Evaluating the Stolt-stretch parameter

Stolt Stretch Theory Review

In order to simplify the references, we start with definitions of the Stolt migration method. The reader familiar with the Stolt stretch theory can skip this section and go on to new theoretical results in the next section.

The basic migration theory reduces post-stack migration to a two-stage process. The first stage is a downward continuation of the wavefield in depth $z$ based on the wave equation

$${\partial^2 P \over \partial x^2} + {\partial^2 P \over \partial z^2} \, = \,{1 \over {v^2(x,z)}}\, {\partial^2 P \over \partial t^2}\;.$$ (1)

The second stage is the imaging condition $t=0$ (here the velocity $v$ is twice as small as the actual wave velocity). Stolt time migration performs both stages in one step, applying the frequency-domain operator

$$\tilde{P_0}\left(k_x, \omega_0\right)= \tilde{P_v}\left(k_x, \ome... ...,\left\vert {{d\omega_v \left(k,\omega_0\right)}\over{d\omega_0}}\right\vert\;,$$ (2)

where

$$\tilde{P_v}\left(k_x, \omega_v\right) = \iint P_v\left(x, t_v\right) \exp{\left(i \omega_v t_v - i k_x x \right)} \,dt_v\, dx \;\;,$$

$$\tilde{P_0}\left(k_x, \omega_0\right) = \iint P_0\left(x, t_0\right) \exp{\left(i \omega_0 t_0 - i k_x x \right)} \,dt_v\, dx \;\;,$$

$P_0\left(x, t_0\right)$ stands for the initial zero-offset (stacked) seismic section defined on the surface $z=0$ , $P_v\left(x, t_v\right)$ is the time-migrated section, and $t_v$ is the vertical traveltime

$$t_v=\int_{0}^{z}{dz' \over {v(x,z')}}\;\;.$$ (3)

The function $\omega_v \left(k,\omega_0 \right)$ in (2) corresponds to the dispersion relation of the wave equation (1) and in the constant velocity case has the explicit expression

$$\omega_v \left(k,\omega_0 \right)= \left(\omega_0\right) \sqrt{\omega_0^2 - v^2 k^2}\;\;.$$ (4)

The choice of the sign in equation (4) is essential for distinguishing between upgoing and downgoing waves. The upgoing part of the wavefield is the one used in migration.

The case of a varying velocity complicates the frequency-domain algorithm and therefore requires special consideration. Stolt (1978) suggested the following change of the time variable (referred to in the literature as Stolt stretch):

$$s(t)={\left({{2 \over v_0^2}\,\int_0^t\eta d \tau}\right)}^{1/2}\;,$$ (5)

where $v_0$ is an arbitrarily chosen constant velocity, and $\eta$ is a function defined by the parametric expressions

$$\eta(\zeta)=\int_0^{\zeta} v(x,z) \,dz \;,\; \tau(\zeta)=\int_0^{\zeta} { {dz} \over {v(x,z)}}\;\;.$$ (6)

Applying equation (5),we can connect seismic time migration to the transformed wave equation

$${\partial^2 P \over \partial x^2} + W\,{\partial^2 P \over \parti... ...t{t}}} \, = \,{(2-W) \over v_0^2 }\, {\partial^2 P \over \partial \hat{t}^2}\;.$$ (7)

The variables $\hat{z}$ and $\hat{t}$ correspond to the transformed depth and time coordinates, which possess the following property: if $\hat{z}=0$ , $\hat{t}=s\left(t_0\right)$ , and if $\hat{t}=0$ , $\hat{z}=v_0 s\left(t_v\right)$ . $W$ is a varying coefficient defined as

$$W=a^2+2b\,(1-a^2)\;,$$ (8)

where

$$b={{\eta(z)}\over{\eta(\zeta)}}\,,\; a={{s(\tau)\,v_0\,v(x,z)}\ov... ...}\,,\; \tau=\int_0^\zeta{{dz}\over{v(x,z)}}=t+ \int_0^z{{dz'}\over{v(x,z')}}\;.$$

