## 3 Methods and Processing
The relation provides the corresponding shift:  
\[\varphi_{c}(r,z;L;\omega)=\int\limits_{0}^{2\pi}\varphi_{s}(r,z;L,\vartheta_ {s};\omega)d\vartheta_{s} \tag{9}\]  
or equivalents:  
\[\varphi_{c}(r,z;L;\omega)=-2i\cdot\sum_{n=0}^{\infty}(-1)^{n}\cdot\varepsilon _{n}\int_{0}^{\infty}\frac{k}{v}J_{0}(kr)J_{2n}(kL)e^{-iv|z|}dk \tag{10}\]  
Radiation from an infinite set of sources, as expressed by (10), can be represented by the following equation:  
\[\sum_{m=1}^{\infty}\varphi_{c}(r,z;mL;\omega) \tag{11}\]  
Considering that the coefficients of the Bessel functions can be replaced by trigonometric functions [41] and Fourier transforms, the elastic wave radiates through radial circular sources as circular wave sources. In the case where the source is placed in a half-space, the horizontal (\(u\)) and transverse (\(w\)) displacements are given by the equations:  
\[ \begin{split} u(r,z;\omega)&=-\frac{i4\pi}{L}k_{ \beta}^{2}\sum_{n=0}^{\infty}\varepsilon_{n}\frac{k_{\beta}^{2}\cdot\gamma_{n} }{F(k_{n})}e^{-iv_{n}|z|}\cdot J_{1}(kr)\\ w(r,z;\omega)&=-\frac{2\pi}{L}k_{\beta}^{2}\sum_{n=0 }^{\infty}\varepsilon_{n}\frac{k_{\beta}^{2}\cdot\left(2k^{2}-k_{\beta}^{2} \right)}{F(k_{n})}e^{-iv_{n}|z|}\cdot J_{0}(kr)\end{split}  \tag{12}\]  
where parameter \(\beta\) symbolizes the velocity of the S-waves; \(\gamma\) is a scalar potential; \(z\) is the depth, and \(F\) is the Fourier transform. A modified form of the Axitra code [50; 52] was used to calculate the synthetic seismograms. Green functions were computed using the discrete wave methods in [49] and [50]. The synthetic seismogram was a linear combination of three elementary focal mechanisms according to this method. We assumed that \(u(t)\) symbolized the observed seismogram, which corresponded to any station since the effect of the instrument had already been removed. The synthetic seismogram corresponding to a double pair of forces is denoted by \(s(t)\). The synthetic seismogram can then be expressed by the linear combination of three elementary mechanisms of generation of normal, reverse, oblique, and strike-slip faults according to [56], Figure 3.  
The following equation defines the mathematical representation:  
\[s(t)=M_{0}\sum_{i=1}^{3}A_{ij}[\varphi-\varphi(\vartheta,\xi),\delta,\lambda ]*\mathrm{G}_{ij}[h,\Delta(\vartheta,\xi),t] \tag{13}\]  
The index \(j=1\), \(2\), and \(3\) indicate the vertical, radial, and tangential components [35]; \(\xi\) is the geometric distribution of the displacement in the slipped fault; \(\varphi\) denotes the radiation. In addition, in (14), \(M_{0}\) symbolizes the seismic moment; \(A_{ij}\) is the coefficients; \(G_{ij}\) is the Green functions, while in parameter \(\Delta\), each station's epicentral distance and the azimuth were integrated. Finally, the parameters \(\varphi\), \(\delta\), \(\lambda\), and h correspond to the strike, dip, slip of the rupture surface and the focus's depth, respectively. The observed seismograms, \(u(t)\), were linearly related to the seismic moment tensor \(M\) and the Green functions and are described by the mathematical equation:  
\[U(w,q,v;\omega)=G\cdot M \tag{14}\]  
where \(w\), \(q\), and \(v\) denote the vertical, radial, and tangential; r denotes the epicentral distances. In (14), the vector \(U\) represents an array column of dimension (\(n\times 1\)), as observed at each station, with \(G\) denoting the matrix, which has elements computed by the Green functions, dimension (\(n\times 5\)); finally, \(M\) is a matrix of dimensions (\(5\times 1\)) with the elements of the seismic moment tensor. The data in matrix \(G\) were calculated for each epicentral distance [40]. To estimate the seismic moment tensor \(M\) data, matrix \(G\) must be inverted. The inversion of this matrix was conducted in two steps. In the first stage, it was assumed that:  
\[M_{zz}=-\left(M_{xx}+M_{yy}\right) \tag{15}\]  
The last equation ensures that there is no volume change.