Separation of Split Shear Waves in Two Non-orthogonal Sets of Vertical Fractures

The phenomenon of S-wave splitting indicates the development of fractures in the shallow crust. Therefore, methods based on S-wave splitting have been established to predict the development of one set of parallel fractures. However, for rocks containing two non-orthogonal sets of vertical fractures, the mechanism of S-wave splitting is more complex, and the available methods cannot be applied. To resolve this inadequacy, we propose a two-way rotation method to separate split S-waves with the aim of restoring the split S-wave polarizations and predicting the fracture azimuths. First, we calculate the stiffness matrix of fractured media based on the linear slip theory and derive the phase velocities and polarizations of split S-waves induced by fractures using the Christoffel equation. Second, we clarify the S-wave splitting mechanism in this media by employing velocity analysis and deconstruct the S-wave polarizations on the horizontal components. Third, we deduce a two-way rotation matrix obtained by the S-wave splitting modes to separate the split S-waves. To solve for the angle parameters related to the fracture azimuths in the two-way rotation matrix, we superpose the subspace polarizations in two dimensions to determine the polarization azimuths of the split S-waves. Numerical model tests demonstrate that the proposed method is stable under noisy conditions. Finally, we apply the proposed method to real near-offset and walkaround VSP data, and the predicted fracture results are verified by imaging logs and prior knowledge.

Christoffel equation in the polarization variables

To study the topology of polarization fields in homogeneous anisotropic media, we formulate the classic Christoffel equation in the polarization variables and solve it for the slowness vectors of plane waves corresponding to a given polarization. This task might emerge in passive seismology when neither ray nor wavefront-normal direction of a body wave recorded by a single three-component seismometer is available, so that velocities or slownesses of plausible wave arrivals have to be inferred from the recorded direction of particle motion. Our analysis shows that, unless the Christoffel equation degenerates and yields an infinite number of different slowness vectors, the finite nonzero number of its real-valued solutions varies from one to four. Unexpectedly, we find a subset of triclinic solids in which the polarization field contains holes — there exist finite-size solid angles of polarization directions unattainable to any plane wave.

3D, 9C seismic modeling and inversion of Weyburn Field data

Inversion of 3D, 9C wide azimuth vertical seismic profiling (VSP) data from the Weyburn Field for 21 independent elastic tensor elements was performed based on the Christoffel equation, using slowness and polarization vectors measured from field data. To check the ability of the resulting elastic tensor to account for the observed data, simulation of the 3C particle velocity seismograms was done using eighth-order, staggered-grid, finite-differencing with the elastic tensor as input. The inversion and forward modeling results were consistent with the anisotropic symmetry of the Weyburn Field being orthorhombic. It was dominated by a very strong, tranverse isotropy with a vertical symmetry axis, superimposed with minor near-vertical fractures with azimuth [Formula: see text] from the inline direction. The predicted synthetic seismograms were very similar to the field VSP data. The examples defined and provided a validation of a complete workflow to recover an elastic tensor from 9C data. The number and values of the nonzero tensor elements identified the anisotropic symmetry present in the neighborhood of a 3C borehole geophone. Computation of parameter correlation matrices allowed evaluation of solution quality through relative parameter independence.

Rotational motions in homogeneous anisotropic elastic media

Rotational motions in homogeneous anisotropic elastic media are studied under the assumption of plane wave propagation. The main goal is to investigate the influences of anisotropy in the behavior of the rotational wavefield. The focus is on P-waves that theoretically do not generate rotational motion in isotropic media. By using the Kelvin–Christoffel equation, expressions are obtained of the rotational motions of body waves as a function of the propagation direction and the coefficients of the elastic modulus matrix. As a result, the amplitudes of the rotation rates and their radiation patterns are quantified and it is concluded that (1) for strong local earthquakes and typical reservoir situations quasi P-rotation rates induced by anisotropy are significant, recordable, and can be used for inverse problems; and (2) for teleseismic wavefields, anisotropic effects are unlikely to be responsible for the observed rotational energy in the P coda.

Estimating effective elasticity tensors from Christoffel equations

We consider the problem of obtaining the orientation and elasticity parameters of an effective tensor of particular symmetry that corresponds to measurable traveltime and polarization quantities. These quantities — the wavefront-slowness and polarization vectors — are used in the Christoffel equation, a characteristic equation of the elastodynamic equation that brings seismic concepts to our formulation and relates experimental data to the elasticity tensor. To obtain an effective tensor of particular symmetry, we do not assume its orientation; thus, the regression using the residuals of the Christoffel equation results in a nonlinear optimization problem. We find the absolute extremum and, to avoid numerical instability of a global search, obtain an accurate initial guess using the tensor of given symmetry closest to the generally anisotropic tensor obtained from data by linear regression. The issue is twofold. First, finding the closest tensor of particular symmetry without assuming its orientation is challenging. Second, the closest tensor is not the effective tensor in the sense of regression because the process of finding it carries neither seismic concepts nor statistical information; rather, it relies on an abstract norm in the space of elasticity tensors. To include seismic concepts and statistical information, we distinguish between the closest tensor of particular symmetry and the effective one; the former is the initial guess to search for the latter.