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# Azimuthal anisotropy at Valhall: The Helmholtz equation approach

Version of Record online: 8 JUN 2013

DOI: 10.1002/grl.50447

©2013. American Geophysical Union. All Rights Reserved.

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#### How to Cite

2013), Azimuthal anisotropy at Valhall: The Helmholtz equation approach, Geophys. Res. Lett., 40, 2636–2641, doi:10.1002/grl.50447.

, , , , , and (#### Publication History

- Issue online: 3 JUL 2013
- Version of Record online: 8 JUN 2013
- Accepted manuscript online: 6 APR 2013 12:00AM EST
- Manuscript Accepted: 3 APR 2013
- Manuscript Revised: 2 APR 2013
- Manuscript Received: 22 FEB 2013

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Additional supporting information may be found in the online version of this article.

Filename | Format | Size | Description |
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2013GL055689R.fs01.pdf | PDF document | 982K | Schematic representation of the phase velocity computation (at station 595). A) The discrete travel times are spatially located at their corresponding stations. B) The travel times are interpolated onto a 50 m X 50 m regular grid and the travel time surface is cropped at the distances where the measurements become too sparse. C) The spatial gradient of the travel time surface is computed to give an estimate of the local phase slowness. D) The discrete amplitude data are interpolated onto the same grid than the travel times and the Laplacian-of-the-amplitude term of Equation 1 is computed and removed from the gradient term to give E) the distribution of local phase velocity as the magnitude of the vectors and F) the distribution of the local direction of wave propagation (black arrows, only every 5th arrows are shown for the clarity of the figure) as the direction of the vectors. The background color of frame F) shows the difference between the local direction of wave propagation and the straight ray approximation between the central station and each point of the grid. |

2013GL055689R.fs02.pdf | PDF document | 291K | A) 1psi azimuthal anisotropy fast direction and amplitude map at 0.7 s measured with the Helmholtz equation. B) 1psi azimuthal anisotropy fast direction and amplitude map at 0.7 s measured with average phase velocities. |

2013GL055689R.fs03.pdf | PDF document | 794K | A) 3psi azimuthal anisotropy fast direction and amplitude map at 0.7 s measured with the Helmholtz equation. B) 3psi azimuthal anisotropy fast direction and amplitude map at 0.7 s measured with average phase velocities. |

2013GL055689R.fs04.pdf | PDF document | 348K | A) 4psi azimuthal anisotropy fast direction and amplitude map at 0.7 s measured with the Helmholtz equation. B) 4psi azimuthal anisotropy fast direction and amplitude map at 0.7 s measured with average phase velocities. |

2013GL055689R.fs05.pdf | PDF document | 160K | Azimuthal distribution of the phase velocity at 0.7 s for the station 1390 measured with average phase velocities. The small blue dots are the average phase velocity measurements. The large red dots with error bars are the average phase velocity averaged over 20 degrees. The thick red curve is the best fits for the mean velocity, 1psi, 2psi, 3psi and 4psi azimuthal parameters for the averaged velocity measurements. The values of the fitted parameters are shown with the subscript ‘av’. For comparison, the gray dots show the Helmholtz tomography measurements, the black dots with error bars are the Helmholtz measurements averaged in 20 degrees bins and the black curve is the best fit for the mean velocity, 1psi, 2psi, 3psi and 4psi azimuthal parameters. The values of the fitted parameters are shown. Note that the phase velocity measurements are less scattered than for the Helmholtz approach. For the average phase velocity method, we need to solve for the initial phase ambiguity. To remove it, we force the linear regression of the measured travel-times vs inter-station distance to pass by the origin point (Distance, time) = (0,0) by removing the y-intercept value. |

2013GL055689R.fs06.pdf | PDF document | 94K | Distributions of the A, B, C and D parameters (in percent). The number in each frame is the median value of each parameter (note that the C and D values are in average about twice smaller than the A and B parameters). |

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