A preconditioning scheme for full waveform inversion

Method

In this paper, we use a time domain approach for solving the scalar acoustic wave equation (parametrized in terms of P-wave velocity $v_p$ ):

$$\frac{\partial ^2 u(\textbf{x},t)}{\partial t ^2} - v_p(\textbf{x})^2\nabla ^2 u(\textbf{x},t) = v_p(\textbf{x})^2s(\textbf{x},t).$$ (3)

with the source term $s(\textbf{x},t)= \delta (\textbf{x}-\textbf{x}_s) S(t)$ where $S(t)$ is the source function at $\textbf{x}_s$ and $u(\textbf{x},t)$ the pressure field. Tarantola (1984) derives the expression of the gradient for the acoustic equation (3) for each component of $\mathbf{m}$ (equal to $v_p$ only in this case).

$$\nabla f(\textbf{m}_n)= \frac{2}{{{\mathbf{v}_p}_n}^3} \sum_{shot... ...ightarrow{\mathbf{u}_n}}{\partial t^2}\cdot \overleftarrow{\mathbf{\delta u}_n}$$ (4)

where $\overleftarrow{\mathbf{\delta u}_n}$ is the backward propagated residual at iteration $n$ such that $\mathbf{\delta u}_n=\mathbf{u}_{obs} - \mathbf{u}_n$ and $\overrightarrow{\mathbf{u}_n}$ is the forward propagated synthetic source. For our iterative method, we opted for the L-BFGS approach of Nocedal (1980). This quasi-Newton approach computes an estimate of the inverse Hessian iteratively by using a user-defined number of solution and gradient vectors. One of the main benefits of this technique is that because the Hessian is never explicitely formed, there is significant computational and memory savings. W ith the L-BFGS solver, the model is updated as follows:

$$\mathbf{m}_{n+1}=\mathbf{m}_n - \alpha_n \mathbf{H}_n^{-1}\nabla f(\mathbf{m}_n),$$ (5)

where $\mathbf{m}_{n+1}$ is the updated solution, $\alpha_n$ the step length computed by a line search that ensures a sufficient decrease of $f(\mathbf{ m})$ and $\mathbf{H}_n \approx \nabla^2 f(\mathbf{m}_n)$ the approximate Hessian. To improve chances of not falling into a local minimum, we follow a multi-scale approach (Bunks et al., 1995) where the source and data are bandpassed prior to inversion. We introduce our preconditioning scheme in the following section.

A preconditioning scheme for full waveform inversion