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Recent Developments in Helioseismic Analysis Methods and Solar Data Assimilation

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Abstract

We review recent advances and results in enhancing and developing helioseismic analysis methods and in solar data assimilation. In the first part of this paper we will focus on selected developments in time-distance and global helioseismology. In the second part, we review the application of data assimilation methods on solar data. Relating solar surface observations as well as helioseismic proxies with solar dynamo models by means of the techniques from data assimilation is a promising new approach to explore and to predict the magnetic activity cycle of the Sun.

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References

  • L. Alvan, A.S. Brun, S. Mathis, Theoretical seismology in 3D: nonlinear simulations of internal gravity waves in solar-like stars. Astron. Astrophys. 565, 42 (2014)

    Article  ADS  Google Scholar 

  • E.R. Anderson, T.L. Duvall Jr., S.M. Jeffries, Modelling of solar oscillation power spectra. Astrophys. J. 364, 699–705 (1990)

    Article  ADS  Google Scholar 

  • H.W. Babcock, The topology of the Sun’s magnetic field and the 22-year cycle. Astrophys. J. 133, 572 (1961)

    Article  ADS  Google Scholar 

  • E. Bélanger, A. Vincent, P. Charbonneau, Predicting solar flares by data assimilation in avalanche models. I. Model design and validation. Sol. Phys. 245, 141–165 (2007)

    Article  ADS  Google Scholar 

  • A. Birch, T.L. Duvall Jr., L. Gizon, J. Jackiewicz, Helioseismology of the “average” supergranule. Bull. Am. Astron. Soc. 38, 224 (2006)

    ADS  Google Scholar 

  • A. Birch, D.C. Braun, Y. Fan, An estimate of the detectability of rising flux tubes. Astrophys. J. Lett. 723, L190–L194 (2010)

    Article  ADS  Google Scholar 

  • M. Bocquet, in M2 OACOS/WAPE/ParisTech lecture notes (2015), pp. 2014–2015. http://cerea.enpc.fr/HomePages/bocquet/teaching.html

  • D.C. Braun, T.L. Duvall Jr., B.J. LaBonte, Acoustic absorption by sunspots. Astrophys. J. Lett. 319, L27–L31 (1987)

    Article  ADS  Google Scholar 

  • D.C. Braun, Comment on “Detection of emerging sunspot regions in the solar interior”. Science 336, 296 (2012)

    Article  ADS  Google Scholar 

  • A.S. Brun, Towards using modern data assimilation and weather forecasting methods in solar physics. Astron. Nachr. 328, 329–338 (2007)

    Article  ADS  Google Scholar 

  • A.S. Brun, M.S. Miesch, J. Toomre, Modeling the dynamical coupling of solar convection with the radiative interior. Astrophys. J. 742, 79 (2011)

    Article  ADS  Google Scholar 

  • A.S. Brun, M.K. Browning, M. Dikpati, H. Hotta, A. Strugarek, Recent advances on solar global magnetism and variability. Space Sci. Rev. (2013). doi:10.1007/s11214-013-0028-0

    Google Scholar 

  • P. Charbonneau, Dynamo models of the solar cycle. Living Rev. Sol. Phys. 7, 3 (2010)

    Article  ADS  Google Scholar 

  • P. Chatterjee, H.M. Antia, Solar flows and their effect on frequencies of acoustic modes. Astrophys. J. 707, 208–217 (2009)

    Article  ADS  Google Scholar 

  • M.C.M. Cheung, M. Rempel, A.M. Title, M. Schüssler, Simulation of the formation of a solar active region. Astrophys. J. 720, 233–244 (2010)

    Article  ADS  Google Scholar 

  • M.C.M. Cheung, M.L. DeRosa, A method for data-driven simulations of evolving solar active regions. Astrophys. J. 757, 147 (2012)

    Article  ADS  Google Scholar 

  • A.R. Choudhuri, P. Chatterjee, J. Jiang, Predicting solar cycle 24 with a solar dynamo model. Phys. Rev. Lett. 98, 13 (2007)

    Article  Google Scholar 

  • K. DeGrave, J. Jackiewicz, M. Rempel, Validating time-distance helioseismology with realistic quiet-Sun simulations. Astrophys. J. 127, 15 (2014)

