Two-dimensional Frequency-domain Full-Waveform Tomography
for visco-acoustic media
BASIC PROGRAM DESCRIPTION
Frequency-domain full-waveform tomography for visco-acoustic media.
Program FWT2D is a massively parallel code for distributed memory platform which performs frequency-domain full-waveform inversion of seismic data [10,11,12,13]. The program is more specifically designed for wide-angle or global offset data (that is, for any acquisition system involving dense wide-aperture acquisitions and for which sources share a significant range of receivers spanning over large offsets). The inversion looks for the P-wave velocity only. However, heterogeneous attenuation can be provided for the forward modeling. Density is set to 1 but the code can be easily modified to take into account heterogeneous density in the forward modeling. The inversion relies on a classic iterative steepest descent algorithm [1,2]. Iterations are performed in a non linear way which means that the final model at a given iteration is used as the starting model for the next iteration. Single or group of frequencies are inverted successively. The classic procedure is to proceed from the low frequencies to the higher ones . Note however that all the frequencies can be inverted simultaneously by defining a single group of nw frequencies (nw is the total number of frequencies to be inverted). The cost function is based on the least-squares norm. The cost function is weighted with an operator which applies a gain with offset to the residuals. The gradient of the cost function is properly scaled by the diagonal terms of the approximate hessian J^t J where J is the sensitivity matrix . The source can be approximated in the program solving a linear inverse problem (see , , ). The step length is computed by parabolic fitting. The forward problem, that is, wave propagation modeling, is performed with a finite-difference frequency-domain method [6,7,14] and relies on a direct solver to solve the associate sparse system of linear equations whose right-hand sides (RHS) are the sources. We use the massively parallel direct solver MUMPS for distributed memory platform to solve this system [15,16]. Note however that the code can also be run in sequential (see MUMPS documentation). Absorbing boundary condition in the FDFD code are combination of PML [9,14] and 45Â° paraxial conditions.
Parallelization is implemented using the following strategy: - MUMPS performs the impedance matrix factorization in parallel. - During the multiple shot (RHS) resolution phase, the solution (computed wavefields) are distributed on the processors: a subdomain of all the solutions is stored on one processors. - The summations over shots and/or receivers required to computed the diagonal Hessian and gradient are computed in parallel, each processor computing a particular subdomain of the diagonal Hessian and gradient accordingly the distribution of the MUMPS solution.
# Version v4.5 - April 25, 2007, Version v4.6 - July 4, 2007
# Copyright 2007
# SEISCOPE imaging group
# UMR Géoazur - UNS - OCA - CNRS - IRD
# Langage: Fortran90 + MPI
# Authors: F. Sourbier, S. Operto and J. Virieux
# Others contributors by alphabetic order: C. Gelis, B. Hustedt, C. Ravaut.
- Stephane Operto
UMR Géoazur - CNRS-IRD-UNSA-UPMC
Observatoire Océanologique, La Darse - BP48, Villefranche-sur-mer 06230
tel: 33 4 93 76 37 52
fax: 33 4 93 76 37 66
Laboratoire Géophysique Interne et Tectonophysique
BP 53, 38041 Grenoble CEDEX 9
tel: 33 4 76 63 52 54
fax: 33 4 76 63 52 52
web: http://www-lgit.obs.ujf-grenoble.fr/ virieuxj
CONDITIONS OF USE
This program was developed by the SEISCOPE imaging group coordinated by J. Virieux and S. Operto. Research activity of the SEISCOPE imaging group is partly funded by the SEISCOPE consortium.
This program is provided freely under request by sending an email to: firstname.lastname@example.org or email@example.com This program or modified versions of it may be distributed under the condition that this code and any modifications made to it in the same file remain under copyright of the original authors, both source and object code are made freely available without charge, and clear notice is given of the modifications. Distribution of this code as part of a commercial system is permissible only by direct arrangement with S. Operto and J. Virieux.
You shall acknowledge (using following references [10-14]) the contribution of this package in any publication of material dependent upon the use of the package. You shall use reasonable endeavours to notify the authors of the package of this publication.
Methodological concepts implemented in this code are based on the original following references:
INVERSE PROBLEM THEORY
 Pratt, R. G., Shin, C. and Hicks, G. J. (1998). Gauss-Newton and full Newton methods in frequency-space seismic waveform inversion. Geophys. J. Int., 133:341-362.
 Pratt, R. G. (1999). Seismic waveform inversion in the frequency domain, part 1: theory and verification in a physical scale model. Geophysics, 64:888-901.
 Sirgue, L. and Pratt, R. G. (2004). Efficient waveform inversion and imaging: a strategy for selecting temporal frequencies. Geophysics, 69: 231-248.
