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SUMMARY We present an original, generic and efficient approach for computing the first and second partial derivatives of ray (group) velocities along seismic ray paths in general anisotropic (triclinic) elastic media. As the ray velocities deliver the ray element traveltimes, this set of partial derivatives constructs the so-called kinematic and dynamic sensitivity kernels which are used in different key seismic modelling and inversion methods, such as two-point ray bending methods and seismic tomography. The second derivatives are useful in the solution of the above-mentioned kinematic problems, and they are essential for evaluating the dynamic properties along the rays (amplitudes and phases). The traveltime is delivered through an integral over a given Lagrangian defined at each point along the ray. In our approach, we use an arclength-related Lagrangian representing a reciprocal of the ray velocity magnitude. Although this magnitude cannot be explicitly expressed in terms of the medium properties and the ray direction components, its derivatives can still be formulated analytically using the corresponding arclength-related Hamiltonian that can be explicitly expressed in terms of the medium properties and the slowness vector components; this requires first to obtain (invert for) the slowness vector components, given the ray direction components. Computation of the slowness vector and the ray velocity derivatives is considerably simplified by using an auxiliary scaled-time-related Hamiltonian obtained directly from the Christoffel equation and connected to the arclength-related Hamiltonian by a simple scale factor. This study consists of two parts. In Part I, we consider general anisotropic (triclinic) media, and provide the derivatives (gradients and Hessians) of the ray velocity, with respect to (1) the spatial location and direction vectors and (2) the elastic model parameters. The derivatives are obtained for both quasi-compressional and quasi-shear waves, where other types of media, characterized with higher symmetries, can be considered particular cases. In Part II, we apply the theory of Part I explicitly to polar anisotropic media (transverse isotropy with tilted axis of symmetry, TTI), and obtain the explicit ray velocity derivatives for the coupled qP and qSV waves and for SH waves. The derivatives for polar anisotropy are simplified (as compared to general anisotropy), obviously yielding more effective computations. The ray velocity derivatives are tested by checking consistency between the proposed analytical formulae and the corresponding numerical ones.

SUMMARY The inverse problem of finding the slowness vector from a known ray direction in general anisotropic elastic media is a challenging task, needed in many wave/ray-based methods, in particular, solving two-point ray bending problems. The conventional resolving equation set for general (triclinic) anisotropy consists of two fifth-degree polynomials and a sixth-degree polynomial, resulting in a single physical solution for quasi-compressional (qP) waves and up to 18 physical solutions for quasi-shear waves (qS). For polar anisotropy (transverse isotropy with a tilted symmetry axis), the resolving equations are formulated for the slowness vectors of the coupled qP and qSV waves (quasi-shear waves polarized in the axial symmetry plane), and independently for the decoupled pure shear waves polarized in the normal (to the axis) isotropic plane (SH). The novelty of our approach is the introduction of the geometric constraint that holds for any wave mode in polar anisotropic media: The three vectors—the slowness, ray velocity and medium symmetry axis—are coplanar. Thus, the slowness vector (to be found) can be presented as a linear combination of two unit-length vectors: the polar axis and the ray velocity directions, with two unknown scalar coefficients. The axial energy propagation is considered as a limit case. The problem is formulated as a set of two polynomial equations describing: (i) the collinearity of the slowness-related Hamiltonian gradient and the ray velocity direction (third-order polynomial equation) and (ii) the vanishing Hamiltonian (fourth-order polynomial equation). Such a system has up to twelve real and complex-conjugate solutions, which appear in pairs of the opposite slowness directions. The common additional constraint, that the angle between the slowness and ray directions does not exceed ${90^{\rm{o}}}$, cuts off one half of the solutions. We rearrange the two bivariate polynomial equations and the above-mentioned constraint as a single univariate polynomial equation of degree six for qP and qSV waves, where the unknown parameter is the phase angle between the slowness vector and the medium symmetry axis. The slowness magnitude is then computed from the quadratic Christoffel equation, with a clear separation of compressional and shear roots. The final set of slowness solutions consists of a unique real solution for qP wave and one or three real solutions for qSV (due to possible triplications). The indication for a qSV triplication is a negative discriminant of the sixth-order polynomial equation, and this discriminant is computed and analysed directly in the ray-angle domain. The roots of the governing univariate sixth-order polynomial are computed as eigenvalues of its companion matrix. The slowness of the SH wave is obtained from a separate equation with a unique analytic solution. We first present the resolving equation using the stiffness components, and then show its equivalent forms with the well-known parametrizations: Thomsen, Alkhalifah and ‘weak-anisotropy’. For the Thomsen and Alkhalifah forms, we also consider the (essentially simplified) acoustic approximation for qP waves governed by the quartic polynomials. The proposed method is coordinate-free and can be applied directly in the global Cartesian frame. Numerical examples demonstrate the advantages of the method.

