Numerical solutions of the acoustic wave equation, especially in anisotropic media, is crucial to seismic modeling, imaging and inversion as it provides efficient, practical, and stable approximate representation of the medium. However, a clean implementation (free of shear wave artifacts and dispersion) of wave propagation, especially in anisotropic media, requires an integral operator, the direct evaluation of which is extremely expensive. Recently, the low-rank method was proposed to provide a good approximation to the integral operator utilizing Fourier transforms. Thus, we propose to split the integral operator into two terms. The first term provides a differential operator that approximates that can be approximated with a standard finite-difference method. We, then, apply the lowrank approximation on the residual term of the finite-difference operator. We implement the two terms in two complementing steps, in which the spectral step corrects for any errors admitted by the finite difference step. Even though we utilize finite-difference approximations, the resulting algorithm admits spectral accuracy. Also, through the finite difference step, the method can deal approximately with the free surface and absorbing boundary conditions in a straight forward manner. Numerical examples show that the method is of high accuracy and efficiency.

Solving the eikonal equation is used widely in traveltime calculation, tomography, Kirchhoff migration etc. The complex eikonal equation governs the traveltimes in an attenuating medium, where the real and imaginary parts of the traveltimes are associated with the phase and energy-absorption, respectively. Attenuating orthorhombic anisotropy can be used to explain the azimuthal variation of velocity- and attenuation-anisotropy measured from surface seismic data. We present an approximate method to solve the acoustic eikonal equation for an attenuating orthorhombic medium. We combine perturbation theory and Shanks transform in different ways to derive the analytic solutions in the case of homogeneous media. We design a fast marching scheme to solve the acoustic eikonal equation numerically. We share some numerical examples to demonstrate the effectiveness of the complex eikonal equation in predicting attenuation.

Full-waveform inversion (FWI) in anisotropic media is challenging, mainly because of the large computational cost, especially in 3D, and the potential trade-offs between the model parameters needed to describe such media. By analyzing the trade-offs and understanding the resolution limits of the inversion, we can constrain FWI to focus on the main parameters the data are sensitive to and push the inversion toward more reliable models of the subsurface. Orthorhombic anisotropy is one of the most practical approximations of the earth subsurface that takes into account the natural horizontal layering and the vertical fracture network. We investigate the feasibility of a multiparameter FWI for an acoustic orthorhombic model described by six parameters. We rely on a suitable parameterization based on the horizontal velocity and five dimensionless anisotropy parameters. This particular parameterization allows a multistage model inversion strategy in which the isotropic, then, the vertical transverse isotropic, and finally the orthorhombic model can be successively updated. We applied our acoustic orthorhombic inversion on the SEG-EAGE overthrust synthetic model. The observed data used in the inversion are obtained from an elastic variable density version of the model. The quality of the inverted model suggests that we may recover only four parameters, with different resolution scales depending on the scattering potential of these parameters. Therefore, these results give useful insights on the expected resolution of the inverted parameters and the potential constraints that could be applied to an orthorhombic model inversion. We determine the efficiency of the inversion approach on real data from the North Sea. The inverted model is in agreement with the geologic structures and well-log information.

Transversely isotropic (TI) models have become standard in depth imaging and are often used in waveform inversion. Here, we develop a robust wave-equation-based tomographic algorithm for building acoustic VTI (transversely isotropic with a vertical symmetry axis) velocity models from P-wave surface reflection and vertical seismic profiling (VSP) data. Wavefield extrapolation is performed with an integral operator to avoid generating shear-wave artifacts. Focusing energy in extended images produced by reverse-time migration (RTM) makes it possible to update the zero-dip NMO velocity Vnmo and the anellipiticity parameter η. To constrain the anisotropy coefficient δ and improve the accuracy in Vnmo and η, we employ borehole information by introducing an additional objective-function term designed to fit VSP data. Image-guided smoothing is applied to both data- and image-domain gradients to steer the inversion towards geologically plausible solutions. Testing on the VTI Marmousi model shows that the joint inversion of surface and VSP data helps estimate all three relevant medium parameters.

Full-waveform inversion (FWI) in anisotropic media is overall challenging, mainly because of the large computational cost, especially in 3D, and the potential trade-offs between the model parameters needed to describe such a media. We propose an efficient 3D FWI implementation for orthorhombic anisotropy under the acoustic assumption. Our modeling is based on solving the pseudo-differential orthorhombic wave equation split into a differential operator and a scalar one. The modeling is computationally efficient and free of shear wave artifacts. Using the adjoint state method, we derive the gradients with respect to a practical set of parameters describing the acoustic orthorhombic model, made of one velocity and five dimensionless parameters. This parameterization allows us to use a multi-stage model inversion strategy based on the continuity of the scattering potential of the parameters as we go from higher symmetry anisotropy to lower ones. We apply the proposed approach on a modified SEG-EAGE overthrust synthetic model. The quality of the inverted model suggest that we may recover only 4 parameters, with different resolution scales depending on the scattering potential of these parameters.

