# FWI+: In-Depth's Full Waveform Inversion

In-Depth’s FWI+ uses the actual acquisition geometry to estimate the misfit function. The algorithm uses refracted energies in the shallow section and naturally transits to reflected energies in the deeper area. It typically use 3~4 frequency bands between 3 Hz and 15 Hz.

It is normally recommended adopting the K-Tomography/FWI interleaving workflow with fault and horizon constrains. This workflow has the key benefit of mitigating against the local minima for either FWI or tomography. L1 norm in data domain is used to match the phase information in FWI. In-Depth’s FWI+ achieves fast convergence rate and it shows less contamination from amplitude/wavelet mismatching.

## FWI-Tomography POCS

The objective function used in the K-Tomography/FWI interleaving workflow can be written as

.

The first term is a modified FWI objective functional, where

- x is our model, such as velocity and anisotropy; d is the field data;
- H is the wave propagator or Helmholtz equation solver;
- R is a projection operator from data space to synthetics space. Here, we use a tau-p domain warping filter as Rd(τ,p)=u(τ+dτ,p+dp) .

The second term is Tomographic objective functional, where

- A is the tomographic kernel,
- b is the residual curvature in migrated domain.

Inversion Method:

- Use POCS to iteratively invert the above two terms, where we assume they are positive definite.
- Each projection step has minimal or positive impact on the other term.
- Conjugate Gradient for each term
- 15-80 iterations for FWI projection
- 200-500 iterations for Tomographic term

In this framework, we view the FWI term as a projection to prevent the tomography term from becoming trapped in a local solution. Hence, the number of iterations used for the tomography term is generally larger than those used for the FWI term.