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 — Mathématiques, Volume 4 (2017), pp. 223-255.

Motivated by a vulcanological problem, we establish a sound mathematical approach for surface deformation effects generated by a magma chamber embedded into Earth’s interior and exerting on it a uniform hydrostatic pressure. Modeling assumptions translate the problem into classical elasto-static system (homogeneous and isotropic) in a half-space with an embedded cavity. The boundary conditions are traction-free for the air/crust boundary and uniformly hydrostatic for the chamber boundary. These are complemented with zero-displacement condition at infinity (with decay rate). After a short presentation of the model and of its geophysical interest, we establish the well-posedness of the problem and provide an appropriate integral formulation for its solution for cavity with general shape. Based on that, assuming that the chamber is centered at some fixed point z and has diameter r>0, small with respect to the depth d, we derive rigorously the principal term in the asymptotic expansion for the surface deformation as ε=r/d0 + . Such a formula provides a rigorous proof of the Mogi point source model in the case of spherical cavities generalizing it to the case of cavities of arbitrary shape.

Motivés par un problème volcanologique, nous établissons une approche mathématique solide pour les effets de déformation de surface engendrés par une chambre magmatique contenue à l’intérieur de la Terre et soumise à une pression hydrostatique uniforme. Des hypothèses de modélisation traduisent le problème en un système élasto-statique (homogène et isotrope) classique dans un demi-espace avec une cavité incluse. Les conditions au bord sont sans traction au bord air/croûte, et uniformément hydrostatiques au bord de la chambre. Elles sont complétées par une condition de déplacement nul à l’infini (avec taux de décroissance). Après une courte présentation du modèle et de son intérêt géophysique, nous établissons que le problème est bien posé et proposons une formulation intégrale appropriée pour sa solution dans le cas d’une cavité de forme générale. En conséquence, supposant que la chambre est centrée en un point z et est de diamètre r>0 petit par rapport à la profondeur d, nous en déduisons rigoureusement le terme principal du développement asymptotique de la déformation de surface sous la forme ε=r/d0 + . Une telle formule permet de donner une preuve rigoureuse du modèle des points sources de Mogi dans le cas des cavités de forme arbitraire, qui généralise celle dans le cas des cavités sphériques.

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DOI: 10.5802/jep.42
Classification: 35C20, 31B10, 35J25, 86A60
Keywords: Lamé operator, asymptotic expansions, single and double layer potentials
Mot clés : Opérateur de Lamé, développement asymptotique, potentiels de simple et double couche

Andrea Aspri 1; Elena Beretta 2; Corrado Mascia 1

1 Dipartimento di Matematica “G. Castelnuovo”, Sapienza – Università di Roma, P.le Aldo Moro, 2 - 00185 Roma, Italy
2 Dipartimento di Matematica, Politecnico di Milano, Via Edoardo Bonardi 9 - 20133 Milano, Italy
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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     title = {Analysis of a {Mogi-type} model describing surface deformations induced by a magma chamber embedded in an elastic half-space},
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Andrea Aspri; Elena Beretta; Corrado Mascia. 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 — Mathématiques, Volume 4 (2017), pp. 223-255. doi : 10.5802/jep.42. https://jep.centre-mersenne.org/articles/10.5802/jep.42/

[1] Y. A. Abramovich & C. D. Aliprantis - An invitation to operator theory, Graduate Studies in Mathematics, vol. 50, American Mathematical Society, Providence, RI, 2002 | MR | Zbl

[2] G. Alessandrini, E. Beretta, E. Rosset & S. Vessella - “Optimal stability for inverse elliptic boundary value problems with unknown boundaries”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 29 (2000) no. 4, p. 755-806 | Numdam | MR | Zbl

[3] G. Alessandrini, M. Di Cristo, A. Morassi & E. Rosset - “Stable determination of an inclusion in an elastic body by boundary measurements”, SIAM J. Math. Anal. 46 (2014) no. 4, p. 2692-2729 | DOI | MR | Zbl

