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Lanczos algorithm

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Journal Article

A fast reduced-rank interpolation method for prestack seismic volumes that depend on four spatial dimensions

Jianjun Gao, Mauricio D. Sacchi, Xiaohong Chen
Journal: Geophysics
Published: 21 December 2012
Geophysics (2013) 78 (1): V21–V30.
DOI: https://doi.org/10.1190/geo2012-0038.1
... data into a level-four block Toeplitz matrix. Rank reduction of this matrix via the Lanczos bidiagonalization algorithm is used to recover missing observations and to attenuate random noise. The computational cost of the Lanczos bidiagonalization is dominated by the cost of multiplying a level-four...
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Journal Article

Solution of 3D time-domain electromagnetic problems using optimal subspace projection

M. Zaslavsky, V. Druskin, L. Knizhnerman
Journal: Geophysics
Published: 20 December 2011
Geophysics (2011) 76 (6): F339–F351.
DOI: https://doi.org/10.1190/geo2011-0088.1
... of computational algorithms that included the conventional time-domain finite-difference, discrete Fourier transform and spectral Lanczos decomposition methods. Numerical examples for the controlled-source EM problem showed that the new algorithm produces accurate solutions on time intervals spanning from...
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Journal Article

A computationally efficient tool for assessing the depth resolution in large-scale potential-field inversion

Valeria Paoletti, Per Christian Hansen, Mads Friis Hansen, Maurizio Fedi
Journal: Geophysics
Published: 16 May 2014
Geophysics (2014) 79 (4): A33–A38.
DOI: https://doi.org/10.1190/geo2014-0017.1
... this with a synthetic and a real-case example. Our main contribution is the combined use of the Lanczos bidiagonalization algorithm, established in the scientific computing community, and the DRP defined in the geoscience community. Our main interest in the Lanczos algorithm comes from the fact...
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Comparison of two rank reduction <span class="search-highlight">algorithms</span>. Computational time for rank re...

Comparison of two rank reduction algorithms. Computational time for rank re...

Published: 21 December 2012
Figure 1. Comparison of two rank reduction algorithms. Computational time for rank reduction of a hypercube of size N 4 via Lanczos bidiagonalization with fast matrix-vector multiplications and the truncated SVD.
Book Chapter

Stable discontinuous staggered grid in the finite-difference modeling of seismic motion

Author(s)
Jozef Kristek, Peter Moczo, Martin Galis
Book: Numerical Modeling of Seismic Wave Propagation: Gridded Two-way Wave-equation Methods
Series: Society of Exploration Geophysicists Geophysics Reprint Series
Publisher: Society of Exploration Geophysicists
Published: 01 January 2012
EISBN: 9781560803089
... number. The algorithm allows for large numbers of time levels without inaccuracy and eventual instability due to numerical noise inevitably generated at the contact of two grids with different spatial grid spacings. The key feature of the algorithm is the application of the Lanczos downsampling filter...
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Abstract We present an algorithm of the spatial discontinuous grid for the 3-D fourth-order velocity–stress staggered-grid finite-difference modelling of seismic wave propagation and earthquake motion. The ratio between the grid spacing of the coarser and finer grids can be an arbitrary odd number. The algorithm allows for large numbers of time levels without inaccuracy and eventual instability due to numerical noise inevitably generated at the contact of two grids with different spatial grid spacings. The key feature of the algorithm is the application of the Lanczos downsampling filter. The algorithm of the discontinuous grid is directly applicable also to the displacement-stress staggered-grid finite-difference scheme.

Journal Article

A parallel finite-difference approach for 3D transient electromagnetic modeling with galvanic sources

Michael Commer, Gregory Newman
Journal: Geophysics
Published: 01 January 2004
Geophysics (2004) 69 (5): 1192–1202.
DOI: https://doi.org/10.1190/1.1801936
...Michael Commer; Gregory Newman Abstract A parallel finite-difference algorithm for the solution of diffusive, three-dimensional (3D) transient electromagnetic field simulations is presented. The purpose of the scheme is the simulation of both electric fields and the time derivative of magnetic...
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Journal Article

