Preface 1. Introduction 2. The elastodynamics and its simple solutions 3. Seismic rays and travel times 4. Dynamic ray tracing. Paraxial ray methods 5. Ray amplitudes 6. Ray synthetic seismograms Appendix. Fourier transform, Hilbert transform and analytical signals References Index.

Seismic Ray Theory

Preface 1. Introduction to rock physics 2. Rock physics interpretation of texture, lithology and compaction 3. Statistical rock physics: combining rock physics, information theory, and statistics to reduce uncertainty 4. Common techniques for quantitative seismic interpretation 5. Case studies: lithology and pore-fluid prediction from seismic data 6. Workflows and guide lines 7. Hands-on References Index.

Quantitative Seismic Interpretation: Applying Rock Physics Tools to Reduce Interpretation Risk

Summary

We use body wave ray theory in conjunction with the Born approximation tocompute 3-D Frechet kernels for finite-frequency seismic traveltimes, measured by cross-correlation of a broad-band waveform with a spherical earth synthetic seismogram. Destructive interference among adjacent frequencies in the broad-band pulse renders a cross-correlation traveltime measurement sensitive only to the wave speed in a hollow banana-shaped region surrounding the unperturbed geometrical ray. The Frechet kernel expressing this sensitivity is expressed as a double sum over all forward-propagating body waves from the source and backward-propagating body waves from the receiver to every single scatterer in the vicinity of this central ray. The kernel for a differential traveltime, measured by cross-correlation of two phases at the same receiver, is simply the difference of the respective single-phase kernels. In the paraxial approximation, an absolute or differential traveltime kernel can be computed extremely economically by implementing a single kinematic and dynamic ray trace along each source-to-receiver ray.

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Fréchet kernels for finite-frequency traveltimes—I. Theory

A wave equation, derived using the acoustic medium assumption for P-waves in transversely isotropic (TI) media with a vertical symmetry axis (VTI media), yields a good kinematic approximation to the familiar elastic wave equation for VTI media. The wavefield solutions obtained using this VTI acoustic wave equation are free of shear waves, which significantly reduces the computation time compared to the elastic wavefield solutions for exploding‐reflector type applications. From this VTI acoustic wave equation, the eikonal and transport equations that describe the ray theoretical aspects of wave propagation in a TI medium are derived. These equations, based on the acoustic assumption (shear wave velocity = 0), are much simpler than their elastic counterparts, yet they yield an accurate description of traveltimes and geometrical amplitudes. Numerical examples prove the usefulness of this acoustic equation in simulating the kinematic aspects of wave propagation in complex TI models.

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An acoustic wave equation for anisotropic media

The study of P-wave reflection coefficients in anisotropic media is important for amplitude variation with offset (AVO) analysis. While numerical evaluation of the reflection coefficient is straightforward, numerical solutions do not provide analytic insight into the influence of anisotropy on the AVO signature. To overcome this difficulty, I present an improved approximation for P-wave reflection coefficients at a horizontal boundary in transversely isotropic media with vertical axis of symmetry (VTI media). This solution has the same AVO-gradient term describing the low-order angular variation of the reflection coefficient as the equations published previously, but is more accurate for large incidence angles. The refined approximation is then extended to transverse isotropy with a horizontal axis of symmetry (HTI), which is caused typically by a system of vertical cracks. Comparison of the approximate reflection coefficients for P-waves incident in the two vertical symmetry planes of HTI media indicates that the azimuthal variation of the AVO gradient is a function of the shear-wave splitting parameter y, and the anisotropy parameter describing P-wave anisotropy for nearvertical propagation in the vertical plane containing the symmetry axis.

P-wave reflection coefficients for transversely isotropic models with vertical and horizontal axis of symmetry

It has become necessary in seismology to consider more complicated models of the Earth9s structure in order to obtain synthetic seismograms that are more consistent with actual field data. Gassmann (1964) and Postma (1955) have presented results dealing with travel-time methods in anisotropic media—in particular, transversely isotropic media. Kinematic properties alone, however, are not enough to conclusively interpret seismic records. Consequently, dynamic properties must be considered producing a need for synthetic seismograms. One of the most efficient methods for obtaining synthetic seismograms is through the use of asymptotic ray theory (Hron and Kanasewich, 1971; Hron, 1973; Hron, Kanasewich and Alpaslan, 1974). A necessary step in the implementation for layered media displaying transverse isotropy is the computation of reflection and transmission coefficients at the interface between two such layers. Reflection coefficients for a free interface and the corresponding surface conversion coefficients must be computed, as well. Theoretical formulas for reflection, transmission, and surface conversion coefficients corresponding to the zero-order approximation of asymptotic theory are presented for the above-mentioned media.

