data=signal + noise

 

noise
Figure 1 In correlation traces, there is a ``noise peak'' due to the direct arrival, and a ``signal peak'' from the reflection of the direct wave.

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 $\tau - p$ 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, $\theta$, reflecting at the surface and then again on a subsurface reflector to emerge at an angle $\theta_e$. The following development is for constant velocity.

 

angles
Figure 2 Thick lines are reflectors. Dashed lines define the normal to a reflector. Solid lines are rays. Dot-dashed lines are vertical references. Tree-like items are trees. $\theta_i$ corresponds to the incident plane wave. $\alpha$ is the incidence angle from the normal of the reflector plane. $\phi$ is geologic dip. $\theta_e$ is the emergence angle that will define the ray-parameter of the signal. $\alpha_o$ shows that for $\phi = 0$,$\theta_e = \theta_i$ in constant velocity.

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

 

$$\theta_i \:=\: \theta_e \:+\: 2 \phi \:.$$ (1)

So by transforming the correlation volume into $\tau - p$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.