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Kinematics of Zero-Offset Velocity Continuation

The kinematic equation for zero-offset velocity continuation is
{{\partial \tau} \over {\partial v}} =
v\,\tau\,\left({{\partial \tau} \over {\partial x}}\right)^2\;.
\end{displaymath} (2)

The typical boundary-value problem associated with it is to find the traveltime surface $\tau_2(x_2)$ for a constant velocity $v_2$, given the traveltime surface $\tau_1(x_1)$ at some other velocity $v_1$. Both surfaces correspond to the reflector images obtained by time migrations with the specified velocities. When the migration velocity approaches zero, post-stack time migration approaches the identity operator. Therefore, the case of $v_1 =
0$ corresponds kinematically to the zero-offset (post-stack) migration, and the case of $v_2 = 0$ corresponds to the zero-offset modeling (demigration). The variable $x$ in equation (2) describes both the surface midpoint coordinate and the subsurface image coordinate. One of them is continuously transformed into the other in the velocity continuation process.

The appropriate mathematical method of solving the kinematic problem posed above is the method of characteristics (Courant and Hilbert, 1989). The characteristics of equation (2) are the trajectories followed by individual points of the reflector image in the velocity continuation process. These trajectories are called velocity rays (Adler, 2002; Fomel, 1994; Liptow and Hubral, 1995). Velocity rays are defined by the system of ordinary differential equations derived from (2) according to the Hamilton-Jacobi theory:

$\displaystyle {{{dx} \over {dv}} = - 2\,v\,\tau\,\tau_x}$ $\textstyle ,$ $\displaystyle {{{d\tau} \over {dv}} = - \tau_v}\;,$ (3)
$\displaystyle {{{d\tau_x} \over {dv}} = v\,\tau_x^3}$ $\textstyle ,$ $\displaystyle {{{d\tau_v} \over {dv}} = \left(\tau + v\,\tau_v\right)\,\tau_x^2}\;,$ (4)

where $\tau_x$ and $\tau_v$ are the phase-space parameters. An additional constraint for $\tau_x$ and $\tau_v$ follows from equation (2), rewritten in the form
\tau_v = v\,\tau\,\tau_x^2\;.
\end{displaymath} (5)

The general solution of the system of equations (3-4) takes the parametric form
$\displaystyle x(v)$ $\textstyle =$ $\displaystyle A - C v^2\;,\quad
\tau^2(v) = B - C^2\,v^2\;,$ (6)
$\displaystyle \tau_x(v)$ $\textstyle =$ $\displaystyle {C \over {\tau(v)}}\;,\quad
\tau_v(v) = {{C^2\,v} \over {\tau(v)}}\;,$ (7)

where $A$, $B$, and $C$ are constant along each individual velocity ray. These three constants are determined from the boundary conditions as
A = x_1 + v_1^2\,\tau_1\,{{\partial \tau_1} \over {\partial x_1}} = x_0\;,
\end{displaymath} (8)

B = \tau_1^2\,\left(1 + v_1^2\,
\left({{\partial \tau_1} \over {\partial x_1}}\right)^2\right) = \tau_0^2\;,
\end{displaymath} (9)

C = \tau_1\,{{\partial \tau_1} \over {\partial x_1}} =
\tau_0\,{{\partial \tau_0} \over {\partial x_0}}\;,
\end{displaymath} (10)

where $\tau_0$ and $x_0$ correspond to the zero velocity (unmigrated section), while $\tau_1$ and $x_1$ correspond to the velocity $v_1$. The simple relationship between the midpoint derivative of the vertical traveltime and the local dip angle $\alpha$ (appendix A),
{{\partial \tau} \over {\partial x}} =
{{\tan{\alpha}} \over v}\;,
\end{displaymath} (11)

shows that equations (8) and (9) are precisely equivalent to the evident geometric relationships (Figure 1)
x_1 + v_1\,\tau_1\,\tan{\alpha} = x_0\;,
\;{\tau_1 \over {\cos{\alpha}}} = \tau_0\;.
\end{displaymath} (12)

Equation (10) states that the points on a velocity ray correspond to a single reflection point, constrained by the values of $\tau_1$, $v_1$, and $\alpha$. As follows from equations (6), the projection of a velocity ray to the time-midpoint plane has the parabolic shape $x(\tau) = A + (\tau^2 - B) / C$, which has been noticed by Chun and Jacewitz (1981). On the depth-midpoint plane, the velocity rays have the circular shape $z^2(x) = (A - x)\,B / C - (A -
x)^2$, described by Liptow and Hubral (1995) as ``Thales circles.''

