Relative Geometrical Spreading Research Articles

Diving waves in acoustic factorized orthorhombic media

AbstractDiving waves propagating in the subsurface are massive sources of low‐frequency information that can be used to constrain the kinematic component of the velocity model. Compared to reflected waves, less is known about the behaviour of diving waves, especially in the presence of azimuthal anisotropy. Anisotropy is needed to place the events to the correct depths and match travel times in synthetics with recorded data. Obtaining more insights into the influence of anisotropy on diving wave propagation can help to find parameters with a low trade‐off for inversion. Here, we derive equations for diving qP‐waves in an acoustic factorized anisotropic model with orthorhombic anisotropy. The effects of the anisotropic parameters in the acoustic factorized orthorhombic model are tested by perturbing , , , and and observing differences in the ray paths, the effective vertical slowness and the relative geometrical spreading. The properties of diving waves in this model are also compared with those in an acoustic isotropic model and acoustic factorized anisotropic models with elliptical‐ and vertical transverse isotropic anisotropy. From our analysis, we found that perturbing and has the most significant influence on these characteristics. The , and parameters were shown to induce minor changes. Compared with the other models, the acoustic factorized orthorhombic model had the most in common with the acoustic factorized anisotropic model with elliptical anisotropy. Although, in general, none of the other models could fully represent the effects of orthorhombic anisotropy.

On singularity points in elastic orthorhombic media

The analysis of singularity points in anisotropic models of low symmetry is very important for seismic modelling and seismic data processing. Ray tracing: the group velocity vector dramatically changes the orientation in vicinity of singularity point. Seismic amplitudes: the relative geometrical spreading of seismic wave tends to zero in singularity point. Seismic wave polarization: the polarization vectors dramatically change in vicinity of singularity point. Multiarrival: one of the waves has triplication associated with singularity point, another wave has a lacuna.The singularity points are considered for elastic anisotropic model with orthorhombic symmetry. The conditions for the presence of singularity points are derived depending on location of these points, in symmetry plane and in-between the symmetry planes. The explicit equations for images of the singularity points in group velocity domain are derived. The conditions for different types of degeneracy of these images are defined. We also perform the analysis of interaction of the group velocity images of different singularity points.

The approximation for relative geometric spreading in the elastic vertical transversely isotropic medium

In seismic data processing, compensation for geometric spreading along the raypath is necessary. The accurate approximation of the relative geometric spreading is critical because the traveltime parameters in the approximation can be estimated from velocity analysis. Two approximation types (direct and indirect) are generalized based on the procedure to derive the explicit expression. We develop perturbation-based approximations of the relative geometric spreading for P and S waves with these two types in the elastic transversely isotropic media with a vertical symmetry axis (VTI). The rational approximation and the general moveout approximation counterparts are compared to verify the superiority of our approach. Our study compares the performance of direct and indirect approximations for optimal selection under different cases. In the numerical examples, we apply the error sensitivity in the layer depth, anellipticity value, and vertical velocity ratio. It is found that the P-wave case is much more accurate than the S-wave case, and the direct type obtains relatively higher accuracy than the indirect one. In addition, the perturbation coefficients for the acoustic VTI model are derived not only with a much simpler expression but also achieve a highly accurate result that is recommended for the P-wave seismic processing.

Extension of the kinematic approximations to the multilayered elastic orthorhombic medium

Understanding the kinematics of horizontally layered reservoir rocks is important to their proper characterization; to accomplish this, it is necessary to specify the explicit model for these kinematic properties. Accurate approximations for traveltime and relative geometric spreading in an elastic homogeneous orthorhombic (ORT) model have been investigated with different forms: shifted hyperbola form (SHF), Taylor series, and the rational form. We have extended these approximations to the multilayered ORT model by adopting composite coefficients and effective model parameters. The multilayered model is characterized without and with the azimuthal variation among layers. There is an overdetermined problem when the azimuthal variation exists; to address that case, the least-squares method is adopted. To check the feasibility of the expansion, we select the SHF approximation specified in the homogeneous elastic ORT model for the calculation in the numerical example. Four groups of examples are analyzed to investigate the influence on the accuracy of the approximation with the change in rotation angle, degree of anisotropy, and the direction of the orientation. The results indicate that, for the multilayer, the accuracy of the approximation is proportional to the degree of anisotropy and the value of the angle of rotation. The relative errors in traveltime and relative geometric spreading in this multilayered extension are very small and can be implemented in practical applications.