[4] H. Ammari, E. Bretin, J. Garnier, H. Kang, H. Lee & A. Wahab - Mathematical methods in elasticity imaging, Princeton Series in Applied Mathematics, Princeton University Press, Princeton, NJ, 2015 | Zbl

[5] H. Ammari & H. Kang - Reconstruction of small inhomogeneities from boundary measurements, Lect. Notes in Math., vol. 1846, Springer-Verlag, Berlin, 2004 | MR | Zbl

[6] H. Ammari & H. Kang - Polarization and moment tensors. With applications to inverse problems and effective medium theory, Applied Mathematical Sciences, vol. 162, Springer, New York, 2007 | Zbl

[7] C. Amrouche, M. Dambrine & Y. Raudin - “An L p theory of linear elasticity in the half-space”, J. Differential Equations 253 (2012) no. 3, p. 906-932 | DOI | MR | Zbl

[8] A. Aspri, E. Beretta & C. Mascia - “Asymptotic expansion for harmonic functions in the half-space with a pressurized cavity”, Math. Methods Appl. Sci. 39 (2016) no. 10, p. 2415-2430 | DOI | MR | Zbl

[9] M. Battaglia, J. Gottsmann, D. Carbone & J. Fernández - “4D volcano gravimetry”, Geophysics 73 (2008) no. 6, p. 3-18 | DOI

[10] M. Battaglia & D. P. Hill - “Analytical modeling of gravity changes and crustal deformation at volcanoes: the Long Valley caldera, California, case study”, Tectonophysics 471 (2009), p. 45-57 | DOI

[11] D. J. Cedio-Fengya, S. Moskow & M. S. Vogelius - “Identification of conductivity imperfections of small diameter by boundary measurements. Continuous dependence and computational reconstruction”, Inverse Problems 14 (1998) no. 3, p. 553-595 | DOI | MR | Zbl

[12] P. G. Ciarlet - Mathematical elasticity. Vol. I. Three-dimensional elasticity, Studies in Mathematics and its Applications, vol. 20, North-Holland Publishing Co., Amsterdam, 1988 | Zbl

[13] G. Currenti, A. Bonaccorso, C. Del Negro, D. Scandura & E. Boschi - “Elasto-plastic modeling of volcano ground deformation”, Earth Planetary Science Letters 296 (2010), p. 311-318 | DOI

[14] G. Currenti, C. Del Negro & G. Ganci - “Finite element modeling of ground deformation and gravity field at Mt. Etna”, Annals Geophysics 51 (2008) no. 1, p. 105-119

[15] B. E. J. Dahlberg, C. E. Kenig & G. C. Verchota - “Boundary value problems for the systems of elastostatics in Lipschitz domains”, Duke Math. J. 57 (1988) no. 3, p. 795-818 | DOI | MR | Zbl

[16] P. M. Davis - “Surface deformation due to inflation of an arbitrarily oriented triaxial ellipsoidal cavity in an elastic half-space, with reference to Kilauea Volcano, Hawaii”, J. Geophysical Research 91 (1986) no. B7, p. 7429-7438 | DOI

[17] G. De Natale & F. Pingue - “Ground Deformation Modeling in Volcanic Areas”, in Monitoring and Mitigation of Volcano Hazards (R. Scarpa & R. I. Tilling, eds.), Springer, 1996, p. 365-388 | DOI

[18] D. Dzurisin - Volcano Deformation. Geodetic Monitoring Techniques, Springer Praxis Books, Springer, 2006

[19] E. B. Fabes, C. E. Kenig & G. C. Verchota - “The Dirichlet problem for the Stokes system on Lipschitz domains”, Duke Math. J. 57 (1988) no. 3, p. 769-793 | DOI | MR | Zbl

[20] Y. Fialko, Y. Khazan & M. Simons - “Deformation due to a pressurized horizontal circular crack in an elastic half-space, with applications to volcano geodesy”, Geophys. J. Int. 146 (2001) no. 1, p. 181-190 | DOI

