EXAMPLES

A simple analytic example of isotropic heterogeneity is the case of a constant velocity gradient. In this case the velocity distribution can be described by the linear function $v\left(z\right)=v\left(0\right)(1+\alpha z)$. Stolt stretch transform is found from (5) as  

$$s(t)=\left({e^{2 \alpha v\left(0\right)\,t} -1 - 2 \alpha v\left(0\right)\,t} \over {2 \alpha^2 v_0^2}\right)^{1/2}\;.$$ (34)

Let $\kappa$ be the logarithm of the velocity change v(z)/v(0). Then an explicit expression for W factor follows from (24):  

$$W={{2\,\kappa}\over{e^{2\,\kappa}-1}}={v^2\left(0\right) \over v_{rms}^2(\kappa)}\;.$$ (35)

For $\kappa \rightarrow 0$ (a small depth or a small velocity gradient), $W \approx 1-\kappa$. For $\kappa \rightarrow \infty$ (a large positive change of velocity) W monotonically approaches zero. Formula (35) can be a useful rule of thumb for a rough estimation of W.

Numerical example of the Stolt stretch parameter computation is illustrated in Figures 2 and 3. The left side of Figure 2 shows a smoothed interval velocity curve from the Gulf of Mexico. The corresponding optimal values of the W factor as a function of vertical time (in the isotropic model) are shown on the right. Though the velocity function is smooth, substantial changes in W occur, making its mean value for the times $t_v \leq 6$ sec equal to 0.631.

The theory of cascaded migrations Beasley et al. (1988); Larner and Beasley (1987) proves that Stolt-type f-k migration for a nonuniform velocity $v\left(t_v\right)$ can be performed as a cascaded process consisting of migrations with the smaller velocities $v_i\left(t_v\right)\,,\;i=1,2,\ldots,n$, such that $v_1^2+v_2^2+\cdots+v_n^2=v^2$ . As shown by Larner and Beasley 1987, it is important to partition the velocity so that for each particular t_v all the velocities in the cascade, except maybe the last one, are constant. The advantage of the cascaded f-k migration method is based on the fact that each small velocity v_i describes a more homogeneous medium than the initial $v\left(t_v\right)$ function. Therefore, the W factor for each migration in a cascade is closer to 1, and the Stolt stretch approximation is more accurate. This fact is illustrated in Figure 3, which shows an optimal partitioning of the velocity and the corresponding values of the W factor. In accordance with the empirical conclusions of Beasley et al. 1988, a cascade of only four migrations was sufficient to increase the value of W to more than 0.8. With a further increase of the number of cascaded migrations, the method becomes as accurate with respect to vertical velocity variations as phase-shift migration. Theoretically, this limit corresponds to the velocity continuation concept Fomel (1994). Note that the theory of cascaded f-k migration is strictly valid for isotropic models. The anisotropic interpretation does not support it, since the intrinsic anisotropy factor is not supposed to change with the velocity partitioning.

 

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Figure 2 Smoothed interval velocity distribution from the Gulf of Mexico (left) and the corresponding W factor as a function of vertical time (right). The mean value of W is 0.631.

 

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Figure 3 Left: Optimal partitioning of the velocity function for the method of cascaded migrations. Right: corresponding mean values of W. Top: four cascaded migrations. Bottom: seven cascaded migrations.