Andres Pech

Geometrical spreading of P-waves in horizontally layered, azimuthally anisotropic media

For processing and inverting reflection data, it is convenient to represent geometrical spreading through the reflection traveltime measured at the earth’s surface. Such expressions are particularly important for azimuthally anisotropic models in which variations of geometrical spreading with both offset and azimuth can significantly distort the results of wide-azimuth amplitude-variationwith-offset (AVO) analysis. Here, we present an equation for relative geometrical spreading in laterally homogeneous, arbitrarily anisotropic media as a simple function of the spatial derivatives of reflection traveltimes. By employing the Tsvankin-Thomsen nonhyperbolic moveout equation, the spreading is represented through the moveout coefficients, which can be estimated from surface seismic data. This formulation is then applied to P-wave reflections in an orthorhombic layer to evaluate the distortions of the geometrical spreading caused by both polar and azimuthal anisotropy. The relative geometrical spreading of P-waves in homogeneous orthorhombic media is controlled by five parameters that are also responsible for time processing. The weak-anisotropy approximation, verified by numerical tests, shows that azimuthal velocity variations contribute significantly to geometrical spreading, and the existing equations for transversely isotropic media with a vertical symmetry axis (VTI) cannot be applied even in the vertical symmetry planes. The shape of the azimuthally varying spreading factor is close to an ellipse for offsets smaller than the reflector depth but becomes more complicated for larger offset-to-depth ratios. The overall magnitude of the azimuthal variation of the geometrical spreading for the moderately anisotropic model used in the tests exceeds 25% for a wide range of offsets. While the methodology developed here is helpful in modeling and analyzing anisotropic geometrical spreading, its main practical application is in correcting the wideazimuth AVO signature for the influence of the anisotropic overburden.

Quartic moveout coefficient: 3D description and application to tilted TI media

Nonhyperbolic (long-spread) moveout provides essential information for a number of seismic inversion/ processing applications, particularly for parameter estimation in anisotropic media. Here, we present an analytic expression for the quartic moveout coefficient A4 that controls the magnitude of nonhyperbolic moveout of pure (nonconverted) modes. Our result takes into account reflection-point dispersal on irregular interfaces and is valid for arbitrarily anisotropic, heterogeneous media. All quantities needed to compute A4 can be evaluated during the tracing of the zero-offset ray, so long-spread moveout can be modeled without timeconsuming multioffset, multiazimuth ray tracing. The general equation for the quartic coefficient is then used to study azimuthally varying nonhyperbolic moveout of P-waves in a dipping transversely isotropic (TI) layer with an arbitrary tilt” of the symmetry axis. Assuming that the symmetry axis is confined to the dip plane, we employed the weak-anisotropy approximation to analyze the dependence of A4 on the anisotropic parameters. The linearized expression for A4 is proportional to the anellipticity coefficient·…†i‐ and does not depend on the individual values of the Thomsen parameters. Typically, the magnitude of nonhyperbolic moveout in tilted TI media above a dipping reflector is highest near the reflector strike, whereas deviations from hyperbolic moveout on the dip line are substantial only for mild dips. The azimuthal variation of the quartic coefficient is

Quartic moveout coefficient for a dipping azimuthally anisotropic layer

Interpretation and inversion of azimuthally varying nonhyperbolic reflection moveout requires accounting for both velocity anisotropy and subsurface structure. Here, our previously derived exact expression for the quartic moveout coefficient A4 is applied to P‐wave reflections from a dipping interface overlaid by a medium of orthorhombic symmetry.The weak‐anisotropy approximaton for the coefficient A4 in a homogeneous orthorhombic layer is controlled by the anellipticity parameters η(1), η(2), and η(3), which are responsible for time processing of P‐wave data. If the dip plane of the reflector coincides with the vertical symmetry plane [x1, x3], A4 on the dip line is proportional to the in‐plane anellipticity parameter η(2) and always changes sign for a dip of 30○. The quartic coefficient on the strike line is a function of all three η–parameters, but for mild dips it is mostly governed by η(1)—the parameter defined in the incidence plane [x2, x3]. Whereas the magnitude of the dip line A4 typically become...

Multicomponent stacking-velocity tomography for transversely isotropic media

Accurate estimation of the velocity field is the most difficult step in imaging of seismic data for anisotropic media. Here, the velocity‐analysis problem is examined for the most common anisotropic model of sedimentary formations—transverse isotropy (TI) with arbitrary orientation of the symmetry axis. We show that supplementing wide‐azimuth reflected PP data with mode‐converted (PS) waves yields more stable estimates of the anisotropic coefficients and, in many cases, helps to constrain the model in depth.An important processing step preceding the inversion is computation of the traveltimes of the pure SS‐waves from those of the PP‐, and PS‐waves based on a technique recently developed by Grechka and Tsvankin. This procedure allows us to replace PS‐wave moveout, which is generally asymmetric with respect to zero offset, with the symmetric (hyperbolic on short spreads) moveout of the pure SS reflections. Then, generalizing the algorithm previously suggested for PP data, we develop a joint tomographic inv...

