Finite-frequency sensitivity kernels for global seismic wave propagation based upon adjoint methods

Qinya Liu, Jeroen Tromp

Research output: Contribution to journalArticlepeer-review

130 Scopus citations


We determine adjoint equations and Fréchet kernels for global seismic wave propagation based upon a Lagrange multiplier method. We start from the equations of motion for a rotating, self-gravitating earth model initially in hydrostatic equilibrium, and derive the corresponding adjoint equations that involve motions on an earth model that rotates in the opposite direction. Variations in the misfit function χ then may be expressed as, where δχ = ∫v Km δ ln md3r + ∫Σ Kd δ ln dd2r + ∫ΣFS Kd · ∇Σδ ln dd2r, where δ ln m = δm/m denotes relative model perturbations in the volume V, δ ln d denotes relative topographic variations on solid-solid or fluid-solid boundaries Σ, and ∇Σ δln d denotes surface gradients in relative topographic variations on fluid-solid boundaries ΣFS. The 3-D Fréchet kernel Km determines the sensitivity to model perturbations δln m, and the 2-D kernels Kd and Kd determine the sensitivity to topographic variations δln d. We demonstrate also how anelasticity may be incorporated within the framework of adjoint methods. Finite-frequency sensitivity kernels are calculated by simultaneously computing the adjoint wavefield forward in time and reconstructing the regular wavefield backward in time. Both the forward and adjoint simulations are based upon a spectral-element method. We apply the adjoint technique to generate finite-frequency traveltime kernels for global seismic phases (P, Pdiff, PKP, S, SKS, depth phases, surface-reflected phases, surface waves, etc.) in both 1-D and 3-D earth models. For 1-D models these adjoint-generated kernels generally agree well with results obtained from ray-based methods. However, adjoint methods do not have the same theoretical limitations as ray-based methods, and can produce sensitivity kernels for any given phase in any 3-D earth model. The Fréchet kernels presented in this paper illustrate the sensitivity of seismic observations to structural parameters and topography on internal discontinuities. These kernels form the basis of future 3-D tomographic inversions.

Original languageEnglish (US)
Pages (from-to)265-286
Number of pages22
JournalGeophysical Journal International
Issue number1
StatePublished - Jul 2008

All Science Journal Classification (ASJC) codes

  • Geophysics
  • Geochemistry and Petrology


  • Computational seismology
  • Inverse theory
  • Seismic tomography
  • Theoretical seismology
  • Wave propagation


Dive into the research topics of 'Finite-frequency sensitivity kernels for global seismic wave propagation based upon adjoint methods'. Together they form a unique fingerprint.

Cite this