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19 - Data Assimilation in Geodynamics: Methods and Applications

from Part III - ‘Solid’ Earth Applications: From the Surface to the Core

Published online by Cambridge University Press:  20 June 2023

By
Alik Ismail-Zadeh ,
Igor Tsepelev  and
Alexander Korotkii
Edited by
Alik Ismail-Zadeh ,
Fabio Castelli ,
Dylan Jones  and
Sabrina Sanchez
Alik Ismail-Zadeh
Affiliation:
Karlsruhe Institute of Technology, Germany
Fabio Castelli
Affiliation:
Università degli Studi, Florence
Dylan Jones
Affiliation:
University of Toronto
Sabrina Sanchez
Affiliation:
Max Planck Institute for Solar System Research, Germany
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Summary

Abstract: In this chapter, we review basic methods for data assimilation used in geodynamic modelling: backward advection (BAD), variational/adjoint (VAR), and quasi-reversibility (QRV). The VAR method is based on a search for model parameters (e.g. mantle temperature and flow velocity in the past) by minimising the differences between present observations of the relevant physical parameters (e.g. temperature derived from seismic tomography, geodetic measurements) and those predicted by forward models for an initial guess temperature. The QRV method is based on introduction of the additional term involving the product of a small regularisation parameter and a higher-order temperature derivative into the backward heat equation. The data assimilation in this case is based on a search of the best fit between the forecast model state and the observations by minimising the regularisation parameter. To demonstrate the applicability of the considered data assimilation methods, a numerical model of the evolution of mantle plumes is considered. Also, we present an application of the data assimilation to dynamic restoration of the thermal state of the mantle beneath the Japanese islands and their surroundings. The geodynamic restoration for the last 40 million years is based on the assimilation of the present temperature inferred from seismic tomography, and constrained by the present plate movement derived from geodetic observations, and paleogeographic and paleomagnetic plate reconstructions. Finally, we discuss some challenges, advantages, and disadvantages of the data assimilation methods.

Keywords

adjoint problembackward advectiondynamic restorationlithospheric slabmantle plumeobjective functionaloptimizationquasi-reversibility methodvariational data assimilation
Type
Chapter
Information
Applications of Data Assimilation and Inverse Problems in the Earth Sciences , pp. 293 - 310
Publisher: Cambridge University Press
Print publication year: 2023

