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# PROOF FOR 15-DEGREE PARTIAL NORMAL MOVEOUTS

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