Skip to main content
SpringerLink
Log in
Book cover

Imaging Convection and Magnetism in the Sun pp 75–103Cite as

Full Waveform Inversion

  • Chapter
  • First Online:

  • 598 Accesses

Part of the book series: SpringerBriefs in Mathematics ((BRIEFSMATH))

Abstract

Inferring interior properties of the Sun from photospheric measurements of the seismic wavefield constitutes the helioseismic inverse problem. Deviations in seismic measurements (such as wave travel times) from their fiducial values estimated for a given model of the solar interior imply that the model is inaccurate. In this section, we implement iterative inversions to obtain the sub-surface structure of perturbations. The model can be successively improved using either steepest descent or Krylov-subspace techniques such as conjugate gradient, or limited-memory Broyden-Fletcher-Goldfarb-Shanno method.

The material for this section is primarily taken from Hanasoge and Tromp (2014) and Hanasoge (2014a).

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  • Beck JG, Duvall TL Jr (2001) Time-distance study of supergranulation. In: Proceedings of the SOHO 10/GONG 2000 Workshop: Helio- and asteroseismology at the dawn of the millennium, 2-6 October 2000, Santa Cruz de Tenerife, Tenerife, Spain. Edited by A. Wilson, Scientific coordination by P. L. Pallé. ESA SP-464, Noordwijk: ESA Publications Division, 2001, p 577–581. http://adsabs.harvard.edu/abs/2001ESASP.464..577B

  • Birch A, Duvall TL, Hanasoge S (2007) Local helioseismology of supergranulation. In: American astronomical society meeting abstracts #210. Bulletin of the American Astronomical Society, vol 39, p 160

    Google Scholar 

  • Birch AC, Gizon L (2007) Linear sensitivity of helioseismic travel times to local flows. Astron Nachr 328:228. DOI 10.1002/asna.200610724

    Article  MATH  Google Scholar 

  • Birch AC, Kosovichev AG, Price GH, Schlottmann RB (2001) The accuracy of the born and ray approximations in time-distance helioseismology. Astrophys J Lett 561:L229–L232. DOI 10.1086/324766

    Article  Google Scholar 

  • Birch AC, Kosovichev AG, Duvall TL 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

    Article  Google Scholar 

  • Bozdaǧ E, Trampert J, Tromp J (2011) Misfit functions for full waveform inversion based on instantaneous phase and envelope measurements. Geophys J Int 185:845–870. DOI 10.1111/j.1365-246X.2011.04970.x

    Article  Google Scholar 

  • Braun DC, Birch AC (2008) Prospects for the detection of the deep solar meridional circulation. Astrophys J Lett 689:L161–L165. DOI 10.1086/ 595884. 0810.0284

    Article  Google Scholar 

  • Cameron R, Gizon L, Duvall TL Jr (2008) Helioseismology of sunspots: confronting observations with three-dimensional MHD simulations of wave propagation. Sol Phys 251:291–308. DOI 10.1007/s11207-008-9148-1. 0802.1603

    Article  Google Scholar 

  • Candès EJ, Romberg J, Tao T (2006) Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans Inf Theory 52(2):489–509

    Article  MathSciNet  MATH  Google Scholar 

  • Christensen-Dalsgaard J, Dappen W, Ajukov SV, Anderson ER, Antia HM, Basu S, Baturin VA, Berthomieu G, Chaboyer B, Chitre SM, Cox AN, Demarque P, Donatowicz J, Dziembowski WA, Gabriel M, Gough DO, Guenther DB, Guzik JA, Harvey JW, Hill F, Houdek G, Iglesias CA, Kosovichev AG, Leibacher JW, Morel P, Proffitt CR, Provost J, Reiter J, Rhodes EJ Jr, Rogers FJ, Roxburgh IW, Thompson MJ, Ulrich RK (1996) The current state of solar modeling. Science 272:1286

    Article  Google Scholar 

  • Dahlen FA, Baig AM (2002) Fréchet kernels for body-wave amplitudes. Geophys J Int 150:440–466. DOI 10.1046/j.1365-246X.2002.01718.x

    Article  Google Scholar 

  • DeGrave K, Jackiewicz J, Rempel M (2014) Validating time-distance helioseismology with realistic quiet-sun simulations. Astrophys J 788:127. DOI 10.1088/0004-637X/788/2/127. 1404.4645

    Article  Google Scholar 

  • Dombroski DE, Birch AC, Braun DC, Hanasoge SM (2013) Testing helioseismic-holography inversions for supergranular flows using synthetic data. Sol Phys 282:361–378. DOI 10.1007/s11207-012-0189-0. 1211.6886

