General expressions for internal deformation due to a moment tensor in an elastic/viscoelastic multilayered half-space

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1 Geophys. J. Int. (28) 75, doi:./j x x General expressions for internal deformation due to a moment tensor in an elastic/viscoelastic multilayered half-space Akinori Hashima, Youichiro Takada, 2 Yukitoshi Fukahata and Mitsuhiro Matsu ura Department of Earth and Planetary Science, University of Tokyo, 7-3- Hongo, Bunkyo-ku, Tokyo 3-33, Japan. ahash@eps.s.u-tokyo.ac.jp 2 Department of Earth Sciences, University of Oxford, Oxford, UK Accepted 28 April 28. Received 28 March 27; in original form 27 February 2 GJI Seismology SUMMARY In the framework of elasticity theory any indigenous source can be represented by a moment tensor. We have succeeded in obtaining general expressions for internal deformation due to a moment tensor in an elastic/viscoelastic multilayered half-space under gravity. First, starting from Stokes classical solution, we obtained the expressions for static displacement fields due to a moment tensor in an infinite elastic medium. Then, performing the Hankel transformation of the static solution in Cartesian coordinates, we derived static displacement potentials for a moment tensor in cylindrical coordinates. Second, representing internal deformation fields by the superposition of a particular solution calculated from the displacement potentials and the general solution for an elastic multilayered half-space out sources, and using the generalized propagator matrix method, we obtained exact expressions for internal elastic deformation fields due to a moment tensor. Finally, applying the correspondence principle of linear viscoelasticity to the elastic solution, we obtained general expressions for quasi-static internal deformation fields due to a moment tensor in an elastic/viscoelastic multilayered half-space. The moment tensor can be generally decomposed into the three independent force systems corresponding to isotropic expansion, crack opening and shear faulting, and so the general expressions include internal deformation fields for these force systems as special cases. As numerical examples we computed the quasi-static internal displacement fields associated dyke intrusion, episodic segmental ridge opening and steady plate divergence in an elastic viscoelastic two-layered half-space. We also demonstrated the usefulness of the source representation moment tensor through the numerical simulation of deformation cycles associated the periodic occurrence of interplate earthquakes in a ridge-transform fault system. Key words: Transient deformation; Elasticity and anelasticity; Mid-ocean ridge processes; Kinematics of crustal and mantle deformation; Mechanics, theory, and modelling. INTRODUCTION Since Steketee (958a, b) introduced the concept of dislocation into seismology, mathematical expressions for static deformation fields caused by shear faulting in an elastic half-space have been obtained by many investigators (Chinnery 96, 963; Maruyama 964; Press 965; Savage & Hastie 966; Mansinha & Smylie 97; Sato & Matsu ura 974; Iwasaki & Sato 979; Matsu ura & Tanimoto 98; Okada 992). The elastic half-space model is a simple and good approximation to the real Earth for instantaneous coseismic crustal deformation. For long-term crustal deformation, however, the elastic half-space model is no longer good approximation, because we cannot neglect the effects of stress relaxation in the viscoelastic asthenosphere underlying the elastic lithosphere (e.g. Nur & Mavko 974; Savage & Prescott 978; Thatcher & Rundle 979, 984; Matsu ura & Iwasaki 983; Matsu ura & Sato 989; Sato & Matsu ura 992, 993; Hashimoto & Matsu ura 2; Hashimoto et al. 24). In general, the viscoelastic solutions of quasi-static problems can be obtained from the associated elastic solutions by applying the correspondence principle of linear viscoelasticity (Lee 955; Radok 957), and so we need to obtain elastic solutions for layered Earth models first. For layered half-space models the mathematical expressions of surface deformation due to shear faulting have been derived two similar but different approaches. On the basis of general source representation (Ben-Menahem & Singh 968a, b), Singh (97), Jovanovich et al. (974a, b) and Rundle (98) have obtained the elastic solutions the up going algorithm of the Thomson Haskell propagator matrix method (Thomson 95; Haskell 953). Applying the correspondence principle of linear viscoelasticity to these elastic solutions, Rundle (978, 982a) has obtained the quasi-static solutions for an elastic viscoelastic layered half-space. On the other hand, Sato (97), Sato & 992 C 28 The Authors Journal compilation C 28 RAS

2 General expressions for internal deformation due to a moment tensor 993 Matsu ura (973) and Matsu ura & Sato (975) have obtained the elastic solutions the down going algorithm of the propagator matrix method. Applying the correspondence principle to these elastic solutions, Matsu ura et al. (98), Iwasaki & Matsu ura (98) and Matsu ura & Sato (989) have obtained the quasi-static solutions for an elastic viscoelastic layered half-space. The solutions derived the up going algorithm and those the down going algorithm are outwardly similar, but they are essentially different in a computational point of view. Fukahata & Matsu ura (25) have revealed that the solutions derived the up going algorithm become unstable above the source depth, and those the down going algorithm become unstable below the source depth. For example, Roth (99) and Ma & Kusznir (992) have obtained the expressions for internal elastic deformation fields by extending the formulation of Singh (97). Their expressions become numerically unstable above the source depth. On the other hand, Matsu ura & Sato (997) have obtained the expressions for internal viscoelastic deformation fields by extending the formulation of Matsu ura et al. (98). Their expressions become numerically unstable below the source depth. Then, one way to avoid the numerical instability is to use the down going algorithm above the source depth and the up going algorithm below the source depth. Pan (997) and Wang (999) took this way to obtain numerically stable expressions for internal elastic deformation fields. Another way is to use a static (zero-frequency) version of the generalized reflection and transmission coefficient matrix method (Kennett 983). Xie & Yao (989) took this way to obtain numerically stable expressions for elastic internal deformation fields. Recently, by introducing the generalized propagator matrix that unifies the up going and down going algorithms, Fukahata & Matsu ura (25) have obtained numerically stable expressions for elastic internal deformation fields. Applying the correspondence principle of linear viscoelasticity to this elastic solution, Fukahata & Matsu ura (26) have obtained general expressions for internal deformation fields due to shear faulting in an elastic/viscolelastic multilayered half-space under gravity. As to isotropic expansion and crack opening in a layered half-space, on the other hand, we have not yet obtained the complete expressions for internal deformation fields. For example, Rundle (978, 982b) and Fernández & Rundle (994) have obtained the expression for elastic surface deformation due to isotropic expansion. Folch et al. (2) and Fernández et al. (2) have obtained expressions for viscoelastic surface deformation due to isotropic expansion. Roth (993) has derived the expressions for internal elastic deformation fields due to crack opening by extending his formulation of shear faulting (Roth 99). Hofton et al. (995) have obtained viscoelastic surface deformation due to crack opening. In the derivation of theses expressions, however, only the up going algorithm of the propagator matrix method is used, and so they are numerically unstable above the source depth. On the other hand, He et al. (23a) have obtained the expressions for elastic internal deformation fields due to isotropic expansion and crack opening by extending the formulation of Xie & Yao (989) for shear faulting, based on the generalized reflection and transmission coefficient matrix method. In the framework of elasticity theory any indigenous source can be represented by a moment tensor (Backus & Mulcahy 976a, b). The moment tensor, which is a second-order symmetric tensor the diagonal elements of force dipoles and the off-diagonal elements of force couples, can be decomposed into three independent force systems corresponding to isotropic expansion, crack opening and shear faulting. By using the generalized reflection and transmission coefficient matrix method, He et al. (23b) have formulated surface deformation due to a moment tensor in an elastic layered half-space. In the present study, extending the formulation of Fukahata & Matsu ura (25, 26) for shear faulting, we obtain general expressions for internal displacement fields due to a moment tensor in an elastic/viscoelastic multilayered half-space. In Section 2, we derive the expressions of static displacement potentials in cylindrical coordinates by performing the Hankel transformation of Stokes classical solution in Cartesian coordinates. In Section 3, we obtain the expressions for internal displacement fields due to a moment tensor in an elastic multilayered half-space. In Section 4, applying the correspondence principle of linear viscoelasticity to the elastic solution, we obtain the expressions for internal displacement fields due to a moment tensor in an elastic/viscoelastic multilayered half-space. In Section 5, as numerical examples, we show the quasi-static internal displacement fields associated dyke intrusion, episodic segmental ridge opening and steady plate divergence in the case of an elastic viscoelastic two-layered half-space. We also show the deformation cycle associated the periodic occurrence of interplate earthquakes in a ridge-transform fault system. 2 STATIC DEFORMATION CAUSED BY A MOMENT TENSOR 2. Displacement potentials in cylindrical coordinates We consider an infinite elastic medium the following constitutive equation: σ kk = 3K ε kk, σ ij = 2με ij () σ ij = σ ij 3 σ kkδ ij, ε ij = ε ij 3 ε kkδ ij, (2) where σ ij and ε ij denote stress and strain tensors, respectively, and K and μ are the bulk modulus and rigidity of the medium, respectively. Infinitesimal deformation of the elastic medium is governed by well-known Navier s equation: ( ρ 2 u i = K + ) ( ) u t 2 3 μ j + μ ( ) ui + f i, (3) x i x j x j x j where ρ is density, and u i (i =, 2, 3) and f i (i =, 2, 3) are displacement and body force vectors, respectively. Stokes (849) has obtained a particular solution (Green s tensor) of Navier s equation for a unit impulsive point force f i = δ ip δ(x ξ)δ(t τ) in Cartesian coordinates C 28 The Authors, GJI, 75, Journal compilation C 28 RAS

