Calculating Path-Dependent Travel Time Prediction Variance and Covariance for a Global Tomographic P-Velocity Model
LOS ALAMOS NATIONAL LAB NM
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Several studies have shown that global 3D models of the compression wave speed in the Earths mantle can provide superior first P travel time predictions at both regional and teleseismic distances. However, given the variable data quality and uneven data sampling associated with this type of model, it is essential that there be a means to calculate high-quality estimates of the path-dependent variance and covariance associated with the predicted travel times of ray paths through the model. In this paper, we show a methodology for accomplishing this by exploiting the full model covariance matrix. Typical global 3D models have on the order of 12 million nodes, so the challenge in calculating the covariance matrix is formidable 0.9 TB storage for 12 of a symmetric matrix, necessitating an Out-Of-Core OOC blocked matrix solution technique. With our approach the tomography matrix G which includes Tikhonov regularization terms is multiplied by its transpose GTG and written in a blocked sub-matrix fashion. We employ a distributed parallel solution paradigm that solves for GTG-1 by assigning blocks to individual processing nodes for matrix decomposition update and scaling operations. We first find the Cholesky decomposition of GTG which is subsequently inverted. Next, we employ OOC matrix multiplication methods to calculate the model covariance matrix from GTG-1 and an assumed data covariance matrix. Given the model covariance matrix we solve for the travel-time covariance associated with arbitrary ray-paths by integrating the model covariance along both ray paths. Setting the paths equal yields the variance for that path.
- Geology, Geochemistry and Mineralogy