Geophys. J. Int. (2008) 174, 672–695
doi: 10.1111/j.1365-246X.2008.03847.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. Piersanti1 and G. Spada2
1 INGV
- Istituto Nazionale di Geofisica e Vulcanologia, Roma, Italy. E-mail: [email protected]
di Fisica, Università degli Studi di Urbino ‘Carlo Bo’, Urbino, Italy
2 Istituto
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 I N T RO D U C T I O N
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
672
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
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Journal compilation Post–Widder algorithm and post-seismic rebound
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
Model U
6000
6000
N=2
N=5
N=10
N=50
5500
5500
5000
ρ (kg m )
ρ (kg m )
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Model L
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ρ (kg m )
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Radius (km)
Model R
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Radius (km)
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Radius (km)
Figure 1. Density profiles of the four layering models for N = 2, 5, 10 and 50.
Model P
Model U
300
300
N=2
N=5
N=10
N=50
250
250
μ (GPa)
μ (GPa)
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Model R
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μ (GPa)
μ (GPa)
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150
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100
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50
3500
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Radius (km)
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6000
Figure 2. Rigidity profiles of the four layering models for N = 2, 5, 1 and 50.
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0
3500
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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,
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’
d=50 km, vertical component
2
0
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40
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0
0
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d=200 km, vertical component
2
0
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d=200 km, horizontal component
2
10
0
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0
10
20
30
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50
d=500 km, horizontal component
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0
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%
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40
d=1000 km, vertical component
2
0
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2
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10
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50
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10
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0
2
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0
10
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50
d=100 km, vertical component
2
10
d=50 km, horizontal component
2
10
10
(1)
0
10
20
30
Number of layers, N
40
50
0
10
10
0
10
20
30
Number of layers, N
40
50
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 = 1021 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, 10 and 1000 yr, respectively. Two
horizontal dashed lines represent 5 and 10 per cent misfit thresholds.
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Journal compilation Post–Widder algorithm and post-seismic rebound
that for the Maxwell linear rheological law reads
μs
,
μ̃ =
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|xe
x(t) = xe δ(t) +
esk t ,
(3)
d
|R(s)|
k=1
ds
s=sk
where† is the adjoint and |. . .| denotes the determinant. The elastic
response is
xe = 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 τk =
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 )
φ=
d=50 km, vertical component
%
%
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10
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40
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2
%
%
0
0
10
20
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0
0
10
20
30
40
50
d=200 km, horizontal component
2
10
0
%
%
20
10
10
50
d=200 km, vertical component
2
10
0
10
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40
0
10
10
50
d=500 km, vertical component
2
0
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50
d=500 km, horizontal component
2
10
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0
%
%
10
d=100 km, horizontal component
2
10
10
0
10
20
30
40
0
10
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50
d=1000 km, vertical component
2
0
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20
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50
d=1000 km, horizontal component
2
10
10
0
%
%
0
10
10
10
10
0
10
10
50
10
10
(8)
d=50 km, horizontal component
2
0
10
φlm (r )Ylm (θ, φ),
10
10
10
l
∞ l=0 m=−l
10
10
(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:
⎡
⎤
u lm (r )
l
∞ ⎢
⎥ m
u=
(7)
⎣ vlm (r )∇θ + tlm (r )∇φ ⎦ Yl (θ, φ)
l=0 m=−l
vlm (r )∇φ − tlm (r )∇θ
− s1k
2
675
0
10
20
30
Number of layers, N
40
50
0
10
10
0
10
20
30
Number of layers, N
40
50
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.
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where ∇ = (∇ r , ∇ θ , ∇ φ ) is the gradient operator in spherical coordinates and Ylm are the spherical harmonic functions
Ylm (θ, φ) = (−1)m Plm (cos θ )eimφ
(9)
Plm (z)
with
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:
n+1 n
d ˜
(−1)n n
f (t) = lim
,
(10)
f
(s)
n→∞
n!
t
ds n
s= n
t
where f˜ (s) indicates the Laplace-transform of f (t). A direct application of eq. (10) 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. (10) has been proposed by Gaver (1966), based on the
d=50 km, vertical component
2
0
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%
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10
10
0
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40
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2
%
%
0
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40
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2
0
%
%
40
50
0
0
10
20
30
40
50
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2
10
0
10
20
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40
0
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50
d=500 km, vertical component
0
10
20
30
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50
d=500 km, horizontal component
2
10
10
0
%
%
30
10
2
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0
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40
0
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50
d=1000 km, vertical component
2
0
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50
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2
10
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2
0
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0
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d=50 km, horizontal component
2
10
0
10
20
30
Number of layers, N
40
50
0
10
10
0
10
20
30
Number of layers, N
40
50
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.
