Analysis of a Model of Elastic Dislocations in Geophysics

  • Andrea Aspri
  • Elena BerettaEmail author
  • Anna L. Mazzucato
  • Maarten V. De Hoop


We analyze a mathematical model of elastic dislocations with applications to geophysics, where by an elastic dislocation we mean an open, oriented Lipschitz surface in the interior of an elastic solid, across which there is a discontinuity of the displacement. We model the Earth as an infinite, isotropic, inhomogeneous, elastic medium occupying a half space, and assume only Lipschitz continuity of the Lamé parameters. We study the well posedness of very weak solutions to the forward problem of determining the displacement by imposing traction-free boundary conditions at the surface of the Earth, continuity of the traction and a given jump on the displacement across the fault. We employ suitable weighted Sobolev spaces for the analysis. We utilize the well-posedness of the forward problem and unique-continuation arguments to establish uniqueness in the inverse problem of determining the dislocation surface and the displacement jump from measuring the displacement at the surface of the Earth. Uniqueness holds for tangential or normal jumps and under some geometric conditions on the surface.



The authors thank C. Amrouche, S. Salsa, and M. Taylor for useful discussion and for suggesting relevant literature. They also thank the anonymous referees for their careful reading of this work and useful suggestions. A. Aspri and A. Mazzucato thank the Departments of Mathematics at NYU-Abu Dhabi and at Politecnico of Milan for their hospitality. A. Mazzucato was partially supported by the US National Science Foundation Grant DMS-1615457. M. V. de Hoop gratefully acknowledges support from the Simons Foundation under the MATH + X program, the National Science Foundation under Grant DMS-1815143, and the corporate members of the Geo-Mathematical Group at Rice University.