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References

Alekseev, A. K., and Navon, I. M. (2001). The analysis of an ill-posed problem using multiscale resolution and second order adjoint techniques. Computer Methods in Applied Mechanics and Engineering, 190, 1937–53.CrossRefGoogle Scholar
Bennett, A. F. (1992). Inverse Methods in Physical Oceanography. Cambridge: Cambridge University Press.Google Scholar
Billen, M. I. (2008). Modeling the dynamics of subducting slabs. Annual Review of Earth and Planetary Sciences, 36, 325–56.CrossRefGoogle Scholar
Bird, P. (2003). An updated digital model of plate boundaries. Geochemistry, Geophysics, Geosystems, 4, 1027. https://doi.org/10.1029/2001GC000252.CrossRefGoogle Scholar
Boussinesq, J. (1903). Theorie Analytique de la Chaleur, vol. 2. Paris: Gauthier-Villars.Google Scholar
Bubnov, V. A. (1976). Wave concepts in the theory of heat. International Journal of Heat and Mass Transfer, 19, 175–84.Google Scholar
Bubnov, V. A. (1981). Remarks on wave solutions of the nonlinear heat-conduction equation. Journal of Engineering Physics and Thermophysics, 40(5), 565–71.Google Scholar
Bunge, H.-P., Richards, M. A., and Baumgardner, J. R. (2002). Mantle circulation models with sequential data-assimilation: Inferring present-day mantle structure from plate motion histories. Philosophical Transactions of the Royal Society A, 360, 2545–67.Google Scholar
Bunge, H.-P., Hagelberg, C. R., and Travis, B. J. (2003). Mantle circulation models with variational data assimilation: Inferring past mantle flow and structure from plate motion histories and seismic tomography. Geophysical Journal International, 152, 280–301.Google Scholar
Busse, F. H., Christensen, U., Clever, R. et al. (1993). 3D convection at infinite Prandtl number in Cartesian geometry: A benchmark comparison. Geophysical and Astrophysical Fluid Dynamics, 75, 39–59.Google Scholar
Cattaneo, C. (1958). Sur une forme de l’equation de la chaleur elinant le paradox d’une propagation instantance. Comptes Rendus de l’Académie des Sciences, 247, 431–33.Google Scholar
Chandrasekhar, S. (1961). Hydrodynamic and Hydromagnetic Stability. Oxford: Oxford University Press.Google Scholar
Christensen, U. R., and Yuen, D. A. (1985). Layered convection induced by phase transitions. Journal of Geophysical Research, 90, 10291–300.Google Scholar
Coltice, N., Husson, L., Faccenna, C., and Arnould, M. (2019). What drives tectonic plates? Science Advances, 5, eaax4295.Google Scholar
Conrad, C. P., and Gurnis, M. (2003). Seismic tomography, surface uplift, and the breakup of Gondwanaland: Integrating mantle convection backwards in time. Geochemistry, Geophysics, Geosystems, 4(3). https://doi.org/10.1029/2001GC000299.Google Scholar
Davaille, A., and Vatteville, J. (2005). On the transient nature of mantle plumes. Geophysical Research Letters, 32, L14309. https://doi.org/10.1029/2005GL023029.Google Scholar
Doglioni, C., Ismail-Zadeh, A., Panza, G., and Riguzzi, F. (2011). Lithosphere-asthenosphere viscosity contrast and decoupling. Physics of the Earth and Planetary Interiors, 189, 1–8.Google Scholar
Drewes, H. (2009). The Actual Plate Kinematic and Crustal Deformation Model APKIM2005 as basis for a non-rotating ITRF. In Drewes, H., ed., Geodetic Reference Frames, IAG Symposia series 134. Berlin: Springer, pp. 95–99.Google Scholar
Forte, A. M., and Mitrovica, J. X. (1997). A resonance in the Earth’s obliquity and precession over the past 20 Myr driven by mantle convection. Nature, 390, 676–80.CrossRefGoogle Scholar
Fukao, Y., Widiyantoro, S., and Obayashi, M. (2001). Stagnant slabs in the upper and lower mantle transition region. Reviews of Geophysics, 39, 291–323.Google Scholar
Furumura, T., and Kennett, B. L. N. (2005). Subduction zone guided waves and the heterogeneity structure of the subducted plate: Intensity anomalies in northern Japan. Journal of Geophysical Research, 110, B10302. https://doi.org/10.1029/2004JB003486.Google Scholar
Ghelichkhan, S., and Bunge, H.-P. (2016). The compressible adjoint equations in geodynamics: derivation and numerical assessment. International Journal on Geomathematics, 7, 1–30.Google Scholar
Ghelichkhan, S., and Bunge, H.-P. (2018). The adjoint equations for thermochemical compressible mantle convection: derivation and verification by twin experiments. Proceedings of the Royal Society A, 474, 20180329.CrossRefGoogle ScholarPubMed
Glišović, P., and Forte, A. M. (2014). Reconstructing the Cenozoic evolution of the mantle: Implications for mantle plume dynamics under the Pacific and Indian plates. Earth and Planetary Science Letters, 390, 146–156.CrossRefGoogle Scholar
