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 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.
Source and receiver rotation by 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.
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.
The next stage of the separation process is to estimate the P/S_{1} 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 ``S_{1}'' 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-s_{1} velocity panel.
rec-obj
Figure 7 Contour plot of the F2 objective function for the receiver separation parameters. |
A similar P/S_{1} 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 S_{1}-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/S_{2} wave separation.