Fitting with extrapolation filters

Independent source gathers from the same shot point have nearly identical ground roll power spectra. Their phases do not match precisely, and so some fitting is required to match the ground roll in the two gathers. Similar to the cross-equalization of Rickett 1997, here I use an extrapolation filter, which attempts to predict traces in one shot gather from the corresponding traces in another.

The filter estimation can be written simply as  

$${\bf d_2 \approx D_1 a} .$$ (1)

Where ${\bold d_2}$ is a trace or window from one shot gather and operator ${\bold D_1}$is convolution of the identical window from another shot gather with the filter ${\bold a}$.

Ground roll is typically the dominant energy in raw shot gathers, so basic (L₂) least squares is suitable to remove it, in contrast to numerous other recent SEP projects, where robust (L₁) solvers have been desired.

The natural output in this case has more than two dimensions, because comparing any two source impacts produces an output the size of a shot gather. Thus for the six recorded impacts there are fifteen possible output gathers.

In this example each impact is compared with the next in the recording order, so that equation (1) is really  

$$\left[ \begin{array} {c} {\bold d_2} \\ {\bold d_2} \\ {\... ...} \\ {\bold a_{45}} \\ {\bold a_{56}} \\ \end{array} \right]$$ (2)

There are more available impact record combinations; for instance a filter a₂₅ could be estimated. The plotted outputs are really the residual after applying the filters estimated by equation (2). To combine them, here I just square and sum the planes, and convolve with a first derivative wavelet to remove the zero frequency and restore the wavefield appearance. The results are shown, before and after the convolution, in Figure 4.

 

mineOut
Figure 4 Output from the mine data. Left panel is five residual panels squared and summed. Right panel is first derivative of left.