Initial-value and modal approaches for transient viscoelastic responses with complex viscosity profiles

An initial-value theory is proposed for studying viscoelastic responses of compressible earth models with complex viscosity profiles. Continuous spectra are caused by both compressibility and the viscosity stratification, and they can cause numerical difficulties for the modal approach. We have studied the different responses for various types of viscosity profiles and found that there are differences in the responses at short wavelengths for viscosity profiles with sharp low-viscosity zones. We found that many modes are required to match the full initial-value solutions in the presence of sharp viscosity stratifications in both the upper and lower mantle. The normal-mode approach is best suited for simple layered models, long wavelengths and timescales greater than several thousand years, while the initial-value approach is indispensable in treating short-timescale problems with sharp low-viscosity zones in the upper mantle and viscosity stratification in the lower mantle.