We present results of a direct method to compute realistic long period (>200 s in this example) synthetic seismograms using higher order perturbation theory (up to the 3rd order). Normal modes are computed for the Earth, taking into account rotation, ellipticity, 3D elastic lateral heterogeneities (SAW12D) for the whole mantle, and anelastic lateral variations in the upper mantle (QR19). Coupling between different modes and branches is included in the computation. We study the sensitivity of the waveform to 3D anelasticity and the biases between 3D elastic and 3D anelastic variations. The resulting seismograms can be computed to higher frequency and may then be used to perform whole mantle joint inversions for elastic and anelastic structure.