Wavelength-dependent Fresnel beam propagator and migration in VTI media
Introduction
Seismic migration methods can be categorized as either wave-equation-based migrations or ray-based migrations. Reverse time migration (RTM) (Baysal et al., 1983; Chang and Mcmechan, 1990), based on the two-way wave-equation, can image complex structures without dip limitation. However, it is expensive in implementation, and additional computational cost is required when angle domain common imaging gathers (ADCIGs) are generated. Ray-based depth migration methods are popular in generating ADCIGs for velocity model building due to the flexibility and efficiency of implementation. The Kirchhoff pre-stack depth migration (KPSDM) (Bleistein and Gray, 2001) constructs an image by a weighted summation of the seismic data along corresponding elliptical traveltime isochrones. Generally, kinematic ray tracing is used in conventional implementation of KPSDM, which has several drawbacks, e.g., ray shadow areas, traveltime interpolation and arched artifacts. Gaussian beam migration (GBM) (Hill, 1990, Hill, 2001; Hale, 1992; Gray and Bleistein, 2009) is a ray-based depth migration method whose accuracy rivals that of wave-equation based migrations and whose efficiency rivals that of KPSDM. The GBM can be extended to general transversely isotropic (TI) case via the anisotropic ray tracing system (Alkhalifah, 1995; Zhu et al., 2007; Han et al., 2014; Liu et al., 2015) and generalized to viscoacoustic media by introducing complex traveltimes to the ray tracing procedure (Keers et al., 2001; Bai et al., 2016; Liu et al., 2017). On the other hand, the Gaussian beam can be used as wave propagator in inversion imaging. Geng and Xie (2014) propose a finite-frequency turning wave tomographic method, in which Green functions are expressed as a summation of Gaussian beams. However, the central rays of these beam-based imaging methods above are still traced under the classic ray theory that is the assumption of high-frequency asymptotic solution of the wave equation. Moreover, the choice of initial beam width is quasi-empirical, and the beam width of GBM has no definite physical meaning.
As the practical seismic data are typical band-limited signals, finite-frequency effect should be considered when performing ray-based migrations. The first Fresnel zone, a spatial extension of the ray, primarily influences the wavefields at the receivers (Červený and Soares, 1992). Therefore, taking the first Fresnel zone into account gives an exact physical meaning of beam width, which considers finite-frequency effects during beam propagation and migration. Sun and Schuster (2001) introduce the stationary phase approximation to the Kirchhoff migration integral, restricting the smearing of the input trace data along the Fresnel zone portions of the migration quasi ellipsoids. And this wave path migration method is extended to 3D case and applied to field data (Sun and Schuster, 2003). Buske et al. (2009) propose a Fresnel volume migration method to reduce the artifacts caused by weighted summation of seismic data along elliptical isochrones. Hu and Stoffa (2009) propose a slowness-driven beam migration by introducing a Fresnel weighting function that works well for low-fold seismic data migration. Liu et al. (2014a) develop a Fresnel beam migration (FBM) method in transversely isotropic media with a vertical symmetry axis (VTI) based on anisotropic traveltimes. Jordi et al. (2016) propose a traveltime-based fat ray tomography method and give a scale-related criterion of fat ray construction. Through these imaging methods above consider finite-frequency effect in the construction of wave propagators, the traveltimes are still computed under the high-frequency assumption.