Glišović, P., and Forte, A. M. (2016). A new back-and-forth iterative method for time-reversed convection modeling: Implications for the Cenozoic evolution of 3-D structure and dynamics of the mantle. Journal of Geophysical Research, 121, 4067–84.Google Scholar
Glišović, P., and Forte, A. M. (2017). On the deep-mantle origin of the Deccan Traps. Science, 355(6325), 613–16.Google Scholar
Glišović, P., and Forte, A M. (2019). Two deep-mantle sources for Paleocene doming and volcanism in the North Atlantic. Proceedings of the National Academy of Sciences USA, 116(27), 13227–32.Google Scholar
Hall, R. (2002). Cenozoic geological and plate tectonic evolution of SE Asia and the SW Pacific: computer-based reconstructions, model and animations. Journal of Asian Earth Sciences, 20, 353–431.CrossRefGoogle Scholar
Hansen, U., Yuen, D. A., and Kroening, S. E. (1990). Transition to hard turbulence in thermal convection at infinite Prandtl number. Physics of Fluids, A2(12), 2157–63.Google Scholar
Hier-Majumder, C. A., Belanger, E., DeRosier, S., Yuen, D. A., and Vincent, A. P. (2005). Data assimilation for plume models. Nonlinear Processes in Geophysics, 12, 257–67.CrossRefGoogle Scholar
Honda, S., Yuen, D. A., Balachandar, S., and Reuteler, D. (1993). Three-dimensional instabilities of mantle convection with multiple phase transitions. Science, 259, 1308–11.Google Scholar
Horbach, A., Bunge, H.-P., and Oeser, J. (2014). 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. International Journal on Geomathematics, 5, 163–94.Google Scholar
Howard, L. N. (1966). Convection at high Rayleigh number. In Goertler, H., and Sorger, P., eds., Applied Mechanics, Proc. of the 11th Intl Congress of Applied Mechanics, Munich, Germany 1964. New York: Springer-Verlag, pp. 1109–15.Google Scholar
Ismail-Zadeh, A., and Tackley, P. (2010). Computational Methods for Geodynamics. Cambridge: Cambridge University Press.Google Scholar
Ismail-Zadeh, A. T., Talbot, C. J., and Volozh, Y. A. (2001a). Dynamic restoration of profiles across diapiric salt structures: Numerical approach and its applications. Tectonophysics, 337, 21–36.Google Scholar
Ismail-Zadeh, A. T., Korotkii, A. I., Naimark, B. M., and Tsepelev, I. A. (2001b). Numerical modelling of three-dimensional viscous flow with gravitational and thermal effects. Computational Mathematics and Mathematical Physics, 41(9), 1331–45.Google Scholar
Ismail-Zadeh, A. T., Korotkii, A. I., and Tsepelev, I. A. (2003a). Numerical approach to solving problems of slow viscous flow backwards in time. In Bathe, K. J., ed., Computational Fluid and Solid Mechanics. Amsterdam: Elsevier Science, pp. 938–41.Google Scholar
Ismail-Zadeh, A. T., Korotkii, A. I., Naimark, B. M., and Tsepelev, I. A. (2003b). Three-dimensional numerical simulation of the inverse problem of thermal convection. Computational Mathematics and Mathematical Physics, 43(4), 587–99.Google Scholar
Ismail-Zadeh, A., Schubert, G., Tsepelev, I., and Korotkii, A. (2004a). Inverse problem of thermal convection: Numerical approach and application to mantle plume restoration. Physics of the Earth and Planetary Interiors, 145, 99–114.Google Scholar
Ismail-Zadeh, A. T., Tsepelev, I. A., Talbot, C. J., and Korotkii, A. I. (2004b). Three-dimensional forward and backward modelling of diapirism: Numerical approach and its applicability to the evolution of salt structures in the Pricaspian basin. Tectonophysics, 387, 81–103.Google Scholar
Ismail-Zadeh, A., Mueller, B., and Schubert, G. (2005). Three-dimensional modeling of present-day tectonic stress beneath the earthquake-prone southeastern Carpathians based on integrated analysis of seismic, heat flow, and gravity observations. Physics of the Earth and Planetary Interiors, 149, 81–98.CrossRefGoogle Scholar
Ismail-Zadeh, A., Schubert, G., Tsepelev, I., and Korotkii, A. (2006). Three-dimensional forward and backward numerical modeling of mantle plume evolution: Effects of thermal diffusion. Journal of Geophysical Research, 111, B06401. https://doi.org/10.1029/2005JB003782.Google Scholar
Ismail-Zadeh, A., Korotkii, A., Schubert, G., and Tsepelev, I. (2007). Quasi-reversibility method for data assimilation in models of mantle dynamics. Geophysical Journal International, 170, 1381–98.Google Scholar
Ismail-Zadeh, A., Schubert, G., Tsepelev, I., and Korotkii, A. (2008). Thermal evolution and geometry of the descending lithosphere beneath the SE-Carpathians: An insight from the past. Earth and Planetary Science Letters, 273, 68–79.Google Scholar
