2. Geometrical spreading Introduction Ray theory Green’s function Relative geometrical spreading in a vertically heterogeneous VTI medium Stack of horizontal homogeneous VTI layers Examples

Ray theory Green’s function Ray theory approximation to the elastodynamic Green’s function (Cerveny, 2001) It should also be taken into account the R/T coefficients at interface and the factor The amplitude (2.1) (2.2) (2.3)

Ray theory Green’s function The realtive geometrical spraeding factor with matrix In Cartesian coordinate system: (2.4) (2.5) (2.6)

Relative geometrical spreading in a vertically heterogeneous medium The relative geometrical spreading can be separated into two factors In-plane factor Out-of-plane factor (2.7) (2.8) (2.9)

Stack of horizontal homogeneous layers (2.10) (2.11)  is the phase angle  is the group angle

Examples (Ursin and Hokstad, 2003) Figure 2.1: P- and SV-traveltimes

Examples Figure 2.2: Inverse relative geometrical spreading

Examples Figure 2.3: P-wave traveltimes and inverse relative geometrical spreading

Examples Figure 2.4: SV-wave traveltimes and inverse relative geometrical spreading