Post-seismic rebound of a spherical Earth: new insights from the application of the Post Widder inversion formula

Transcription

1 Geophys. J. Int. (2008) 174, doi:.1111/j x x Post-seismic rebound of a spherical Earth: new insights from the application of the Post Widder inversion formula D. Melini, 1 V. Cannelli, 1 A. Piersanti 1 and G. Spada 2 1 INGV - Istituto Nazionale di Geofisica e Vulcanologia, Roma, Italy. melini@ingv.it 2 Istituto di Fisica, Università degli Studi di Urbino Carlo Bo, Urbino, Italy Accepted 2008 May 7. Received 2008 April 24; in original form 2007 July 9 GJI Seismology SUMMARY The post-seismic response of a viscoelastic Earth to a seismic dislocation can be computed analytically within the framework of normal-modes, based on the application of propagator methods. This technique, widely documented in the literature, suffers from several shortcomings; the main drawback is related to the numerical solution of the secular equation, whose degree increases linearly with the number of viscoelastic layers so that only coarse-layered models are practically solvable. Recently, a viable alternative to the standard normal-mode approach, based on the Post Widder Laplace inversion formula, has been proposed in the realm of postglacial rebound models. The main advantage of this method is to bypass the explicit solution of the secular equation, while retaining the analytical structure of the propagator formalism. At the same time, the numerical computation is much simplified so that additional features such as linear non-maxwell rheologies can be simply implemented. In this work, for the first time, we apply the Post Widder Laplace inversion formula to a post-seismic rebound model. We test the method against the standard normal-mode solution and we perform various benchmarks aimed to tune the algorithm and to optimize computation performance while ensuring the stability of the solution. As an application, we address the issue of finding the minimum number of layers with distinct elastic properties needed to accurately describe the post-seismic relaxation of a realistic Earth model. Finally, we demonstrate the potentialities of our code by modelling the post-seismic relaxation after the 2004 Sumatra Andaman earthquake comparing results based upon Maxwell and Burgers rheologies. Key words: Numerical solutions; Transient deformation; Rheology: mantle. 1 INTRODUCTION Analytical models of coseismic and post-seismic response have been for decades a valuable tool to investigate the physics of Earth s interior and to model the seismic quasi-static displacements. In more recent years, completely numerical models, based on techniques like the finite element method, became widely employed; these methods allow to overcome most of the intrinsic limitations of analytical models, for instance including lateral heterogeneities (Wu 2004). Nevertheless, analytical models have not completely lost their relevance, since they are often used as a benchmarking and calibration tool for numerical codes. In the finite element approach, a crucial point is represented by the mesh generation, which is often extremely time-consuming and oriented to the details of the particular problem being solved, while analytical models can be easily applied automatically to problems involving a large number of seismic sources, as done by Casarotti et al. (2001) or Melini & Piersanti (2006). In this paper, we focus on the semi-analytical model originally developed by Piersanti et al. (1995) by extending the work of Sabadini et al. (1984), which allows to compute the post-seismic relaxation of a spherical, incompressible, self-gravitating, layered viscoelastic Earth. The solution is based on the standard normal-modes approach (hereafter NM) originally introduced by Peltier (1974) in the realm of viscoelastic Earth models; while this is a widely employed solution scheme, it suffers from several limitations. The main shortcoming is the numerical instability connected with the solution of the so-called secular equation, which may imply a loss of accuracy in the computation of the solution, if not a complete degeneration of the harmonic terms. Since the degree of the secular equation scales with the number of layers, only coarse models can be safely employed in practice. In fact, the application of post-seismic models based upon NM has been limited so far to a few viscoelastic layers. Moreover, if a compressible rheology is considered, it has been extensively shown (Vermeersen et al. 1996b) that the secular equation becomes transcendental and the number of associated roots is infinite, so that its solution even with purely numerical methods poses various difficulties. Several methods have been proposed in the literature as workarounds to the shortcomings of the NM approach. Riva & Vermeersen (2002) developed a rescaling procedure aimed at the 672 C 2008 The Authors

2 Post Widder algorithm and post-seismic rebound 673 elimination of stiffness in matrix propagators; in the seismological context, an elegant way of integrating the displacement stress vectors with the method of second-order minors have been proposed by Friederich & Dalkolmo (1995). The issue of computing the Laplace inverse has been addressed, among others, by Rundle (1982) by means of a Prony-series approach and by Tanaka et al. (2006) with a direct numerical integration in the complex plane. Recently, a new solution scheme has been proposed (Spada & Boschi 2006) to overcome these difficulties in the realm of both surface and tidal loading problem, which shares a large part of the analytical formulation with post-seismic relaxation and therefore may suffer from the same problems (Spada 2008). This solution scheme is based on the application of the so-called Post Widder formula (Post 1930; Widder 1930; hereafter PW), which provides a convenient way of evaluating the Laplace inverse of a function, avoiding the computation of Bromwich path integrals and thus bypassing the Residue Theorem and any root-finding procedure. Spada & Boschi (2006) have shown that the application of the PW method to postglacial rebound allows to overcome many modelling limitations while, at the same time, leading to a substantial simplification of the codes; Spada (2008) has recently shown that the PW formula permits a straightforward implementation of general (possibly transient) linear rheologies in addition to the Maxwell law. These improvements, however, came at the cost of a consistent increase of the computation power Model P N=2 N=5 N= N= Model U ρ (kg m ) ρ (kg m ) Radius (km) Radius (km) 6000 Model L 6000 Model R ρ (kg m ) 4500 ρ (kg m ) Radius (km) Radius (km) Figure 1. Density profiles of the four layering models for N = 2, 5, and 50. μ (GPa) Model P N=2 N=5 N= N=50 μ (GPa) Model U Radius (km) Radius (km) 300 Model L 300 Model R μ (GPa) μ (GPa) Radius (km) Radius (km) Figure 2. Rigidity profiles of the four layering models for N = 2, 5, 1 and 50.

