# C-wave registration

When a C-wave is acquired, it is common to have a corresponding P-wave dataset, and to compare images from these two datasets. To make the comparison, it is necessary to compute both images either in depth, or in a common time-scale; the latter is often the first step. Of course, a given reflection event will arrive at later times in the C-image than in the P-image, so it is necessary to compress the C-image to P-wave time. This process is called C-wave registration[1]. It may be accomplished by finding the (time-dependent) compression factor ${\displaystyle {\frac {t_{C}}{t_{P}}}}$ which maximizes the correlation between the two images.

This results in an estimate of vertical travel-time ratio, since

${\displaystyle V_{P0}t_{P}=V_{S0}t_{S}=V_{S0}(t_{C}-t_{P})}$

whence

${\displaystyle {\frac {V_{P0}}{V_{S0}}}={\frac {t_{C}}{t_{P}}}-1}$

If the C-wave data comes from split-spread acquisition, then normally, the two sides of the spread have opposite polarity, because of the asymmetry of the C-wave reflection coefficient. Hence, it is necessary to reverse the algebraic sign of one side of the gather, before making the image. However, even so the split-spread gather may not be symmetric (unlike with split-spread P-wave gathers), because of lateral variation in the distribution of velocities, so it may be advisable to make images with only one side of the gathers, at least in the early iterations of imaging. This travel-time asymmetry is due to so-called Diodic_velocity.

## Reference

1. Thomsen, L., 1999. Converted-Wave Reflection Seismology over inhomogeneous, anisotropic media, GEOPHYSICS, 64(3), 678-690.