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sequence:
f k (t) =
k ln 2
t
2k
k
k
k
(k + j) ln 2
j
˜
f
(−1)
t
j
j=0
(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:
2M
ln 2 k ln 2
f (t, M) =
ζk f˜
(12)
t k=1
t
with
ζk = (−1) M+k
min(k,M)
j=[(k+1)/2]
j M+1
M!
M
j
2j
j
j
,
k− j
(13)
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:
2M
ln 2 k ln 2
k ln 2
x(t, M) =
R−1
b+p .
(14)
ζk Q
t k=1
t
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-
d=50 km, vertical component
2
0
%
%
10
10
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40
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2
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50
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50
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2
10
0
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0
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50
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0
10
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50
d=500 km, horizontal component
2
10
10
0
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10
2
10
0
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20
30
40
0
10
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50
d=1000 km, vertical component
2
0
10
20
30
40
50
d=1000 km, horizontal component
2
10
10
0
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%
20
10
10
50
10
10
10
10
d=100 km, horizontal component
2
0
10
0
10
10
10
0
10
10
50
10
10
d=50 km, horizontal component
2
10
10
677
0
10
20
30
Number of layers, N
40
50
0
10
10
0
10
20
30
Number of layers, N
40
50
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.
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ogy (Yuen & Peltier 1982; Pollitz 2003) that we consider in Section 5
below can be simply implemented computing the equivalent elastic
rigidity as
μ̃ B =
s + μ1 /η1
(s + μ2 /η2 )(s + μ1 /η1 ) + μ1 s/η2
(15)
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
f (t) − f (t, M) 10−0.90M
= (16)
f (t)
(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
http://myweb.lmu.edu/dmsmith/FMLIB.html (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 = 10, t 3 = 102 , t 4 = 103 , t 5 = 104 and t 6 = 105 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 = 10 km (S3).
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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
M U L T I L AY E R E D M O D E L S
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
679
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.
35
N=1
N=2
N=4
N=6
N=10
PREM
30
25
ur (mm)
20
15
10
5
0
5
10
0
10
1
10
2
10
t (yr)
3
10
4
5
10
10
10
N=1
N=2
N=4
N=6
N=10
PREM
20
40
φ
u (mm)
30
50
60
70
10
5
1
x 10
0
10
1
10
2
10
t (yr)
3
10
4
5
10
10
3
4
3
φ (Nm/kg)
2
1
N=1
N=2
N=4
N=6
N=10
PREM
0
1
2
3
4
10
1
0
10
1
10
2
10
t (yr)
3
10
4
10
5
10
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.
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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 23 N uniform layers.
(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, 10 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 = 1019 Pa s and
upper and lower mantle viscosities η UM = 1021 Pa s and η LM = 3 ×
1021 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
40
N=5
N=10
N=20
N=40
N=50
N=200
PREM
30
10
r
u (mm)
20
0
10
20
10
0
10
1
10
2
10
t (yr)
3
10
4
5
10
10
10
N=5
N=10
N=20
N=40
N=50
N=200
PREM
20
40
φ
u (mm)
30
50
60
70
10
8
1
x 10
0
10
1
10
2
10
t (yr)
3
10
4
5
10
10
3
6
φ (Nm/kg)
4
N=5
N=10
N=20
N=40
N=50
N=200
PREM
2
0
2
4
10
1
0
10
1
10
2
10
t (yr)
3
10
4
5
10
10
Figure 9. Time-dependent displacement and incremental geopotential obtained with model U, varying the number of layers. See also caption of Fig. 8.
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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 = 10, t 3 = 102 , t 4 =
103 , t 5 = 104 and t 6 = 105 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 = 10 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 10 ≤ N ≤ 15 layers
to fit the discretized PREM in nearly all conditions, but has some
instabilities in the near-field. Model L requires N ∼ 10 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 ∼ 10
layers in all conditions. It can also be noted that a definite source–
681
observer distance exists, in the range between 500 and 1000 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. 10, we plot the ratios u r /u rPREM ,
u φ /u PREM
and φ/φ PREM as a function of the number of layers N,
φ
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 = 10 is
adequate, while for the coseismic response at least N = 20 is
25
20
ur / ur
PREM
15
10
5
0
1
2
3
4
5
6
7 8 9 10
Number of layers, N
20
30
40
50
60
1
2
3
4
5
6
7 8 9 10
Number of layers, N
20
30
40
50
60
1.05
PREM
1
uφ / uφ
0.95
0.9
0.85
0.8
2
t=0
t=10 yr
2
t=10 yr
1.8
t=103 yr
φ/φ
PREM
1.6
4
t=10 yr
1.4
1.2
1
0.8
1
2
3
4
5
6
7 8 9 10
Number of layers, N
20
30
40
50
60
Figure 10. 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.