3 674 D. Melini et al. requirements; incidentally, this is the reason why, despite its age, the PW Laplace inversion has not been used in practical application until the wide availability of high-performance computer systems. In this work, we apply the PW method to the post-seismic rebound model by Piersanti et al. (1995). Following Spada & Boschi (2006), we have suitably modified the NM analytical formulation of the model to apply the PW formula. Particular attention has been put on the benchmark of the PW code to assure its coherence with independent solutions and to the optimization of the algorithm parameters since the PW method leads to a substantial increase of the computation times, so that the optimal trade-off between stability and performance has to be carefully established. As a practical application of our code, we investigate the effect of elastic layering structure on post-seismic relaxation as done by Spada & Boschi (2006) for the postglacial uplift problem. Finally, we illustrate the new capabilities offered by the PW method by a forward modelling of the effect of the Burgers rheology on the post-seismic relaxation following the 2004 Sumatra Andaman earthquake. 2 VISCOELASTIC NORMAL MODES The theoretical framework of the NM technique applied to postseismic viscoelastic deformations has been presented in a number of manuscripts (Pollitz 1992; Piersanti et al. 1995; Vermeersen et al. 1996a; Soldati et al. 1998). Here we only focus on those parts that are relevant for the illustration of the PW inversion method; the reader is referred to Piersanti et al. (1995) and Boschi et al. (2000) for the details. As shown by Smylie & Mansinha (1971), the equilibrium equations and the Poisson equation for a spherical, incompressible, selfgravitating viscoelastic body can be reduced to a system of algebraic equations. For both spheroidal and torsional components, and for any harmonic degree l and order m, the Laplace-transformed solution reads x(s) = QR 1 b + p, (1) where s is Laplace variable, and the unknown vector x includes information upon displacements and incremental gravitational potential (Peltier 1974; Sabadini et al. 1984). As discussed by Piersanti et al. (1995) and Boschi et al. (2000), the arrays Q and R in eq. (1) are determined by propagating the fundamental matrix of the system through the mantle, while vectors b and p account for boundary conditions at the Earth s surface, at the core mantle boundary, and at the source radius. By the Correspondence Principle of linear viscoelasticity (e.g. Fung 1965) all the variables in eq. (1) implicitly depend on the Laplace variable s through a complex shear modulus 2 d=50 km, vertical component 2 d=50 km, horizontal component d=0 km, vertical component d=0 km, horizontal component d=200 km, vertical component d=200 km, horizontal component d=500 km, vertical component d=500 km, horizontal component d=00 km, vertical component d=00 km, horizontal component Number of layers, N Number of layers, N Figure 3. Misfit of radial and horizontal deformations obtained with model P with respect to results obtained with a discretized PREM model, as a function of number of stratification layers N, for various source observer distances. The seismic source is a pointlike pure thrust with dip δ = 20 and scalar moment M 0 = 21 N m, buried at 70 km. The same scalar moment is assumed in all the following figures, unless explicitly given. The observer is located on the direction with azimuth 90 with respect to the strike direction. Solid, dashed and dotted lines represent the responses at t = 0, and 00 yr, respectively. Two horizontal dashed lines represent 5 and per cent misfit thresholds.

4 Post Widder algorithm and post-seismic rebound 675 that for the Maxwell linear rheological law reads μ = μs s + μ/η, (2) where μ and η are the elastic shear modulus and viscosity for a given layer, respectively (Fung 1965). Within the traditional NM method (Wu & Peltier 1982), Laplace inversion of eq. (1) is normally performed by the Residue Theorem. Following Boschi et al. (2000), for an impulsive source time-history, the solution is [ ] K QR b + R p R x e x(t) = x e δ(t) + d k=1 R(s) e sk t, (3) ds s=s k where is the adjoint and... denotes the determinant. The elastic response is x e = lim x(s), (4) s and with s k,(k = 1,... K ) we indicate the (isolated) roots of the secular equation R(s) =0, (5) that determine the characteristic decay times of NM by τ k = 1 s k. As discussed by Pollitz (1992) and Spada et al. (2004), for a stable density stratification and incompressible rheology, the roots of the secular equations are found on the negative real axis of the complex plane. Assuming a Maxwell rheology, for the poloidal problem their number is K = 4L, where L is the number of mantle layers with distinct characteristic Maxwell times, while for the toroidal problem, K = 2L. The time-domain solution vector in eq. (3) corresponds to an impulsive source; the result can be easily generalized to the case of an arbitrary source time-history f (t) by a time convolution between x(t) and f (t). In what follows, we will consider sources with an Heaviside time-history, f (t) = H(t). The displacement vector u = (u r, u θ, u φ ) (6) and the perturbation to gravitational potential φ can be explicitly obtained from the harmonic coefficients u lm, v lm, t lm and φ lm, included in the time-domain solution vector x(t) and in its toroidal analogue, as follows: l u lm (r) u = v lm (r) θ + t lm (r) φ Y m l (θ,φ) (7) l=0 m= l v lm (r) φ t lm (r) θ φ = l l=0 m= l φ lm (r)y m l (θ,φ), (8) 2 d=50 km, vertical component 2 d=50 km, horizontal component d=0 km, vertical component d=0 km, horizontal component d=200 km, vertical component d=200 km, horizontal component d=500 km, vertical component d=500 km, horizontal component d=00 km, vertical component d=00 km, horizontal component Number of layers, N Number of layers, N Figure 4. Misfit of radial and horizontal deformations obtained with model R with respect to results obtained with the reference model, as a function of number of stratification layers N. See also caption of Fig. 3.

5 676 D. Melini et al. where =( r, θ, φ ) is the gradient operator in spherical coordinates and Yl m are the spherical harmonic functions Y m l (θ,φ) = ( 1) m P m l (cos θ)e imφ (9) with Pl m (z) being the associated Legendre functions; further details are found in Piersanti et al. (1995) and Soldati et al. (1998). An approximated expression of the gravitational acceleration variation g at the deformed surface r = a + u r can be obtained from φ lm coefficients, as discussed by Soldati et al. (1998). 3 POST WIDDER ALGORITHM At the core of the semi-analytical NM approach outlined above is the explicit computation of eq. (3), which demands knowledge of the roots of the secular equation (5). Since its degree scales with the number of mechanically distinct layers (Wu & Ni 1996; Spada et al. 2004) and since for high polynomial degrees the root-finding algorithms become unstable due to numerical noise and roots coalescence (Vermeersen & Sabadini 1997; Spada 2008), the range of practically solvable Earth models is actually limited. Moreover, to explicitly compute the elastic limit x e in eq. (4) and the derivative of R(s) in eq. (3), one must keep track of the single polynomial coefficients in s of Q and R, which implies a rapidly increasing complexity of the code as L increases. As recently discussed by Spada & Boschi (2006), the limitations of the NM approach can be overcome using a numerical implementation of the PW formula (Post 1930; Widder 1930), which provides the Laplace inverse by sampling the values of the transform and its derivatives on the real positive axis. The method is particularly attractive since it allows to skip the numerical solution of eq. (5) while retaining the same analytical and elegant structure of the NM approach. Since for a stably stratified incompressible Earth the roots of the secular equation are placed along the real negative axis (Vermeersen & Mitrovica 2000), the sampling region is singularity-free, which makes the PW formula a valid alternative to the normal modes approach. In its original formulation (Post 1930; Widder 1930), the PW formula reads: ( ) ( 1) n n+1 [ n d n f (t) = lim n n! t ds n ] f (s), () s= n t where f (s) indicates the Laplace-transform of f (t). A direct application of eq. () is not practical because it involves the nth derivative of the Laplace transform which is not available analytically in general, while a numerical estimation would become increasingly unstable for high values of n due to the propagation of round-off errors in the finite differentiation and consequent catastrophic cancellation (Abate & Valkó 2004). For practical applications, a discretized version of eq. () has been proposed by Gaver (1966), based on the 2 d=50 km, vertical component 2 d=50 km, horizontal component d=0 km, vertical component d=0 km, horizontal component d=200 km, vertical component d=200 km, horizontal component d=500 km, vertical component d=500 km, horizontal component d=00 km, vertical component d=00 km, horizontal component Number of layers, N Number of layers, N Figure 5. Misfit of radial and horizontal deformations obtained with model L with respect to results obtained with the reference model, as a function of number of stratification layers N. See also caption of Fig. 3.