Rapid convergence of some seismic processing algorithms

J. H. Rosenbaum, G. F. Boudreaux
Journal: Geophysics
Published: 01 December 1981
Geophysics (1981) 46 (12): 1667–1672.
DOI: https://doi.org/10.1190/1.1441174
... values. Related examples include the time-domain evaluation of the Hilbert transform and the extrapolation to the real axis of spectra evaluated in the complex domain. A formula of this type was first presented by Lanczos (1956). The validity of the algorithms depends upon the fact that summation...
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Journal Article

Resolving seismic anisotropy; sparse matrix methods for geophysical inverse problems

Don W. Vasco, John E. Peterson, Ernest L. Majer
Journal: Geophysics
Published: 01 June 1998
Geophysics (1998) 63 (3): 970–983.
DOI: https://doi.org/10.1190/1.1444408
...Don W. Vasco; John E. Peterson; Ernest L. Majer Abstract Two techniques for the singular value decomposition (SVD) of large sparse matrices are directly applicable to geophysical inverse problems: subspace iteration and the Lanczos method. Both methods require very little in-core storage...
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Journal Article

Magnetic variations in the reconnaissance of sedimentary basins; field procedure and generalized inversion of short-period data from the Rio Grande Rift

J. F. Hermance, Gregory A. Neumann
Journal: Geophysics
Published: 01 December 1990
Geophysics (1990) 55 (12): 1567–1576.
DOI: https://doi.org/10.1190/1.1442808
... site spacings than those used for conventional MT surveys. Procedures for acquiring and processing MV field data are straightforward, as are methods for inverting data to obtain plausible geophysical models.Using a new 2-D generalized inverse algorithm which employs singular value damping...
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A novel approach to the model appraisal and resolution analysis of regularized geophysical inversion

Michael S. Zhdanov, Ekaterina Tolstaya
Journal: Geophysics
Published: 26 September 2006
Geophysics (2006) 71 (6): R79–R90.
DOI: https://doi.org/10.1190/1.2336347
... of the nonlinear inverse problem at a point m 0 . Numerical calculation of the resolution density is a very challenging computational problem. In Appendix A , we present an efficient algorithm for determining this parameter based on the SLDM ( Lanczos, 1961 ; Druskin and Knizhnerman, 1994 ; Golub...
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Journal Article

Computational efficient multidimensional singular spectrum analysis for prestack seismic data reconstruction

Jinkun Cheng, Mauricio Sacchi, Jianjun Gao
Journal: Geophysics
Published: 11 February 2019
Geophysics (2019) 84 (2): V111–V119.
DOI: https://doi.org/10.1190/geo2018-0343.1
... ), randomized MSSA (RMSSA) ( Oropeza and Sacchi, 2011 ), SSA via Lanczos bidiagonalization ( Gao et al., 2013 ), and the proposed computational efficient SSA, respectively. The four algorithms exhibit similar capabilities in the attenuation of random noise. However, the proposed computational efficient SSA...
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Journal Article

Smooth fitting of geophysical data using continuous global surfaces

Stephen D. Billings, Gerry N. Newsam, Rick K. Beatson
Journal: Geophysics
Published: 01 January 2002
Geophysics (2002) 67 (6): 1823–1834.
DOI: https://doi.org/10.1190/1.1527082
... ( v ) by their tridiagonal Lanczos representation. This requires two separate Lanczos procedures applied to C , one initiated with the vector f and the other by the vector u . The algorithm is very efficient because, given h k ( v ) and g k ( v ), the estimates at the next iteration, h...
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Time-domain modeling of electromagnetic diffusion with a frequency-domain code

Wim A. Mulder, Marwan Wirianto, Evert C. Slob
Journal: Geophysics
Published: 13 November 2007
Geophysics (2008) 73 (1): F1–F8.
DOI: https://doi.org/10.1190/1.2799093
.... The Lanczos method constructs the small matrix iteratively. Druskin and Knizhnerman (1994) show that accurate results can be obtained by performing m iterations, where m = O ( n T log n ) . As before, T is the length of time for which the solution needs to be computed, and n...
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Journal Article

