In the normal incidence case of Backus averaging, the layer properties of thickness, , compliance, , and mass density, , are replaced by the corresponding properties of the equivalent homogeneous medium, , ,and :

(1) | ||

(2) | ||

(3) |

that is, the thickness of is the sum thickness of the layers, , and the equivalent medium mechanical properties are the thickness-weighted averages of those of the layered medium.

However, these layers can also be described in terms of the layer properties of one-way travel-time, , and impedance, , with slowness, acting as the means for changing the independent variable between depth and time. It is well known that:

(4) | ||

(5) |

and from these:

(6) | ||

(7) |

and thus:

(8) |

but , so:

(9) |

and by similar reasoning the slowness equivalent:

(10) |

and, since = :

(11) |

4/20/1999