On the computation of long period seismograms in a 3-D earth using normal mode based approximations

Tomographic inversions for large-scale structure of the earth's mantle involve a forward modelling step of wave propagation through 3-D heterogeneity. Until now, most investigators have worked in the framework of the simplest theoretical assumptions, namely the infinite frequency 'ray theory' in the case of body wave traveltime inversions, or the 'path-average' approximation (PAVA) to normal mode perturbation theory, in the case of surface waves and long-period waveforms. As interest is shifting to mapping shorter wavelength structures, the need for a more accurate theoretical account of the interaction of seismic waves with mantle heterogeneity, coupled with improvements in path coverage, has been realized. Here we discuss different levels of approximations used in the context of normal mode perturbation theory, when modelling time domain seismic waveforms. We compare the performance of asymptotic approximations, which collapse the effects of 3-D structure onto the great circle vertical plane: the 1-D PAVA and a 2-D approximation called non-linear asymptotic coupling theory (NACT), which both are zeroth order asymptotic approximations. We then discuss how off-vertical plane effects can be introduced using higher order asymptotics. These computationally efficient approximations are compared to the linear Born formalism (BORN), which computes scattering integrals over the entire surface of the sphere. We point out some limitations of this linear formalism in the case of spatially extended anomalies, and show how that can be remedied through the introduction of a non-linear term (NBORN). All these approximations are referenced to a precise 3-D numerical computation afforded by the spectral element method. We discuss simple geometries, and explore a range of sizes of anomalies compared to the wavelength of the seismic waves considered, thus illustrating the range of validity and limitations of the various approximations considered.