Hunting for microseismic reflections using multiplets |

. | (1) |

We solve this by minimizing the objective function

(2) |

subject to the constraint

. | (3) |

Let and be two spherical surface coordinate parameters over which we will minimize. Taking partial derivatives of the constraints yields:

(4) |

and

, | (5) |

which says that is perpendicular to the two partial derivatives. Next, taking partial derivatives of the sum of squares expression gives

(6) |

and

. | (7) |

Therefore is also perpendicular to both partial derivatives and consequently must be parallel to . This means that

, | (8) |

where is the eigenvalue of the matrix that will make the least squares expression a minimum. Transposing we get

, | (9) |

which is a classic eigenvector problem for the matrix . Since the right singular vectors of are the same as the eigenvectors of , we used the LAPACK routine

Hunting for microseismic reflections using multiplets |

2012-05-10