Theoretical Amplitudes and Velocities of Surface Waves


Program objectives

This program allows the generation of theoretical amplitudes and velocities for the Rayleigh and Love waves. The program calculates theoretical amplitudes and velocities for any layered model, for any frequency and any mode. This program will accept one liquid layer at the surface. In such case, the amplitudes of Rayleigh wave are that at the top of the solid array. For Love wave, the liquid layer is ignored. The method for the phase velocities computation is described in detail by Abo-Zena (1979). The group velocity is calculated from the phase velocity, by derivation of the phase velocity (Ben-Menahem and Singh, 1981). Finally, the amplitudes of the surface waves are computed according to Aki and Richards (1980). You can compute the theoretical amplitudes and velocities for the Rayleigh and Love waves by means of the present program, as it will be described below.

Program description

The LR program and its files are enclosed into a ZIP file named "LR.zip". When you have got the ZIP file and you have uncompressed this file, you have three files named LR.exe, LR1.dat and LR2.dat. The file LR.exe contains a program (in FORTRAN code for PC) for the computation of the theoretical amplitudes and velocities for the surface waves. All the program capabilities are controlled by parameters enclosed in the file named LR.dat. The files LR1.dat and LR2.dat are two sample files that can be used to run the LR program. The file LR.dat must be in the free format and must contain the parameters:

TMIN, TMAX, NT, LRVEL, LRAMP
NZ
THICK(1), ALPHA(1), SVEL(1), RHO(1)
THICK(2), ALPHA(2), SVEL(2), RHO(2)
...
...
...
THICK(NZ), ALPHA(NZ), SVEL(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.
LRVEL = With value 1 the phase velocities will be computed for the period interval given above. With value 2 the phase and group velocities will be computed.
LRAMP = With value 1 the surface wave amplitudes will be computed for the period interval given above. With value 0 the surface wave amplitudes will not be computed.
NZ = Number of layers of the earth model considered (maximum 100).
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 earth model considered. The NZ layer is the semi-infinite medium and its thickness must be given as zero.

Running the program

The LR program can handle earth models with or without a liquid layer at the surface. Figure 1 and 2 show the phase and group velocities resulting for the sample file LR1.dat, in which a liquid layer at the free surface has not been considered. This file must be renamed to LR.dat and after run the LR program.

Fig. 1. Phase velocities resulting for the sample file LR1.dat, in which a liquid layer at the free surface has not been considered.

Fig. 2. Group velocities resulting for the sample file LR1.dat, in which a liquid layer at the free surface has not been considered.

With LRAMP = 1 the surface wave amplitudes are computed, for the period interval given in the input-data file LR.dat. Figure 3 and 4 show the surface wave amplitudes resulting for the sample file LR1.dat, in which a liquid layer at the free surface has not been considered.

Fig. 3. Surface wave amplitudes resulting for the fundamental mode, at the period range given in the input-data file LR1.dat.

Fig. 4. Surface wave amplitudes resulting for several modes at the period of 16 seconds.

Figure 5 and 6 show the phase and group velocities resulting for the sample file LR2.dat, in which a liquid layer at the free surface has been considered. In such case, the amplitudes of Rayleigh wave are that at the top of the solid array. For Love wave, the liquid layer is ignored. The file LR2.dat must be renamed to LR.dat and after run the LR program.

Fig. 5. Phase velocities resulting for the sample file LR2.dat, in which a liquid layer at the free surface has been considered.

Fig. 6. Group velocities resulting for the sample file LR2.dat, in which a liquid layer at the free surface has been considered.

With LRAMP = 1 the surface wave amplitudes are computed, for the period interval given in the input-data file LR.dat. Figure 7 and 8 show the surface wave amplitudes resulting for the sample file LR2.dat, in which a liquid layer at the free surface has been considered.

Fig. 7. Surface wave amplitudes resulting for the fundamental mode, at the period range given in the input-data file LR2.dat.

Fig. 8. Surface wave amplitudes resulting for several modes at the period of 1 second.

References

Abo-Zena A. (1979). Dispersion function computations for unlimited frequency values. Geophys J R Astr Soc, 58, 91105.
Aki K. and Richards P. G. (1980). Quantitative Seismology. Theory and methods. W. H. Freeman and Company, San Francisco.
Ben-Menahem A. and Singh S. J. (1981). Seismic Waves and Sources. Springer-Verlag, Berlin.