Approximations for traveltime, slope, curvature, and geometric spreading of elastic waves in layered transversely isotropic media

Each seismic body wave, including quasi-, P-, S-, and converted wave modes, carries useful subsurface information. For the processing, imaging, amplitude analysis, and forward modeling of each wave mode, we need approximate equations of the traveltime, slope (ray parameter), and curvature as a function of offset. Considering the large offset coverage of modern seismic acquisitions, we have developed new approximations designed to be accurate at zero and infinitely large offsets over layered transversely isotropic media with vertical symmetry axis. Our approximation for traveltime is a modified version of the extended generalized moveout approximation that comprises six parameters. Our direct approximations for the ray parameter and curvature use new algebraically simple equations with three parameters. We define these parameters for each wave mode without ray tracing so that we have similar approximate equations for all wave modes that only change based on the parameter definitions. However, our approximations are unable to reproduce S-wave triplications that may occur in some strongly anisotropic models. Using our direct approximation of traveltime derivatives, we also obtain a new expression for the relative geometric spreading. We determine the high accuracy of our approximations using numerical tests on a set of randomly generated multilayer models. Using synthetic data, we show simple applications of our approximations for the normal moveout correction and relative geometric spreading compensation of different wave modes.

Eigenrays in 3D heterogeneous anisotropic media, Part II: Dynamics

ABSTRACTThis paper is the second in a sequel of two papers and dedicated to the computation of paraxial rays and dynamic characteristics along the stationary rays obtained in the first paper. We start by formulating the linear, second‐order, Jacobi dynamic ray tracing equation. We then apply a similar finite‐element solver, as used for the kinematic ray tracing, to compute the dynamic characteristics between the source and any point along the ray. The dynamic characteristics in our study include the relative geometric spreading and the phase correction due to caustics (i.e. the amplitude and the phase of the asymptotic form of the Green's function for waves propagating in 3D heterogeneous general anisotropic elastic media). The basic solution of the Jacobi equation is a shift vector of a paraxial ray in the plane normal to the ray direction at each point along the central ray. A general paraxial ray is defined by a linear combination of up to four basic vector solutions, each corresponds to specific initial conditions related to the ray coordinates at the source. We define the four basic solutions with two pairs of initial condition sets: point–source and plane‐wave. For the proposed point–source ray coordinates and initial conditions, we derive the ray Jacobian and relate it to the relative geometric spreading for general anisotropy. Finally, we introduce a new dynamic parameter, similar to the endpoint complexity factor, presented in the first paper, used to define the measure of complexity of the propagated wave/ray phenomena. The new weighted propagation complexity accounts for the normalized relative geometric spreading not only at the receiver point, but along the whole stationary ray path. We propose a criterion based on this parameter as a qualifying factor associated with the given ray solution. To demonstrate the implementation of the proposed method, we use several isotropic and anisotropic benchmark models. For all the examples, we first compute the stationary ray paths, and then compute the geometric spreading and analyse these trajectories for possible caustics. Our primary aim is to emphasize the advantages, transparency and simplicity of the proposed approach.

Traveltime and relative geometric spreading approximation in elastic orthorhombic medium

Wavefield properties such as traveltime and relative geometric spreading (traveltime derivatives) are highly essential in seismic data processing and can be used in stacking, time-domain migration, and amplitude variation with offset analysis. Due to the complexity of an elastic orthorhombic (ORT) medium, analysis of these properties becomes reasonably difficult, where accurate explicit-form approximations are highly recommended. We have defined the shifted hyperbola form, Taylor series (TS), and the rational form (RF) approximations for P-wave traveltime and relative geometric spreading in an elastic ORT model. Because the parametric form expression for the P-wave vertical slowness in the derivation is too complicated, TS (expansion in offset) is applied to facilitate the derivation of approximate coefficients. The same approximation forms computed in the acoustic ORT model also are derived for comparison. In the numerical tests, three ORT models with parameters obtained from real data are used to test the accuracy of each approximation. The numerical examples yield results in which, apart from the error along the y-axis in ORT model 2 for the relative geometric spreading, the RF approximations all are very accurate for all of the tested models in practical applications.

