Ekkehart Tessmer

3-D elastic modeling with surface topography by a Chebychev spectral method

The 3-D numerical Chebychev modeling scheme accounts for surface topography. The method is based on spectral derivative operators. Spatial differencing in horizontal directions is performed by the Fourier method, whereas vertical derivatives are carried out by a Chebychev method that allows for the incorporation of boundary conditions into the numerical scheme. The method is based on the velocity‐stress formulation. The implementation of surface topography is done by mapping a rectangular grid onto a curved grid. Boundary conditions are applied by means of characteristic variables. The study of surface effects of seismic wave propagation in the presence of surface topography is important, since nonray effects such as diffractions and scattering at rough surfaces must be considered. Several examples show this. The 3-D modeling alogrithm can serve as a tool for understanding these phenomena since it computes the full wavefield.

Solution of the equations of dynamic elasticity by a Chebychev spectral method

We present a spectral method for solving the two‐dimensional equations of dynamic elasticity, based on a Chebychev expansion in the vertical direction and a Fourier expansion for the horizontal direction. The technique can handle the free‐surface boundary condition more rigorously than the ordinary Fourier method. The algorithm is tested against problems with known analytic solutions, including Lamb’s problem of wave propagation in a uniform elastic half‐space, reflection from a solid‐solid interface, and surface wave propagation in a haft‐space containing a low‐velocity layer. Agreement between the solutions is very good. A fourth example of wave propagation in a laterally heterogeneous structure is also presented. Results indicate that the method is very accurate and only about a factor of two slower than the Fourier method.

3-D Elastic Modeling With Surface Topography By a Chebychev Spectral Method

The 3-D numerical Chebychev modeling scheme accounts for surface topography. The method is based on spectral derivative operators. Spatial differencing in horizontal directions is performed by the Fourier method, whereas vertical derivatives are camed out by a Chebychev method that allows for the incorporation of boundary conditions into the numerical scheme. The method is based on the velocity-stress formulation. The implementation of surface topography is done by mapping a rectangular grid onto a curved grid. Boundary conditions are applied by means of characteristic variables. The study of surface effects of seismic wave propagation in the presence of surface topography is important, since nonray effects such as diffractions and scattering at rough surfaces must be considered. Several examples show this. The 3-D modeling alogrithm can serve as a tool for understanding these phenomena since it computes the full wavefield.

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.

Using the rapid expansion method for accurate time-stepping in modeling and reverse-time migration

Reverse-time migration is based on seismic forward modeling algorithms, where spatial derivatives usually are calculated by finite differences or by the Fourier method. Time integration in general is done by finite-difference time stepping of low orders. If the spatial derivatives are calculated by high-order methods and time stepping is based on low-order methods, there is an imbalance that might require that the time-step size needs to be very small to avoid numerical dispersion. As a result, computing times increase. Using the rapid expansion method (REM) avoids numerical dispersion if the number of expansion terms is chosen properly. Comparisons with analytical solutions show that the REM is preferable, especially at larger propagation times. For reverse-time migration, the REM needs to be applied in a time-stepping manner. This is necessary because the original implementation based on very large time spans requires that the source term is separable in space and time. This is not appropriate for reverse-time migration where the sources have different time histories. In reverse-time migration, it might be desirable to use the Poynting vector information to estimate opening angles to improve the quality of the image. In the solution of the wave equation, this requires that one calculates not only the pressure wavefield but also its time derivative. The rapid expansion method can be extended easily to provide this time derivative with negligible extra cost.

Tube wave modeling by the finite-difference method with varying grid spacing

A finite-difference approach of a P-SV modeling scheme is applied to compute seismic wave propagation in heterogeneous isotropic media, including fluid-filled boreholes. The discrete formulation of the equation of motion requires the definition of the material parameters at the grid points of the numerical mesh. The grid spacing is chosen as coarse as possible with respect to the accurate representation of the shortest wavelength. If we assume frequencies lower than 250 Hz then the grid spacing is usually chosen in the range of a few meters. One encounters difficulties because of the large-scale difference between the grid spacing and the size of the borehole, usually several centimeters.

Localization of Seismic Events By Diffraction Stacking

The localization of seismic events is of great importance for hydro frac and reservoir monitoring. For deposits with weak 4-D signatures the passive seismic method may provide an alternative option for reservoir characterization. We introduce a new localization technique which does not require any picking of events in the individual seismograms of the recording network. The localization is performed by a modified diffraction stack of the squared amplitudes of the input seismograms resulting in the image section. The method is target oriented and is best suited for large networks of surface and/or downhole receivers. The source location is obtained from the maximum of the image section for the time window under consideration. Since the focusing analysis is performed only in this section, no optimized search procedures are required. The source time is determined in a second processing step after the source location. Initial tests with 2-D homogeneous media indicate the high potential of the method. Since the maximum of the image section is distinct even very weak events can be detected.

Reverse-time migration in TTI media

There is increasing need to perform migration with anisotropic velocity models. Reverse-time migration clearly is more computer resources demanding compared to other migration methods. Therefore, to accelerate computations, the anisotropic elasto-dynamic equations of motions are often simplified to account for qP-wave propagation only in, e.g., TTI media. This reduces the computational effort considerably. However, these so-called pseudo-acoustic wave equations show numerical instabilities if the orientations of the symmetry axes inside the medium do not coincide. Such modeling schemes are not useful for situations where structural changes of the anisotropy occur. However, using a true anisotropic modeling scheme allows reverse-time migration without these restrictions. qPwaves only propagation as with pseudo-acoustic wave equations is possible as well. Synthetic examples of reverse-time migration calculated using the 2D Fourier-method demonstrate the method’s ability to cope very well with arbitrary orientations of symmetry axes.

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.

3-D Elastic Modeling by a Chebychev Spectral Method

The implementation of the free surface boundary condition into a composite spectral Chebyshev-Fourier method for solving the three dimensional equations of motion is presented. In this method spatial derivatives with respect to the vertical direction are calculated with a Chebyshev spectral method, while differencing with respect to the horizontal directions is performed with the Fourier method. The technique can handle free surface boundary conditions in a more rigorous manner than the ordinary Fourier method. It therefore appears wellsuited for realistic full wave modeling, in particular of near surface layer problems. Isotropic media and transversely isotropic media with a vertical axis of symmetry are considered. A comparison of the isotropic modeling results with the analytic solution for Lamb’s problem shows the high accuracy of the alogorithm.