(50) |

(51) |

(52) |

An extrapolator in equation () will be stable if either decreases or remains constant with depth. Therefore for stable extrapolation,

(53) |

While the helical factorization begins with equation (), we really solve the implicit system,

(54) | ||

(55) |

(56) |

Equating equations () and () produces a formula for in terms of the helical factorization, :

(57) | ||

(58) | ||

(59) |

Bulletproof stability requires to be symmetric non-negative definite. For the helical factorization this matrix is given by,

(60) |

In the constant velocity case, the matrix represents a stationary filtering operation. Therefore the composite matrices, and commute with each other. Under this scenario, the matrix becomes the zero matrix, which clearly satisfies the non-negative definite criterion required for stability. Constant velocity extrapolation with equation () is therefore unconditionally stable.

Unfortunately, however, if the velocity varies laterally, and no longer commute with each other, and so the composite matrices and do not commute either. Consequently stable extrapolation cannot be guaranteed. Furthermore, there are no obvious steps we can take to ensure that equation () remains non-negative definite in areas of strong lateral velocity variations. In practice, equation () does indeed encounter stability problems in some areas. Section illustrates this problem with some examples.

Essentially the problem revolves around the fact that I factor and , rather than itself. We can ensure our factorization is symmetric non-negative definite, but not the extrapolator itself.

5/27/2001