3 994 A. Hashima et al. (x, x 2, x 3 )as R/β G ip (x, t; ξ, ) = 3ζ iζ p δ ip sδ(t s)ds + ρr 3 R/α R = (x ξ ) 2 + (x 2 ξ 2 ) 2 + (x 3 ξ 3 ) 2, ζ i = x i ξ i R = R x i. ζ iζ p ρα 2 R δ(t R/α) ζ iζ p δ ip ρβ 2 R δ(t R/β) (4) Here, α and β are the P- and S-wave velocities of the medium, and δ ij is the Kronecker delta. The dynamic displacement field u d i (x, t) caused by the moment tensor M pq(τ) acting for τ at a point ξ = is written in the form of convolution integral: t u d i (x, t) = G ip,q (x, t τ;, )M pq (τ)dτ, (7) where G ip,q represents the partial derivatives of Green s tensor G ip respect to the source coordinate ξ q. The exact expression of u d i (x, t), which can be obtained by substituting eq. (4) into eq. (7), is given in Aki & Richards (98). On the assumption that M pq (t) becomes a constant M pq at t, taking the limit of t for the exact dynamic solution, we can directly obtain the corresponding static solution u s i (x) as u s i (x) = 8πμR γ (3ζ iζ 2 p ζ q ζ i δ pq ζ p δ qi ζ q δ ip ) + 2ζ q δ ip ]M pq (8) γ = (3K + μ)/(3k + 4μ). (9) By using the relations in eq. (6), the above solution can be rewritten in vector form: u s (x) = (γ ) ( s) + s] () 8πμ s = M R. () Now we perform the transformation of variables from Cartesian coordinates (x, x 2, x 3 ) to cylindrical coordinates (r, ϕ, z): x = r cos ϕ, x 2 = r sin ϕ, x 3 = z. (2) Then, the elements of moment tensor M in cylindrical coordinates are related those in Cartesian coordinates as M rr M rϕ M rz cos ϕ sinϕ M M 2 M 3 cos ϕ sinϕ M ϕr M ϕϕ M ϕz = sinϕ cos ϕ M 2 M 22 M 23 sinϕ cos ϕ (3) M zr M zϕ M zz M 3 M 32 M 33 and the distance R in eq. (5) is expressed as R = r 2 + z 2. (4) In static problems, as Takeuchi (959) has shown, a proper representation of solution vectors u s in terms of displacement potentials, Φ s, Φ s 2 and Ψs,isgivenby u s = Φ s γ z Φs 2 + (2 γ ) +. (5) Φ s 2 Ψ s Comparing the above general representation the specific expression for a moment tensor in eq. (), we obtain a set of equations to be solved for the displacement potentials: ( Φ s z γ ) zφs 2 + 2Φ s 2 = (γ z ) 8πμ s 2 s z (6) ( ) 2 z 2 2 Ψ s = 8πμ ( s) z (7) ( ) 2 2 z 2 2 Φ s 2 = 8πμ ( s) z, (8) where the subscript z denotes the z-component of the corresponding vector. From eqs (6) (8) and eq. (), using a basic formula of Hankel transform R = r 2 + z = e ξ z J (ξr)dξ, (9) 2 (5) (6) C 28 The Authors, GJI, 75, Journal compilation C 28 RAS

