Igor Ravve

The Ricker and Gabor wavelets and their Hilbert transforms are widely used in many physical/engineering fields. In seismology, they are commonly utilized to describe source functions involved in the computation of synthetic seismograms in acoustic or elastic media, or inversely, the extraction of source wavelets from real seismic traces. We provide analytical formulae of the Ricker and Gabor wavelets’ Hilbert transforms, yielding to analytic expressions of the envelope, instantaneous frequency, and phase attributes of these wavelets. We analyze the plausibility of these wavelets by revealing the regions of positive instantaneous frequency and nonzero envelope.

SUMMARY: Reflector-normal angles and reflector-curvature parameters are the principal geometric attributes used in seismic interpretation for characterizing the orientations and shapes, respectively, of geological reflecting surfaces. Commonly, the input data set for their computation consists of fine 3-D grids of scalar fields representing either the seismic-driven reflectivities (e.g. amplitudes of 3-D seismic migrated volumes) or model-driven reflectivities, computed, for example, from the derived elastic impedance parameters. Conventionally, the computation of curvature parameters at each gridpoint is based on analysing the local change in the inline/crossline dips, considering the potential existence of a local quadratic reflecting surface in the vicinity of that point. This assumption breaks down for subsurface points in the vicinity of either complex reflecting surfaces (e.g. brittle/rough/tilted synclines/anticlines, ridges/troughs and saddles) and/or sharp, discontinuous geological features (e.g. fault edges/tips, pinch-outs, fracture systems, channels and small geobodies), where the values of the computed curvature become extremely high. However, while these high values can indicate the existence of non-reflecting objects, they do not deliver their specific geometric characteristics. In this study, we present a novel method that better characterizes the shapes of these complex geological features by extending the assumption of local surfaces (2-D surfaces in 3-D space) into local hypersurfaces (3-D hypersurfaces in 4-D space), with their corresponding (three rather than two) principal (and effective) curvature parameters. We demonstrate the advantages of our method by comparing the conventional dip-based surface curvature parameters with the hypersurface curvature parameters, using a synthetic model/image with different types and shapes of geological features and a seismic image of real data containing a complex fault and hidden buried channels.

Reflector-normal angles and reflector-curvature parameters are the principal geometric attributes used in seismic interpretation for characterizing the orientations and shapes, respectively, of geological reflecting surfaces. Commonly, the input dataset for their computation consists of fine 3D grids of scalar fields representing either the seismic-driven reflectivities (e.g., amplitudes of 3D seismic migrated volumes) or model-driven reflectivities, computed, for example, from the derived elastic impedance parameters. Conventionally, the computation of curvature parameters at each grid point is based on analyzing the local change in the inline/crossline dips, considering the potential existence of a local quadratic reflecting surface in the vicinity of that point. This assumption breaks down for subsurface points in the vicinity of either complex reflecting surfaces (e.g., brittle/rough/tilted synclines/anticlines, ridges/troughs, and saddles) and/or sharp discontinuous geological features (e.g., fault edges/tips, pinch-outs, fracture systems, channels, small geobodies), where the values of the computed curvature become extremely high. While these high values can indicate the existence of non-reflecting objects, they do not deliver their specific geometric characteristics. We present a novel method that better characterizes the shapes of these complex geological features by extending the assumption of local surfaces (2D surfaces in 3D space) into hypersurfaces (3D hypersurfaces in 4D space), with their corresponding principal and effective curvature parameters. We demonstrate the advantages of our method by comparing the conventional dip-based surface curvature parameters with the hypersurface curvature parameters, using a synthetic model/image with different types and shapes of geological features and two seismic images of real data containing complex faults and hidden buried channels.

