Richard L. Gibson

AN ENERGY-CONSERVING DISCONTINUOUS MULTISCALE FINITE ELEMENT METHOD FOR THE WAVE EQUATION IN HETEROGENEOUS MEDIA

Seismic data are routinely used to infer in situ properties of earth materials on many scales, ranging from global studies to investigations of surficial geological formations. While inversion and imaging algorithms utilizing these data have improved steadily, there are remaining challenges that make detailed measurements of the properties of some geologic materials very difficult. For example, the determination of the concentration and orientation of fracture systems is prohibitively expensive to simulate on the fine grid and, thus, some type of coarse-grid simulations are needed. In this paper, we describe a new multiscale finite element algorithm for simulating seismic wave propagation in heterogeneous media. This method solves the wave equation on a coarse grid using multiscale basis functions and a global coupling mechanism to relate information between fine and coarse grids. Using a mixed formulation of the wave equation and staggered discontinuous basis functions, the proposed multiscale methods have the following properties. • The total wave energy is conserved. • Mass matrix is diagonal on a coarse grid and explicit energy-preserving time discretization does not require solving a linear system at each time step. • Multiscale basis functions can accurately capture the subgrid variations of the solution and the time stepping is performed on a coarse grid. We discuss various subgrid capturing mechanisms and present some preliminary numerical results.

Modeling and velocity analysis with a wavefront-construction algorithm for anisotropic media

Wavefront construction methods allow for efficient simulation of seismic wave propagation in anisotropic media using ray theory. This approach models an individual wavefront as a mesh that is adaptively refined as the wave travels through an earth model. The algorithm, implemented here for quasi-compressional waves, provides an effective tool for modeling and analyzing a large vertical seismic-profile data set from the Gulf of Mexico. Seismic velocities in this region generally increase linearly with depth, but simulations show that an isotropic vertical-gradient model cannot accurately reproduce traveltimes of first arrivals. However, a simple velocity-analysis inversion based on the wavefront-construction method allows a straightforward estimation of improved isotropic and transversely isotropic models that significantly reduce the misfit between computed traveltimes and the observations. The goal of this analysis is to produce a simple model suitable for use as a starting point for migrations and for a...

An improved mesh generation scheme for the wavefront construction method

Wavefront construction is an effective tool for the rapid calculation of ray fields in anisotropic media. The method explicitly tracks the propagation of a wavefront through a model, mapping it to a computational mesh that is interpolated when accuracy criteria based on paraxial ray methods are violated. Takeoff angles are used often to define the initial ray directions, but uniform sampling in the two angles leads to oversampling of the ray field in the direction of the axis. Such sampling can lead also to numerical instability associated with vanishing derivatives with respect to the azimuthal angle. We suggest a new wavefront mesh definition using the cubed-sphere mesh, which is a coordinate system used to solve partial differential equations in spherical geometries. When using this mesh, ray directions are assigned by mapping points on a regular discretization of the faces of a cube surrounding the source to corresponding rays. This scheme produces a nearly uniform distribution of rays with minimal effort and using the cubed-sphere coordinates as ray parameters to calculate partial derivatives completely eliminates the singularities that arise when takeoff angles are used as ray parameters. Numerical results for quantities related to seismic amplitudes confirm that this new mesh does provide more stable and reliable results.

Static load balancing using non-uniform mesh partitioning based on ray density prediction for the parallel wavefront construction method

Abstract The Wavefront Construction (WFC) method, which was developed based on ray theory, is one of the most efficient seismic modeling tools. WFC propagates a wavefront represented by rays in a computational mesh that is refined whenever an accuracy criterion is violated. Since WFC interpolates new rays during wave propagation, the wavefront mesh is considered highly adaptive. Recently, a parallel WFC was developed using the Standard Template Adaptive Parallel Library. However, due to wavefront density adaptivity, the parallel implementation exhibits inefficient performance owing to load imbalances between processors. In this paper we apply a static load balancing approach based on the prediction of future load for a synthetic salt dome model, to improve performance. This approach utilizes a preliminary conventional ray simulation to estimate the cost (future load) of each cell in the WFCs initial wavefront mesh. Then it applies a non-uniform mesh decomposition that results in a more efficient parallel WFC. Compared to the original implementation, our implementation shows better and more stable scalability in most WFC simulations conducted on the salt model. This paper contributes to understanding the behavior of wavefront mesh adaptability and predicting earth model complexities, and serves as a guide for achieving the ultimate goal, a fully load-balanced parallel WFC.