A synthetic poststack data set generated from the standard 3D French model was chosen for
the initial application of MD.
This data set was used to calibrate the
parameters and determine the best way to apply migration deconvolution to a 3D data.
to reduce computation time, only a portion of the model data set
was used (Figure 1.2).
The migration and traveltime parameters are given in Table 1. In the
Kirchhoff migration program, obliquity, time derivative and geometry
spreading factors are included.
Poststack Kirchhoff migration images of the
French data are shown in Figures 1.3 and 1.4,
which correspond to the inline and crossline sections,
Table 1. The migration and traveltime parameters for the French model; where XORGR, YORGR and ZORGR are the origin coordinates of the migration image; DXRI, DYRI, DZRI are the image interval in X, Y and Z directions; NXRI, NYRI and NZRI are the number of image samples in X, Y and Z directions; XORGT and YORGT are the origin coordinates for the traveltime calculation; DXOT and DYOT are the traveltime intervals on the surface in X and Y directions; NXOT and NYOT are the number of traveltime grids on the surface in X and Y directions; DXRT, DYRT and DZRT are the traveltime intervals in the image cube; NXRT, NYRT and NZRT are the number of traveltime grids in the image cube.
The Kirchhoff migration image was used as input for migration deconvolution and the deconvolved migrated image is shown in the lower part of Figures 1.3 and 1.4. Compared with the migration image, the deconvolved migration images are noticeably improved in spatial resolution. However some artifacts are founded at the top of the image and the problem is likely caused by the traveltimes used in the calculation of the migration Green's function.
In these numerical experiments, the traveltimes are calculated by tetrahedral ray tracing, which involve an analytic ray tracing within a tetrahedralized parameterization of an unsmoothed velocity model. This method is efficient for generating traveltime, however it causes a problem in migration deconvolution. The rapid spatial variation in traveltimes causes the migration Green's function vary too faster in space than the variation of the lobes in the migration Green's function spectrum. This has been observed to poorly condition the matrix of linear equations and will introduce alias-like artifacts in the final MD image. To deal with this problem, both spatial smoothing and interpolation of the traveltime field was used.
For traveltime smoothing, a local averaging filter is applied to the traveltime field. Figure 1.5 shows MD images with smoothed traveltimes, where the smoothing filter is a 2D 5-point horizontal median filter followed by a 1D 4-point vertical averaging filter. In addition we find that interpolation is an effective method in smoothing traveltimes. Here traveltimes are interpolated from a coarse grid to a fine grid, which will smooth the traveltimes in space. Figure 1.6 shows the MD results using traveltime interpolation. In practice, the two methods are used together to ensure stability.
In order to closely examine the effect of migration deconvolution, Figure 1.7 shows a horizontal section indicated by the arrow on the LHS of Figure 1.2. The horizontal section cuts the half-sphere, and as shown in Figure 1.2 it is homogeneous inside the half-sphere, therefore the reflections should appear at the boundary of the sphere. Comparing of the Kirchhoff migration image with the MD image in Figure 1.7 shows that migration deconvolution improves the horizontal spatial resolution.
The French model test shows that migration deconvolution can improve the spatial resolution of the migrated image. The test also revealed that MD is sensitive to the spatial variation in traveltimes. To overcome this sensitivity, smoothing of the traveltimes is important for successful migration deconvolution. If the velocity model is not smoothed during ray tracing then smoothing should be applied to the computed traveltimes. For tetrahedral ray tracing in a complex velocity model, time interpolation and a smoothing filter are necessary steps for migration deconvolution.