Green s functions of coseismic strain changes and investigation of effects of Earth s spherical curvature and radial heterogeneity

Transcription

1 Geophys. J. Int. (2006) 167, doi: /j X x Green s functions of coseismic strain changes and investigation of effects of Earth s spherical curvature and radial heterogeneity Wenke Sun, Shuhei Okubo and Guangyu Fu Earthquake Research Institute, University of Tokyo, Tokyo , Japan. sunw@eri.u-tokyo.ac.jp Accepted 2006 May 19. Received 2006 March 27; in original form 2005 August 15 SUMMARY A set of Green s functions is presented for calculating the coseismic strain caused by four independent seismic sources in a spherically symmetric, non-rotating, perfectly elastic, and isotropic (SNREI) earth model. Coesponding expressions are derived assuming that the seismic sources are located at the polar axis. The proper combination of these expressions allows calculation of the coseismic strain components resulting from an arbitrary seismic source at any Earth position. Calculations of strain components are made for the near field resulting from the four independent sources at a depth of 32 km inside the 1066A earth model. Results agree well with those calculated for a half-space earth model, thus confirming the validity of the theory presented in this research. A case study is performed and earth model effects are investigated. Furthermore, this paper investigates effects of spherical curvature and the stratified structure of the Earth in computing coseismic strain changes. Curvature effects are small for three types of seismic sources, but extremely large for the horizontal tensile opening on the vertical fault plane. Because a general coseismic deformation comprises four independent solutions, the large curvature effect on the horizontal tensile opening source contributes to the general result. Effects of Earth s stratified structure are large for all depths and epicentral distances. They reach a discrepancy greater than 30 per cent almost everywhere, and 100 per cent in a very far field. Results show that effects of crustal structure mainly exist in the near field; they do not affect results for a far field. Key words: dislocation theory, earthquake, spherical earth model, strain. GJI Seismology 1 INTRODUCTION Advances in modern geodetic techniques such as GPS, InSAR, and altimetry allow better detection of coseismic deformations such as displacement, gravity change, and strain. Such geophysical geodetic information is useful for studying seismic mechanisms, Earth structure, and earthquake prediction. A quasi-static dislocation theory is necessary to properly apply the observed geophysical phenomena to interpret or invert the seismic parameters. Numerous studies have been undertaken by many scientists to study coseismic deformation in a half-space earth model. Among them are Steketee (1958), Maruyama (1964), Press (1965), Jovanovich et al. (1974a,b) and Okada (1985). Those studies presented analytical expressions for calculating surface displacement, tilt, and strain resulting from various dislocations buried in a semi-infinite (half-space) medium. Especially, Okada (1985) summarized previous studies and presented a complete set of analytical formulae for calculating these geodetic deformations. Okubo (1991, 1992) proposed closed-form expressions to describe potential and gravitic changes resulting from dislocations. Because of their mathematical simplicity, these dislocation theories (e.g. Okada 1985; Okubo 1991, 1992) have been applied widely to study or invert seismic faults. However, the validity of these theories is strictly limited to a near field because the Earth s curvature and radial heterogeneity are ignored. Modern geodesy can detect and observe far-field crustal deformation. Consequently, even a global co-seismic deformation, a dislocation theory for a more realistic earth model, is demanded to interpret far-field deformation. Efforts to develop formations for such an earth model have been advanced through numerous studies (e.g. Ben-Menahem & Singh 1968; Ben-Menahem & Israel 1970; Smylie & Mansinha 1971). Such studies have revealed that the Earth s curvature effects are negligible for shallow events, whereas vertical layering might have considerable effects on deformation fields. However, Sun & Okubo s (2002) recent study comparing discrepancies between a half-space and a homogeneous sphere (accounting for self-gravitation) and between a homogeneous sphere and a stratified sphere indicates that both curvature and vertical layering exert marked effects on coseismic deformation. Amelung & Wolf (1994) studied the spherical effect problem for surface loading. They compared spherical earth models with incremental gravitational force (IGF) and plate earth models without IGF and found that eors resulting from neglect of sphericity and the IGF partially compensate each C 2006 The Authors 1273

