Alfred Behle

A spectral scheme for wave propagation simulation in 3-D elastic‐anisotropic media

This work presents a new scheme for wave propagation simulation in three‐dimensional (3-D) elastic-anisotropic media. The algorithm is based on the rapid expansion method (REM) as a time integration algorithm, and the Fourier pseudospectral method for computation of the spatial derivatives. The REM expands the evolution operator of the second‐order wave equation in terms of Chebychev polynomials, constituting an optimal series expansion with exponential convergence. The modeling allows arbitrary elastic coefficients and density in lateral and vertical directions. Numerical methods which are based on finite‐difference techniques (in time and space) are not efficient when applied to realistic 3-D models, since they require considerable computer memory and time to obtain accurate results. On the other hand, the Fourier method permits a significant reduction of the working space, and the REM algorithm gives machine accuracy with half the computational effort as the usual second-order temporal differencing sch...

Multi-domain Chebyshev-Fourier method for the solution of the equations of motion of dynamic elasticity

Abstract A multi-domain approach for the solution of the equations of elasticity in two spatial dimensions is presented. The equations of momentum conservation and the stress-strain relations are recast as a system of five coupled equations in time in which the particle velocities and the stresses are the unknowns. Solution schemes for both 2D Cartesian and polar coordinates are derived. In both cases the solution is assumed periodic in one coordinate (the x or θ directions) and non-periodic in the other direction. The numerical algorithm uses a Fourier expansion in the periodic direction and domain decomposition and a modified Chebyshev expansion in the remaining direction. The multi-domain approach is tested against problems with known solutions. In all cases it appears as accurate as solutions with a single domain. The multi-domain concept adds flexibility and improves efficiency. It allows use of different grid sizes in different regions depending on the material properties and allows a relatively uniform grid spacing in the polar coordinate case.

Fast modeling of seismic waves in laterally inhomogeneous media

A fast (coarse grid) method is introduced to calculate the seismic wave field in laterally inhomogeneous media. The new method is based on an incorporation of the inverse dis persion relation into the equation of motion for a homogeneous medium. An inverse Fourier transform yields then a modified equation of motion in which grid dispersion is efficiently suppressed. At interfaces the solutions of the associated equations of motions are subjected to the standard boundary conditions. The theory is developed for the 2JJ acoustic case. Two numerical examples are presented: The first one is a comparative calculation and points out that the new method yields sufficient accurate results. The second one deals with a large scale hydrocarbon cap model. This example shows that an incorporation of a Znd-order inverse dispersion relation together with a Znd-order differencing of the modified wave equation enables a forward modeling in a grid which dimensions are comparable to the ones associated with the Fourier method.

EXPLORATION ORIENTED SEISMIC MODELING AND INVERSION

Exploration oriented seismic modeling requires accurate and efficient methods. A variety of direct modeling methods has been extended and improved within the framework of the EOS-1 project. The aim is to calculate seismograms of the full wavefield in complex subsurface models which are related to exploration targets. General Finite-Element Method (FEM) programs for 2D and 3D wave propagation simulation have been implemented. A hybrid method using spectral elements has been developed for the 2D acoustic and elastic case and for the 3D acoustic case. Curved elements have been successfully introduced into the 2D elastic scheme for the description of irregular interfaces. The method shows a fast convergence rate and the high accuracy typical of spectral methods. In the classical FEM as well as in the spectral-element method (SPEM) a domain decomposition method based on a sub-structuring concept has successfully been implemented. The efficiency is superior to that of the previous spectral element code, both in computer memory usage and computer time. In higher order finite-difference modeling (FDM) emphasis is put on schemes with varying grid spacing. This allows to represent areas in which a high spatial resolution is required or the wave propagation velocities are low by a fine computational mesh without the need to extend this fine mesh to other regions. This results in a reduction of the computational effort and memory requirement. 2D and 3D Fourier Spectral Modeling (FSM) schemes for acoustic and for elastic media have been developed for regular and staggered grid techniques. A combination of a regular grid in the vertical direction and a staggered grid in the horizontal direction improves the results obtained for an elastic half-space with a free surface. A 2D acoustic Fourier modeling scheme in generalized curvilinear coordinates has been developed. With this scheme, there are significantly fewer spurious diffractions than with ordinary Cartesian-coordinate modeling. Stable Chebyshev Spectral Modeling (CSM) schemes have been developed for 2D and 3D elastic media. Boundary conditions can be easily implemented in the CSM. This allows the high-accuracy simulation of the seismic response of composite fluid/solid media as well as of a free surface, which is not possible for FSM. Both, 2D and 3D codes have been extended to general anisotropy and interesting case studies have been performed for anisotropic media with a free surface. General anisotropy cannot be modeled by finite-difference methods. Therefore the Chebyshev Spectral Method is superior to FDM in this respect. The elastic schemes allow to take surface topography into account, something which is also not possible in FD-schemes. 3D acoustic and elastic modeling schemes in cylindrical coordinates have been developed for borehole modeling. The computational domain is divided into cylindrical subdomains in order to improve the stability conditions and to compensate for increasing angular grid spacing with increasing radius. A general and consistent constitutive equation for anisotropic viscoelastic media has been studied. The direction-dependent quality factors obtained in this way have been compared to those measured in seismograms obtained by numerical modeling. A modeling code has been developed for 3D viscoelastic media. A time integration algorithm for viscoelastic media modeling based on the approximation of the evolution operator by polynomial interpolation has been developed and tested. The new approach is two times faster than second-order differencing in time. The seismic forward modeling methods developed in the EOS-1 project were successfully applied to simulate seismic wave propagation in models of the subsurface in actual exploration areas. Various FORTRAN packages for analytic reference solutions have been developed, originally for the use in several project groups. They are a valuable tool for the evaluation of numerical methods. In order to solve the inverse problem, an inversion method based on integral equations has been developed and applied to synthetic data. Further, a new algorithm for the stochastic improvement of the tomographic inversion has been developed and tested on synthetic data. Stochastic conditions are taken into account to guide the inversion process. The algorithm was extended to the tomographic reconstruction of the 3D velocity fields from a set of cross-well views.

Iterative integral equation inversion

2. The basic integral equation Inversion of reflection seismograms aims at a reconstruction of the velocity-depth function. By interpreting the depth dependent propagation speed as a perturbation of a reference velocity a nonlinear Fredholm integral equation of the first kind for the perturbation potential can be derived. The most idealized version of the seismic inverse problem is a situation where an acoustic plane wave normally incident upon a medium whose properties vary only with depth. We also assume the density to be constant.

Three‐dimensional wave propagation simulation in elastic — Anisotropic media

Numerical methods which are based on finite-difference techniques (in time and space) are not efficient when applied to realistic 3-D models, since they rquire considerable computer memory and time to obtain accurate results. On the other hand, the Fourier method permits a significant reduction of the working space, and the REM algorithm gives machine accuracy with the same computational effort as the usual second-order temporal differencing scheme.