Theory of FWI

Mathematically, regular FWI can be formulated as a minimization problem with the following objective function:

$$\begin{split} J_{FWI} &=\vert\vert \mathbf{d} - f\left(\mathbf{m}\right) \vert\vert _2^2 + constraint, \end{split}$$ (1)

where $\mathbf{m}$ represents the model, $\mathbf{d}$ is the observed dataset, $f\left(.\right)$ is the corresponding modeling operator, and $\vert\vert.\vert\vert _2^2$ stands for the L2-norm. In most FWI implementations, $\mathbf{m}$ consists of a gridded velocity distribution that explains both propagation and reflection of the seismic data and forward modeling is done via a finite-difference implementation of the two-way wave equation (Virieux and Operto, 2009). Note that in most FWI implementations, density variations are neglected. Minimizing this misfit function is likely to suffer from ill-posedness and non-uniqueness because of limited input data and non-linearity of the forward modeling operator. Adding regularization to the objective function can be one effective way to mitigate the ill-posedness and non-uniqueness of this inverse problem.