Abstract
Inversions for local helioseismology are an important and necessary step for obtaining three-dimensional maps of various physical quantities in the solar interior. Frequently, the full inverse problems that one would like to solve prove intractable because of computational constraints. Due to the enormous seismic data sets that already exist and those forthcoming, this is a problem that needs to be addressed. To this end, we present a very efficient linear inversion algorithm for local helioseismology. It is based on a subtractive optimally localized averaging (SOLA) scheme in the Fourier domain, utilizing the horizontal-translation invariance of the sensitivity kernels. In Fourier space the problem decouples into many small problems, one for each horizontal wave vector. This multichannel SOLA method is demonstrated for an example problem in time–distance helioseismology that is small enough to be solved both in real and Fourier space. We find that both approaches are successful in solving the inverse problem. However, the multichannel SOLA algorithm is much faster and can easily be parallelized.
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Backus, G.E., Gilbert, J.F.: 1968, The resolving power of gross Earth data. Geophys. J. 16, 169 – 205. doi: 10.1111/j.1365-246X.1968.tb00216.x .
Basu, S., Antia, H.M., Tripathy, S.C.: 1999, Ring diagram analysis of near-surface flows in the Sun. Astrophys. J. 512, 458 – 470. doi: 10.1086/306765 .
Birch, A.C., Kosovichev, A.G., Duvall, T.L. Jr.: 2004, Sensitivity of acoustic wave travel times to sound-speed perturbations in the solar interior. Astrophys. J. 608, 580 – 600. doi: 10.1086/386361 .
Couvidat, S., Birch, A.C.: 2009, Helioseismic travel-time definitions and sensitivity to horizontal flows obtained from simulations of solar convection. Solar Phys. 257, 217 – 235. doi: 10.1007/s11207-009-9371-4 .
Couvidat, S., Birch, A.C., Kosovichev, A.G., Zhao, J.: 2004, Three-dimensional inversion of time–distance helioseismology data: ray-path and Fresnel-zone approximations. Astrophys. J. 607, 554 – 563. doi: 10.1086/383342 .
Couvidat, S., Gizon, L., Birch, A.C., Larsen, R.M., Kosovichev, A.G.: 2005, Time–distance helioseismology: inversion of noisy correlated data. Astrophys. J. Suppl. 158, 217 – 229. doi: 10.1086/430423 .
Duvall, T.L. Jr., Jefferies, S.M., Harvey, J.W., Pomerantz, M.A.: 1993, Time–distance helioseismology. Nature 362, 430 – 432. doi: 10.1038/362430a0 .
Gizon, L., Birch, A.C.: 2002, Time–distance helioseismology: the forward problem for random distributed sources. Astrophys. J. 571, 966 – 986. doi: 10.1086/340015 .
Gizon, L., Birch, A.C.: 2004, Time–distance helioseismology: noise estimation. Astrophys. J. 614, 472 – 489. doi: 10.1086/423367 .
Gizon, L., Birch, A.C.: 2005, Local helioseismology. Living Rev. Solar Phys. 2, 6. http://www.livingreviews.org/lrsp-2005-6 .
Gizon, L., Duvall, T.L. Jr., Larsen, R.M.: 2000, Seismic tomography of the near solar surface. J. Astrophys. Astron. 21, 339.
Gizon, L., Birch, A.C., Spruit, H.C.: 2010, Local helioseismology: three-dimensional imaging of the solar interior. Annu. Rev. Astron. Astrophys. 48, 289 – 338. doi: 10.1146/annurev-astro-082708-101722 .
Gizon, L., Schunker, H., Baldner, C.S., Basu, S., Birch, A.C., Bogart, R.S., Braun, D.C., Cameron, R., Duvall, T.L., Hanasoge, S.M., Jackiewicz, J., Roth, M., Stahn, T., Thompson, M.J., Zharkov, S.: 2009, Helioseismology of sunspots: a case study of NOAA region 9787. Space Sci. Rev. 144, 249 – 273. doi: 10.1007/s11214-008-9466-5 .
González Hernández, I., Kholikov, S., Hill, F., Howe, R., Komm, R.: 2008, Subsurface meridional circulation in the active belts. Solar Phys. 252, 235 – 245. doi: 10.1007/s11207-008-9264-y .
Haber, D.A., Hindman, B.W., Toomre, J., Bogart, R.S., Larsen, R.M., Hill, F.: 2002, Evolving submerged meridional circulation cells within the upper convection zone revealed by ring-diagram analysis. Astrophys. J. 570, 855 – 864. doi: 10.1086/339631 .
Hansen, P.C.: 1998, Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion, SIAM, Philadelphia. ISBN 0-89871-403-6.
Jackiewicz, J., Gizon, L., Birch, A.C.: 2008a, High-resolution mapping of flows in the solar interior: fully consistent OLA inversion of helioseismic travel times. Solar Phys. 251, 381 – 415. doi: 10.1007/s11207-008-9158-z .