Kinematical characteristics of reflected waves in anisotropic elastic media play an important role in the seismic imaging workflow. Considering compressional and converted waves, we derive new, azimuthally dependent, slowness‐domain approximations for the kinematical characteristics of reflected waves (radial and transverse offsets, intercept time and traveltime) for layered orthorhombic media with varying azimuth of the vertical symmetry planes. The proposed method can be considered an extension of the well‐known ‘generalized moveout approximation’ in the slowness domain, from azimuthally isotropic to azimuthally anisotropic models. For each slowness azimuth, the approximations hold for a wide angle range, combining power series coefficients in the vicinity of both the normal‐incidence ray and an additional wide‐angle ray. We consider two cases for the wide‐angle ray: a ‘critical slowness match’ and a ‘pre‐critical slowness match’ studied in Parts I and II of this work, respectively. For the critical slowness match, the approximations are valid within the entire slowness range, up to the critical slowness. For the ‘pre‐critical slowness match’, the approximations are valid only within the bounded slowness range; however, the accuracy within the defined range is higher. The critical slowness match is particularly effective when the subsurface model includes a dominant high‐velocity layer where, for nearly critical slowness values, the propagation in this layer is almost horizontal. Comparing the approximated kinematical characteristics with those computed by numerical ray tracing, we demonstrate high accuracy.

Sedimentary layers affected by vertical compaction and strong lateral tectonic stresses are often characterized by low anisotropic symmetry (e.g., tilted orthorhombic [TOR]/monoclinic or even triclinic). Considering all types of pure-mode and converted waves, we derive the normal moveout (NMO) series coefficients of near normal-incidence reflected waves in arbitrarily anisotropic horizontally layered media, for a leading error term of order six. The NMO series can be either a function of the invariant horizontal slowness (slowness domain) or the surface offset (offset domain). The NMO series coefficients, referred to also as effective parameters, are associated with the corresponding azimuthally varying NMO velocity functions. We distinguish between local (single-layer) and global (overburden multilayer) effective parameters, which are related by forward and inverse Dix-type transforms. We derive the local effective parameters for an arbitrary anisotropic (triclinic) layer, which is the main contribution of this paper. With some additional geologic constraints, the local effective parameters can then be converted into the interval elastic properties. To demonstrate the applicability of our method, we consider a synthetic layered model in which each layer is characterized with TOR symmetry. The corresponding global effective model loses the symmetries of the individual layers and is characterized by triclinic symmetry.

Considering all types of pure-mode and converted waves, we derive the azimuthally dependent, fourth-order normal moveout (NMO) velocity functions, and hence the corresponding effective anellipticity functions, for horizontally layered orthorhombic media. We emphasize that this paper does not suggest a new nonhyperbolic traveltime approximation; rather, it provides exact expressions of the NMO series coefficients, computed for normal-incidence rays, which can then be further used within known azimuthally dependent traveltime approximations for short to moderate offsets. We do not assume weak anisotropy or acoustic approximation for P-waves. At each layer, the elastic parameters, thickness, and azimuth of the orthorhombic vertical symmetry planes are considered to be different. We distinguish between two different azimuths: slowness azimuth (part 1 of this paper) and offset azimuth (part 2 of this paper). In part 1, the slowness-azimuth domain NMO is approximated as a series of either infinitesimal horizontal slowness (slowness-azimuth/slowness domain) or infinitesimal offsets (slowness-azimuth/offset domain). Similarly, in part 2, we distinguish between two offset-azimuth domains: offset-azimuth/slowness and offset-azimuth/offset. Note that the azimuthally dependent NMO velocity functions of each of the four cases are different. The validity of the method is tested by introducing our derived azimuthally dependent, fourth-order effective anellipticity, into the well-known azimuthally dependent, asymptotic nonhyperbolic traveltime approximation, in which we compare the traveltime approximation versus exact numerical ray tracing for short to moderate offsets. It is clearly shown that for these types of azimuthally anisotropic layered models, the fourth-order terms are essential even for relatively small horizontal-slowness values or short offsets.