    Google Scholar 

  • M.L. DeRosa, A.S. Brun, J.T. Hoeksema, Solar magnetic field reversals and the role of dynamo families. Astrophys. J. 757, 96 (2012)

    Article  ADS  Google Scholar 

  • M. Dikpati, P. Charbonneau, A Babcock-Leighton flux transport dynamo with solar-like differential rotation. Astrophys. J. 518, 508–520 (1999)

    Article  ADS  Google Scholar 

  • M. Dikpati, P.A. Gilman, Simulating and predicting solar cycles using a flux-transport dynamo. Astrophys. J. 649, 498–514 (2006)

    Article  ADS  Google Scholar 

  • M. Dikpati, J.L. Anderson, Evaluating potential for data assimilation in a flux-transport dynamo model by assessing sensitivity and response to meridional flow variation. Astrophys. J. 756, 20 (2012)

    Article  ADS  Google Scholar 

  • M. Dikpati, J.L. Anderson, D. Mitra, Ensemble Kalman filter data assimilation in a Babcock-Leighton solar dynamo model: an observation system simulation experiment for reconstructing meridional flow-speed. Geophys. Res. Lett. 41, 5361 (2014)

    Article  ADS  Google Scholar 

  • T.L. Duvall Jr., A dispersion law for solar oscillations. Nature 300, 242–243 (1982)

    Article  ADS  Google Scholar 

  • T.L. Duvall Jr., S.M. Jefferies, J.W. Harvey, M.A. Pomerantz, Time-Distance Helioseismology. Nature 362, 430–432 (1993)

    Article  ADS  Google Scholar 

  • T.L. Duvall Jr., A.C. Birch, L. Gizon, Direct measurement of travel-time kernels for helioseismology. Astrophys. J. 646, 553–559 (2006)

    Article  ADS  Google Scholar 

  • T.L. Duvall Jr., A.C. Birch, The vertical component of the supergranular motion. Astrophys. J. 725, L47–L51 (2010)

    Article  ADS  Google Scholar 

  • T.L. Duvall Jr., S.M. Hanasoge, Subsurface supergranular vertical flows as measured using large distance separations in time-distance helioseismology. Sol. Phys. 287, 71–83 (2013)

    Article  ADS  Google Scholar 

  • T. Felipe, D. Braun, A. Crouch, A. Birch, Scattering of the f-mode by small magnetic flux elements from observations and numerical simulations. Astrophys. J. 757, 148–160 (2012)

    Article  ADS  Google Scholar 

  • A. Fournier, G. Hulot, D. Jault, W. Kuang, A. Tangborn, N. Gillet, E. Canet, J. Aubert, F. Lhuillier, An introduction to data assimilation and predictability in geomagnetism. Space Sci. Rev. 155, 247–291 (2010)

    Article  ADS  Google Scholar 

  • L. Gizon, A.C. Birch, Time-distance helioseismology: noise estimation. Astrophys. J. 614, 472–489 (2004)

    Article  ADS  Google Scholar 

  • L. Gizon, H. Schunker, C.S. Baldner, S. Basu, A.C. Birch, R.S. Bogart, D.C. Braun, R. Cameron, T.L. Duvall Jr., S.M. Hanasoge, J. Jackiewicz, M. Roth, T. Stahn, M.J. Thompson, S. Zharkov, Helioseismology of sunspots: a case study of NOAA region 9787. Space Sci. Rev. 114, 249–273 (2009)

    Article  ADS  Google Scholar 

  • R. Giering, T. Kaminski, Recipes for adjoint code construction. ACM Trans. Math. Softw. 24, 437–474 (1998)

    Article  MATH  Google Scholar 

  • D. Gough, B.W. Hindman, Helioseismic detection of deep meridional flow. Astrophys. J. 714, 960–970 (2010)

    Article  ADS  Google Scholar 

  • D.H. Hathaway, Solar cycle forecasting. Space Sci. Rev. 144, 401–412 (2009)

    Article  ADS  Google Scholar 

  • F. Hill, Rings and trumpets – three-dimensional power spectra of solar oscillations. Astrophys. J. 333, 996–1013 (1988)

