Vector process cases
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2D and 3D seismic signals
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Knowing the correlation coefficient r, it is possible to build a spectral density matrix :math:`{S}_{X}(\omega ,t)\in {\mathit{Mat}}_{ℂ}(M,M)` describing a 2D or 3D vector process from the DSP scalar :math:`{S}_{X}(\omega ,t)\in ℝ`. In the case of two horizontal components, :math:`M=2`, the DSP matrix is written as:

:math:`{S}_{X}(\omega ,t)=\left[\begin{array}{cc}{S}_{X}(\omega ,t)& \rho \\ \rho & {S}_{X}(\omega ,t)\end{array}\right]`

For an earthquake, the vertical component is generally considered to be uncorrelated to the two horizontal components. GENE_ACCE_SEISME makes it possible to generate 2D signals with correlated horizontal components (defined by the correlation coefficient COEF_CORR, which can be zero) and a vertical component that is not correlated to the other two, but to which the horizontal/vertical ratio applies.




Variable seismic field in space
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In the context of a temporal study with a seismic excitation that varies in space, the seismic movement is described by a space-time field. After discretization, the problem comes down to the simulation of a vector process, described by its matrix DSP as for the 2D problem.

In the literature, coherence functions :math:`\gamma` are proposed to define the spatial correlation of seismic movement. Two types of coherence functions are available for operator GENE_ACCE_SEISME: the Mita & Luco exponential coherence function and the empirical Abrahamson function. More details on consistency functions can be found in the documentation for operator DYNA_ISS_VARI, in particular R4.05.04. A spectral density matrix :math:`{S}_{X}(\omega ,t)\in {\mathit{Mat}}_{ℂ}(M,M)` is constructed, where :math:`M` is the number of nodes for which the field must be evaluated. The components of this matrix are written as:

:math:`{S}_{\mathit{ij}}(\omega ,t)=\gamma (\omega ,{d}_{\mathit{ij}}){S}_{X}(\omega ,t)`

where :math:`{d}_{\mathit{ij}}` is the horizontal distance between two nodes :math:`i` and :math:`j` in the mesh.