**Synthetic Seismograms Generator**

**Program objectives**

This program allows the generation of synthetic seismograms of surface waves.
For it, the relationship between *group delay or group time* and
*Fourier phase spectrum* given by (Papoulis, 1962)
is taken into account, where
and f is the *frequency*.
Using this formula, the phase spectrum of a *time signal* f(t) can be computed by
where (Bath, 1974)
and U is the *group velocity*.
Thus, the phase spectrum can be computed for any frequency by the formula
Then, if an *amplitude spectrum* is considering joint to this phase spectrum, a complete
*Fourier spectrum* can be generated by
computing its corresponding time signal f(t) by means of the *inverse Fourier transform*
given by (Bath, 1974)
This time signal f(t), obtained from the above-described Fourier spectrum by means of the
inverse Fourier transform, is the *synthetic seismogram* of the surface wave which
dispersion curve U has been considered.

**Program description**

The SYNTHETIC program and its data files are enclosed into a ZIP file named
"synthetic.zip".
When you have got the ZIP file and you have uncompressed this file, you have five files
named **synthetic.exe**, **dsynthetic.dat**, **group1.dat**, **group2.dat**
and **spectrum2.dat**. The file **synthetic.exe** contains a program (in FORTRAN
code for PC) for the computation of the synthetic seismogram of the surface wave which
dispersion curve U will be considered. Figure 1 shows the results of this program.
All the program capabilities are controlled by parameters enclosed in the file named
**dsynthetic.dat**. This file is in the free format and contains the parameters:

IOPT, DUR, N, DIST

The description of all parameters is as follows:

IOPT = With value 0 the synthetic seismogram is computed from the phase spectrum, calculated
from the dispersion curve by integration, joint to a theoretical amplitude spectrum computed
by a simple formula. With value 1 the synthetic seismogram is computed from the phase spectrum,
calculated as before, joint to an amplitude spectrum supplied by the user in an input-data
file.

DUR = Length of the synthetic seismogram in seconds. This value is ignored with IOPT = 1.

N = Number of the samples for the synthetic seismogram (must be an even number with maximum
value = 100000). This number is ignored with IOPT = 1.

DIST = Epicentral distance used to calculate the group times from the group velocities.

**Running the program**

The SYNTHETIC program have two options, with IOPT = 0 the synthetic seismogram is computed from
the phase spectrum and a theoretical amplitude spectrum (computed by a simple formula).
With IOPT = 1 the synthetic seismogram is computed from the phase spectrum and an amplitude
spectrum supplied by the user in an input-data file. When you choose IOPT = 0, the program
computes the phase spectrum from the dispersion curve contained in the file **group.dat**.
The file named **group1.dat** is a sample of this file supplied into the ZIP file. Figure 1a
shows the dispersion curve contained in the file named **group1.dat** in blue line. When you
choose IOPT = 0, the program computes the amplitude spectrum by a simple formula, given the
results shown in Figure 1b in blue line. When the phase spectrum and the amplitude spectrum
have been computed, the program performs the inverse Fourier transform to get the time signal
shown in Figure 1c (blue line), which is the synthetic seismogram corresponding to the surface
wave with the group velocity curve given by the dispersion curve enclosed in the file
**group.dat**. At the final, you have two new files named **spectrum.dat** and
**synthetic.dat**. The file named **spectrum.dat** contains the amplitude spectrum
shown in Figure 1b (blue line). The file named **synthetic.dat** contains the synthetic
seismogram shown in Figure 1c (blue line). If you choose IOPT = 1, the program computes the
phase spectrum as in the option IOPT = 0, but the amplitude spectrum must be supplied by the
user in the input-data file named **spectrum.dat**. The file named **spectrum2.dat** is a
sample of this file supplied into the ZIP file. Figure 1 shows the results of the program
running with the option IOPT = 1 (red line). The group velocity curve considered with this
option is the dispersion curve enclosed in the file **group2.dat**. At the final, you have
a new file named **synthetic.dat**. This file contains the synthetic seismogram shown in
Figure 1c (red line).
**Fig. 1** (a) Dispersion curves contained in the files **group1.dat** (blue line) and
**group2.dat** (red line). (b) Theoretical amplitude spectrum computed by a simple formula
(blue line) and amplitude spectrum contained in the file **spectrum2.dat** (red line).
(c) The synthetic seismogram in blue line is computed from the phase spectrum, calculated
from the dispersion curve by integration, joint to a theoretical amplitude spectrum computed
by a simple formula. The synthetic seismogram in red line is computed from the phase spectrum,
calculated as before, joint to an amplitude spectrum supplied by the user in an input-data
file.

**References**

Bath M., 1974. *Spectral analysis in Geophysics*. Elsevier, Amsterdam.

Papoulis A., 1962. *The Fourier integral and its applications*. McGraw-Hill, New York.