On the choice of wavenumbers in viscoelastic seismic modelling with discrete wavenumber-frequency methods

Seismic wave propagation through an attenuating medium has a unique description in the linear viscoelastic model which, by the correspondence principle, is a direct extension of the elastic model. Horizontal wavenumbers are constant across a horizontal boundary, but the vertical wavenumbers are ambiguously defined when incidence is from a layer with relatively low velocity or strong attenuation. The direct choice of vertical wavenumber in a horizontally stratified model implies increasing amplitudes with depth and reflection coefficients which are qualitatively different from the corresponding elastic coefficients. This is unacceptable. It can be avoided by choosing a non-attenuating source layer for modelling near vertical reflections in low-loss media. However, when layers exhibit strong attenuation, the 'conjugated viscoelastic model' should be used instead. This has the advantages that we stay within the framework of the viscoelastic model, increasing amplitudes are avoided and the reflection coefficients are realistic smooth deviations from the corresponding elastic coefficients.