Reflected waveform inversion (RWI) provides a method to reduce the nonlinearity of the standard full waveform inversion (FWI). However, the drawback of the existing RWI methods is inability to utilize diving waves and the extra sensitivity to the migrated image. We propose a combined FWI and RWI optimization problem through dividing the velocity into the background and perturbed components. We optimize both the background and perturbed components, as independent parameters. The new objective function is quadratic with respect to the perturbed component, which will reduce the nonlinearity of the optimization problem. Solving this optimization provides a true amplitude image and utilizes the diving waves to update the velocity of the shallow parts. To insure a proper wavenumber continuation, we use an efficient scattering angle filter to direct the inversion at the early stages to direct energy corresponding to large (smooth velocity) scattering angles to the background velocity update and the small (high wavenumber) scattering angles to the perturbed velocity update. This efficient implementation of the filter is fast and requires less memory than the conventional approach based on extended images. Thus, the new FWI procedure updates the background velocity mainly along the wavepath for both diving and reflected waves in the initial stages. At the same time, it updates the perturbation with mainly reflections (filtering out the diving waves). To demonstrate the capability of this method, we apply it to a real 2D marine dataset.
Recently, full waveform inversion (FWI) is gaining considerable attention (Virieux and Operto, 2009) as it promises us high resolution velocity models, and an opportunity to utilize more information from the data (like diving waves). However, the promise comes with a considerable cost and many limitations. Specifically, the objective function of FWI conventionally given by the least square misfit between the observed and modeled data, is far from being smooth or convex, especially for high frequency data and specifically for reflections. To slightly mitigate the reflection data limitation, Xu et al. (2012) and Zhou et al. (2012) developed a method based mainly on the work of Plessix et al. (1995) to invert for smooth velocity models using the modeled reflection energy from an image. We refer to the method as reflection waveform inversion (RWI). Biondi and Almomin (2014) propose to extend the equation, split the velocity into a background and perturbed one, and optimize them with a nested approach. Wu and Alkhalifah (2014) implemented RWI using the spectral method as a wavefield extrapolator and utilized a modified objective function that integrates RWI into FWI. In addition, Alkhalifah (2014) proposed a new scattering angle filter applied to the gradient to help control the wavenumber components injected to the model, which will help us avoid falling in a local minima for FWI or RWI. In (Wu and Alkhalifah, 2015), we propose to split the velocity into background and perturbed components and optimize both simultaneously. In order to fully utilize the features of this method, we suggest a small weight to the gradient of v (background) to enhance the convergence of w (perturbation) at the early stage.