2 1274 W. Sun, S. Okubo and G. Fu other. Sabadini & Vermeersen (1997) investigated the influence of lithospheric and mantle stratification on global coseismic and post-seismic deformation based on the normal mode technique. They found that the mantle viscosity structure has a major influence on post-seismic deformation in a far field. Saito (1967) presented a theory to calculate the amplitudes of free oscillations caused by a point source in a spherically symmetric sphere. He expressed his results in terms of normal mode solutions and source functions. Stratified sphere models, such as the 1066A model (Gilbert & Dziewonski 1975) or the PREM model (Dziewonski & Anderson 1981), are the most realistic: they reflect both sphericity and the stratified structure of the Earth. For such an earth model, Rundle (1982) studied viscoelastic gravitational deformation by a rectangular thrust fault in a layered Earth. Pollitz (1992) solved the problem of regional displacement and strain fields induced by dislocation in a viscoelastic, non-gravitational model. Pollitz et al. (2004) also explained strain accumulation in San Francisco Bay. Sun & Okubo (1993, 1998) and Sun et al. (1996) presented methods to calculate coseismic displacements and changes in gravity in spherically symmetric earth models. Okubo (1993) proposed a reciprocity theorem for connecting solutions of dislocation and tidal, shear, and load deformations. That study found that deformation on the Earth s surface caused by dislocations at source radius r = r s are expressible simply using a linear combination of the tide, shear and load deformations at r = r s. Ma & Kusznir (1994) modified elastic dislocation theory to derive subsurface displacements for faults in a three-layer elastic gravitational medium and its application to examine coseismic and post-seismic surface and subsurface displacement during continental extensional faulting. Piersanti et al. (1995, 1997) and Sabadini et al. (1995) studied displacement and its rates induced by dislocation in viscoelastic, stratified earth models, accounting for sphericity and self-gravitation using a self-consistent approach. They produced near-field and far-field surface displacement and velocity results for various viscosity profiles in the mantle. Pollitz (1996) investigated coseismic displacement and strain from earthquake faulting on a layered spherical Earth using normal mode technique, ignoring the Earth s self-gravitation. He asserted that the effect of sphericity is generally less than about 2 per cent of maximum deformation within 100 km of an earthquake source at crustal depths, and that up to 20 per cent eor would be introduced if the Earth s layered structure were ignored. Antonioli et al. (1998) discussed stress diffusion following large strike-slip earthquakes to observe the spherical and layered effects by comparing results of spherical and plate earth models. Nostro et al. (1999, 2001) studied the effects of the Earth s sphericity and layering through coseismic and post-seismic deformations. Banerjee et al. (2005) and Hearn & Burgmann (2005) dealt with the same issue for the 2004 Sumatra earthquake and a strike-slip source, respectively. All of those studies used the normal mode approach; some investigations were performed for either specified modes (Antonioli et al. 1998; Nostro et al. 1999) or specified seismic source, for example, strike-slip fault (Antonioli et al. 1998). On the other hand, the computing accuracy of the normal mode method is suspect because it has intrinsic numerical difficulties and because it requires artificial assumptions of compressibility and layer numbers (Tanaka et al. 2006). For those reasons, this study presents a new method to calculate the coseismic strain tensor for a more realistic earth model (e.g. SNREI). This method is applicable to any layered symmetric earth model (e.g. 1066A or PREM) without artificial assumptions. Principally, the strain tensor is easily obtainable by taking derivatives of a displacement field. These displacement solutions have been known since the 1960s (Saito 1967; Takeuchi & Saito 1972), particularly the displacement Green s functions given by Sun et al. (1996). Computing the displacement gradient is sufficient, theoretically, bridge the knowledge gap from the displacement field to strain. However, it is difficult to do so in practice, particularly using a spherical earth model. Therefore, this research is intended to present new practical expressions of strain components derived from displacement solutions in a spherical Earth caused by a dislocation point source. These expressions, called Green s functions, are aanged in a combination of four types of fundamental sources. Implementing those expressions in a computer code is feasible. These formulations are useful to calculate strain components caused by any kind of dislocation at an arbitrary position on the Earth. In addition, the strain Green s functions are useful to investigate effects of sphericity and a stratified Earth structure in computing coseismic strain changes by applying different earth models and dislocation theories. 2 DISPLACEMENT GREEN S FUNCTIONS OF ELASTIC RESPONSES OF A SPHERE TO A POINT DISLOCATION According to our previous work (Sun et al. 1996), a dislocation model is assumed in Fig. 1. It is defined at radial distance r s on an infinitesimal fault ds by slip vector ν, normal n, slip angle λ, and dip angle δ in the e 1, e 2, e 3 coordinate system; unit vectors e 1 and e 2 are taken, respectively, in the equatorial plane in the directions of longitude ϕ = 0 and π/2, and e 3 along polar axis r. The relative movement of the two fault sides is defined as (U/2) ( U/2) = U. Therefore, displacement vector Uν and normal n of the source are expressible as Uν = U(ν 1 e 1 + ν 2 e 2 + ν 3 e 3 ) n = n 1 e 1 + n 2 e 2 + n 3 e 3. (1) Note that for a tensile opening, the slip vector and the normal become equal: ν = n. If dislocation occurs in a spherical Earth, such as in a homogeneous sphere or a SNREI earth (spherically symmetric, non-rotating, perfectly elastic and isotropic, Dahlen 1968), the excited displacement u(r, θ, ϕ) (radial distance, colatitude and longitude) is describable in the form of displacement Green s functions along with spherical coordinates (e r, e θ, e ϕ ) as: u(a,θ,ϕ) = i, j [ u ij r e r + ( u s,ij θ + u t,ij ) θ eθ + ( u s,ij ϕ ) ] + u t,ij UdS ϕ eϕ νi n j, (2) a 2