New acoustic approximation for transversely isotropic media with a vertical symmetry axis

Seismic data processing in the elastic anisotropic model is complicated due to multiparameter dependency. Approximations to the P-wave kinematics are necessary for practical purposes. The acoustic approximation for P-waves in a transversely isotropic medium with a vertical symmetry axis (VTI) simplifies the description of wave propagation in elastic media, and as a result, it is widely adopted in seismic data processing and analysis. However, finite-difference implementations of that approximation are plagued with S-wave artifacts. Specifically, the resulting wavefield also includes artificial diamond-shaped S-waves resulting in a redundant signal for many applications that require pure P-wave data. To derive a totally S-wave-free acoustic approximation, we have developed a new acoustic approximation for pure P-waves that is totally free of S-wave artifacts in the homogeneous VTI model. To keep the S-wave velocity equal to zero, we formulate the vertical S-wave velocity to be a function of the model parameters, rather than setting it to zero. Then, the corresponding P-wave phase and group velocities for the new acoustic approximation are derived. For this new acoustic approximation, the kinematics is described by a new eikonal equation for pure P-wave propagation, which defines the new vertical slowness for the P-waves. The corresponding perturbation-based approximation for our new eikonal equation is used to compare the new equation with the original acoustic eikonal. The accuracy of our new P-wave acoustic approximation is tested on numerical examples for homogeneous and multilayered VTI models. We find that the accuracy of our new acoustic approximation is as good as the original one for the phase velocity, group velocity, and the kinematic parameters such as vertical slowness, traveltime, and relative geometric spreading. Therefore, the S-wave-free acoustic approximation could be further applied in seismic processing that requires pure P-wave data.

S‐wave kinematics in acoustic transversely isotropic media with a vertical symmetry axis

ABSTRACTAcoustic transversely isotropic models are widely used in seismic exploration for P‐wave processing and analysis. In isotropic acoustic media only P‐wave can propagate, while in an acoustic transversely isotropic medium both P and S waves propagate. In this paper, we focus on kinematic properties of S‐wave in acoustic transversely isotropic media. We define new parameters better suited for S‐wave kinematics analysis. We also establish the travel time and relative geometrical spreading equations and analyse their properties. To illustrate the behaviour of the S‐wave in multi‐layered acoustic transversely isotropic media, we define the Dix‐type equations that are different from the ones widely used for the P‐wave propagation.

Generalized non‐hyperbolic approximation for qP‐wave relative geometrical spreading in a layered transversely isotropic medium with a vertical symmetry axis

ABSTRACTCompensation for geometrical spreading along the ray‐path is important in amplitude variation with offset analysis especially for not strongly attenuative media since it contributes to the seismic amplitude preservation. The P‐wave geometrical spreading factor is described by a non‐hyperbolic moveout approximation using the traveltime parameters that can be estimated from the velocity analysis. We extend the P‐wave relative geometrical spreading approximation from the rational form to the generalized non‐hyperbolic form in a transversely isotropic medium with a vertical symmetry axis. The acoustic approximation is used to reduce the number of parameters. The proposed generalized non‐hyperbolic approximation is developed with parameters defined by two rays: vertical and a reference rays. For numerical examples, we consider two choices for parameter selection by using two specific orientations for reference ray. We observe from the numerical tests that the proposed generalized non‐hyperbolic approximation gives more accurate results in both homogeneous and multi‐layered models than the rational counterpart.

Geometric spreading in orthorhombic media

Geometric spreading is an important factor that needs to be taken into account in the analysis of seismic amplitudes. In particular, when using any modification of amplitude variation with offset or amplitude versus azimuth methods, the effect of geometric spreading is crucial to isolate the effect of reflection from a particular interface. The relative geometric spreading controls the amplitude of seismic waves passing through a velocity model. In the case of an anisotropic medium, geometric spreading becomes very complicated. Usually, geometric spreading is computed from ray tracing. I have derived simple analytical formulas to compute the relative geometric spreading of P-waves in a stack of acoustic orthorhombic layers with azimuthal variations in symmetry planes. I also analyzed the kinematic properties of the derived equations and performed sensitivity analysis with respect to three anelliptic parameters. A simple and accurate approximation for the relative geometric spreading is derived and tested against well-known approximation. My approximations give insight into the role that anelliptic parameters play into the azimuthal distribution of amplitudes and can be used for amplitude analysis in multilayered orthorhombic models.