[21] G. Fichera - “Sull’esistenza e sul calcolo delle soluzioni dei problemi al contorno, relativi all’equilibrio di un corpo elastico”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 4 (1950) no. 248, p. 35-99 | MR | Zbl

[22] A. Friedman & M. Vogelius - “Identification of small inhomogeneities of extreme conductivity by boundary measurements: a theorem on continuous dependence”, Arch. Rational Mech. Anal. 105 (1989) no. 4, p. 299-326 | DOI | MR | Zbl

[23] M. E. Gurtin - The Linear Theory of Elasticity, Encyclopedia of Physics, Vol. VI a/2, Springer-Verlag, 1972

[24] B. B. Guzina & M. Bonnet - “Topological derivative for the inverse scattering of elastic waves”, Quart. J. Mech. Appl. Math. 57 (2004) no. 2, p. 161-179 | DOI | MR | Zbl

[25] T. Kato - Perturbation theory for linear operators, Springer-Verlag, Berlin, 1995 | Zbl

[26] R. Kress - Linear integral equations, Applied Mathematical Sciences, vol. 82, Springer-Verlag, Berlin, 1989 | MR

[27] V. D. Kupradze - Potential methods in the theory of elasticity, Israel Program for Scientific Translations, Jerusalem; Daniel Davey & Co., Inc., New York, 1965 | Zbl

[28] M. Lisowski - “Analytical volcano deformation source models”, in Volcano Deformation. Geodetic Monitoring Techniques (D. Dzurisin, ed.), Springer Praxis Books, Springer, 2006, p. 279-304

[29] A. Manconi, T. R. Walter & F. Amelung - “Effects of mechanical layering on volcano deformation”, Geophys. J. Int. 170 (2007) no. 2, p. 952-958 | DOI

[30] D. F. McTigue - “Elastic stress and deformation near a finite spherical magma body: resolution of the point source paradox”, Journal Geophysical Research 92 (1987) no. 12, p. 931-940

[31] R. D. Mindlin - “Force at a Point in the Interior of a SemiInfinite Solid”, Journal of Applied Physics 7 (1936), p. 195-202 | Zbl

[32] R. D. Mindlin - “Force at a Point in the Interior of a Semi-Infinite Solid”, in Proceedings of The First Midwestern Conference on Solid Mechanics (University of Illinois, Urbana, Ill.), 1954

[33] K. Mogi - “Relations between the eruptions of various volcanoes and the deformation of the ground surfaces around them”, Bull. Earthq. Res. Inst. U. Tokyo 36 (1958), p. 99-134

[34] A. Morassi & E. Rosset - “Stable determination of cavities in elastic bodies”, Inverse Problems 20 (2004) no. 2, p. 453-480 | DOI | MR | Zbl

[35] Y. Okada - “Surface deformation due to shear and tensile faults in a half-space”, Bull. Seismol. Soc. Am. 75 (1985), p. 1135-1154

[36] P. Segall - Earthquake and volcano deformation, Princeton University Press, Princeton, NJ, 2010 | DOI | Zbl

[37] P. Segall - “Volcano deformation and eruption forecasting”, Geological Society Special Publication 380 (2013) no. 1, p. 85-106 | DOI

[38] D. Volkov - “A double layer surface traction free Green’s tensor”, SIAM J. Appl. Math. 69 (2009) no. 5, p. 1438-1456 | DOI | MR | Zbl

[39] C. A. Williams & G. Wadge - “The effects of topography on magma chamber deformation models: application to Mt. Etna and radar interferometry”, Geophysicl Research Letters 25 (1998) no. 10, p. 1549-1552 | DOI

[40] X.-M. Yang & P. M. Davis - “Deformation from inflation of a dipping finite prolate spheroid in an elastic half-space as a model for volcanic stressing”, Journal Geophysical Research 93 (1988) no. B5, p. 4249-4257 | DOI

[41] K. Yosida - Functional analysis, Grundlehren Math. Wiss., vol. 123, Springer-Verlag, Berlin-New York, 1980 | Zbl

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