Using the Post-Widder formula to compute the Earth's viscoelastic Love numbers

Abstract

The post-glacial or post-seismic relaxation of a Maxwell viscoelastic earth, 1-D or slightly \ud laterally heterogeneous, can be calculated in a normal-mode approach, based on an application \ud of the propagator technique. This semi-analytical approach, widely documented in the liter- \ud ature, allows to compute the response of an earth model whose rheological parameters vary \ud quite strongly with depth, at least as accurately and efficiently as by 1-D numerical integra- \ud tion (Runge–Kutta). Its main drawback resides in the need to identify the roots of a secular \ud polynomial, introduced after reformulating the problem in the Laplace domain, and required \ud to transform the solution back to the time domain. Root finding becomes increasingly difficult, \ud and ultimately unaffordable, as the complexity of rheological profiles grows: the secular poly- \ud nomial gradually gets more ill behaved, and a larger number of more and more closely spaced \ud roots is to be found. Here, we apply the propagator method to solve the Earth’s viscoelastic \ud momentum equation, like in the above-mentioned normal-mode framework, but bypass root \ud finding, using, instead, the Post–Widder formula to transform the solution, found again in the \ud Laplace domain, back to the time domain. We test our method against earlier normal-mode \ud results, and prove its effectiveness in modelling the relaxation of earth models with extremely \ud complex rheological profiles

Application of a Numerical Inverse Laplace Integration Method to Surface Loading on a Viscoelastic Compressible Earth Model

“…In order to compute responses for an incompressible earth model, the Lamé's constant k is set to a large value ( = 100 l) without setting up the differential equation system for the incompressible case (WU and PELTIER, 1982). Figure 1 (bottom) shows good agreement between the result for the 200-layer PREM model obtained by SPADA and BOSCHI (2006) and that for the same model obtained by the presented approach.…”

“…For incompressible models, both have been developed (e.g., HAGER, 1994, 1995;MARTINEC, 2000;ZHONG et al, 2003;SPADA and BOSCHI, 2006). For compressible models, only time-domain approaches (e.g., HANYK et al, 1995;STEFFEN et al, 2006) have been used.…”

“…The numerical root-finding algorithms do not work for identifying these roots associated with dilatation modes (HAN and WAHR, 1995). Incidentally, the identification of roots is also difficult for incompressible models that include a viscoelastic lithosphere (SPADA and BOSCHI, 2006).…”

“…The viscosity is 10 40 Pa s down to the depth of 120 km, which accounts for the elastic lithosphere. The viscosity in the mantle (3480 km < r < 6251 km) is shown in Figure 2, which is obtained by a polynomial interpolation of the convex viscosity profile (RICARD and WUMING, 1991) used in previous studies (e.g., HANYK et al, 1995;VERMEERSEN and SABADINI, 1997;SPADA and BOSCHI, 2006). In the solid core, the viscosity is 10 25 Pa s, which effectively behaves as an elastic body.…”

Effects of Earth's layered structure, gravity and curvature on coseismic deformation

“…The resulting 3-D coseismic displacement (U z , U r and U t ) are computed using the Okada (1985) formulae for the homogeneous model and the code EDGRN/EDCMP for the layered model. Spada & Boschi (2006) pointed out that a uniformly layered structure with ∼40 layers approximates the results obtained using a layered PREM discretization to within 1 per cent. In our work, two layered structures are considered, which are adopted from the seismic reference models PREM (Dziewonski & Anderson 1981) and AK135 , respectively.…”