    Article  ADS  Google Scholar 

  • F. Hill, P.B. Stark, R.T. Stebbins, E.R. Anderson, H.M. Antia, T.M. Brown, T.L. Duvall Jr., D.A. Haber, J.W. Harvey, D.H. Hathaway, R. Howe, R. Hubbard, H.P. Jones, J.R. Kennedy, S.G. Korzennik, A. Kosovichev, J.W. Leibacher, K.G. Libbrecht, J.A. Pintar, E.J. Rhodes Jr., J. Schou, M.J. Thompson, S. Tomczyk, C.G. Toner, R. Toussaint, W.E. Williams, The solar acoustic spectrum and eigenmode parameters. Science 272, 1292–1295 (1996)

    Article  ADS  Google Scholar 

  • J. Hirzberger, L. Gizon, S.K. Solanki, T.L. Duvall Jr., Structure and evolution of supergranulation from local helioseismology. Sol. Phys. 251, 417–437 (2008)

    Article  ADS  Google Scholar 

  • C.P. Hung, L. Jouve, A.S. Brun, A. Fournier, O. Talagrand, Estimating the deep solar meridional circulation using magnetic observations and a dynamo model: a variational approach, Astrophys. J. (2015) (to appear)

  • S. Ilonidis, J. Zhao, A. Kosovichev, Detection of emerging sunspot regions in the solar interior. Science 333, 993–996 (2011)

    Article  ADS  Google Scholar 

  • S. Ilonidis, J. Zhao, T. Hartlep, Helioseismic investigation of emerging magnetic flux in the solar convection zone. Astrophys. J. 777, 138 (2013)

    Article  ADS  Google Scholar 

  • J. Jackiewicz, L. Gizon, A.C. Birch, High-resolution mapping of flows in the solar interior: fully consistent OLA inversion of helioseismic travel times. Sol. Phys. 251, 381–415 (2008)

    Article  ADS  Google Scholar 

  • S.M. Jefferies, S.V. Vorontsov, C. Giebink, Toward improving the seismic visibility of the solar tachocline, in Proc. SOHO 18/GONG 2006/HELAS I: Beyond the Spherical Sun, ed. by K. Fletcher, M. Thompson (ESA, Noordwijk, 2006), ESA SP-624

    Google Scholar 

  • L. Jouve, A.S. Brun, On the role of meridional flows in flux transport dynamo models. Astron. Astrophys. 474, 239–250 (2007)

    Article  ADS  Google Scholar 

  • L. Jouve, A.S. Brun, O. Talagrand, Assimilating data into an \(\alpha \varOmega\) dynamo model of the Sun: a variational approach. Astrophys. J. 735, 31 (2011)

    Article  ADS  Google Scholar 

  • E. Kalnay, Atmospheric Modeling, Data Assimilation and Predictability (Cambridge University Press, Cambridge, 2003)

    Google Scholar 

  • P.J. Käpylä, M.J. Mantere, A. Brandenburg, Cyclic magnetic activity due to turbulent convection in spherical wedge geometry. Astrophys. J. Lett. 755, L22 (2012)

    Article  ADS  Google Scholar 

  • I. Kitiashvili, A.G. Kosovichev, Application of data assimilation method for predicting solar cycles. Astrophys. J. Lett. 688, L49 (2008)

    Article  ADS  Google Scholar 

  • S.G. Korzennik, A mode-fitting methodology optimized for very long helioseismic time series. Astrophys. J. 626, 585–615 (2005)

    Article  ADS  Google Scholar 

  • S.G. Korzennik, M.C. Rabello-Soares, J. Schou, T.P. Larson, Accurate characterization of high-degree modes using MDI observations. Astrophys. J. 772, 87 (2013), 28 pp

    Article  ADS  Google Scholar 

  • T.P. Larson, J. Schou, Improvements in global mode analysis. J. Phys. Conf. Ser. 118, 012083 (2008)

    Article  ADS  Google Scholar 

  • T.P. Larson, J. Schou, Variations in global mode analysis, in Solar-Stellar Dynamos as Revealed by Helio- and Asteroseismology, ed. by M. Dikpati, T. Arentoft, I. Gonzalez Hernandez, C. Lindsey, F. Hill. Astron. Soc. Pacific Conf. Series, vol. 416 (2009), pp. 311–314

    Google Scholar 

  • T.P. Larson, J. Schou, HMI global helioseismology data analysis pipeline. J. Phys. Conf. Ser. 271, 012062 (2011)

    Article  ADS  Google Scholar 

  • E.M. Lavely, M.H. Ritzwoller, The effect of global-scale, steady-state convection and elastic-gravitational asphericities on helioseismic oscillations. Philos. Trans. R. Soc. Lond. A 339, 431–496 (1992)

    Article  ADS  Google Scholar 

  • R.B. Leighton, A magneto-kinematic model of the solar cycle. Astrophys. J. 156, 1 (1969)

    Article  ADS  Google Scholar 

  • F. Lhuillier, J. Aubert, G. Hulot, Earth’s dynamo limit of predictability controlled by magnetic dissipation. Geophys. J. Int. 186, 492–508 (2011)

    Article  ADS  Google Scholar 

  • C. Lindsey, D.C. Braun, Helioseismic imaging of sunspots at their antipodes. Sol. Phys. 126, 101–115 (1990)

    Article  ADS  Google Scholar 

  • M.S. Miesch, A.S. Brun, M.L. DeRosa, J. Toomre, Structure and evolution of giant cells in global models of solar convection. Astrophys. J. 673, 557 (2008)

    Article  ADS  Google Scholar 

  • H. Moradi, C. Baldner, A.C. Birch, D.C. Braun, R.H. Cameron, T.L. Duvall Jr., L. Gizon, D. Haber, S.M. Hanasoge, B.W. Hindman, J. Jackiewicz, E. Khomenko, R. Komm, P. Rajaguru, M. Rempel, M. Roth, R. Schlichenmaier, H.J. Schunker, H.C. Spruit, K.G. Strassmeier, M.J. Thompson, S. Zharkov, Modeling the subsurface structure of sunspots. Sol. Phys. 267, 1–62 (2010)

    Article  ADS  Google Scholar 

  • N.J. Nelson, B.P. Brown, A.S. Brun, M.S. Miesch, J. Toomre, Magnetic wreaths and cycles in convective dynamos. Astrophys. J. 762, 73 (2013)

    Article  ADS  Google Scholar 

  • P. Petit, B. Dintrans, S.K. Solanki, M. Aurière, F. Lignières, J. Morin, F. Paletou, J. Ramirez, C. Catala, R. Fares, Toroidal vs. poloidal magnetic fields in sun-like stars: a rotation threshold. Mon. Not. R. Astron. Soc. 388, 80–88 (2008)

    Article  ADS  Google Scholar 

  • M.C. Rabello-Soares, S.G. Korzennik, J. Schou, Analysis of MDI high-degree mode frequencies and their rotational splittings. Sol. Phys. 251, 197 (2008)

    Article  ADS  Google Scholar 

  • J. Reiter, E.J. Rhodes Jr., A.G. Kosovichev, J. Schou, P.H. Scherer, T.P. Larson, A method for the estimation of p-mode parameters from averaged solar oscillation power spectra. Astrophys. J. 803, 92 (2015), 42 pp

    Article  ADS  Google Scholar 

  • M. Rempel, M.C.M. Cheung, Numerical simulations of active region scale flux emergence: from spot formation to decay. Astrophys. J. 785, 90 (2014)

    Article  ADS  Google Scholar 

  • E.J. Rhodes Jr., J. Reiter, J. Schou, T. Larson, P. Scherrer, J. Brooks, P. McFaddin, B. Miller, J. Rodriguez, J. Yoo, Temporal changes in the frequencies and widths of the solar p-mode oscillations. J. Phys. Conf. Ser. 271, 012029 (2011)

    Article  ADS  Google Scholar 

  • M. Roth, M. Stix, Meridional circulation and global solar oscillations. Sol. Phys. 251, 77–89 (2008)

    Article  ADS  Google Scholar 

  • S. Sanchez, A. Fournier, J. Aubert, The predictability of advection-dominated flux-transport solar dynamo models. Astrophys. J. 781, 8 (2014)

    Article  ADS  Google Scholar 

  • A. Schad, J. Timmer, M. Roth, A unified approach to the helioseismic inversion problem of the solar meridional flow from global oscillations. Astrophys. J. 734, 97–105 (2011)

