Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

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

  • Original Paper
  • Published:
GEM - International Journal on Geomathematics Aims and scope Submit manuscript

Abstract

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.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  • Artemieva, I.M.: The continental lithosphere: reconciling thermal, seismic, and petrologic data. Lithos 109(1–2), 23–46 (2009)

    Article  Google Scholar 

  • Boehler, R.: High-pressure experiments and the phase diagram of lower mantle and core materials. Rev. Geophys. 1998, 221–245 (2000)

    Article  Google Scholar 

  • Braess, D.: Finite elements: theory, fast solvers, and applications in solid mechanics. Cambridge University Press, Cambridge (2001)

    Google Scholar 

  • Braun, J.: The many surface expressions of mantle dynamics. Nat. Geosci. 3(12), 825–833 (2010)

    Article  Google Scholar 

  • Bunge, H.-P., Richards, M.A., Baumgardner, J.R.: The effect of depth dependent viscosity on the planform of mantle convection. Nature 379, 436–438 (1996)

    Article  Google Scholar 

  • Bunge, H.-P., Hagelberg, C.R., Travis, B.J.: Mantle circulation models with variational data assimilation: inferring past mantle flow and structure from plate motion histories and seismic tomography. Geophys. J. Int. 152(2), 280–301 (2003)

    Article  Google Scholar 

  • Bunge, H.-P., Richards, M.A., Baumgardner, J.R.: Mantle-circulation models with sequential data assimilation: inferring present-day mantle structure from plate-motion histories. Philos. Trans. A. Math. Phys. Eng. Sci. 360(1800), 2545–67 (2002)

    Article  Google Scholar 

  • Bunge, H.-P., Richards, M.A., Lithgow-Bertelloni, C., Baumgardner, J.R., Grand, S.P., Romanomicz, B.A.: Time scales and heterogeneous structure in geodynamic earth models. Science 280(5360), 91–95 (1998)

    Article  Google Scholar 

  • Bunge, H.-P., Richards, M.A., Baumgardner, J.R.: Study of three-dimensional mantle convection at \(10^{8}\) Rayleigh number: Effects of depth-dependent phase change formulation. J. Geophys. Res. 102 (B6), 11991–12007 (1997)

  • Burstedde, C., Stadler, G., Alisic, L., Wilcox, L.C., Tan, E., Gurnis, M., Ghattas, O.: Large-scale adaptive mantle convection simulation. Geophys. J. Int. 192(3), 889–906 (2013)

    Article  Google Scholar 

  • Conrad, C.P., Gurnis, M.: Seismic tomography, surface uplift, and the breakup of Gondwanaland: integrating mantle convection backwards in time. Geochem. Geophys. Geosyst. 4(3) (2003) . doi:10.1029/2001GC000299

  • Davies, D.R., Goes, S., Davies, J.H., Schuberth, B.S.A., Bunge, H.-P., Ritsema, J.: Reconciling dynamic and seismic models of Earth’s lower mantle: the dominant role of thermal heterogeneity. Earth Planet. Sci. Lett. 353–354, 253–269 (2012)

    Article  Google Scholar 

  • Dziewonski, A.M., Anderson, D.L.: Preliminary reference Earth model. Phys. Earth Planet. Int. 25(4), 297–356 (1981)

    Article  Google Scholar 

  • Fichtner, A., Bleibinhaus, F., Capdeville, Y.: Full seismic waveform modelling and inversion. Springer, Berlin, Heidelberg (2011)

  • Fichtner, A., Bunge, H.-P., Igel, H.: The adjoint method in seismology: I. Theory. Phys. Earth Planet. Int. 157(1–2), 86–104 (2006a)

    Article  MATH  Google Scholar 

  • Fichtner, A., Bunge, H.-P., Igel, H.: The adjoint method in seismology—II. Applications: traveltimes and sensitivity functionals. Phys. Earth Planet. Int. 157(1–2), 105–123 (2006b)