 Shin, C., Yoon, K., Marfurt, K.J., Park, K., Yang, D., Lim, H., Chung, S., and Shin, S. (2001). Efficient calculation of a partial derivative wavefield using reciprocity for seismic imaging and inversion. Geophysics, 66(6): 1856-1863.
FINITE-DIFFERENCE FREQUENCY-DOMAIN WAVE PROPAGATION MODELING
 Marfurt, K. (1984). Accuracy of finite-difference and finite-elements modeling in scalar and elastic wave equation. Geophysics, 49: 533-549.
 Jo, C. H., Shin, C. and Suh, J. H. (1996). An optimal 9-point, finite-difference, frequency-space, 2D scalar wave extrapolator. Geophysics, 61(2): 529-537.
 Stekl, I. and Pratt, R. G. (1998). Accurate viscoelastic modeling by frequency-domain finite differences using rotated operators. Geophysics, 63(5): 1779-1794.
 Saenger, E. H., Gold, N. and Shapiro, A. (2000). Modeling the propagation of elastic waves using a modified finite-difference grid. wave motion, 31:77-92.
 Berenger, J. -P. (1994). A perfectly matched layer for absorption of electromagnetic waves. Journal of Computational Physics, 114:185-200.
 B. Hustedt, S. Operto and J. Virieux (2004). Mixed-grid and staggered-grid finite- difference methods for frequency-domain acoustic wave modelling. Geophysical Journal International, 157:1269-1296.
 C. Ravaut, S. Operto, L. Improta, J. Virieux, A. Herrero and P. Dell’Aversana (2004). Multiscale imaging of complex structures from multifold wide-aperture seismic data by ferquency-domain full-waveform tomography: application to a thrust belt. Geophysical Journal International, 159: 1032-1056.
 S. Operto, C. Ravaut, L. Improta, J. Virieux, A. Herrero and P. Dell’Aversana (2004). Quantitative imaging of complex structures from dense wide-aperture seismic data by multiscale traveltime and waveform inversions: a case study. Geophysical Prospecting, 52: 625-651.
 J. X. Dessa, S. Operto, S. Kodaira, A. Nakanishi, G. Pascal (2004). J. Virieux and Y. Kaneda, Multiscale seismic imaging of the eastern Nankai trough by full waveform inversion> Geophys. Res. Letters, 31, L18606, doi: 10.1029 /2004GL020453,2004.
 S. Operto, J. Virieux, J. X. Dessa and G. Pascal (2006). Crustal seismic imaging from multifold ocean bottom seismometer data by frequency-domain full-waveform tomography: application to the eastern-Nankai trough. Journal of Geophysical Research, 111(B09306):doi:10.1029/2005JB003835.
M. Malinowsky and S. Operto, Quantitative imaging of the Permo-Mesozoic complex and its basement by frequency domain waveform tomography of wide-aperture seismic data from the Polish basin, Geophysical Prospecting, 56, 805-825, 2008
F. Sourbier, S. Operto, J. Virieux, P. Amestoy and J. -Y. L’Excellent, FWT2D: a massively parallel program for frequency-domain Full-Waveform Tomography of wide-aperture seismic data - Part 1: algorithm. Computer & Geosciences, 35 (2009), 487-495.
F. Sourbier, S. Operto, J. Virieux, P. Amestoy and J. -Y. L’Excellent, FWT2D: a massively parallel program for frequency-domain Full-Waveform Tomography of wide-aperture seismic data - Part 2: numerical examples and scalability analysis. Computer & Geosciences, 35 (2009), 496-514.
EXTERNAL CALLS (MUMPS, SCALAPACK, BLACS, BLAS)
The resolution of the forward problem (wave propagation modeling) relies on the massively parallel direct solver MUMPS for distributed memory platform. Please visit the MUMPS web page to recover the package and for the conditions of use. http://mumps.enseeiht.fr/ or http://graal.ens-lyon.fr/MUMPS/ MUMPS requires the SCALAPACK (SCALABLE LAPACK) (http://www.netlib.org/scalapack/sca...), BLACS (Basic Linear Algebra Communication Subprograms)(http://www.netlib.org/blacs/) and BLAS (Basic Linear Algebra subprograms (http://www.netlib.org/blas/) librairies for parallel execution.
Two reference on MUMPS:
 Amestoy, P. R., Duff, I. S., Koster, J., and L’Excellent, J. Y. (2001). A fully asynchronous multifrontal solver using distributed dynamic scheduling. SIAM Journal of Matrix Analysis and Applications, 23(1): 15-41
Amestoy, P. R., Guermouche, A., L’Excellent, J. Y., and Pralet, S. (2006). Hybrid scheduling for the parallel solution of linear systems. Parallel computing, 32:136-156.
Note that MUMPS and fwt2d.v4.5 can be compiled and and run in sequential even if MPI calls are performed in the program. In that case, only BLAS libraries are required. Please see the MUMPS documentation for more information about the MUMPS installation.