Evaluating the Stolt-stretch parameter

EVALUATING THE $W$ PARAMETER

A remarkable connection between the Stolt stretch equation and different three-parameter traveltime approximations leads to a constructive estimate of the $W$ parameter. The first useful observation is a formal similarity between equation (11) and Malovichko's approximation for the reflection traveltime curve in vertically inhomogeneous media (Sword, 1987; Castle, 1988; de Bazelaire, 1988; Malovichko, 1978) defined by

$$t_0= \left(1-{1\over {S\left(t_v\right)}}\right) \,t_v+ {1\over {... ...t(t_v\right)}\, {{\left(x-x_0\right)^2} \over {v_{rms}^2\left(t_v\right)}}}}\;.$$ (13)

In equation (13), $v_{rms}$ is the effective (root mean square) velocity along the vertical ray

$$v_{rms}^2\left(t_v\right)={\eta(z)\over t_v}= {1 \over t_v}\,\int_{0}^{t_v} v^2(t) \,dt\;,$$ (14)

and $S$ is f the parameter of heterogeneity, defined by the equation:

$$S\left(t_v\right)={1 \over{v_{rms}^4 t_v}}\,\int_{0}^{t_v} v^4(t) \,dt\;.$$ (15)

In terms of the $S$ parameter, the variance of the squared velocity distribution along the vertical ray is

$$\sigma^2={1 \over t_v}\,\int_{0}^{t_v} v^4(t) \,dt - v_{rms}^4=v_{rms}^4 (S-1)\;.$$ (16)

As follows from equality (16), $S\geq 1$ for any type of velocity distribution ($S$ equals 1 in a constant velocity case). For most of the distributions occurring in practice, $S$ ranges between 1 and 2.

Since reflection from a horizontal reflector in vertically-heterogeneous media is kinematically equivalent to diffraction from a point, we can regard equation (13), which is known as the most accurate three-parameter approximation of the NMO curve, as an approximation of the summation path for the post-stack Kirchhoff migration operator. In this case, it has the same meaning as equation (11). An important difference between the two equations is the fact that equation (13) is written in the initial coordinate system and includes coefficients varying with depth, while equation (11) applies the transformed coordinate system and constant coefficients. Using this fact, we compare the accuracy of the approximations and derive the following explicit expression, which relates Stolt's $W$ factor to Malovichko's parameter of heterogeneity:

$$W=1-{{v_0^2\,s^2\left(t_v\right)} \over{v_{rms}^2\left(t_v\right)... ...(t_v\right)} \over {v_{rms}^2\left(t_v\right)}} -S\left(t_v\right) \right)\;\;.$$ (17)

The details of the derivation are given in the appendix. Expression (17) is derived so as to provide the best possible value of $W$ for a given depth (or vertical time $t_v$ ). To get a constant value for a range of depths, one should take an average of the right-hand side of (17) in that range. The error associated with Stolt stretch can be approximately estimated from (A-1) as the difference between the fourth-order terms:

$$\delta={{l^4 \over 8}\,{{W\left(t_v\right)-W} \over {t_v s^2\left(t_v\right) v_{rms}^2\left(t_v\right) v_0^2}}}\;,$$ (18)

where $W\left(t_v\right)$ is the right-hand side of (17), and $W$ is the constant value of $W$ chosen for Stolt migration.

Evaluating the Stolt-stretch parameter