    Article  MATH  Google Scholar 

  • Fletcher, R., Reeves, C.M.: Function minimization by conjugate gradients. Comp. J. 7, 149–154 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  • Forte, A.M., Quéré, S., Moucha, R., Simmons, N.A., Grand, S.P., Mitrovica, J.X., Rowley, D.B.: Joint seismic-geodynamic-mineral physical modelling of African geodynamics: A reconciliation of deep-mantle convection with surface geophysical constraints. Earth Planet. Sci. Lett. 295(3–4), 329–341 (2010)

    Article  Google Scholar 

  • Fournier, A., Hulot, G., Jault, D., Kuang, W., Tangborn, A., Gillet, N., Canet, E., Aubert, J., Lhuillier, F.: An introduction to data assimilation and predictability in geomagnetism. Space Sci. Rev. 155(1–4), 247–291 (2010)

    Article  Google Scholar 

  • Freeden, W., Gervens, T., Schreiner, M.: Constructive approximation on the sphere (With applications to geomathematics). Oxford Science Publication, Clarendon Press, Oxford (1998)

    MATH  Google Scholar 

  • Freeden, W., Maier, T., Zimmermann, S.: A survey on wavelet methods for (geo) applications. Rev. Matemática 16(1), 277–310 (2003)

  • Freeden, W.: A General Construction Principle of Wavelets. Min. Sky 53–70 (2001)

  • Glatzmaier, G.A.: Numerical simulations of mantle convection: time-dependent, three-dimensional, compressible, spherical shell. Geophys. Astrophys. Fluid Dyn. 43(2), 223–264 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  • Gmeiner, B., Gradl, T., Gaspar, F., Rüde, U.: Optimization of the multigrid-convergence rate on semi-structured meshes by local Fourier analysis. Comput. Math. Appl. 65(4), 694–711 (2013)

    Article  MathSciNet  Google Scholar 

  • Grand, S.P., van der Hilst, R.D., Widiyantoro, S.: Global seismic tomography: a snapshot of convection in the Earth. GSA Today 7(4), 1–7 (1997)

    Google Scholar 

  • Gurnis, M., Turner, M., Zahirovic, S., DiCaprio, L., Spasojevic, S., Müller, R., Boyden, J., Seton, M., Manea, V.C., Bower, D.J.: Plate tectonic reconstructions with continuously closing plates. Comput. Geosci. 38(1), 35–42 (2012)

    Article  Google Scholar 

  • Hadamard, J.: Sur les problèmes aux dérivées partielles et leur signification physique. Princet. Univ. Bull. 13(49–52), 28 (1902)

    Google Scholar 

  • Hager, B.H., Richards, M.A.: Long-wavelength variations in Earth’s geoid: physical models and dynamical implications. Philos. Trans. R. Soc. Lond Ser. A Math. Phys. Sci. 328(1599), 309–327 (1989)

    Article  Google Scholar 

  • Hager, B.H., O’Connell, R.J.: Subduction zone dip angles and flow driven by plate motion. Tectonophysics 50(2–3), 111–133 (1978)

    Article  Google Scholar 

  • Hager, B.H., O’Connell, R.J.: Kinematic models of large-scale flow in the Earth’s mantle. J. Geophys. Res. Solid Earth 84(B3), 1031–1048 (1979)

    Article  Google Scholar 

  • Heine, C., Müller, R.D., Steinberger, B., DiCaprio, L.: Integrating deep Earth dynamics in paleogeographic reconstructions of Australia. Tectonophysics 483(1–2), 135–150 (2010)

    Article  Google Scholar 

  • Iaffaldano, G., Bunge, H.-P., Bücker, M.: Mountain belt growth inferred from histories of past plate convergence: a new tectonic inverse problem. Earth Planet. Sci. Lett. 260(3–4), 516–523 (2007)

