Anelliptic approximations for phase and group velocities of qP waves in transversely isotropic (TI) media have been widely applied in various seismic data processing and imaging tasks. We have revisited previously proposed approximations and suggested two improvements. The first improvement involves finding an empirical connection between anelliptic parameters along different fitting axes based on laboratory measurements of anisotropy of rock samples of different types. The relationship between anelliptic parameters observed was strongly linear suggesting a novel set of anisotropic parameters suitable for the study of qP-wave signatures. The second improvement involves suggesting a new functional form for the anelliptic parameter term to achieve better fitting along the horizontal axis. These modifications led to improved three-parameter and four-parameter approximations for phase and group velocities of qP-waves in TI media. In several model comparisons, the new three-parameter approximations appeared to be more accurate than previous approximations with the same number of parameters. These modifications also served as a foundation for an extension to orthorhombic media, where qP velocities involved nine independent elastic parameters. As determined by previous researchers, qP-wave propagation in orthorhombic media could be adequately approximated using just six combinations of those nine parameters. We have developed novel six-parameter approximations for phase and group velocities for qP-waves in orthorhombic media. The orthorhombic phase-velocity approximation provides a more accurate alternative to previously known approximations and can find applications in full-wave modeling, imaging, and inversion. The group-velocity approximation is also highly accurate and can find applications in ray tracing and velocity analysis.

Reconstructing the details of subsurface structures deep beneath complex overburden structures, such as sub-salt, remains a challenge for seismic imaging. Over the past years, the Marchenko redatuming approach has proven to reliably retrieve full-wavefield information in the presence of complex overburden effects. When used for redatuming, current practical Marchenko schemes cannot make use of a priori subsurface models with sharp contrasts because of their requirements regarding initial focusing functions, which for sufficiently complex media can result in redatumed fields with significant waveform inaccuracies. Using a scattering framework, we present an alternative form of the Marchenko representation that aims at retrieving only the unknown perturbations to both focusing functions and redatumed fields. From this framework, we propose a two-step practical focusing-based redatuming scheme that first solves an inverse problem for the background focusing functions, which are then used to estimate the perturbations to focusing functions and redatumed fields. In our scheme, initial focusing functions are significantly different from previous approaches since they contain complex waveforms encoding the full transmission response of the a priori model. Our goal is the handling of not only highly complex media but also realistic data - band-limited, unevenly sampled, free-surface-multiple contaminated data. To that end, we combine the versatility of Rayleigh-Marchenko redatuming with the proposed scattering-based scheme allowing an extended version of the method able to handle single-sided band-limited multicomponent data. This Scattering-Rayleigh-Marchenko strategy accurately retrieves wavefields while requiring minimum pre-processing of the data. In support of the new methods, we present a comprehensive set of numerical tests using a complex 2D subsalt model. Our numerical results show that the scattering approaches retrieve accurate redatumed fields that appropriately account for the complexity of the a priori model. We show that the improvements in wavefield retrieval translate into measurable improvements in our subsalt images.

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