Replacement-medium concept of multiple suppression

In seismology wavelengths are so long that we tend to forget it is physically possible to have a directional wave source and a directional receiver. Suppose we had, or were somehow able to simulate, a source that radiated only down and a receiver that received only waves coming up. Then suppose that we were somehow able to downward continue this source and receiver beneath the sea floor. This would eliminate a wide class of multiple reflections. Sea-floor multiples and peglegs would be gone. That would be a major achievement. One minor problem would remain, however. The data might now lie along a line that would not be flat, but would follow the sea floor. So there would be a final step, an easy one, which would be to upward continue through a replacement medium that did not have the strong disruptive sea-floor reflection coefficient. The process just described would be called impedance replacement. It is analogous to using a replacement medium in gravity data reduction. It is also analogous to time shifting seismograms for some replacement velocity.

The migration operation downward continues an upcoming wave. This is like downward continuing a geophone line in which the geophones can receive only upcoming waves. In reality, buried geophones see both upcoming and downgoing waves. The directionality of the source or receiver is built into the sign chosen for the square-root equation that is used to extrapolate the wavefield. With the reciprocal theorem, the shots could also be downward continued. Likewise shots physically radiate both up and down, but we can imagine shots that radiate either up or down, and mathematically the choice is a sign. So the results of four possible experiments at the sea floor, all possibilities of upward and downward directed shots and receivers, can be deduced.

Extrapolating all this information across the sea-floor boundary requires an estimate of the sea-floor reflection coefficient. This coefficient enters the calculation as a scaling factor in forming linear combinations of the waves above the sea floor. The idea behind the reflection-coefficient estimation can be expressed in two ways that are mathematically equivalent:

• The waves impinging on the boundary from above and below should have a cross-correlation that vanishes at zero lag.
• There should be minimum power in the wave that impinges on the boundary from below.

After the geophones are below, you must start to think about getting the shots below. To invoke reciprocity, it is necessary to invert the directionality of the shots and receivers. This is why it was necessary to include the auxiliary experiment of upward-directed shots and receivers.

EXERCISES:

1. Refer to Figure 14.

   a.

   What graphical measurement shows that the interval velocity for simple sea-floor multiples equals the interval velocity for peglegs?

   b.

   What graphical measurements determine the sediment velocity?

   c.

   With respect to the velocity of water, deduce the numerical value of the (inverse) Snell parameter p.

   d.

   Deduce the numerical ratio of the sediment velocity to the water velocity.

2. Consider the upcoming wave U observed over a layered medium of layer impedances given by ( I₁ , I₂ , I₃ , ... ), and the upcoming wave U' at the surface of the medium ( I₂ , I₂ , I₃ , ... ). Note that the top layer is changed.

   a.

   Draw raypaths for some multiple reflections that are present in the first medium, but not in the second.

   b.

   Presuming that you can find a mathematical process to convert the wave U to the wave U', what multiples are removed from U' that would not be removed by the Backus operator?

   c.

   Utilizing techniques in FGDP, chapter 8, derive an equation for U' in terms of U, I₁, and I₂ that does not involve I₃, $I_4 \ ...$.