Diffracted multiples

Consider now a diffractor sitting on top of the water-bottom reflector at the lateral position x_d and depth z_d (Figure ). The moveout of the primary diffraction is given by  

$$t_d=t_s+t_g=\frac{1}{V}\left[\sqrt{(x_d-m_D+h_D)^2+z_d^2}+\sqrt{(m_D+h_D-x_d)^2+x_d^2}\right]$$ (3)

where t_s and t_g are traveltimes from the source and receiver to the diffractor, respectively, m_D is the horizontal position of the CMP gather and h_D is the half-offset between the source and the receiver. This is the equation of a hyperbola with apex at the horizontal position directly above the diffractor.

 

rayprimdiff Figure 5 Image source construction for the primary diffraction.

Consider now the multiple that hits the diffractor on its second bounce on the dipping reflector (source-side multiple), as shown in Figure . This multiple is no longer the equivalent of any primary, but we can compute its traveltime in the following way:

1.

Form the image source.

2.

Find the point at the surface such that the lines joining that point with both the image source and the diffractor intercept with the same angle with respect to the vertical (Snell's law).

3.

Compute the raypath using the law of cosines and divide by the velocity to get the traveltime.

The coordinates of the image source (X_is,Z_is) are given by

$$\begin{eqnarray} X_{is}&=&X_s-2Z_0\sin\varphi,\\ Z_{is}&=&2Z_0\cos\varphi,\end{eqnarray}$$ (4) (5)

where  

$$Z_0=Z_D-h_D\sin\varphi$$ (6)

is the perpendicular distance between the shot and the reflector.

 

raymuldiff Figure 6 Diffracted multiple. Notice that the last leg of the multiple does not satisfy Snell's law.

From the description of numeral 3. above, the surface coordinate of the multiple bounce at the surface is:  

$$X_{ms}=\frac{X_{is}z_d+x_{d}Z_{is}}{Z_{is}+z_{d}}.$$ (7)

The aperture angle of the first bounce of the multiple is $\beta_s+\varphi$where $\beta_s$ is the takeoff angle of the multiple with respect to the vertical. The aperture angle at the surface bounce is $\beta_s+2\varphi$ and can be easily computed as

$$\tan(\beta_s+2\varphi)=\frac{x_d-X_{ms}}{z_d}.$$ (8)

The traveltime of the first leg of the multiple, in terms of $\beta_s$ and the vertical distance between the source and the reflector, $Z_s=\frac{Z_D-h_D\sin\varphi}{\cos\varphi}$, is:  

$$t_{m_1}=\frac{Z_s\cos\varphi}{V\cos(\varphi+\beta_s)}.$$ (9)

Similarly, repeated application of the law of sines gives the traveltime of the other three legs of the multiple

$$\begin{eqnarray} t_{m_2}&=&\frac{Z_s\cos\varphi\cos\beta_s}{V\cos(\varphi+\beta_... ...varphi\cos\beta_s}{V\cos(2\varphi+\beta_s)\cos(3\varphi+\beta_s)}.\end{eqnarray}$$ (10) (11)

The total arrival time of the diffracted multiple is therefore:

$$\begin{eqnarray} t_m&=&\frac{Z_s\cos\varphi}{V\cos(\varphi+\beta_s)}\left(1+\fra... ...phi)\cos\varphi}{\cos(\beta_s+3\varphi)}+\frac{z_d}{V\cos\beta_r},\end{eqnarray}$$ (12)

where $\tan\beta_r=\frac{m_D+h_D-x_d}{zd}$ and $\beta_r$ is the emergence angle of the diffracted multiple with respect to the vertical. Figure  compares the moveout of the diffracted multiple with that of the water-bottom multiple. Obviously, the apex of the diffracted multiple is not at zero offset.

 

moveouts3 Figure 7 Moveout curves of primary, water-bottom multiple, diffraction and diffracted multiple from a dipping interface on a CMP gather.