3-D electromagnetic anisotropy modeling using finite differences

Tsili Wang, Sheng Fang
Journal: Geophysics
Published: 01 January 2001
Geophysics (2001) 66 (5): 1386–1398.
DOI: https://doi.org/10.1190/1.1486779
... Power Challenge R10000 processor, most of which was spent on the Lanczos decomposition to solve the system of equations. We have developed a finite-difference simulation algorithm for modeling multicomponent EM responses to a 3-D anisotropic medium. The algorithm takes into account full-tensor...
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Investigating the causes of permeability anisotropy in heterogeneous conglomeratic sandstone using multiscale digital rock

Peng Chi, Jianmeng Sun, Weichao Yan, Xin Luo, Ping Feng
Journal: Geophysics
Published: 07 May 2024
Geophysics (2024) 89 (4): MR197–MR207.
DOI: https://doi.org/10.1190/geo2023-0596.1
... the cycle-in-cycle GAN to enhance the resolution of CT images. The training methods for unpaired CT images simplify the acquisition of data sets for training and testing. Recently, Karimpouli et al. (2023) have proposed two strategies for HR 3D digital rock reconstruction using 2D algorithms based...
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A matrix-transform numerical solver for fractional Laplacian viscoacoustic wave equation

Hanming Chen, Hui Zhou, Ying Rao, Pengyuan Sun, Jianlei Zhang, Yangkang Chen
Journal: Geophysics
Published: 19 June 2019
Geophysics (2019) 84 (4): T283–T297.
DOI: https://doi.org/10.1190/geo2018-0271.1
... and the matrices T m and V m from the Lanczos decomposition in equation  10 . We adopt the iterative algorithm proposed by Ilić et al. (2008) and Yang et al. (2011) to implement the Lanczos decomposition. The iterative flow is shown in Table  1 . Through the repeated iterations...
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Finite-difference modeling of electromagnetic tool response for logging while drilling

Tsili Wang, Jack Signorelli
Journal: Geophysics
Published: 01 January 2004
Geophysics (2004) 69 (1): 152–160.
DOI: https://doi.org/10.1190/1.1649383
.... The algorithm allows the simulation of inhomogeneous media with arbitrary conductivity and magnetic permeability distributions. With the use of an irregular finite-difference grid, a complex LWD antenna with cavities and slots is modeled accurately. Our results show that ferrite inserted in antenna slots...
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Journal Article

A dual-grid nonlinear inversion technique with applications to the interpretation of dc resistivity data

Carlos Torres-Verdín, Vladimir L. Druskin, Sheng Fang, Leonid A. Knizhnerman, Alberto Malinverno
Journal: Geophysics
Published: 01 January 2000
Geophysics (2000) 65 (6): 1733–1745.
DOI: https://doi.org/10.1190/1.1444858
..., except that they are implemented with an approximate forward problem that is at least an order of magnitude faster to solve than the algorithm used to simulate the measurements. Any given auxiliary minimization in the cascade is fully self-contained and yields a solution of the unknown model parameters...
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Journal Article

Fourth‐Order Staggered‐Grid Finite‐Difference Seismic Wavefield Estimation Using a Discontinuous Mesh Interface (WEDMI)

Shiying Nie, Yongfei Wang, Kim B. Olsen, Steven M. Day
Published: 25 September 2017
Bulletin of the Seismological Society of America (2017) 107 (5): 2183–2193.
DOI: https://doi.org/10.1785/0120170077
.... , 2013 ). However, several authors and users of the FD discontinuous grid implementations found that the methods suffer inherently from stability problems after a large number of timesteps (e.g., S. Aoi, written comm., 2016). We also note that the Lanczos filter that is already applied to several...
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Journal Article

Electromagnetic induction in a generalized 3D anisotropic earth, Part 2: The LIN preconditioner

Chester J. Weiss, Gregory A. Newman
Journal: Geophysics
Published: 01 January 2003
Geophysics (2003) 68 (3): 922–930.
DOI: https://doi.org/10.1190/1.1581044
... of the iterative sequence on the i th Lanczos vector v i , [see algorithm 8.1 in Freund and Nachtigal (1994) ]. Thus, at each step of the QMR sequence (where the Krylov subspace is spanned by the Lanczos vectors v 0 … v i ), we solve the system (4) Since we seek a preconditioning operator...
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