Surface deformation due to a strike-slip fault in an elastic gravitational layer overlying

We extend a technique previously used to model surface displacements resulting from thrust faulting in an elastic-gravitational l yer over a viscoelastic-gravitational h f-space to the case of strike-slip faulting. The method involves the calculation of the Green's functions for a strike-slip point source contained in an elastic-gravitational layer over a viscoelastic-gravitational half-space. The correspondence principle of linear viscoelasticity is applied to introduce time dependence. The resulting Green' s functions are then integrated over the source region to obtain the near-field displacements. Several sample calculations are presented involving 90 ø and 30 ø dipping faults and ruptures completely and partially through the elastic layer. We also illustrate the time dependent deformation due to a buried fault. Results show that the use of a viscoelastic half-space underlying an elastic layer introduces a long wavelength component into the deformation field [Cohen and Kramer, 1984], even in cases of non vertical strike-slip fault and inclusion of gravitational effect, that cannot be modeled by purely elastic techniques. Calculations have shown that vertical postseismic displacement isinsignificant and that the horizontal movement is about he same magnitude as the coseismic strike-slip displacement. The inclusion of gravity affects the horizontal displacement due to vertical strike-slip faulting in far field and the vertical displacement for dipping strike-slip faulting in near-field. The computed results have been fit to the Global Positioning System measurements of the Landers earthquake taken shortly after the main shock, assuming a relaxation time of the order of days. This relaxation time is considerably shorter than times of the order of years to decades found in previous studies. The major differences between this detailed three-dimensional nd simplified two-dimensional model are the decay of magnitude in displacement field and the distinct displacement pattern in the regions beyond the fault tip. The displacement field due to the cyclic earthquakes was constructed by considering the finite fault length and inclusion of gravity. It is found that the displacement field is dominated by the plate motion in the case of short recurrence time. On the other hand, a "looping" and migrating pattern in the displacement field is found in the case of very long recurrence time, which is not seen in those 2D simplified models. Introduction An important goal of modern crustal deformation studies is to understand the transient post event ground deformation sometimes seen following large earthquakes. Measurements of surface displacements reveal the presence of transient strain patterns in the crust after such events. Stress relaxation in a non elastic region situated below the surface lastic zone is believed to be one of the driving mechanisms of transient strain [Nut and Mavko, 1974]. A second possible xplanation is continued slip at depth on the fault [Fitch and Scholz, 1971]. The postseismic displacements can be explained by either of these two mechanisms using certain assumptions. By inverting the geodetic data and comparing the assumptions to the geology or seismicity •Now at Institute of Earth Sciences, Academia Sinica, Nankang, Taipei, Taiwan. Copyright 1996 by the American Geophysical Union. Paper number 95JB03118. 0148-0227/96/95JB-03118505.00 constraints, analytical models can be used to provide a means of distinguishing between these mechanisms. Elastic models involving dislocation sources in a homogeneous half-space [e.g., Chinnery, 1961] or in a layered half-space [e.g., Jovanovich et al., 1974] cannot explain transient deformations following an earthquake. Time dependence can be introduced by including viscoelastic relaxation or time dependent slip. Such models have been studied by Nut and Mavko [ 1974], Spence and Turcotte [1979], Savage and Prescott [1978], and Matsu'ura and Tanimoto [ 1980] among others. Rundle and Jackson [1977] used an analytic approximation of the Green's function to calculate displacements due to a strike-slip point source in an elastic layer over a viscoelastic half-space. Cohen and Kramer [ 1984], Li and Rice [1987], Lehner and Li [1982], and Spence and Turcotte [1979], used two-dimensional model with various approaches to study the displacement field or cyclic deformations due to infinite long vertical strike-slip faults. Yang and ToksOz [ 1981 ] studied the time dependent deformations due to finite vertical strike-slip faults with three-dimensional finite element method. They reported the significant effects in the region beyond the fault tip 3199 3200 YU ET AL.: DEFORMATION DUE TO STRIKE-SLIP FAULT which cannot be found in a simplified 2D model. Rundle [1978, 1980] used the exact Green's functions for finite quasi-static sources in an elastic layer over a linear viscoelastic half-space to calculate the viscoelastic crustal deformation for a thrust fault. Hofion et al. [1995] studied the surface deformation due to dike emplacement by exact calculation of viscoelastic-gravitational Green's