6 Post Widder algorithm and post-seismic rebound 677 sequence: f k (t) = k ln 2 t ( ) ( ) 2k k k k j=0( 1) j f j [ ] (k + j)ln2 t (11) with f k (t) f (t) for k. Eq. (11) does not involve the derivatives of f, but its principal shortcoming is its slow convergence, which scales with k as f (t) f k (t) c/k (Valkó & Abate 2004). Moreover, the alternating nature of the series in eq. (11) may lead to loss of precision due to cancellation of significant digits. To overcome these problems, several acceleration schemes have been proposed, which are reviewed by Valkó & Abate (2004). One of the most employed is the Salzer acceleration scheme, originally proposed by Stehfest (1970), which is based on a re-arrangement of the terms in eq. (11). Accordingly, the approximate Laplace inverse is: f (t, M) = ln 2 t with ζ k = ( 1) M+k 2M k=1 min(k,m) j=[(k+1)/2] ( ) k ln 2 ζ k f t j M+1 M! ( M j (12) )( )( ) 2 j j, (13) j k j where [N] is the greatest integer less or equal to N, and we note that the weights ζ k depend only on M and k. This is also known as the Gaver Stehfest algorithm; two of its most remarkable features are that it is linear and involves only real algebra. Using eq. (12), we can write the Gaver Stehfest approximation of the solution vector x(t) for the spheroidal case as: x(t, M) = ln 2 t 2M k=1 ( k ln 2 ζ k [Q t ) ( ) ] k ln 2 R 1 b + p. (14) t The solution of a post-seismic rebound problem through the application of eq. (14) has potentially many advantages. As mentioned above, the numerical problem of the solution of the secular equation is completely bypassed but, at the same time, the solution scheme based on propagator matrices is still valid. Since using the PW formula the relaxation times associated to NM remain undetermined, estimates of the characteristic timescales of mantle relaxation should be obtained by interpolation of the response, as done by Hanyk (1999) in the context of glacio-isostatic adjustment. Since within the PW approach the matrices in eq. (14) have simply to be evaluated in the time-domain, the resulting codes are extremely simplified and, at the same time, more flexible. Indeed, for each sampling point s k = k ln 2/t, it is sufficient to compute an equivalent rigidity according to eq. (2) and proceed with the evaluation of eq. (14) using formally elastic analytical expressions. With this prescription, it is straightforward to extend the code to non-maxwell rheologies (Spada 2008). For instance, the transient Burgers rheol- 2 d=50 km, vertical component 2 d=50 km, horizontal component d=0 km, vertical component d=0 km, horizontal component d=200 km, vertical component d=200 km, horizontal component d=500 km, vertical component d=500 km, horizontal component d=00 km, vertical component d=00 km, horizontal component Number of layers, N Number of layers, N Figure 6. Misfit of radial and horizontal deformations obtained with model U with respect to results obtained with the reference model, as a function of number of stratification layers N. See also caption of Fig. 3.

7 678 D. Melini et al. ogy (Yuen & Peltier 1982; Pollitz 2003) that we consider in Section 5 below can be simply implemented computing the equivalent elastic rigidity as s + μ 1 /η 1 μ B = (15) (s + μ 2 /η 2 )(s + μ 1 /η 1 ) + μ 1 s/η 2 instead of using eq. (2), where μ 1, η 1 represent the shear modulus and viscosity of the Maxwell element, while μ 2, η 2 pertain to the Kelvin Voigt element of this rheological model (Peltier et al. 1981). While the impact of this change in the code is simply a different expression for the computing of the equivalent elastic rigidity in the evaluation of eq. (14), it is known from previous studies that within the NM method the implementation of rheological laws such as eq. (15) implies a significant increase of algebraic complexity that can be eventually tackled only using algebraic manipulators (Yuen et al. 1986). This point has been recently addressed by Spada & Boschi (2006). With respect to alternative solution schemes proposed in literature, the PW approach has several advantages. It basically requires a sampling of the Laplace-transformed solution vector on a pre-determined set of points on the real axis, as with the Pronyseries method proposed by Rundle (1982), but it does not require an estimation of the roots of the secular equation nor any assumption on the functional form of the solution in the time-domain. With respect to schemes involving an explicit Laplace inversion, as done by Tanaka et al. (2006), the PW method has the further advantage of not requiring numerical integrations on the complex plane. As pointed out by Abate & Whitt (2006), the Gaver Stehfest algorithm usually requires high numerical precision, because the oscillating terms in eq. (12) may lead to catastrophic cancellation. In particular, when the order of the approximation is M, such precision can be estimated as about D = 2.2M and the relative error on the computation of the numerical transform is ɛ = f (t) f (t, M) f (t) 0.90M (16) (Abate & Valkó 2004), valid only for functions having all the singularities of their Laplace transform on the real negative axis and indefinitely differentiable in t. The highest precision available with commercial FORTRAN compilers corresponds to the IEEE extended-precision format (REAL 16), which has a number of significant digits D 30, depending on the particular implementation (see IEEE Task P754, 1985). Since a certain number of digits should be kept as guard digits to avoid the propagation of round-off errors, such precision may be not sufficient to successfully apply the PW method. It is therefore convenient to use one of the publicly available multiprecision libraries, which allow to carry out the entire computation at any desired precision level. The drawback is, of course, a massive performance degradation in comparison with the usage of native hardware floating-point. For this work we adopted the Fortran 90 multiprecision library FMLIB, freely available on (Smith 1989). In order to determine the algorithm parameters that minimize the computation load and simultaneously ensure stability, we perform an Figure 7. Minimum number of layers needed to reproduce the observables computed with the reference model within a 5 per cent threshold. The x axis corresponds to source observer distance along the direction with azimuth 90 with respect to the strike, while the y axis represents observation time: t 1 = 0, t 2 =, t 3 = 2, t 4 = 3, t 5 = 4 and t 6 = 5 yr, respectively. Three point seismic sources are considered: a pure thrust with dip δ = 20 at depth z = 70 km (S1), a pure thrust with δ = 8 at z = 20 km (S2) and a pure vertical strike-slip at z = km (S3).

