On the Mathematical Analysis of an Elastic-gravitational Layered Earth Model for Magmatic Intrusion: The Stationary Case

Thumbnail Image
Full text at PDC
Publication Date
Advisors (or tutors)
Journal Title
Journal ISSN
Volume Title
Google Scholar
Research Projects
Organizational Units
Journal Issue
In the early eighties Rundle (1980, 1981a,b, 1982) developed the techniques needed for calculations of displacements and gravity changes due to internal sources of strain in layered linear elastic-gravitational media. The approximation of the solution for the half space was obtained by using the propagator matrix technique. The Earth model considered is elastic-gravitational, composed of several homogeneous layers overlying a bottom half space. Two dislocation sources can be considered, representing magma intrusions and faults. In recent decades theoretical and computational extensions of that model have been developed by Rundle and co-workers (e.g., Fernandez and Rundle, 1994a,b; Fernandez et al., 1997, 2005a; Tiampo et al., 2004; Charco et al., 2006, 2007a,b). The source can be located at any depth in the media. In this work we prove that the perturbed equations representing the elastic-gravitational deformation problem, with the natural boundary and transmission conditions, leads to a well-posed problem even for varied domains and general data. We present constructive proof of the existence and we show the uniqueness and the continuous dependence with respect to the data of weak solutions of the coupled elastic-gravitational field equations.
UCM subjects
Unesco subjects
AKI, k., and RICHARDS, P.G., Quantitative Seismology, (University Science Books, Sausalito, California., 2002). BRÉZIS, H., Analyses fonctionelle: théorie et applications, (Alianza, Madrid., 1984). BUSTIN, A., HYNDMAN, R.D., LAMBERT, A., RISTAU, J., He, J., DRAGERT, H., and VAN DER KOOIJ, M. (2004), Fault parameters of the Nisqually earthquake determined from moment tensor solutions and the surface deformation from GPS and InSAR, Bull. Seismol. Soc. Am. 94, 2, 363–376. CHARCO, M., FERNÁNDEZ, J., LUZÓ N, F., and RUNDLE, J. B. (2006), On the relative importance of self-gravitation and elasticity in modelling volcanic ground deformation and gravity changes, J. Geophys. Res. 111, B03404, doi:10.1029/2005JB003754. CHARCO, M., LUZÓ N, F., FERNÁNDEZ, J., and TIAMPO, K.F. (2007), Topography and selfgravitation interaction in elastic-gravitational modelling, Geochem. Geophy. Geosystems (G3), 8, Q01001, doi:10.1029/2006GC001412. CHARCO, M., FERNÁNDEZ, J., LUZÓ N, F., TIAMPO, K.F., and RUNDLE, J.B. (2007b), Some insights about topographic, elastic and self-gravitation interaction in modelling ground deformation and gravity changes in active volcanic areas, Pure appl. geophys. 164/4, 865–878. DÍAZ, J.I., and TALENTI, G. (2004), A free boundary problem related to the location of volcanic gas sources, Pure Appl. Geophys. 161, 1509–1517. DIXON, T.H., MAO, A., BURSIK, M., HEFLIN, M., LANGBEIN, J., STEIN, R., and WEBB, F. (1997), Continuous monitoring of surface deformation at Long Valley Caldera, California, with GPS, J. Geophys. Res. 102, 12017–12034. DZURISIN, D. (2003), A comprehensive approach to monitoring volcano deformation as a window on the eruptive cycle, Rev. Geophys. 41(1), 1001, doi:10.1029/2001RG000107 (Correction: 2003,41(2), 1009, doi:10.1029/2003RG000134). FERNÁNDEZ, J., and RUNDLE, J.B. (1994a), Gravity changes and deformation due to a magmatic intrusion in a two-layered crustal model, J. Geophys. Res. 99, 2737–2746. FERNÁNDEZ, J., and RUNDLE, J.B. (1994b), FORTRAN program to compute displacement, potential and gravity changes resulting from a magma intrusion in a multilayered Earth model, Comp. Geosci. 20, 461–510. FERNÁNDEZ, J., RUNDLE, J.B., GRANELL, R.D.R, and YU, T.-T. (1997), Programs to compute deformation due to a magma intrusion in elastic-gravitational layered Earth models, Comp. Geosci. 23, 231–249. FERNÁNDEZ, J., CHARCO, M., TIAMPO, K.F., JENTZSCH, G., and RUNDLE, J. B. (2001), Joint interpreta- tion of displacement and gravity data in volcanic areas. A test example: Long Valley- Caldera, California, J. Volcanol. Geotherm. Res. 28, 1063-1066. FERNÁNDEZ, J., TIAMPO, K.F., RUNDLE, J.B., and JENTZSCH, G. (2005a), On the interpretation of vertical gravity gradients produced by magmatic intrusions, J. Geodyn. 