Model smoothing is another strategy to valid ray-based propagators and migration methods for complex subsurface. As the models should be smooth enough on the scale of the width of the first Fresnel zone in ray based methods (Kravtsov and Orlov, 1990), several approaches take finite-frequency effects via model smoothing before or during the wave propagation under the frame of ray theory. Lomax (1994) calculate finite-frequency ray paths by a wavelength smoothing scheme within a local plane that is perpendicular to the central ray with the width of one wavelength on both sides of the central ray. Yarman et al. (2013) calculate band-limited ray paths through a band-limited ray tracing algorithm that is then used as a finite-frequency wave propagator in KPSDM. This algorithm is implemented through computing equivalent ray parameters by maximizing the Kirchhoff boundary integral within the intersection of the first Fresnel zone with model boundaries. Zelt and Chen (2016) calculate frequency-dependent traveltimes by performing a wavelength-dependent velocity smoothing (WDS) operator followed by a finite-difference eikonal solver. Chen and Zelt (2017) compare full wavefields synthetics obtained from finite-difference modeling with the frequency-dependent traveltimes calculated using WDS in the isotropic case. A frequency-dependent traveltime tomography algorithm is developed based on the frequency-dependent traveltime fields and then applied to near surface velocity estimation (Zelt and Chen, 2016) and starting model building for full waveform inversion (Chen and Zelt, 2016; Chen et al., 2017). Model smoothing strategies of these methods have definite physical meanings, whereas physical backgrounds of conventional smoothing approaches (e.g., Gaussian smoothing) are not definite (Ettrich and Gajewski, 1996; Gray, 2000; Žáček, 2002).
In this paper, we extend the WDS to VTI case, in which both WDS vertical velocity and anisotropic parameters are obtained. Based on the VTI WDS models, we calculate anisotropic frequency-dependent traveltimes for Fresnel beam propagator and migration. Beyond previous studies of Fresnel beams (Liu et al., 2014a) and fat rays (Sun and Schuster, 2001, Sun and Schuster, 2003; Jordi et al., 2016), we use finite-difference traveltimes to construct a VTI wavelength-dependent Fresnel beam propagator. This paper is organized as follows. We first illustrate the methodology of the VTI WDS operator and introduce an anisotropic frequency-dependent traveltime calculator based on an anisotropic dynamic programming approach. Then, we develop a wavelength-dependent Fresnel beam propagator using frequency-dependent traveltimes, which is finally used in the construction of a wavelength-dependent Fresnel beam migration (WDFBM) method. Finite-frequency traveltime fields, WDFBM images and ADCIGs are compared and analyzed to illustrate the accuracy and effectiveness of our method.
Section snippets
WDS in VTI media
In anisotropic media, the velocity of a given type of seismic wave varies with wave propagating directions. The P-wave phase velocity expression can be deduced from the Christoffel equation of TI media (White, 1983). The P-wave phase slowness can be simplified by the following equationwhere sp is the P-wave phase slowness, s0 is the vertical P-wave slowness, θ is the phase angle to the symmetry axis, ε and δ are the anisotropic Thomsen parameters. It is the group
Numerical examples
In this section, we demonstrate the accuracy of WDS-based finite-frequency traveltime calculator and WDFBM by using the VTI fault model and the SEG/Hess synthetic data set. For comparison, the same data set are performed with different smoothing approaches and migration methods.
Discussions
As the conventional ray-based seismic wave propagators and migration methods are under the asymptotic ray theory framework, model smoothing is required. Gaussian smoothing, one of the commonly used smoothing operators, causes inaccurate traveltimes and wave propagating directions for the band-limited seismic wave propagation and migration. The VTI WDS operator quantitatively measures the smoothing level with definite physical backgrounds, which considers the band-limited characteristics of wave
Conclusion
The proposed WDS operator for VTI media captures the characteristics of band-limited wave propagation with explicit physical background. Based on WDS models, the frequency-dependent traveltimes computation can be performed with the anisotropic dynamic programming scheme. The VTI wavelength-dependent Fresnel beam propagator is constructed on the basis of the frequency-dependent traveltimes according to the reference frequency, then the migration images and ADCIGs are generated by the WDFBM.
Acknowledgments
The authors acknowledge Hess Corporation and SEG for providing the SEG/Hess synthetic data. The authors wish to acknowledge the financial support of the Great and Special Project (2016ZX05026001-001) and the National Natural Science Foundation of China (41604092, 41574115).
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