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Image and physical space

In this chapter, I compute discontinuities of a seismic image, not discontinuities of the depicted subsurface. The computations are based on the set of image pixels and do not directly relate to the physical subsurface that the pixels represent. Consequently, partial derivatives indicate change per sample and not change per physical unit, such as meters or seconds. Two differently sampled images of a single physical field f(x,y,z) will result in two different discontinuity images.

The pixel space is defined by the uniform samples of a given discrete image. Its variables count samples. The vector space has the natural norm . Convolutions such as expression (2) conveniently compute approximations of the partial derivatives of the image. All derivative expressions share the unit amplitude change over sample. The gradient magnitude expression (1) combines the partial derivatives of the sample space.

A change of variables relates the pixel coordinates to the underlying physical coordinates (x,y,z):
where (x0, y0, z0) are the axes offsets and (dx, dy, dz) are the axes increments between samples. The partial z-derivative of the physical space and the pixel space are related by  
where f is a physical space function and is the corresponding pixel space function.

The physical space is more intractable than the pixel space. For example, the gradient magnitude of a time-migrated image in pixel space is well defined by expression (1). The same gradient magnitude cannot be formed in the corresponding physical space, since the units of the partial spatial derivatives and the partial temporal derivative differ.

To compute the gradient magnitude of the physical space, a time-migrated image would have to be converted from time to depth. A true time-depth conversion requires a subsurface velocity model and the combination of an inverse time migration and a depth migration. Such a costly transformation is unsuitable for a mere discontinuity attribute computation. An alternative pseudo-depth conversion, , is a change in variables and leads to a partial derivative weight

where ft is the time-migrated and fz is the pseudo-depth function. v is a constant approximate subsurface velocity. The expression

links the partial z-derivative of the time migrated pixel function to the partial z-derivative of the pseudo-depth pixel space .

Seismic images are usually oversampled in time (small dt), and barely sampled adequately in space. Consequently, the time-depth conversion factor dz/dt usually increases the weight of the pixel z-derivative. In the case of the Gulf salt dome image of Figure 7, the relation between the two partial derivatives is

since dz = 0.05 km, dt = 0.004 s, v = 3.0 km/s.

Figure 16 shows the gradient magnitude image of the first seismic test case computed in pixel time space. Figure 17 shows the corresponding magnitude image based on the pseudo-depth space. Both images show little difference, except that the salt body appears noisier in the unweighted image.

In general, I choose the unphysical pixel space to compute this chapter's discontinuity attributes. Why? The computation of a discontinuity attribute is not based on a physical process and it makes little sense to talk about its correctness. Its usefulness results purely from its interpretational clarity and its computational simplicity.

In my experience, a large weight of the temporal pixel derivative decreases the quality of the discontinuity map. I believe the deterioration is due to the importance of the horizontal derivative terms in annihilating the prevalent horizontal sedimentary layers. Empirically, the rather successful discontinuity attribute of Figure 23 demonstrates the importance of horizontal comparison. The attribute is horizontal correlation.

Furthermore, the pixel space measurements are conceptually and computationally simple. Traditionally, enhancement techniques in the image processing literature use the same pixel space approach Castleman (1996).

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