An anelliptic approximation for geometric spreading in transversely isotropic and orthorhombic media

The relative geometric spreading along the raypath contributes to the amplitude decay of the seismic wave propagation that needs to be considered for amplitude variation with offset or other seismic data processing methods that require the true amplitude processing. Expressing the P-wave geometric spreading factor in terms of the offset-traveltime-based parameters is a more practical and convenient way because these parameters can be estimated from the nonhyperbolic velocity analysis. We have developed an anelliptic approximation for the relative geometric spreading of P-wave in a homogeneous transversely isotropic medium with vertical symmetry axis (VTI) and an orthorhombic (ORT) medium under the acoustic anisotropy assumption. The coefficients in our approximation are only defined within the symmetry planes and computed from fitting with the exact parametric expression. For an ORT model, due to the symmetric behavior in different symmetry planes, the other coefficients in the approximation can be easily obtained by making corresponding changes in indices from the computed coefficients in one symmetry plane. From the numerical examples, we found that for a homogeneous VTI model, the anelliptic approximation is more accurate than the generalized nonhyperbolic moveout approximation form for larger offset. For a homogeneous ORT model, our anelliptic approximation is more accurate than its traveltime-based counterparts. Using the Dix-type equations for the effective parameters, our anelliptic form approximation is extended to a multilayered VTI and ORT models and has accurate results in both models.

Three-dimensional generalized non-hyperbolic approximation for relative geometrical spreading

Three-dimensional generalized non-hyperbolic approximation for relative geometrical spreading

S-wave singularities in tilted orthorhombic media

Quasi S-wave propagation in low-symmetry anisotropic media is complicated due to the existence of point singularities (conical points) — points in the phase space at which slowness sheets of the split S-waves touch each other. At these points, two eigenvalues of the Christoffel tensor (associated with the quasi S-waves) degenerate into one and polarization directions of the S-waves, which lay in the plane orthogonal to the polarization of the quasi longitudinal wave, are not uniquely defined. In the vicinity of these points, slowness sheets of the S-waves have complicated shapes, leading to rapid variations in polarization directions, multipathing, and cusps and discontinuities of the shear wavefronts. In a tilted orthorhombic medium, the point singularities can occur close to the vertical, distorting the traveltime parameters that are defined at the zero offset. We have analyzed the influence of the singularities on these parameters by examining the derivatives of the slowness surface up to the fourth order. Using two orthorhombic numerical models of different shear anisotropy strength and with different number of singularity points, we evaluate the complexity of the slowness sheets in the vicinity of the conical points and analyze how the traveltime parameters are affected by the singularities. In particular, we observe that the hyperbolic region associated with the singularity points in a model with moderate to strong shear anisotropy spans over a big portion of the slowness surfaces and the traveltime parameters are strongly affected outside the hyperbolic region. In general, the fast shear mode is less affected by the singularities; however, the effect is still very pronounced. Moreover, the hyperbolic region associated with the singularity points on the slow S-wave affects the slowness surface of the fast mode extensively. In addition, we evaluate a relation between the slowness surface Gaussian curvature and the relative geometric spreading, which has anomalous behavior due to the singularities.

Kinematical characteristics of the factorized velocity model

ABSTRACTThe factorized velocity model that incorporates both vertical heterogeneity and constant anisotropy is one of the complicated analytical models used in seismic data processing and interpretation. In this paper, I derive the analytic equations for offset, traveltime and relative geometrical spreading for the quasi‐compressional (qP‐) waves that can be used for modelling and inversion of the traveltime parameters. I show that the presence of anelliptic anisotropy usually dominates over the vertical heterogeneity with respect to the non‐hyperbolicity of the factorized velocity model.