8 Post Widder algorithm and post-seismic rebound 679 extensive set of benchmarks and check the PW solution against an independent NM solution obtained with the model by Piersanti et al. (1995) (see Appendix A). 4 FITTING PREM WITH MULTILAYERED MODELS One of the key issues in modelling post-seismic displacements is the effect of lithospheric and mantle layering. For the response of the Earth to surface loads, this topic has been addressed by Spada & Boschi (2006), which pointed out that a uniformly layered stratification with 40 layers approximates the results obtained using a finely layered PREM discretization to within 1 per cent. In the case of post-seismic rebound, several complications arise. The number of harmonic terms needed to obtain a satisfactory convergence for a point seismic source is much larger than in the postglacial rebound case (Riva & Vermeersen 2002; Casarotti 2003); indeed, while the forcing terms of a surface load are integrated over the load itself leading to a smoothing of small wavelength components of the Green s function, for a seismic source an explicit integration over the source is not viable so that the solution virtually contains all harmonics. Moreover, the forcing term corresponding to a seismic dislocation contains δ terms in addition to the δ terms (where δ is the Dirac delta and δ is its derivative; see e.g. Piersanti et al. 1995), which are not present in surface load forcing terms, introducing additional stiffness in the solution. Therefore, for high harmonic degrees, the solution may become sensitive to short wavelength layering structure; moreover, the post-seismic relaxation is also dependent on the depth of the seismic source, since the predominant relaxation mode will be that of the viscoelastic layer closest to the source (Nostro et al. 1999, 2001). In what follows, we attempt to characterize the dependence of post-seismic relaxation on elastic layering structure, while keeping constant the viscosity profile. We have computed physical observables with models of increasing radial resolution and studied their convergence to results obtained with a reference model based on a PREM discretization. We define four different approaches to the layered model definition, as follows: (i) Model U: A uniform, equally spaced stratification with N layers from Earth surface to CMB. u r (mm) N=1 N=2 N=4 N=6 N= PREM t (yr) u φ (mm) N=1 N=2 N=4 N=6 N= PREM t (yr) 5 x φ (Nm/kg) N=1 N=2 N=4 N=6 N= PREM t (yr) Figure 8. Time-dependent displacement and incremental geopotential obtained with model P, varying the number of layers. The seismic source is a 1-D finite pure thrust with length 80 km, depth z = 20 km and dip angle δ = 8. Observation point is at 250 km from the source, on the direction with azimuth 90 relative to the strike.

9 680 D. Melini et al. (ii) Model L: An 80 km homogeneous lithosphere and a uniformly stratified mantle with N equal-spaced layers. (iii) Model R: An 80 km layered lithosphere, with N/3 uniform layers, and a layered mantle with 2 N uniform layers. 3 (iv) Model P: An 80 km lithosphere stratified with the corresponding layers in table III of Dziewonski & Anderson (1981) and a uniformly layered mantle with N layers. For each layer, we assign a constant density and rigidity computed by volume-averaging the corresponding PREM values. Density and rigidity profiles of the four models are shown in Figs 1 and 2 for N = 2, 5, and 50. The viscosity profile is the same for each of the four models and assumes an 80 km thick elastic lithosphere, a 200 km thick low-viscosity asthenosphere with η A = 19 Pa s and upper and lower mantle viscosities η UM = 21 Pa s and η LM = 3 21 Pa s, respectively (Boschi et al. 2000). For each layering we compute surface displacements and compare them with values obtained with a reference model that closely approximates PREM. This reference model is built by considering all the layers listed in Table III of Dziewonski & Anderson (1981) for r > 3480 km (i.e. outside the core) and a uniform fluid core with density obtained by volume-averaging PREM core layers. In what follows, we will refer to this reference model as discretized PREM. In Figs 3 6 we show the misfit of the displacements with respect to the results obtained with discretized PREM for each of the four models as a function of the N, for various source observer distances and times. Clearly, a complete investigation of the convergence details is not possible due to the high-dimensional parameter space (source observer distance and azimuth, source depth, observation time). Anyway, our study shows that models P and R yield a more fast and regular convergence to PREM values, model U gives a slow and sometimes unstable convergence while model L does not improve the convergence when resolution is increased. These results confirm the well-known fact that a detailed lithosphere layering is important in modelling displacements; indeed, model P, which has a lithosphere layering that closely follows PREM, gives the best convergence; model R, which has a layered lithosphere, gives also a satisfactory convergence while models U and L show a poor convergence. In particular, since in model U we define a set of uniform, equally spaced layers, a very large N is needed to get a sufficiently stratified lithosphere. Incidentally, we note that this is in agreement with all the previous findings about post-seismic rebound, assessing the importance of shallow layers in determining surface u r (mm) N=5 N= N=20 N=40 N=50 N=200 PREM t (yr) u φ (mm) N=5 N= N=20 N=40 N=50 N=200 PREM t (yr) 8 x φ (Nm/kg) N=5 N= N=20 N=40 N=50 N=200 PREM t (yr) Figure 9. Time-dependent displacement and incremental geopotential obtained with model U, varying the number of layers. See also caption of Fig. 8.

10 Post Widder algorithm and post-seismic rebound 681 displacement and gravity variation (Vermeersen et al. 1996a; Antonioli et al. 1998; Nostro et al. 2001). Our findings are also in agreement with those obtained by Sabadini & Vermeersen (1997), who pointed out that elastic lithosphere stratification has a major influence in post-seismic displacements while the same resolution is not needed for mantle seismic sources. In Fig. 7 the convergence of our models to reference results is summarized by plotting the minimum number of layers needed to reproduce displacements and geopotential field obtained with the discretized PREM to within 5 per cent. We compute the misfit for all the previously considered models as a function of distance along the direction with azimuth 90 with respect to the strike direction, for six different observation times: t 1 = 0, t 2 =, t 3 = 2, t 4 = 3, t 5 = 4 and t 6 = 5 yr, respectively. We use three different point seismic sources: a pure thrust with dip δ = 20 (depth z = 70 km), a pure thrust with δ = 8 (z = 20 km) and a pure strikeslip with δ = 90 (z = km). We see that in most cases model U requires a large number of layers (N > 40) to reproduce reference results; model R requires approximately N 15 layers to fit the discretized PREM in nearly all conditions, but has some instabilities in the near-field. Model L requires N layers to fit reference results, except in the near-field where with the maximum number of layers (N 50) it still fails to reproduce reference results. With model P, a good convergence is obtained with N layers in all conditions. It can also be noted that a definite source observer distance exists, in the range between 500 and 00 km, at which the required number of layers shows some steep increases because the displacement or the gravity field crosses zero, and consequently the relative errors become quite large. In Figs 8 and 9, the convergence of models P and U to the discretized PREM is shown for a fixed source observer distance as a function of time. We employed a finite 1-D thrust source with length 80 km, dip δ = 8 and depth z = 20 km. The observation point is located at 250 km from the source, at an azimuth of 90 with respect to the strike direction. The time-dependent displacement and incremental gravitational potential obtained with different N values are compared with results obtained with the reference model. Model P reproduces well the PREM results with a few layers, while with model U N = 200 layers are needed to reach convergence, both in the coseismic and post-seismic limit. To assess more precisely the minimum number of layers needed to fit the PREM discretization within a specified threshold, we selected model P, which turns out to be the model with fastest convergence, and computed displacements as a function of N in a fixed point for specified observation times. In Fig., we plot the ratios u r /ur PREM, u φ /u PREM φ and φ/φ PREM as a function of the number of layers N, for the same 1-D finite source used above. Since layering only involves the shear modulus and density, the coseismic response shows the slowest convergence. For the post-seismic responses N = is adequate, while for the coseismic response at least N = 20 is u r / u r PREM Number of layers, N u φ / u φ PREM Number of layers, N φ / φ PREM t=0 t= yr t= 2 yr t= 3 yr t= 4 yr Number of layers, N Figure. Displacement and incremental geopotential obtained with model P as a function of the number of layers. Seismic source and observation points are the same of Fig. 8.

