4. Vector process cases

4.1. 2D and 3D seismic signals

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

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

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.

4.2. Variable seismic field in space

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 \(\gamma\) 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 \({S}_{X}(\omega ,t)\in {\mathit{Mat}}_{ℂ}(M,M)\) is constructed, where \(M\) is the number of nodes for which the field must be evaluated. The components of this matrix are written as:

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

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