4 General expressions for internal deformation due to a moment tensor 995 we can obtain explicit expressions for the displacement potentials in cylindrical coordinates in the form of semi-infinite integral respect to wave number ξ: Φ s (r,ϕ,z) = μ 4 (2 γ )(M rr + M ϕϕ ) 2γ M zz ]J (ξr) + sgn(z)m rz J (ξr) + 2 γ } (M rr M zz )J 2 (ξr) e ξ z dξ (2) 4 Φ s 2 (r,ϕ,z) = μ sgn(z) 4 (M rr + M ϕϕ ) 2M zz ]J (ξr) + M rz J (ξr) + sgn(z) } 4 (M rr M ϕϕ )J 2 (ξr) ξe ξ z dξ (2) Ψ s (r,ϕ,z) = sgn(z)m ϕz J (ξr) + M rϕ J 2 (ξr)]e ξ z dξ, (22) μ where J n (ξr) is the nth order Bessel function and sgn(z) denotes the sign function that takes the value of for z > and for z <. It should be noted that all of these potentials satisfy the Laplace equation except for z =. We rewrite eqs (2) (22) in the following more concise form: Φ s (r,ϕ,z) = χ s (z; ξ) j(r,ϕ; ξ)dξ Φ s 2 (r,ϕ,z) = Ψ s (r,ϕ,z) = χ s 2 (z; ξ) j(r,ϕ; ξ)ξdξ. χ s (z; ξ) j (r,ϕ; ξ)dξ (23) Here, χ s, χs 2 and χ s are the z-dependent coefficient vectors defined by χ s (z; ξ) = 4γ 2 + γ sgn(z) 2 γ ]e ξ z χ s 2 (z; ξ) = 3sgn(z) sgn(z)]e ξ z χ s (z; ξ) = sgn(z) ]e ξ z and j and j are the r- and ϕ-dependent source vectors defined by m J (ξr) m 2 J (ξr) j(r,ϕ; ξ) = m 3 cos ϕ J (ξr) + m 4 sinϕ J (ξr) m 5 cos 2ϕ J 2 (ξr) + m 6 sin2ϕ J 2 (ξr) ] m3 sinϕ J (ξr) + m 4 cos ϕ J (ξr) j (r,ϕ; ξ) = 2m 5 sin2ϕ J 2 (ξr) + 2m 6 cos 2ϕ J 2 (ξr) (24) (25) (26) m = M + M 22 + M 33 9K, m 2 = M M M 33, m 3 = M 3 2μ μ, m 4 = M 23 μ, m 5 = M M 22 4μ, m 6 = M 2 2μ. (27) Each component of the source vectors j and j represents a horizontal mode solution corresponding to some force system composed of the elements of moment tensor M pq. Substituting the expressions of displacement potentials in eq. (23) into eq. (5), we can obtain the explicit expressions for displacement fields due to a moment tensor in cylindrical coordinates. 2.2 Specific representation of source vectors There are many ways to decompose a moment tensor into several independent force systems. It is natural to decompose a moment tensor into two force systems corresponding to isotropic expansion and displacement discontinuity across an internal surface. The displacement discontinuity vector can be decomposed into the normal and tangential components, which correspond to crack opening and shear faulting, respectively. Therefore, we can decompose a moment tensor into three independent force systems corresponding to isotropic expansion, crack C 28 The Authors, GJI, 75, Journal compilation C 28 RAS

5 996 A. Hashima et al. O O O x x x x 2 a x 2 u x 2 u x 3 x 3 x 3 (a) (b) (c) Figure. Three basic physical processes represented by a moment tensor: (a) isotropic expansion, (b) shear faulting, and (c) crack opening. opening and shear faulting. This decomposition is natural and always possible by solving the eigenvalue problems of moment tensor. In this section we show the explicit expressions of the source vectors j and j for isotropic expansion, crack opening and shear faulting Isotropic expansion A fractional volume change Θ = ΔV/V in an infinitesimal sphere a radius a (Fig. a) can be expressed in terms of a moment tensor as M pq = a3 KΘδ pq. (28) 3 Substituting eq. (28) into eqs (25) and (26) eq. (27), we obtain the expressions of j and j for isotropic expansion as J (ξr) j(r,ϕ; ξ) = 4 ( ) 9 πa3 Θ, j (r,ϕ; ξ) =. (29) Shear faulting and crack opening A displacement discontinuity u a unit direction vector = (ν, ν 2, ν 3 ) on an infinitesimal fault plane dσ a unit normal vector n = (n, n 2, n 3 ) can be expressed in terms of a moment tensor as M pq = ΔudΣ Kn k ν k δ pq + μ (n p ν q + n q ν p 23 )] n kν k δ pq. (3) The above expression includes both cases of shear faulting (n = ) and crack opening (n = ). We first consider the case of shear faulting on an inclined plane its strike parallel to the x -axis, a dip angle θ, and a slip angle λ (Fig. b). In this case the components of the direction vector and the normal vector n are given as n = (, sinθ, cos θ), ν = (cos λ, sinλ cos θ, sinλsinθ). (3) Substituting eqs (3) and (3) into eqs (25) and (26) eq. (27), we obtain the expressions of j and j for shear faulting as j(r,ϕ; ξ) = ΔudΣ 4 sin2θsinλj (ξr) (cos λ cos θ cos ϕ sinλ cos 2θsinϕ) J (ξr) (32) 4 (sinλsin2θ cos 2ϕ + 2 cos λsinθsin2ϕ) J 2(ξr) j (r,ϕ; ξ) = ΔudΣ (cos λ cos θsinϕ + sinλ cos 2θ cos ϕ) J (ξr) 2 (sinλsin2θsin2ϕ 2 cos λsinθ cos 2ϕ) J. (33) 2(ξr) Note that the first component of the source vector j is always zero for shear faulting. Sato (97) has obtained the representation of source vectors equivalent to those in eqs (32) and (33). In the case of crack opening on a plane its strike parallel to the x -axis and a dip angle θ (Fig. c), since the direction vector is parallel to the normal vector n, wehave n = ν = (, sinθ, cos θ). (34) C 28 The Authors, GJI, 75, Journal compilation C 28 RAS

6 General expressions for internal deformation due to a moment tensor 997 In the same way as in the case of shear faulting, substituting eqs (3) and (34) into eqs (25) and (26) eq. (27), we obtain the expressions of j and j for crack opening as 3 J (ξr) ] j(r,ϕ; ξ) = ΔudΣ 6 (3 cos2 θ )J (ξr) sin2θ cos ϕ J (ξr), j (r,ϕ; ξ) = ΔudΣ. (35) sin2θsinϕ J (ξr) sin 2 θsin2ϕ J 2 (ξr) 2 sin2 θ cos 2ϕ J 2 (ξr) 3 ELASTIC SOLUTION FOR A LAYERED HALF-SPACE We consider the elastic medium composed of n parallel layers overlying a half-space as shown in Fig. 2. Here, every layer and interface is numbered in ascending order from the free surface, and the positive z-axis is taken as directed into the medium. The depth of the jth interface is denoted by H j, and the thickness of the jth layer by h j. We use bulk modulus K j and rigidity μ j,orγ j = (3K j + μ j )/(3K j + 4μ j ) and μ j, to represent the elastic property of the jth layer. A point moment tensor MH(t) is located at (,, d) in the mth layer. Here, we use the Heaviside step function H(t) to represent an instantaneous source time process at t =. Then, in a layer the source ( j = m), the elastic displacement field u E (r, ϕ, z, t; j) can be represented by the sum of a particular solution u s (r, ϕ, z d, t; j) for an infinite medium the source, which is obtained by substituting the displacement potentials in eq. (23) into eq. (5), and the general solution u g (r, ϕ, z, t; j) for the layer out sources, which is obtained by superposing mode solutions of the Laplace equation. In a layer out sources ( j m), on the other hand, the elastic displacement field u E is given by the general solution u g. Thus we can generally represent the elastic displacement field as u E (r,ϕ,z, t; j) = u g (r,ϕ,z, t; j) + δ jm u s (r,ϕ,z d, t; m). (36) 3. Representation of internal deformation fields We represent the general solution u g of the jth layer (H j z H j ) by using displacement potentials, Φ g j, Φg 2 j and Ψ g j, in the same form as eq. (5): u g (r,ϕ,z; j) = Φ g j γ j(z H j ) Φ g 2 j + (2 γ j) +. (37) Φ g 2 j General expressions for the displacement potentials are given by the superposition of all the mode solutions of the Laplace equation. From among all the mode solutions we choose some specific ones corresponding to the horizontal modes in the particular solution to satisfy boundary Ψ g j H () H m H j ( j) h j z H j r (r,,z) x H m d (m) x 2 d H m M pq u u z u r H m H n (n) z x 3 (a) (b) Figure 2. The coordinate system, source geometry, and a layered structure model. Every layer and interface is numbered in ascending order from the free surface. The direction of the x 3 -axis in Cartesian coordinates or the z-axis in cylindrical coordinates is taken to be vertically downward. The depth of the jth interface is denoted by H j, and the thickness of the jth layer by h j = H j H j. The source is located at (,, d) onthez-axis. C 28 The Authors, GJI, 75, Journal compilation C 28 RAS