  1. 1.
    Alessandrini, G., Rondi, L., Rosset, E., Vessella, S.: The stability for the Cauchy problem for elliptic equations. Inverse Probl. 25, 123004, 2009. 47ppADSMathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Amrouche, C., Bonzom, F.: Exterior problems in the half-space for the Laplace operator in weighted Sobolev spaces. J. Differ. Equ. 246, 1894–1920, 2009ADSMathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Amrouche, C., Dambrine, M., Raudin, Y.: An \(L^p\) theory of linear elasticity in the half-space. J. Differ. Equ. 253, 906–932, 2012ADSzbMATHCrossRefGoogle Scholar
  4. 4.
    Amrouche, C., Girault, V., Giroire, J.: Weighted Sobolev spaces for Laplace’s equation in \({\mathbb{R}}^n\). Journal de Mathématiques Pures et Appliquées73, 579–606, 1994MathSciNetzbMATHGoogle Scholar
  5. 5.
    Amrouche, C., Nečasová, S.: Laplace equation in the half-space with a nonhomogeneous Dirichlet boundary condition. Mathematica Bohemica126, 265–274, 2001MathSciNetzbMATHGoogle Scholar
  6. 6.
    Amrouche, C., Nečasová, S., Raudin, Y.: Very weak, generalized and strong solutions to the Stokes system in the half-space. J. Differ. Equ. 244, 887–915, 2008ADSMathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Árnadóttir, T., Segall, P.: The 1989 Loma Prieta earthquake imaged from inversion of geodetic data. J. Geophys. Res. 99, 21,835–21,855, 1994ADSCrossRefGoogle Scholar
  8. 8.
    Aspri, A., Beretta, E., Mascia, C.: Analysis of a Mogi-type model describing surface deformations induced by a magma chamber embedded in an elastic half-space. Journal de l’École Polytechnique - Mathématiques4, 223–255, 2017MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Aspri, A., Beretta, E., Rosset, E.: On an elastic model arising from volcanology: an analysis of the direct and inverse problem. J. Differ. Equ. 265, 6400–6423, 2018ADSMathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Beretta, E., Francini, E., Vessella, S.: Determination of a linear crack in an elastic body from boundary measurements—Lipschitz stability. SIAM J. Math. Anal. 40(3), 984–1002, 2008MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Beretta, E., Francini, E., Kim, E., Lee, J.-Y.: Algorithm for the determination of a linear crack in an elastic body from boundary measurements. Inverse Probl. 26(8), 085015, 2010ADSMathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Bergh, J., Löfström, J.: Interpolation Spaces—An Introduction. Springer, Berlin 1976zbMATHCrossRefGoogle Scholar
  13. 13.
    Bonafede, M., Rivalta, E.: The tensile dislocation problem in a layered elastic medium. Geophys. J. Int. 136, 341–356, 1999ADSCrossRefGoogle Scholar
  14. 14.
    Cambiotti, G., Zhou, X., Sparacino, F., Sabadini, R., Sun, W.: Joint estimate of the rupture area and slip distribution of the 2009 L’Aquila earthquake by a Bayesian inversion of GPS data. Geophys. J. Int. 209, 992–1003, 2017ADSCrossRefGoogle Scholar
  15. 15.
    Cohen, S.: Convenient formulas for determining dip–slip fault parameters from geophysical observables. Bull. Seismol. Soc. Am. 86, 1642–1644, 1996Google Scholar
  16. 16.
    Colli Franzone, P., Guerri, L., Magenes, E.: Oblique double layer potentials for the direct and inverse problems of electrocardiology. Math. Biosci. 68, 23–55, 1984MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Deloius, B., Nocquet, J.-M., Vallée, M.: Slip distribution of the February 27, 2010 Mw= 8.8 Maule earthquake, central Chile, from static and high-rate GPS, InSAR, and broadband teleseismic data. Geophys. Res. Lett.37(17) (2010)Google Scholar
  18. 18.
    Eshelby, J.D.: Dislocation theory for geophysical applications. Philos. Trans. R. Soc. A274, 331–338, 1973ADSzbMATHCrossRefGoogle Scholar
  19. 19.
    Evans, E.L., Meade, B.J.: Geodetic imaging of coseismic slip and postseismic afterslip: sparsity promoting methods applied to the great Tohoku earthquake. Geophys. Res. Lett. 39, 1–7, 2012Google Scholar
  20. 20.
    Fuchs, M.: The Green-matrix for elliptic systems which satisfy the Legendre–Hadamard condition. Manuscripta Mathematica46, 97–115, 1984MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Fuchs, M.: The Green matrix for strongly elliptic systems of second order with continuous coefficients. Zeitschrift für Analysis und ihre Anwendungen6, 507–531, 1986MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Fukahata, Y., Wright, T.J.: A non-linear geodetic data inversion using ABIC for slip distribution on a fault with an unknown dip angle. Geophys. J. Int. 173, 353–364, 2008ADSCrossRefGoogle Scholar
  23. 23.
    Hanouzet, B.: Espaces de Sobolev avec poids. Application au problème de Dirichlet dans un demi-espace. Rendiconti del Seminario Matematico della Università di Padova46, 227–272, 1971MathSciNetzbMATHGoogle Scholar
  24. 24.
    Jiang, Z., Wang, M., Wang, Y., Wu, Y., Che, S., Shen, Z.K., Bürgmann, R., Sun, J., Yang, Y., Liao, H., Li, Q.: GPS constrained coseismic source and slip distribution of the 2013 Mw6. 6 Lushan, China, earthquake and its tectonic implications. Geophys. Res. Lett. 41, 407–413, 2014ADSCrossRefGoogle Scholar
  25. 25.
    Johnson, K.M., Hsu, Y.J., Segall, P., Yu, S.B.: Fault geometry and slip distribution of the 1999 Chi-Chi, Taiwan, earthquake imaged from inversion of GPS data. Geophys. Res. Lett. 28, 2285–2288, 2001ADSCrossRefGoogle Scholar
  26. 26.
    Koch, H., Lin, C.-L., Wang, J.-N.: Doubling inequalities for the Lamé system with rough coefficients. Proc. Am. Math. Soc. 144, 5309–5318, 2016zbMATHCrossRefGoogle Scholar
  27. 27.
    Kondrat’ev, V.A., Oleinik, O.A.: Boundary-value problems for the system of elasticity theory in unbounded domains. Korn’s inequalities. Russian Math. Surveys43, 65–119, 1988MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Kupradze, V.D.: Potential Methods in the Theory of Elasticity. Israel Program for Scientific Translations, Jerusalem 1965zbMATHGoogle Scholar
