Yuriy Ivanov

Weak‐anisotropy approximation for P‐wave reflection coefficient at the boundary between two tilted transversely isotropic media

ABSTRACT Existing and commonly used in industry nowadays, closed‐form approximations for a P‐wave reflection coefficient in transversely isotropic media are restricted to cases of a vertical and a horizontal transverse isotropy. However, field observations confirm the widespread presence of rock beds and fracture sets tilted with respect to a reflection boundary. These situations can be described by means of the transverse isotropy with an arbitrary orientation of the symmetry axis, known as tilted transversely isotropic media. In order to study the influence of the anisotropy parameters and the orientation of the symmetry axis on P‐wave reflection amplitudes, a linearised 3D P‐wave reflection coefficient at a planar weak‐contrast interface separating two weakly anisotropic tilted tranversely isotropic half‐spaces is derived. The approximation is a function of the incidence phase angle, the anisotropy parameters, and symmetry axes tilt and azimuth angles in both media above and below the interface. The expression takes the form of the well‐known amplitude‐versus‐offset “Shuey‐type” equation and confirms that the influence of the tilt and the azimuth of the symmetry axis on the P‐wave reflection coefficient even for a weakly anisotropic medium is strong and cannot be neglected. There are no assumptions made on the symmetry‐axis orientation angles in both half‐spaces above and below the interface. The proposed approximation can be used for inversion for the model parameters, including the orientation of the symmetry axes. Obtained amplitude‐versus‐offset attributes converge to well‐known approximations for vertical and horizontal transverse isotropic media derived by Rüger in corresponding limits. Comparison with numerical solution demonstrates good accuracy.

Reflected waves in finely layered tilted orthorhombic media

Upscaling in seismics is a homogenization of finely layered media in the zero-frequency limit. An upscaling technique for arbitrary anisotropic layers has been developed by Schoenberg and Muir. Applying this technique to a stack of layers of orthorhombic (ORT) symmetry whose vertical symmetry planes are aligned, results in an effective homogeneous layer with orthorhombic symmetry. If the symmetry planes in a horizontal orthorhombic layer are rotated with respect to vertical, the medium is referred to as tilted orthorhombic (TOR) medium, and the stack composed of TOR layers in zero-frequency limit will produce an effective medium of a lower symmetry than orthorhombic. We consider a P-wave that propagates through a stack of thin TOR layers, then it is reflected (preserving the mode) at some interface below the stack, and then propagates back through the same stack. We propose to use a special modified medium for the upscaling in case of this sequential down- and up-propagation: each TOR layer in the stack is replaced by two identical TOR layers whose tilt angles have the opposite algebraic sign. In this modified medium, one-way propagation of a seismic wave (any wave mode) is equivalent to propagation of a pure-mode reflection in the original medium. We apply this idea to study the contribution from an individual layer from the stack and show how the approach can be applied to a stack of TOR layers. To demonstrate the applicability of the model, we use well log data for the upscaling. The model we propose for the upscaling can be used in well-seismic ties to correct the effective parameters obtained from well log data for the presence of tilt, if latter is confirmed by additional measurements (for example, borehole imaging).

New Approximation for qP-wave Velocities in TI Media

A new approximation for qP-wave velocities is developed for shales (TI medium). Proposed approximation is based on the linear relation between the velocity curvatures taken at zero and 90 degrees angles. We found that this relation is controlled by one parameter only. The resulting approximation is using the average value of this parameter from a large dataset of shale properties. Comparing with other known approximations for few shale samples, the proposed approximation shows the best accuracy.

NMO Velocity Ellipse in Tilted Elastic Orthorhombic Medium

We demonstrate the way to calculate a NMO velocity ellipse for direct and reflected (including converted) wave modes in a homogeneous horizontal layer with tilted orthorhombic symmetry (the tilt is defined by three Euler angles). For elliptic orthorhombic model, we present explicit expressions for the shift of the minimum of a traveltime surface. We use numerical examples to demonstrate how the position of a NMO velocity ellipse behaves depending on the orientation of the symmetry planes in an orthorhombic medium.

Zero- and Infinite-frequency Limits of P-wave Traveltime Parameters in Tilted Orthorhombic Media

We analyze and compare zero- and infinite-frequency limits of traveltime parameters of the reflected P-wave in tilted orthorhombic media. Weak-anisotropy approximations are obtained for vertical and normal moveout velocities and anellipticity parameters as a function of tilt angle. We show that they are equal in both frequency limits. We derive the approximate relation between zero- and infinite- frequency limits of the vertical P-wave velocity and demonstrate that vertical P-wave velocity in infinite-frequency is always higher than the one in zero-frequency limit. We use real well log data to demonstrate the correspondence between the frequency limits.