Stolt Stretch

Stolt stretch Claerbout (1985); Levin (1985); Stolt (1978) is a method of extending constant-velocity frequency-domain migration to the case of a vertically variable velocity. The method consists of stretching the time axis according to the formula  

$$\tau(t_z)= \left({{2 \over V_0^2}\,\int_0^{t_z}\,t\,V_{rms}^2(t)\,dt}\right)^{1/2}\;,$$ (57)

double Fourier transform, and migration according to the dispersion relation  

$$\omega_m(k,\omega_0)= \left(1-{1\over W}\right)\,\omega_0+ {... ...gn}(\omega_0)}\over W}\, \sqrt{\omega_0^2 - W\,V_0^2\,k_x^2}\;,$$ (58)

where V₀ is a constant frame velocity, $\omega_0$ and $\omega_m$are the frequencies before and after the migration, corresponding to the stretched time coordinate, k_x is the wavenumber, and W is a constant parameter (W=1 in the constant velocity case). Fomel 1995 has shown that the optimal choice of the Stolt stretch parameter W for a particular traveltime depth t_z is given by the expression  

$$W=1-{{V_0^2\,\tau^2\left(t_z\right)} \over{V_{rms}^2\left(t_... ... {V_{rms}^2\left(t_z\right)}} -S_2\left(t_z\right) \right)\;\;.$$ (59)

This expression remains valid in the case of a vertically heterogeneous VTI medium if the values of V_rms and S₂ are computed according to formulas (44) and (46). The method of cascaded migrations Larner and Beasley (1987) can improve the performance of Stolt migration in the case of variable velocity Beasley et al. (1988). However, this method affects only the isotropic part of the model and cannot change the contribution of the anisotropic parameters. Therefore, in the anisotropic case, it is important to incorporate anisotropic parameters into the Stolt stretch correction.