3 Green s functions of coseismic strain changes 1275 Figure 1. A dislocation model expressed by the slip vector ν, normal n, slip angle λ, and dip angle δ in the e 1, e 2, e 3 coordinate system. Relative movement of the two fault sides is defined as (U/2) ( U/2) = U. The bottom row represents four seismic sources used in this study. where n u ij r (a,θ,ϕ) = u s,ij θ (a,θ,ϕ) = u s,ij ϕ n=0 m= n (a,θ,ϕ) = u t,ij θ (a,θ,ϕ) = n y n,ij n=0 m= n n n=0 m= n n n=0 m= n y n,ij 1,m (a)y m n (θ,ϕ), 3,m (a) Y n m(θ,ϕ), θ y n,ij 3,m (a) 1 sin θ y t,n,ij 1,m (a) 1 sin θ Yn m(θ,ϕ), ϕ Yn m(θ,ϕ), ϕ n u t,ij ϕ (a,θ,ϕ) = y t,n,ij 1,m (a) Y n m(θ,ϕ), (7) θ n=0 m= n with Yn m(θ,ϕ) = Pm n eimϕ (8) Yn m (θ,ϕ) = ( 1) m P n m e i m ϕ. In those equations, Pn m is the associated Legendre s function and a is the Earth s radius; the superscript s represents spheroidal deformation and t stands for toroidal deformation. Note that factor a 2 in eq. (2) comes from a normalization of y-symbols, in which variables y n,ij k,m (a) and yt,n,ij k,m (a) in eqs (3) (7) become dimensionless. The y-variables yn,ij k,m (a) and yt,n,ij k,m (a) are obtainable by solving the linearized first order equations of equilibrium, stress strain relation, and Poisson s equation for excited deformation (Saito 1967; Takeuchi & Saito 1972): Ẏ s = A s Y s (9) Ẏ t = A t Y t, where Y s = (y n,ij 1,m (3) (4) (5) (6),..., yn,ij 6,m )T, Y t = (y t,n,ij 1,m, yt,n,ij 2,m )T, and dot represents the derivative with respect to r. Used in eq. (4), y n,ij 6,m is the variable, and yn,ij 5,m is defined in Takeuchi & Saito (1972). As and A t are the coefficient matrices depending on the earth model. The related with y n,ij 1,m solutions Y s and Y t satisfy the following discontinuity across the radius of the source r = r s : S s,t = [ Y s,t (r s + 0) Y s,t (r s 0) ] δ(r r s ). Vectors S s = (s n,ij 1,m,..., sn,ij 6,m )T and S t = (s t,n,ij 1,m, st,n,ij 2,m )T represent spheroidal and toroidal source functions of spherical harmonic degree n and order m for a point dislocation; they are given in Takeuchi & Saito (1972). Note that only spherical orders m 2 are involved because the source is taken on the polar axis. To solve the non-homogeneous eq. (9) with free boundary conditions n, m, i, j : y n,ij 2,m (a) = yn,ij 4,m (a) = yn,ij 6,m (a) = yt,n,ij 2,m (a) = 0, (11) several methods have been proposed and discussed by Smylie & Mansinha (1971), Takeuchi & Saito (1972) and Okubo (1993). Details of their application are available from Sun (1992) and Sun & Okubo (1993). (10)

4 1276 W. Sun, S. Okubo and G. Fu Because i = 1, 2, 3 and j = 1, 2, 3, the combination of i and j is nine: the total solutions of all y should be nine. Geophysical consideration shows that y n,ij k,m (a) is symmetric with respect to i and j. Hence the number of independent solutions reduces to six. Furthermore, the intrinsic symmetry within fault geometry indicates that respective components y n,11 k,m (a) and yn,13 k,m (a) can be calculated easily using a rotational transformation about the polar axis to y n,22 k,m (a) and yn,23 k,m (a). Consequently, four independent solutions yn,ij(a) exist. Therefore, if any four independent solutions are obtained, other solutions among the nine are also obtained easily. In this study, we choose (y n,12 k,m, yn,32 k,m, yn,22 k,m, yn,33 and the coesponding toroidal solution, for four independent solutions excited, respectively, by a vertical strike-slip, a vertical dip-slip, a horizontal opening along a vertical fault and a vertical opening along a horizontal fault. Once numerical solutions of eq. (9) are obtained, the coseismic strain tensor can be derived and calculated numerically. k,m k,m ), 3 GREEN S FUNCTIONS OF COSEISMIC STRAIN FOR FOUR INDEPENDENT SOURCES According to conventional theory of elasticity, the components of coseismic strain are expressible in spherical coordinates as (Takeuchi & Saito 1972) the following. e = u r r e θθ = 1 u θ r θ + 1 r u r, e = 1 u ϕ r sin θ ϕ + 1 r u θ cot θ + 1 r u r, e = 1 u ϕ r θ 1 r u ϕ cot θ + 1 u θ r sin θ ϕ, e ϕr = 1 u r r sin θ ϕ + u ϕ r 1 r u ϕ, (12) (13) (14) (15) (16) e rθ = u θ r 1 r u θ + 1 u r r θ. (17) Using the displacement components defined in the section above, the expressions of the strain components are derived in the following. The last two components e rθ and e rϕ vanish on the free Earth surface. Therefore, only the remaining four components are considered hereafter. It is noteworthy that the seismic factor UdS/a 2 is omitted hereafter for convenience in all expressions of strain components. In other words, the unit seismic factor UdS/a 2 = 1 is used. In a practical calculation of coseismic deformations, the factor UdS/a 2 should be multiplied in all strain components. Inserting the displacement Green s functions into eqs (12) (17), omitting the easy but tedious mathematical derivations, and taking into account the responses to the four independent sources, the above four strain components are reducible into the following 16 components, expressed as an epicentral distance-related variation (a so-called Green s function) multiplied by a ϕ-related factor: e 12 (a,θ,ϕ) = sin 2ϕê12 (a,θ), (18) e 32 (a,θ,ϕ) = sin ϕê32 (a,θ), e 22,0 (a,θ,ϕ) = ê 22,0 (a,θ), e 33 (a,θ,ϕ) = ê33 (a,θ), e 12 θθ (a,θ,ϕ) = sin 2ϕê12 θθ (a,θ), e 32 θθ (a,θ,ϕ) = sin ϕê32 θθ (a,θ), e 22,0 θθ (a,θ,ϕ) = ê 22,0 θθ (a,θ), e 33 θθ (a,θ,ϕ) = ê33 θθ (a,θ), e 12 (a,θ,ϕ) = sin 2ϕê12 (a,θ), e 32 (a,θ,ϕ) = sin ϕê32 (a,θ), e 22,0 (a,θ,ϕ) = ê22,0 (a,θ), e 33 (a,θ,ϕ) = ê33 (a,θ), e 12 (a,θ,ϕ) = cos 2ϕê12 (a,θ), (19) (20) (21) (22) (23) (24) (25) (26) (27) (28) (29) (30)

