Canales' 1984 implementation is in the f-x domain, so he makes the implicit assumption that the data is time-stationary, i.e., that each trace is the convolution of a single time-invariant wavelet with the earth's random reflectivity sequence. Computationally, this approach is very efficient. However, since ground roll is highly dispersive, and thus nonstationary, we instead choose a t-x method utilizing nonstationary PEF's Claerbout (1998a); Clapp et al. (1999); Crawley (1999).
Consider the recorded data to be the linear superposition of coherent signal plus coherent noise:
(1) |
Also assume that both the signal and noise are predictable, i.e., made up of of one or more local plane wave segments. The prediction error (residual) of the convolution of the signal and the noise with the corresponding nonstationary PEF's and is then uncorrelated. Writing these ideas as convolutional ``fitting goals'', we have
(2) |
Rewriting the constraint equation (1), , allows us to eliminate from equation (2), and suggests a regularized least squares optimization problem for the unknown signal
(3) |
We now consider some issues involving the calculation of the nonstationary signal and noise PEF's, and .
(4) |
The noise model should be a first-order estimate of the noise - if it was perfect, the job would already be done. Ideally the noise model should contain the spatial correlation of all noise events, which will realistically differ in amplitude from the actual noise by a small, arbitrary time- and space-variant scale factor. Of course, a small amount of signal will contaminate the noise model, but as the ground roll will have much higher amplitude than any embedded signal, we assume that the L_{2} PEF estimation of will effectively ignore the embedded signal, though this is not always the case.
If the ground roll and signal are sufficiently well-separated in temporal frequency, application of a lowpass filter to the data may produce a satisfactory noise model. The degree to which the noise and signal are separated is dependent on many variables - geography, depth to target, geology, and source type, to name a few. Similarly, if a hyperbolic Radon transform focuses primaries well, then it may be possible to mute the primaries in space, then inverse transform to obtain a noise model. We have tested both approaches, and find that lowpass filtering is the most reliable, although this assertion is data-dependant. Clapp and Brown (1999) applied an L_{1} iterative hyperbolic Radon transform to a multiple-infested 2-D seismic line, and found that the primaries were focused far better than for the usual L_{2} least squares DRT.
Using Spitz' 1999 choice for the signal PEF, , rewrite equation (3)
(5) |
(6) |
(7) |