** Next:** RATE OF CONVERGENCE
** Up:** Jedlicka: Cascaded normal moveout
** Previous:** PARTIAL NORMAL MOVEOUT

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 *N*^{th} 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