Don Vasco, Lawrence Berkeley National Laboratory
Osni Marques, NERSC, Lawrence Berkeley National Laboratory
Lane Johnson, University of California, Berkeley

Research Objectives

The Center for Computational Seismology is developing computational methods for solving geophysical inverse problems. Inverse methods image the Earth by taking observations gathered at the Earth's surface and inferring the internal structure of our planet. Our prototype problem uses several million seismic arrival times to estimate the full three-dimensional velocity structure of the Earth. In addition to producing images of Earth structure, we seek to estimate the reliability of the images by computing our resolution of Earth structure and the associated uncertainties of our estimates.

Computational Approach

The estimation of structure and the model assessment can be reduced to large-scale problems in linear algebra. Ideally we would invert the matrix relating the seismic travel times to the Earth's structure. However, the matrix is approximately 1 million rows by 300,000 columns, precluding a formal inversion. We use an iterative block Lanczos code to estimate successive singular values and singular vectors associated with our data matrix.

Accomplishments

In the past year we were successful in imaging the Earth on a 6° by 6° scale (the lateral dimension of a typical cell). We were able to image subducting tectonic plates, ocean ridges, and velocity anomalies at the base of the Earth's liquid core. We ported the block Lanczos code to the T3E and applied it to assess the resolution of the 6° by 6° grid, some 1 million equations by 100,000 unknowns. We were able to calculate up to 9,000 Lanczos values and vectors. Our estimates of the full three-dimensional Earth velocity structure are the first ever, and our model assessment is the first complete analysis of resolution and uncertainty.

Recently, two research groups have produced models of mantle structure (velocity variations in only the mantle) on a finer scale of 1°-3°. Such small cells are necessary for imaging features like the thin subducting slabs. We are now extending our work to the finer 3° by 3° grid in the mantle while retaining our current 6° grid in the Earth's core. Thus, we will be able to estimate velocity structure and assess its reliability on a finer scale. The new finer grid will have almost 4 times the number of parameters. In order to estimate resolution and uncertainty, we must calculate between 10,000 and 30,000 Lanczos values and vectors.

An image of the compressional velocity variation within the Earth's mantle, that is, the velocity at which seismic waves propagate through the mantle. Cold colors signify higher average velocities, while warm colors signify lower velocities. The high velocities circling the Pacific are thought to represent subducting crust associated with plate tectonics.

Significance

To date, seismic research has focused on specific regions of the Earth, such as the mantle, using only data most sensitive to these regions. Ours is the first study to attempt to utilize all seismic arrival times to solve for the structure of the entire Earth-crust, mantle, and core. We are also the first to conduct a thorough assessment of our model by calculating model parameter resolution and uncertainty. No one to date has adequately characterized the reliability of the finer-scale models, so, it is not clear if the seismic data actually allow us to image such fine details. Our work will provide the first definitive answer to this question.

The Lanczos code being used here has other applications such as structural engineering, web search engines, and numerical applications.

Publications

D. W. Vasco, L. R. Johnson, and O. Marques, "Global Earth structure: Inference and assessment," Geophys. J. Int. 137, 381 (1999).

D. W. Vasco and L. R. Johnson, "Whole Earth structure estimated from seismic arrival times," J. Geophys. Res. 103, 2633 (1998).

D. W. Vasco, J. E. Peterson, and E. L. Majer, "Resolving seismic anisotropy: Sparse matrix methods for geophysical inverse problems," Geophysics 63, 970 (1998).