Multigrid methods and the surface consistent equations of Geophysics

John Millar

The surface consistent equations are a large linear system that is frequently used in signal enhancement for land seismic surveys. Different signatures may be consistent with a particular dynamite (or other) source. Each receiver and the conditions around the receiver will have different impact on the signal.

Seismic deconvolution operators, amplitude corrections and static shifts of traces are calculated using the surface consistent equations, both in commercial and scientific seismic processing software.

The system of equations is singular, making direct methods such as Gaussian elimination impossible to implement. Iterative methods such as Gauss-Seidel and conjugate gradient are frequently used. A limitation in the nature of the methods leave the long wavelengths of the solution poorly resolved.

To reduce the limitations of traditional iterative methods, we employ a multigrid method. Multigrid methods re-sample the entire system of equations on a more coarse grid. An iterative method is employed on the coarse grid. The long wavelengths of the solutions that traditional iterative methods were unable to resolve are calculated on the reduced system of equations. The coarse estimate can be interpolated back up to the original sample rate, and refined using a standard iterative procedure.

Multigrid methods provide more accurate solutions to the surface consistent equations, with the largest improvement concentrated in the long wavelengths. Synthetic models and tests on field data show that multigrid solutions to the system of equations can significantly increase the resolution of the seismic data, when used to correct both static time shifts and in calculating deconvolution operators.

The first chapter of this thesis is a description of the physical model we are addressing.

It reviews some of the literature concerning the surface consistent equations, and provides background on the nature of the problem.

Chapter 2 contains a review of iterative and multigrid methods. A finite difference approximation to Laplaces equation is solved to outline some of the methods used later in the thesis. This includes defining what is meant by long wavelength errors, which is a principal concept in this study.

Chapter 3 begins with a standard definition of the surface consistent equations. The ability of each method to estimate a solution to a synthetic surface consistent system is tested. The error remaining due to limitations in the solving method is shown to reduce the signal quality of a seismic gather. How the errors manifest in the data is demonstrated using static, amplitude and deconvolution corrections. The results from a multigrid method are compared to those of a Gauss-Seidel and bi-conjugate gradient method.

Finally, a surface consistent deconvolution of the CREWES Blackfoot data set is presented. The multigrid solutions to the equations are shown to provide deconvolution operators that improve the resolution of the seismic trace over Gauss-Seidel and conjugate gradient methods.