A plane wave propagating in a viscoelastic medium is generally inhomogeneous, meaning that the direction in which the spatial rate of amplitude attenuation is maximum is generally different from the direction of travel. The angle between these two directions, which we call the “attenuation angle,” is an acute angle. In order to trace the ray corresponding to a plane wave propagating between a source point and a receiver point in a layered viscoelastic medium, one must know both the initial propagation angle (the angle that the raypath makes with the vertical) and the initial attenuation angle at the source point. In some recent literature on the computation of ray‐synthetic seismograms in anelastic media, values for the initial attenuation angle are chosen arbitrarily; but this approach is fundamentally unsatisfactory, since different choices lead to different results for the computed waveforms. Another approach, which is more deterministic and physically acceptable, is to deduce the value of the initial attenuation angle from the value of the complex ray parameter at the saddle point of the complex traveltime function. This value can be obtained by applying the method of steepest descent to evaluate approximately the integrals giving the exact wave field at the observation point. This well‐known technique results in the ray‐theory limit. The initial propagation angle can also be determined from the saddle point. Among all possible primary rays between source and receiver, each having different initial propagation and attenuation angles, the ray determined by the saddle point, which we call a “stationary ray,” has the smallest traveltime, a result which is consistent with Fermat’s principle of least time. Such stationary rays are complex rays, i.e., the spatial (e.g., Cartesian) coordinates of points on stationary raypaths are complex numbers, whereas the arbitrarily determined rays mentioned above are usually traced as real rays. We compare examples of synthetic seismograms computed with stationary rays with those from some arbitrarily determined rays. If the initial value of the attenuation angle is arbitrarily chosen to be a constant for all initial propagation angles, the differences between the two types of seismograms are generally small or negligible in the subcritical zone, except when the constant is relatively large in value, say, within 10 degrees or so of its upper bound of 90 degrees. In that case, the differences are significant but still not large. However, if the surface layer is highly absorptive, the differences can be quite large and pronounced. For larger offsets, i.e., in the supercritical zone, large phase discrepancies can exist between the waveforms for the stationary rays and those for the arbitrarily determined rays, even if the constant initial attenuation angle is not large and even for moderate absorptivity in the surface layer.

In isotropic anelastic media, the phase velocity of an inhomogeneous plane body wave, which is a function of Q and the degree of inhomogeneity γ, is significantly less than the corresponding homogeneous wave phase velocity typically only if γ is very large (unless Q is unusually low). Here we investigate inhomogeneous waves in anisotropic anelastic media, where phase velocities are also functions of the direction of phase propagation θ and find that (1) the low phase velocities can occur at values of γ which are substantially less than the isotropic values and that they occur over a limited range of oblique directions θ, and (2) for large positive values of γ, there are ranges of oblique directions θ in which the inhomogeneous waves cannot propagate at all because there is no physically acceptable solution to the dispersion relation. We show examples of how the waves of case 1 can occur in practice and cause a number of anomalous wave propagation effects. The waves of case 2, though, do not arise in practice (they do not correspond to any points on the horizontal slowness plane). We also show that in the decomposition of a cylindrical wave into plane waves, inhomogeneous plane waves occur whose amplitudes grow in the direction of phase propagation and that this direction is away from the receiver to which they are contributing. The energy in these waves does, however, travel toward the receiver, and their amplitudes decay in the direction of energy propagation. We also show that if the commonly used definition for the quality factor in an isotropic medium, Q = −Re(μ)/Im(μ) where μ is a complex modulus, is applied to an anisotropic anelastic medium in order to study absorption anisotropy, a generally unreliable measure of the anelasticity of inhomogeneous wave propagation in a given arbitrary direction is obtained. The more fundamental definition based on energy loss (i.e., 2π/Q = ΔE/E) should be used in general, and we present some basic formulas for this quantity, as well as others, for plane waves in transversely isotropic anelastic media.

