The U/D imaging concept says that reflectors exist in the earth at places where the onset of the downgoing wave is time-coincident with an upcoming wave. Figure 17 illustrates the concept.
udimage
Figure 17 Upcoming and downgoing waves observed with buried receivers. A disturbance leaves the surface at t = 0 and is observed passing the buried receivers G_{1}...G_{5} at progressively later times. At the depth of a reflector, z_{3}, the G_{3} receiver records both the upcoming and downgoing waves in time coincidence. Shallower receivers also record both waves. Deeper receivers record only D. The fundamental principle of reflector mapping states that reflectors exist where U and D are time-coincident. (Riley) |
It is easy to confuse the survey-sinking concept with the U/D concept because of the similarity of the phrases used to describe them: ``downward continue the shots'' sounds like ``downward continue the downgoing wave.'' The first concept refers to computations involving only an upcoming wavefield U(s,g,z,t). The second concept refers to computations involving both upcoming U(x,z,t) and downgoing D(x,z,t) waves. No particular source location enters the U/D concept; the source could be a downgoing plane wave.
In profile migration methods, the downgoing wave is usually handled theoretically, typically as an impulse whose travel time is known analytically or by ray tracing. But this is not important: the downgoing wave could be handled the same way as the upcoming wave, by the Fourier or finite-difference methods described in previous chapters. The upcoming wave could be expressed in Cartesian coordinates, or in the moveout coordinate system to be described below.
The time coincidence of the downgoing and upcoming waves can be quantified in several ways. The most straightforward seems to be to look at the zero lag of the cross-correlation of the two waves. The image is created by displaying the zero-lagged cross-correlation everywhere in (x,z)-space.
The time coincidence of the upcoming wave and the earliest arrival of a downgoing wave gives evidence of the existence of a reflector, but in principle, more can be learned from the two waves. The amplitude ratio of the upcoming to the downgoing wave gives the reflection coefficient.
In the Fourier domain, the product represents the zero lag of the cross-correlation. The reflection coefficient ratio is given by .This ratio has many difficulties. Not only may the denominator be zero, but it may have zeroes in the wrong part of the complex plane. This happens when the downgoing wave is causal but not minimum phase. The phase of the complex conjugate of a complex number equals the phase of the inverse of the number. Thus the ratio U/D and the product both have the same phase. It seems you can invent other functional forms that compromise the theoretical appeal of U/D with the stability of .
Don C. Riley [1974] proposed another form of the U/D principle, namely, that the upcoming waves must vanish for all time before the first arrival of the downgoing wave. Riley's form found use in wave-equation dereverberation.