7 998 A. Hashima et al. conditions. Thus, by using the same source vectors j and j in eqs (25) and (26), we may write the displacement potentials of u g in the following integral form: Φ g j (r,ϕ,z) = Φ g 2 j (r,ϕ,z) = Ψ g j (r,ϕ,z) = χ g j (z; ξ) j(r,ϕ; ξ)dξ χ g 2 j (z; ξ) j(r,ϕ; ξ)ξdξ χ g j (z; ξ) j (r,ϕ; ξ)dξ the z-dependent coefficient vectors χ g j, χg 2 j and χ g j defined by ( ) ( ) χ g j (z; ξ) = ( δ jn) A + j A + 2 j A + 3 j A + 4 j e ξ(z H j ) + A j A 2 j A 3 j A 4 j e ξ(z H j ) ( ) ( ) χ g 2 j (z; ξ) = ( δ jn) B + j B + 2 j B + 3 j B + 4 j e ξ(z H j ) + B j B 2 j B 3 j B 4 j e ξ(z H j ) ( ) ( ) χ g j (z; ξ) = ( δ jn) C + j C + 2 j e ξ(z H j ) + C j C 2 j e ξ(z H j ) (38). (39) Here, A ± kj (k =, 2, 3, 4), B ± kj (k =, 2, 3, 4) and C ± kj (k =, 2) are the ξ-dependent potential coefficients to be determined from boundary conditions. It should be noted that the terms exponentially increasing depth z must be removed from eq. (39) for the substratum ( j = n), because every displacement and stress components do not diverge at z from causality. Substituting eq. (23) and (38) into eq. (5) and (37), respectively, we obtain expressions for ur(ϕ,z) s (r, ϕ, z, t; m) and ug r(ϕ,z) (r, ϕ, z, t; j) in the same integral form. We combine them by eq. (36) to obtain the formal expressions for elastic displacement fields ur(ϕ,z) E (r, ϕ, z, t; j) due to a moment tensor MH(t) at a point (,, d) inthemth layer: H(t) ur E (r,ϕ,z, t; j) = y E (z; ξ; j) r j(r,ϕ; ξ) + y E (z; ξ; j) r H(t) uϕ E (r,ϕ,z, t; j) = u E z (r,ϕ,z, t; j) = H(t) y E (z; ξ; j) r y E 2 (z; ξ; j)j(r,ϕ; ξ)ξdξ ϕ j(r,ϕ; ξ) y E (z; ξ; j) r j (r,ϕ; ξ) ] ϕ j (r,ϕ; ξ) dξ ] dξ. (4) The corresponding expressions for stress components σ zr(ϕ,z) E (r, ϕ, z, t; j) are calculated from the displacement potentials in eqs (23) and (38) by the definition of stress components in cylindrical coordinates as σzr E(r,ϕ,z, t; j) = μ j H(t) σzϕ E (r,ϕ,z, t; j) = μ j H(t) σzz E(r,ϕ,z, t; j) = μ j H(t) 2y E 3 (z; ξ; j) 2y E 3 (z; ξ; j) r r j(r,ϕ; ξ) + y E 2 (z; ξ; j) ] r ϕ j (r,ϕ; ξ) ξdξ ] ϕ j(r,ϕ; ξ) y E 2 (z; ξ; j) r j (r,ϕ; ξ) ξdξ 2y E 4 (z; ξ; j)j(r,ϕ; ξ)ξ 2 dξ. (4) Here, for the source vectors j(r, ϕ; ξ) and j (r, ϕ; ξ), we have already obtained the definite expressions in eqs (25) and (26). Thus, our problem is to determine the z-dependent deformation vectors, yk E (k =, 2, 3, 4) and y E k (k =, 2) from boundary conditions at the Earth s surface and layer interfaces. 3.2 Deformation matrices and boundary conditions We define the 4 4 and 2 2 deformation matrices Y E and Y E composed of the deformation vectors yk E(k =, 2, 3, 4) and y k E (k =, 2), respectively, as Y E (z; j) = y E y2 E y3 E y4 E (z; j) (z; j) (z; j), Y E (z; j) = (z; j) y E y E 2 ] (z; j). (42) (z; j) Hereafter we omit the ξ- and t-dependence in notation for simplicity. The formal expressions of the deformation matrices, which are composed of the deformation matrices for the particular solution, Y s and Y s, and those for the general solution, Y g and Y g, are given as follows: Y E (z; j) = Y g (z; j) + δ jm exp( z d ξ)y s (z d; m) Y E (z; j) = Y g (z; j) + δ jm exp( z d ξ)y s (z d; m), (43) C 28 The Authors, GJI, 75, Journal compilation C 28 RAS

