Two algorithms for calculation of theoretical seismograms for anisotropic media (Q2729327)

The article is devoted to numerical modeling of the elastic wave propagation in nonhomogeneous anisotropic media. The following direct dynamic problem can be posed: Find the components of the elastic displacements vector \(U\) which satisfy the system NEWLINE\[NEWLINE \begin{aligned} &\rho U_{tt} = c_{11}U_{xx} + c_{12}V_{xy} + c_{13}W_{xz}+ c_{66}(U_{yy} + V_{yx}) + \left(c_{55}(W_x + U_z)\right)_x + F_x(t,x,y,z), \\ &\rho V_{tt} = c_{66}(U_{xy} + V_{xx}) + c_{12}V_{yy} + c_{12}U_{xy} + c_{23}W_{yz} + \left(c_{44}(V_z + W_y)\right)_z + F_y(t,x,y,z), \\ &\rho W_{tt} = c_{55}(W_{xx} + U_{xz}) + c_{44}(V_{yz} + W_{yy}) + \left(c_{13}U_z + c_{23}V_y + c_{33}W_z\right)_z + F_z(t,x,y,z) \end{aligned} NEWLINE\]NEWLINE under the initial conditions NEWLINE\[NEWLINE U_{t = 0} = U_t|_{t = 0} = 0, \quad V|_{t = 0} = V_t|_{t = 0} = 0, \quad W|_{t = 0} = W_t|_{t = 0} = 0 NEWLINE\]NEWLINE and the boundary conditions NEWLINE\[NEWLINE \begin{gathered} c_{55}(W_x + U_z)|_{z = 0} = 0,\quad c_{44}(V_z + W_y)vert_{z = 0} = 0,\quad (c_{13}U_x + c_{23}V_y + c_{33}W_z)|_{z = 0} = 0. \end{gathered} NEWLINE\]NEWLINE The coefficients \(c_{ij}(z)\) and the density \(\rho(z)\) (the parameters of the anisotropic medium of orthotropic type) are arbitrary continuous positive functions; \(F_x\), \(F_y\), \(F_z\) are the components of the force vector describing the action of a source localized.NEWLINENEWLINE For solving the problem, the authors use the finite integral transformations and expose two algorithms for finding a numerical solution. A common feature of both algorithms is the reduction of the multidimensional problem to a series of one-dimensional problems by means of the finite integral Fourier transform with respect to the coordinates \(x\) and \(y\). The first algorithm is based on the explicit finite-difference method of second-order approximation with respect to time and fourth-order approximation with respect to the spatial variable. The second algorithm is based on employing the Laguerre transformation with respect to the time variable and the finite-difference approximation with respect to the spatial variable \(z\). The authors note that both algorithms proposed can be parallelized for multicomputers.