39/5, 475–492. doi: 10.1016/j.jog.2005.04.005. FERNÁNDEZ, J., ROMERO, R., CARRASCO, D., TIAMPO, K.F., RODRÍGUEZ-VELASCO, G., APARICIO, A., ARAN˜ A, V., and GONZA´ LEZ-MATESANZ, F.J. (2005b), Detection of displacements in Tenerife Island Canaries, using radar interferometry, Geophys. J. Int. 160, 33–45. doi:10.111/j.1365-246x.200502487x. GILBARG, D., and TRUDINGER, N.S., Elliptic Partial Differential Equations of Second Order, (Springer-Verlag, Berlin., 1977) GUDMUNSSON, S., and SIGMUNDSSON, F. (2002), Three-dimensional surface motion maps estimated from combined interferometric systhetic aperture radar and GPS data, J. Geo-phys. Res. 107, B10, 2250. doi:10.1029/2001JB000283. LANARI, R., BERARDINO, P., BORGSTRO¨M, S., DEL GAUDIO, C., DE MARTINO, P., FORNARO, G., GUARINO, S., RICCIARDI, G.P., SANSOSTI, E., and LUNDGREN, P. (2004), The use of IFSAR and classical geodetic techniques for caldera unrest episodes: Application to the Campi Flegrei uplift event of 2000, J. Volcanol. Geotherm. Res. 133, 247–260. LARSON, K.M., CERVELLI, P., LISOWSKI, M., MIKLIUS, A., SEGALL, P., and OWEN, S. (2001), Volcano monitoring using the Global Positioning System: Filtering strategies, J. Geo-phys. Res. 106, B9, 19453–19464. LIONS, J.L. (1981), Some methods in the mathematical analysis of systems and their control, Science, Beijing. LOVE, A.E.H., Some problems in Geodynamics, (Cambridge University Press, New York., 1911). LUNDGREN, P., and STRAMONDO, S., (2002), Slip distribution of the 1997 Umbria-Marche earthquake sequence: Joint inversion of GPS and sysnthetic aperture radar interferometry data, J. Geophys. Res. 107, B11, 2316. doi:10.1029/2000JB000103. MANZO, M., RICCIARDI, G.P., CASU, F., VENTURA, G., ZENI, G., BORGSTRO¨M, S., BERARDINO, P., DEL GAUDIO, C., and LANARI, R. (2006), Surface deformation analysis in the Ischia Island (Italy) based on spaceborne radar interferometry, J. volcanol. Geother. Res. 151, 399–416. NATIONAL RESEARCH COUNCIL OF THE NATIONALACADEMIES, Living on an active Earth. Per- spectives on Earthquake Science, (The National Academic Press, Washington, D.C., 2003, 33pp.) PRITCHARD, M.E., and SIMONS, M. (2002), A satellite geodetic survey of large-scale deformation of volcanic centres in the central Andes, Nature 418, 167–171. PUGLISI, G., and COLTELLI, M. (2001), SAR Interferometry applications on active volcanoes: state of the art and perspective for volcano monitoring, Il Nuovo Cimento 24C, 133–145. RUNDLE, J.B. (1980), Static elastic-gravitational deformation of a layared half space by point couple, J. Geophys. Res., 85, 5355–5363. RUNDLE, J.B. (1981a), Numerical Evaluation of static elastic-gravitational deformation of a layared half space by point couple sources, Rep., Sand 80–2048. RUNDLE, J.B. (1981b), Vertical displacements from a rectangular fault in layered elasticgravitational media, J. Phys. Earth 29, 173–186. RUNDLE, J.B. (1982), Viscoeslastic-gravitational deformation by a rectangular thrust fault in a layered Earth, J. Geophys. Res. 87, 9 ,7787–7796. (Correction: J. Geophys. Res. 88, 10.647–10.653). RUNDLE, J.B. (1983), Correction to ‘‘Deformation, gravity and potential changes due to volcanic loading of the crust’’, J. Geophys. Res, 88, 10, 647–10, 653. SAGIYA, T., MIYAZAKI, S., and TADA, T. (2000), Continuous GPS array and present-day crustal deformation of Japan, Pure appl. Geophys. 157, 2303–2322. SAMSONOV, S., and TIAMPO, K., (2006), Analytical optimization of InSAR and GPS dataset for derivation of threedimensional surface motion, (Journal, etc. missing) SEGALL, P., and DAVIS, J. (1997), GPS applications for geodynamics and earthquakes studies, Annual Rev. Earth Planet. Sci. 25, 301-336. SIGURDSSON, H., HOUGHTON, B., MCNUTT, S. R., RYMER, H., and STIX, J. (2000), Encyclopedia of Volcanoes, (Academic Press, 1417, 34pp.) TAMISIEA, M.E., MITROVICA, J.X., and DAVIS, J. L. (2007), GRACE Gravity Data Constrain Ancient Ice Geometries and Continental Dynamics over Laurentia, Science 316, 5826, TIAMPO, K.F., FERNÁNDEZ, J., JENTZSCH, G., CHARCO, M., and RUNDLE, J.B. (2004), Inverting for the parameters of a volcanic source using a genetic algorithm and a model for magmatic intrusion in elastic-gravitational layered Earth models, Comp. Geosci. 30/9- 10 985–1001. WRIGHT, T.J. (2002), Remote monitoring of the earthquake cycle using satellite radar interfer- ometry, Phil. Trans. R. Soc. Lond. A 360, 2873–2888. YOSHIYUKI, T., SHUHEI, O., MORITO, M., ISAO, K., and TOSHIHIRO, K., (2001), First detection of absolute gravity change caused by earthquake, Geophys. Res. Lett. 28, 2979–2981.