11 682 D. Melini et al. 18 o N 12 o N 6 o N 0 o 6 o S CHMI BNKK PHKT ARAU SAMP NTUS BAKO Another important issue is to assess whether a compressible rheology might be more important than a fine-layered stratification in modelling coseismic and post-seismic displacements; this issue has been recently addressed by Tanaka et al. (2006, 2007). According to their results, for a shallow earthquake compressibility mostly affects the elastic solution and it is more pronounced for vertical displacements, where its maximum impact is about 50 per cent, than on the horizontal displacements, where it accounts for 6 per cent at most. For a deep source, the effect of compressibility turns out to be more significant on all timescales. According to the results by Tanaka et al., we can conclude that using a detailed elastic layering which closely follows PREM can be as important as including compressibility when modelling horizontal displacements occurring after a giant earthquake, which are expected to peak in the sublitospheric shallow astenosphere as first pointed out by Sabadini et al. (1984). Compressibility is likely to become a critical issue when modelling interferometric and gravitational data, which are much more sensitive to vertical displacements. However, an accurate modelling of these observables also requires self-gravitation, and for fully compressible self-gravitating models the PW method is probably not viable since the secular equation has singularities on the real positive axis (Hanyk et al. 1999). 90 o E 95 o E 0 o E 5 o E 1 o E 115 o E Figure 11. Seismic source geometry model of the 2004 Sumatra earthquake and GPS sites considered in this study. needed. Considering that model P has 5 additional lithospheric layers besides the N mantle layers, we can conclude that 25 layers represents the minimum resolution needed to reproduce the PREM results within the chosen precision for the whole relaxation process. On the basis of what we discussed above, one interesting issue is to ascertain if a detailed stratification is really needed, considering the current (or near future) accuracies of GPS and gravitational data. Typical accuracies of GPS measurements are of the order of a few millimetres, often comparable to regional coseismic and postseismic signals following a large earthquake, and therefore even with a very coarse-layered model the approximation level may be well within the associated observational accuracy. However, for giant earthquakes, offsets of the order of centimetres have been recorded at near-field GPS sites with relative errors as small as a few percent (see e.g. Banerjee et al. 2007); in this case, a detailed layering is required in order to model the displacement field within experimental uncertainties. For what concerns the geoid determination, a detailed monthly observation is made available by GRACE, which has an estimated accuracy of 2 3 mm and a spatial resolution of 400 km (Tapley et al. 2004). Also in this case, the expected error on the geoid is comparable with the expected signal from a giant earthquake. Nevertheless, Panet et al. (2007) have shown that a suitable application of stacking and data filtering techniques effectively allows to detect small perturbations in the gravity solutions. Since the accuracy of the reconstructed signal is dependent on its strength and on the modelling accuracy of other geophysical signals, it is not possible to definitely ascertain what level of layering detail is effectively needed to keep the approximation within experimental uncertainties, and each case has to be considered separately on the basis of specific signal to noise ratios. 5 CASE STUDY: POST-SEISMIC RELAXATION FOLLOWING 2004 SUMATRA EARTHQUAKE In the following we discuss a practical application of the PW algorithm to the post-seismic relaxation following the 2004 Sumatra Andaman megathrust event, the second-greatest in the instrumental age (Banerjee et al. 2005; Lay et al. 2005; Vigny et al. 2005; Boschi et al. 2006; Pollitz et al. 2006). We model the seismic source as composed by four 1-D fault segments (see Fig. 11), which approximate the geometry of the slip distribution (Ammon et al. 2005). The contribution of each segment is obtained by superimposition of contributions of point sources with a discretization step of 6 km. The cumulative seismic moment of the source has been set to M 0 = Nm, corresponding to a body wave magnitude m b = 9.3 (Park et al. 2005; Tsai et al. 2005). With this model we compute the time-dependent displacement u and geoid perturbation H = φ/g, where φ is the perturbation to gravity potential and g is the reference gravity field. The post-seismic evolution has been computed with three different viscosity models which combine Maxwell and Burgers rheologies for asthenosphere and upper mantle, as summarized in Table 1. Elastic layering is that of model P with N = 20 and is common to all three rheologies. In Fig. 12 the components of displacement and geoid perturbations are shown in the coseismic (t = 0) case, which is common Table 1. Summary of rheologies employed in the study of post-seismic relaxation of the 2004 Sumatra earthquake. Viscosities are expressed in Pa s. r (km) Rheology 1 Rheology 2 Rheology Elastic Maxwell Burgers Maxwell η = 19 η 1 = 19 η = 19 η 2 = Maxwell Maxwell Burgers η = 21 η = 21 η 1 = 21 η 2 = Maxwell, η = 3 21

12 Post Widder algorithm and post-seismic rebound 683 Figure 12. Coseismic displacement field and geoid perturbation following the 2004 Sumatra earthquake. to the three rheological models. Figs show the post-seismic displacement and geoid perturbation obtained with the three rheology models of Table 1 for times t = 0.25, 0.5, 0.75 and 2 yr. These figures show the incremental post-seismic relaxation relative to the previous time step; for the first time step the reference field is the coseismic response plotted in Fig. 12. From the post-seismic fields we observe that the models with transient rheology show a fast postseismic relaxation immediately after the event, consistently with the presence of a low-viscosity element included in the mechanical analogue of the Burgers body (see Table 1). For longer timescales the response is comparable to that of a Maxwell rheology. The rheological model with a transient mantle shows a post-seismic relaxation affecting a much larger area, because in these conditions stress diffusion is enhanced (Yuen et al. 1986). The effect of a transient rheology on post-seismic relaxation appears to be more strong on the horizontal components, and therefore it is reasonable to expect in GPS time-series a clear signature of the rheological model even for short timescales, as pointed out by Pollitz et al. (2006). The variation of the Earth gravity field following the 2004 Sumatra earthquake has been strong enough to be detected by the GRACE satellite mission. The coseismic effect has been first evidenced by a suitable reprocessing of raw satellite ranging data (Han et al. 2006), while from subsequent analyses (Panet et al. 2007) it has been possible to extract the post-seismic signal also from monthly gravity solutions; a qualitative comparison of modelled geoid perturbation with observed data is therefore viable. The coseismic geoid perturbation determined by Panet et al. (2007) is dominated by a strong negative anomaly located on the Andaman Sea, with a weaker positive anomaly appearing at smaller wavelengths on the westward region. The amplitude of these anomalies is millimetric, but they turn out to depend on the spatial scale of the wavelet functions used to perform the analysis. The coseismic geoid perturbation resulting from our modelling (Fig. 12) shows a marked minimum centred at latitude 5 N and longitude 90 E with peak amplitude of 2 mm, comparable with the observed minimum but shifted westward of about 7. On the eastern part of the investigated region, our results show a positive lobe in the geoid perturbation map, falling outside the area considered by Panet et al. (2007); at small wavelengths we do not reproduce the positive anomaly evidenced by Panet et al. (2007). The observed post-seismic evolution of the geoid height shows a positive transient located in the Andaman Sea, which stabilizes after 3 4 months, superimposed to a broader slowly relaxing positive signal (Panet et al. 2007). Our modelling results (Fig. 16), for rheologies 2 and 3 which are characterized by a transient effect, show a fast signal which stabilizes within 4 months superimposed on a slower relaxation, as observed in data. For rheology 2, we find a positive post-seismic signal close to that observed but shifted eastward and a negative lobe on the western part that is not seen in the data, even if for t > 4 months a negative feature is present on the southwestern part of the investigated region (fig. 6 of Panet et al. 2007). From this comparison, we can summarize that our forward model reproduces the observed geoid time-evolution features if a Burgers astenosphere is employed. However, the small-scale pattern is not well reproduced, even if it is qualitatively in agreement and has the same orders of magnitude of the observational data. While a detailed understanding of the differences between observed and modelled feature would require further analyses, we can argue that

