Water-bottom multiples

Consider a model with a dipping water-bottom in 2D. The raypath of the primary reflection can be easily computed using the concept of the image source as illustrated in Figure .

 

rayprim Figure 1 Construction of the primary reflection from a dipping interface.

The moveout of the primary reflection in the CMP domain is given by:  

$$t_p=\sqrt{\left(\frac{2Z_D}{V}\right)^2+\left(\frac{2h_D}{V_{NMO}}\right)^2}=\sqrt{t_0^2+\left(\frac{2h_D}{V_{NMO}}\right)^2}$$ (1)

where t_p is the time of the primary, $\varphi$ is the dip angle of the reflector, Z_D is the perpendicular distance between the surface and the reflector at the CMP location, h_D is half the source-receiver offset, V is the propagation velocity above the dipping reflector, $V_{NMO}=V/\cos\varphi$ is the normal moveout velocity and t₀ is the traveltime of the zero-offset reflection. This is obviously the equation of a hyperbola, as illustrated in Figure .

The raypath of the multiple reflection can be considered as a cascaded of two primary reflections as SRME methods do (Figure ), but the traveltime of the multiple, t_m, can be computed more easily as the traveltime of an equivalent primary from a reflector dipping at twice the dip angle of the actual reflector as illustrated in Figure . That is,  

$$t_m=\sqrt{\left(\frac{2\hat{Z}_D}{V}\right)^2+\left(\frac{2h_D}{\hat{V}_{NMO}}\right)^2}$$ (2)

where $\hat{Z}_D$ is the perpendicular distance between the surface and the equivalent reflector with twice the dip at the CMP location and $\hat{V}_{NMO}$ is now $V/\cos(2\varphi)$. Figure  corresponds to a CMP showing the primary and the multiple reflection. Obviously, they are both hyperbolas since the multiple has the same kinematics as a primary from a reflector dipping at twice the dip as indicated above.

 

raymul1 Figure 2 Decomposition of the water-bottom multiple as a cascaded of two primary reflections.

 

raymul2 Figure 3 Multiple reflection as a primary from an equivalent reflector with twice the dip angle.

 

moveouts1 Figure 4 Moveout curves of primary and water-bottom multiple from a dipping interface on a CMP gather.