Ismail-Zadeh, A., Honda, S., and Tsepelev, I. (2013). Linking mantle upwelling with the lithosphere descent and the Japan Sea evolution: a hypothesis. Scientific Reports, 3, 1137. https://doi.org/10.1038/srep01137.Google Scholar
Ismail-Zadeh, A., Korotkii, A., and Tsepelev, I. (2016). Data-Driven Numerical Modeling in Geodynamics: Methods and Applications. Heidelberg: Springer.Google Scholar
Jolivet, L., Tamaki, K., and Fournier, M. (1994). Japan Sea, opening history and mechanism: A synthesis. Journal of Geophysical Research, 99, 22232–59.Google Scholar
Kalnay, E. (2003). Atmospheric Modeling, Data Assimilation and Predictability. Cambridge: Cambridge University Press.Google Scholar
Karato, S. (2010). Rheology of the Earth’s mantle: A historical review. Gondwana Research, 18, 17–45.Google Scholar
Kaus, B. J. P., and Podladchikov, Y. Y. (2001). Forward and reverse modeling of the three-dimensional viscous Rayleigh–Taylor instability. Geophysical Research Letters, 28, 1095–8.Google Scholar
Kirsch, A. (1996). An Introduction to the Mathematical Theory of Inverse Problems, New York: Springer-Verlag.Google Scholar
Lattes, R., and Lions, J. L. (1969). The Method of Quasi-Reversibility: Applications to Partial Differential Equations. New York: Elsevier.Google Scholar
Li, D., Gurnis, M., and Stadler, G. (2017). Towards adjoint-based inversion of time-dependent mantle convection with nonlinear viscosity. Geophysical Journal International, 209(1), 86–105.Google Scholar
Liu, D. C., and Nocedal, J. (1989). On the limited memory BFGS method for large scale optimization. Mathematical Programming, 45, 503–28.Google Scholar
Liu, L., and Gurnis, M. (2008). Simultaneous inversion of mantle properties and initial conditions using an adjoint of mantle convection. Journal of Geophysical Research, 113, B08405. https://doi.org/10.1029/2008JB005594.Google Scholar
Liu, L., Spasojevic, S., and Gurnis, M. (2008). Reconstructing Farallon plate subduction beneath North America back to the Late Cretaceous. Science, 322, 934–38.Google Scholar
Liu, L., Gurnis, M., Seton, M. et al. (2010). The role of oceanic plateau subduction in the Laramide orogeny. Nature Geoscience, 3, 353–7.Google Scholar
Liu, M., Yuen, D. A., Zhao, W., and Honda, S. (1991). Development of diapiric structures in the upper mantle due to phase transitions. Science, 252, 1836–9.Google Scholar
Malevsky, A. V., and Yuen, D. A. (1993). Plume structures in the hard-turbulent regime of three-dimensional infinite Prandtl number convection. Geophysical Research Letters, 20, 383–6.Google Scholar
Maruyama, S., Isozaki, Y., Kimura, G., and Terabayashi, M. (1997). Paleogeographic maps of the Japanese Islands: Plate tectonic synthesis from 750 Ma to the present. The Island Arc, 6, 121–42.Google Scholar
Massimi, P., Quarteroni, A., Saleri, F., and Scrofani, G. (2007). Modeling of salt tectonics. Computational Methods in Applied Mathematics, 197, 281–93.Google Scholar
McLaughlin, D. (2002). An integrated approach to hydrologic data assimilation: Interpolation, smoothing, and forecasting. Advances in Water Resources, 25, 1275–86.Google Scholar
Moore, W. B., Schubert, G., and Tackley, P. (1998). Three-dimensional simulations of plume–lithosphere interaction at the Hawaiian Swell. Science, 279, 1008–11.Google Scholar
Morse, P. M., and Feshbach, H. (1953). Methods of Theoretical Physics. New York: McGraw-Hill.Google Scholar
Moucha, R., and Forte, A. M. (2011). Changes in African topography driven by mantle convection. Nature Geoscience, 4, 707–12.Google Scholar
Northrup, C. J., Royden, L. H., and Burchfiel, B. C. (1995). Motion of the Pacific plate relative to Eurasia and its potential relation to Cenozoic extension along the eastern margin of Eurasia. Geology, 23, 719–22.Google Scholar
Obayashi, M., Sugioka, H., Yoshimitsu, J., and Fukao, Y. (2006). High temperature anomalies oceanward of subducting slabs at the 410-km discontinuity. Earth and Planetary Science Letters, 243, 149–58.Google Scholar
Obayashi, M., Yoshimitsu, J., and Fukao, Y. (2009). Tearing of stagnant slab. Science, 324, 1173–5.Google Scholar
Olson, P., and Singer, H. (1985). Creeping plumes. Journal of Fluid Mechanics, 158, 511–31.Google Scholar
Peng, D., and Liu, L. (2022). Quantifying slab sinking rates using global geodynamic models with data-assimilation, Earth-Science Reviews, 230, 104039.Google Scholar
Ratnaswamy, V., Stadler, G., and Gurnis, M. (2015). Adjoint-based estimation of plate coupling in a non-linear mantle flow model: Theory and examples. Geophysical Journal International, 202, 768–86.Google Scholar
