Finite‐difference modeling of sonic wavefields with a dipping interface

Geophysics ◽  
1997 ◽  
Vol 62 (4) ◽  
pp. 1270-1277
Author(s):  
Hsui‐lin Liu ◽  
Michael Prange ◽  
Francois Daube
Keyword(s):  
Wave Propagation ◽  
Reflection Coefficient ◽  
Finite Difference ◽  
Amplitude Variation ◽  
Sh Waves ◽  
Complex Wave ◽  
Hard Formation ◽  
Sonic Logs ◽  
Horizontal Interface ◽  
Velocity Bias

Formation bedding can cause complex wave propagation in a borehole and introduce velocity bias in sonic logs. Because of the lack of symmetry, little is known about sonic wavefields propagating through a dipping bed. In this paper, we investigate effects on the sonic dipole and monopole wavefields across a dipping interface using a 3-D finite‐difference method. For dipole wavefields propagating from a soft to a hard formation across a dipping interface, the transmission is reduced greatly when compared with a horizontal interface. The different transmissions of SV‐ and SH‐waves through the dipping interface result in significant azimuthal amplitude variation and generate large cross‐coupled components. This apparent anisotropy should be taken into account when estimating formation shear anisotropy in a dipping formation. For monopole wavefields, the azimuthal averaging caused by a dipping interface reduces the reflection across an interface. This may affect fracture evaluation using the Stoneley reflection coefficient in a dipping formation.

Boundary conditions for the numerical solution of wave propagation problems

Geophysics ◽  
1978 ◽  
Vol 43 (6) ◽  
pp. 1099-1110 ◽  
Author(s):  
Albert C. Reynolds
Keyword(s):  
Wave Propagation ◽  
Reflection Coefficient ◽  
Finite Difference ◽  
Boundary Conditions ◽  
Numerical Solution ◽  
Neumann Boundary ◽  
Synthetic Seismograms ◽  
Neumann Boundary Conditions ◽  
Numerical Calculations ◽  
Reflection Coefficients

Many finite difference models in use for generating synthetic seismograms produce unwanted reflections from the edges of the model due to the use of Dirichlet or Neumann boundary conditions. In this paper we develop boundary conditions which greatly reduce this edge reflection. A reflection coefficient analysis is given which indicates that, for the specified boundary conditions, smaller reflection coefficients than those obtained for Dirichlet or Neumann boundary conditions are obtained. Numerical calculations support this conclusion.

Hybrid modeling of elastic‐wave propagation in two‐dimensional laterally inhomogeneous media

Geophysics ◽  
1987 ◽  
Vol 52 (6) ◽  
pp. 765-771 ◽  
Author(s):  
B. Kummer ◽  
A. Behle ◽  
F. Dorau
Keyword(s):  
Integral Equation ◽  
Wave Propagation ◽  
Finite Difference ◽  
Elastic Wave ◽  
Boundary Integral Equation ◽  
Integral Equation Method ◽  
Boundary Integral ◽  
Two Dimensional ◽  
Hybrid Scheme ◽  
Finite Difference Techniques

We have constructed a hybrid scheme for elastic‐wave propagation in two‐dimensional laterally inhomogeneous media. The algorithm is based on a combination of finite‐difference techniques and the boundary integral equation method. It involves a dedicated application of each of the two methods to specific domains of the model structure; finite‐difference techniques are applied to calculate a set of boundary values (wave field and stress field) in the vicinity of the heterogeneous domain. The continuation of the near‐field response is then calculated by means of the boundary integral equation method. In a numerical example, the hybrid method has been applied to calculate a plane‐wave response for an elastic lens embedded in a homogeneous environment. The example shows that the hybrid scheme enables more efficient modeling, with the same accuracy, than with pure finite‐difference calculations.

