Azimuthal Variation of Surface Wave Velocity


Program objectives

The surface-wave velocity is affected by the anisotropy, as it is well known. The azimuthal dependence of surface-wave propagation in a slightly anisotropic medium has been described by Smith and Dahlen (1973). Thus, with this hypothesis the problem of the azimuthal variation of surface-wave velocity can be simplified and easily solved (Corchete, 2012). If a slight anisotropy is considered, the azimuthal dependence of the surface-wave phase velocities can be written in the form

This azimuthal variation of surface-wave velocity can be obtained with the present program, as it will be described below.

Program description

The INVANS program and its data files are enclosed into a ZIP file named "Invans.zip". When you have got the ZIP file and you have uncompressed this file, you have three files named invans.exe, invans.dat and invans_mod.dat. The file invans.exe contains a program (in FORTRAN code for PC) for the computation of the azimuthal variation of surface-wave velocity. All the program capabilities are controlled by parameters enclosed in the file named invans.dat. The file invans.dat must be in the free format and must contain the parameters:

TMIN, TMAX, NT
NZ1, NZ2, NZ
THICK(1), ALPHA(1), BETA(1), RHO(1)
THICK(2), ALPHA(2), BETA(2), RHO(2)
...
...
...
THICK(NZ), ALPHA(NZ), BETA(NZ), RHO(NZ)

The description of all parameters is as follows:

TMIN, TMAX, NT = Period range (in seconds) in which the computations will be performed. TMIN is the minimum and TMAX is the maximum of this interval. The maximum value for NT is 100.
NZ = Number of layers of the isotropic earth model considered (maximum 100).
NZ1, NZ2 = Interval of layers in which the slight anisotropy is considered (maximum 99). The semi-infinite medium is always considered isotropic.
THICK(i), ALPHA(i), SVEL(i), RHO(i) = Thickness (km), P-wave velocity (km/s), S-wave velocity (km/s) and density (g/cm3); for the ith layer of the isotropic earth model considered. The NZ layer is the semi-infinite medium and its thickness must be given as zero.

The file invans_mod.dat must be in the free format and must contain the anisotropic model given by the parameters:

C1111(1), C1122(1), C1133(1), C1123(1), C1113(1), C1112(1)
C2222(1), C2233(1), C2223(1), C2213(1), C2212(1)
C3333(1), C3323(1), C3313(1), C3312(1)
C2323(1), C2313(1), C2312(1)
C1313(1), C1312(1)
C1212(1)
...
...
...
C1111(MZ), C1122(MZ), C1133(MZ), C1123(MZ), C1113(MZ), C1112(MZ)
C2222(MZ), C2233(MZ), C2223(MZ), C2213(MZ), C2212(MZ)
C3333(MZ), C3323(MZ), C3313(MZ), C3312(MZ)
C2323(MZ), C2313(MZ), C2312(MZ)
C1313(MZ), C1312(MZ)
C1212(MZ)

where MZ is NZ2-NZ1+1 the total number of layers with anisotropy. The notation used here for the elastic coefficients is detailed by Babuska and Cara (1991).

Running the program

Figures 1 to 10 show the azimuthal variation of surface-wave velocity resulting for the sample files invans.dat and invans_mod.dat. In this example, Love and Rayleigh waves have been considered. The phase velocity computed for the isotropic model, contained in the file invans.dat, is shown with dashed line in the following figures.
Fig. 1. Azimuthal variation of surface-wave velocity resulting for the sample files invans.dat and invans_mod.dat, computed for the fundamental mode.
Fig. 2. Azimuthal variation of surface-wave velocity resulting for the sample files invans.dat and invans_mod.dat, computed for the first higher mode.
Fig. 3. Azimuthal variation of surface-wave velocity resulting for the sample files invans.dat and invans_mod.dat, computed for the second higher mode.
Fig. 4. Azimuthal variation of surface-wave velocity resulting for the sample files invans.dat and invans_mod.dat, computed for the third higher mode.
Fig. 5. Azimuthal variation of surface-wave velocity resulting for the sample files invans.dat and invans_mod.dat, computed for the fourth higher mode.
Fig. 6. Azimuthal variation of surface-wave velocity resulting for the sample files invans.dat and invans_mod.dat, computed for the fifth higher mode.
Fig. 7. Azimuthal variation of surface-wave velocity resulting for the sample files invans.dat and invans_mod.dat, computed for the sixth higher mode.
Fig. 8. Azimuthal variation of surface-wave velocity resulting for the sample files invans.dat and invans_mod.dat, computed for the seventh higher mode.
Fig. 9. Azimuthal variation of surface-wave velocity resulting for the sample files invans.dat and invans_mod.dat, computed for the eighth higher mode.
Fig. 10. Azimuthal variation of surface-wave velocity resulting for the sample files invans.dat and invans_mod.dat, computed for the ninth higher mode.

References

Babuska V. and Cara M. (1991). Seismic Anisotropy in the Earth. Kluwer Academic Publishers, Dordrecht, The Netherlands.

Corchete V. (2012). Review of the methodology for the inversion of surface-wave phase velocities in a slightly anisotropic medium. Computers and Geosciences, 41, 56-63.

Smith M. L. and Dahlen F. A. (1973). The azimuthal dependence of Love and Rayleigh wave propagation in a slightly anisotropic medium. J. Geophys. Res., 78, 3321-3333.