The mathematical theory which is typically used to model the intrinsic anelasticity of the earth is the linear theory of viscoelasticity. The effects of anelasticity on wave propagation, such as absorption and dispersion, are often described using one‐dimensional (1-D) plane waves of the form [Formula: see text] with k complex and frequency‐dependent. These waves are solutions of the 1-D viscoelastic wave equation. The reflection and transmission of plane waves in a layered viscoelastic medium is, however, a 2-D or 3-D problem. The solutions to the 2-D or 3-D viscoelastic wave equation are the so‐called general plane waves, which are classified as homogeneous or inhomogeneous depending upon whether or not the planes of constant phase, i.e., wavefronts, coincide with the planes of constant amplitude (the 1-D plane waves mentioned above are strictly homogeneous).

Abstract. We present new and exact mathematical formulas for the P-SV particle displacement reflection and transmission coefficients for elastic plane waves incident upon a nonwelded contact (or linear slip) interface separating two solid half-spaces. We represent the nonwelded contact at the interface with displacement discontinuity boundary conditions, that is, the traction is continuous across the interface, but the particle displacement is not: the discontinuity in the displacement is proportional to the traction. The formulas are derived by algebraically solving these boundary conditions. The formulas can be expressed in the form of those for the welded contact case, with some terms being modified due to nonwelded contact, plus additional terms due purely to nonwelded contact. Such formulas are useful because they provide insight into the nature of the coe•cients and can be used to derive approximate formulas which have applications in the inversion and interpretation of seismic data. The exact formulas can also be applied to the case of a viscous nonwelded interface by simply replacing one of their parameters (the tangential specific compliance) with a modified parameter which includes the specific viscosity. We also apply the exact formulas to the specific case in which the incidence and transmission media are identical (e.g., a joint, fault, or fracture in a single homogeneous medium). In this case, reflected elastic waves are produced, unlike the case of welded contact, in which no reflections are produced because of the lack of an impedance contrast. This effect may partially explain occasional reports in the literature of anomalously large seismic amplitudes observed in areas where impedance contrasts are small. We present some numerical examples illustrating the effects of nonwelded contact on reflection and transmission coefficients. We show that energy is conserved at all incidence angles at a nonwelded interface between two different elastic solids. We also present some examples of applications of the exact formulas: (1) we derive approximate formulas for a weakly nonwelded interface between two identical solids, (2) we use them to study the sensitivity of the coefficients to the normal and parallel specific compliances (measures of the amount of nonweldedness), finding that they are generally more sensitive to the normal than the parallel compliance (except for the SS reflection), that the sensitivities vary considerably with incidence angle, and that the PP and SS transmission coefficients are basically insensitive to the compliances for a weakly nonwelded interface, and (3) we further approximate the formulas in example 1 to the case of small propagation angles, resulting in formulas that could be useful in amplitude versus offset studies.

Many geological structures of interest are known to exhibit fracturing. Fracturing directly affects seismic wave propagation because, depending on its scale, fracturing may give rise to scattering and/or anisotropy. A fracture may be described mathematically as an interface in nonwelded contact (i.e., as a displacement discontinuity). This poses a difficulty for finite‐difference modeling of seismic wave propagation in fractured media, because the standard heterogeneous approach assumes welded contact. In the past, this difficulty has been circumvented by incorporating nonwelded contact into the medium parameters using equivalent medium theory. We present an alternate method based on the homogeneous approach to finite differencing, whereby nonwelded contact boundary conditions are imposed explicitly. For simplicity, we develop the method in the SH‐wave case. In the homogeneous approach, nonwelded contact boundary conditions are discretized by introducing auxiliary, so‐called fictitious, grid points. Wavefield values at fictitious grid points are then used in the discrete equation of motion, so that the time‐evolved wavefield satisfies the correct boundary conditions. Although not as general as the heterogeneous approach, the homogeneous approach has the advantage of being relatively simple and manifestly satisfying nonwelded contact boundary conditions. For fractures aligned with the numerical grid, the homogeneous and heterogeneous approaches yield identical results. In particular, in both approaches nonwelded contact results in a larger maximum stable time step size than in the welded contact case.