Power-gradient velocity model

A proposed power-gradient velocity model incorporates several well-known velocity models as special cases. The model covers a wide range of possible velocity distributions and has four parameters, giving more flexibility in velocity-model manipulation. For this nonlinear model, the kinematic characteristics — offset-traveltime parameteric equations, traveltime parameters, relative geometric spreading, and propagator phase — are computed. The characteristics are investigated with respect to a parameter responsible for nonlinearity of velocity distribution. The inversion of travel-time parameters was studied in three- and four-parameter frameworks.

Improved geometric-spreading approximation in layered transversely isotropic media

A new approximation (direct approximation) for the relative geometric spreading of qP-waves in a layered transversely isotropic medium uses three traveltime parameters: two-way vertical traveltime, normal-moveout velocity, and the heterogeneity coefficient. These traveltime parameters, which can be estimated in velocity analysis, enter the parameters of geometric-spreading approximation. The new approximation is based on the acoustic approximation for a single-layer vertical transversely isotropic (VTI) medium with coefficients defined from Taylor expansion and asymptotic behavior at infinite offset. The acoustic approximation is used to reduce the number of parameters and to make the expression for the direct geometric-spreading approximation similar to a well-known traveltime approximation. The same traveltime parameters also allow for an independent approximation of the radiation pattern. Two numerical tests (single-layer and multilayered VTI models) prove that the new approximation is more accurate than standard approximations based on shifted hyperbola or two rational approximations.

Paraxial ray methods for anisotropic inhomogeneous media

ABSTRACTA new formalism of surface‐to‐surface paraxial matrices allows a very general and flexible formulation of the paraxial ray theory, equally valid in anisotropic and isotropic inhomogeneous layered media. The formalism is based on conventional dynamic ray tracing in Cartesian coordinates along a reference ray. At any user‐selected pair of points of the reference ray, a pair of surfaces may be defined. These surfaces may be arbitrarily curved and oriented, and may represent structural interfaces, data recording surfaces, or merely formal surfaces. A newly obtained factorization of the interface propagator matrix allows to transform the conventional 6 × 6 propagator matrix in Cartesian coordinates into a 6 × 6 surface‐to‐surface paraxial matrix. This matrix defines the transformation of paraxial ray quantities from one surface to another. The redundant non‐eikonal and ray‐tangent solutions of the dynamic ray‐tracing system in Cartesian coordinates can be easily eliminated from the 6 × 6 surface‐to‐surface paraxial matrix, and it can be reduced to 4 × 4 form. Both the 6 × 6 and 4 × 4 surface‐to‐surface paraxial matrices satisfy useful properties, particularly the symplecticity. In their 4 × 4 reduced form, they can be used to solve important boundary‐value problems of a four‐parametric system of paraxial rays, connecting the two surfaces, similarly as the well‐known surface‐to‐surface matrices in isotropic media in ray‐centred coordinates. Applications of such boundary‐value problems include the two‐point eikonal, relative geometrical spreading, Fresnel zones, the design of migration operators, and more.

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-variation-with-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 wide-azimuth AVO signature for the influence of the anisotropic overburden.

Geometrical spreading in a layered transversely isotropic medium with vertical symmetry axis

Compensation for geometrical spreading is important in prestack Kirchhoff migration and in amplitude versus offset/amplitude versus angle (AVO/AVA) analysis of seismic data. We present equations for the relative geometrical spreading of reflected and transmitted P‐ and S‐wave in horizontally layered transversely isotropic media with vertical symmetry axis (VTI). We show that relatively simple expressions are obtained when the geometrical spreading is expressed in terms of group velocities. In weakly anisotropic media, we obtain simple expressions also in terms of phase velocities. Also, we derive analytical equations for geometrical spreading based on the nonhyperbolic traveltime formula of Tsvankin and Thomsen, such that the geometrical spreading can be expressed in terms of the parameters used in time processing of seismic data. Comparison with numerical ray tracing demonstrates that the weak anisotropy approximation to geometrical spreading is accurate for P‐waves. It is less accurate for SV‐waves, but has qualitatively the correct form. For P waves, the nonhyperbolic equation for geometrical spreading compares favorably with ray‐tracing results for offset‐depth ratios less than five. For SV‐waves, the analytical approximation is accurate only at small offsets, and breaks down at offset‐depth ratios less than unity. The numerical results are in agreement with the range of validity for the nonhyperbolic traveltime equations.