**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.