SYNTHETIC DATA EXAMPLES

I modeled the response of a simple one layered model to a unit displacement source on the free surface. The surface layer is the orthorhombic medium used in my earlier examples. The modeling algorithm is the phase shift scheme I described in an earlier report (Nichols, 1991). A synthetic nine-component gather is shown in Figure . It is displayed in the $\tau-p$ domain. The X-x section clearly has many different wavetypes present in the gather. The first event is the P-p wave the second is the P-s and S-p arrivals. The last two events are the two S-s wave arrivals. The two shear waves are split even at zero slowness because this is an orthorhombic medium.

 

orig
Figure 4 Synthetic nine-component shot gather shown in the $\tau-p$ domain. Many different wavetypes can be seen on each component.

Source and receiver rotation by $20^\circ$ gives the gather shown in Figure . The horizontal component sections have been diagonalized at zero slowness where the rotation is the exact operator for S-wave separation. The two shear waves have been reasonably separated at all but the highest slownesses. However the horizontal component sections in the rotated reference frame still contain P-p energy and converted wave energy.

 

rotated
Figure 5 Nine component gather after source and receiver rotation by $20^\circ$. The two shear wavetypes have been totally separated at zero slowness and approximately separated at other slownesses.

Figure  shows the constant velocity stack panels for the X-x and Z-z components of the data. There is coherent energy at three separate places on the panel. These correspond to the different reflection events, P-P, P-s/S-p and S-s.

 

vel-rot
Figure 6 Constant velocity stack panels for the rotated data. The top panel is the X-x component and the bottom panel is the Z-z component.

The next stage of the separation process is to estimate the P/S₁ separation coefficients for the receiver operator. Figure  is a contour plot of the ``F2'' objective function for a range of separation coefficients. The values of the coefficients at the minimum of this plot were used in a receiver separation operator that was applied to the synthetic data. The resulting gather is displayed in Figure . The first column of this plot should now correspond to an ``S₁'' receiver. Therefore there should be no upcoming P-waves visible on the first column. Similarly the last column should now correspond to a ``P-receiver'' there are no upcoming S-waves visible in this column. S-p waves are still visible in the bottom right section because the Z source generates both P- and S-waves. Figure  shows the constant velocity stacks after receiver separation. There is almost no energy at the S-s wave velocity on the Z-p velocity panel and no energy at the P-p wave velocity on the X-s₁ velocity panel.

 

rec-obj Figure 7 Contour plot of the F2 objective function for the receiver separation parameters.

A similar P/S₁ separation operator was estimated for the source operator and applied to the data. The result is shown in Figure . Although the separation is not complete at the highest slownesses the P- and S₁-waves have been completely separated for a wide range of slownesses around zero slowness. In particular the separation of the two converted wavetypes has been particularly successful, compare the original data in Figure  with the final separation result in Figure . The regions where the separation is unsuccessful correspond the to regions in which the first order approximation breaks down. In this example I have not yet applied the final stage of the process, the P/S₂ wave separation.

 

rec-sep
Figure 8 Nine-component gather after application of the receiver P/S₁ operator. The upcoming P waves have been removed from the first column and the upcoming S₁-waves have been removed from the third column.

 

vel-recsep
Figure 9 Constant velocity stack panels for the data after receiver P/S₁ separation. The top panel is the X-s₁ component and the bottom panel is the Z-p component. There is little upcoming P-wave energy on the X-s panel and little upcoming S₁-wave energy on the Z-p panel.

 

src-sep
Figure 10 Nine component gather after application of the source and receiver P/S₁ operators. The downgoing P waves have been removed from the first row and the downgoing S₁-waves have been removed from the third row.

 

vel-srcsep
Figure 11 Constant velocity stack panels for the data after source and receiver P/S₁ separation. The top panel is the S₁-s₁ component and the bottom panel is the P-p component. The separation is not complete but most of the unwanted energy has been removed.