Ribe, N. M., and Christensen, U. (1994). Three-dimensional modeling of plume–lithosphere interaction. Journal of Geophysical Research, 99, 669–82.Google Scholar
Rowley, D. B. (2008). Extrapolating oceanic age distributions: Lessons from the Pacific region. Journal of Geology, 116, 587–98.Google Scholar
Samarskii, A. A., and Vabishchevich, P. N. (2007). Numerical Methods for Solving Inverse Problems of Mathematical Physics. Berlin: De Gruyter.Google Scholar
Samarskii, A. A., Vabishchevich, P. N., and Vasiliev, V. I. (1997). Iterative solution of a retrospective inverse problem of heat conduction. Mathematical Modeling, 9, 119–27.Google Scholar
Schubert, G., Turcotte, D. L., and Olson, P. (2001). Mantle Convection in the Earth and Planets. Cambridge: Cambridge University Press.Google Scholar
Schuh-Senlis, M., Thieulot, C., Cupillard, P., and Caumon, G. (2020). Towards the application of Stokes flow equations to structural restoration simulations. Solid Earth, 11, 1909–30.Google Scholar
Seno, T., and Maruyama, S. (1984). Paleogeographic reconstruction and origin of the Philippine Sea. Tectonophysics, 102, 53–84.CrossRefGoogle Scholar
Shephard, G., Müller, R., Liu, L. et al. (2010). Miocene drainage reversal of the Amazon River driven by plate–mantle interaction. Nature Geoscience, 3, 870–75.Google Scholar
Spasojevic, S., Liu, L., and Gurnis, M. (2009). Adjoint models of mantle convection with seismic, plate motion, and stratigraphic constraints: North America since the Late Cretaceous. Geochemistry, Geophysics, Geosystems, 10, Q05W02. https://doi.org/10.1029/2008GC002345.Google Scholar
Steinberger, B., and O’Connell, R.J. (1997). Changes of the Earth’s rotation axis owing to advection of mantle density heterogeneities. Nature, 387, 169–73.Google Scholar
Steinberger, B., and O’Connell, R. J. (1998). Advection of plumes in mantle flow: implications for hotspot motion, mantle viscosity and plume distribution. Geophysical Journal International, 132, 412–34.Google Scholar
Tikhonov, A. N. (1963). Solution of incorrectly formulated problems and the regularization method. Soviet Mathematics Doklady, 4, 1035–8.Google Scholar
Tikhonov, A. N., and Arsenin, V. Y. (1977). Solution of Ill-Posed Problems. New York: Halsted Press.Google Scholar
Tikhonov, A. N., and Samarskii, A. A. (1990). Equations of Mathematical Physics. New York: Dover Publications.Google Scholar
Trompert, R. A., and Hansen, U. (1998). On the Rayleigh number dependence of convection with a strongly temperature-dependent viscosity. Physics of Fluids, 10, 351–60.Google Scholar
Tsepelev, I. A. (2011). Iterative algorithm for solving the retrospective problem of thermal convection in a viscous fluid. Fluid Dynamics, 46, 835–42.Google Scholar
Turcotte, D. L., and Schubert, G. (2002). Geodynamics, 2nd ed. Cambridge: Cambridge University Press.Google Scholar
Vasiliev, F. P. (2002). Methody optimizatsii. Moscow: Factorial Press.Google Scholar
Vernotte, P. (1958). Les paradoxes de la theorie continue de l’equation de la chaleur. Comptes Rendus de l’Académie des Sciences, 246, 3154–5.Google Scholar
Wang, K., Hyndman, R. D., and Yamano, M. (1995). Thermal regime of the Southwest Japan subduction zone: Effects of age history of the subducting plate. Tectonophysics, 248, 53–69.Google Scholar
Worthen, J., Stadler, G., Petra, N., Gurnis, M., and Ghattas, O. (2014). Towards an adjoint-based inversion for rheological parameters in nonlinear viscous mantle flow. Physics of the Earth and Planetary Interiors, 234, 23–34.Google Scholar
Yamano, M., Kinoshita, M., Goto, S., and Matsubayashi, O. (2003). Extremely high heat flow anomaly in the middle part of the Nankai Trough. Physics and Chemistry of the Earth, 28, 487–97.Google Scholar
Yamazaki, T., Takahashi, M., Iryu, Y. et al. (2010). Philippine Sea Plate motion since the Eocene estimated from paleomagnetism of seafloor drill cores and gravity cores. Earth Planets Space, 62, 495–502.Google Scholar
Yu, N., Imatani, S., and Inoue, T. (2004). Characteristics of temperature field due to pulsed heat input calculated by non-Fourier heat conduction hypothesis. JSME International Journal Series A, 47(4), 574–80.Google Scholar
Zhong, S. (2005). Dynamics of thermal plumes in three-dimensional isoviscous thermal convection. Geophysical Journal International, 162, 289–300.Google Scholar
Zou, X., Navon, I. M., Berger, M. et al. (1993). Numerical experience with limited-memory quasi-Newton and truncated Newton methods. SIAM Journal of Optimization, 3(3), 582–608.Google Scholar

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