Stolt Stretch Theory Review

Stolt time migration can be summarized as the following sequence of transformations:

$$p_0(x,t_0) \rightarrow P_0(k_x,\omega_0) \rightarrow P(k_x,\omega) \rightarrow p(x,t) \; ,$$ (1)

where  

$$P_0(k_x,\omega_0) = P(k_x,\omega(k_x,\omega_0)) \left\vert \frac{d\omega(k_x,\omega_0)}{d\omega_0} \right\vert$$ (2)

The function $\omega(k,\omega_0)$ is the dispersion relation and has the following expression in the constant velocity case:

$$\fbox {$ \omega(k,\omega_0) = \mbox{sign}(\omega_0) \sqrt{\omega_0^2 - v^2 k_x^2} $}$$ (3)

The approximation suggested by Stolt 1978 for extending the method to v(z) media involves a change of the time variable (Stolt-stretch):  

$$s(t) = \sqrt{\frac{2}{v_0^2} \int_0^t \tau v_{rms}^2(\tau) d\tau} \; ,$$ (4)

where s(t) is the stretched time variable, v₀ is an arbitrarily chosen constant velocity, and v_rms(t) is the root mean square velocity along the vertical ray, defined by

$$v_{rms}(t) = \frac{1}{t} \int_0^t v^2(\tau) d\tau \;.$$ (5)

This change of variable yields a transformed wave-equation for the wavefield extrapolation, in which Stolt replaces a slowly varying complicated function of several parameters (denoted by W) by its average value. Making this approximation yields a new dispersion relation in the transformed coordinate system:  

$$\fbox {$ \hat{\omega}(k_x,\hat{\omega}_0) = \left( 1 - \fra... ...at{\omega}_0)}{W} \sqrt{\hat{\omega}_0^2 - W v_0^2 k_x^2 } $}$$ (6)

This factor W contains all the information about the heterogeneities of the medium. However, it has to be determined a priori, that is, before migration. This empirical choice for W was one of the drawbacks of the Stolt-stretch method. Fomel 1995 derived an analytical formulation of the Stolt-stretch parameter, based on Malovichko's formula for approximating traveltimes in vertically inhomogeneous media Malovichko (1978):

$$t_0 = \left( 1 - \frac{1}{S(t)}\right) t + \frac{1}{S(t)} \sqrt{t^2 + S(t) \frac{(x-x_0)^2}{v_{rms}^2(t)}} \; ,$$ (7)

where the function S(t) defines the so-called parameter of heterogeneity:

$$S(t) = \frac{1}{v_{rms}^4 t} \int_0^{t} v^4(t) dt$$ (8)

Fomel proved that, for a given depth (or vertical traveltime), the optimal value of W is  

$$\fbox {$ W(t) = 1 - \frac{v_0^2 s^2(t)}{v_{rms}^2(t) t^2} \left( \frac{v^2(t)}{v_{rms}^2(t)} - S(t)\right) $} \; ,$$ (9)

where v_rms(t) is the root mean square velocity along the vertical ray, and $t = \int_0^z \frac{dz}{v(z)}$ the vertical traveltime. The value of W used during Stolt migration is the average along the vertical profile of these W(t). In the case of an homogeneous constant-velocity model, W is equal to 1.0, whereas it has to be less than 1.0 if the velocity increases monotonically with depth.

We can sum up the application of Stolt-stretch algorithm with the optimal parameter W by the following sequence of steps: