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Here, we extend prediction filter theory to N plane waves. Assume that the
data have N plane waves with different dipping angles, the dataset can then be
expressed by
| |
(22) |
Fourier transform along the time axis will give us
| |
(23) |
where .
Trace is represented by
| |
(24) |
where propagator .Assuming trace is known, trace W(f,x) can
be predicted by a N points prediction filter
| |
(25) |
Inserting equation (23) into equation (25)
| |
(26) |
For each
| |
(27) |
Equation (27) can be expressed in matrix form
| |
(28) |
Equation (28) is a Van der Monde system. This system guarantees that
there is one solution. So every is a function of .
This means that the prediction filter relies on frequency in the case of
frequency-independent grid; whereas in the case of frequency-dependent
grids, we can still get a prediction filter which is independent from
frequency.
Next: Estimating the prediction filter
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Stanford Exploration Project
11/11/1997