Simultaneous inversion of the background velocity and the perturbation in full-waveform inversion

Abstract

The gradient of standard full-waveform inversion (FWI) attempts to map the residuals in the data to perturbations in the model. Such perturbations may include smooth background updates from the transmission components and high wavenumber updates from the reflection components. However, if we fix the reflection components using imaging, the gradient of what is referred to as reflected-waveform inversion (RWI) admits mainly transmission background-type updates. The drawback of existing RWI methods is that they lack an optimal image capable of producing reflections within the convex region of the optimization. Because the influence of velocity on the data was given mainly by its background (propagator) and perturbed (reflectivity) components, we have optimized both components simultaneously using a modified objective function. Specifically, we used an objective function that combined the data generated from a source using the background velocity, and that by the perturbed velocity through Born modeling, to fit the observed data. When the initial velocity was smooth, the data modeled from the source using the background velocity will mainly be reflection free, and most of the reflections were obtained from the image (perturbed velocity). As the background velocity becomes more accurate and can produce reflections, the role of the image will slowly diminish, and the update will be dominated by the standard FWI gradient to obtain high resolution. Because the objective function was quadratic with respect to the image, the inversion for the image was fast. To update the background velocity smoothly, we have combined different components of the gradient linearly through solving a small optimization problem. Application to the Marmousi model found that this method converged starting with a linearly increasing velocity, and with data free of frequencies below 4 Hz. Application to the 2014 Chevron Gulf of Mexico imaging challenge data set demonstrated the potential of the proposed method.

Reflection full waveform inversion

“…Although this true-amplitude migration provides relatively accurate AVO, in theory, it does not guarantee the modeled data during de-migration can match the observed data in terms of amplitude. Zhou et al (2015) and Wu and Alkhalifah (2015) built a joint objective function by updating the long-wavelength background velocity and the short-wavelength velocity interface alternately or simultaneously. This is achieved by separating the gradient of FWI into the background velocity component and the velocity interface component.…”

Selective data extension for full-waveform inversion: An efficient solution for cycle skipping

“…The only difference is in the computation of the adjoint source. Due to the simple form of this objective function, it can be easily combined with other method, such as RWI (Wu and Alkhalifah, 2015;Wu, 2016a, 2016b). In addition, the selective function MðζÞ is continuous and its derivative is discontinuous, as shown in Figure 3.…”

“…Zhou et al (2015) propose similar ideas that invert for velocity and impedance. In previous work, we proposed to invert for the background and perturbation simultaneously (Wu and Alkhalifah, 2015;Alkhalifah and Wu, 2016b), which can use diving waves, first-order reflection, and even multiscattering energy (Alkhalifah and Wu, 2016a) together. RWI can update along the wavepath of diving and reflected waves that can reduce the cycle-skipping problem in some way.…”

Full-waveform inversion with reflected waves for 2D VTI media

“…A spatial-and temporal-correlation-based objective function that can handle phase delays larger than half a period is presented by Chi et al (2015). Wu and Alkhalifah (2015) propose a new optimization approach that estimates both the background and perturbed model simultaneously.…”