Jackiewicz, J., Gizon, L., Birch, A.C.: 2008b, The forward and inverse problems in time-distance helioseismology. J. Phys. Conf. Ser. 118(1), 012033. doi: 10.1088/1742-6596/118/1/012033 .
Jackiewicz, J., Gizon, L., Birch, A.C., Thompson, M.J.: 2007a, A procedure for the inversion of f-mode travel times for solar flows. Astron. Nachr. 328, 234 – 239. doi: 10.1002/asna.200610725 .
Jackiewicz, J., Gizon, L., Birch, A.C., Duvall, T.L. Jr.: 2007b, Time–distance helioseismology: sensitivity of f-mode travel times to flows. Astrophys. J. 671, 1051 – 1064. doi: 10.1086/522914 .
Jacobsen, B., Moller, I., Jensen, J., Efferso, F.: 1999, Multichannel deconvolution, MCD, in geophysics and helioseismology. Phys. Chem. Earth, Part A, Solid Earth Geod. 24, 215 – 220. doi: 10.1016/S1464-1895(99)00021-6 .
Jensen, J.M., Jacobsen, B.H., Christensen-Dalsgaard, J.: 1998, MCD inversion for sound speed using time–distance data. In: Korzennik, S. (ed.) Structure and Dynamics of the Interior of the Sun and Sun-Like Stars SP-418, ESA, Noordwijk, 635 – 640.
Jensen, J.M., Duvall, T.L. Jr., Jacobsen, B.H., Christensen-Dalsgaard, J.: 2001, Imaging an emerging active region with helioseismic tomography. Astrophys. J. Lett. 553, L193 – L196. doi: 10.1086/320677 .
Kosovichev, A.G.: 1996, Tomographic imaging of the Sun’s interior. Astrophys. J. Lett. 461, L55 – L57. doi: 10.1086/309989 .
Kosovichev, A.G., Duvall, T.L. Jr.: 1997, Acoustic tomography of solar convective flows and structures. In: Pijpers, F.P., Christensen-Dalsgaard, J., Rosenthal, C.S. (eds.) SCORe’96: Solar Convection and Oscillations and their Relationship, Astrophys. Space Science Lib. 225, Kluwer, Dordrecht, 241 – 260.
Louis, A.K., Maass, P.: 1990, A mollifier method for linear operator equations of the first kind. Inverse Probl. 6, 427 – 440.
Paige, C.C., Saunders, M.A.: 1982, LSQR, an algorithm for sparse linear equations and sparse least squares. ACM Trans. Math. Softw. 8, 43 – 71.
Pijpers, F.P., Thompson, M.J.: 1992, Faster formulations of the optimally localized averages method for helioseismic inversions. Astron. Astrophys. 262, 33 – 36.
Pijpers, F.P., Thompson, M.J.: 1994, The SOLA method for helioseismic inversion. Astron. Astrophys. 281, 231 – 240.
Schuster, T.: 2007, The Method of Approximate Inverse: Theory and Applications, Lecture Notes in Mathematics 1906, Springer, Berlin. ISBN 978-3-540-71226-8.
Švanda, M., Gizon, L., Hanasoge, S.M., Ustyugov, S.D.: 2011, Validated helioseismic inversions for 3D vector flows. Astron. Astrophys. 530, A148. doi: 10.1051/0004-6361/201016426 .
Tikhonov, A.N.: 1963, On the solution of incorrectly formulated problems and the regularization method. Sov. Math. Dokl. 4, 1035 – 1038.
Woodard, M.F.: 2007, Probing supergranular flow in the solar interior. Astrophys. J. 668, 1189 – 1195. doi: 10.1086/521391 .
Zhao, J., Kosovichev, A.G.: 2004, Torsional oscillation, meridional flows, and vorticity inferred in the upper convection zone of the Sun by time–distance helioseismology. Astrophys. J. 603, 776 – 784. doi: 10.1086/381489 .
Zhao, J., Kosovichev, A.G., Duvall, T.L. Jr.: 2001, Investigation of mass flows beneath a sunspot by time–distance helioseismology. Astrophys. J. 557, 384 – 388. doi: 10.1086/321491 .
Zhao, J., Georgobiani, D., Kosovichev, A.G., Benson, D., Stein, R.F., Nordlund, Å.: 2007, Validation of time–distance helioseismology by use of realistic simulations of solar convection. Astrophys. J. 659, 848 – 857. doi: 10.1086/512009 .
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M. Švanda is on leave from the Astronomical Institute, Academy of Sciences of the Czech Republic and the Faculty of Mathematics and Physics, Charles University in Prague, Czech Republic.
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Jackiewicz, J., Birch, A.C., Gizon, L. et al. Multichannel Three-Dimensional SOLA Inversion for Local Helioseismology. Sol Phys 276, 19–33 (2012). https://doi.org/10.1007/s11207-011-9873-8
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DOI: https://doi.org/10.1007/s11207-011-9873-8