GEM - International Journal on Geomathematics

, Volume 5, Issue 2, pp 163–194 | Cite as

The adjoint method in geodynamics: derivation from a general operator formulation and application to the initial condition problem in a high resolution mantle circulation model

  • André HorbachEmail author
  • Hans-Peter Bunge
  • Jens Oeser
Original Paper


The adjoint method is a computationally efficient way to compute the gradient of a physical observable or an associated objective function relative to its parameters. In geodynamics the observable can be thought of as a representation of the present day heterogeneity structure in the Earth’s mantle, inferred in some form through seismic imaging, while a crucial derivative of interest is that relative to an earlier convective system state. Since mantle convection is governed by coupled, non-linear conservation equations for mass, momentum and energy, computation of the derivative consists of iterative solutions to the forward and the adjoint problem, rendering the approach superior to finite difference approximations, which become impractical at the resolution of modern geodynamic models. Moreover, similarities in the forward and adjoint equations allow one to apply existing numerical codes that solve the forward problem to the adjoint equations with little adaptation. Bunge et al. (Geophys J Int 152(2):280–301 (2003)), have derived the adjoint equations for mantle convection using the concept of Lagrangian multipliers. Here we introduce a more general approach using an operator formulation in Hilbert spaces, in order to connect to recent work in seismology (Fichtner et al. Phys Earth Planet Int 157(1–2):86–104 (2006a)), where the approach was used to derive the adjoint equations for the scalar wave equation. We demonstrate the practicality of the method for use in a high resolution mantle circulation model with more than 80 million finite elements by restoring a representation of present day mantle heterogeneity derived from the global seismic shear wave study of Grand et al. (GSA Today 7(4):1–7 1997) back in time for the past 40 million years. An important result is our finding of a strong global minimum for the unknown initial condition, regardless of the assumed first guess for the initial heterogeneity structure, which we attribute to the uniqueness theorem by Serrin. Paleo mantle convection modelling will improve our ability to test assumptions about the internal structure and dynamics of the Earth’s mantle against the geologic record.


Adjoint method Inverse problem Geodynamics Seismic tomography Regularisation Mantle convection 

Mathematics Subject Classifications

65M32 65N21 49N45 86A22 86A17 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of Earth and Environmental SciencesLudwig-Maximilians University MunichMunichGermany

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