Parameter estimation in orthorhombic media using multicomponent wide-azimuth reflection data

Orthorhombic models with a horizontal symmetry plane adequately describe seismic signatures recorded over many naturally fractured reservoirs. The inversion of wide-azimuth traveltimes of PP and SS (the fast S1 and slow S2) reflections are discussed for Tsvankins anisotropic parameters and the azimuths of the vertical symmetry planes of orthorhombic media. If shear waves are not excited, SS traveltimes can be found from PP and PS (converted-wave) data, which makes the method applicable to offshore surveys. The feasibility of parameter estimation is strongly dependent on reflector dip and orientation. For a horizontal reflector beneath a single orthorhombic layer, the vertical velocities and reflector depth cannot be found from conventional-spread reflection traveltimes alone. If the reflector is dipping, the inversion is ambiguous when the dip plane is close to one of the vertical symmetry planes of the orthorhombic layer above it. The parameter estimation becomes possible if the dip direction deviates b...

Velocity analysis for tilted transversely isotropic media: A physical modeling example

Transverse isotropy with a tilted symmetry axis (TTI media) has been recognized as a common feature of shale formations in overthrust areas, such as the Canadian Foothills. Since TTI layers cause serious problems in conventional imaging, it is important to be able to reconstruct the velocity model suitable for anisotropic depth migration. Here, we discuss the results of anisotropic parameter estimation on a physical‐modeling data set. The model represents a simplified version of a typical overthrust section from the Alberta Foothills, with a horizontal reflector overlaid by a bending transversely isotropic layer. Assuming that the TTI layer is homogeneous and the symmetry axis stays perpendicular to its boundaries, we invert P-wave normal‐moveout (NMO) velocities and zero‐offset traveltimes for the symmetry‐direction velocity V0 and the anisotropic parameters e and δ. The coefficient e is obtained using the traveltimes of a wave that crosses a dipping TTI block and reflects from the bottom of the model. T...

P‐wave quartic reflection moveout in weakly anisotropic dipping layer

Deviations of P-wave reflection traveltimes from a hyperbola, called the nonhyperbolic or quartic moveout, need to be properly handled while processing the long-spread seismic data. As observed nonhyperbolic moveout is usually attributed to the presence of anisotropy, we devote our paper to deriving and analyzing a general formula that describes azimuthally varying quartic moveout coefficient in a homogeneous, weakly anisotropic medium above a dipping, mildly curved reflector.

Geometrical spreading of P‐waves in azimuthally anisotropic media

Geometrical spreading is highly sensitive to elastic anisotropy and may strongly influence the AVO signature of reflected waves recorded over anisotropic formations. For purposes of processing and inversion of reflection data, it is convenient to express geometrical spreading through the reflection traveltime measured at the earth’s surface. Here, we obtain the inverse geometricalspreading factor L−1 for horizontally layered anisotropic media with a horizontal symmetry plane as a simple function of the traveltime derivatives with respect to offset and azimuth. By employing the Tsvankin-Thomsen nonhyperbolic moveout equation, the factor L−1 is represented through the moveout coefficients which can be estimated from surface seismic data.

Quartic Moveout Coefficient : 3-D Description And Application to Tilted TI Media

Nonhyperbolic (long-spread) moveout provides essential information for a number of seismic inversion/ processing applications, particularly for parameter estimation in anisotropic media. Here, we present an analytic expression for the quartic moveout coefficient A4 that controls the magnitude of nonhyperbolic moveout of pure (nonconverted) modes. Our result takes into account reflection-point dispersal on irregular interfaces and is valid for arbitrarily anisotropic, heterogeneous media. All quantities needed to compute A4 can be evaluated during the tracing of the zero-offset ray, so long-spread moveout can be modeled without timeconsuming multioffset, multiazimuth ray tracing. The general equation for the quartic coefficient is then used to study azimuthally varying nonhyperbolic moveout of P-waves in a dipping transversely isotropic (TI) layer with an arbitrary tilt ν of the symmetry axis. Assuming that the symmetry axis is confined to the dip plane, we employed the weak-anisotropy approximation to analyze the dependence of A4 on the anisotropic parameters. The linearized expression for A4 is proportional to the anellipticity coefficient η≈ 2− δ and does not depend on the individual values of the Thomsen parameters. Typically, the magnitude of nonhyperbolic moveout in tilted TI media above a dipping reflector is highest near the reflector strike, whereas deviations from hyperbolic moveout on the dip line are substantial only for mild dips. The azimuthal variation of the quartic coefficient is governed by the tilt ν and reflector dip φ and has a much more complicated character than the NMO–velocity ellipse. For example, if the symmetry axis is vertical (VTI media, ν= 0) and the dip φ > 30◦, A4 goes to zero on two lines with different azimuths where it changes sign. If the symmetry axis is orthogonal to the reflector (this model is typical for thrust-and-fold belts), the strike-line quartic coefficient is defined by the well-known expression for a horizontal VTI layer (i.e., it is independent of dip), while the dip-line A4 is proportional to cos4 φ and rapidly decreases with dip. The high sensitivity of the quartic moveout coefficient to the parameter η and the tilt of the symmetry axis can be exploited in the inversion of wide-azimuth, long-spread P-wave data for the parameters of TI media.