# Tutorial 09: Acoustic Full Waveform Inversion

```{note}
This tutorial is available as a Python script [`examples/09_acoustic_fwi.py`](https://github.com/GeoBrain-Project/geobrain/blob/main/examples/09_acoustic_fwi.py) and an interactive Jupyter notebook [`examples/notebooks/09_acoustic_fwi.ipynb`](https://github.com/GeoBrain-Project/geobrain/blob/main/examples/notebooks/09_acoustic_fwi.ipynb).
```

Recover the Marmousi2 velocity model from seismic data using gradient-based
Full Waveform Inversion (FWI) with PyTorch automatic differentiation.

## What You Will Learn

- Generate observed seismic data from a true velocity model
- Start inversion from a smoothed initial model
- Iterate the forward-misfit-backward-update loop
- Monitor convergence and compare inverted vs. true models

## Key Concepts

**Full Waveform Inversion (FWI)** iteratively updates the velocity model to
minimize the waveform misfit between simulated and observed seismic data.
GeoBrain's differentiable acoustic propagator provides exact gradients via
automatic differentiation, avoiding the need for an adjoint-state
implementation.

The inversion begins from a smoothed version of the true velocity model, which
provides a low-frequency starting point that avoids cycle-skipping. At each
iteration the forward solver generates synthetic shot gathers, the L2 waveform
misfit is computed against the observed data, and PyTorch's autograd
back-propagates through the time-stepping loop to obtain the gradient with
respect to every velocity cell. A simple gradient-descent update (or Adam
optimizer) then nudges the model toward the true solution.

## Code

```python
from geobrain.physics.wave import (
    AcousticModel, AcousticPropagator, GridConfig, BoundaryConfig,
)

# Enable gradient on velocity
model = AcousticModel(
    grid=grid, boundary=boundary,
    vp=vp_init, rho=rho,
    vp_grad=True,
    device='cuda',
)

propagator = AcousticPropagator(model, survey, device='cuda')

for epoch in range(n_epochs):
    result = propagator.forward(checkpoint_segments=4)
    loss = ((result['p'] - d_obs) ** 2).sum()
    loss.backward()
    # Update vp with gradient descent
```

## Results

The smoothed initial model removes sharp interfaces from the true Marmousi2
velocity field while retaining the large-scale trend, giving the optimizer a
physically reasonable starting point.

```{figure} ../../examples/figs/09_initial_model.png
:width: 100%
:name: fig-09-initial-model

Smoothed initial velocity model for FWI.
```

After several hundred iterations the inverted model recovers the major
structural features and velocity contrasts present in the true model.

```{figure} ../../examples/figs/09_fwi_results.png
:width: 100%
:name: fig-09-fwi-results

FWI results: true model, initial model, and inverted model comparison.
```

## Full Example

See [`examples/09_acoustic_fwi.py`](https://github.com/GeoBrain-Project/geobrain/blob/main/examples/09_acoustic_fwi.py).