Applications of General Three-Dimensional Geographical Curve-Linear Library in Computational Seismology

We developed a general library for handling a class of objects we call Geographical Curvilinear grids (GCLgrids). A GCLgrid can be viewed as topologically equivalent to a rectangular box divided into regular cubes with nodes at each corner of each cube. The actual grid can be arbitrarily distorted provided it can be mapped back into a uniform grid with no singularities. A simple example is a spherical shell divided along latitude, longitude, and depth. Two algorithms are the core of this library. First, we use Newton's method to search for a location in space from a starting point. This algorithm converges reasonably fast if the grid is not extremely distorted. Second, we interpolate the grid using methods known from finite analysis. A Jacobian matrix for an 8-node cube is computed to transform a distorted cube into a standard one. Shape functions are used to describe the actual object mapped to the standard cube. Once the transformation matrices are computed, we can interpolate n-element vectors as quickly as scalar data. We apply this technique to two computational seismology problems. First, in a plane wave migration procedure, we transform the seismic data from time to depth domain, and this approach provides a general solution to this complex mapping problem. Secondly, this approach allows us to utilize 3D travel time tables to locate events using 3D velocity models. This library may also have use in numerical modeling of regional scale geodynamic problems.