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Up: Jedlicka: Cascaded normal moveout
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Let us write
| |
(15) |
for the nontruncated case and
| |
(16) |
for the truncated case.
We will estimate the difference and show that this difference tends to zero as .We can write
| |
(17) |
| (18) |
| (19) |
As we did in the previous section, we can rewrite this equation into a
system of equations:
| |
(20) |
| (21) |
| (22) |
| (23) |
By summing up all these equations we get
| |
(24) |
Now we need to prove that there exists a constant K independent of N such that
| |
(25) |
for all .From equation (16) we can see that
| |
(26) |
Similarly from equation (24) we can see that
| |
(27) |
Putting both equations together we have
| |
(28) |
or
| |
(29) |
where the last equation follows from equation (14).
If we define
| |
(30) |
then from equation (24) we have
| |
(31) |
which tends to zero as .This proves that the Nth partial normal moveout applied N times is equal
to the normal moveout for large N.
The convergence of cascaded NMO to a semicircle can be clearly seen
on Figure a. The proof of convergence holds for .
The behavior for x/v = t can be seen from sequences at the sides of
Figure b. The figure convinces us that there is convergence even
for x/v=t, but that it is extremely slow.
Next: RATE OF CONVERGENCE
Up: Jedlicka: Cascaded normal moveout
Previous: PARTIAL NORMAL MOVEOUT
Stanford Exploration Project
1/13/1998