5 Green s functions of coseismic strain changes 1277 e 32 (a,θ,ϕ) = cos ϕê32 (a,θ), e 22,0 (a,θ,ϕ) = ê22,0(a,θ) = 0, (31) (32) e 33 (a,θ,ϕ) = ê33 (a,θ) = 0, Therein, the epicentral distance-related variations ê ij kl (a,θ) are the strain Green s functions from: ê 12 (a,θ) = 2λ a(λ + 2μ) ê 32 (a,θ) = 2λ a(λ + 2μ) ê 22,0 λ (a,θ) = a(λ + 2μ) ê 33 (a,θ) = λ a(λ + 2μ) ê 12 θθ (a,θ) = 2 a ê 32 θθ (a,θ) = 2 a ê 22,0 θθ (a,θ) = 1 a ê 33 θθ (a,θ) = 1 a ê 12 (a,θ) = 2 a n=2 n=2 n=1 n=0 n=0 ( 2y n,12 1,2 (a) n(n + 1)yn,12 3,2 (a)) P 2 n, (34) ( 2y n,32 1,1 (a) n(n + 1)yn,32 3,1 (a)) P 1 n, (35) ( y n,22 1,0 (a) + n(n + 1)yn,22 3,0 (a)) P n, (36) ( y n,33 1,0 (a) + n(n + 1)yn,33 3,0 (a)) P n, (37) [ y n,12 3,2 (a) d2 Pn 2 y n,12 dθ 2 2y t,n,12 1,2 (a) n=1 1,2 P2 n ( 1 dpn 2 cos θ sin θ dθ sin 2 θ P2 n [ y n,32 3,1 (a) d2 Pn 1 y n,32 dθ 2 y t,n,32 1,1 (a) n=0 n=0 n=2 + 2yt,n,12 1,1 (a)p1 n ( 1 dpn 1 cos θ sin θ dθ sin 2 θ P1 n (33) )], (38) )], (39) [ ] y n,22 3,0 (a) d2 P n + y n,22 dθ 2 1,0 (a)p n, (40) [ ] y n,33 3,0 (a) d2 P n + y n,33 dθ 2 1,0 (a)p n, (41) { y n,12 3,2 (a) ( 4P 2 n sin θ sin θ [ dp 2 n 1,2 (a) sin θ dθ ) cos θ dp2 n dθ y n,12 1,2 (a)p2 n cot θ P 2 n ] }, (42) { ê 32 (a,θ) = 2 y n,32 3,1 (a) ( ) P 1 n cos θ dp1 n y n,32 1,1 a sin θ sin θ dθ (a)p1 n n=1 [ ] + yt,n,32 1,1 (a) } dp 1 n cot θ P 1 n sin θ dθ, (43) ê 22,0 (a,θ) = 1 a ê 33 (a,θ) = 1 a ê 12 (a,θ) = 2 a n=0 n=0 n=2 +y t,n,12 1,2 [ cot θy n,22 3,0 (a) dp ] n + y n,22 1,0 dθ (a)p n, (44) [ cot θy n,33 3,0 (a) dp ] n + y n,33 1,0 dθ (a)p n, (45) { 4y n,12 3,2 (a) [ dp2 n sin θ [ ] + cot θ P 2 n dθ ] } cot θ(a) dp2 n 4P2 n d2 Pn 2, (46) dθ sin 2 θ dθ 2

6 1278 W. Sun, S. Okubo and G. Fu Figure 2. Homogeneous half-space and homogeneous sphere. Figure 3. Coseismic strain components e ij (a, θ, ϕ) resulting from the four independent sources at a depth of 32 km in a homogeneous earth model. ê 32 (a,θ) = 2 a n=1 +y t,n,32 1,1 (a) { 2y n,32 3,1 (a) ( dp1 n sin θ ( ) + cot θ P 1 n dθ ) } cot θ dp1 n P1 n d2 Pn 1. (47) dθ sin 2 θ dθ 2 The case of e 22 kl (a, θ, ϕ) is special because the source function contains two parts with m = 0 and m = 2. The formulas shown above give solutions for the case of m = 0; solutions of the m = 2 case can be derived from e 12 kl (a, θ, ϕ) because the following relation holds. j = 1, 6:s n,22 j,±2 = isn,12 j,±2 j = 1, 2:s t,n,22 j,±2 e 22,2 e 22,2 e 22,2 e 22,2 = is t,n,12 j,±2. Consequently, (a,θ,ϕ) = ê 12 (a,θ) cos 2ϕ θθ (a,θ,ϕ) = êθθ 12 (a,θ) cos 2ϕ (a,θ,ϕ) = ê12 (a,θ) cos 2ϕ (a,θ,ϕ) = ê12 (a,θ) sin 2ϕ. (48) (49)