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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).
CHMI
o
18 N
BNKK
12oN
PHKT
ARAU
o
6 N
SAMP
NTUS
o
0
6oS
90oE
BAKO
95oE
100oE
105oE
110oE
115oE
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 C A S E S T U D Y: P O S T - S E I S M I C
R E L A X AT I O N F O L L O W I N G 2 0 0 4
S U M AT R A E A RT H Q UA K E
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 = 1.3 × 1023 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
6291–6371
Rheology 3
Elastic
6091–6291
Maxwell
η = 1019
Burgers
η 1 = 1019
η 2 = 5 × 1017
Maxwell
η = 1019
5701–5701
Maxwell
η = 1021
Maxwell
η = 1021
Burgers
η 1 = 1021
η 2 = 5 × 1017
3480–5701
Maxwell, η = 3 × 1021
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Figure 12. Coseismic displacement field and geoid perturbation following the 2004 Sumatra earthquake.
to the three rheological models. Figs 13–16 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
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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
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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 ∼10 mm at
sites where the predicted signal is stronger, and one order of magnitude less on sites BAKO and NTUS where the predicted response
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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 × 1017 Pa s, η 2 = 1019 Pa s (see Table 1). This is in agreement
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of post-seismic velocities and curvatures with a transient Burgers
asthenosphere with similar viscosity values.
6 C O N C LU S I O N S
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
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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.
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687
Figure 16. Post-seismic evolution of the perturbation to geoid height following the 2004 Sumatra earthquake. See also caption of Fig. 13.
AC K N OW L E D G M E N T S
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 email request to [email protected].
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uN at BNKK
uE at BNKK
uN at BAKO
uE at BAKO
11
8
10
7
mm
9
6
8
R1
R2
R3
350
5
7
400
450
500
550
350
400
500
550
6
350
400
uE at PHKT
450
500
550
4
350
400
500
550
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550
400
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500
Days after Jan 1, 2004
550
uN at CHMI
450
uE at CHMI
mm
uN at PHKT
450
350
400
450
500
550
350
400
uN at ARAU
450
500
550
350
400
uE at ARAU
450
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550
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400
uN at NTUS
450
uE at NTUS
30
mm
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20
350
400
450
500
550
350
400
u at SAMP
450
500
550
500
550
15
350
400
450
500
Days after Jan 1, 2004
550
350
u at SAMP
E
mm
N
350
400
450
500
550
350
Days after Jan 1, 2004
400
450
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
200
R1
R2
R3
200
100
0
100
mm/yr
mm/yr
100
0
200
300
400
500
600
BNKK
BAKO
PHKT
CHMI
ARAU
NTUS
700
SAMP
BNKK
BAKO
Postseismic curvature N
PHKT
CHMI
ARAU
NTUS
SAMP
NTUS
SAMP
Postseismic curvature E
1000
3000
2500
500
2000
0
mm/yr2
mm/yr2
1500
500
1000
500
1000
0
1500
500
2000
BNKK
BAKO
PHKT
CHMI
ARAU
NTUS
SAMP
1000
BNKK
BAKO
PHKT
CHMI
ARAU
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).
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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 = 103 yr) displacement
radial coefficients u l,−2 and v l,−2 obtained by a NM approach with
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Figure A1. Post-seismic (t = 103 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, 10 and 50 and
D = 2.2M. According to Abate & Whitt (2006), in order to
achieve a numerical accuracy of 1 part over 104 (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 10 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 = 110, 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 10 km) it may be of order
104 (Riva & Vermeersen 2002; Casarotti 2003), using a large M
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Figure A2. Relative errors on the radial coefficients u l=1500,m=−2 and v l=1500,m=−2 computed with the PW method with respect to the reference NM values.
The left- and right-hand panels show, respectively, the elastic response and the post-seismic response at t = 103 yr. See also caption of Fig. A1 for details of
seismic source and stratification.
Figure A3. Contour lines in the (D, M) plane corresponding to a 1 per cent maximum relative error on the harmonic coefficients of degree l = 2000. Different
contours represent different observation times ranging from t = 0 to 106 yr; the region on the left-hand side of the contours corresponds to divergence while
the region on the right-hand side corresponds to convergence. See also caption of Fig. A1 for details of seismic source and stratification.
