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

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