The adjoint method applied to time-distance helioseismology

Shravan M. Hanasoge, Aaron Birch, Laurent Gizon, Jeroen Tromp

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34 Scopus citations


For a given misfit function, a specified optimality measure of a model, its gradient describes the manner in which one may alter properties of the system to march toward a stationary point. The adjoint method, arising from partial-differential-equation-constrained optimization, describes a means of extracting derivatives of a misfit function with respect to model parameters through finite computation. It relies on the accurate calculation of wavefields that are driven by two types of sources, namely, the average wave-excitation spectrum, resulting in the forward wavefield, and differences between predictions and observations, resulting in an adjoint wavefield. All sensitivity kernels relevant to a given measurement emerge directly from the evaluation of an interaction integral involving these wavefields. The technique facilitates computation of sensitivity kernels (Fréchet derivatives) relative to three-dimensional heterogeneous background models, thereby paving the way for nonlinear iterative inversions. An algorithm to perform such inversions using as many observations as desired is discussed.

Original languageEnglish (US)
Article number100
JournalAstrophysical Journal
Issue number1
StatePublished - Sep 1 2011

All Science Journal Classification (ASJC) codes

  • Astronomy and Astrophysics
  • Space and Planetary Science


  • Sun: dynamo
  • Sun: helioseismology
  • Sun: interior
  • Sun: oscillations
  • magnetohydrodynamics (MHD)
  • waves


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