13 684 D. Melini et al. Figure 13. Post-seismic evolution of the u r component of displacement following the 2004 Sumatra earthquake for observation times t = 0.25, 0.5, 0.75 and 2 yr, respectively, using the three rheology models summarized in Table 1. For each time step we plot the incremental field relative to the previous time step; for the first step the incremental field is relative to the coseismic case (Fig. 12). these are most probably to be attributed to our coarse source approximation, which may affect the modelled predictions especially in the near-field. The 2004 Sumatra earthquake, with its exceptional energy release, produced static offsets recorded at continuously operating GPS sites up to thousands of kilometres away from the source (Banerjee et al. 2007). At near-field GPS stations, it has been possible to record the post-seismic signal due the slow post-seismic recovery of ductile layers, which represents an unique opportunity to test asthenosphere rheological models (Pollitz et al. 2006). We selected a set of seven GPS sites from those first investigated by Vigny et al. (2005), whose location is shown in Fig. 11, and computed the expected time-series at those sites with the rheological models of Table 1. In Fig. 17 we show predicted time-series for the north (N) and east (E) components of displacement at the selected GPS sites. For each rheology, we assess the effect of elastic stratification by comparing the results obtained with a detailed elastic layering (model P, N = 20, solid line) with those obtained with a coarse stratification, defined by assigning homogeneous PREMaveraged elastic parameters to each of the three layers of Table 1 (N = 3, dashed line). A dotted vertical line marks the occurrence time of the Nias earthquake (2005 March 28, M w = 8.7). From the results of Fig. 17, we can draw two main observations. First, the impact of elastic layering is mostly limited to coseismic jumps, while it does not affect the evolution of post-seismic displacement, which is only driven by the rheological features. Next, typical differences due to the rheological model between predicted displacements at the epoch of Nias earthquake turn out to be of order mm at sites where the predicted signal is stronger, and one order of magnitude less on sites BAKO and NTUS where the predicted response

14 Post Widder algorithm and post-seismic rebound 685 Figure 14. Post-seismic evolution of the u N = u θ component of displacement following the 2004 Sumatra earthquake. See also caption of Fig. 13. is weaker. Coseismic offsets produced by the Nias earthquake have been observed on GPS recordings and turn out to be comparable or larger (Banerjee et al. 2007) with respect to the difference between rheological models; therefore we may conclude that the signature of the rheological details is likely to have been swamped out by the coseismic signal of the Nias earthquake. In Fig. 18 we compare the post-seismic velocity and curvature at each site, computed with a quadratic fit of the time-series, with those obtained from observed time-series (data are taken from fig. 14 of Pollitz et al. 2006). The fit has been carried out in a time window ranging from the 2004 Sumatra earthquake (2004 December 26) to the 2005 Nias earthquake (2005 March 28). A pure Maxwell rheology (Rheology 1) seems not fully adequate to account for the observed post-seismic signals. Models with Burgers transient rheology give a better fit to the data; the best agreement is obtained with Rheology 2 which assumes a Burgers astenosphere with η 1 = 5 17 Pa s, η 2 = 19 Pa s (see Table 1). This is in agreement with results by Pollitz et al. (2006), who obtained the best fitting of post-seismic velocities and curvatures with a transient Burgers asthenosphere with similar viscosity values. 6 CONCLUSIONS We have shown that the Gaver Stehfest algorithm is a viable method to apply the PW Laplace inversion to the problem of post-seismic relaxation of a spherical, layered, incompressible viscoelastic Earth. This approach allows us to bypass the solution of the secular equation, whose degree increases linearly with the model complexity, thus enabling us to stably solve fine-layered models. Its main shortcoming with respect to the standard NM approach is the requirement of high precision floating-point arithmetic, which at the present stage cannot be provided by native hardware formats so that high-level multiprecision libraries are needed. This results in a consistent loss of computational efficiency that, for the most complex models, may

15 686 D. Melini et al. Figure 15. Post-seismic evolution of the u E = u φ component of displacement following the 2004 Sumatra earthquake. See also caption of Fig. 13. be a severe limit for the range of practical applications. Nevertheless, through a careful fine-tuning of the algorithm parameters, we showed that it is possible to keep a stable convergence while retaining the computation times within reasonable limits. An advantage of the PW Laplace inversion is that the whole computation is carried out in the time domain using the equivalent elastic problem. In this way, the code is substantially simplified with respect to the application of standard NM techniques so that a straightforward implementation of a non-maxwell rheology is possible, as we have shown. Using the PW algorithm illustrated in the first part of our work, we have addressed the problem of finding the minimum number of stratification layers needed to fit the results of a discretized PREM model to within a specified error threshold. We have investigated the behaviour of different layering schemes for lithosphere and mantle, and found the optimal model to be a layering with a lithosphere closely following the PREM discretization and a uniformly layered mantle; with this scheme, we estimated that observables computed with the discretized PREM model are reproduced within 1 per cent with 20 mantle layers and five lithospheric layers. These conclusions have been obtained without varying the viscosity layering, to avoid the introduction of further degrees of freedom which would have complicated the interpretation of our results. Finally, we have applied our code to the post-seismic relaxation of the 2004 Sumatra Andaman earthquake to show the effect of a transient asthenosphere or mantle on the time-dependent displacement and geoid perturbation when compared to a purely Maxwell rheology model. The results of our forward modelling have been compared with the post-seismic geoid signals observed by GRACE and with data from continuously operating GPS stations. We have found that a Burgers transient asthenosphere fits significantly better the post-seismic signals observed in GPS time-series, and gives a geoid perturbation with a temporal evolution that qualitatively agrees with satellite observations.

