, False]  # phase only

Therefore, only src_rec_file_forward_RL_PH.csv and src_rec_file_forward_LV_PH.csv are read in this example. The helper script also copies the two group-velocity tables so the example can be extended by changing vel_type to [True, True].

At this point the phase files provide the source-receiver geometry. Their travel-time and velocity columns will be replaced by synthetic values during the forward calculation.

2. Generate the target model and synthetic data

Run the radial-anisotropy checkerboard forward calculation:

mpirun -np 8 ../../bin/SURFATT_cb_fwd \
  -i input_params.yml \
  -n 2/3/2 \
  -s 2 \
  -r \
  -m 0.2 \
  -p 0.08/0.08

The command-line options define the synthetic model:

The forward stage proceeds as follows:

1. SurfATT builds the 1-D starting model and extrudes it over the 3-D grid. The initial radial-anisotropy ratio is $\zeta_0=1$, so $V_{SV}=V_{SH}$.

2. Alternating perturbations are applied in the $(V_s,\zeta)$ parameterization:

   $$V_s^{\mathrm{target}}=V_s^0(1\pm0.08), \qquad \zeta^{\mathrm{target}}=1(1\pm0.10).$$

3. Each checkerboard cell is converted from $(V_s,\zeta)$ to $(V_{SV},V_{SH})$ for the wave-propagation calculation.

4. Rayleigh- and Love-wave phase travel times are calculated through the target model.

5. The synthetic travel times are written back to src_rec_file_forward_RL_PH.csv and src_rec_file_forward_LV_PH.csv. These two files become the observations for the inversion.

The most important products of the forward stage are:

Target checkerboard

The horizontal slices below show the prescribed $V_{SV}$ and $\zeta$ perturbations at depths of 1 and 5 km. Both parameters use the same alternating 2 × 3 lateral pattern, while the sign reverses between the shallow and deep checkerboard layers.

Target $V_{SV}$ and $\zeta$ perturbations at 1 and 5 km depth.

The vertical section at latitude $25^\circ$ shows the two depth layers directly. The upper anomalies occupy approximately 0–2 km, while the deeper checkerboard extends through the lower part of the 0–8 km model.

Target checkerboard section at latitude $25^\circ$.

3. Invert the synthetic observations

After the forward command finishes successfully, run:

mpirun -np 8 ../../bin/SURFATT_tomo -i input_params.yml

The inversion uses the same input_params.yml, but it does not receive the target model. It only reads the synthetic Rayleigh and Love observations and reconstructs the model from the unperturbed starting model.

For each inversion iteration, SurfATT:

1. predicts Rayleigh- and Love-wave phase travel times from the current model;
2. calculates residuals relative to the synthetic observations;
3. computes adjoint sensitivity kernels for $V_{SV}$ and $\gamma=V_{SH}/V_{SV}$;
4. smooths and conditions the gradients using the postproc settings;
5. uses LBFGS with line search to update $V_{SV}$ and $\gamma$; and
6. records the misfit, current model, gradients, and predicted travel times.

The run stops when it reaches niter: 40, when the recent mean-misfit reduction falls below min_derr: 0.0015%, or when the line search can no longer find an acceptable update.

4. Check the inversion products

The main inversion outputs are:

The original target_model.h5 remains in OUTPUT_FILES/, allowing a direct target-versus-recovery comparison. The inversion itself never reads this file.

Visualization

Open plot_model.ipynb after the inversion to reproduce these figures or inspect other depths and sections. For a quantitative assessment, compare target_model.h5 and final_model.h5 using the same color scale and examine:

The recovered horizontal slices retain the alternating polarity and locations of the target anomalies in both $V_{SV}$ and $\zeta$. The recovered amplitudes are smoother and weaker than the target amplitudes, particularly for $\zeta$, because of the imposed gradient smoothing and the finite sensitivity of the Rayleigh–Love dataset.

Recovered $V_{SV}$ and $\zeta$ perturbations at 1 and 5 km depth.

The vertical section confirms that the inversion resolves both checkerboard layers. Smoothing broadens the anomaly boundaries and reduces the recovered $\zeta$ amplitude, but the depth-dependent sign changes remain visible.

Recovered checkerboard section at latitude $25^\circ$.