Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Surface-Focused Seismic Holography of Sunspots: II. Expectations from Numerical Simulations Using Sound-Speed Perturbations


Helioseismic observations of sunspots show that wave travel times, at fixed horizontal phase speed, depend on the temporal frequency of the waves employed in the data analysis. This frequency variation has been suggested to be consistent with near-surface (vertical length scales of order one Mm or smaller) changes in wave propagation properties relative to the quiet Sun. We investigate this suggestion by employing numerical simulations of acoustic-wave propagation through models with horizontally and vertically inhomogeneous structure. Standard methods of surface-focused helioseismic holography are applied to the resulting simulated wave fields. We find that the travel-time shifts measured using holography from the simulations with deep sound-speed perturbations (relative to a plane-parallel quiet-Sun model) do not show a systematic frequency dependence at phase speeds above about 20 km s−1. However, shallow sound-speed perturbations, similar to those proposed to model the acoustic scattering properties of sunspots observed with Hankel analysis, produce systematic frequency dependence at these phase speeds. In both cases, positive travel-time shifts can be caused by positive sound-speed perturbations. The details of the travel-time shifts are, however, model dependent.


  1. Bogdan, T.J., Hindman, B.W., Cally, P.S., Charbonneau, P.: 1996, Absorption of p-Modes by Slender Magnetic Flux Tubes and p-Mode Lifetimes. Astrophys. J. 465, 406.

  2. Braun, D.C., Birch, A.C.: 2006, Surface effects in sunspot seismology. Astrophys. J. 647, 187.

  3. Braun, D.C., Birch, A.C.: 2008, Surface-focused seismic holography of sunspots: I. Observations. Solar Phys. 251, 267. doi:10.1007/s11207-008-9152-5.

  4. Braun, D., Lindsey, C.: 2000, Phase-sensitive holography of solar activity. Solar Phys. 192, 307.

  5. Cally, P.S.: 2000, Modelling p-mode interaction with a spreading sunspot field. Solar Phys. 192, 395.

  6. Cally, P.S., Crouch, A.D., Braun, D.C.: 2003, Probing sunspot magnetic fields with p-mode absorption and phase-shift data. Mon. Not. Roy. Astron. Soc. 346, 381.

  7. Cameron, R., Gizon, L., Daiffallah, K.: 2007, SLiM: a code for the simulation of wave propagation through an inhomogeneous, magnetised solar atmosphere. Astron. Nachr. 328, 313.

  8. Cameron, R., Gizon, L., Duvall, T.L. Jr.: 2008, Helioseismology of sunspots: confronting observations with three-dimensional MHD simulations of wave propagation. Solar Phys. 251, 291. doi:10.1007/s11207-008-9148-1.

  9. Chou, D.Y.: 2000, Acoustic imaging of solar active regions – (Invited Review). Solar Phys. 192, 241.

  10. Chou, D.Y., Sun, M.T., Chang, H.K.: 2000, A study of sunspots with phase time and travel time of p-mode waves in acoustic imaging. Astrophys. J. 532, 622.

  11. Crouch, A.D., Cally, P.S.: 2003, Mode conversion of solar p modes in non-vertical magnetic fields – i. Two-dimensional model. Solar Phys. 214, 201.

  12. Duvall, T.L., Jefferies, S.M., Harvey, J.W., Pomerantz, M.A.: 1993, Time-distance helioseismology. Nature 362, 430.

  13. Duvall, T.L., D’Silva, S., Jefferies, S.M., Harvey, J.W., Schou, J.: 1996, Downflows under sunspots detected by helioseismic tomography. Nature 379, 235.

  14. Duvall, T.L., Kosovichev, A.G., Scherrer, P.H., Bogart, R.S., Bush, R.I., de Forest, C., Hoeksema, J.T., Schou, J., Saba, J.L.R., Tarbell, T.D., Title, A.M., Wolfson, C.J., Milford, P.N.: 1997, Time-distance helioseismology with the MDI instrument: initial results. Solar Phys. 170, 63.

  15. Fan, Y., Braun, D.C., Chou, D.Y.: 1995, Scattering of p-modes by sunspots. II. Calculations of phase shifts from a phenomenological model. Astrophys. J. 451, 877.

  16. Gizon, L., Birch, A.C.: 2002, Time-distance helioseismology: the forward problem for random distributed sources. Astrophys. J. 571, 966.

  17. Gizon, L., Birch, A.C.: 2005, Local helioseismology. Living Rev. Solar Phys. 2, 6.

  18. Hanasoge, S.: 2007, Theoretical studies of wave propagation in the sun. PhD thesis, Stanford University.

  19. Hanasoge, S.M., Duvall, T.L. Jr., Couvidat, S.: 2007, Validation of helioseismology through forward modeling: realization noise subtraction and Kernels. Astrophys. J. 664, 1234.

  20. 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, Astrophysics and Space Science Library 225, 241.

  21. Lindsey, C., Braun, D.: 1990, Helioseismic imaging of sunspots at their antipodes. Solar Phys. 126, 101.

  22. Lindsey, C., Braun, D.C.: 2000, Basic principle of solar acoustic holography. Solar Phys. 192, 261.

  23. Scherrer, P.H., Bogart, R.S., Bush, R.I., Hoeksema, J.T., Kosovichev, A.G., Schou, J., Rosenberg, W., Springer, L., Tarbell, T.D., Title, A., Wolfson, C.J., Zayer, I., MDI Engineering Team: 1995, The solar oscillations investigation – Michelson Doppler imager. Solar Phys. 162, 129.

  24. Spruit, H.C., Bogdan, T.J.: 1992, The conversion of p-modes to slow modes and the absorption of acoustic waves by sunspots. Astrophys. J. 391, L109.

  25. Thompson, M.J., Zharkov, S.: 2008, Recent developments in local helioseismology. Solar Phys., 251, 225. doi:10.1007/s11207-008-9143-6.

  26. Werne, J., Birch, A., Julien, K.: 2004, The need for control experiments in local helioseismology. In: Dansey, D. (ed.) Proceedings of the SOHO 14/GONG 2004 Workshop (ESA SP-559) Helio- and Asteroseismology: Towards a Golden Future SP-559, ESA, Noordwijk, 172.

  27. Zhao, J., Kosovichev, A.: 2003, Helioseismic observation of the structure and dynamics of a rotating sunspot beneath the solar surface. Astrophys. J. 591, 446.

Download references

Author information

Correspondence to A. C. Birch.

Rights and permissions

Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (, which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

Reprints and Permissions

About this article

Cite this article

Birch, A.C., Braun, D.C., Hanasoge, S.M. et al. Surface-Focused Seismic Holography of Sunspots: II. Expectations from Numerical Simulations Using Sound-Speed Perturbations. Sol Phys 254, 17–27 (2009).

Download citation


  • Helioseismology, forward modeling
  • Helioseismology, theory
  • Waves, propagation