Aliasing and Shadow Zones

Seismic data are often insufficiently and irregularly sampled in space because of logistical and/or cost constraints on the data acquisition. The poor spatial sampling causes an incomplete illumination of the reflector that degrades the resolution of the image and may create artifacts. In some situations the artifacts can be avoided by applying well-established safeguards (e.g. antialiased migration) during the imaging process, but at the expenses of image resolution. In others, like the challenges presented by shadow zones in subsalt exploration, present imaging technology is still wanting. The development of solutions to the partial illumination problem is an active area of algorithmic research, and is likely to remain so in the near future. Chapter 9 in 3-D Seismic Imaging discusses the different cases in which poor illumination becomes a problem and presents several research directions that my students and I have explored to address the challenge.

Spatially aliased data

Spectrum of aliased data

The simplest case of spatial sampling problems are encountered when the data sampling is regular but too sparse; in this cases the data are spatially aliased. The two images on the left illustrate the concept of spatial aliasing of seismic data for a subset of a marine data set recorded in the Gulf of Mexico over a salt diapir. The image on the left shows a window of the data. The flat reflectors are sufficiently sampled, but the salt-flank reflections are clearly aliased. The image in the middle show the Fourier spectrum of the data window on the left. The vertical axis of the spectrum is the temporal frequency and the horizontal axis is the spatial wavenumber. The spectrum is replicated on both sides of the main spatial band to facilitate the visual identification of the aliased energy, which corresponds to the signal energy encircled by the blue-and-yellow ellipses.

Migrated images of aliased data

When imaging aliased data like the ones showed above, we face a trade-off between aliasing artifacts and resolution. We can avoid imaging artifacts by excluding from the imaging process the high-frequency and steeply dipping events that are undersampled in the data. However, this antialiasing process also decreases the image resolution. This trade-off is illustrated in the two panels shown on the right. The two panels compare the result of migrating the data without any antialiasing safeguard with the image obtained when antialiasing is applied. The image without antialiasing has strong artifacts. These artifacts are stronger in the shallow area and tend to decrease in amplitudes with depth, but they are present everywhere. Antialiasing successfully removed the artifacts, though at the expense of reducing the frequency contents of some coherent reflections.

3-D prestack data sets are multi offsets; that is, they are acquired with the distance between sources and receivers spanning a wide range from few hundred meters to more than 10 kilometers. Multi-offset data provide redundant illumination of the subsurface reflectors, because different data offsets record events that were reflected with different aperture angle in the subsurface. Near offsets correspond to reflection close to normal incidence, whereas long-offset data correspond to wide-aperture reflections. This data redundancy is routinely exploited for increasing the signal-to-noise ratio of images, as well as for estimating propagation velocity. The simplest way to exploit data redundancy is to average the images obtained from the different subsets (stacking). For example, common-offset cube can be independently imaged, but if each of them is irregularly and insufficiently sampled in space the partial images would be affected by artifacts. These artifacts can be attenuated by stacking and the stacked image is more interpretable than the offset cubes.

However, there are situations in which simple stacking is not sufficient to produce an interpretable image, or, in other cases, the interpretation of the partial images corresponding to different offsets or reflection angle may provide additional information on petrophysical properties of the subsurface, such as in Aperture Versus Offset analysis (AVO). When simple stacking is not sufficient, the redundancy in the data can be exploited by applying advanced data and/or image regularization techniques that are usually based on inversion theory. The routine application of these methods is hampered by their computational and implementation costs, but they are slowly making their way into industrial practices.

The following example shows an application of the data-regularization method presented by Vlad and Biondi (2002) . This method exploits the redundancy in the data domain between offset cubes. It interpolates the recorded data by solving a least-squares inverse problem that maximizes consistency between offset cubes after the application of Normal Moveout (NMO) and Azimuth Moveout (AMO), which make the kinematic of offset cubes consistent with each other. The NMO and AMO operators are discussed in Chapter 3 in 3-D Seismic Imaging.

