**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.