Example

Figure landCmp shows a CMP gather from a land seismic survey, and a time slice through several CMP gathers. There is a visible ``noise cone'' defined by some low velocity. Inside the cone there is very little coherency along the midpoint axis.

As an experiment with noisy data interpolation, I reduced the data to every second CMP, then performed NMO and stack to produce the top left panel of Figure stacks. Then I further reduced the data to every fourth CMP and interpolated with and without noise estimation, and again did NMO and stack to try to reproduce the stack in the top left of Figure stacks. The top right panel, the result without any noise estimation, has significant problems with reflector continuity. In particular, strong reflectors in the center of the section, at 1.2 seconds and at .9 seconds, have undesirable gaps in the midpoint direction. The bottom left stack, which is really a stack of the interpolated and extracted signal $\bold s$,is a much more coherent stack, but is slightly lower in temporal frequency content. The bottom right stack is the same, after matched filtering to restore the temporal spectrum. The spectra of the original (top left) stack and the interpolated stack without matched filtering (bottom left) are shown in the left side of Figure specs. After matched filtering, the interpolated, noise suppressed stack has the same temporal frequencies as the stack of the original data; without the filtering step, it is boosted at low frequencies and suppressed at high frequencies. The right side shows the spectra of the original (top left) stack and the interpolated, matched filtered stack (bottom right). The effects of signal and noise separation are shown in Figure outCmp. The left panel shows the signal component estimated from one of the input CMP gathers, the right shows signal component for an interpolated CMP gather.

 

landCmp
Figure 6 Land seismic data. Left panel shows a CMP gather, right shows a time slice through several CMP gathers. The data are incoherent along the midpoint axis, close to zero offset.

 

stacks
Figure 7 Land data interpolation. Top left panel is a stack of the known data. Top right panel is the stack after throwing away half the CMP gathers and reinterpolating them without attempting to compensate for noise. Bottom left is the stack of the signal component extracted from the known data and interpolated into the missing data. Bottom right is the same stack after matched filtering to get back the temporal frequency content of the known data.

 

outCmp
Figure 8 Interpolation and noise removal. Left side shows the estimated signal component of an input CMP gather, right side shows the signal component of an interpolated gather.

 

specs
Figure 9 Spectra of stacked data. Left panel shows spectra of the original stack and the stack of the interpolated data, without matched filtering. Right panel shows spectra of the original stack and the matched filtered stack. Solid lines represent original data.