8 General expressions for internal deformation due to a moment tensor 999 where 4γ m 2 + γ m ( 3 z ξ) sgn(z) + γ m zξ 2 γ m ( + z ξ) sgn(z)(4γ Y s (z; m) = m ) 4sgn(z)( γ m ) + 3γ m zξ + γ m ( z ξ) γ m zξ sgn(z)(4γ m ) sgn(z)( 4γ m ) + 3γ m zξ γ m ( z ξ) sgn(z) + γ m zξ 4γ m + γ m ( 3 z ξ) γ m zξ γ m ( + z ξ) ] sgn(z) Y s (z; m) = sgn(z) and Y g (z; j) = E j (z H j )A j, Y g (z; j) = E j (z H j )A j (45) A + j + A j A + 2 j + A 2 j A + 3 j + A 3 j A + 4 j + A 4 j B A j( n) = j + B j B + 2 j + B 2 j B + 3 j + B 3 j B + 4 j + ( B 4 j C + A + j A j A + 2 j A 2 j A + 3 j A 3 j A + 4 j A 4 j, A j( n) = j + C j C + 2 j + C ) 2 j C + j C j C + 2 j C, (46) 2 j B + j B j B + 2 j B 2 j B + 3 j B 3 j B + 4 j B 4 j a n A n A 2n A 3n A 4n b A n = n a n = B n B 2n B 3n B ( ) ( ) 4n A n A 2n A 3n A 4n, A n = cn C n C 2n =. (47) c n C n C 2n b n B n B 2n B 3n B 4n Here, E j and E j are the purely structure-dependent matrices (independent of source properties), whose explicit expressions are given in Appendix A. The deformation matrices defined in eq. (43) must satisfy the stress-free condition, including the gravitational effects associated surface uplift and subsidence (McConnell 965; Matsu ura & Sato 989), at the Earth s surface (z = ): σ zr(ϕ) (r,ϕ,; ) =, σ zz (r,ϕ,; ) ρ gu z (r,ϕ,; ) =, (48) where ρ is the density of the surface layer, and g is the acceleration of gravity at the Earth s surface. From eqs (4) and (4) the stress-free condition can be written in terms of the deformation vectors as y E 3 (; ) =, y E 2 (; ) =, 2μ ξy E 4 (; ) ρ gy E 2 (; ) =, (49) or in terms of the deformation matrices as Y E (; ) = GY, Y E (; ) = Y (5) y y Y = 2, Y = ( ) y. Here, G is a purely structure-dependent matrix, whose explicit expression is given in Appendix A. At each layer interface (z = H j )every components of the displacement and stress vectors must satisfy the condition of continuity, ur(ϕ,z) (r,ϕ,h j +; j + ) = u r(ϕ,z) (r,ϕ,h j ; j) σ zr(ϕ,z) (r,ϕ,h j +; j + ) = σ zr(ϕ,z) (r,ϕ,h j ; j), (52) which can be written in terms of the deformation matrices as Y E (H + j ; j + ) = D j Y E (H j ; j), Y E (H + j ; j + ) = D j Y E (H j ; j). (53) Here, D j and D j are purely structure-dependent matrices, whose explicit expressions are given in Appendix A. In the above expressions, we ignored the gravitational effects associated vertical displacements at the layer interfaces, because the density difference between the upper and lower layers is small in comparison that at the Earth s surface. If we want to include the gravitational effects at the layer interfaces, the matrices D j and D j in eq. (53) should be slightly modified in a similar way to the case of the Earth s surface (Iwasaki & Matsu ura 982). In addition, using the down going expressions of the generalized propagator matrices defined by Fukahata & Matsu ura (25), F j (z) E j (z)e j (), F j(z) E j (z)e j () = E j(z), (54) we can relate the deformation matrices at the top (z = H + j ) and the bottom (z = H j )ofthejth layer out sources ( j m, n)as Y E( H j ; j m ) = F j (h j )Y E( H + j ; j), Y E( H j ; j m ) = F j (h j)y E( H + j ; j). (55) C 28 The Authors, GJI, 75, Journal compilation C 28 RAS (44) (5)

9 A. Hashima et al. Here, F j (z) and F j (z) are purely structure-dependent matrices. The explicit expression for F j(z) is given in Appendix A together E j () and its inverse E j (). For the source layer ( j = m), taking the direct source effects into account, we obtain the following relations: Y E (Hm ; m) = F m(h m )Y E (H + m ; m) F m(h m d)δ m Y E (Hm ; m) = F m (h m)y E (H + m ; m) F m (H (56) m d)δ m Δ m = Y s ( ; m) Y s ( + ; m), Δ m = Y s ( ; m) Y s ( + ; m). (57) The explicit expressions for Δ m and Δ m are given by ( ) 4γ Δ m = 2 m 4 + 4γ m 4γ m + 4γ m, Δ m = 2. (58) For the substratum ( j = n), from eqs (43) and (45), we obtain Y E (H + n ; n) = E n()a n + δ mn exp( H n d ξ)y s (H n d; n) Y E (H + n ; n) = A n + δ. (59) mn exp( H n d ξ)y s (H n d; n) By using the relations (55), (56) and (59) and the boundary conditions (5) and (53), the deformation matrices can be connected from the top to the bottom of the layered half-space in the down going way as A n = PY Q m, A n = P Y Q m P = E n n D n j F n j (h n j )]G, P = D n jf n j(h n j )] (6) () n j= Q m( n) = E n () n m j= j= D n j F n j (h n j )]D m F m (H m d)δ m n m Q m( n) = D n j F n j (h n j)]d m F m(h m d)δ m j= and Q m(=n) = e (d Hn )ξ E n ()Ys (H n d; n). (63) Q m(=n) = e (d Hn )ξ Y s (H n d; n) The matrix equations (6) (63) are formally the same as those for shear faulting in Fukahata & Matsu ura (25). If we take z = H j as the reference depth in the representation of displacement potentials Φ g j, Φg 2 j and Ψg j instead of z = H j, we obtain the up going expressions of the generalized propagator matrices: F j (z) E j ( z)e j (), F j (z) E j( z)e j () = E j( z). (64) The upgoing propagator matrices F j (z) and F j (z) are related the down going propagator matrices F j(z) and F j (z) in eq. (54), respectively, as F j (z) = F j ( z) = F j (z), F j(z) = F j( z) = F j (z). (65) Using the up-going propagator matrices, we can connect the deformation matrices from the bottom to the top in the up going way, and obtain the matrix equations parallel to eq. (6): Y = PA n + Q m, Y = P A n + Q m n ] n P = G F j ( h j )D j En (), P = F j ( h ] j)d j j= Q m( n) m ] = G F j ( h j )D j Fm (H m d)δ m j= j= m Q m( n) = F j ( h ] j)d j F m (H m d)δ m j= (6) (62) (66) (67) (68) C 28 The Authors, GJI, 75, Journal compilation C 28 RAS

10 and Q n(=m) n ] = e (d Hn )ξ G F j ( h j )D j Y s (H n d; n) j= j= n ] Q n(=m) = e (d H n )ξ F j ( h j )D j Y s (H n d; n) General expressions for internal deformation due to a moment tensor. (69) The matrix equations (66) (69) are formally the same as those for shear faulting in Fukahata & Matsu ura (25). 3.3 Internal deformation fields As shown in Fukahata & Matsu ura (25), we can solve eq. (6) for the deformation vectors at the Earth s surface (y, y 2 and y )by eliminating the potential coefficients in the substratum (a n, b n and c n )as ( ) y = (P22 + P 42 ) ( q m + ) qm 3 (P2 + P 32 ) ( q m 2 + ) ] qm 4 y δ 2 (P 2 + P 4 ) ( q m + ) qm 3 + (P + P 3 ) ( q m 2 + ), y qm = ( q m δ 4 + ) q m 2 (7) δ = (P + P 3 )(P 22 + P 42 ) (P 2 + P 32 )(P 2 + P 4 ), δ = P + P 2. (7) On the other hand, eliminating the deformation vectors y, y 2 and y in eq. (66), we obtain the potential coefficients a n, b n and c n as ( ) an = δ ( P 42 P 44 ) q m 3 + ( ] P 32 P 34 ) q m 4 b n ( P 4 P 43 ) q m 3 (, c P 3 P 33 ) q m n = q m 4 δ 2 (72) δ = ( P 3 P 33 )( P 42 P 44 ) ( P 32 P 34 )( P 4 P 43 ), δ = P 2 P 22. (73) Here, P ( ) ij and P ( ) ij are the ij-components of P ( ) and P ( ) (, respectively, and q ) i and q ( ) i are the ith rows of Q ( ) and Q ( ), respectively. We have derived two different types of solutions Y ( ) in eq. (7) and A ( ) n in eq. (72). Using either type of solutions, we can obtain the expressions for the deformation matrices Y E and Y E at any depth, but they are not always numerically stable, because the ξ-dependent exponential factors in the deformation matrices diverge at ξ in some cases (Fukahata & Matsu ura 25). In the substratum (H n z) we choose the second type of solutions to obtain the proper (numerically stable) expressions: Y E (z; n) = E n (z H n )A n + δ mn exp( z d ξ)y s (z d; n) Y E (z; n) = E n (z H n )A n + δ mn exp( z d ξ)y s (z d; n). (74) In the layers overlying the substratum, if the computation point is shallower than the source depth ( z < d), we choose the first type of solutions to obtain the proper expressions: j Y E (z; j) = F j (z H j ) D j k F j k (h j k )]GY k=, (75) j Y E (z; j) = F j (z H j ) D j k F j k (h j k)]y k= and if the computation point is deeper than the source depth (d < z < H n ), we choose the second type of solutions to obtain the proper expressions: Y E (z; j) = F j (z H j )D j Y E (z; j) = F j (z H j)d j n k= j+ n k= j+ ] Fk ( h k )D k En ()A n F k ( h ] k)d k A n. (76) Substituting the solutions Y and Y in eq. (7) or A n and A n in eq. (72) into eqs (74) (76), we can finally obtain the definite expressions for Y E and Y E in the following concise forms: Y E (z; j) = exp( qξ)s jm (z) + δ jn δ mn exp( z d ξ)y s (z d; n) Y E (z; j) = exp( qξ)s jm (z) + δ (77) jnδ mn exp( z d ξ)y s (z d; n) z d ( j n or m n) q =. (78) z + d 2H n ( j = m = n) The explicit expressions for S jm and S jm are given in the appendix of Fukahata & Matsu ura (26), where the source-dependent matrices Y s and Δ in Q m and Q m should be replaced Y s in eq. (44) and Δ m in eq. (58), respectively. Furthermore, it should be noted that the C 28 The Authors, GJI, 75, Journal compilation C 28 RAS