    Article  Google Scholar 

  • Duvall TL, Hanasoge SM (2013) Subsurface supergranular vertical flows as measured using large distance separations in time-distance helioseismology. Sol Phys 287:71–83. DOI 10.1007/s11207-012-0010-0. 1207.6075

    Article  Google Scholar 

  • Duvall TL, Hanasoge SM, Chakraborty S (2014) Additional evidence supporting a model of shallow, high-speed supergranulation. Sol Phys 289:3421–3433. DOI 10.1007/s11207-014-0537-3. 1404.2533

    Article  Google Scholar 

  • Duvall TL Jr (2003) Nonaxisymmetric variations deep in the convection zone. In: Proceedings of SOHO 12/GONG+ 2002. Local and global helioseismology: the present and future, 27 October-1 November 2002, Big Bear Lake, CA, USA. Edited by H. Sawaya-Lacoste, ESA SP-517, Noordwijk, Netherlands: ESA Publications Division, 2003, p 259–262. http://adsabs.harvard.edu/abs/2003ESASP.517..259D

  • Duvall TL Jr, Gizon L (2000) Time-distance helioseismology with f modes as a method for measurement of near-surface flows. Sol Phys 192:177–191. DOI 10.1023/A:1005239503637

    Article  Google Scholar 

  • Duvall TL Jr, D’Silva S, Jefferies SM, Harvey JW, Schou J (1996) Downflows under sunspots detected by helioseismic tomography. Nature 379:235. DOI 10.1038/379235a0

    Article  Google Scholar 

  • Fichtner A, Kennett BLN, Igel H, Bunge HP (2009) Full seismic waveform tomography for upper-mantle structure in the Australasian region using adjoint methods. Geophys J Int 179:1703–1725. DOI 10.1111/j.1365-246X. 2009.04368.x

    Article  Google Scholar 

  • Giles MB, Pierce NA (2000) An introduction to the adjoint approach to design. Flow Turbul Combust 65:393–415

    Article  MATH  Google Scholar 

  • Giles PM (2000) Time-distance measurements of large-scale flows in the solar convection zone. Ph.D. thesis, Stanford University, CA, USA

    Google Scholar 

  • Giles PM, Duvall TL Jr, Scherrer PH, Bogart RS (1997) A flow of material from the suns equator to its poles. Nature 390:52

    Article  Google Scholar 

  • Gizon L (2003) Probing flows in the upper solar convection zone. Ph.D. thesis, Stanford University, CA, USA

    Google Scholar 

  • Gizon L (2004) Helioseismology of time-varying flows through the solar cycle. Sol Phys 224:217–228. DOI 10.1007/s11207-005-4983-9

    Article  Google Scholar 

  • Gizon L, Birch AC (2002) Time-distance helioseismology: the forward problem for random distributed sources. Astrophys J 571:966–986. DOI 10. 1086/340015

    Google Scholar 

  • Gizon L, Duvall TL, Schou J (2003) Wave-like properties of solar supergranulation. Nature 421:43–44. arXiv:astro-ph/0208343

    Google Scholar 

  • Gizon L, Schunker H, Baldner CS, Basu S, Birch AC, Bogart RS, Braun DC, Cameron R, Duvall TL, Hanasoge SM, Jackiewicz J, Roth M, Stahn T, Thompson MJ, 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

    Article  Google Scholar 

  • Gizon L, Birch AC, Spruit HC (2010) Local helioseismology: three-dimensional imaging of the solar interior. Annu Rev Astron Astrophys 48:289–338. DOI 10.1146/annurev-astro-082708-101722. 1001.0930

    Article  Google Scholar 

  • Hanasoge S, Birch A, Gizon L, Tromp J (2012a) Seismic probes of solar interior magnetic structure. Phys Rev Lett 109(10):101101. DOI 10.1103/ PhysRevLett.109.101101. 1207.4352

    Article  Google Scholar 

  • Hanasoge SM (2007) Theoretical studies of wave interactions in the sun. Ph.D. thesis, Stanford University, California, USA

    Google Scholar 

  • Hanasoge SM (2014a) Full waveform inversion of solar interior flows. Astrophys J 797(23)

    Google Scholar 

  • Hanasoge SM (2014b) Measurements and kernels for source-structure inversions in noise tomography. Geophys J Int 196:971–985. DOI 10.1093/ gji/ggt411. 1310.0857

    Article  Google Scholar 

  • Hanasoge SM, Tromp J (2014) Full waveform inversion for time-distance helioseismology. Astrophys J 784(69):69. DOI 10.1088/0004-637X/784/1/ 69. 1401.7603

