Constant offset imaging

Because a constant offset section is an offset continuation of a zero offset section, I assume that the diffraction curve on a constant offset section is well approximated by a flat top hyperbola in the vicinity of its apex when the lateral velocity variations are confined to a reasonable scale. Consequently, a constant offset time-migration can collapse the energy distributed over this portion of the diffraction curve.

For a given time image position (t₀,x₀), the migration curve of the constant-offset Kirchhoff summation is:

 

$$t = \sqrt{t^2_0+\displaystyle{(m-h-x_0)^2 \over v^2_m}}+ \sqrt{t^2_0+\displaystyle{(m+h-x_0)^2 \over v^2_m}},$$ (7)

where v_m is a migration velocity, m and h are midpoint and half offset, respectively. For a given half offset h, this equation defines traveltime t as a function of midpoint m. The coordinates of the apex of the migration curve are:

 

$$\left\{ \begin{array} {lll} t_a & = & 2\sqrt{t^2_0+\displaystyle{h^2 \over v^2_m}} \\ \\ m_a & = & x_0.\end{array}\right.$$ (8)

Thus, the Kirchhoff time-migration focuses the diffracted energy along the diffraction curve whose apex is at (t_a,m_a) to the point (t₀,x₀) of the time-migrated image. The correct time-to-depth conversion should map the image sample at (t₀,x₀) to the true position (x,z) of the scatterer.

Similarly to profile imaging, I define image rays for constant offset imaging to be the propagation paths of the energy in the signals at the apex positions of diffraction curves on a constant offset section. At the apex of the diffraction curve, the derivative of the traveltime with respective to the midpoint location is zero. We know that the traveltime is $\tau(x,z;m-h)+\tau(x,z;m+h)$. If we define p(x,z;x_s) to be the partial derivative of $\tau(x,z;x_s)$ with respect to x_s, then image rays for constant offset imaging satisfy the following relation:

p(x,z;m-h)+p(x,z;m+h)=0.

Figure  shows an example of such an image ray. By tracing the image ray passing through the scatterer at subsurface position (x,z), we can find the coordinates of the apex position of the diffraction curve, as follows:

 

$$\left\{ \begin{array} {lll} t_a & = & \tau(x,z;m-h)+\tau(x,z... ... \ \vert \ \ p(x,z;m-h)+p(x,z;m+h)=0\right\}.\end{array}\right.$$ (10)

Combining equations (10) and (7) gives us the mapping functions of time-to-depth conversion, as follows:

 

$$\left\{ \begin{array} {lll} t_0 =\sqrt{\displaystyle{1 \over... ... \ \vert \ \ p(x,z;m-h)+p(x,z;m+h)=0\right\}.\end{array}\right.$$ (11)

Because these two mapping functions contain an implicit function, the image-ray corrections for constant offset imaging is more difficult than one for profile imaging.

 

imraycos
Figure 2 Ray paths in a constant offset experiment. The solid line shows an image ray.