Figure 1.
Zero-offset reflection in a constant velocity medium (a scheme).
[pdf] [png] [xfig]

For an example of kinematic continuation by velocity rays, let us consider the case of a point diffractor. If the diffractor location in the subsurface is the point ${x_d,z_d}$, then the reflection traveltime at zero offset is defined from Pythagoras's theorem as the hyperbolic curve

\tau_0(x_0) = {{\sqrt{z_d^2 + (x_0 - x_d)^2}} \over v_d}\;,
\end{displaymath} (13)

where $v_d$ is half of the actual velocity. Applying equations (6) produces the following mathematical expressions for the velocity rays:
$\displaystyle x(v)$ $\textstyle =$ $\displaystyle x_d\,{v^2 \over v_d^2} +
x_0\,\left(1 - {v^2 \over v_d^2}\right)\;,$ (14)
$\displaystyle \tau^2(v)$ $\textstyle =$ $\displaystyle \tau_d^2 + {{(x_0 - x_d)^2} \over v_d^2}\,
\left(1 - {v^2 \over v_d^2}\right)\;,$ (15)

where $\tau_d = {z_d \over v_d}$. Eliminating $x_0$ from the system of equations (14) and (15) leads to the expression for the velocity continuation ``wavefront'':
\tau(x)=\sqrt{\tau_d^2 + {{(x - x_d)^2} \over {v_d^2 - v^2}}}\;.
\end{displaymath} (16)

For the case of a point diffractor, the wavefront corresponds precisely to the summation path of the residual migration operator (Rothman et al., 1985). It has a hyperbolic shape when $v_d > v$ (undermigration) and an elliptic shape when $v_d < v$ (overmigration). The wavefront collapses to a point when the velocity $v$ approaches the actual effective velocity $v_d$. At zero velocity, $v=0$, the wavefront takes the familiar form of the post-stack migration hyperbolic summation path. The form of the velocity rays and wavefronts is illustrated in the left plot of Figure 2.

Figure 2.
Kinematic velocity continuation in the post-stack migration domain. Solid lines denote wavefronts: reflector images for different migration velocities; dashed lines denote velocity rays. a: the case of a point diffractor. b: the case of a dipping plane reflector.
[pdf] [png] [sage]

Another important example is the case of a dipping plane reflector. For simplicity, let us put the origin of the midpoint coordinate $x$ at the point of the plane intersection with the surface of observations. In this case, the depth of the plane reflector corresponding to the surface point $x$ has the simple expression

z_p(x) = x\,\tan{\alpha}\;,
\end{displaymath} (17)

where $\alpha$ is the dip angle. The zero-offset reflection traveltime $\tau_0(x_0)$ is the plane with a changed angle. It can be expressed as
\tau_0(x_0) = p\,x_0\;,
\end{displaymath} (18)

where $p = {{\sin{\alpha}}\over v_p}$, and $v_p$ is half of the actual velocity. Applying formulas (6) leads to the following parametric expression for the velocity rays:
$\displaystyle x(v)$ $\textstyle =$ $\displaystyle x_0\,(1 - p^2\,v^2)\;,$ (19)
$\displaystyle \tau(v)$ $\textstyle =$ $\displaystyle p\,x_0\,\sqrt{1 - p^2\,v^2}\;.$ (20)

Eliminating $x_0$ from the system of equations (19) and (20) shows that the velocity continuation wavefronts are planes with a modified angle:
\tau(x)={{p\,x} \over {\sqrt{1 - p^2\,v^2}}}\;.
\end{displaymath} (21)

The right plot of Figure 2 shows the geometry of the kinematic velocity continuation for the case of a plane reflector.

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