Edward S. Krebes

On the reflection and transmission of viscoelastic waves- Some numerical results

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 exp[i(ωt-kx)] 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).

On computing ray‐synthetic seismograms for anelastic media using complex rays

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 ...

Inhomogeneous plane waves and cylindrical waves in anisotropic anelastic media

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.

Finite‐difference modeling of SH‐wave propagation in nonwelded contact media

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. Wavefi...

A standard finite-difference scheme for the time-domain computation of anelastic wavefields

We show that the finite‐differencing technique based on the consecutive application of the central difference operator to spatial derivatives, a standard well‐known technique that has been commonly used in the seismological literature for solving the elastic equation of motion, can also be used to obtain a stable time‐domain, finite‐difference scheme for solving the anelastic equation of motion. We compare the results of the scheme for a heterogeneous medium with those of the time‐domain finite‐difference scheme previously developed by Emmerich and Korn and find that they agree very closely. We show, analytically, that in the case of a homogeneous medium, the two schemes give identical numerical results for certain zero initial conditions. The scheme based on the standard technique uses more computer time and memory than the scheme of Emmerich and Korn. However, from a theoretical viewpoint, it is easier to analyze, as it is developed solely with a familiar standard method.

Applying finite element analysis to the memory variable formulation of wave propagation in anelastic media

Finite‐element methods are applied to solution of seismic wave motion in linear viscoelastic media using the memory variable formalism. The displacements are represented as a superposition of a set of basis functions. It is shown that if memory variables are represented using the spatial derivatives of those basis functions, rather than the basis functions themselves, the equations to be solved are simpler and require less computer memory. Using this formulation, results for SH waves in one and two dimensions are calculated using a simple explicit finite‐element‐in‐space/finite‐difference‐in‐time scheme. These results agree with those found with a “method of lines” solution. Results in a homogeneous medium also agree with the frequency domain solutions of Kjartansson’s constant-Q method.

High-precision potential-field and gradient-component transformations and derivative computations using cubic B-splines

Potential-field and gradient-component transformations and derivative computations are necessary for many techniques of data enhancement, direct interpretation, and inversion. We advance new unified formulas for fast interpolation, differentiation, and integration and propose flexible high-precision algorithms to perform 3D and 2D potential-field- and gradient component transformations and derivative computations in the space domain using cubic B-splines. The spline-based algorithms are applicable to uniform or nonuniform rectangular grids for the 3D case and to regular or irregular grids for the 2D case. The fast Fourier transform (FFT) techniques require uniform grid spacing. The spline-based horizontal-derivative computations can be done at any point in the computational domain, whereas the FFT methods use only grid points. Comparisons between spline and FFT techniques through two gravity-gradient examples and one magnetic example show that results computed with the spline technique agree better with t...

Reflection and transmission of ultrasound from a planar interface

Information about the local forces at an interface between two solids is of considerable technological interest. We have used a simple model of a homogeneous solid. We consider a cubic crystal with springs connecting the nearest and the next nearest neighbors. Two such crystals can be joined by a different set of springs at the interface. The force constants of the springs at the interface describe the local forces at the interface. If a longitudinal wave is incident on such an interface at an angle less than the critical angle, it will be partly reflected and transmitted as a longitudinal and transverse wave. We have derived the equations for the transmitted and reflected amplitudes. These can be expressed in terms of the macroscopic properties of solids on either side of the interface, and in terms of the force constants at the interface. It is shown that at low frequencies our equations reduce to those derived from macroscopic theory in which it is assumed that the bond at the interface is perfect and ...

The Homogeneous Finite-Difference Formulation of the P-SV-Wave Equation of Motion

Two different approaches to finite-difference modeling of the elastodynamic equations have been used: the heterogeneous and the homogeneous. In the heterogeneous approach, boundary conditions at interfaces are treated implicitly; in the homogeneous, they are explicitly discretized. We present a homogeneous finite-difference scheme for the 2-D P-SV-wave case. This scheme represents a generalization of earlier such schemes, being able to model media with arbitrary non-uniformities, provided only that all interfaces are aligned with the numerical grid. We perform a detailed comparison of the generalized homogeneous scheme with the analogous heterogeneous scheme, and show the two schemes to be identical for media with a spatially constynt Poissons ratio. For media where Poissons ratio is spatially varying, the schemes differ by terms first-order in the spatial step size. However, a comparison of the numerical results produced by the two schemes shows that the resulting differences are negligible for a wide range of values of the Poissons ratio contrast.

Examination of the relative influence of current gathering on fixed loop and moving source electromagnetic surveys

A physical model study was conducted of the responses provided by moving-source and fixed-loop frequency-domain electromagnetic prospecting systems when operated over the same target conductor located in a conductive host environment. The results indicate that the fixed-loop responses display enhancement due to the current gathering effect that exceeds that seen in the moving-source responses by at least an order of magnitude for all the source-to-receiver separations tested for the moving-source system. The results also indicate that as transmitter frequency is increased the current gathering effect displays an abrupt onset in the responses provided by both systems, but that this onset begins at a frequency which is a decade lower for the fixed-loop system than the corresponding frequency of onset for the moving-source system. The current gathering enhancement effects show a clear reduction with increase of target depth for the fixed-loop system but an increase with depth for the moving-source system. The model parameters employed in these studies are shown to be well related to typical conditions found in full-scale surveys.