Stig Hestholm

3-D finite-difference elastic wave modeling including surface topography

Three‐dimensional finite‐difference (FD) modeling of seismic scattering from free surface topography has been pursued. We have developed exact 3-D free surface topography boundary conditions for the particle velocities. A velocity‐stress formulation of the full elastic wave equations together with the boundary conditions has been numerically modeled by an eighth‐order FD method on a staggered grid. We give a numerical stability criterion for combining the boundary conditions with curved‐grid wave equations, where a curved grid represents the physical medium with topography. Implementation of this stability criterion stops instabilities from arising in areas of steep and rough topographies. We have simulated scattering from teleseismic P-waves using a plane, vertically incident wavefront and real topography from a 40 × 40 km area centered at the NORESS array of seismic receiver stations in southeastern Norway. Synthetic snapshots and seismograms of the wavefield show clear conversion from P-waves to Rg (sh...

Elastic wave modeling with free surfaces: Stability of long simulations

Three‐dimensional (3D) elastic wave propagation modeling in the velocity‐stress formulation using finite differences (FDs) have been investigated for a homogeneous medium covered by a representative, relatively steep surface topography consisting of a 1D square root function. This scenario using various numerical implementations is explored. The behavior with regard to stability of long simulations is expected to be indicative of each numerical implementations robustness for other types of topographies/media. Employing various combinations of the FD order was found only to change the time of the first incidence of instability. On the other hand, nonequidistant grids in the horizontal and vertical directions are found to be extremely useful for long‐term stability of 3D wave propagation modeling with our free‐surface boundary condition for single‐valued topographies. In particular dz ≥ (3/2) dx is found completely stable for all tested vP/vS ratios. Such relationships of using dz > dx are also favorable f...

Effects of free-surface topography on moving-seismic-source modeling

A curved-grid velocity-stress formulation for viscoelastic wave modeling is used with an arbitrary number of relaxation mechanisms to model a desired Q -behavior. These equations are discretized by high-order staggered finite differences (FDs) in the interior of the medium, and we gradually reduce the FD order to two at the stress-free surface, where we implement our boundary conditions for an arbitrary topographic surface. A moving source is simulated along the surface of a relatively general and locally steep surface topography and, for comparison, along a plane surface. The topography consists of a significant hill surrounded by a valley. Similar two-layered geologic models are used with both topographic surfaces, with the upper layer being a lossy sedimentary layer having a relatively strong contrast with the lower, higher-velocity half-space. Local topographic highs create varying amplitude amplifications at different times during motion of the source. A pronounced wavefield accumulation is evident a...

Quick and accurate Q parameterization in viscoelastic wave modeling

We introduce a procedure for including the attenuation factor Q in a consistent manner in seismic modeling and show 3D examples. The Q fitting over a chosen frequency band involves two algorithms: The first creates starting values of relaxation times, and the second does nonlinear inversion using the results of the first as initial values. The resulting Q function gives a good approximation to a constant Q over the chosen frequency band. The algorithm is combined with a finite-difference (F-D) code that includes topographies in 3D seismic media. The velocity-stress formulation for viscoelastic wave modeling is used with an arbitrary number of relaxation mechanisms to model a desired Q behavior. These equations are discretized by high-order F-Ds in the interior of the medium, and we gradually reduce the F-D order to two at the stress-free surface, where we implement our free-surface boundary conditions. The seismic F-D algorithm is applied to a marine seismic experiment, with and without viscoelasticity, to emphasize the importance of including physical attenuation and dispersion in seismic modeling. Their inclusion, even for marine surveys, is clearly important for lossy ocean bottoms. Our procedure for more accurate modeling of physical dispersion and attenuation may increase future motivation to include viscoelasticity in seismic inversion.

Instabilities in applying absorbing boundary conditions to high-order seismic modeling algorithms

The problem of artificial reflections from grid boundaries in the numerical discretization of elastic and acoustic wave equations has long plagued geophysicists. Even if modern computers have made it possible to extend the synthetics over more wavelengths (equivalent to larger propagation distances), efficient absorption methods are still needed to minimize interference from unwanted reflections from the numerical grid boundaries. In this study, we examine applicabilities and stabilities of the optimal absorbing boundary condition (OABC) of Peng and Toksoz (1994, 1995) for 2-D and 3-D acoustic and elastic wave modeling. As a basis for comparison, we use exponential damping (ED) (Cerjan et. al., 1985), in which velocities and stresses are multiplied by progressively decreasing terms when approaching the boundaries of the numerical grid.

3-D versus 2-D finite-difference seismic synthetics including real surface topography

Abstract We have pursued and compared two-dimensional (2-D) and three-dimensional (3-D) finite-difference (F-D) modeling of scattering from free surface topography. A velocity–stress formulation of the full elastic wave equations are combined with exact boundary conditions for the surface topography and numerically discretized by an eighth-order F-D method on a staggered grid. We have simulated scattering in 2-D and 3-D from teleseismic P-waves using a plane, vertically incident P-wave and real topography from a 60×60 km2 area including the NORESS array in southeastern Norway. Many field observations that are not easily explained by simpler 2-D cases are shown to better match qualitative effects from 3-D surface topography modeling. These include strong amplifications at hills, complex wave pattern caused by scattering, and directivity of scattered waves. Snapshots and seismograms show clear conversion from P- to Rg- (short period fundamental mode Rayleigh) waves in an area of rough topography in the vicinity of the array site. All results are consistent with numerous observations. By parallellization of the original software, possibilities have been opened for modeling with higher resolution and/or larger areas than before.

Modeling seismic waves in orthorombic, viscoelastic media by finite‐differences

Realistic modeling of seismic wave propagation in heavily fractured rocks enforces us to account for anistropy and attenuation. For finite-difference (F-D) methods, this may be achieved by using relaxation functions in an anisotropic formulation of the stress-strain relations. When the relaxation functions are given by a set of standard linear solids (SLS), an efficient implementation is possible based on the introduction of so-called memory variables. The equations neccessary for a F-D scheme in an orthorombic, viscoelastic medium are given and results for an explosion source in a simple two layer model are shown and discussed.