SUMMARY: Considering general anisotropic (triclinic) media and both, quasi-compressional (qP) and quasi-shear (qS) waves, in Part I of this study, we obtained the ray (group) velocity gradients and Hessians with respect to the ray locations, directions and the elastic model parameters along ray trajectories. Ray velocity derivatives for anisotropic elastic media with higher symmetries were considered particular cases of general anisotropy. In this part, Part II, we follow the computational workflow presented in Part I, formulating the ray velocity derivatives directly for polar anisotropic media (transverse isotropy with tilted axis of symmetry, TTI) for the coupled qP waves (quasi-compressional waves) and qSV waves (quasi-shear waves polarized in the ‘axial’ plane) and for SH waves (shear waves polarized in the ‘normal’ plane). The acoustic approximation for qP waves is considered a special case. In seismology, the medium properties, normally specified at regular 3-D fine gridpoints, are the five material parameters: the axial compressional and shear wave velocities, the three (unitless) Thomsen parameters and two geometric parameters: the polar angles defining the local direction (the tilt) of the medium symmetry axis. All the parameters are assumed spatially (smoothly) varying, so that their spatial gradients and Hessians can be reliably numerically computed. Two case examples are considered; the first represents compacted shale/sand rocks (with positive anellipticity) and the second, unconsolidated sand rocks with strong negative anellipticity (manifesting a qSV triplication). The ray velocity derivatives obtained in this part are first tested by comparing them with the corresponding numerical (finite difference) derivatives. Additionally, only for validation purpose, we show that exactly the same results (ray velocity derivatives) can be obtained if we transform the given polar anisotropic model parameters (five material and two geometric) into the 21 stiffness tensor components of a general anisotropic (triclinic) medium, and apply the theory derived in Part I. Since in many practical wave/ray-based applications in polar anisotropic media only the spatial derivatives of the axial compressional wave velocity are taken into account, we analyse the effect (sensitivity) of the spatial derivatives of the other parameters on the ray velocity and its derivatives (which, in turn, define the corresponding traveltime derivatives along the ray).

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.

In this part, we apply the same finite-element approach, used in Part III for the vanishing first traveltime variation (to obtain the stationary rays), for the second traveltime variation, in order to compute the dynamic characteristics along the stationary ray. The finite-element solver involves application of the weak formulation and the Galerkin method to the linear second-order Jacobi ordinary differential equation (derived in Part V), yielding an original linear algebraic equation set for dynamic ray tracing. In our formulation, the resolving matrix of the linear equation set coincides with the global traveltime Hessian computed for the kinematic ray tracing, making the solution of the dynamic problem straightforward. The proposed method is unconditionally stable (the solution does not explode when the intervals between the nodes are increased) and is more accurate than the commonly used numerical integration (e.g., Runge-Kutta) methods, in particular, for stationary rays passing through heterogeneous anisotropic models with complex wave phenomena.

In Part V of this study, we presented an original Lagrangian approach for computing the dynamic characteristics along stationary rays, by solving the linear, second-order Jacobi differential equation, considering four sets of initial conditions as the basic solutions. We then focused on the computation of the geometric spreading and identification of caustics, where only the two point-source basic solutions with their corresponding initial conditions are required. Solutions of the Jacobi equation represent the normal shift vectors of the paraxial rays and define the geometry of the ray tube with respect to the stationary central ray. Rather than the Lagrangian approach, the dynamic characteristics are traditionally computed with the Hamiltonian approach, formulated normally in terms of two first-order differential equations, where the solution variables are the paraxial shifts and paraxial slowness changes along the ray. In this part (Part VI), we compare and relate the two approaches. We first combine the two first-order Hamiltonian dynamic equations, eliminating the paraxial variations of the slowness vector. This leads to a second-order differential equation in terms of the Hamiltonian shift alone, whose ray-normal counterpart coincides with the normal shift of the corresponding Lagrangian solution, while the ray-tangent component does not affect the Jacobian and the geometric spreading. Comparing the proposed Lagrangian approach to the dynamic ray tracing with the "classical" Hamiltonian approach, we demonstrate that they are fully compatible for general anisotropy. We then derive the two-way relationships between the Hamiltonian's and Lagrangian's Hessians, which are the core computational elements of dynamic ray theory. Finally, we demonstrate the relationships between these two types of the Hessians numerically, for a general triclinic medium.