Migrating FEAVO-affected data

I will use in my heuristic a constant background velocity model with a FEAVO-causing heterogeneity and a single horizontal reflector of reflectivity one. Since the small size of the heterogeneity localizes the effects it produces, it makes possible to indicate only the presence of focusing in a binary fashion (yes/no). The heterogeneity can be thought of either causing absorption or lens-like focusing by a small amount.

The presence of focusing will be reported to the appropriate midpoint, not to the receiver location. This is done because the midpoint coordinate is orthogonal to the offset coordinate in the prestack data space.

The figures below feature straight rays which may seem to indicate constant velocity. However, this takes place only for ease of drawing; the reasoning that follows does not require constant velocity. I will also consider a single signed offset h. The mental experiments to be performed below will be identical both for any other offset and for the prestack dataset taken together.

Figures

 

f1 Figure 1 Single offset seismic experiment - part 1: propagation from sources to reflector, and graph with focusing at the reflector

and

 

f2 Figure 2 Single offset seismic experiment - part 1: graph with focusing at the reflector (bottom), propagation from the reflector to the receivers, and graph with focusing as recorded by receivers (top)

show a seismic experiment decomposed into two separate steps. In the first step (Figure ), the wavefield propagates from the sources to the reflector. Only a single midpoint local area is affected by focusing. In the second step (Figure ), the wavefield propagates to the surface. Already existing focusing is preserved, and a new pass through the heterogeneity causes a second focusing area to appear. The focusing areas detected by the receivers at the surface are reported at the appropriate midpoints (not receiver locations) in the upper graph. The two midpoint local areas in which focusing is present are located at the intersection of the two arms of a midpoint-offset ``Kjartansson V'' Kjartansson (1979) with a line of constant offset h.

At this point one could make the argument that migration by definition recovers amplitudes at the reflector, and it cannot solve illumination problems, so effect 1 in Figure will not be removed by migration, and one should instead try a regularized inversion that will smooth out small irregularities. Let us however examine more closely what happens with the energy from that offset when we do shot-profile migration using all shots.

Downward continuation of the shots wavefields is properly described by Figure and will produce at the reflector, as expected, the same focusing as the real experiment. Downward continuing the receivers (Figure )

 

f3 Figure 3 Downward continuing the receivers in shot-profile migration: graph with focusing as recorded in the data (top), downward continuation with the correct velocity through the heterogeneity, and graph with residual focusing at the reflector (bottom)

with the correct velocity (which includes both the background and the heterogeneity) eliminates focusing area 2, but leaves focusing area 1 intact. What the objection stated in the previous paragraph failed to take into account, though, is the imaging condition (Figure

 

f4 Figure 4 The wavefields need only be similar, not uniform, in order for the imaging condition to produce a uniform-amplitude (no focusing) reflector.

). The imaging condition, taken as a black box, is not sensitive to the particular values of either the shot wavefield or the receiver wavefield, but to how close they are to each other. If at one location they are identical (for this idealized experiment we can use the word ``identical''), it reports a reflection there with a probability (a.k.a. reflection coefficient) of 1. In our case (Figure ) the shot wavefield and the receiver wavefield are identical in every point, and therefore the uniform value of 1 for reflectivity is recovered. Shot-profile migration with all sources therefore completely eliminates FEAVO from the image if the correct velocity is provided. Survey-sinking migration, being mathematically equivalent to it Biondi (2003), also eliminates FEAVO if the velocity is provided.