Finite Difference Modelling of Seismic Wave Phenomena within the Earth’s Upper Mantle

2001 ◽  
pp. 48-56 ◽  
Author(s):  
Trond Ryberg ◽  
Marc Tittgemeyer ◽  
Friedemann Wenzel
Keyword(s):  
Finite Difference ◽  
Seismic Wave ◽  
Upper Mantle ◽  
Wave Phenomena

Finite‐difference modeling of faults and fractures

Geophysics ◽  
1995 ◽  
Vol 60 (5) ◽  
pp. 1514-1526 ◽  
Author(s):  
Richard T. Coates ◽  
Michael Schoenberg
Keyword(s):  
Finite Difference ◽  
Seismic Wave ◽  
Computation Time ◽  
Anisotropic Materials ◽  
Model Material ◽  
Material Behavior ◽  
Interface Condition ◽  
Medium Theory ◽  
Difference Grid ◽  
Receiver Array

For the purposes of seismic propagation, a slip fault may be regarded as a surface across which the displacement caused by a seismic wave is discontinuous while the stress traction remains continuous. The simplest assumption is that this slip and the stress traction are linearly related. Such a linear slip interface condition is easily modeled when the fault is parallel to the finite‐difference grid, but is more difficult to do for arbitrary nonplanar fault surfaces. To handle such situations we introduce equivalent medium theory to model material behavior in the cells of the finite‐ difference grid intersected by the fault. Virtually identical results were obtained from modeling the fault by (1) an explicit slip interface condition (fault parallel to the grid) and (2) using the equivalent medium theory when the finite‐difference grid was rotated relative to the fault and receiver array. No additional computation time is needed except for the preprocessing required to find the relevant cells and their associated moduli. The formulation is sufficiently general to include faults in and between arbitrary anisotropic materials with slip properties that vary as a function of position.

Mimetic seismic wave modeling including topography on deformed staggered grids

Geophysics ◽  
2014 ◽  
Vol 79 (3) ◽  
pp. T125-T141 ◽  
Author(s):  
Josep de la Puente ◽  
Miguel Ferrer ◽  
Mauricio Hanzich ◽  
José E. Castillo ◽  
José M. Cela
Keyword(s):  
Finite Difference ◽  
Seismic Wave ◽  
Seismic Waves ◽  
Numerical Solutions ◽  
Surface Condition ◽  
Finite Difference Methods ◽  
Staggered Grid ◽  
Grid Deformation ◽  
Large Dispersion ◽  
Difference Methods

Finite-difference methods for modeling seismic waves are known to be inaccurate when including a realistic topography, due to the large dispersion errors that appear in the modelled surface waves and the scattering introduced by the staircase approximation to the topography. As a consequence, alternatives to finite-difference methods have been proposed to circumvent these issues. We present a new numerical scheme for 3D elastic wave propagation in the presence of strong topography. This finite-difference scheme is based upon a staggered grid of the Lebedev type, or fully staggered grid (FSG). It uses a grid deformation strategy to make a regular Cartesian grid conform to a topographic surface. In addition, the scheme uses a mimetic approach to accurately solve the free-surface condition and hence allows for a less restrictive grid spacing criterion in the computations. The scheme can use high-order operators for the spatial derivatives and obtain low-dispersion results with as few as six points per minimum wavelength. A series of tests in 2D and 3D scenarios, in which our results are compared to analytical and numerical solutions obtained with other numerical approaches, validate the accuracy of our scheme. The resulting FSG mimetic scheme allows for accurate and efficient seismic wave modelling in the presence of very rough topographies with the advantage of using a structured staggered grid.

scholarly journals Unsplit complex frequency-shifted PML implementation using auxiliary differential equations for seismic wave modeling

Geophysics ◽  
2010 ◽  
Vol 75 (4) ◽  
pp. T141-T154 ◽  
Author(s):  
Wei Zhang ◽  
Yang Shen
Keyword(s):  
Free Surface ◽  
Finite Difference ◽  
Differential Equations ◽  
Seismic Wave ◽  
Numerical Scheme ◽  
Complex Frequency ◽  
Order Accuracy ◽  
First Order ◽  
Time Marching ◽  
Interior Domain

The complex-frequency-shifted perfectly matched layer (CFS-PML) technique can efficiently absorb near-grazing incident waves. In seismic wave modeling, CFS-PML has been implemented by the first-order-accuracy convolutional PML technique or second-order-accuracy recursive convolution PML technique. Both use different algorithms than the numerical scheme for the interior domain to update auxiliary memory variables in the PML and thus cannot be used directly with higher-order time-marching schemes. We work with an unsplit-field CFS-PML implementation using auxiliary differential equations (ADEs) to update the auxiliary memory variables. This ADE CFS-PML results in complete first-order differential equations. Thus, the numerical scheme for the interior domain can be used to solve ADE CFS-PML equations. We have implemented ADE CFS-PML in the finite-difference time-domain method and in anonstaggered-grid finite-difference method with the fourth-order Runge-Kutta scheme, demonstrating its straightforward implementation in different numerical time-marching schemes. We have also theoretically analyzed the role of the scalingfactor of CFS-PML; it transforms the PML to a transversely isotropic material, reducing the effective wave speed normal to the PML layer and bending the wavefront toward the normal direction of the PML layer. Our numerical tests indicate that the optimal value reduces the points per dominant wavelength at the outermost boundary to three, about half the value required by the numerical scheme. We also have found that the PML equations should be derived taking the free-surface boundary condition into account in finite-difference methods. Otherwise, the free surface in the PML layer causes instability or ineffective absorption of surface waves. Tests show that we can use a narrow-slice mesh with ADE CFS-PML to simulate full wave propagation efficiently in models with complex structure.

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