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Convergence test at 1% tolerance - strike-slip
50
45
40
Order of the Gaver sequence (M)
may result in a drastic increase of the computation time, that may
ultimately lead to practical unusability of the PW method.
To overcome these difficulties, we investigated whether convergence can be obtained for all harmonic degrees with a small order
M of the Gaver sequence and a system precision D ≥ 2.2M. Indeed,
in agreement with Abate & Whitt (2006), the desired precision can
be achieved with low values of M provided that a number of digits
larger than D = 2.2M are retained. Since our goal is to minimize the
computation time while keeping the error within a specified threshold, we want to investigate if the additional CPU load required to
carry out the computation with a higher precision is worth the performance increase from the decreased number of sampling points.
In Fig. A2, we show the dependence on the number of significant
digits D of the relative error on u 1500,−2 and v 1500,−2 . The relative
error is computed with respect to the reference value obtained with
(PW )
(NM)
(NM)
the normal-mode approach, that is, u = |u lm − u lm |/|u lm | and
(PW )
(NM)
(NM)
v = |v lm − v lm |/|v lm |, using a Gaver sequence of order M =
10. In order to achieve a relative error of 1 per cent at least a system
precision D = 26 is needed, greater than the value recommended by
Abate & Whitt (2006) which is D = 22; this result is substantially
independent from the observation time t.
From the discussion above, the most crucial stability problems
are found for large harmonic degrees, where the stiffness of propagator matrices leads to precision loss, so that the actual accuracy
may result in smaller than the nominal precision D. For this reason
we have calibrated the algorithm parameters (D, M) by requiring
agreement with the NM solution for l = L max to within 1 per cent.
The value of L max has been shown to depend mainly on the radial
distance between source and observer (Casarotti 2003): for small
distances, L max increases because the solution is characterized by
smaller wavelengths that need large harmonic degrees to be reproduced. We considered a set of seismic sources and computed
the harmonic coefficients u lm , v lm , t lm and φ lm , with l = L max , for
a range of post-seismic observation times ranging from t = 0 to
106 yr, as a function of D and M. In all cases we have found a steep
transition from stability to divergence of the solution, similarly to
what we show in Fig. A2. In Fig. A3 we represent the transition
from stability to divergence on the (D, M) plane, by computing the
maximum relative error on radial coefficients max = max( u , v ,
t , φ ) and plotting contour lines corresponding to max (D, M) =
1 per cent, for one of the examined seismic sources. From Fig. A3
we see that the transition from stability to divergence is almost independent of the post-seismic observation time.
In Fig. A4 we summarize the results of the benchmarks. For each
radial coefficient (u lm , v lm , φ lm and t lm ) a line obtained by interpolating the contours of Fig. A3 marks the boundary between convergence and divergence regions in the (D, M) planes. The transition
is confined in a narrow band, and appears to be only weakly dependent on the source characteristics. The relationship D = 2.2M given
by Abate & Whitt (2006) is also shown, and for low values of M
(less than ∼25) it suggests a system precision D not sufficient to
reproduce correctly the largest harmonic degrees, thus leading to
the degenerations observed in Fig. A1. Since the numerical errors
become larger with increasing harmonic degree, choosing a set of
parameters such that the PW algorithm is stable for l = L max ensures convergence on all the lower degrees; therefore if we set the
(D, M) parameters according to the results of Fig. A4, stability is
ensured also for l ≤ L max . We fixed these parameters to (D = 40,
M = 8), represented by the white circle in Fig. A4; with this choice,
the computation of each harmonic degree at a given time requires
about 0.16 s with a performance improvement of a factor 10 with
respect to the algorithm parameters (D = 110, M = 50) employed
D = 2.2 M
35
30
z=10 km
25
convergence region
20
z=70 km
15
ulm
vlm
10
tlm
5
20
40
60
80
100
120
System precision (D)
Convergence test at 1% tolerance - dip-sli[p
50
45
D = 2.2 M
40
Order of the Gaver sequence (M)
692
35
z=10 km
30
25
z=70 km
convergence region
20
15
ulm
vlm
10
tlm
5
20
40
60
80
100
120
System precision (D)
Figure A4. Convergence and divergence regions in the (D, M) plane based
on the comparison of the largest harmonic degree coefficients computed
with NM and PW methods. Employed seismic sources are a point strike-slip
with dip δ = 60◦ (top panel) and a point thrust with dip δ = 45◦ (bottom
panel). Solid lines correspond to a source depth z = 70 km, dashed lines to
a source depth z = 10 km. A dot–dashed line represents the relation D =
2.2M. White and grey circles mark the points (D = 40, M = 8) and (D =
30, M = 8), respectively.