7 Green s functions of coseismic strain changes 1279 Figure 4. Coseismic strain caused by four independent sources at a depth of 32 km in the half-space earth model, as calculated using Okada s formulation (Okada 1985). Therefore, the solutions e 22 kl (a, θ, ϕ) are finally obtainable as the following. e 22 (a,θ,ϕ) = e22,0 (a,θ,ϕ) + ê 12 (a,θ) cos 2ϕ eθθ 22 (a,θ,ϕ) = e22,0 θθ (a,θ,ϕ) + êθθ 12 (a,θ) cos 2ϕ e 22 e 22 (a,θ,ϕ) = e22,0(a,θ,ϕ) + ê12 (a,θ) cos 2ϕ (a,θ,ϕ) = ê12 (a,θ) sin 2ϕ. (50) Expressions (18) (50) illustrate the main results of this study. They are useful to calculate strain components excited by four types of sources buried in a spherically symmetric earth model. In combination, these components allow calculation of a strain field that is excited by an arbitrary seismic source (see the following section). 4 COSEISMIC STRAIN CAUSED BY AN ARBITRARY INCLINED DISLOCATION A general formula is necessary to apply those four previous independent solutions to a practical event. For this purpose, we consider an inclined dislocation on the polar axis with the fault line in the direction of the Greenwich meridian. A dislocation vector ν and its normal n are expressible in terms of slip angle λ and dip angle δ of the fault (Fig. 1) as n = e 2 sin δ + e 3 cos δ ν = e 1 cos λ + e 2 cos δ sin λ + e 3 sin δ sin λ. (51) We have a shear dislocation problem if the dislocation vector ν runs parallel to the fault plane. In this case, the excited coseismic strain vector can be expressed using the above strain components of the four independent sources as the following. e s (a,θ,ϕ) = e ij UdS (a,θ,ϕ)ν i n j a { 2 [ ]} 1 UdS = cos λ[e 12 sin δ e 13 cos δ] + sin λ 2 (e33 e 22 ) sin 2δ e 32 cos 2δ. (52) a 2

8 1280 W. Sun, S. Okubo and G. Fu Figure 5. Comparison of coseismic horizontal strain components e ij θθ (a, θ, ϕ) calculated for a homogeneous sphere (the red line) and a half-space (the blue line) earth models. The epicentral distance is divisible into the following segments: (a) 0 2, (b) 2 5, (c) 5 9, (d) 9 50 and (e) The four panels in respective subfigures depict the results for the four seismic sources: the vertical strike-slip, vertical dip-slip, horizontal opening at a vertical fault, and vertical opening at a horizontal fault. Therein, ( e 13 (a,θ,ϕ) = e 32 a,θ,ϕ+ π ). 2 For a tensile opening, the slip vector and the normal become the same: ν = n = e 2 sin δ + e 3 cos δ. (53) (54)

9 Green s functions of coseismic strain changes 1281 Figure 5. (Continued.) Table 1. Comparison between the y-solutions calculated for a homogeneous sphere with self-gravitation (ȳ n,12 1,2 (G), ȳn,12 3,2 (G)) and that with non-gravitation (ȳ n,12 1,2 (NG), ȳn,12 3,2 (NG)). Degree n ȳ n,12 1,2 (G) ȳn,12 1,2 (NG) ȳn,12 3,2 (G) ȳn,12 3,2 (NG) E E E E E E E E E E E E E E E E E E E E-03 In this case, the excited coseismic strain is obtainable as e t (a,θ,ϕ) = e ij UdS (a,θ,ϕ)ν i n j a 2 = (e 33 cos 2 δ + e 22 sin 2 δ + e 32 sin 2δ) UdS. a 2 Once the three strain components e, e θθ and e are obtained, dilatation (volume change) can also be calculated as = e + e θθ + e. (55) (56) 5 NUMERICAL RESULTS FOR A HOMOGENEOUS EARTH MODEL AND EFFECTS OF THE EARTH S SPHERICAL CURVATURE To bolster the validity of the expressions described above, a numerical test is made by considering two earth models: a half-space model and a homogeneous sphere (Fig. 2). Both earth models have identical media parameters, which are equivalent to those of the top layer of the

10 1282 W. Sun, S. Okubo and G. Fu Figure 6. Coseismic strain components e ij (a, θ, ϕ) resulting from the four independent sources at a depth of 32 km in the 1066A earth model. Figure 7. Coseismic strain components e ij θθ(a, θ, ϕ) resulting from the four independent sources at a depth of 32 km in the 1066A earth model.

11 Green s functions of coseismic strain changes 1283 Figure 8. Coseismic strain components e ij (a, θ, ϕ) resulting from the four independent sources at a depth of 32 km in the 1066A earth model. Figure 9. Coseismic strain components e ij = e 33 (a, θ, ϕ) = 0. (a, θ, ϕ) resulting from the two independent sources at a depth of 32 km in the 1066A earth model. e22,0 (a, θ, ϕ) 1066A earth model (Gilbert & Dziewonski 1975). Numerical calculations are performed for the two models and the results are compared. The coseismic strain theory described above can be considered valid if the results agree well in the near field. For this purpose, Okada s theory (1986) is used for the half-space earth model; the cuent theory is used for the homogeneous model. The Green s functions in eqs (34) (47) show that the important computations are the y-solutions and Legendre functions and their derivatives. The y-solutions y n,ij k,m (a) and yt,n,ij k,m (a) have already been discussed in previous papers (e.g. Sun & Okubo 1993; Sun et al. 1996). They are easily obtainable through proper numerical integration of eq. (9). The Legendre functions and their derivatives are calculable using recuence formulas. For this purpose, some necessary formulas are derived in this study and are listed in Appendix A. Green s functions are then obtainable through series summations. The computation for the near field is more difficult than that for the far field because of the slow convergence of the series of the former. Therefore, some skills are required to accelerate computation (see Sun & Okubo 1993). On the other hand, coseismic deformations in the near field are useful for comparison with results of a half-space model because they dominate those of the far field. Therefore, the following discussions are limited to the near field.