Since the $W$ parameter varies slowly with $x$ and $\hat{z}$ , Stolt suggested to replace it with its average value. Thus equation (7) is then approximated by an equation with constant coefficients, which has the dispersion relation

$$\widehat{\omega}_v\left(k,\widehat{\omega}_0 \right)= \left(1-{1\... ...idehat{\omega}_0\right)}\over W}\, \sqrt{\widehat{\omega}_0^2 - W v_0^2 k^2}\;.$$ (9)

As outlined above, Stolt's approximate method for migration in heterogeneous media consists of the following steps:

1. stretching the time variable according to equation (5),
2. interpolating the stretched time to a regular grid,
3. double Fourier transform,
4. f-k time migration by the operator (2) with the dispersion relation (9),
5. inverse Fourier transform,
6. inverse stretching (that is, shrinking) of the vertical time variable on the migrated section.

The value of $W$ must be chosen prior to migration. According to Stolt's original definition (8), the depth variable $z$ gradually changes in the migration process from zero to $\zeta$ , causing the coefficient $b$ in (8) to change monotonically from 0 to 1. If the velocity $v$ monotonically increases with depth, then $\eta''(z)={\partial v \over \partial z}\geq 0$ , and the average value of $b$ is

$$\bar{b}={1 \over {\zeta \eta(\zeta)}}\, {\int_0^\zeta \eta(z) dz}... ...\eta(\zeta)}}\, {\int_0^\zeta {\eta(\zeta) {z \over \zeta} }dz}= {1 \over 2}\;.$$ (10)

As follows from equations (8) and (10), in the case of monotonically increasing velocity, the average value of $W$ has to be less than 1 ($W$ equals 1 in a constant-velocity case). Analogously, in the case of a monotonically decreasing velocity, $W$ is always greater than 1. In practice, $W$ is included in migration routines as a user-defined parameter, and its value is usually chosen to be somewhere in the range of 1/2 to 1. The next section describes a straightforward way to determine the most appropriate value of $W$ for a given velocity distribution.

A useful tool for that purpose is Levin's equation for the traveltime curve. Levin (1985) applied the stationary phase technique to the dispersion relation (9) to obtain an explicit equation for the summation curve of the integral migration operator analogous to the Stolt stretch migration. The equation evaluates the summation path in the stretched coordinate system, as follows:

$$s\left(t_0\right)= \left(1-{1\over W}\right) s\left(t_v\right)+ {... ...W}\, \sqrt{s^2\left(t_v\right) + {W\, {{\left(x-x_0\right)^2} \over v_0^2}}}\;,$$ (11)

where $x_0$ is the midpoint location on a zero-offset seismic section, and $x$ is the space coordinate on the migrated section. Equation (11) shows that, with the stretch of the time coordinate, the summation curve has the shape of a hyperbola with the apex at $\left\{x,s\left(t_v\right)\right\}$ and the center (the intersection of the asymptotes) at $\left\{x,{\left(1-{1\over W}\right)}\,s\left(t_v\right)\right\}$ . In the case of homogeneous media, $W=1$ , $s(t)\equiv t$ , and equation (11) reduces to the known expression for a hyperbolic diffraction traveltime curve. It is interesting to note that inverting equation (11) for $s\left(t_v\right)$ determines the impulse response of the migration operator:

$$\hat{z}-\hat{z_0}= \left({1\over Q}-1\right) R \pm {1\over Q}\, \sqrt{R^2 - {Q\,{\left(x-x_0\right)^2}}}\;,$$ (12)

where $R=v_0 \hat{t}$ , and $Q=2-W$ . Equation (12) can be interpreted as the wavefront from a point source in the $\{x,\hat{z},\hat{t}\}$ domain of equation (7). Wavefronts from a point source in the stretched coordinates for $W<2$ have an elliptic shape, with the center of the ellipse at $\{x,\hat{z_0}+\left( {1 \over Q}-1\right)\, R \}$ and the semi-axes $a_x={R \over \sqrt{Q}}$ and $a_z={r \over Q}$ . The ellipses stretch differently for $W<1$ and $W>1$ , as shown in Figure 1. In the upper part that corresponds to the upgoing waves, the ellipses look nearly spherical, since the radius of the front curvature at the top apex equals the distance from the source.

stofro
Figure 1. Wavefronts from a point source in the stretched coordinate system. Left: velocity decreases with depth (W=1.5). Right: velocity increases with depth (W=0.5).

Evaluating the Stolt-stretch parameter