3-D moveout velocity analysis and parameter estimation for orthorhombic media

Abstract

Orthorhombic symmetry describes several azimuthally anisotropic models typical for fractured formations, such as those containing two orthogonal crack systems or parallel vertical cracks in a VTI (transversely isotropic with a vertical symmetry axis) background. Here, we present a methodology for inverting multiazimuth P-wave reflection traveltimes for the parameters of vertically inhomogeneous orthorhombic media. Our approach is based on the general analytic representation of normal‐moveout (NMO) velocity as an ellipse in the horizontal plane. A minimum of three differently oriented common‐midpoint (CMP) lines (or a “wideazimuth” 3-D survey) is needed to reconstruct the ellipse and thus obtain NMO velocity in any azimuthal direction. Then, the orientation and the semiaxes of the NMO ellipse, which are dependent on both anisotropy and heterogeneity, can be inverted for the medium parameters. Our analytic and numerical study shows that for the model of a homogeneous orthorhombic layer above a dipping reflector, the exact P-wave NMO velocity is determined by the symmetry‐plane orientation and five parameters: the NMO velocities from a horizontal reflector measured in the symmetry planes [[Formula: see text]] and three anisotropic coefficients η(1,2,3)introduced by analogy with the Alkhalifah‐Tsvankin parameter η for VTI media. The importance of the medium parameterization in terms of the η coefficients goes well beyond the NMO-velocity function. By generating migration impulse responses, we demonstrate that the parameters [Formula: see text] and η(1,2,3)are sufficient to perform all time processing steps (normal‐moveout and dip‐moveout corrections, prestack and poststack time migration) in orthorhombic models. The velocities [Formula: see text] and the orientation of the vertical symmetry planes can be found using the azimuthally dependent NMO velocity from a horizontal reflector. Then the NMO ellipse of at least one dipping event is additionally needed to obtain the coefficients η(1,2,3)that control the dip dependence of normal moveout. We discuss the stability of the inversion procedure and specify the constraints on the dip and azimuth of the reflector; for instance, for all three η coefficients to be resolved individually, the dip plane of the reflector should not coincide with either of the symmetry planes. To carry out parameter estimation in vertically inhomogeneous orthorhombic media, we apply the generalized Dix equation of Grechka, Tsvankin and Cohen, which operates with the matrices responsible for interval NMO ellipses rather than with the NMO velocities themselves. Our algorithm is designed to find the interval values of [Formula: see text] and η(1,2,3)using moveout from horizontal and dipping reflectors measured at different vertical times (i.e., only surface P-wave data are needed). Application to a synthetic multiazimuth P-wave data set over a layered orthorhombic medium with depth‐varying orientation of the symmetry planes verifies the accuracy of the inversion method.