16 Post Widder algorithm and post-seismic rebound 687 Figure 16. Post-seismic evolution of the perturbation to geoid height following the 2004 Sumatra earthquake. See also caption of Fig. 13. ACKNOWLEDGMENTS We thank Dr L.L.A. Vermeersen and an anonymous reviewer for their helpful and incisive comments. This work was partly supported by the MIUR-FIRB research grant Sviluppo di nuove tecnologie per la protezione e la difesa del territorio dai rischi naturali. FORTRAN codes are available by request to melini@ingv.it. REFERENCES Abate, J. & Valkó, P.P., Multi-precision Laplace transform inversion, Int. J. Numer. Meth. Engng., 60, , doi:.02/nme.995. Abate, J. & Whitt, W., A unified framework for numerically inverting Laplace transforms, INFORMS J. Comput., 18(4), Ammon, C.J. et al., Rupture process of the 2004 Sumatra-Andaman earthquake, Science, 308, Antonioli, A., Piersanti, A. & Spada, G., Stress diffusion following large strike-slip earthquakes: a comparison between spherical and flatearth models, Geophys. J. Int., 133(1), Banerjee, P., Pollitz, F.F. & Bürgmann, R., The size and duration of the Sumatra Andaman earthquake form far-field static offsets, Science, 308, Banerjee, P., Pollitz, F.F., Nagarajan, B. & Bürgmann, R., Coseismic slip distributions of the 26 December 2004 Sumatra Andaman and 28 March 2005 Nias earthquakes from GPS static offsets, Bull. seism. Soc. Am., 97(1A), S86 S2. Boschi, L., Tromp, J. & O Connell, R.J., On Maxwell singularities in postglacial rebound, Geophys. J. Int., 136, Boschi, L., Piersanti, A. & Spada, G., Global post-seismic deformation: deep earthquakes, J. geophys. Res., 5(1), Boschi, E., Casarotti, E., Devoti, R., Melini, D., Piersanti, A., Pietrantonio, G. & Riguzzi, F., Coseismic deformation induced by the Sumatra earthquake, J. Geodyn., 42, Casarotti, E., Postseismic stress diffusion in a spherical viscoelastic Earth model, PhD thesis, University of Bologna, Italy. Casarotti, E., Piersanti, A., Lucente, F.P. & Boschi, E., Global postseismic stress diffusion and fault interaction at long distances, Earth planet. Sci. Lett., 191(1 2),

17 688 D. Melini et al. u N at BNKK u E at BNKK u N at BAKO u E at BAKO mm mm mm mm R1 R2 R u N at PHKT u N at ARAU u at SAMP N u E at PHKT u E at ARAU u at SAMP E u N at CHMI u N at NTUS u E at CHMI u E at NTUS Days after Jan 2004 Days after Jan 1, , Days after Jan 1, 2004 Days after Jan 1, 2004 Figure 17. Modelled time-series at regional GPS sites shown in Fig. 11, for each rheological model of Table 1. Solid curves correspond to an elastic stratification with N = 20 layers while dashed curves correspond to an elastic stratification with N = 3 layers. A vertical line marks the occurrence of the March 28, 2005 Nias earthquake. Postseismic velocity N Postseismic velocity E 300 R R2 R mm/yr 0 mm/yr BNKK BAKO PHKT CHMI ARAU NTUS SAMP 700 BNKK BAKO PHKT CHMI ARAU NTUS SAMP Postseismic curvature N Postseismic curvature E mm/yr mm/yr BNKK BAKO PHKT CHMI ARAU NTUS SAMP 00 BNKK BAKO PHKT CHMI ARAU NTUS SAMP Figure 18. Post-seismic velocity and curvature at each GPS site for the three rheological models of Table 1. The triangles represent observed values (from fig. 14 of Pollitz et al. 2006). Dziewonski A.M. & Anderson D.L., Preliminary reference earth model, Phys. Earth planet. Inter., 25, Ellis, T.M.R., Philips, I. & Lahey, T.M., Fortran 90 Programming, Addison-Wesley, Harlow, England. Friederich, W. & Dalkolmo, J., Complete synthetic seismograms for a spherically symmetric Earth by a numerical computation of the Green s function in the frequency domain, Geophys. J. Int., 122, Fung, Y.C., Foundations of Solid Mechanics, Prentice-Hall, Englewood Cliffs, New Jersey. Gaver, D.P., Observing stochastic processes and approximate transform inversion, Oper. Res., 14, Han, S., Shum, C., Bevis, M., Ji, C. & Kuo, C., Crustal dilatation observed by GRACE after the 2004 Sumatra-Andaman earthquake, Science, 313, Hanyk, L., Viscoelastic response of the Earth: initial-value approach, PhD thesis, Charles University, Prague, 137 pp. Hanyk, L., Matyska, C. & Yuen, D., Secular gravitational instability of a compressible viscoelastic sphere, Geophys. Res. Lett., 26(5), IEEE Task P854, ANSI IEEE , in Standard for Binary Floating-Point Arithmetc, IEEE, New York. Lay, T. et al., The Great Sumatra-Andaman Earthquake of 26 December 2004, Science, 308,

