General formulations of global co-seismic deformations caused by an arbitrary dislocation in a spherically symmetric earth model-applicable to deformed earth surface and space-fixed point

Abstract

S U M M A R YBased on the authors' previous work, co-seismic deformations for a spherical symmetric earth model are summarized and reformulated. Unified expressions presented herein accommodate physical deformations: displacement, potential, gravity, geoid and strain changes. The corresponding Green's functions are derived by combining spheroidal and toroidal deformations. Sign errors in previous publications are corrected in these new formulas. These expressions are developed basically for a deformed earth surface because most traditional geodetic measurements are performed on the terrain surface. However, through development of space geodetic techniques, such as the satellite gravity missions, co-seismic gravity changes can be detected from space. In this case, the above dislocation theory (e.g., the co-seismic gravity change) cannot be applied directly to the observed data because the data do not include surface crustal deformation (the free air gravity change). Correspondingly, the contribution by the vertical displacement part must be removed from the traditional expressions. For this purpose, we present the corresponding expressions applicable to space observations. Some numerical technical problems are discussed. In addition, a smoothing technique is necessary to damp the high-frequency contribution so that the theory can be applied reasonably. Global co-seismic deformations caused by the 2004 Sumatra-Andaman earthquake (M9.3) are studied as an application of the new Green's function. That earthquake caused a global deformation detected by GPS, strain metres and even a satellite gravity mission. These global deformations are calculated based on the derived Green's functions and the seismic-wave derived earth model. A segment-summation scheme is used considering the slip distribution on a limited fault plane. The results are useful for interpreting observed deformations, especially those in the far field. The earthquake reveals global co-seismic deformations and effects of spherical curvature and the earth's layered structure. Comparisons between results for a spherical earth mode and a half-space model show a large discrepancy at an epicentral distance of about 1000 km, implying that effects of spherical curvature and layer structure are considerably large. In addition, the theoretical results are compared with the real observed strain steps, horizontal displacements and gravity changes caused by that earthquake. Good agreement validates the results of the current theoretical work. Finally, we discuss the application the above theory to the GRACE data through several case studies.

Effects of Earth's layered structure, gravity and curvature on coseismic deformation

“…In this case, the methods by Okubo (1991Okubo ( , 1992, Sun et al (2009), Pollitz (1996, 1997), Wang (1999 Note that the inversion study in Section 6 is just to investigate the effect of different earth models and theoretical frame, not for a real fault inversion. Indeed, the case study is not particularly innovative since it presents an inversion of static GPS offsets focusing on a layered direct model.…”

“…For source depths less than 100 km, the larger sphere is nearly equivalent with a half-space. Again, four independent point sources, that is, strike-slip, dip-slip, horizontal tensile and vertical tensile (Sun et al 2009), are considered for this investigation, which are located at depth of 20 and 100 km, respectively. Fig.…”

“…Okubo (1993) presented a reciprocity theorem between the dislocations and the conventional external forces in a SNREI (spherically symmetric, non-rotating, elastic and isotropic) earth model. Sun & Okubo (1993, 1998 and Sun et al (2009) introduced the dislocation Love numbers for calculating Green's functions for coseismic deformation, geoid and gravity changes based on the SNREI earth model. By comparing a layered spherical earth model (1066A) with a homogeneous half-space, Sun & Okubo (2002) found that the layering and curvature effects on the coseismic surface deformation (vertical displacement) can reach together the order of 20 per cent.…”

An analytical approach to estimate curvature effect of coseismic deformations

“…Sun & Okubo (1993) proposed an approach to compute changes in potential and gravity for a spherical earth model based on 1066A (Gilbert & Dziewonski 1975) model and PREM (Dziewonski & Anderson 1981) model. Sun et al (2009) defined the dislocation Love numbers and Green's functions for four independent sources (strike-slip, dip-slip, horizontal tensile and vertical tensile) to calculate coseismic deformation based on the SNREI earth model. Sun (2003Sun ( , 2004a also presented asymptotic solutions for calculating deformation in a spherically symmetric Earth.…”

On the response of the Earth to a fault system: its evaluation beyond the epicentral reference frame

“…We review the static first-order perturbation theory for calculating the displacement field and the local incremental gravitational potential of spherically symmetric self-gravitating elastic Earth models due to displacement dislocations with the aim of overcoming some complexities of already developed formalisms for dealing with this problem (Smylie and Manshina, 1971;Takeuchi and Saito, 1972;Ben-Menahem and Singh, 1981;Dahlen and Tromp, 1998). At present, the method for calculating static perturbations due to displacement dislocations deviates somehow from that used for other forcings, like surface and internal loads and tidal forces (Farrell, 1972;Sabadini et al, 1982;Wu and Peltier, 1982), although both methods rely on the spherical harmonic expansion of the perturbations.…”