Dispersive body waves

Dispersive body waves is an important aspect of seismic theory. When a wave propagates through subsurface materials both energy dissipation and velocity dispersion takes place. Energy dissipation is frequency dependent and causes decreased resolution of the seismic images when recorded in seismic prospecting. The attendant dispersion is a necessary consequence of the energy dissipation and causes the high frequency waves to travel faster than the low-frequency waves. The consequence for the seismic image is a frequency dependent time-shift of the data, and so correct timings for lithological identification cannot be obtained.Fig.1.a.Wave equation with Kramer Kronig relation b=0.07Fig.1.b.Wave equation with Kramer Kronig relation b=0.1Fig.2.a.Phase only inversion with b=0.1 and fh=80 Hz (red graph)Fig.2.b.Phase only inversion with b=0.1 and fh=30 Hz (red graph) Dispersive body waves is an important aspect of seismic theory. When a wave propagates through subsurface materials both energy dissipation and velocity dispersion takes place. Energy dissipation is frequency dependent and causes decreased resolution of the seismic images when recorded in seismic prospecting. The attendant dispersion is a necessary consequence of the energy dissipation and causes the high frequency waves to travel faster than the low-frequency waves. The consequence for the seismic image is a frequency dependent time-shift of the data, and so correct timings for lithological identification cannot be obtained. When we know the energy dissipation (attenuation), we can calculate the time shift due to dispersion because there is a relation between attenuation and the dispersion in a seismic media.Dispersion equations are obtained from the application of an integral transform in the frequency domain that are of the Kramers-Krönig type. This effect is described in the article ‘Dispersive body waves’ by Futterman (1962). For a better understanding of dispersion waves in seismic medias I would recommend the book 'Seismic inverse Q-filtering' by Yanghua Wang (2008). He discusses the theory of Futterman and starts with the wave equation: