Wiener Deconvolution


Program objectives

The Wiener Deconvolution is a technique used to obtain the phase-velocity dispersion curve and the attenuation coefficients, by a two-stations method, from two pre-processed traces (instrument corrected) registered in two different stations, located at the same great circle that the epicentre. This filtering technique is described in detail by Hwang and Mitchell (1986). You can compute this filtering technique by means of the present programs, as it will be described below.

Program description

The Wiener Deconvolution programs and their files, which are needed to run this application, are enclosed into a ZIP file named "wiener.zip". When you have got the ZIP file and you have uncompressed this file, you have four files named wiener1.exe, wiener2.exe, segnal1.dat and segnal2.dat. The traces contained in the files segnal1.dat and segnal2.dat must have the same origin time, duration and sampling ratio. The file segnal1.dat must contain the trace recorded at the station more near to the epicentre (station 1). The files wiener1.exe and wiener2.exe contain programs (in FORTRAN code for PC) for the computation of the Wiener Deconvolution. These computer programs will be used to obtain the phase-velocity dispersion curve and the attenuation coefficients, by a two stations method, from the two pre-processed traces contained in the files segnal1.dat and segnal2.dat. The phase-velocity dispersion curve obtained will be contained in the file named phase.dat, the attenuation coefficients will be contained in the file named gamma.dat. The program wiener1.exe must be run the first, to obtain the cross-correlation and the auto-correlation, as it is shown in Figure 1. This program accepts input data files with a maximum number of 50000 records. The correlations can be windowed as it is described by Hwang and Mitchell (1986).

Fig. 1 (a) Observed seismogram corresponding to the pre-processed trace (instrument corrected), contained in the file segnal1.dat. (b) Observed seismogram corresponding to the pre-processed trace (instrument corrected), contained in the file segnal2.dat. (c) Cross-correlation of the traces plotted in Figures 1a and 1b. (d) Auto-correlation of the trace plotted in Figure 1a.

The program wiener2.exe must be run after the above-mentioned program, to obtain the phase-velocity dispersion curve and the attenuation coefficients, from the cross-correlation and the auto-correlation the shown in Figure 1. This program asks for the input of some data during its running. The data asked are:

Minimum period = The minimum value of the period to be considered for the computation.
Maximum period = The maximum value of the period to be considered for the computation.
Epicentral distance for station 1 = The epicentral distance (in km) for the station more near to the epicentre. In this station is recorded the trace contained of the file segnal1.dat .
Epicentral distance for station 2 = The epicentral distance (in km) for the station more far to the epicentre. In this station is recorded the trace contained of the file segnal2.dat .
Integer number = This number is used in the computation of the phase velocities, to take into account the multivaluation of the phase (Bath, 1974).

For the data shown in Figure 1, these parameters have the values:

>> Minimum period = 1
>> Maximum period = 100
>> Epicentral distance for station 1 = 361.6
>> Epicentral distance for station 2 = 1208.9
>> Integer number = 2

The integer number asked by the program can be changed during the running of the program iteratively. For it, the program presents the following text:

Continue (s/n)?

If you wish to run other iteration with a different value for the integer number, you can input "n" or "N", then the program asks again for a new value for this number. Then, you can see the new values of the phase velocity listed on the screen. The multivaluation of the phase does necessary to search the value of this number, to obtain a reasonable dispersion curve for the phase velocity. This number can be any integer positive or negative. For the data shown in Figure 1 this number is 2. When you consider that the phase velocities have been determined, you can input "s" or "S" to answer the above-mentioned question. Then, the program finishes computing the group-velocity dispersion curve and the quality factor. These data are contained in the files group.dat and Q.dat, respectively. The group velocity is calculated from the phase velocity, by derivation of the phase velocity (Ben-Menahem and Singh, 1981). The quality factor is calculated from the group velocity and the attenuation coefficients (Ben-Menahem and Singh, 1981). Figure 2 shows the results of the program wiener2.exe.

Fig. 2 (a) Phase-velocity dispersion curve contained in the file phase.dat. (b) Group-velocity dispersion curve contained in the file group.dat. (c) Attenuation coefficients contained in the file gamma.dat. (d) Quality factor contained in the file Q.dat.

References

Bath M., 1974. Spectral Analysis in Geophysics. Elsevier, Amsterdam.

Ben-Menahem A. and Singh S. J., 1981. Seismic Waves and Sources. Springer-Verlag, Berlin.

Hwang H. J. and Mitchell B. J., 1986. Interstation surface wave analysis by frequency-domain Wiener deconvolution and modal isolation. Bulletin of the Seismological Society of America, 76, No. 3, 847-864.