Fractal heterogeneities in sonic logs and low-frequency scattering attenuation |

The energy spectrum,
, of von Kármán's autocorrelation
function in equation 7 is real and even:

(25) | |||

(26) |

Values of defined by equation 22 are

(27) | |||

(28) |

Coefficient , defined by equation 8, is an increasing function of exponent and has to be calculated numerically, except for some specific values:

The dispersion relation of equation 21 solves for an explicit solution of attenuation and dispersion:

(29) | |||

(30) |

When , the derivation produces simple expressions as detailed in Appendix B. The use of with the Kramers-Krönig relation can be used to determine the real part of . In the context of the second-order approximation, scattering attenuation in a von Kármán isotropic medium is

(31) |

For , the scattering attenuation reduces to the Rayleigh diffusion regime:

(32) |

Fractal heterogeneities in sonic logs and low-frequency scattering attenuation |

2013-03-02