    Article  Google Scholar 

  • Hanasoge SM, Couvidat S, Rajaguru SP, Birch AC (2008) Impact of locally suppressed wave sources on helioseismic traveltimes. Mon Not R Astron Soc 391:1931–1939. DOI 10.1111/j.1365-2966.2008.14013.x. 0707.1369

    Article  Google Scholar 

  • Hanasoge SM, Duvall TL, DeRosa ML (2010a) Seismic constraints on interior solar convection. Astrophys J Lett 712:L98–L102. DOI 10.1088/2041-8205/ 712/1/L98. 1001.4508

    Article  Google Scholar 

  • Hanasoge SM, Birch A, Gizon L, Tromp J (2011) The adjoint method applied to time-distance helioseismology. Astrophys J 738:100. DOI 10. 1088/0004-637X/738/1/100. 1105.4263

    Google Scholar 

  • Hanasoge SM, Duvall TL Jr, Sreenivasan KR (2012b) Anomalously weak solar convection. Proc Natl Acad Sci 109(30):11,928–11,932

    Google Scholar 

  • Hirzberger J, Gizon L, Solanki SK, Duvall TL (2008) Structure and evolution of supergranulation from local helioseismology. Sol Phys 251:417–437. DOI 10.1007/s11207-008-9206-8

    Article  Google Scholar 

  • Jackiewicz J, Gizon L, Birch AC, Duvall TL Jr (2007) Time-distance helioseismology: sensitivity of f-mode travel times to flows. Astrophys J 671:1051–1064. DOI 10.1086/522914. 0708.3554

    Article  Google Scholar 

  • Jackiewicz J, Birch AC, Gizon L, Hanasoge SM, Hohage T, Ruffio JB, Švanda M (2012) Multichannel three-dimensional SOLA inversion for local helioseismology. Sol Phys 276:19–33. DOI 10.1007/s11207-011-9873-8. 1109.2712

    Article  Google Scholar 

  • Jain K, Komm RW, González Hernández I, Tripathy SC, Hill F (2012) Subsurface flows in and around active regions with rotating and non-rotating sunspots. Sol Phys 279:349–363. DOI 10.1007/ s11207-012-0039-0. 1205.2356

    Article  Google Scholar 

  • Khan A, Boschi L, Connolly J (2009) On mantle chemical and thermal heterogeneities and anisotropy as mapped by inversion of global surface wave data. J Geophys Res Solid Earth (1978–2012) 114(B9). DOI 10.1029/ 2009JB006399

    Google Scholar 

  • Komm R, González Hernández I, Hill F, Bogart R, Rabello-Soares MC, Haber D (2013) Subsurface meridional flow from HMI using the ring-diagram pipeline. Sol Phys 287:85–106. DOI 10.1007/s11207-012-0073-y

    Article  Google Scholar 

  • Kosovichev AG, Duvall TLJ, Scherrer PH (2000) Time-distance inversion methods and results - (invited review). Sol Phys 192:159–176

    Article  Google Scholar 

  • Luo Y, Modrak R, Tromp J (2013) Strategies in adjoint tomography. In: Freeden W, Nashed MZ, Sonar T (eds) Handbook of geomathematics, 2nd edn. Springer, Berlin

    Google Scholar 

  • Rickers F, Fichtner A, Trampert J (2013) The Iceland–Jan Mayen plume system and its impact on mantle dynamics in the North Atlantic region: evidence from full-waveform inversion. Earth Planet Sci Lett 367:39–51

    Article  Google Scholar 

  • Schou J, Antia HM, Basu S, Bogart RS, Bush RI, Chitre SM, Christensen-Dalsgaard J, di Mauro MP, Dziembowski WA, Eff-Darwich A, Gough DO, Haber DA, Hoeksema JT, Howe R, Korzennik SG, Kosovichev AG, Larsen RM, Pijpers FP, Scherrer PH, Sekii T, Tarbell TD, Title AM, Thompson MJ, Toomre J (1998) Helioseismic studies of differential rotation in the solar envelope by the solar oscillations investigation using the Michelson Doppler imager. Astrophys J 505: 390–417. DOI 10.1086/306146

    Article  Google Scholar 

  • Schou J, Scherrer PH, Bush RI, Wachter R, Couvidat S, Rabello-Soares MC, Bogart RS, Hoeksema JT, Liu Y, Duvall TL, Akin DJ, Allard BA, Miles JW, Rairden R, Shine RA, Tarbell TD, Title AM, Wolfson CJ, Elmore DF, Norton AA, Tomczyk S (2012) Design and ground calibration of the helioseismic and magnetic imager (HMI) instrument on the solar dynamics observatory (SDO). Sol Phys 275: 229–259. DOI 10.1007/s11207-011-9842-2

