ACS PRF | ACS
All e-Annual Reports
45207-AC8
Attenuation Analysis for Azimuthally Anisotropic Media
Effective attenuation anisotropy of thin-layered media
One of the well-known factors responsible for the anisotropy of seismic attenuation is interbedding of thin attenuative layers with different properties. We applied the Backus averaging formalism to obtain the complex stiffness matrix of an effective medium formed by an arbitrary number of anisotropic, attenuative constituents.
The main focus was on effective VTI (transversely isotropic with a vertical symmetry axis) models that include isotropic and VTI constituents. Also, we analyzed azimuthally anisotropic models composed of HTI (TI with a horizontal symmetry axis) constituents. Assuming that the stiffness contrasts, as well as the intrinsic velocity and attenuation anisotropy, are weak, we developed explicit first-order (linear) and second-order (quadratic) approximations for the attenuation-anisotropy parameters εQ, δQ, and γQ. Whereas the first-order approximation for each parameter is given simply by the volume-weighted average of its interval values, the second-order terms include coupling between various factors related to both heterogeneity and intrinsic anisotropy. Interestingly, the effective attenuation for P- and SV-waves is anisotropic even for a medium composed of isotropic layers with identical attenuation, provided there is a velocity variation among the constituent layers.
Extensive numerical testing shows that the second-order approximations for εQ, δQ, and γQ are close to the exact solution for most plausible subsurface models. The accuracy of the first-order approximations depends on the magnitude of the quadratic terms, which is largely governed by the strength of the velocity (rather than attenuation) anisotropy and velocity contrasts. If some of the constituents are azimuthally anisotropic with misaligned vertical symmetry planes, the effective velocity and attenuation functions may have different principal azimuthal directions or even different symmetries.
Plane-wave attenuation anisotropy in orthorhombic media
Orthorhombic models are often used in the interpretation of azimuthally varying seismic signatures recorded over fractured reservoirs. We developed an analytic framework for describing the attenuation coefficients in orthorhombic media with orthorhombic attenuation (i.e., the symmetry of both the real and imaginary parts of the stiffness tensor is identical) under the assumption of ``homogeneous'' wave propagation.
The analogous form of the Christoffel equation in the symmetry planes of orthorhombic and VTI (transversely isotropic with a vertical symmetry axis) media helps to obtain the symmetry-plane attenuation coefficients by adapting the existing VTI equations. To take full advantage of this equivalence with transverse isotropy, we introduced a parameter set similar to the VTI attenuation-anisotropy parameters εQ, δQ, and γQ. This notation, based on the same principle as Tsvankin's velocity-anisotropy parameters for orthorhombic media, leads to concise linearized equations for the symmetry-plane attenuation coefficients of all three modes (P, S1, and S2).
The attenuation-anisotropy parameters also allowed us to simplify the P-wave attenuation coefficient AP outside the symmetry planes under the assumptions of small attenuation and weak velocity and attenuation anisotropy. The approximate coefficient AP has the same form as the linearized P-wave phase-velocity function, with the velocity parameters εQ, δQ, and γQ.
The reduction in the number of parameters responsible for the P-wave attenuation and the simple approximation for the coefficient AP provide a basis for inverting P-wave attenuation measurements from orthorhombic media. The attenuation processing has to be preceded by anisotropic velocity analysis that can be performed (in the absence of pronounced velocity dispersion) using existing algorithms for nonattenuative media.
Far-field radiation from seismic sources in 2D attenuative anisotropic media
Anisotropic attenuation may strongly influence the energy distribution along the wavefront, which has serious implications for AVO (amplitude-variation-with-offset) analysis and amplitude-preserving migration. We carried out an asymptotic (far-field) study of 2D radiation patterns for media with anisotropic velocity and attenuation functions.
An important parameter for wave propagation in attenuative media is the angle between the wave and attenuation vectors, which is called the “inhomogeneity angle.” Application of saddle-point integration helped us to evaluate the inhomogeneity angle and test the common assumption of “homogeneous” wave propagation that ignores the misalignment of the wave and attenuation vectors. For transversely isotropic media, the inhomogeneity angle vanishes in the symmetry directions and remains small if the model has weak attenuation and weak velocity and attenuation anisotropy. However, reflection and transmission at medium interfaces can substantially increase the inhomogeneity angle, which has a serious impact on both the attenuation coefficients and radiation patterns.
The combined influence of angle-dependent velocity and attenuation results in pronounced distortions of radiation patterns, with the contribution of attenuation anisotropy rapidly increasing as the wave propagates away the source. Our asymptotic solution also establishes the relationships between the phase and group parameters when wave propagation cannot be treated as homogeneous.