    Article  ADS  Google Scholar 

  • A. Schad, J. Timmer, M. Roth, Measuring the solar meridional flow from perturbations of eigenfunctions of global oscillation. Astron. Nachr. 333, 991–994 (2012)

    Article  ADS  Google Scholar 

  • A. Schad, J. Timmer, M. Roth, Global helioseismic evidence for a deeply penetrating meridional flow consisting of multiple flow cells. Astrophys. J. Lett. 778, L38–L44 (2013)

    Article  ADS  Google Scholar 

  • A. Schad, A new approach for the global helioseismic investigation of the solar meridional flow. Doctoral dissertation, Albert-Ludwig, University of Freiburg, Germany (2013)

  • J. Schou, On the analysis of helioseismic data. Doctoral Dissertation, Aarhus University, Denmark (1992)

  • J. Schou, Observations of medium- and high-degree modes: methods and sandtraps, in Structure and Dynamics of the Interior of the Sun and Sun-Like Stars, ed. by S.G. Korzennik, A. Wilson (ESA, Noordwijk, (1998), pp. 47–52, ESA SP-418

    Google Scholar 

  • J. Schou, P.H. Scherrer, R.I. Bush, R. Wachter, S. Couvidat, M.C. Rabello-Soares, R.S. Bogart, J.T. Hoeksema, Y. Liu, T.L. Duvall Jr., D.J. Akin, B.A. Allard, J.W. Miles, R. Rairden, R.A. Shine, T.D. Tarbell, A.M. Title, C.J. Wolfson, D.F. Elmore, A.A. Norton, S. Tomczyk, Design and ground calibration of the Helioseismic and Magnetic Imager (HMI) instrument on the Solar Dynamics Observatory (SDO). Sol. Phys. 275, 229–259 (2012)

    Article  ADS  Google Scholar 

  • C.J. Schrijver, M.L. DeRosa, Photospheric and heliospheric magnetic fields. Sol. Phys. 212, 165–200 (2003)

    Article  ADS  Google Scholar 

  • H.C. Spruit, Origin of the torsional oscillation pattern of solar rotation. Sol. Phys. 213, 1–21 (2003)

    Article  ADS  Google Scholar 

  • R.F. Stein, Å. Nordlund, On the formation of active regions. Astrophys. J. Lett. 753, L13 (2012)

    Article  ADS  Google Scholar 

  • A. Strugarek, P. Charbonneau, Predictive capabilities of avalanche models for solar flares. Sol. Phys. 289, 4137 (2014)

    Article  ADS  Google Scholar 

  • M. Švanda, L. Gizon, S.M. Hanasoge, S.D. Ustyugov, Validated helioseismic inversions for 3D vector flows. Astron. Astrophys. 530, 148 (2011)

    Article  Google Scholar 

  • M. Švanda, Inversions for average supergranular flows using finite-frequency kernels. Astrophys. J. Lett. 759, 29–33 (2012)

    Article  ADS  Google Scholar 

  • A. Svedin, M.C. Cuéllar, A. Brandenburg, Data assimilation for stratified convection. Mon. Not. R. Astron. Soc. 433, 2278 (2013)

    Article  ADS  Google Scholar 

  • O. Talagrand, The use of adjoint equations in numerical modeling of the atmospheric circulation, in Automatic Differentiation of Algorithms: Theory, Implementation, and Application, ed. by A. Griewank, G.F. Corliss (Society for Industrial and Applied Mathematics, Philadelphia, 1991), pp. 169–180

    Google Scholar 

  • S.V. Vorontsov, J. Christensen-Dalsgaard, J. Schou, V.N. Strakhov, M.J. Thompson, Solar internal rotation as seen from SOHO MDI data, in From Solar Min to Max: Half a Solar Cycle with SOHO, ed. by A. Wilson (2002a), pp. 111–114, ESA SP-508

    Google Scholar 

  • S.V. Vorontsov, J. Christensen-Dalsgaard, J. Schou, V.N. Strakhov, M.J. Thompson, Helioseismic measurement of solar torsional oscillations. Science 296, 101–103 (2002b)

    Article  ADS  Google Scholar 

  • S.V. Vorontsov, S.M. Jefferies, Modeling solar oscillation power spectra. I. Adaptive response function for Doppler velocity measurements. Astrophys. J. 623, 1202–1214 (2005)

