Estimating the model resolution matrix for large seismic tomography problems based on Lanczos bidiagonalization with partial reorthogonalization

SUMMARY The PROPACK package developed by Larsen is able to efficiently and accurately estimate singular values and vectors for large matrices based on the Lanczos bidiagonalization with partial reorthogonalization. We incorporate the Pavtial Reorthogonalization Package (PROPACK) package into the double-difference seismic tomography code tomoDD and estimate the model resolution matrix for large seismic tomography problems. Compared to previous Least Squares QR (LSQR)-based methods for estimating the model resolution matrix the PROPACK-based method calculates the full resolution matrix and thus gives a complete description of how well the model is resolved. Several observations are drawn from the application to data from the 2001 eruption of Mt Etna: for this example, it is reasonable to use ray-sampling density information to characterize the model resolution qualitatively; the model resolution resulting from just velocity inversion bears a close linear relationship to that from simultaneous inversion but always overestimates resolution; and the inversion system using differential times has a greater ability to resolve the source region structure than the system using absolute times.

[1]  Douglas R. Toomey,et al.  Tomographic inversion of local earthquake data from the Hengill‐Grensdalur Central Volcano Complex, Iceland , 1989 .

[2]  H. Simon.  The Lanczos algorithm with partial reorthogonalization , 1984 .

[3]  R. G. Pratt,et al.  Traveltime tomography in anisotropic media—II. Application , 1992 .

[4]  S. Minkoff.  A computationally feasible approximate resolution matrix for seismic inverse problems , 1996 .

[5]  Christopher C. Paige,et al.  The computation of eigenvalues and eigenvectors of very large sparse matrices , 1971 .

[6]  J. Virieux,et al.  Explicit, approximate expressions for the resolution and a posteriori covariance of massive tomographic systems , 1999 .

[7]  Robert W. Clayton,et al.  Adaptation of Back Projection Tomography to Seismic Travel Time Problems , 1988 .

[8]  G. McMechan,et al.  Estimation of resolution and covariance for large matrix inversions , 1995 .

[9]  Akira Hasegawa,et al.  Tomographic imaging of P and S wave velocity structure beneath northeastern Japan , 1992 .

[10]  Clifford H. Thurber,et al.  Parameter estimation and inverse problems , 2005 .

[11]  Ari Tryggvason,et al.  Three-dimensional imaging of the P- and S-wave velocity structure and earthquake locations beneath Southwest Iceland , 2002 .

[12]  W. Holbrook,et al.  Inferring crustal structure in the Aleutian island arc from a sparse wide‐angle seismic data set , 2004 .

[13]  Clifford H. Thurber,et al.  Development and Applications of Double-difference Seismic Tomography , 2006 .

[14]  Jack Dongarra,et al.  LINPACK Users' Guide , 1987 .

[15]  R. Larsen.  Lanczos Bidiagonalization With Partial Reorthogonalization , 1998 .

[16]  E. Papadimitriou,et al.  3D crustal structure from local earthquake tomography around the Gulf of Arta (Ionian region, NW Greece) , 1999 .

[17]  S. Trickett.  F-xy eigenimage noise suppression , 2003 .

[18]  G. Nolet,et al.  Comment on ‘Estimation of resolution and covariance for large matrix inversions’ by J. Zhang and G. A. McMechan , 1996 .

[19]  E. Kissling,et al.  Model parametrization in seismic tomography: a choice of consequence for the solution quality , 2001 .

[20]  P. Charvis,et al.  Seismic structure of Cocos and Malpelo Volcanic Ridges and implications for hot spot‐ridge interaction , 2003 .

[21]  L. Boschi.  Measures of resolution in global body wave tomography , 2003 .

[22]  Luis Rivera,et al.  On the use of the checker-board test to assess the resolution of tomographic inversions , 1993 .

[23]  Don W. Vasco,et al.  Resolution, uncertainty, and whole Earth tomography , 2002 .

[24]  J. Berryman.  Analysis of Approximate Inverses in Tomography II. Iterative Inverses , 2000 .

[25]  Clifford H. Thurber,et al.  Local earthquake tomography with flexible gridding , 1999 .

[26]  C. Chiarabba,et al.  Tomographic images and 3D earthquake locations of the seismic swarm preceding the 2001 Mt. Etna eruption: Evidence for a dyke intrusion , 2002 .

[27]  R. Roberts,et al.  Comment on ‘Explicit, approximate expressions for the resolution and a posteriori covariance of massive tomographic systems’ by G. Nolet, R. Montelli and J. Virieux , 2001 .

[28]  Reply to comment by Z. S. Yao, R. G. Roberts and A. Tryggvason on ‘Explicit, approximate expressions for the resolution and a posteriori covariance of massive tomographic systems’ , 2001 .

[29]  R. Roberts,et al.  Calculating resolution and covariance matrices for seismic tomography with the LSQR method , 1999 .

[30]  Michael A. Saunders,et al.  LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares , 1982, TOMS.

[31]  Malcolm Sambridge,et al.  Non-linear arrival time inversion: constraining velocity anomalies by seeking smooth models in 3-D , 1990 .

[32]  J. Reid,et al.  Tracking the Progress of the Lanczos Algorithm for Large Symmetric Eigenproblems , 1981 .

[33]  B. Parlett.  A new look at the Lanczos algorithm for solving symmetric systems of linear equations , 1980 .

[34]  James G. Berryman,et al.  Analysis of Approximate Inverses in Tomography I. Resolution Analysis of Common Inverses , 2000 .

[35]  Clifford H. Thurber,et al.  Double-Difference Tomography: The Method and Its Application to the Hayward Fault, California , 2003 .

[36]  Clifford H. Thurber,et al.  Hypocenter-velocity structure coupling in local earthquake tomography , 1992 .

[37]  D. Vasco,et al.  Global Earth structure: inference and assessment , 2002 .

[38]  Sean C. Solomon,et al.  Tomographic imaging of the shallow crustal structure of the East Pacific Rise at 9°30′N , 1994 .

[39]  G. Soldati,et al.  The resolution of whole Earth seismic tomographic models , 2005 .

[40]  D. Oldenburg,et al.  A comparison of automatic techniques for estimating the regularization parameter in non-linear inverse problems , 2004 .

[41]  Clifford H. Thurber,et al.  Fine‐scale structure of the San Andreas fault zone and location of the SAFOD target earthquakes , 2004 .

[42]  Claudio Chiarabba,et al.  Magma Ascent and the Pressurization of Mount Etna's Volcanic System , 2003, Science.

[43]  Larry J. Ruff,et al.  How good are our best models? Jackknifing, bootstrapping, and earthquake depth , 1989 .