12 1284 W. Sun, S. Okubo and G. Fu Figure 10. Homogeneous sphere and heterogeneous sphere. Assuming four independent point sources at a depth of 32 km in the homogeneous earth model, the y-solutions y n,ij k,m (a) and yt,n,ij k,m (a) are first computed. Then the strain Green s functions ê ij kl (a,θ) are obtainable using eqs (34) (47). Computations are performed under the condition of UdS/a 2 = 1; the Earth s radius a = 6371 km is taken for all Green s functions. This assumption requires that, for a practical computation of a strain component, parameters UdS and a should use the same kilometre unit so that the final strain components become dimensionless. Once these Green s functions ê ij kl (a,θ) are calculated, the coseismic strain components in eqs (18) (33) are computed easily. Fig. 3 depicts the coseismic strain components e ij (a, θ, ϕ) caused by the four independent sources at a depth of 32 km in the earth model: the vertical strike-slip, vertical dip-slip, horizontal opening at a vertical fault, and vertical opening at a horizontal fault. The horizontal axis indicates colatitude and longitude to 2 ; the vertical axis represents the strain magnitude with the unit of km 1. As expected, the strain component e 12 (a, θ, ϕ) for the vertical strike-slip source appears as a quadratic distribution pattern, the strain component e32 (a, θ, ϕ) for the vertical dip-slip source shows an anti-symmetrical distribution pattern along the longitude line, whereas the two tensile components e 22,0 (a, θ, ϕ) and e 33 (a, θ, ϕ) have a spherically symmetric distribution. For comparison, the coesponding strain components resulting from the four sources in a half-space earth model are also calculated using Okada s formulation (Okada 1985), and are plotted in Fig. 4. Comparison of Figs 3 and 4 illustrates that the two results agree well in both distribution pattern and magnitude. To observe the curvature effect in detail, we plot in Fig. 5 the results of horizontal strain components e ij θθ (a, θ, ϕ) for the four types of seismic sources at the depth of 50 km. The purpose is to compare the coseismic horizontal strain components e ij θθ (a, θ, ϕ) (it is the same for other components) calculated for a homogeneous sphere (the red line) and a half-space (the blue line) earth models. The epicentral distance 90 is divided into the following segments: (a) 0 2, (b) 2 5, (c) 5 9, (d) 9 50 and (e) Distances greater than 90 are omitted because the horizontal components decay quickly by almost 10 orders. The four panels in each subfigure give results of the four seismic sources: the vertical strike-slip, vertical dip-slip, horizontal opening at a vertical fault, and vertical opening at a horizontal fault. Comparison shows that the two curves almost overlap each other; the difference between them (except the horizontal opening source) is difficult to identify, especially in the near field (Fig. 5a). The discrepancy among all components becomes larger as the epicentral distance increases: results for the horizontal tensile source (Fig. 5: third panel) reveal a great relative difference of over 30 per cent. Note that all those results are calculated for the four independent solutions. The small discrepancy for the two shear sources does not mean that the spherical effect is small because a coseismic deformation for an arbitrary seismic source (especially an arbitrary dip slip fault) comprises four independent ones. Furthermore, the big difference for the horizontal tensile source contributes to the coseismic deformation (Sun & Okubo 2002). Therefore, we might conclude that for pure shear and vertical tensile sources, the spherical effect is small: less than 1 per cent in near field. The spherical effect becomes large as the epicentral distance increases. The discrepancy for the horizontal tensile source is extremely large: over 30 per cent of the maximum deformation. Pollitz (1996) investigated the same problem using the normal mode approach without considering the Earth s self-gravitation. Based on the results of shear sources, he concluded that the effect of sphericity is less than about 2 per cent within 100 km of an earthquake source at crustal depths. This is fundamentally equivalent to the above conclusion obtained in this study. It is noteworthy that, because the homogeneous half-space model is non-gravitating, whereas the homogeneous sphere is self-gravitating, the difference obtained by comparison in this section contains both the effect of sphericity and the effect of self-gravitation. To distinguish the effect of self-gravitation, an independent comparison should be made between two homogeneous spheres with and without self-gravitation. For this purpose, we compute the strain changes by decreasing the gravity values by a factor of , so that the earth model can be considered as a (approximated) non-gravitation model. The computation is performed for the vertical strike-slip source at depth of 50 km in this earth model. The computed results are compared with that of the same homogeneous sphere with self-gravitation and it is found no difference between the two results. Actually the difference is too small to be seen by plotting it in a figure. Instead the y-solutions which are used to compute the strain tensor (34) (47) can be used to observe their difference. Comparing the y-solutions ȳ n,12 1,2 (G) and ȳn,12 3,2 (G) (with self-gravitation) and that of ȳ n,12 1,2 (NG) and ȳn,12 3,2 (NG) (without self-gravitation) for several harmonic degrees in Table 1 show that there is no any difference for low harmonic degrees, but only very small difference for high harmonic degrees. This phenomenon implies that the effect of self-gravitation is relative small with respective to the spherical effect. However, it might be expected that in the post-seismic regime the effect increases considerably as pointed out by Nostro et al. (1999).