18 Post Widder algorithm and post-seismic rebound 689 Melini, D. & Piersanti, A., Impact of global seismicity on sea level change assessment, J. geophys. Res., 111, B Melini, D., Piersanti, A., Spada, G., Soldati, G. & Casarotti, E., Earthquakes and relative sealevel changes, Geophys. Res. Lett., 31(9), L Nostro, C., Piersanti, A., Antonioli, A. & Spada, G., Spherical versus flat models of coseismic and postseismic deformations, J. geophys. Res., 4(B6), Nostro, C., Piersanti, A. & Cocco, M., Normal fault interaction caused by coseismic and postseismic stress changes, J. geophys. Res., 6(B9), Panet, I. et al., Coseismic and post-seismic signatures of the Sumatra 2004 December and 2005 March earthquakes in GRACE satellite gravity, Geophys. J. Int., 171, Park, J. et al., Earth s free oscillations excited by the 26 December 2004 Sumatra-Andaman earthquake, Science, 308, Peltier, W.R., The impulse response of a Maxwell Earth, Rev. Geophys., 12, Peltier, W.R., Wu, P. & Yuen, D., The viscosities of the planetary mantle, in Anelasticity of the Earth, American Geophysical Union Geodynamics Series, pp , eds Stacey, F.D., Paterson, M.S. & Nicholas, A., American Geophysical Union, Washington, DC. Piersanti, A., Spada, G., Sabadini, R. & Bonafede, M., Global postseismic deformation, Geophys. J. Int., 120(3), Pollitz, F.F., Postseismic relaxation theory on the spherical Earth, Bull. seism. Soc. Am., 82, Pollitz, F.F., Transient rheology of the uppermost mantle beneath the Mojave Desert, California, Earth planet. Sci. Lett., 215, Pollitz, F.F., Bürgmann, R. & Banerjee, P., Post-seismic relaxation following the great 2004 Sumatra-Andaman earthquake on a compressible self-gravitating Earth, Geophys. J. Int., 167, , doi:.1111/j x x. Post, E.L., Generalized differentiation, Trans. Amer. Math. Soc., 32, Riva, R. & Vermeersen, L.L.A., Approximation method for highdegree harmonics in normal mode modelling, Geophys. J. Int., 151, Rundle, J.B., Viscoelastic-gravitational deformation by a rectangular thrust fault in a layered Earth, J. geophys. Res., 87(B9), Sabadini, R. & Vermeersen, L.L.A., Influence of lithospheric and mantle stratification on global post-seismic deformation, Geophys. Res. Lett., 24(16), Sabadini, R., Yuen, D.A. & Boschi, E., The effects of post-seismic motions on the moment of inertia of a stratified viscoelastic Earth with an astenosphere, Geophys. J. R. astr. Soc., 79, Smith, D., Efficient multiple-precision evaluation of elementary functions, Math. Comput., 52, Smylie, D.E. & Mansinha, L., The elasticity theory of dislocations in real Earth models and changes in the rotation of the Earth, Geophys. J. R. astr. Soc., 23, Soldati, G., Piersanti, A. & Boschi, E., Global postseismic gravity changes of a viscoelastic Earth, J. geophys. Res., 3, Spada, G., Rimbalzo post-glaciale e dinamica rotazionale di un pianeta viscoelastico stratificato, PhD thesis, University of Bologna, Bologna, Italy, 303 pp. Spada, G., ALMA: a Fortran program for computing the viscoelastic Love numbers of a spherically symmetric planet, Comput. Geosci., 34, , doi:.16/j.cageo Spada, G. & Boschi, L., Using the Post-Widder formula to compute the Earth s viscoelastic Love numbers, Geophys. J. Int., 166, , doi:.1111/j x x. Spada, G., Yuen, D.A., Sabadini, R. & Boschi, E., Lower mantle viscosity constrained by seismicity around deglaciated areas, Nature, 351, Spada, G. et al., Modeling Earth s post-glacial rebound, EOS Trans. Am. geophys. Un., 85, Stehfest, H., Algorithm 368: numerical inversion of Laplace transforms, Commun. ACM, 13(1), Tanaka, Y., 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, Geophys. J. Int., 164, Tanaka, Y., Okuno, J. & Okubo, S., A new method for the computation of global viscoelastic post-seismic deformation in a realistic Earth model (II) horizontal displacement, Geophys. J. Int., 170, Tapley, B., Bettadpur, S., Ries, J., Thompson, P. & Watkins, M., GRACE measurements of mass variability in the Earth System, Science, 305, Tsai, V.C., Nettles, M., Ekström, G. & Dziewonski, A.M., Multiple CMT source analysis of the 2004 Sumatra earthquake, Geophys. Res. Lett., 32, L Valkó, P.P. & Abate, J., Comparison of sequence accelerators for the Gaver method of numerical Laplace transform inversion, Comp. Math. Appl., 48, , doi:.16/j.camwa Vermeersen, L.L.A. & Mitrovica, J.X., Gravitational stability of spherical self-gravitating relaxation models, Geophys. J. Int., 142, Vermeersen, L.L.A. & Sabadini, R., A new class of stratified viscoelastic modes by analytical techniques, Geophys. J. Int., 129, Vermeersen, L.L.A., Sabadini, R. & Spada, G., 1996a. Analytical viscoelastic relaxation modes, Geophys. Res. Lett., 23, Vermeersen, L.L.A., Sabadini, R. & Spada, G., 1996b. Compressible rotational deformation, Geophys. J. Int., 126, Vigny, C. et al., Insight into the 2004 Sumatra-Andaman earthquake from GPS measurements in southeast Asia, Nature, 436, Widder, D.W., The inversion of the Laplace integral and the related moment problem, Trans. Amer. Math. Soc., 36, Wolf, D., Glacio-isostatic adjustment in Fennoscandia revisited, J. geophys. Res., 59, Wu, P., Using commercial finite element packages for the study of earth deformations, sea levels and the state of stress, Geophys. J. Int., 158, Wu, P. & Ni, Z., Some analytical solutions for the viscoelastic gravitational relaxation of a two-layer non-self-gravitating incompressible spherical Earth, Geophys. J. Int., 126, Wu, P. & Peltier, W.R., Viscous gravitational relaxation, Geophys. J. R. astr. Soc., 70, Yuen, D. & Peltier, W., Normal modes of the viscoelastic earth, Geophys. J. R. astr. Soc., 69, Yuen, D., Sabadini, R., Gasperini, P. & Boschi, E., On transient rheology and glacial isostasy, J. geophys. Res., 91(B11), APPENDIX A: ALGORITHM BENCHMARKS Here we discuss the results of a suite of benchmarks aimed to tune the PW algorithm parameters. Basically, these are the order M of the Gaver sequence and the number of significant digits D used for the numerical computation. Large values of M and D are expected to provide a stable and accurate inverse Laplace transform (Abate & Whitt 2006) but, in turn, require a non-linearly increasing computation time. Abate and Whitt (2006) have shown that, if the Laplace inverse of a C -class analytical function f (t) is to be evaluated with j significant digits, the computation is to be performed with M = 1.1 j and D = 2.2M, where x denotes the least integer greater than or equal to x. Since in our case we are dealing with numerical expressions that may be affected by the propagation of round-off errors, we carried out an extensive set of numerical benchmarks to verify the validity of the prescription given by Abate & Whitt (2006) in our case. In Fig. A1 we compare the post-seismic (t = 3 yr) displacement radial coefficients u l, 2 and v l, 2 obtained by a NM approach with

19 690 D. Melini et al. Figure A1. Post-seismic (t = 3 yr) radial coefficients u l, 2 and v l, 2 as a function of harmonic degree l for a point thrust fault with dip δ = 45 at a depth z = 70 km. The results of NM and PW methods are compared, for increasing orders of the Gaver sequence (M); the system precision is set to D = 2.2M. The stratification model is summarized in Table A1. those computed by the PW algorithm, with M = 5, and 50 and D = 2.2M. According to Abate & Whitt (2006), in order to achieve a numerical accuracy of 1 part over 4 (j = 4) we should employ M = 5, D = 11. From Fig. A1 we observe that, using M = 5, only the first 300 harmonic coefficients can be computed before the onset of numerical degeneration. This behaviour is to be attributed to the ill-conditioned form of the propagators that build the array R in eq. (14), which contain both regular (r l ) and irregular (r l ) powers of radius. For large l, this may significantly affect the nominal precision of the computations thus producing the effects shown in Fig. A1; incidentally, according to Riva and Vermeersen (2002), this problem could be fixed separating regular from irregular powers of harmonic degree in the multiplications. Increasing the order of the Gaver sequence from M = 5 to shifts the onset of numerical instabilities from l 300 to 1200; with M = 50 we can reproduce correctly all the 2000 harmonics computed by the NM method. However, using large M values turns out to be extremely time-consuming; for M = 50 and D = 1, the computation of radial coefficients for a single harmonic degree and time step requires about 1.7 s on a 1.6 GHz Intel Itanium2 CPU. Since the maximum degree L max needed for convergence of the truncated harmonic series in eqs (7) and (8) increases with decreasing source depth, and for shallow sources (d km) it may be of order 4 (Riva & Vermeersen 2002; Casarotti 2003), using a large M