    Article  Google Scholar 

  • Schunker H, Braun DC, Cally PS, Lindsey C (2005) The local helioseismology of inclined magnetic fields and the showerglass effect. Astrophys J Lett 621: L149–L152. DOI 10.1086/429290

    Article  Google Scholar 

  • Snieder R (2004) Extracting the Green’s function from the correlation of coda waves: a derivation based on stationary phase. Phys Rev E 69(4):046610. DOI 10.1103/PhysRevE.69.046610

    Article  Google Scholar 

  • Stein RF, Nordlund Å (2000) Realistic solar convection simulations. Sol Phys 192:91–108. DOI 10.1023/A:1005260918443

    Article  Google Scholar 

  • Švanda M (2012) Inversions for average supergranular flows using finite-frequency kernels. Astrophys J Lett 759:L29. DOI 10.1088/2041-8205/759/2/L29. 1209.6147

    Article  Google Scholar 

  • Švanda M, Gizon L, Hanasoge SM, Ustyugov SD (2011) Validated helioseismic inversions for 3D vector flows. Astron Astrophys 530:A148. DOI 10.1051/0004-6361/201016426. 1104.4083

    Article  Google Scholar 

  • Swisdak M, Zweibel E (1999) Effects of large-scale convection on p-mode frequencies. Astrophys J 512:442–453. DOI 10.1086/306764. arXiv:astro-ph/9809135

    Google Scholar 

  • Tape C, Liu Q, Maggi A, Tromp J (2009) Adjoint tomography of the Southern California crust. Science 325(5943):988–992. DOI 10.1126/science.1175298. http://www.sciencemag.org/cgi/content/abstract/325/5943/988. http://www.sciencemag.org/cgi/reprint/325/5943/988.pdf

  • Tromp J, Luo Y, Hanasoge S, Peter D (2010) Noise cross-correlation sensitivity kernels. Geophys J Int 183:791–819, DOI 10.1111/j.1365-246X. 2010.04721.x

    Article  Google Scholar 

  • Woodard M (2014) Detectability of large-scale solar subsurface flows. Sol Phys 289:1085–1100. DOI 10.1007/s11207-013-0386-5

    Article  Google Scholar 

  • Woodard M, Schou J, Birch AC, Larson TP (2013) Global-oscillation eigenfunction measurements of solar meridional flow. Sol Phys 287:129–147. DOI 10.1007/s11207-012-0075-9

    Article  Google Scholar 

  • Woodard MF (2007) Probing supergranular flow in the solar interior. Astrophys J 668:1189–1195. DOI 10.1086/521391

    Article  Google Scholar 

  • Zhao J, Kosovichev AG (2003) On the inference of supergranular flows by time-distance helioseismology. In: Proceedings of SOHO 12/GONG+ 2002. Local and global helioseismology: the present and future, 27 October-1 November 2002, Big Bear Lake, CA, USA. Edited by H. Sawaya-Lacoste, ESA SP-517, Noordwijk, Netherlands: ESA Publications Division, 2003, p 417–420. http://adsabs.harvard.edu/abs/2003ESASP.517..417Z

  • Zhao J, Kosovichev AG, Duvall TL Jr (2001) Investigation of mass flows beneath a sunspot by time-distance helioseismology. Astrophys J 557:384–388. DOI 10.1086/321491

    Article  Google Scholar 

  • Zhao J, Bogart RS, Kosovichev AG, Duvall TL Jr, Hartlep T (2013) Detection of equatorward meridional flow and evidence of double-cell meridional circulation inside the sun. Astrophys J Lett 774:L29. DOI 10. 1088/2041-8205/774/2/L29. 1307.8422

    Google Scholar 

  • Zhu H, Bozdağ E, Duffy TS, Tromp J (2013) Seismic attenuation beneath Europe and the North Atlantic: implications for water in the mantle. Earth Planet Sci Lett 381(0):1–11. Doi http://dx.doi.org/10.1016/j.epsl.2013.08.030. http://www.sciencedirect.com/science/article/pii/S0012821X13004676

Download references

Author information

Authors and Affiliations

  1. Department of Astronomy and Astrophysics, Tata Institute of Fundamental Research, Mumbai, India

    Shravan Hanasoge

Authors
  1. Shravan Hanasoge
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2015 The Author

About this chapter

Cite this chapter

Hanasoge, S. (2015). Full Waveform Inversion. In: Imaging Convection and Magnetism in the Sun. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-27330-3_4

Download citation

Publish with us

Policies and ethics

Search

Navigation