13 Green s functions of coseismic strain changes 1285 Figure 11. Comparison of coseismic strain changes e ij kl (a, θ, ϕ) calculated for four types of dislocations at a depth of 32 km buried in the homogeneous sphere: 1066A earth model and with its new top layer. Panels from top to bottom respectively depict results for (a) e ij (a, θ, ϕ), (b) e ij θθ (a, θ, ϕ), (c) eij (a, θ, ϕ) and (d) e ij (a, θ, ϕ). Subplots in (a), (b), (c) and (d) are results for the four respective sources: vertical strike-slip, vertical dip-slip, horizontal tensile opening and vertical tensile opening. The curve with blue colour shows results calculated for the 1066A earth model, the red line is for homogeneous sphere results, and the green line shows results calculated for a new model 1066A earth model with a new top layer of 11 km with parameters equivalent to those at 30 km. 6 NUMERICAL RESULTS FOR A SNREI EARTH MODEL AND EFFECTS OF EARTH S RADIAL HETEROGENEITY Next, we test the above coseismic strain theory by performing a numerical calculation. Assuming the four independent point sources at a depth of 32 km in the 1066A earth model (Gilbert & Dziewonski 1975), we first calculate the strain Green s functions ê ij kl (a,θ)in

14 1286 W. Sun, S. Okubo and G. Fu Figure 11. (Continued.) eqs (34) (47). The computations are performed under the condition of UdS/a 2 = 1; Earth s radius a = 6371 km is assumed for all Green s functions. Fig. 6 depicts the coseismic strain components e ij (a, θ, ϕ) resulting from the four independent sources at a depth of 32 km in the 1066A earth model. Results for other strain components e ij θθ (a, θ, ϕ), eij (a, θ, ϕ), and eij (a, θ, ϕ) are shown respectively in Figs 7 9. They show a similar distribution pattern to that of component e ij (a, θ, ϕ), but with different magnitude. To study the effect of stratified structure, we consider a homogeneous sphere and a heterogeneous sphere (1066A model); we compare results calculated from the two models as illustrated in Fig. 10. Numerical calculations were made in the previous and this section. Fig. 11 shows all strain components e ij kl (a, θ, ϕ) within the epicentral distance of 1 calculated for the four types of seismic sources at a 32 km depth. The curve with blue colour shows results calculated for the 1066A earth model; the red line is for homogeneous sphere results. The figure shows that discrepancies between the two models were greater than 30 per cent almost everywhere, including the epicentre. Discrepancies caused by stratified structure are much greater than those of sphericity. Consequently, it is infeed that the half-space dislocation theory might create eor of 30 per cent. We further consider a new model 1066A earth model with a new top layer of 11 km with parameters equivalent to those at 30 km to investigate the effect of stratified structure. The coesponding calculated numerical results are plotted in Fig. 11 using the green line. A great difference is apparent through comparison of both the homogeneous and 1066A earth models. It confirms again the importance of the Earth s structure in computing coseismic strain changes. The coseismic strain change decays fast as epicentral distance increases. Therefore, it is hard to observe the difference between the results in the far field. In addition, because the strain deformations vary by alternating sign, it is impossible to plot them using a logarithmic scale. We compute and plot the strain changes for different distance segments in the following to overcome this difficulty. To control the article length, we consider here only the components e ij θθ (a, θ, ϕ) of the strain changes (it is similar for other components). Therefore, in Fig. 12, we plot the global coseismic strain components e ij θθ (a, θ, ϕ) calculated for seismic sources at depth of 50 km in a homogeneous sphere (the red line), 1066A earth model (the blue line) and 1066A with a new top layer of 11 km with parameters equivalent to those at 30 km (the green line). The epicentral distance is divisible into the following segments: (a) 0 2, (b) 2 5, (c) 5 9, (d) 9 50, (e) and (f) The four panels in each subfigure give results of the four seismic sources (from top to bottom): the vertical strike-slip, vertical dip-slip, horizontal opening at a vertical fault, and vertical opening at a horizontal fault. The unit is nanostrain. It is seen from Fig. 12 that the discrepancy is generally very large for all components and at any distance: it is greater than 30 per cent. This discrepancy seems larger than that obtained by Pollitz (1996), who asserted that up to 20 per cent eor would be introduced if the Earth s layered structure were ignored. The difference between Pollitz s (1996) conclusions and ours is partially attributable to Pollitz s (1996) lack of using self-gravitation. Furthermore, the two studies use different earth models. A further observation in Fig. 12 shows that the result (green line) for the 1066A model with the new top layer seems close to that (red line) for the homogeneous earth model in the near field (Fig. 12a). Then it (green line) increasingly approximates the result of the 1066A model as the epicentral distance increases; it finally overlaps at great distance (from 2 to 180 ). That fact implies that the effect of different crustal structure mainly exists in near field, but does not affect results in the far field. The contrast of the crust and the structure under the crust seems to play an important role in the coseismic strain deformation. The source depth is also an important factor (see the following section). On the other hand, results for a homogeneous sphere (the red line) and the layered Earth (the blue line) show great differences everywhere from 30 per cent in the near field to over 100 per cent in the far field. It might be concluded that the effect of a stratified structure of an earth model is outstanding not only in a near field, but also in a far field. In addition, Antonioli et al. s (1998) comparison between a flat and spherical model has elucidated the role played by the earth structure

15 Green s functions of coseismic strain changes 1287 Figure 12. Comparison of global coseismic horizontal strain components e ij θθ (a, θ, ϕ) calculated for seismic sources at depth of 50 km in a homogeneous sphere (the red line), 1066A earth model (the blue line), and the 1066A with a new top layer of 11 km with parameters equivalent to those at 30 km (the green line). The epicentral distance is divided into the following segments: (a) 0 2, (b) 2 5, (c) 5 9, (d) 9 50, (e) and (f) The four panels in each subfigure give results of the four seismic sources (from top to bottom): the vertical strike-slip, vertical dip-slip, horizontal opening at a vertical fault, and vertical opening at a horizontal fault. The unit is nanostrain. in determining coseismic and post-seismic displacements. The importance of layer structure has also been show by Nostro et al. (2001) by discussing the coseismic and post-seismic stress changes for different earth models. 7 DEPTH DEPENDENCE OF THE EFFECTS OF EARTH S CURVATURE AND RADIAL HETEROGENEITY In the above two sections, we have discussed the effects of spherical curvature and stratification of the Earth. In this section, we consider the depth dependence of the effects. For this purpose, we compare the calculated results for different source depths in the half-space model and