Density map of a 3-D land data set

Migration depth slices before/after data regularization

The image on the left shows the data density (fold) as a function of the surface coordinates for a constant-offset cube of a land data set recorded in South America. Dark regions in the fold map represent area of poor coverage. The presence of dark obliques stripes in the fold map is an indication that the partial images from this data set will be strongly affected by artifacts. The movie on the right shows the effects of this poor sampling and the improvements achieved by applying data regularization. The two panels show depth slices (slices of the image cube taken at constant depth) for two different reflection-aperture angles: near angles on the left panel and far angles on the right panel. One frame of the movie shows the images obtained by migrating the original data and the other one shows the corresponding images obtained by imaging the regularized data. Two thin meandering channels are visible in the top-left and bottom-right corners of the image obtained by imaging after data regularization. These channels are overwhelmed by artifacts in the images obtained without data regularization.

The final example illustrates the most challenging illumination problem; it occurs when the reflectors are poorly illuminated not because of holes in the acquisition geometry, which are known from the data geometry, but because the seismic wavefield is strongly distorted by a complex velocity model, which is not known a priori and cannot be perfectly estimated from the data. In this cases, the migration and velocity estimation problems are tightly intertwined, and it is often impossible to unravel the effects of poor illumination from the effects of velocity errors. Furthermore, to apply inversion-based methods we must perform the regularization in the image space, and we thus must invert an operator that connects the image space with the data space. The application, and consequently the inversion, of such an operator is computationally more demanding than the application of operators defined in the data space by at least one order of magnitude.

Stacked section and ADCIG in shadow zone The shadow zone in the image are caused by salt edge

The images on the left show a common, and practically important, instance of poor illumination induced by a complex overburden. The reflectors below a complex salt body are only partially illuminated from surface data because the salt body, in particular the edge and the steeply dipping flanks, prevent some of the reflected energy to reach the surface. The image on the left shows the ``stack'' of the migration results and an Angle Domain Common Image Gather (ADCIG); that is, a section of the image cube that display reflectivity as a function of depth and the reflection-aperture angle. The vertical yellow line superimposed onto the stack indicates the location of the ADCIG. The holes in the middle of the angular range of the ADCIG are caused by the incomplete illumination of the reflectors. The image on the right show a snapshot of the wavefield, and of the corresponding rayfield, for a source located under the salt edge at the same horizontal location as the ADCIG. It clearly shows that the salt edge obstructs the wavepaths from the surface to the reflectors below the edge; some components of the wavefield are refracted downward, and never reaches the surface, creating the shadow zones identified in the images.

Comparison of migration and inversion results

Prucha and Biondi (2002) introduced an inversion-based method to improve the imaging results under the salt edge. Their method is based on the regularized inversion of the wave-equation operator connecting the reflectivity image to the surface data. This operator is singular, as demonstrated by the existence of the shadow zones. To prevent the inversion process from diverging, data redundancy is exploited by imposing smoothness across the aperture-angle axis in the image space. This image-regularization scheme was dubbed geophysical regularization, because this condition is consistent with our knowledge of the physics of reflections; i.e., that in the pre-critical angle region the reflection coefficient is a smooth function of the reflection angle.

The movie on the right demonstrates the effectiveness of the inversion method to reduce imaging artifacts. The left and middle panels show the results of migration (with and without a diagonal normalization factor), whereas the panel on the right show the image obtained by inversion. The movie flips between the images and the velocity model to facilitate the visual connection between the imaged reflectors and the discontinuities in the velocity model that created the reflections. Notice that the regularized inversion not only improves the imaging of ``true'' reflectors (indicated with yellow arrows), but also attenuates ``false'' reflectors (indicated with red arrows) that are coherent in the migrated images and thus could mislead the geological interpretation of the subsalt structure. It is likely that to achieve further improvements in the image we need to introduce additional constraints provided by a geological interpretation of the seismic images. Prucha and Biondi (2002) demonstrated the potential advantages of introducing a a geological regularization in addition to a geophysical regularization. However, the methodology for introducing geological knowledge into seismic imaging is a research area where first-order progress are needed.