Abstract
We describe an algorithm for rapidly computing the surface displacements induced by a general polygonal load on a layered, isotropic, elastic half-space. The arbitrary surface pressure field is discretized using a large number, n, of equally-sized circular loading elements. The problem is to compute the displacement at a large number, m, of points (or stations) distributed over the surface. The essence of our technique is to reorganize all but a computationally insignificant part of this calculation into an equivalent problem: compute the displacements due to a single circular loading element at a total of m n stations (where m n is the product m × n). We solve this “parallel” problem at high computational speed by utilizing the sparse evaluation and massive interpolation (SEMI) method. Because the product m n that arises in our parallel problem is normally very large, we take maximum possible advantage of the acceleration achieved by the SEMI algorithm.
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Becker, J.M., and M. Bevis (2004), Love’s problem, Geophys. J. Int. 156, 2, 171–178, DOI: 10.1111/j.1365-246X.2003.02150.x.
Bevis, M., E. Kendrick, A. Cser, and R. Smalley Jr. (2004), Geodetic measurement of the local elastic response to the changing mass of water in Lago Laja, Chile, Phys. Earth Planet. In. 141, 2, 71–78, DOI: 10.1016/j.pepi.2003.05.001.
Bevis, M., D. Alsdorf, E. Kendrick, L.P. Fortes, B. Forsberg, R. Smalley Jr., and J. Becker (2005), Seasonal fluctuations in the mass of the Amazon River system and Earth’s elastic response, Geophys. Res. Lett. 32, 16, L16308, DOI: 10.1029/2005GL023491.
Bevis, M., E. Kendrick, R. Smalley Jr., I. Dalziel, D. Caccamise, I. Sasgen, M. Helsen, F.W. Taylor, H. Zhou, A. Brown, D. Raleigh, M. Willis, T. Wilson, and S. Konfal (2009), Geodetic measurements of vertical crustal velocity in West Antarctica and the implications for ice mass balance, Geochem. Geophys. Geosyst. 10, 10, Q01005, DOI: 10.1029/2009GC002642.
Bevis, M., J. Wahr, S.A. Khan, F.B. Madsen, A. Brown, M. Willis, E. Kendrick, P. Knudsen, J.E. Box, T. van Dam, Caccamise II, B. Johns, T. Nylen, R. Abbott, S. White, J. Miner, R. Forsberg, H. Zhou, J. Wang, T. Wilson, D. Bromwich, and O. Francis (2012), Bedrock displacements in Greenland manifest ice mass variations, climate cycles and climate change, Proc. Natl. Acad. Sci. USA 109, 30, 11944–11948, DOI: 10.1073/pnas.1204664109.
Blewitt, G., D. Lavallée, P. Clarke, and K. Nurutdinov (2001), A new global mode of Earth deformation: Seasonal cycle detected, Science 294, 5550, 2342–2345, DOI: 10.1126/science.1065328.
Boussinesq, J.V. (1885), Application des Potentiels à l’Etude de l’Equilibre et du Mouvement des Solides Elastiques, Gauthier-Villars, Paris, 508 pp. (in French).
Davis, J.L., P. Elósegui, J.X. Mitrovica, and M.E. Tamisiea (2004), Climate-driven deformation of the solid Earth from GRACE and GPS, Geophys. Res. Lett. 31, 24, L24605, DOI: 10.1029/2004GL021435.
Dong, D., P. Fang, Y. Bock, M.K. Cheng, and Miyazaki (2002), Anatomy of apparent seasonal variations from GPS-derived site position time series, J. Geophys. Res. 107, B4, 2075, DOI: 10.1029/2001JB000573.
Dziewonski, A.M., and D.L. Anderson (1981), Preliminary reference Earth model, Phys. Earth Planet. In. 25, 4, 297–356, DOI: 10.1016/0031-9201(81)90046-7.
Hager, B.H. (1991), Weighing the ice sheets using space geodesy: A way to measure changes in ice sheet mass, EOS Trans. AGU 72, 17, 91.
Heki, K. (2001), Seasonal modulation of interseismic strain buildup in northeastern Japan driven by snow loads, Science 293, 5527, 89–92, DOI: 10.1126/science.1061056.
Khan, S.A., J. Wahr, L.A. Stearns, G.S. Hamilton, T. van Dam, K.M. Larson, and O. Francis (2007), Elastic uplift in southeast Greenland due to rapid ice mass loss, Geophys. Res. Lett. 34, 21, L21701, DOI: 10.1029/2007 GL031468.
Krabill, W., W. Abdalati, E. Frederick, S. Manizade, C. Martin, J. Sonntag, R. Swift, R. Thomas, W. Wright, and J. Yungel (2000), Greenland ice sheet: High-elevation balance and peripheral thinning, Science 289, 5478, 428–430, DOI: 10.1126/science.289.5478.428.
Lamb, H. (1901), On Boussinesq’s problem, Proc. Lond. Math. Soc. 34, 1, 276–284, DOI: 10.1112/plms/s1-34.1.276.
Love, A.E.H. (1929), The stress produced in a semi-infinite solid by pressure on part of the boundary, Philos. Trans. Roy. Soc. Lond. A 228, 377–420, DOI: 10.1098/rsta.1929.0009.
Mooney, W.D., G. Laske, and T.G. Masters (1998), CRUST 5.1: A global crustal model at 5° × 5°, J. Geophys. Res. 103, B1, 727–747, DOI: 10.1029/97JB02122.
Pan, E., M. Bevis, F. Han, H. Zhou, and R. Zhu (2007), Surface deformation due to loading of a layered elastic half-space: A rapid numerical kernel based on a circular loading element, Geophys. J. Int. 171, 1, 11–24, DOI: 10.1111/j.1365-246X.2007.03518.x.
Sasgen, I., D. Wolf, Z. Marinec, V. Klemann, and J. Hagerdorn (2005), Geodetic signatures of glacial changes in Antarctica: Rates of geoid height change and radial displacement due to present and past ice-mass variations, GFZ Scientific Technical Rep. STR05/01, GeoForschungsZentrum, Potsdam, Germany.
Terazawa, K. (1916), On the elastic equilibrium of a semi-infinite solid under given boundary conditions, with some applications, J. Coll. Sci. Imp. Univ. Tokyo 37, 7, 1–64.
van Dam, T., J. Wahr, P.C.D. Milly, A.B. Shmakin, G. Blewitt, D. Lavallée, and K.M. Larson (2001), Crustal displacements due to continental water loading, Geophys. Res. Lett. 28, 4, 651–654, DOI: 10.1029/2000GL012120.
Zhou, H. (2008), Layered Cartesian half-space models for Earth’s elastic response to contemporary surface loading phenomena, Ph.D. Thesis, Ohio State University, Columbus, USA, 160 pp.
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Bevis, M., Pan, E., Zhou, H. et al. Surface Deformation due to Loading of a Layered Elastic Half-space: Constructing the Solution for a General Polygonal Load. Acta Geophys. 63, 957–977 (2015). https://doi.org/10.1515/acgeo-2015-0034
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DOI: https://doi.org/10.1515/acgeo-2015-0034