Table A1. Stratification model used for benchmarking. With r, ρ, μ and η we
indicate radius, density, shear modulus and viscosity, respectively. The elastic parameters are obtained by volume-averaging the corresponding PREM
values.
Layer
1
2
3
4
r (km)
ρ (kg m−3 )
μ (GPa)
η (Pa s)
6291–6371
6091–6291
3471–6091
0–3471
3115
3400
4695
10 930
56.0
67.5
186.0
–
∞
1019
1021
–
Lithosphere
Asthenosphere
Mantle
Fluid core
above. An extremely significant performance improvement could
be certainly obtained using the hardware REAL∗16 numerical format instead of a multiprecision library. Most commercial FORTRAN implementations have a REAL∗16 format with a precision
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Figure A5. Convergence of the displacement vector u and gravity acceleration variation g as a function of the maximum harmonic degree L max . Seismic
source is a point thrust mechanism with dip 45◦ and depth z = 5 km. Displacement and gravity are computed at 100 km from the fault, on the direction with
azimuth 90◦ with respect to the strike direction.
D ∼ 30 (Ellis et al. 1994), represented by the grey circle in Fig. A4.
In this way, the working point lies in the transition zone between
convergence and degeneration; since the transition between stability and degeneration is very steep, as we see from Fig. A2,
working with those parameters is unsafe because numerical degeneration may occur in critical conditions. Moreover, the computation of matrix products between ill-conditioned terms may
lead to an accuracy degradation resulting in an effective precision
smaller than the nominal D. This does not happen when employing multiprecision numerical libraries, which generally have an accuracy control through the use of extra ‘guard-digits’. For these
reasons we decided to follow a conservative approach and use D
= 40 even if it may be excessively large. However, in large numerical simulations where performance is an important issue, the
use of hardware floating-point instead of multiprecision may be
considered.
In all the benchmarks presented above we considered only lithospheric sources, because our reference NM code has this limitation;
to ensure that the PW solution is stable also for deep sources we
performed further tests. In Fig. A5 the displacement u and the perturbation to gravity acceleration g are plotted as a function of
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Journal compilation L max . The solutions first show some oscillations, then a stable value
is reached, and finally for large harmonic degrees (l ≥ 40 000) numerical degeneration is observed. These are due to the instabilities
arising from the iterative algorithms used for spherical harmonics
evaluation that suffer from numerical error propagation effects. It
is therefore crucial to make sure that there is a sufficiently wide
stable zone, where the truncation point L max can be established. To
this aim, we computed u and g as a function of L max for seismic sources at depths ranging from 5 to 700 km. For each depth,
we identify the convergence point as the harmonic degree where
the solution is contained within 0.5 per cent of its limit value, and
the degeneration point as the harmonic degree where the solution
first departs by the same amount from the limit value. The convergence and degeneration points are shown as a function of depth
in Figs A6 and A7, for source–observer distances of 100 and 1000
km, respectively. It is evident that there is a wide separation between
convergence and degeneration points, which is almost constant with
varying source depth and observation time. This result demonstrates
the stability of the PW solution even for deep sources, and can be
practically employed to set the value of L max as a function of the
source depth.
694
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Coseisimic response (t=0)
5
Harmonic degree
10
4
10
3
10
2
10
0
100
200
300
400
Source depth (km)
500
600
700
500
600
700
500
600
700
2
Postseismic response (t=5 × 10 y)
5
10
4
Harmonic degree
10
3
10
2
10
1
10
0
100
200
300
400
Source depth (km)
Postseismic response (t=104 y)
5
10
4
Harmonic degree
10
3
10
2
10
1
10
0
100
200
300
400
Source depth (km)
Figure A6. Harmonic degrees at which the solutions reach convergence (circles) and degeneration (triangles). Observables are computed for a thrust point
source with δ = 45◦ , at a distance of 100 km on the direction with azimuth 90◦ with respect to the strike direction. Convergence is assumed when the solution
lies within 0.5 per cent of the limit value.
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Coseisimic response (t=0)
5
10
4
Harmonic degree
10
3
10
2
10
1
10
0
100
200
300
400
Source depth (km)
500
600
700
500
600
700
500
600
700
Postseismic response (t=5 × 102 y)
5
10
4
Harmonic degree
10
3
10
2
10
1
10
0
100
200
300
400
Source depth (km)
4
Postseismic response (t=10 y)
5
10
4
Harmonic degree
10
3
10
2
10
1
10
0
100
200
300
400
Source depth (km)
Figure A7. Harmonic degrees at which the solutions reach convergence (circles) and degeneration (triangles) for an observation distance of 1000 km. See also
caption of Fig. A6.
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