Stolt stretch

The great strength and the great weakness of the Stolt migration method is that it uses Fourier transformation over depth. This is a strength because it makes his method much faster than all other methods. And it is a weakness because it requires a velocity that is a constant function of depth. The earth velocity typically ranges over a factor of two within the seismic section, and the effect of velocity on migration tends to go as its square. To ameliorate this difficulty, Stolt suggested stretching the time axis to make the data look more like it had come from a constant-velocity earth. Stolt proposed the stretching function  

$$\tau (t) \eq \sqrt{ {2 \over v_0^2 } \ \ \int_0^t \ \ {t\ v_{\rm RMS}^2 (t) } \ dt \ }$$ (6)

where  

$$v_{\rm RMS}^2 (t) \eq {1 \over t }\ \int_0^t \ v^2 (t) \ dt$$ (7)

At late times, which are associated with high velocities, Stolt's stretch implies that $\tau$ grows faster than t. The $\tau$-axis is uniformly sampled to allow the fast Fourier transform. Thus at late time the samples are increasingly dense on the t-axis. This is the opposite of what earth Q and the sampling theorem suggest, but most people consider this a fair price.

The most straightforward derivation of (6) and (7) is based on the idea of matching the curvature of ideal hyperbola tops to the curvature on the stretched data. The equation of an ideal hyperbola in $(x, \tau )$-space is  

$$v_0^2 \, \tau^2 \eq x^2 \ +\ z^2$$ (8)

Simple differentiation shows that the curvature at the hyperbola top is  

$$\left. {d^2 \tau \over dx^2} \right\vert _{x=0} \eq {1 \over \tau \ v_0^2}$$ (9)

It can be shown that in a stratified medium, equation (9) applies, except that the velocity must be replaced by the RMS velocity:  

$$\left. {d^2 t \over dx^2} \right\vert _{x=0} \eq {1 \over t \ v_{\rm RMS}^2}$$ (10)

We seek a stretched time $\tau (t)$.We would like to match the curves t (x) and $\tau (x)$ for all x. But that would overdetermine the problem. Instead we could just match the derivatives at the hyperbola top, i.e., the second derivative of $\tau [t(x)]$ with respect to x at x=0. With the substitutions (9) and (10), this would give an expression for $\tau\,d \tau /dt$ which, after integrating and taking its square root, yields (6) and (7).

A different derivation of the stretch gives a more accurate result at steeper angles. Instead of matching hyperbola curvature at the top, we go some distance out on the flank and match the slope and value. It is the flanks of the hyperbola that actually migrate, not the tops, so this result is more accurate. Algebraically the derivation is also easier, because only first derivatives are needed. Differentiating equation (8) with respect to x for a reflector at any depth z_j gives  

$${d \tau \over dx } \eq {x \over \tau \ v_0^2 }$$ (11)

There is an analogous expression in a stratified medium. To obtain it, solve $x = \int \, v\, \sin\theta \,dt =$ $p \int v^2 dt$for p = dt/dx:  

$${dt \over dx } \eq {x \over \int_0^t \ v^2 (p,t) \ dt }$$ (12)

Expressions (11) and (12) play the same role as (9) and (10), but (11) and (12) are valid everywhere, not just at the hyperbola top. Differentiating $\tau (t)$ gives  

$${d \tau \over dx } \eq {d \tau \over dt }\ {dt \over dx }$$ (13)

Inserting (11) and (12) into (13) gives  

$${x \over \tau \ v_0^2 } \eq {d \tau \over dt }\ {x \over \int_0^t \ v^2 (p,t) \ dt }$$ (14)

 

$$\tau \ d \tau \eq \left[ \ {1 \over v_0^2} \ \int_0^t \ v^2 (p,t ' ) \ dt ' \ \right] \ dt$$ (15)

Integrating (15) gives $\tau^2 /2$ on the left. Then, taking the square root gives (6) but with a new definition for RMS velocity:  

$$v_{\rm RMS}^2 (t) \eq {1 \over t }\ \int_0^t \ v^2 (p,t) \ dt$$ (16)

The thing that is new is the presence of the Snell parameter p. In a stratified medium characterized by some velocity, say, v ' (z), the velocity v(p,t) is defined for the tip of the ray that left the surface at an angle with a stepout p. In practice, what value of p should be used? The best procedure is to look at the data and measure the $p\,=\,dt/dx$ of those events that you wish to migrate well. A default value is $p \ =\ 2(\sin$ 30$^\circ$)/(2.5 km/sec) = .4 millisec/meter. The factor of 2 is from the exploding-reflector model.