16 1288 W. Sun, S. Okubo and G. Fu Figure 12. (Continued.) the 1066A model, meaning that the effects are combined ones, with contributions from both sphericity and Earth stratification. Fig. 13 gives coseismic strain components e ij (a, θ, ϕ) (a) and eij θθ (a, θ, ϕ) (b) calculated for the 1066A earth model (blue) and the homogeneous half-space (red) caused by seismic sources at respective depths of 3, 10, 100 and 300 km. The four panels in each subfigure depict results of the four seismic sources (from top to bottom): the vertical strike-slip, vertical dip-slip, horizontal opening at a vertical fault, and vertical opening at a horizontal fault. Comparisons show that no difference pertains for the very shallow source (3 km), but a large discrepancy appears for the deep depths. Generally, the effects of spherical curvature and stratification increase as the source depth increases. The spherical effect is verified as small in Section 5. Therefore, the large discrepancy for depths is considered mainly to arise from the contribution of Earth stratification.

17 Green s functions of coseismic strain changes 1289 Figure 13. Comparison of coseismic strain components e(a, ij θ, ϕ) (a) and e ij θθ (a, θ, ϕ) (b) calculated for the 1066A earth model (blue) and the homogeneous half-space (red) caused by seismic sources at respective depths of 3, 10, 100 and 300 km. The four panels in each subfigure show results of the four seismic sources (from top to bottom): the vertical strike-slip, vertical dip-slip, horizontal opening at a vertical fault, and vertical opening at a horizontal fault. 8 SUMMARY This study presents a new set of Green s functions for calculating coseismic strain changes for an SNREI earth model. Four independent point sources are assumed to be located at the polar axis. A proper combination of these expressions is useful to calculate coseismic strain components caused by an arbitrary seismic source at any geographical position of the Earth. Numerical computations are performed for calculating strain components in the near field caused by four independent sources at a depth of 32 km inside the 1066A earth model. Results agree well with those calculated for a half-space earth model. Results confirm the validity of the theory presented herein. Then, numerical computations and comparisons are also made to investigate effects of spherical curvature and stratified structure of the Earth using Okada s (1985) and the cuent dislocation theories. Curvature effects are found to be small for three types of seismic sources (two shear sources and the vertical tensile opening on the horizontal fault plane), but they are very large for the horizontal tensile opening on the vertical fault plane. A general coseismic deformation comprises four independent solutions (Sun & Okubo 2004). Therefore, the large curvature effect on the horizontal tensile opening source would contribute the general result. Stratified effects are very large for all depths and epicentral distances. They reach a discrepancy of greater than 30 per cent almost everywhere and reach at a discrepancy of 100 per cent in the very far field. Behaviour of the effects is complicated; it is difficult to find and present a simple expression because effects are influenced by multiple factors including source type, source depth, the earth model, and media parameters. However, it is clear that the stratification effect is rather large, even for a near field. For reliable application, a dislocation theory for a heterogeneous sphere, such as the 1066A or the PREM earth model, is recommended. ACKNOWLEDGMENTS Comments by two anonymous reviewers greatly improved the manuscript and are deeply appreciated. This research was supported financially by a JSPS Grant-in-Aid for Scientific Research (C ).

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Tanaka, T., Okuno, J. & Okubo, S., A new method for the computation of global viscoelastic post-seismic deformation in a realistic earth model (I) vertical displacement and gravity variation, Geophysical Journal International, 164, 273, doi: /j x x. Takeuchi, H. & Saito, M., Seismic surface waves, Methods Comput. Phys., 11, APPENDIX : APPENDIX A: LEGENDRE FUNCTIONS The following Legendre s functions, where the high order degree derivatives are newly derived, are used in this study: P 0 = 1, P 1 = cos θ, P n = 2n 1 n cos θ P n 1 n 1 P n 2, n P n+1 = 2n + 1 n + 1 cos θ P n n n + 1 P n 1, (A1) (A2) (A3) (A4)

19 dp n dθ = n sin θ [cos θ P n P n 1 ], Green s functions of coseismic strain changes 1291 d 2 P n = n [ cos θ Pn 1 (1 + n sin 2 θ)p dθ 2 sin 2 n ], (A6) θ P 1 n = dp n dθ = n sin θ [cos θ P n P n 1 ], dpn 1 = n [ (1 + n sin 2 θ)p dθ sin 2 n cos θ P n 1 ], (A8) θ d 2 Pn 1 = n [ ( 2 + n 2 sin 2 θ) cos θ P dθ 2 sin 3 n θ + (1 n n 2 + (1 + n + n 2 ) cos 2 θ)p n 1 ], (A9) P 2 n = n sin 2 θ [ ((n + 1) sin 2 θ + 2 cos 2 θ)p n + 2 cos θ P n 1 ], (A10) dpn 2 = n [ (n(1 n) sin 2 θ + 4) cos θ P dθ sin 3 n + ((n 2 + n + 2) sin 2 θ 4)P n 1 ], (A11) θ d 2 Pn 2 = n [ ( 4(n + 1) + n(n + 1) 2 sin 2 θ dθ 2 sin 4 θ + 4 (n 2) cos 2 θ n 2 (n 1) cos 2 θ sin 2 θ)p n + (2(1 + 3n n 2 ) + (8 9n n 2 ) sin 2 θ + 2(n 2 3n + 5) cos 2 θ) cos θ P n 1 ]. (A5) (A7) (A12)