Summary We present an efficient and stable procedure for estimating second- and fourth-order azimuthally-dependent effective parameters from full-azimuth residual moveouts. The residual moveouts are automatically picked at depth image points along full-azimuth angle domain reflection angle gathers. It is assumed that the azimuthally varying residual moveouts are due to fracture systems within compacted sand/shale sediment layers which were not accounted for in the seismic migration. The extracted (up to eight) effective parameters can then be used to obtain local (layer) effective parameters, characterizing the intensity and orientation of the fracture systems at each layer. Finally, the local effective parameters can be inverted to obtain interval anisotropic (e.g., orthorhombic) model parameters to be used in orthorhombic seismic migration.

We study the azimuthally dependent hyperbolic moveout approximation for small angles (or offsets) for quasi-compressional, quasi-shear, and converted waves in onedimensional multi-layer orthorhombic media. The vertical orthorhombic axis is the same for all layers, but the azimuthal orientation of the horizontal orthorhombic axes at each layer may be different. By starting with the known equation for normal moveout velocity with respect to the surface-offset azimuth and applying our derived relationship between the surface-offset azimuth and phase-velocity azimuth, we obtain the normal moveout velocity versus the phase-velocity azimuth. As the surface offset/azimuth moveout dependence is required for analysing azimuthally dependent moveout parameters directly from time-domain rich azimuth gathers, our phase angle/azimuth formulas are required for analysing azimuthally dependent residual moveout along the migrated local-angle-domain common image gathers. The angle and azimuth parameters of the local-angle-domain gathers represent the opening angle between the incidence and reflection slowness vectors and the azimuth of the phase velocity ψphs at the image points in the specular direction. Our derivation of the effective velocity parameters for a multi-layer structure is based on the fact that, for a one-dimensional model assumption, the horizontal slowness ph and the azimuth of the phase velocity ψphs remain constant along the entire ray (wave) path. We introduce a special set of auxiliary parameters that allow us to establish equivalent effective model parameters in a simple summation manner. We then transform this set of parameters into three widely used effective parameters: fast and slow normal moveout velocities and azimuth of the slow one. For completeness, we show that these three effective normal moveout velocity parameters can be equivalently obtained in both surface-offset azimuth and phase-velocity azimuth domains.

Summary Considering reciprocity where the traveltime is an even function of the offset or horizontal-slowness, the fourth-order normal moveout (NMO) series are governed by the normal-incidence time and eight effective parameters: three second-order and five fourth-order. Local effective parameters are related to the individual layers, while the global effective parameters are related to the overburden multi-layer model. Local and global parameters are related by forward and inverse Dix-type transforms. The NMO formulae are different in the slowness and offset domains, but the eight parameters are the same in both cases. We suggest a new set of intuitive normalized effective parameters, classified into two “azimuthally isotropic” and six “azimuthally anisotropic” parameters. We provide feasible ranges for the normalized parameters, thus allowing their used for controlled inversion.

