Decoupled Fréchet kernels based on a fractional viscoacoustic wave equation

Abstract

We formulate the Fréchet kernel computation using the adjoint-state method based on a fractional viscoacoustic wave equation. We first numerically prove that both the 1/2- and the 3/2-order fractional Laplacian operators are self-adjoint. Using this property, we show that the adjoint wave propagator preserves the dispersion and compensates the amplitude, while the time-reversed adjoint wave propagator behaves identically as the forward propagator with the same dispersion and dissipation characters. Without introducing rheological mechanisms, this formulation adopts an explicit Q parameterization, which avoids the implicit Q in the conventional viscoacoustic/viscoelastic full waveform inversion ( Q-FWI). In addition, because of the decoupling of operators in the wave equation, the viscoacoustic Fréchet kernel is separated into three distinct contributions with clear physical meanings: lossless propagation, dispersion, and dissipation. We find that the lossless propagation kernel dominates the velocity kernel, while the dissipation kernel dominates the attenuation kernel over the dispersion kernel. After validating the Fréchet kernels using the finite-difference method, we conduct a numerical example to demonstrate the capability of the kernels to characterize both velocity and attenuation anomalies. The kernels of different misfit measurements are presented to investigate their different sensitivities. Our results suggest that rather than the traveltime, the amplitude and the waveform kernels are more suitable to capture attenuation anomalies. These kernels lay the foundation for the multiparameter inversion with the fractional formulation, and the decoupled nature of them promotes our understanding of the significance of different physical processes in the Q-FWI.

Understanding Subsurface Fracture Evolution Dynamics Using Time‐Lapse Full Waveform Inversion of Continuous Active‐Source Seismic Monitoring Data

“…Beyond the systematic V p reductions observed, time‐lapse amplitude changes are also apparent and could be inverted for seismic attenuation changes with uncertainty estimation (Xing & Zhu, 2022). Recent studies have shown seismic amplitude reduction is associated with tensile fracture initiation, a possible precursor to the damage on the rock surface (Modiriasari et al., 2017), and is sensitive to fluid saturation changes (Zhu et al., 2017).…”

Fractional laplacians viscoelastic wave equation low-rank temporal extrapolation

“…This kind of CQ wave equation also brings accuracy improvements in Q-compensated reverse-time migration (Q-RTM) methods, due to approximately decoupling the amplitude loss and phase distortion operators Sun et al, 2015;Li et al, 2016;Wang Y. et al, 2018;Chen et al, 2020a). Another benefit of the FLCQ wave equations is the explicit Q term in the equations, which facilitates developing full waveform inversion methods (Chen et al, 2017;2020b;Xing and Zhu, 2020;Yang et al, 2020;Xing and Zhu, 2022).…”

Propagating Seismic Waves in VTI Attenuating Media Using Fractional Viscoelastic Wave Equation

“…Bai and Tsvankin (2016) developed a detailed numerical demonstration and analysis of vertical transversely isotropic (VTI) attenuation by 2D time‐domain finite‐difference modeling. Although mechanical models have been widely adopted in the literature of seismic forward modeling and inverse problems, it is worth pointing out two essential issues (Xing & Zhu, 2022): first, these mechanical model‐based modeling approaches introduce the memory variables (most often L = 3 relaxation elements for seismology studies), which require significant computation time and memory, especially in 3D (e.g., Robertsson et al., 1994; Savage et al., 2010; Zhu et al., 2013); second, these approaches bring about difficulties in inverse problems as the Q is implicitly parameterized by a set of relaxation times (Fichtner and Van Driel, 2014).…”

“…In seismology, Zhu and his collaborators (Zhu & Carcione, 2014; Zhu & Harris, 2014) developed the decoupled fractional Laplacian (DFL) viscoelastic/viscoacoustic wave equations, which have benefited the attenuation‐compensation seismic imaging (Li et al., 2016; Wang et al., 2017; Zhao et al., 2018; Zhu et al., 2014; Zhu & Sun, 2017) and inversion (Chen et al., 2020; Xing and Zhu, 2020, 2022; Xue et al., 2017; Yang et al., 2020). Subsequently, several decoupled‐form viscous wave equations are proposed.…”