Unwrapped phase inversion with an exponential damping

Abstract

Full-waveform inversion (FWI) suffers from the phase wrapping (cycle skipping) problem when the frequency of data is not low enough. Unless we obtain a good initial velocity model, the phase wrapping problem in FWI causes a result corresponding to a local minimum, usually far away from the true solution, especially at depth. Thus, we have developed an inversion algorithm based on a space-domain unwrapped phase, and we also used exponential damping to mitigate the nonlinearity associated with the reflections. We construct the 2D phase residual map, which usually contains the wrapping discontinuities, especially if the model is complex and the frequency is high. We then unwrap the phase map and remove these cycle-based jumps. However, if the phase map has several residues, the unwrapping process becomes very complicated. We apply a strong exponential damping to the wavefield to eliminate much of the residues in the phase map, thus making the unwrapping process simple. We finally invert the unwrapped phases using the backpropagation algorithm to calculate the gradient. We progressively reduce the damping factor to obtain a high-resolution image. Numerical examples determined that the unwrapped phase inversion with a strong exponential damping generated convergent long-wavelength updates without low-frequency information. This model can be used as a good starting model for a subsequent inversion with a reduced damping, eventually leading to conventional waveform inversion.

A robust full-waveform inversion based on a shifted correlation of the envelope of wavefields

“…Another group of methods are based on measuring the quality or the difference of an extended image (Shen et al, 2003;Sava and Biondi, 2004;Biondi and Symes, 2004;Zhang and Schuster, 2013;Alkhalifah and Wu, 2017). For data without low enough frequency information, some have proposed to generate artificial low frequencies by approximating the data in a transformed domain (Shin and Cha, 2008;Hu, 2014;Choi and Alkhalifah, 2015;Li and Demanet, 2016). For reflection dominated data, 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 modeled reflected energy from an image.…”

Waveform inversion with exponential damping using a deconvolution-based objective function

“…Exponential damping has been used widely for Laplacedomain FWI Cha, 2008, 2009) and the unwrapped phase inversion in the frequency-domain (Choi and Alkhalifah, 2015) to mitigate the cycle-skipping problem. In the Laplace-and frequency-domain FWI, the exponentially damped wavefields can be properly handled using the complex logarithm or phase value.…”

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

“…Another group of methods is based on measuring the quality or the difference of extended image (Shen et al, 2003;Biondi and Symes, 2004;Sava and Biondi, 2004;Zhang and Schuster, 2013;Alkhalifah and Wu, 2017). For data without low enough frequency information, some researchers have proposed to generate artificial low frequencies by approximating the data in a transformed domain (Shin and Cha, 2008;Hu, 2014;Choi and Alkhalifah, 2015;Li and Demanet, 2016). For reflectiondominated data, Xu et al (2012) and Zhou et al (2012) develop a method based mainly on the work of Plessix et al (1995) to invert for smooth velocity models using modeled reflected energy from an image.…”