functions. Ma and Kusznir [1992, 1993, 1994] calculated displacements for a fully relaxed model by setting the elastic Lfirne parameters equal to their relaxed values. However, they neither calculated deformation during transient relaxation nor did they show the viscoelastic behavior. Pollitz [1992] modeled the effects of postseismic relaxation on a spherical symmetric Earth with the normal mode formalism. We extend the work of Rundle [1980, 1981 ] to include the case of strike-slip faulting in an elasticgravitational ayer overlying a viscoelastic-gravitational half-space. The viscoelastic region must possess instantaneous elastic properties. We also assume small perturbation strains which imply that materials in the viscoelastic region obey linear-constitutive laws. Furthermore, a Maxwell rheology is assumed, which implies that the viscoelastic region behaves as an elastic solid over short time periods and as a Newtonian fluid over long timescales. We include gravitational effects in our calculations. For deformation at the surface of an elastic half-space, gravitational effects become significant over wavelengths greater than 1000 km [Rundle, 1980] but has little relevance to the near-field deformation for a vertical strike-slip fault because vertical displacements are insignificant. In viscoelastic models, stresses in some regions of Earth decrease as flow occurs: the initial elastic stresses induce flow in the medium, generating a change in the displacements and gravitational stresses as a result. Equilibrium is eventually attained between the gravitational and elastic stresses in the flowing region. However, for a vertical strike-slip fault, gravitational effects are insignificant in the near-field, and the inclusion of gravitational effects is important only in cases of non vertical dip angle. In this paper, surface displacements following a dipping strike-slip faulting are modeled using Green's functions. The solutions for the elastic-gravitational problem are computed, then the correspondence principle, which relates the elasticgravitational solution to the Laplace-transformed viscoelasticgravitational solution, is applied. Finally, the Green's functions are integrated over the finite source region to obtain the time dependent, near-field displacements. To show an example of using our model, a comparison between calculated results and the Global Positioning System (GPS) measurements of the Landers earthquake is made. The magnitude and decay pattern of survived data fit the result derived from this model, although the relaxation times obtained are considerably shorter than those found in previous studies of viscoelastic rebound [e.g., Thatcher and Rundle, 1984]. However, $hen et al. [1994] do find the same relaxation period in all the GPS baselines measured shortly after the Landers earthquake. The difference in these two findings may be caused by various relaxation mechanisms in diverse regions. where u is the perturbed displacement vector in the deformed cylindrical coordinate system (r, O, z), q• is the gravitational potential in this coordinate system, er, e& and e z are the unit vectors, o' is Poison's ratio, PO is the density, and/a is the rigidity. Rundle [1981] found that for displacements resulting from a dip-slip event in a layered elastic-gravitational medium, selfgravitation effects arising from the nonzero values of G O , the gravitational constant, were generally much smaller than gravitational effects relating to the surface acceleration g. As z -->+0% all perturbed quantities are presumed to tend to zero, so setting G O = 0 implies 4 is constant. Making use of this, Rundle [1981 ] used the equation v2u+ VV.u+Pøgv(u.e,)PøgezV.u=0. (3) 1-2o g p Using the vector base, Pm = erJm ( kr)exp( im0 ) (4a) B ( 0 e 1 0--•)(kr xp(i 0) = r + e0 j )e rn m Ok r • m (4b) ( 1 O 0--•--•-r) (kr) p(im0) C m = • .•-er e0 Jm ex (4c) where drn(kr) a e cylindrical Bessel functions, i = x/-1 and k corresponds to the wave number in dynamic problems [BenMenahem and Sing& 1968], we can expand u in (4a) -(4c): t• = Z kdk[Wm(z) Pm +Um(z) Bm + Vm(z)Cm] ' m=00 The solution given by l'm(Z ) is not considered further here as it is identical to the solution in the nongravitating case. Urn(Z) and Wrn(z) are given by Rundle [1981]: Um(z)) ( 1 1 a'z ( i )1 +k(p•(k) ) + (p•ik))e-a2z(6) W m(z)) = Pt(k e+Pik e-a'z 1 ea2Z where + a 1 = + k 2 + kr I + a2 = + k 2 kq 1 2• Pog -20 o)' p krl •" 1 + ,•'p•:(k) = +l(aj+ k) ajk•"' (7) Solution to the Infinite Space Problem The equations to be solved [Love, 1911] are: where, j=l, 2. The gravitational wave number kg found by setting a2(k) = 0 is defined as V2u+ 1 1-2o P0g ( ) P0 P0 g W-u+ V u.e z -•V4•ezV.u=0 (1) V24• =-4xP0G0 V .u (2) kg =n•]-•. (8) For k < kg, a2 is purely imaginary, and for k > kg, a2 is real. YU ET AL.' DEFORMATION DUE TO STRIKE-SLIP FAULT 3201 Solution to the Layered Half-Space Problem Rundle [ 1980] used a polar coordinate system (r, O, z) and the z axis oriented down into the medium at the surface of a layered, elastic-gravitational half-space. The elastic module in the nth layer are denoted by )•n and gn and the density by On The solution in the nth layer is given by