Separation of Migration and Tomography Modes of Full‐Waveform Inversion in the Plane Wave Domain

Abstract

Full‐waveform inversion (FWI) includes both migration and tomography modes. The migration mode acts like a nonlinear least squares migration to map model interfaces with reflections, while the tomography mode behaves as tomography to build a background velocity model. The migration mode is the main response of inverting reflections, while the tomography mode exists in response to inverting both the reflections and refractions. To emphasize one of the two modes in FWI, especially for inverting reflections, the separation of the two modes in the gradient of FWI is required. Here we present a new method to achieve this separation with an angle‐dependent filtering technique in the plane wave domain. We first transform the source and residual wavefields into the plane wave domain with the Fourier transform and then decompose them into the migration and tomography components using the opening angles between the transformed source and residual plane waves. The opening angles close to 180° contribute to the tomography component, while the others correspond to the migration component. We find that this approach is very effective and robust even when the medium is relatively complicated with strong lateral heterogeneities, highly dipping reflectors, and strong anisotropy. This is well demonstrated by theoretical analysis and numerical tests with a synthetic data set and a field data set.

A review on reflection-waveform inversion

“…One simple angle filtering function can be found in Fig. 2d of Yao et al (2018a). The analysis above shows that the scattering angle filtering based on the time-lag gather has limitation: firstly, assuming the velocity is isotropic; secondly, relatively large memory and computation cost is caused due to generating the time-lag gather and high-dimension Fourier transform; Thirdly, the coordinate transform needs resampling, and thus simple interpolation, e.g., spline interpolation, could introduce alias.…”

“…A natural thought for the angle measurement is in the plane-wave domain. Thus, Yao et al (2018a) proposed to Fourier transform the source and residual wavefields into the wavenumber domain and then achieve the scattering angle filtering by measuring the angle of the elements in the wavefields. The gradient expressed in Eq.…”

Reflection Full Waveform Inversion With Second‐Order Optimization Using the Adjoint‐State Method

“…To handle more complex geology, reflection tomography has been extended by using a wave‐equation forward modeling engine, based on either a data‐domain (Clement et al., 2001; Xu et al., 2012; Zhou et al., 2012) or an image‐domain (Sava & Biondi, 2004; Shen & Symes, 2008) objective function. On the other hand, with the popularity of FWI technique, people have proposed novel strategies to highlight the tomographic term, such as wavefield decomposition or scattering angle filtering (e.g., Alkhalifah & Wu, 2015; Wang et al., 2016; Yao et al., 2018). Latest attempts include simultaneous inversion of the background and perturbation models in the data domain and mixed data‐image domain optimization with appropriate weighting to mitigate nonlinearity (Biondi & Almomin, 2013; Wu & Alkhalifah, 2015; Zhou et al., 2015), or by exploiting nonlinear descent direction (Geng et al., 2018).…”

Efficient reflection waveform inversion using a locally normalized zero‐lag correlative objective function

“…However, for most acquisition systems, the low‐frequency data are often unavailable. Therefore, in recent years, research has focused on recovering low‐wavenumber updates regardless of frequencies contained in data using diving or refracted waves (Luo and Schuster 1991; Leeuwen and Mulder 2010; Bozdağ et al ., 2011; Wu et al ., 2014; Alkhalifah 2015; Alkhalifah and Wu 2016; Liu et al ., 2018,2020a,2020b; Yao et al ., 2018; Zhang et al ., 2018; Yi et al ., 2019; He and Liu 2020; He et al ., 2020). The diving or refracted waves, however, can only provide shallow low‐wavenumber updates since the maximum offset is mostly limited.…”