C-AVO

The exact solution for C-wave reflectivity (for the case of a plane P-wave incident upon an planar boundary between isotropic media) is given by Aki and Richards (1980)[1]. As with the corresponding result for P-P reflections, this is too complicated to be useful in exploration seismics.

However, when linearized in terms of fractional jumps in elastic properties, the approximate result[2] is analogous to the well-known equations for P-AVO, and easy to analyse:

${\displaystyle R_{C}(\theta )=\sin(\theta )\left[R_{C1}+R_{C3}\sin ^{2}(\theta )+R_{C5}\sin ^{4}(\theta )\right]}$

where:

${\displaystyle R_{C1}=-{\frac {1}{2}}\left({\frac {\Delta \rho }{\rho }}+{\frac {2}{\Gamma }}{\frac {\Delta \mu }{\mu }}\right)}$

${\displaystyle R_{C3}=-{\frac {1}{2\Gamma ^{2}}}\left({\frac {\Delta \rho }{2\rho }}-(2+\Gamma ){\frac {\Delta \mu }{\mu }}\right)}$

${\displaystyle R_{C5}={\frac {1}{4\Gamma ^{3}}}\left({\frac {(1+\Gamma )}{\Gamma }}\right)^{2}{\frac {\Delta \mu }{\mu }}}$

with

${\displaystyle \Gamma ={\frac {V_{P}}{V_{S}}}}$

This expression is similar to the corresponding expression for P-AVO:, except for the leading factor ${\displaystyle \sin(\theta )}$, which
a) makes the reflected amplitude zero at normal incidence.
b) makes the reflectivity assymetric with interchange of source and receiver positions. This immediately shows that C-waves do not obey the Scalar Reciprocity Theorem, although they do (of course) obey the Vector Reciprocity Theorem. This polarity reversal is why it is necessary to reverse the algebraic sign of one side of a split-spread before making an image using both sides[3].

Note that the the coefficients only involve the jumps (${\displaystyle \Delta }$) in shear modulus ${\displaystyle \mu }$ and density ${\displaystyle \rho }$; the jump in P-velocity ${\displaystyle \Delta }$ does not appear (even though the incident wave is a P-wave).

This is the theory. However, in practice, it may sometimes happen that the C-wave reflectivity at normal incidence is not zero. The reasons for this are not clear, but such cases require a deep revision of the theory outlined above.

References

1. Aki, K, and PG Richards (1980). Quantitative Seismology: Theory and Methods. WH Freeman and Co.
2. Thomsen, L., 2014. Seismic Anisotropy in Exploration and Exploitation, the SEG/EAGE Distinguished Instructor Short Course #5 Lecture Notes, 2nd Edition, Soc. Expl. Geoph., Tulsa
3. Thomsen, L., 1999. Converted-Wave Reflection Seismology over inhomogeneous, anisotropic media, GEOPHYSICS, 64(3), 678-690.