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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 problem associated with it is to find the traveltime surface $\tau(x)$ for a constant velocity v, given the traveltime surface $\tau_1(x_1)$ at some other velocity v1. 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 v1 = 0 corresponds kinematically to the zero-offset (post-stack) migration, and the case of v = 0 corresponds to the zero-offset modeling (demigration).

The appropriate mathematical method of solving the kinematic problem posed above is the method of characteristics Courant (1962). The characteristics of equation (2) are the trajectories followed by individual points of the reflector image in the velocity continuation process, which I have called velocity rays Fomel (1994). Velocity rays are defined by the system of ordinary differential equations derived from (2) according to the classic rules of mathematical physics:
{{{dx} \over {dv}} = - 2\,v\,\tau\,\tau_x} & , &
{{{d\tau} \ove...
 ...d\tau_v} \over {dv}} = \left(\tau + v\,\tau_v\right)\,\tau_x^2}\;.\end{eqnarray} (3)
An additional constraint for the quantities $\tau_x$ and $\tau_v$follows from equation (2), rewritten in the form  
\tau_v = v\,\tau\,\tau_x^2\;.\end{displaymath} (5)
One can easily solve the system of equations (3) and (4) by the classic mathematical methods for the ordinary differential equations. The general solution of the system takes the parametric form
x(v) & = A - C v^2\;,\;
\tau^2(v) & = B - C^2\,v^2\;,
 ... \over {\tau(v)}}\;,\;
\tau_v(v) & = {{C^2\,v} \over {\tau(v)}}\;,\end{eqnarray} (6)
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^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 x0 correspond to the zero velocity (unmigrated section). Equations (8), (9), and (10) have a clear geometric meaning illustrated in Figure 1. Noting 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)
we can see that equations (8) and (9) are precisely equivalent to the evident geometric relationships  
x + v\,\tau\,\tan{\alpha} = x_0\;,
\;{\tau \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 $\tau$,v, 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).

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 xd,zd, 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 is half of the actual velocity. Applying formulas (6), we can deduce that the velocity rays in this case have the following mathematical expressions:
x(v) & = & x_d\,{v^2 \over v_d^2} + 
x_0\,\left(1 - {v^2 \over ...
 ...x_0 - x_d)^2} \over v_d^2}\,
\left(1 - {v^2 \over v_d^2}\right)\;,\end{eqnarray} (14)
where $\tau_d = {z_d \over v_d}$.Eliminating x0 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 vd > v (undermigration) and an elliptic shape when vd < v (overmigration). The wavefront collapses to a point when the velocity v coincides with the actual effective velocity vd. 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. Left: the case of a point diffractor. Right: the case of a dipping plane reflector.
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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 plane reflector has the simple expression  
z_p(x) = x\,\tan{\alpha}\;,\end{displaymath} (17)
where $\alpha$ is the dip angle. The zero-offset reflection traveltime 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 vp is the half of the actual velocity. Applying formulas (6) leads to the following parametric expression for the velocity rays:
x(v) & = & x_0\,(1 - p^2\,v^2)\;,
\tau(v) & = & p\,x_0\,\sqrt{1 - p^2\,v^2}\;.\end{eqnarray} (19)
Eliminating x0 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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