11 2 A. Hashima et al. generalized propagator matrices F(z), F (z) and E(z) in Fukahata & Matsu ura (26) are different from those in Fukahata & Matsu ura (25) and the present paper but only in the ξ-dependent exponential factor exp( z ξ). 4 VISCOELASTIC SOLUTION FOR A LAYERED HALF-SPACE We now consider the case where the lth layer of the layered half-space is viscoelastic. The rheological property of the viscoelastic layer is assumed to be Maxwell in shear and elastic in bulk: σ kk = 3K l ε kk, t σ ij + μ l σ ij = 2μ l η l t ε ij, (79) where η l denotes the viscosity of the lth layer. Performing the Laplace transformation of eq. (79), we obtain a relation between the Laplace transforms of stress and strain tensors, which is formally identical the constitutive equation of elastic media in eq. (): σ kk = 3K l ε kk, σ ij = 2ˆμ l (s) ε ij. (8) Here, s is the Laplace transform variable, the tilde denotes the Laplace transform of the corresponding physical quantity, and ˆμ l (s) isthe Laplace operator defined by ˆμ l (s) = μ ls τ l = η l. (8) s + /τ l μ l The Laplace operator ˆγ l (s) corresponding to γ i = (3K i + μ i )/(3K i + 4μ i ), which is used in the description of elastic solution, is obtained from eq. (8) as ˆγ l (s) = γ ls + /υ l υ l = 3τ l s + /υ l 4γ l. (82) Here, it should be noted that the parameters τ l and υ l in eqs (8) and (82) have the dimension of time. Then, applying the correspondence principle of linear viscoelasticity (Lee 955; Radok 957), we can directly obtain viscoelastic solution ũr(ϕ,z) V in the s-domain from the associated elastic solution u r(ϕ,z) E by replacing μ l and γ l ˆμ l (s) and ˆγ l (s) and the source time function H(t) its Laplace transform /s. Hereafter, for convenience, we represent the μ l - and γ l -dependence of the deformation matrices Y E and Y E explicitly. Since the source vectors j and j do not contain any elastic modulus, the viscoelastic solution ũr(ϕ,z) V in the s-domain can be written as ũr V (r,ϕ,z, s; j) = ũϕ V (r,ϕ,z, s; j) = ũz V (r,ϕ,z, s; j) = ỹ V (z, s; ξ; j) r ỹ V (z, s; ξ; j) r j(r,ϕ; ξ) + ỹ V (z, s; ξ; j) r ϕ ỹ V 2 (z, s; ξ; j)j(r,ϕ; ξ)ξdξ ] ϕ j (r,ϕ; ξ) dξ ] dξ j(r,ϕ; ξ) ỹ V (z, s; ξ; j) r j (r,ϕ; ξ) ỹ V (2) (z, s; ξ; j) = s ye (2) (z; ξ; j; ˆμ l, ˆγ l ), ỹ V (z, s; ξ; j) = s y E (z; ξ; j; ˆμ l). (84) Here, the s-dependent deformation vectors yk E(z; j; ˆμ l;ˆγ l ) and y E k (z; j; ˆμ l) are the kth rows of the s-dependent deformation matrices Y E (z; j; ˆμ l, ˆγ l ) and Y E (z; j; ˆμ l ), respectively, which are directly obtained from Y E and Y E in eqs (74) (76) by replacing μ l and γ l ˆμ l (s) and ˆγ l (s): that is, if the substratum is viscoelastic (l = n), Y E (z; j; ˆμ n, ˆγ n ) = exp( qξ)s jm (z; ˆμ n, ˆγ n ) + δ jn δ mn exp( z d ξ)y s (z d; n; ˆγ n ) Y E (z; j; ˆμ n ) = exp( qξ)s jm (z; ˆμ, (85) n) + δ jn δ mn exp( z d ξ)y s (z d; n) and otherwise Y E (z; j; ˆμ l, ˆγ l ) = exp( qξ)s jm (z; ˆμ l, ˆγ l ) + δ jn δ mn exp( z d ξ)y s (z d; n; γ n ) Y E (z; j; ˆμ l ) = exp( qξ)s jm (z; ˆμ. (86) l) + δ jn δ mn exp( z d ξ)y s (z d; n) From eqs (44) and (82) we can see that the matrix Y s (z d; n; ˆγ n ) in eq. (85) has the form of a rational function of first-degree polynomials in s: Y s (z d; n; ˆγ n ) = s + /υ n sy s (z d; n; γ n ) + ] Y s (z d; n; γ n = ). (87) υ n The matrix S ( ) jm, defined by the products of plural matrices, contains the γ l -dependent matrices Δ m (if m = l), E n () and E n () (if n = l), and F l ] and the μ l -dependent matrices G and G (if l = ), D ( ), D ( ), D ( ) l l l and D ( ) l ]. For the γ l-dependent matrices, replacing γ l ˆγ l (s), we obtain the corresponding s-dependent matrices in the form of B(ˆγ l ) = sb(γ l ) + ] B(γ l = ). (88) s + /υ l υ l (83) C 28 The Authors, GJI, 75, Journal compilation C 28 RAS