Summary Considering all types of pure-mode and converted waves, we derive the azimuthally-dependent normal moveout (NMO) series coefficients of near normal-incidence reflection waves in general anisotropic (triclinic) horizontally layered media, for a leading error term of order six. The NMO series can be either a function of the invariant horizontal-slowness (slowness domain) or the surface-offset (offset domain). The NMO series coefficients of different orders, also referred to as effective parameters, are associated with the corresponding azimuthally-dependent NMO velocity functions. We distinguish between local (single-layer) and global (overburden multilayer) effective parameters, where the local and global effective parameters are related by forward and inverse Dix-type transforms. We first consider the case in which the reciprocity assertion for incidence and reflected waves holds, i.e. pure-mode waves for general anisotropic horizontally-layered media, and converted waves for anisotropic horizontally-layered models sharing a common horizontal symmetry plane. Considering reciprocity, the odd-power coefficients of the NMO series cancel, and the remaining coefficients are zero-offset time, three second-order and five fourth-order effective parameters. Next we consider converted waves in general anisotropic media, where reciprocity no longer holds. Twelve additional parameters are required: two firstorder, four third-order and six fifth-order effective parameters.

We consider a case where a 3D depth migration has been performed in the local angle domain (LAD) using rich-azimuth seismic data (e.g., conventional land surveys). The subsurface geologic model is characterized by considerable azimuthally anisotropic velocity variations. The background velocity field used for the migration can consist of azimuthally independent, e.g., vertical transverse isotropy, and/or azimuthally dependent (e.g., orthorhombic), velocity layers. The resulting 3D full-azimuth reflection angle gathers generated by the LAD migration represent in situ high-resolution amplitude preserved reflectivities associated with opening angles between incident and reflected slowness vectors in the specular directions. Residual moveouts (RMOs) automatically picked on these 3D image gathers along major horizons can indicate considerable residual periodic azimuthal variations. This situation is typical in depth imaging applied to unconventional shale plays, where the background velocity model doesn’t yet account for the aligned stress/fracture systems that exist in some of the target layers. We use the azimuthally dependent, phase-angle RMOs to update the interval parameters of the background model, accounting for the azimuthal anisotropy effect. Until now, this problem was mainly treated in the unmigrated time-offset domain, which is limited in describing the actual in situ changes of the velocity field with azimuths. The subsurface full-azimuth phase-angle domain RMOs provide better physical parameters to analyze the in situ azimuthal variations of the anisotropic media. Our method is grounded in a newly derived generalized Dix-based theory, where locally the background and updated models are assumed to be 1D anisotropic velocity models. At each lateral location, the orthorhombic axis [Formula: see text] points in the vertical direction across all layers, but the azimuthal orientations of the orthorhombic layers change from layer to layer. An effective model for such a layered structure (background or updated) is represented by a single layer with a vertical time identical to that of the whole package, effective fast and slow normal moveout (NMO) velocities, and an effective azimuthal orientation of the slow NMO velocity. Our approach begins with computation of these effective parameters for the background model and conversion of the high-resolution RMOs into a dense set of updated, effective, azimuthally dependent NMO velocities, which are then converted into three effective parameters of the updated model. Next, we apply a generalized Dix-based inversion approach to estimate the local NMO parameters for each updated layer. Finally, we convert the local parameters into interval azimuthally varying anisotropic model parameters (e.g., TTI, orthorhombic, or tilted orthorhombic) within each layer. The 1D Dix-based approach presented in this work should not be considered an alternative to more accurate 3D global inversion approaches, such as global anisotropic tomography. However, the proposed method can be effectively used for moderately laterally varying models, and some of the principal physical rules derived for the 1D model can be further used to improve the formulation and geophysical constraints applied to 3D global inversion methods.

