# C-wave

A "C-wave" is a wave which converts from P to S at the reflector (in global seismology, a PS-wave)[1]. Other converted modes are possible, and may even occur in exploration seismic datasets, but the C-wave is usually the most prominent arrival in 4C-OBS datasets, because conversion of downgoing energy at the sea-floor (in transmission) is minimal, because of the elastic properties of the near-seafloor sediments. The C-wave raypath is asymmetrical (see figure), so many concepts from P-wave seismics do not apply.

For modest offsets, in 1D (vertically inhomogeneous) anisotropic media, the moveout is hyperbolic , with a short-spread moveout velocity given by:

${\displaystyle V_{CNMO}^{2}={\frac {V_{PNMO}^{2}}{1+\Gamma _{0}}}}$

where the vertical velocity ratio is the ratio of P- and S- vertical velocities.

${\displaystyle \Gamma _{0}={\frac {V_{P0}}{V_{S0}}}}$

If the medium is isotropic, this reduces to

${\displaystyle V_{CNMO}^{2}=V_{PNMO}V_{SNMO}}$,

but this is not typical in most exploration settings, so one should rely on the more general formula above. (The vertical velocity ratio ${\displaystyle \Gamma _{0}}$ is readily found, as a function of depth, from the C-wave registration process.)

Because of the asymmetry of the raypath, the moveout typically deviates from hyperbolic at shorter offsets than is typical with P-waves. Also because of the asymmetry, the Scalar Reciprocity Theorem does not apply, so in general if source and receiver positions are interchanged, the data will be different. Instead, the Vector Reciprocity Theorem applies. Furthermore, the C-wave conversion coefficient is an odd function of the incidence angle, so the algebraic sign of the reflectivity reverses if source and receiver positions are interchanged (as in a split-spread gather).

Typically, the moveout velocity must be determined from a Common-Conversion-Point gather, a set of traces with different source-receiver offsets, but the same C-wave depth point (CDP). Because of the asymmetry, the CDP is not the source-receiver midpoint (the MDP), but must be calculated, as a function of source-receiver offset and depth. (This calculation involves the effective velocity ratio ${\displaystyle \Gamma _{eff}}$.) Hence, construction of the gather is not a simple sort of traces, as with the MDP, but is an iterative process, almost a migration.

In an OBS context, it is common to acquire split spreads. If ${\displaystyle V_{P}}$ and/or ${\displaystyle V_{S}}$ vary laterally, the traveltimes (hence the moveout velocities) of split-spread gathers will be different on the two sides of the gather (unlike with pure P-waves); this is called "Diodic velocity". In this case, the two sides of the gather should be analyzed separately, as in off-end acquistion. A tentative CCP gather may be formed based on a first estimate of the position of the CCP, and an estimate of ${\displaystyle V_{CNMO}}$ may be found by conventional semblance techniques. With this estimate, the position of the CCP may be refined, and the process may be iterated.

At the beginning of the process, it is expected that images made from positive source-receiver offsets will differ from those made with negative offsets, and should not be superposed. The different images result from diodic velocity structure, and from initially imprecise estimates of ${\displaystyle V_{CNMO}}$ and ${\displaystyle \Gamma _{eff}}$. From the differences in these images, an iterative process can be constructed to refine the estimates of ${\displaystyle V_{CNMO}}$ and ${\displaystyle \Gamma _{eff}}$.

## Reference

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