12 General expressions for internal deformation due to a moment tensor 3 For the μ l -dependent matrices, replacing μ l ˆμ l (s), we obtain the corresponding s-dependent matrices in the form of B(ˆμ l ) = sb(μ l ) + ] B(μ l = ). (89) s + /τ l τ l When μ l appears in the form of /μ l, the limit of /μ l at μ l diverses. In this case, instead of eq. (89), we use the following formula to obtain the corresponding s-dependent matrices: B(ˆμ l ) = sb(μ l ) + } B(μ l ) B( μ l )]. (9) s 2τ l Therefore, as shown in Fukahata & Matsu ura (26), the s-dependent deformation matrices Y E (z; j; ˆμ l, ˆγ l ) and Y E (z; j; ˆμ l ) can be expressed in the form of a rational function of s as M Y E i= (z; j; ˆμ l, ˆγ l ) = A is i M M i= b is, i Y E i= (z; j; ˆμ l ) = A i si M i= b i si. (9) Then, the s-dependent deformation vectors yk E(z; j; ˆμ l, ˆγ l ) and y E k (z; j; ˆμ l), which are the kth rows of Y E (z; j; ˆμ l, ˆγ l ) and Y E (z; j; ˆμ l ), can also be expressed in the form of a rational function: y E k (z; j; ˆμ l, ˆγ l ) = M i= a kis i M i= b is i, y E k (z; j; ˆμ l) = M i= a ki si M i= b i si (92) a km = y E k b (z; j; μ a km l,γ l ), = y E M b k (z; j; μ l). (93) M Here, it should be noted that the degrees M and M of the polynomials for the deformation vectors depend on the case. For example, when the source is located in one of the elastic layers (m l) overlying the viscoelastic substratum (l = n), the degrees of polynomials are M = 3 and M =. When the viscoelastic layer out sources intervenes between elastic layers (l m, n), the degrees of polynomials are M = 6 and M = 2 (Matsu ura et al. 98). If the number of viscoelastic layers is more than one, this procedure becomes more complicated (Sato & Matsu ura 993). General treatment in such a case is given in Fukahata & Matsu ura (26). Given the explicit expressions for the s-dependent deformation vectors, yk E(z; j; ˆμ l, ˆγ l ) and y E k (z; j; ˆμ l) in the form of rational functions, we can obtain the viscoelastic solution in the time domain by using the algorithm developed by Matsu ura et al. (98) as follows. First, division algorithm and partial fraction resolution, we rewrite ỹk V (z, s; ξ; j) and ỹ V k (z, s; ξ; j) in eq. (84) as ỹk V (z, s; ξ; j) = M ] s ye k (z; ξ; j; μ l,γ l ) v ki (ξ) s s ς i= i (ξ) ỹ V k (z, s; ξ; j) = s y E k (z; ξ; j; μ M ] (94) l) v ki (ξ) s s ς i (ξ) v ki (ξ) = b M (ξ)ς i (ξ) v ki (ξ) = b M (ξ)ς i (ξ) M j=( j i) M j=( j i) i= ς i (ξ) ς j (ξ) M j= j= a kj (ξ)ς j i (ξ) M ς i (ξ) ς j (ξ) a kj (ξ)ς j i (ξ), (95) where ς i (ξ) and ς i (ξ) denote the roots of algebraic equations M i= b i(ξ)s i = and M i= b i (ξ)si =, respectively, which are always real-negative. Next, performing the inverse Laplace transformation of eq. (94), we obtain the expressions for the deformation vectors in the time domain as M yk V (z, t; ξ; j) = H(t)yE k (z; ξ; j; μ l,γ l ) H(t) v ki (ξ) e ] ς i (ξ)t i=. (96) y V k (z, t; ξ; j) = H(t)y E k (z; ξ; j; μ M l) H(t) v ki (ξ) e ς (ξ)t] i i= Here, the first terms on the right-hand side of the above equations represent the instantaneous elastic deformation, and the second terms the transient deformation due to viscoelastic stress relaxation. Finally, replacing ỹk V (z, s; ξ; j)andỹ V k (z, s; ξ; j) in eq. (83) yv k (z, t; ξ; j) and (z, t; ξ; j) in eq. (96), respectively, we obtain the expressions for viscoelastic displacements in the time domain: y V k u V r(ϕ,z) (r,ϕ,z, t; j) = u E r(ϕ,z) (r,ϕ,z, t; j) + ut r(ϕ,z) (r,ϕ,z, t; j), (97) where ur(ϕ,z) E are the instantaneous elastic responses obtained in Section 3, and ur(ϕ,z) T are the transient responses due to viscoelastic stress relaxation, represented by the superposition of exponentially decaying modes the ξ-dependent relaxation times of / ς i (ξ) or C 28 The Authors, GJI, 75, Journal compilation C 28 RAS

13 4 A. Hashima et al. / ς i (ξ) : u T r (r,ϕ,z, t; j) = H(t) uϕ T (r,ϕ,z, t; j) = H(t) uz T (r,ϕ,z, t; j) = H(t) M i= M i= M i= v i (ξ) r j(r,ϕ; ξ)t i(t; ξ)dξ + v i (ξ) r ϕ j(r,ϕ; ξ)t i(t; ξ)dξ + v 2i (ξ)j(r,ϕ; ξ)t i (t; ξ)ξdξ M i= M i= v i (ξ) r ϕ j (r,ϕ; ξ)t i (t; ξ)dξ ] v i (ξ) r j (r,ϕ; ξ)t i (t; ξ)dξ ] T i (t; ξ) = exp ς i (ξ)t], T i (t; ξ) = expς i (ξ)t]. (99) (98) 5 NUMERICAL EXAMPLES In the previous sections we derived the expressions for the quasi-static internal displacement fields due to a moment tensor in an elastic/viscoelastic multilayered half-space under gravity. In order to obtain the displacement fields due to a finite-dimensional source we numerically integrate the point source solutions over the source region. In the present section, as numerical examples, we show the quasi-static internal displacement fields associated dyke intrusion, episodic segmental ridge opening and steady plate divergence in the case of an elastic viscoelastic two-layered half-space. The values of structural parameters used for computation are given in Table. The coordinate system and crack geometry are shown in Fig. 3, where the rectangle represents a vertical tensile crack length L and width W. We also demonstrate the usefulness of the source representation moment tensor through the numerical simulation of deformation cycles associated the periodic occurrence of interplate strike-slip earthquakes in a ridge-transform fault system. Further numerical examples for shear faulting are given in Fukahata & Matsu ura (26). At the moment of crack opening or shear faulting, the step-response of the composite system is completely elastic. As time passes, deviatoric stress in the viscoelastic substratum gradually relaxes. Then, after the completion of the viscoelastic stress relaxation, a certain amount of permanent displacements remains. At the final stage the elastic viscoelastic composite system behaves just like an elastic plate floating on water. In the following numerical examples we show how the patterns of displacement fields change time and depend on the source extent. Table. The structural parameters of the elastic viscoelastic two-layered model used for computation No. Thickness V p (km s ) V s (km s ) ρ (kg m 3 ) η (Pa s) h O L x y W H z Figure 3. The coordinate system and crack geometry in numerical computation. The rectangle represents a vertical tensile crack length L and width W in the elastic surface layer overlying a viscoelastic half-space. 5. Dyke intrusion First, we consider the sudden opening of a -km-long vertical crack embedded in a 2-km-thick elastic surface layer different depths to the top. The bottom of the crack reaches down to the elastic viscoelastic layer interface. The depth to the crack top is taken to be 5,, 5 and C 28 The Authors, GJI, 75, Journal compilation C 28 RAS