We present a new subsurface angle-domain seismic imaging system for generating and extracting high-resolution information about subsurface angle-dependent reflectivity. The system enables geophysicists to use all recorded seismic data in a continuous fashion directly in the subsurface local angle domain (LAD), resulting in two complementary, full-azimuth, common-image-angle gather systems: directional and reflection. The complete set of information from both types of angle gathers leads to accurate, high-resolution, reliable velocity model determination and reservoir characterization. The directional angle decomposition enables the implementation of specular and diffraction imaging in real 3D isotropic/anisotropic geological models, leading to simultaneous emphasis on continuous structural surfaces and discontinuous objects such as faults and small-scale fractures. Structural attributes at each subsurface point, e.g., dip, azimuth and continuity, can be derived directly from the directional anglegathers. The reflection-angle gathers display reflectivity as a function of the opening angle and opening azimuth. These gathers are most meaningful in the vicinity of actual local reflecting surfaces, where the reflection angles are measured with respect to the derived background specular direction. The reflection-angle gathers are used for automatic picking of full-azimuth angle-domain residual moveouts (RMO) which, together with the derived background orientations of the subsurface reflection horizons, provide a complete set of input data to isotropic/anisotropic tomography. The full-azimuth, angle-dependent amplitude variations are used for reliable and accurate amplitude versus angle and azimuth (AVAZ) analysis and reservoir characterization. The proposed system is most effective for imaging and analysis below complex structures, such as subsalt and subbasalt, high-velocity carbonate rocks, shallow low-velocity gas pockets, and others. In addition, it enables accurate azimuthal anisotropic imaging and analysis, providing optimal solutions for fracture detection and reservoir characterization.

We use residual moveouts measured along continuous full azimuth reflection angle gathers, in order to obtain effective horizontal transversely isotropic model parameters. The angle gathers are generated through a special angle domain imaging system, for a wide range of reflection angles and full range of phase velocity azimuths. The estimation of the effective model parameters is performed in two stages. First, the background horizontal transversely isotropic (HTI)/vertical transversely isotropic (VTI) layered model is used, along with the values of reflection angles, for converting the measured residual moveouts (or traveltime errors) into azimuthally dependent normal moveout (NMO) velocities. Then we apply a digital Fourier transform to convert the NMO velocities into azimuthal wavenumber domain, in order to obtain the effective HTI model parameters: vertical time, vertical compression velocity, Thomsen parameter delta and the azimuth of the medium axis of symmetry. The method also provides a reliability criterion of the HTI assumption. The criterion shows whether the medium possesses the HTI type of symmetry, or whether the azimuthal dependence of the residual traveltime indicates to a more complex azimuthal anisotropy. The effective model used in this approach is defined for a 1D structure with a set of HTI, VTI and isotropic layers (with at least one HTI layer). We describe and analyse the reduction of a multi-layer structure into an equivalent effective HTI model. The equivalent model yields the same NMO velocity and the same offset azimuth on the Earth's surface as the original layered structure, for any azimuth of the phase velocity. The effective model approximates the kinematics of an HTI/VTI layered structure using only a few parameters. Under the hyperbolic approximation, the proposed effective model is exact.

Anisotropy in subsurface geological models is primarily caused by two factors: sedimentation in shale/sand layers and fractures. The sedimentation factor is mainly modelled by vertical transverse isotropy (VTI), whereas the fractures are modelled by a horizontal transversely isotropic medium (HTI). In this paper we study hyperbolic and non-hyperbolic normal reflection moveout for a package of HTI/VTI layers, considering arbitrary azimuthal orientation of the symmetry axis at each HTI layer. We consider a local 1D medium, whose properties change vertically, with flat interfaces between the layers. In this case, the horizontal slowness is preserved; thus, the azimuth of the phase velocity is the same for all layers of the package. In general, however, the azimuth of the ray velocity differs from the azimuth of the phase velocity. The ray azimuth depends on the layer properties and may be different for each layer. In this case, the use of the Dix equation requires projection of the moveout velocity of each layer on the phase plane. We derive an accurate equation for hyperbolic and high-order terms of the normal moveout, relating the traveltime to the surface offset, or alternatively, to the subsurface reflection angle. We relate the azimuth of the surface offset to its magnitude (or to the reflection angle), considering short and long offsets. We compare the derived approximations with analytical ray tracing.

