Reflection and transmission coefficients for SH wave in plane wave domain
Ritesh Kumar Sharma, Robert James Ferguson
The calculation of reflection and transmission coefficients at any interface is an important problem of elastodynamic theory. Historically, reflection and transmission coefficients have been obtained in several domains according to their importance. It is known that for isotropic case reflection and transmission coefficients depend on the acoustic impedance contrast and angle of incidence. The angle of incidence can be computed by using the cross product of two known unit normals. In this approach, derived reflection and transmission coefficients will be in plane wave domain. By deriving reflection and transmission coefficients in plane wave domain for 3D media, determination of dip and azimuth of interface are avoided, and, thereby, we avoid ray tracing ray tracing and exposure to caustics especially in anisotropic media. The problem that I solve, in this regard, is the problem of the special case of dipping interface and how to rotate the plane wave coordinate system from that determined by the computational grid, and the system determined by a dipping interface. Classical reflection and transmission coefficients in plane wave coordinates are worked out for reflectors aligned with the computational grid. For non-aligned reflectors, those with dip and azimuth, computation of effective reflection and transmission coefficients is not straight forward, for this the coordinate system must be rotated. To do this, a normal for each individual plane wave based on local velocity and vector cross product of this normal with the normal to reflector are computed. This cross product yields a ray parameter that presently is used to compute corresponding reflection and transmission coefficients for a given plane wave. The importance of this approach is the automatic adaptation of the reflection and transmission coefficients expression to a special case of dipping interface. These coefficients can then be used to scale the amplitude component of plane wave extrapolation across a reflector as is done in seismic forward modeling. Another importance of reflection and transmission coefficients in plane wave domain, is their use in Rayleigh Sommerfeld Modeling(RSM) of seismic data. In line traces and cross line traces are required in order to model the plane wave inputs. Presently, the problem associated with data acquisition is studied here by changing the number of cross line traces.