Figure 1

Figure 1 shows the simple model of a plane wave emergent on the
the surface layer and then exciting what we will call a source wavelet
that can be used to image reflectors at depth. As the correlated
traces in panel **(c)** show however, the output traces
will have correlation peaks at lags corresponding to the arrival of
the direct wave and the reflection. For the purpose of subsurface
imaging then, we will refer to the direct wave correlation as noise,
*n*, and the reflection correlation as signal, *s*. The signal will
have hyperbolic move out while the noise will exhibit linear move out.

Because of this fact, reflection energy will have parabolic shape in the domain while the linear incident waves will coalesce to a point, that may be buried within reflection parabolas that will prevent a simple mute. However, if we were able to find the points corresponding to the direct waves for every event, we could make a noise model to use in an adaptive subtraction scheme.

From Figure 1 one can see that every distinct
incidence plane wave of ray-parameter *p*_{i} will contribute to the
imaging of a reflector with the offset shown. If the incident wave is
less vertical, the reflection ray will emerge past the second
receiver. This leads us to the conclusion that all energy due to
reflections has uniquely parameterized direct rays
that excited it. Figure 2 shows the geometry of a
single incoming plane wave, defined by its ray's angle to the vertical,
, reflecting at the surface and then again on a subsurface
reflector to emerge at an angle . The following development
is for constant velocity.

Figure 2

Therefore, we would like an expression for the angle of the direct wave as a function of the emergence angle of the reflected wave, . By inspection of the of the geometry, that relation is:

(1) |

So by transforming the correlation volume into space, parabolic summations that would correspond to an event should
be mappable back to its source plane wave. This kinematic mapping can
then act as the training model for a PEF estimation that is the operator *N ^{-1}* for the
coherent noise attenuation exercise outlined as the subtraction
method in Guitton et al. (2001). Implementation of this
methodology will follow presently.

Rays traveling close to 90^{o} from the azimuth of a
recording line will not result in a recorded reflection along the same
line. However, because it will arrive at all the receivers in the
line at nearly the same time, these events should be easy to separate
with a simple velocity dependent mute.

6/8/2002