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![]() | Time migration velocity analysis by velocity continuation | ![]() |
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The change of variable transforms
equation (1) to the form
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t2
Figure 7. Synthetic seismic data before (left) and after (right) transformation to the ![]() |
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Figure 7 shows a simple synthetic model of seismic
reflection data generated from the model in Figure 2
before and after transforming the grid, regularly spaced in , to a
grid, regular in
. The left plot of Figure 8
shows the Fourier transform of the data. Except for the nearly
vertical event, which corresponds to a stack of parallel layers in the
shallow part of the data, the data frequency range is contained near
the origin in the
space. The right plot of Figure
8 shows the phase-shift filter for continuation from
zero imaging velocity (which corresponds to unprocessed data) to the
velocity of 1 km/sec. The rapidly oscillating part (small frequencies
and large wavenumbers) is exactly in the region, where the data
spectrum is zero. It corresponds to physically impossible reflection
events.
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t2-fft
Figure 8. Left: the real part of the data Fourier transform. Right: the real part of the velocity continuation operator (continuation from 0 to 1 km/s) in the Fourier domain. |
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The described algorithm is very attractive from the practical point of view because of its efficiency (based on the FFT algorithm). The operation count is roughly the same as in the Stolt migration implemented with equation (4): two forward and inverse FFTs and forward and inverse grid transform with interpolation (one complex-number transform in the case of Stolt migration). The velocity continuation algorithm can be more efficient than the Stolt method because of the simpler structure of the innermost loop (step 4 in the algorithm).
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![]() | Time migration velocity analysis by velocity continuation | ![]() |
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