Generalization of von Neumann analysis for a model of two discrete half-spaces: The acoustic case

Geophysics ◽  
2007 ◽  
Vol 72 (5) ◽  
pp. SM35-SM46 ◽  
Author(s):  
Matthew M. Haney
Keyword(s):  
Wave Propagation ◽  
Numerical Model ◽  
Finite Difference ◽  
Stability Criterion ◽  
Reflection And Transmission Coefficients ◽  
Reflection And Transmission ◽  
Von Neumann ◽  
Transmission Coefficients ◽  
Strong Material ◽  
Von Neumann Analysis

Evaluating the performance of finite-difference algorithms typically uses a technique known as von Neumann analysis. For a given algorithm, application of the technique yields both a dispersion relation valid for the discrete time-space grid and a mathematical condition for stability. In practice, a major shortcoming of conventional von Neumann analysis is that it can be applied only to an idealized numerical model — that of an infinite, homogeneous whole space. Experience has shown that numerical instabilities often arise in finite-difference simulations of wave propagation at interfaces with strong material contrasts. These interface instabilities occur even though the conventional von Neumann stability criterion may be satisfied at each point of the numerical model. To address this issue, I generalize von Neumann analysis for a model of two half-spaces. I perform the analysis for the case of acoustic wave propagation using a standard staggered-grid finite-difference numerical scheme. By deriving expressions for the discrete reflection and transmission coefficients, I study under what conditions the discrete reflection and transmission coefficients become unbounded. I find that instabilities encountered in numerical modeling near interfaces with strong material contrasts are linked to these cases and develop a modified stability criterion that takes into account the resulting instabilities. I test and verify the stability criterion by executing a finite-difference algorithm under conditions predicted to be stable and unstable.

Reflection of SH waves from anisotropic transition layer

1974 ◽  
Vol 64 (6) ◽  
pp. 1979-1991 ◽  
Author(s):  
V. Thapliyal
Keyword(s):  
Reflection Coefficient ◽  
Incident Wave ◽  
Transition Layer ◽  
Seismic Energy ◽  
Considerable Effect ◽  
Anisotropy Factor ◽  
Sh Wave ◽  
Sh Waves ◽  
Obliquely Incident ◽  
Order Discontinuity

abstract The effects of anisotropy on the reflection of SH-waves (horizontally polarized shear waves) from a transition layer are studied. The transition layer is sand-wiched between two isotropic homogeneous half-spaces and is constituted by a medium which is both anisotropic and inhomogeneous. The SH-wave potentials are obtained for an anisotropic inhomogeneous medium in which both the anisotropy factor (ratio of the horizontal rigidity to the vertical rigidity) and vertical velocity vary with depth. An expression for the reflection coefficient of SH waves is obtained when the material mentioned above forms a finite transition zone between two isotropic homogeneous half-spaces. For further generalization, a second-order discontinuity along with the first-order on eis being assumed in the material properties, at the boundaries of the transition layer. The mathematical and numerical analyses show that the anisotropy factor, found at the top of the transition layer (N0/M0) produces considerable effect on the reflection coefficient for an obliquely incident SH wave. It has been noted that the greater the thickness of the transition layer, the greater is the dependence of the reflection coefficient upon the value of the anisotropy (N0/M0). The minima and maxima of the reflection of seismic energy are found dependent on the value of anisotropy. For greater values of the anisotropy, these maxima and minima shift toward the lower values of the wavelength of the propagating wave (or toward the higher values of the thickness of the transition layer). In fact, the values of the reflection coefficient at which these maxima and minima of seismic energy occur are found greater for the higher values of anisotropy. The effects of anisotropy are found more pronounced for the larger angles of incidence. This remains so until the angle of refraction becomes imaginary. However, no effects of the anistropy factor are found on the reflection coefficients for a normally incident wave. The results, mentioned above, are therefore discussed only for the obliquely incident wave. A geophysically interesting situation has been chosen for studying, quantitatively, the effects of the anisotropy factor on the reflection of SH waves.

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