    Article  Google Scholar 

  • Ismail-Zadeh, A., Schubert, G., Tsepelev, I., Korotkii, A.: Inverse problem of thermal convection: numerical approach and application to mantle plume restoration. Phys. Earth Planet. Inter. 145(1–4), 99–114 (2004)

    Article  Google Scholar 

  • Jarvis, G.T., McKenzie, D.P.: Convection in a compressible fluid with infinite Prandtl number. J. Fluid Mech. 96(03), 515–583 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  • Jordan, T.H.: Composition and development of the continental tectosphere. Nature 5671, 544–548 (1978)

    Article  Google Scholar 

  • Kress, R.: Linear integral equations. Springer, New York (1999)

  • Li, K., Jackson, A., Livermore, P.W.: Variational data assimilation for the initial-value dynamo problem. Phys. Rev. E 84(5), 056321 (2011)

    Article  Google Scholar 

  • Liu, L., Gurnis, M.: Simultaneous inversion of mantle properties and initial conditions using an adjoint of mantle convection. J. Geophys. Res. 113(B8), B08405 (2008)

    Google Scholar 

  • McNamara, A.K., Zhong, S.: Thermochemical structures beneath Africa and the Pacific Ocean. Nature 437(7062), 1136–1139 (2005)

    Article  Google Scholar 

  • Menemenlis, D., Wunsch, C.: Linearization of an oceanic general circulation model for data assimilation and climate studies. J. Atmos. Ocean. Technol. 1995, 1420–1443 (1997)

    Article  Google Scholar 

  • Mitrovica, J.X., Forte, A.M.: A new inference of mantle viscosity based upon joint inversion of convection and glacial isostatic adjustment data. Earth Planet. Sci. Lett. 225(1–2), 177–189 (2004)

    Article  Google Scholar 

  • Moucha, R., Forte, A.M., Mitrovica, J.X., Rowley, D.B., Quéré, S., Simmons, N.A., Grand, S.P.: Dynamic topography and long-term sea-level variations: there is no such thing as a stable continental platform. Earth Planet. Sci. Lett. 271(1–4), 101–108 (2008a)

    Article  Google Scholar 

  • Moucha, R., Forte, A.M., Rowley, D.B., Mitrovica, J.X., Simmons, N.A., Grand, S.P.: Mantle convection and the recent evolution of the Colorado Plateau and the Rio Grande Rift valley. Geology 36(6), 439 (2008b)

    Article  Google Scholar 

  • Müller, R.D., Sdrolias, M., Gaina, C., Roest, W.R.: Age, spreading rates, and spreading asymmetry of the world’s ocean crust. Geochem. Geophys. Geosyst. 9(4), Q04006 (2008)

    Article  Google Scholar 

  • Oeser, J., Bunge, H.-P., Mohr, M.: Cluster Design in the Earth Sciences: TETHYS. In: Gerndt, M., Kranzlmüller, D.: (eds.) High Perform. Comput. Commun., vol. 4208 of Lecture Notes in Computer Science. Springer, pp 31–40 (2006)

  • Paulson, A., Zhong, S., Wahr, J.: Inference of mantle viscosity from GRACE and relative sea level data. Geophys. J. Int. 171(2), 497–508 (2007)

    Article  Google Scholar 

  • Piazzoni, A., Steinle-Neumann, G., Bunge, H.-P., Dolejš, D.: A mineralogical model for density and elasticity of the Earth’s mantle. Geochem. Geophys. Geosyst. 8(11), (2007) . doi:10.1029/2007GC001697

  • Richards, M.A., Engebretson, D.C.: Large-scale mantle convection and the history of subduction. Nature 355(6359), 437–440 (1992)

    Article  Google Scholar 

  • Schuberth, B.S.A., Bunge, H.-P., Ritsema, J.: Tomographic filtering of high-resolution mantle circulation models: Can seismic heterogeneity be explained by temperature alone? Geochem. Geophys. Geosyst. 10(5), (2009a) . doi:10.1029/2009GC002401