To examine the feasibility of elastic full waveform inversion (FWI) for azimuthally rotated orthorhombic (rORT) media, we analyze the sensitivity of the 9-component (9C) land data set acquired on the surface on each of the ORT parameters. The trade-off analysis supports that the parameter set that includes deviation parameters offers the best choice for a 9C data set. Compared to the data from an explosive source, using the 9C land data, ORT parameters show different trade-off patterns for the different source and receiver components. For this reason, finding an optimal component considering trade-offs is another important issue to better recover subsurface rotated orthorhombic anisotropy.

The multi-component wavefield contains both compressional and shear waves. Separating wave-modes has many applications in seismic workflows. Conventionally, anisotropic wave-mode separation is implemented by either directly filtering in the wavenumber domain or nonstationary filtering in the space domain, which are computationally expensive. These methods could be categorized into the pseudo-derivative family and only work well within Finite Difference (FD) methods. In this paper, we establish a relationship between group-velocity direction and polarity direction and propose a method, which could go beyond modeling by FD. In particular, we are interested in performing wave-mode separation in a Spectral Element Method (SEM), which is widely used for seismic wave propagation on various scales. The separation is implemented pointwise, independent of its neighbor points, suitable for running in parallel. Moreover, no correction for amplitude and phase changes caused by the derivative operator is required. We have verified our scheme using numerical examples.

This paper shows the effect of anisotropic stress state on intermediate strain properties of cylindrical samples containing spherical glass particles. Tests were carried out with the modified resonant column device available at Ruhr-Universität Bochum. Dry samples were subjected to two anisotropic stress states: (a) cell pressure, σ′h, constant and vertical stress, σ′v, increased (stress state GB-I) and (b) σ′v/σ′h equal to 2 (stress state GB-II). The experimental results revealed that the effect of stress state GB-II on the modulus and damping ratio was more significant and obvious than stress state GB-I. The effect of the anisotropic stress state was explained through the impact of confining pressure and anisotropic stress components on the stiffness and damping ratio. The results showed that: (a) G(γ) increased, η(γ) decreased and their strain non-linearity decreased with an increase in the confining pressure component σ′vσ′h; (b) G(γ) decreased, η(γ) increased and their strain non-linearity increased with an increase in the anisotropic stress component, σ′v/σ′h. The analysis of results revealed that reference shear strain was also affected by anisotropic stress state. Therefore, an empirical relationship was developed to predict the reference shear strain, as a function of confining pressure and anisotropic stress components. Additionally, the damping ratio was written as a function of the minimum damping ratio and the reference shear strain.

The P-wave eikonal equation for orthorhombic (ORT) anisotropic media is a highly nonlinear partial differential equation requiring the solution of a sixth-order polynomial to obtain traveltimes, resulting in complex and time-consuming numerical solutions. To alleviate this complexity, we approximate the solution of this equation by applying a multiparametric perturbation approach. We also investigated the sensitivity of traveltime surfaces inORT mediawith respect to three anelliptic parameters. As a result, a simple and accurate P-wave traveltime approximation valid for ORT media was derived. Two different possible anelliptic parameterizations were compared. One of the parameterizations includes anelliptic parameters defined at zero offset: η1, η2, and ηxy. Another parameterization includes anelliptic parameters defined for all symmetry planes: η1, η2, and η3. The azimuthal behavior of sensitivity coefficients with different parameterizations was used to analyze the crosstalk between anelliptic parameters. © 2016 Society of Exploration Geophysicists.

Full-waveform inversion in anisotropic media using reflected waves suffers from the strong non-linearity of the objective function and trade-offs between model parameters. Estimating long-wavelength model components by fixing parameter perturbations, referred to as reflection-waveform inversion (RWI), can mitigate nonlinearity-related inversion issues. Here, we extend RWI to acoustic VTI (transversely isotropic with a vertical symmetry axis) media. To minimize trade-offs between the model parameters, we employ a new hierarchical two-stage approach that operates with the P-wave normal-moveout velocity and anisotropy coefficents ζ and η. First, is estimated using a fixed perturbation in ζ, and then we invert for η by fixing the updated perturbation in . The proposed 2D algorithm is tested on a horizontally layered VTI model.