Mathematical Geosciences

, Volume 41, Issue 2, pp 163–192 | Cite as

Anisotropic Mean Traveltime Curves: A Method to Estimate Anisotropic Parameters from 2D Transmission Tomographic Data

  • Juan L. Fernández MartínezEmail author
  • Luis M. Pedruelo González


We present the mathematical deduction and properties of the mean traveltime curves for homogeneous elliptical anisotropic media. These curves generalize their isotropic counterparts which have been introduced in the past as a simple data quality analysis technique at the pre-inversion stage for 2D transmission experiments, allowing the inference of prior velocity models to gain stability at the tomographic inversion. Also, the anisotropy parameters (maximum velocity, anisotropic direction and ratio) are shown to affect the shape of these curves. The degree of asymmetry of the anisotropic mean traveltime curves (displacement of the mean time and standard deviation minima from the middle of the gathering line) is related to the direction of anisotropy which can then be visually estimated. Least squares’ fitting of the anisotropic theoretical models to their experimental counterparts is an effective method to estimate at the pre-inversion stage a macroscopic elliptical anisotropic velocity model, valid at the scale of the experiment, and able to match the experimental mean traveltime distribution.

Sensitivity analysis has shown that the mean curve is less prone to errors than the standard deviation curve. Parameter identification from the standard deviation curve becomes unstable for noise levels higher than 5%; data errors produce smearing of the value of the estimated anisotropy ratio and wrong directions of anisotropy biased towards zero degrees. Also, identification from the mean traveltime curve becomes stable when the maximum velocity is well constrained. Finally, this methodology is illustrated with the application to the Grimsel data set. Performing MTC analysis is always recommended since it does not need high numerical requirements, and as shown in the sensibility analysis section, errors in data can be misinterpreted as geological anisotropies.


Inverse problems Transmission tomography Mean traveltime curves Weak elliptical anisotropy 


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Copyright information

© International Association for Mathematical Geosciences 2008

Authors and Affiliations

  • Juan L. Fernández Martínez
    • 1
    Email author
  • Luis M. Pedruelo González
    • 1
  1. 1.Departamento de MatemáticasUniversidad de OviedoOviedoSpain

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