Local tomography is interactive, ray-based, residual-interval-parameter analysis for updating background anisotropic velocity parameters. The method operates directly on image gathers generated by anisotropic curved-ray Kirchhoff time migration. A locally 1D, spatially varying, vertical transversely isotropic model is assumed. The background anisotropy parameters are the instantaneous (interval) vertical compression velocity [Formula: see text] and the two Thomsen anisotropy parameters, [Formula: see text] and [Formula: see text]. The interval velocity [Formula: see text] is updated from short-offset reflection events, and [Formula: see text] is updated from available long-offset data. The medium parameters are updated from the top down both vertically and by layers, one parameter at a time. The picked residual-anisotropy parameters correspond to the residual-moveout (RMO) curves that best fit the migrated reflection events. The method is based on splitting the contribution to the computed RMO at a given point into two parts: from overburden residual parameters and from the actual picked residual parameter. This approach allows for direct residual-interval-parameter analysis to be applied in the same way we perform the commonly used residual-effective-parameter analysis. The local tomography enables a controlled interactive estimation of the long-wavelength anisotropy parameters. The reliable anisotropy parameters estimated by the local approach are used as a background (guiding) model for a global tomography. This makes it possible to successfully apply a global constrained inversion that is performed simultaneously for all parameters of all output intervals using detailed RMO information.

This study presents an enhancement to the existing parametric traveltime approximations in VTI elastic layered media by accounting for long-offset wave propagation in a more correct manner. The method is applicable for all types of pure-mode and converted waves, and it is highly accurate for all waves, except SV with strong anisotropy, dominated by cusps. We suggest a parametric approach where both the offset and the traveltime are functions of the horizontal slowness whose finite upper bound (critical slowness) corresponds to infinite offset and traveltime. We distinguish between the layer with the fastest horizontal velocity (“fast” layer) and the other (“slow”) layers. In vertically varying 1D layered media, the horizontal slowness is constant for all layers in both incident and reflected waves, for all wave types. Thus, for a power series approximation, the coefficients representing the contributions of the individual VTI layers to the total offset and traveltime can be summed. For nearly critical slowness, the contributions of the “fast” and “slow” layers are decoupled; this separation gives a better physical insight into the nature of long-offset wave propagation. The accuracy of the method is demonstrated by comparing the approximated offsets and traveltimes with numerical ray tracing.

We derive azimuthally dependent fourth-order effective velocity and moveout parameters for compressional waves propagating in a layered model that consists of orthorhombic layers. The subsurface layered medium is considered as a locally 1D model, where each layer can be characterized by orthorhombic parameters. The orthorhombic layers have a common vertical axis but different azimuthal orientations of horizontal axes. For a 1D vertically varying anisotropic model, the azimuth of the phase velocity is the same for all layers, while the azimuths of the ray velocity are generally different. We extend the existing studies on the moveout in an azimuthally anisotropic model, accounting for the azimuthal deviation between the phase and ray velocities. We compute the lag between the azimuth of the surface offset (source-receiver vector) and the phase velocity azimuth. An effective model that replaces the multi-layer model with a single azimuthally anisotropic layer is derived, whose moveout and offset azimuth are identical to those of the layer package up to the fourth-order terms. We verify the accuracy of the approximation for small to moderate reflection angles vs. exact analytical ray tracing. The proposed approximation is in particular important when analyzing residual moveouts measured along full-azimuth common image reflection angle gathers.

Orthorhombic models comprise subsurface anisotropy caused by vertical azimuthally-aligned fractures and layering, or by two orthogonal sets of vertical fractures, with or without layering. In this paper we derive new relations for hyperbolic and non-hyperbolic moveout approximations for pure compression waves, considering a 1D model that consists of a set of orthorhombic layers. The layers have a common vertical axis but different orientations of horizontal orthorhombic axes. For 1D models, the azimuth of the phase velocity is the same for all layers, while the azimuths of the ray velocity are generally different. We extend the existing studies on moveout in an orthorhombic model, accounting for the azimuthal deviation between the phase and ray velocities. We then formulate the azimuthally-dependent NMO velocity for a package of layers. Finally, we compare the derived full quartic moveout term with its acoustic approximation, and verify the accuracy of the approximation vs. exact analytical ray tracing.

