REAEME.md

Joint FWI and LSRTM

This example demonstrates how to use seistorch perform joint Full Waveform Inversion (FWI) and Least Squares Reverse Time Migration (LSRTM) on Marmousi model.

Workflow

Generate the geometry files for inversion.

python generate_model_geometry.py

Simulate the observed data.

sh forward.sh

Perform the joint inversion using the source encoding strategy.

sh sefwi.sh

Plot the inverted velocity model and reflectivity model.

python show_results.py

Theory

The joint FWI and LSRTM is based on a wave equations that incorporates a vector reflectivity model. The wave equation is given by: $$ \frac{\partial^2 p}{\partial t^2}-\left(v^2 \nabla^2 p+v \nabla v \cdot \nabla p-2 v^2 \mathbf{R} \cdot \nabla p\right)=S $$ where $p$ is the pressure wavefield, $v$ is the velocity model, $\mathbf{R}=(r_x, r_z)$ is the vector reflectivity model, and $S$ is the source term.

The joint inversion problem is formulated as: $$ \min_{v, \mathbf{R}} \left{ \sum_{s,r}^{} \left| \mathbf{d}_i - \mathbf{F}_i(v, \mathbf{R}) \right|_2^2 \right} $$ where $\mathbf{d}_i$ is the observed data, $\mathbf{F}_i(v, \mathbf{R})$ is the forward modeling operator.

Inversion

For joint FWI and LSRTM in 2D case, there are 3 parameters to be inverted: velocity model, $v$, and vector reflectivity model, $\mathbf{R}=(r_x, r_z)$. The ground truth reflectivity model is calculated by the following formula: $$\mathbf {R}=\frac{1}{2} \frac{\nabla z}{z}=-\frac{1}{2}z\nabla \frac{1}{z}$$ where $z$ is acoustic impedance, which is calculated by the product of velocity and density, i.e., $z=\rho v$.

A geometry of 93 sources and 461 receivers is used here for inversion. The geometry files of the inversion can be generated by the following commands:

python generate_model_geometry.py

The generated ground truth velocity model and initial velocity model are shown in the following figure:

Then simulate the observed data using the following commands:

sh forward.sh

The script sefwi.sh is used to perform the joint inversion with a source encoding strategy. The inversion parameters can be found here and here.

Finally, perform the joint inversion using the following commands:

sh sefwi.sh

The inverted velocity models and reflectivity models are shown in the following figure:

A vertical profile comparison of the inverted velocity model and reflectivity model with the ground truth is shown in the following figure:

There still has room for improvement in the inversion results. The inversion results can be improved by multi-scale inversion and hyperparameter tuning.

References

Whitmore et al., Full Wavefield Modeling with Vector Reflectivity, 2020, EAGE.

Wu et al., Joint Full Waveform Inversion and Least Squares Reverse Time Migration, 2024, IEEE TGRS.