Reflection and transmission coefficients for transversely isotropic media

The ray series method is used to study propagation of seismic waves in the three-dimensional media consisting of generally inhomogeneous layers separated by curved interfaces. The investigation is carried out with the help of the so-called ray centered corrdinate system which was proposed by Popov and Psencik (1978a). It is shown that in this system the principal components of the amplitude coefficients in the ray series for S waves do not rotate about the ray with respect to the basis vectors when the wave progresses, even though they rotate with respect to the unit vectors n^ and b^ along the direction of the normal and binormal to the ray, respectively. This considerably simplifies the final expressions for the amplitude coefficients for S waves, whose two principal components are decoupled in the ray-centered coordinate system. The ray-centered coordinate system is also applied to the eikonal equation in order to produce a dynamic ray tracing system consisting of three nonlinear ordinary differential equations of the first order determining the second derivatives of the time field and, in this manner, even the basic geometrical properties of the wave fronts (e.g., principal curvatures and geometrical spreading) along the ray. Several different modifications of the dynamic ray tracing system are presented. It is demonstrated that in the case of generally inhomogeneous two-dimensional media the dynamic ray tracing system reduces, under certain not too restrictive conditions, to the single first order differential equation of the Riccati type. Finally, the phase matching method is used to determine discontinuities of individual quantities in the dynamic ray tracing system when the wave is impinging on a curved interface separating two generally inhomogeneous media. Since all basic equations are presented in a computationally convenient matrix formulation, they can be readily employed for any numberical evaluation of dynamic properties of seismic waves propagating through structurally complicated media. As the paper describes all basic features of asymptotic ray theory (the name under which the ray series method is known on this continent), it can serve as a starting point for anyone wishing to develop computer programs for the computation of synthetic seismograms.

The ray series method and dynamic ray tracing system for three-dimensional inhomogeneous media

When the Alekseev-Mikhailenko method was used for the exact numerical solution of Lamb's problem, a new strong nongeometrical arrival, denoted by S *, was detected in the synthetic seismograms computed for a P point source located in the proximity of the free surface. Strong dependence of the amplitude of the S * arrival on the depth of the source is seen in computed seismograms. It is shown that under favorable circumstances, i.e., when the source is less than one wavelength from the free surface, the S * arrival may be stronger than ordinary body waves at the same depth. This may be particularly important for studies of the seismic wave fields in oil exploration, where explosive sources are close to the surface. We show that the S * arrival, which features a linear polarization and propagates with the shear wave velocity, may be interpreted as a result of interaction between inhomogeneous plane waves in the integral representation of the P point source, and the free surface. Mathematically, the S * arrival corresponds to the saddle point contribution of the integral along the branch cut originating at the horizontal slowness p = α/β ( α, β indicate P - and S -phase velocities, respectively). This branch cut must be considered when the saddle point approximation to the Weyl-Sommerfeld integral for a shallow P point source is used. Existence of another nongeometrical effect, namely the nonzero vertical component of the converted PS wave reflected from the free surface at normal incidence, is also shown. In our opinion, this arrival can be easily explained by higher order terms in the corresponding ray series. The prominence of both nongeometrical effects suggests they should be incorporated into any synthetic seismogram computations carried out for shallow explosive sources.

Numerical modeling of nongeometrical effects by the Alekseev-Mikhailenko method

In the application of either ray theory or approximate wave methods, a rational decision must be made as to the phases which contribute significantly to ground motion. In multilayered media, many phases travel by different paths but arrive with the same travel time (kinematic analogs). Some of these have the same amplitude and phase characteristics (dynamic analogs), and the number of phases in each dynamic analog must be determined. Given the number of layers and the maximum number of path elements, equations are given to yield the desired quantities. With the use of amplitude distance curves, it is possible to determine which phases in refracted, reflected, or head waves make significant contributions to a synthetic seismogram.

Criteria for selection of phases in synthetic seismograms for layered media

The linear theory of viscoelasticity is used to study the effects of anelasticity on SH body waves propagating through a layered medium. The mathematical properties of SH waves in a viscoelastic medium are outlined. Reflection and transmission coefficients for SH plane waves impinging upon a plane boundary separating two anelastic media are calculated and compared with the coefficients for the perfectly elastic case. Synthetic seismograms for teleseismic SH body waves are computed for a plane-layered crustal model in both the elastic and anelastic cases, using a ray theory approach.

F. Hron

Papers

Reflection and transmission coefficients for transversely isotropic media

The ray series method and dynamic ray tracing system for three-dimensional inhomogeneous media

Numerical modeling of nongeometrical effects by the Alekseev-Mikhailenko method

Criteria for selection of phases in synthetic seismograms for layered media

Ray-synthetic seismograms for SH waves in anelastic media