Focal mechanism analysis using synthetic seismograms

According to the representation theorem, the elastic displacement due to a point source is given by the following equation:
                                          (1)
In this equation,  is the component of the displacement,  is the source time function which indicates how the energy is released during the earthquake process, , is the Green's function that describes the propagation path effects between the source located at  and the station at .  are the moment tensor components . In this equation all components of the moment tensor are assumed to have the same time functionality. Computation of the Green's function is the most important step for producing synthetic seismograms.
1. Simulation of an Earthquake and its Linear Inversion for a Completely Shear Source (Pure Double-Couple)
1-1 Earthquake Simulation: For the simulation purpose, a source with definite rake, dip and strike was assumed. The synthetic seismograms were then calculated using the wavenumber integration method based on a 12km source depth.
1-2 Linear Inversion of the Seismograms for Determination of the Earthquake source parameters under Pure Double-Couple Conditions: The moment tensor is an overall indication of the earthquake source i.e. the volume variations and a variety of shear source in different directions are included in the moment tensor. Consequently, an earthquake due to a pure double-couple can be considered as a special case of the moment tensor.
For big earthquakes, i.e. Mw>=6.0, the source time function also should be included in the inversion.  In such cases, the unknown model parameters are the 6 components of the moment tensor and the source time function components.
In this section, the inversion was carried out under pure double-couple conditions. In other words it was assumed that the sum of moment tensor diagonal components was zero and the eigenvalues of the moment tensor were assumed to be 1, 0, and.
Different parameters may affect the solution in the linear inversion method e.g. the frequency band, the number of components involved and the way the stations are distributed. The effects of the most important parameters are discussed below.
A. The Effect of Frequency Band on the Inversion Solution: The linear inversion method was applied for different frequencies. In most cases, the earthquake specifications obtained were in complete agreement with the simulated earthquake which indicates that in the tested frequency bands, all the frequencies were below the corner frequency of the earthquake source. The solution obtained was very similar to the original mechanism. The fault plane resulted was perpendicular to the original plane that indicates the major fault plane and minor fault plane.
B. The Effect of the Number of Components Involved: The linear inversion method was applied for three different cases. First, the vertical components were used. Second, only the radial components were applied. Third, the inversion was varied solely by tangential components. In the first and second cases, the source coordinates were obtained precisely. But for the third one, the solution was completely wrong even though the depth was determined precisely.
C. The Effect of the Station Distribution Pattern: The effect of the station distribution was examined based on the configuration of seismographic stations in different quadrants. It was concluded that the solution can be obtained even though the data are from one quadrant but the more quadrant are involved the less error there will be.
2. Simulation of an Earthquake and its Linear Inversion for a Non-Pure Double-Couple (without volume variation)
2.1 Earthquake Simulation: In this mode, a five layered crustal model was used for simulation of synthetic seismograms in eight stations distributed in the four quadrants with respect to the source.
2.2 Linear Inversion of the Seismograms for Determination of the Earthquake source parameters under Non-Pure Double-Couple Conditions: In contrast to the linear inversion in 1.2, in this case, the inversion was used in conditions which were more similar to the reality. In the natural mode, the elastic waves traverse layers in the earth about which we don’t have enough information. Therefore, in earthquake mechanism determinations, models are used which are much more simplistic than the real one.
 
Like the other case, each of the parameters affecting the solution in the linear inversion method will be examined briefly
A. The Effect of Frequency Band on the Inversion Solution: The linear inversion method was applied for different frequencies. In every case, a parabolic function with the
corner frequency of 0.2Hz was used. In all the cases, the earthquake specifications were restored and, as in the other case, the depth was determined within 2 or 3 km of the original depth.
B. The Effect of the Number of Components Involved: Again the linear inversion method was applied for three different cases. And the results indicated that given the mechanism, the epicenter distances and the depth of the earthquake, the vertical components are of greater amplitudes, and therefore stabilize the solution to the inversion method.
C. The Effect of the Station Distribution Pattern: Again the effect of the station distribution was examined based on the configuration of seismographic stations in different quadrants and accordingly the solution could be obtained even though the data were from one quadrant but the more quadrants there were involved the less error there was.
Conclusion: Two different types of sources (pure and non-pure double-couple) were
used to produce synthetic seismograms based on the wavenumber integration method
for a given velocity model. In both cases, the source model was obtained precisely depending on the conditions. Generally, it can be concluded that if the velocity
model is not precise its effect can be seen on CLVD as well as the depth. In addition, if the stations are distributed in at least two quadrants, more precise solutions will be gained.