    Article  ADS  Google Scholar 

  • S.V. Vorontsov, Solar p modes of high degree l: Coupling by differential rotation. Mon. Not. R. Astron. Soc. 378, 1499–1506 (2007)

    Article  ADS  Google Scholar 

  • S.V. Vorontsov, S.M. Jefferies, C. Giebink, J. Schou, Toward eliminating systematic errors in intermediate-degree p-mode measurements, in Solar-Stellar Dynamos as Revealed by Helio- and Asteroseismology, ed. by M. Dikpati, T. Arentoft, I. Gonzalez Hernandez, C. Lindsey, F. Hill. Astron. Soc. Pacific Conf. Series, vol. 416 (2009), pp. 301–305

    Google Scholar 

  • S.V. Vorontsov, Effects of differential rotation and meridional circulation in solar oscillations of high degree \(l\). Mon. Not. R. Astron. Soc. 418, 1146–1155 (2011)

    Article  ADS  Google Scholar 

  • S.V. Vorontsov, V.A. Baturin, S.V. Ayukov, V.K. Gryaznov, Helioseismic calibration of the equation of state and chemical composition in the solar convective envelope. Mon. Not. R. Astron. Soc. 430, 1636–1652 (2013a)

    Article  ADS  Google Scholar 

  • S.V. Vorontsov, S.M. Jefferies, Modeling solar oscillation power spectra. II. Parametric model of spectral lines observed in Doppler-velocity measurements. Astrophys. J. 778, 75 (2013b) 10 pp

    Article  ADS  Google Scholar 

  • J. Warnecke, P.J. Käpylä, M.J. Käpylä, A. Brandenburg, On the cause of solar-like equatorward migration in global convective dynamo simulations. Astrophys. J. Lett. 796, L12 (2014)

    Article  ADS  Google Scholar 

  • M.F. Woodard, Effect of subsurface inhomogeneities on the statistics of solar oscillation power spectra. Sol. Phys. 180, 19–28 (1998)

    Article  ADS  Google Scholar 

  • M.F. Woodard, Theoretical signature of solar meridional flow in global seismic data. Sol. Phys. 197, 11–20 (2000)

    Article  ADS  Google Scholar 

  • M.F. Woodard, J. Schou, A.C. Birch, T.P. Larson, Global-oscillation eigenfunction measurements of solar meridional flow. Sol. Phys. 287, 129–147 (2013)

    Article  ADS  Google Scholar 

  • A.R. Yeates, D. Nandy, D.H. Mackay, Exploring the physical basis of solar cycle predictions: flux transport dynamics and persistence of memory in advection versus diffusion-dominated solar convection zones. Astrophys. J. 673, 544–556 (2008)

    Article  ADS  Google Scholar 

  • J. Zhao, A.G. Kosovichev, On the inference of supergranular flows by time-distance helioseismology, in GONG+ 2002. Local and Global Helioseismology: The Present and Future, ed. by H. Sawaya-Lacoste, (ESA, Noordwijk 2003), pp. 417–420, ESA SP-517

    Google Scholar 

  • J. Zhao, K. Nagashima, R.S. Bogart, A.G. Kosovichev, T.L. Duvall Jr., Systematic center-to-limb variation in measured helioseismic travel times and its effect on inferences of solar interior meridional flows. Astrophys. J. Lett. 749, 5 (2012)

    Article  ADS  Google Scholar 

  • J. Zhao, R.S. Bogart, A.G. Kosovichev, T.L. Duvall Jr., T. Hartlep, Detection of equatorward meridional flow and evidence of double-cell meridional circulation. Astrophys. J. Lett. 774, 29 (2013)

    Article  ADS  Google Scholar 

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Acknowledgements

LJ would like to thank Sacha Brun for helpful comments on the data assimilation part in this paper. MR and AS have received funding from the European Research Council under the European Union’s Seventh Framework Program (FP/2007-2013)/ERC Grant Agreement no. 307117. The authors thank M.F. Woodard for providing Fig. 11.

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Schad, A., Jouve, L., Duvall, T.L. et al. Recent Developments in Helioseismic Analysis Methods and Solar Data Assimilation. Space Sci Rev 196, 221–249 (2015). https://doi.org/10.1007/s11214-015-0199-y

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