  29. 29.
    Li, Y., Nirenberg, L.: Estimates for elliptic systems from composite material. Dedicated to the memory of Jürgen K. Moser. Comm. Pure Appl. Math. 56(7), 892–925, 2003MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Lin, C.-L., Nakamura, G., Wang, J.-N.: Optimal three-ball inequalities and quantitative uniqueness for the Lamé system with Lipschitz coefficients. Duke Math. J. 155(1), 198–204, 2010zbMATHCrossRefGoogle Scholar
  31. 31.
    Lions, J.L., Magenes, E.: Non-Homogeneous Boundary Value Problems and Applications, vol. I. Springer, Berlin 1972zbMATHCrossRefGoogle Scholar
  32. 32.
    Martin, P.A., Päivärinta, L., Rempel, S.: A normal crack in an elastic half-space with stress-free surface. Math. Methods Appl. Sci. 16, 563–579, 1993ADSMathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Melrose, Rb: The Atiyah-Patodi-Singer index theorem. Research Notes in Mathematics, 4. A K Peters Ltd., Wellesley, MA (1993)Google Scholar
  34. 34.
    Miller, K.: Nonunique continuation for uniformly parabolic and elliptic equations in selfadjoint divergence form with Hölder continuous coefficients. Bull. Am. Math. Soc. 79, 350–354, 1973zbMATHCrossRefGoogle Scholar
  35. 35.
    Mindlin, R.D.: Force at a point in the interior of a semiinfinite solid. J. Appl. Phys. 7, 195–202, 1936ADSzbMATHGoogle Scholar
  36. 36.
    Mindlin, R.D.: Force at a point in the interior of a semi-infinite solid. Proceedings of The First Midwestern Conference on Solid Mechanics, April, University of Illinois, Urbana, Ill., 1954Google Scholar
  37. 37.
    Mitrea, D., Mitrea, M., Taylor, M.: Layer potentials, the Hodge laplacian, and global boundary problems in nonsmooth Riemannian manifolds. Mem. Am. Math. Soc.150(713), 2001MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Morassi, A., Rosset, E.: Stable determination of cavities in elastic bodies. Inverse Probl. 20, 453–480, 2004ADSMathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Nikkhoo, M., Walter, T.R.: Triangular dislocation: an analytical, artefact-free solution. Geophys. J. Int. 201, 1119–1141, 2015ADSCrossRefGoogle Scholar
  40. 40.
    Okada, Y.: Internal deformation due to shear and tensile fault in a half-space. Bull. Seismol. Soc. Am. 82(2), 1018–1040, 1992Google Scholar
  41. 41.
    Plis, A.: On non-uniqueness in Cauchy problem for an elliptic second order differential equation. Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 11, 95–100, 1963MathSciNetzbMATHGoogle Scholar
  42. 42.
    Rivalta, E., Mangiavillano, W., Bonafede, M.: The edge dislocation problem in a layered elastic medium. Geophys. J. Int. 149, 508–523, 2002ADSCrossRefGoogle Scholar
  43. 43.
    Segall, P.: Earthquake and Volcano Deformation. Princeton University Press, Princeton 2010zbMATHCrossRefGoogle Scholar
  44. 44.
    Serpelloni, E., Anderlini, L., Belardinelli, M.E.: Fault geometry, coseismic-slip distribution and Coulomb stress change associated with the 2009 April 6, M W 6.3, L’Aquila earthquake from inversion of GPS displacements. Geophys. J. Int. 188, 473–489, 2012ADSCrossRefGoogle Scholar
  45. 45.
    Simons, M., Fialko, Y., Rivera, L.: Coseismic deformation from the 1999 M w 7.1 Hector Mine, California, earthquake as inferred from InSAR and GPS observations. Bull. Seismol. Soc. Am. 92(4), 1390–1402, 2002CrossRefGoogle Scholar
  46. 46.
    Trasatti, E., Kyriakopoulos, C., Chini, M.: Finite element inversion of DInSAR data from the Mw 6.3 L’Aquila earthquake, 2009 (Italy). Geophys. Res. Lett. 38, 5, 2011CrossRefGoogle Scholar
  47. 47.
    Triebel, H.: Spaces of Kudrjavcev Type I. Interpolation, embedding, and structure. J. Math. Anal. Appl. 56, 253–271, 1976MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Triki, F., Volkov, D.: Stability estimates for the fault inverse problem. Inverse Probl. 35, 075007, 2019ADSMathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Volkov, D., Voisin, C., Ionescu, R.: Reconstruction of faults in elastic half space from surface measurements. Inverse Probl. 33, 055018, 2017ADSMathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    Volterra, V.: Sur l’equilibre des corps elastiques multiplement connexes. Annales scientifiques de l’ École Normale Supérieure24, 401–517, 1907zbMATHCrossRefGoogle Scholar
  51. 51.
    Walker, R.T., Bergman, E.A., Szeliga, W., Fielding, E.J.: Insights into the 1968–1997 Dasht-e-Bayaz and Zirkuh earthquake sequences, eastern Iran, from calibrated relocations, InSAR and high-resolution satellite imagery. Geophys. J. Int. 187, 1577–1603, 2011ADSCrossRefGoogle Scholar
  52. 52.
    Zielke, O., Mai, P.M.: Subpatch roughness in earthquake rupture investigations. Geophys. Res. Lett. 43, 1893–1900, 2016ADSCrossRefGoogle Scholar
  53. 53.
    Zhou, X., Cambiotti, G., Sun, W., Sabadini, R.: The coseismic slip distribution of a shallow subduction fault constrained by prior information: the example of 2011 Tohoku (\(\text{ M }_w\) 9.0) megathrust earthquake. Geophys. J. Int. 199, 981–995, 2014ADSCrossRefGoogle Scholar
  54. 54.
    Van Zwieten, G.J., Hanssen, R.F., Gutiérrez, M.A.: Overview of a range of solution methods for elastic dislocation problems in geophysics. J. Geophys. Res. Solid Earth118, 1721–1732, 2013ADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Johann Radon Institute for Computational and Applied Mathematics (RICAM)LinzAustria
  2. 2.Dipartimento di MatematicaPolitecnico di MilanoMilanItaly
  3. 3.Department of MathematicsNYU-Abu DhabiAbu DhabiUAE
  4. 4.Department of MathematicsPenn State UniversityUniversity ParkUSA
  5. 5.Department of Computational and Applied MathematicsRice UniversityHoustonUSA
  6. 6.Department of Earth, Environmental, and Planetary SciencesRice UniversityHoustonUSA

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