14 General expressions for internal deformation due to a moment tensor 5 Distance from Crack km] m Depth km] 2 4 t = yr t = yr t = yr t = yr t = yr t = yr t = yr 4 t = yr (a) (b) Figure 4. Temporal change of internal displacement fields due to the sudden opening of a -km-long vertical crack embedded in the 2-km-thick elastic surface layer different depths to the top. The vertical cross-sections of the internal displacement fields at the centre of the -km-long crack are shown in the cases of (a) W = 5 km, (b) W = km, (c) W = 5 km and (d) W = 2 km. The double solid line represents the vertical section of the crack. The values of structural parameters used for computation are given in Table. km, which correspond to the cases (a), (b), (c) and (d) in Fig. 4, respectively. This numerical example may be compared to dyke intrusion. In each case we show the vertical cross-sections of internal displacement fields at t =,, and yr after the crack opening. In either case of (a), (b) and (c) where the crack is buried under the Earth s surface, we can observe divergent horizontal displacements due to crack opening at the early stage. As time passes, the region surrounding the crack gradually subsides because of the viscoelastic stress relaxation in the substratum. This deformation pattern can be interpreted as the downward bending of the elastic plate by crack opening at the bottom. In the case (d) where the top of the crack reaches to the Earth s surface, the displacement fields show essentially different patterns from the previous three cases. At the early stage we can observe the broad uplift of the Earth s surface in addition to the divergent horizontal displacements. At the latter stage, the uplift pattern holds in the upper half of the elastic surface layer, but gradually changes to subsidence in the lower half as stress relaxation proceeds in the viscoelastic substratum. Then, the deformation pattern becomes nearly symmetric respect to the mean depth of the elastic surface layer. 5.2 Episodic segmental ridge opening Second, we consider the sudden opening of a -km-long vertical crack that cuts through the 2-km-thick elastic surface layer overlying a viscoelastic substratum. This case may be compared to episodic segmental ridge opening, observed in Iceland for example. In Fig. 5, we show the patterns of surface horizontal displacement fields (top panel) and the vertical cross-sections of internal displacement fields (bottom panel) at the time t =,, and yr after the crack opening. From the patterns of surface horizontal displacement fields we can observe that the deformation area gradually expands in the direction perpendicular to the crack strike time. Such diffusive crustal motion C 28 The Authors, GJI, 75, Journal compilation C 28 RAS

15 6 A. Hashima et al. Distance from Crack km] m Depth km] 2 4 t = yr t = yr t = yr t = yr t = yr t = yr t = yr 4 t = yr (c) (d) Figure 4. (Continued.) has been observed after the episodic segmental ridge opening in Iceland (Foulger et al. 992). From the vertical cross-sections of internal displacement fields we can observe the broad uplift of the Earth s surface at the early stage. As time passes, while the divergent horizontal displacements remain, the upward displacements in the elastic surface layer gradually disappear because of the viscoelastic stress relaxation in the substratum. 5.3 Steady plate divergence Third, we consider the sudden opening of an infinitely long crack that divides the elastic surface layer into two plates. Here, the thickness of the elastic surface layer h is taken to be km. The values of the other structural parameters are the same as those in the previous cases. This case may be compared to steady plate divergence, observed in mid-ocean ridges, as explained later. For the computation of displacement fields due to an infinitely long crack, we use the solution for a line source, which is obtained by integrating the solution for a point source derived in the previous section along the x-axis (Sato & Matsu ura 993; Fukahata & Matsu ura 25) as H(t) ux L (y, z, t; j) = H(t) u L y (y, z, t; j) = H(t) uz L (y, z, t; j) = y V (z, t; ξ; j) l x(y; ξ)dξ y V (z, t; ξ; j) l y(y; ξ)dξ y V 2 (z, t; ξ; j) l z(y; ξ)dξ () C 28 The Authors, GJI, 75, Journal compilation C 28 RAS

16 General expressions for internal deformation due to a moment tensor 7 Depth km] Distance along Crack km] Distance from Crack km] m t = yr t = yr t = yr t = yr Figure 5. Temporal change of the surface and internal displacement fields due to the sudden opening of a -km-long vertical crack that cuts through the 2-km-thick elastic surface layer overlying the viscoelastic substratum. The horizontal displacement fields at the surface and the vertical cross-section of internal displacement fields at the centre of the -km-long crack are shown at the top and the bottom of each diagram, respectively. The double solid lines represent the horizontal and vertical sections of the crack. The values of structural parameters used for computation are given in Table. m sinξ y m ( ) cos ξ y m3 cos ξ y m l x(y; ξ) = 2, l y (y; ξ) = 2 2 sinξ y 2m 6 sinξ y m 4 cos ξ y, l m z(y; ξ) = 2 2 cos ξ y m 4 sinξ y, () m 5 sinξ y m 5 cos ξ y where the deformation vectors y V, yv and yv 2 are the same as those in the case of point sources, and m i (i =,..., 6) are defined in eq. (27). In Fig. 6 we show the vertical cross-sections of internal displacement fields at the time t =, 4τ, 2τ and 2 τ. Here, τ = η 2 /μ 2 denotes the characteristic time of stress relaxation in the viscoelastic substratum. The displacement fields at the early stage (t =, 4τ) are similar to those in the case of the -km-long crack (Fig. 5). As time passes (t = 2τ), the upward displacements in the elastic surface layer gradually disappear, and the divergent horizontal displacements become dominant. Unlike the case of the -km-long crack, the deformation area gradually expands time to infinite distance. Then, at the final stage (t = 2 τ), we can observe the horizontally divergent rigid plate displacements, accompanied by upward displacements in the viscoelastic substratum. In Fig. 7 we show the profiles of the horizontal and vertical components of the surface displacement fields in Fig. 6 at t =, 2τ,4τ,τ, 2τ, τ and 2 τ. From Fig. 7(a) we can observe that the initial divergent horizontal displacement field gradually expands from the source area to the distant region time and finally tend to the divergent rigid plate displacements the half value of crack opening. From Fig. 7(b), on the other hand, we can observe that the initial vertical displacements characterized by uplifts in the source area gradually decay to zero time. Hofton & Foulger (996) have computed the temporal change in the profiles of surface displacement fields due to crack opening in a very similar situation (sudden opening of an infinitely long vertical crack cutting through the -km-thick elastic surface layer slightly different elastic constants), and so we can directly compare their results our results in Fig. 7. In their results, the horizontal displacements overshoot the half value of crack opening at t = 2τ, and the transition to the divergent rigid plate displacements is irregular. As for the vertical displacements (their profiles are probably upside down), the significant deformation of the elastic plate remains at t. These discrepancies between their results and our results must come from the mathematical expressions used for numerical computation. Hofton & Foulger (996), based on Hofton et al. (995), used the up going algorithm of the propagator matrix, which is numerically unstable at the Earth s surface. In addition, the use of an approximation technique for inverse Laplace transformation (Rundle 982a) would also cause to the discrepancies. C 28 The Authors, GJI, 75, Journal compilation C 28 RAS