  • Schuberth, B.S.A., Bunge, H.-P., Steinle-Neumann, G., Moder, C., Oeser, J.: Thermal versus elastic heterogeneity in high-resolution mantle circulation models with pyrolite composition: High plume excess temperatures in the lowermost mantle. Geochem. Geophys. Geosyst. 10(1), (2009b) . doi:10.1029/2008GC002235

  • Schuberth, B.S.A., Zaroli, C., Nolet, G.: Synthetic seismograms for a synthetic Earth: long-period P- and S-wave traveltime variations can be explained by temperature alone. Geophys. J. Int. 188(3), 1393–1412 (2012)

    Article  Google Scholar 

  • Serrin, J.: Mathematical principles of classical fluid mechanics. Handb. der Phys. VIII, 125–263 (1959)

    MathSciNet  Google Scholar 

  • Seton, M., Müller, R., Zahirovic, S., Gaina, C., Torsvik, T., Shephard, G., Talsma, A., Gurnis, M., Turner, M., Maus, S., Chandler, M.: Global continental and ocean basin reconstructions since 200Ma. Earth Sci. Rev. 113(3–4), 212–270 (2012)

    Article  Google Scholar 

  • Spasojevic, S., Liu, L., Gurnis, M.: Adjoint models of mantle convection with seismic, plate motion, and stratigraphic constraints: North America since the Late Cretaceous. Geochem. Geophys. Geosyst. 10(5), (2009) . doi:10.1029/2008GC002345

  • Steinberger, B., O’Connell, R.: Changes of the Earth’s rotation axis owing to advection of mantle density heterogeneities. Nature 387 (6629), 169–173 (1997)

  • Steinberger, B., O’Connell, R.: Advection of plumes in mantle flow: implications for hotspot motion, mantle viscosity and plume distribution. Geophys. J. Int. 132 (2), 412–434 (1998)

  • Steinle-Neumann, G., Stixrude, L., Cohen, R.E., Gülseren, O.: Elasticity of iron at the temperature of the Earth’s inner core. Nature 413(6851), 57–60 (2001)

    Article  Google Scholar 

  • Tackley, P.J.: Dynamics and evolution of the deep mantle resulting from thermal, chemical, phase and melting effects. Earth Sci. Rev. 110(1–4), 1–25 (2012)

    Article  Google Scholar 

  • Talagrand, O., Courtier, P.: Variational assimilation of meteorological observations with the adjoint vorticity equation. I: theory. Q. J. R. Meteorol. Soc. 113(478), 1311–1328 (1987)

    Article  Google Scholar 

  • Tarantola, A.: Linearized inversion of seismic reflection data. Geophys. Prospect. 32(6), 998–1015 (1984)

    Article  Google Scholar 

  • Tromp, J., Tape, C., Liu, Q.: Seismic tomography, adjoint methods, time reversal and banana-doughnut kernels. Geophys. J. Int. 160(1), 195–216 (2005)

    Article  Google Scholar 

  • Zhang, Z., Stixrude, L., Brodholt, J.: Elastic properties of MgSiO3-perovskite under lower mantle conditions and the composition of the deep Earth. Earth Planet. Sci. Lett. 379, 1–12 (2013)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Earth and Environmental Sciences, Ludwig-Maximilians University Munich, Theresienstr. 41, 80333 , Munich, Germany

    André Horbach, Hans-Peter Bunge & Jens Oeser

Authors
  1. André Horbach
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hans-Peter Bunge
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jens Oeser
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to André Horbach.

Additional information

Dedicated to Willi Freeden’s 65th Birthday.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Horbach, A., Bunge, HP. & Oeser, J. 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. Int J Geomath 5, 163–194 (2014). https://doi.org/10.1007/s13137-014-0061-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13137-014-0061-5

Keywords

Mathematics Subject Classifications

Search

Navigation