LEAST-SQUARES INTERPOLATION

The forward modeling for sparsely sampled data can be formulated by applying a subsampling operator to the exploding reflector modeling operator as follows:

$${\bf d} = {\bf S_2S\tilde W m}$$ (8)

where data ${\bf d}$ and image ${\bf m}$ have different lengths in the space direction because of the subsampling operator ${\bf S_2}$,and the exploding reflector modeling operator ${\bf S\tilde W}$,the same operator as equation (2), can be any modeling operator such as Kirchhoff, phase-shift, or finite-difference. For a given sparsely sampled wavefield, we can estimate a dense image by minimizing the error between the given wavefield and the wavefield modeled by the modeling equation(7), which is as follows:

$$\min_{\bf m} \vert\vert {\bf d - S_2 S \tilde Wm} \vert\vert$$ (9)

Experiments were done using the Kirchhoff and phase-shift operators. For a simple syncline reflector shown in Figure (a), a synthetic model was generated using the Kirchhoff modeling operator and the subsampled by zapping every other trace. The result of modeling is shown in Figure (b). By applying the conjugate operator, ${\bf W'S'}$, to the subsampled data, we get an interpolated image, as shown in Figure (c). It is very difficult to distinguish the effect of the subsampling operator from the artifacts of the Kirchhoff operator, but we can see that this image is very similar to the image obtained without the subsampling operator shown in Figure (c). The image obtained by the least-squares interpolation is shown in Figure (d); it closely resembles the image obtained from the data without subsampling.

Another experiment done using the Gazdag operator is shown in Figure . This result also closely resembles that for the images without subsampling. The reason may be that the wavefront healing effect during the extrapolation.

 

Kirint
Figure 10 Least-squares Kirchhoff interpolation: (a) The original image used to generate the subsampled wavefield shown in Figure (b), (b) The subsampled wavefield. (c) The interpolated subsurface image obtained by Kirchhoff migration with a subsampling operator, (d) The interpolated subsurface image obtained by least-squares Kirchhoff interpolation (after 10 iterations).

 

Gazint
Figure 11 Least-squares interpolation with Gazdag operator: (a) The original wavefield generated by Gazdag modeling for the synthetic model shown in Figure (a), (b) The subsampled wavefield obtained by zapping every other trace in Figure (a), (c) The interpolated subsurface image obtained by Gazdag migration, (d) The interpolated subsurface image obtained by least-squares Gazdag migration (after 10 iterations).