This work presents a new seismic imaging system for generating amplitude preserved, three-dimensional directional gathers. The proposed method is based on directional angle decomposition that enables the implementation of both specular and diffraction imaging in real 3D isotropic/anisotropic geological models, leading to simultaneous emphasis on both continuous structural surfaces and discontinuous objects, such as faults and small-scale fractures. Structural attributes at each subsurface point, e.g., dip, azimuth and continuity, can be derived directly from the directional angle gathers. The proposed approach is most effective for imaging and analysis below complex structures, such as subsalt and subbasalt, high-velocity carbonate rocks, shallow velocity anomalies, and others.

This work presents a new seismic imaging system for generating amplitude preserved, three dimensional directional gathers. The proposed directional angle decomposition enables the implementation of both specular and diffraction imaging in real 3D isotropic/anisotropic geological models, leading to simultaneous emphasis on both continuous structural surfaces and discontinuous objects, such as faults and small-scale fractures. Structural attributes at each subsurface point, e.g., dip, azimuth and continuity, can be derived directly from the directional angle gathers. The proposed system is most effective for imaging and analysis below complex structures, such as sub-salt and sub-basalt, high-velocity carbonate rocks, shallow velocity anomalies, and others.

Summary The simplest assumption on an anisotropic medium with azimuthally dependent traveltime is an HTI. We use residual moveouts measured along continuous full azimuth reflection angle gathers, in order to obtain effective HTI model parameters. The angle gathers are generated through a special angle domain imaging system, for a wide range of reflection angles and full range of phase velocity azimuths. The estimation of the effective model parameters is performed in two stages. First, the background HTI/VTI layered model is used, along with the values of reflection angles, for converting the measured residual moveouts (or traveltime errors) into azimuthally dependent NMO velocities. Then, we apply a Fourier transform to convert the NMO velocities into azimuthal wavenumber domain, in order to obtain the effective HTI model parameters: vertical time, vertical compression velocity, Thomsen parameter delta, and the azimuth of the medium axis of symmetry. The method also provides a reliability criterion of the HTI assumption. The effective model used in this approach is defined for a 1D model with a set of HTI, VTI, and isotropic layers. We describe and analyze the reduction of a multi-layer structure into an equivalent effective HTI model. The equivalent model yields the same NMO velocity and the same offset azimuth on the earth surface as the original layered structure, for any azimuth of the phase velocity. The effective model approximates the kinematics of an HTI/VTI layered structure using only a few parameters. Under the hyperbolic approximation, the proposed effective model is exact.

This work presents a new seismic imaging system for generating and extracting high-resolution information about subsurface angle dependent reflectivity, with simultaneous emphasis on both continuous structural surfaces and discontinuous objects, such as faults and small-scale fractures. The system enables full-azimuth, angledependent seismic imaging using reflection data recorded through seismic acquisition surveys, especially wideazimuth and long offset data. Geometrical attributes, such as dip-azimuth and continuity of the local reflecting surfaces, can be automatically extracted directly from the full-azimuth angle gathers. Azimuthal anisotropy can be detected, leading to an accurate anisotropy model representation.

Summary Curved Rays Tomography updates background anisotropy velocity parameters in the time-migrated domain. Th e tomography uses image gathers generated by Anisotropy Curved Rays Kirchhoff Time Migration. A locally varying 1D Vertical Transverse Isotropy (VTI) model is assu med. The background anisotropy parameters are the instantaneous (interval) vertical compression veloc ity V and the two Thomsen anisotropy parameters δ and e . Interval velocity (or alternatively δ ) is updated from short offsets reflection events, while e is updated from the available long offset data. Two complementary approaches are presented in this study: local and global. In t he local approach, the medium parameters are updated from top down, layer by layer, one parameter at a time. The residual anisotropy parameters, that best fit the residual m oveout curves, are picked. The residual moveout includes overburden and current layer components. In the glo bal approach, all parameters are inverted simultaneousl y. Due to a large number of offsets, the problem becomes o verdefined, and we solve it by a constrained least-squ ares minimization. The cost function accounts for data a nd model variances, which reflect the reliability of t he data and control parameter variations, respectively. The updated parameters are constrained to a feasible range.