Introduction

Modeling seismic wave propagation in attenuating media accounts for the effective anelastic characteristics of the real Earth (Carcione, 2007). Numerous studies have shown that many of the hydrocarbon prospecting areas, such as those where gas accumulations are present, strongly attenuate seismic waves (Dvorkin and Mavko, 2006). Seismic attenuation can be expressed as a combined effect of energy loss and velocity dispersion.

Attenuation effects can be modeled by incorporating the quality factor, $Q$, in the time-domain wave equation. One of the classic approaches involves a superposition of mechanical elements (e.g., Maxwell and standard linear solid elements) to characterize $Q$, and is known as approximate constant-$Q$ models (Zhu et al., 2013; Carcione, 2007; Blanch et al., 1995; Liu et al., 1976). The approximate constant-$Q$ approach suffers from large computational and memory requirements. Kjartansson (1979) initially proposed a constant-$Q$ model that assumes a linear relationship between the attenuation coefficient and frequency. This model was proven accurate in capturing a nearly constant $Q$ behavior within the seismic frequency band. However, early implementations of the constant-$Q$ model involved a fractional time derivative, which required storing the whole history of the wavefield (Caputo and Mainardi, 1971). This requirement rendered the memory cost too high for practical applications, even when the fractional operator was truncated after a certain time period (Carcione, 2009; Carcione et al., 2002; Podlubny, 1999). To overcome this issue, fractional-Laplacian operators (Chen and Holm, 2004) have been introduced to approximate the constant-$Q$ viscoacoustic wave equation (Zhu and Harris, 2014; Carcione, 2010). The fractional-Laplacian approach is attractive because it can be conveniently formulated in the wavenumber domain using Fourier transforms and without introducing any extra equations or variables (Carcione, 2010). Using this approach, Zhu and Harris (2014) developed a decoupled wave equation that accounts separately for amplitude attenuation and phase dispersion effects, thus allowing for correct compensation for both factors during back propagation by reversing the sign of the attenuation operator and keeping the sign of the dispersion operator unchanged (Zhu, 2014).

Zhu et al. (2014) further proposed to use the fractional-Laplacian $Q$-compensated wave equation for reverse-time migration. Zhang et al. (2010) applied an analogous approach derived from normalization transforms. Alternative strategies for compensating for attenuation in seismic migration include methods based on one-way wave-equation migration (WEM) and its dispersion relation (Valenciano and Chemingui, 2012), as well as methods based on Kirchhoff migration and reverse-time migration (RTM), with time-variant $Q$ filters (Cavalca et al., 2013). Dutta and Schuster (2014) used least-squares RTM for attenuation compensation based on standard linear solid (SLS) model and its adjoint operator (Blanch and Symes, 1995), with a simplified stress-strain relation, which incorporated a single relaxation mechanism (Robertsson et al., 1994; Blanch et al., 1995).

The fractional-Laplacian approach was previously implemented using either a pseudo-spectral method, by averaging the fractional power of the Laplacian operator as an approximation (Zhu and Harris, 2014), or a finite-difference approach (Lin et al., 2009). In this paper, we propose to apply a low-rank approximation scheme (Fomel et al., 2013; Sun and Fomel, 2013) to implement decoupled fractional Laplacians of Zhu and Harris (2014) in wave extrapolation, with the goal of accurately capturing spatially-varying fractional power. The advantage of the low-rank approach is its ability to directly approximate the mixed-domain wave extrapolation operator with a separable representation, which minimizes the number of fast Fourier transforms (FFTs) per time step. Additionally, we derive the adjoint of the forward modeling operator, which correctly compensates for velocity dispersion but not amplitude loss. The proposed operator and its adjoint can be used in least-squares RTM to recover the true reflectivity of the attenuating medium through iterations of migration and modeling (Sun et al., 2014). In this paper, we implement $Q$-RTM using an operator that compensates for amplitude loss during back propagation of the viscoacoustic data (Zhu et al., 2014). When used with the cross-correlation imaging condition, the attenuation-compensated operator is capable of producing images with improved illumination in the attenuating zone. We apply the low-rank $Q$-RTM to synthetic data generated from a constant-$Q$ model to demonstrate the effectiveness of attenuating compensation by the proposed method.


2019-07-17