From a seismic section composed of N traces, it is possible to build for each frequency a complex square matrix of dimension NxN composed of all auto-spectra and cross-spectra between the various traces. After having applied an average operator, the matrix called spectral matrix is of a rank higher to 1. The averages available are the average in distance and the average in frequency.

The average in frequency consists in applying a windowed function on the correlation functions (smoothed periodogram method). This average preferentially applies on large bandwidth signals. The windowed function is usually a Hanning's function at a given power P.

The average in distance consists in applying a moving average filter on the diagonal terms of the spectral matrix. The filtering is done on a distance of L terms. L is an odd number : 2M +1

The spectral matrix is decomposed into its eigenelements in the frequency domain. The data are projected on the different eigenvectors and then displayed in the time - distance domain.

SMF filtering is applied on the noise shot in order to extract the slow Rayleigh wave. It is possible to choose the parameters of the SMF filter: the power P of the Hanning function for an average in frequency and the number M for an average in distance. After estimation, the slow wave is put back in its initial position in the time - distance domain.

The video shows the efficiency of SMF filter with different selected parameters, P ranging from 0 to 32 for an averaging in frequency and M ranging from 1 to 7 for an averaging in distance. We show successively :

• V1 : the initial data projected on the first eigenvector
• V2 : the initial data projected on the second eigenvector
• The residual section.

Panel on the right shows the two dimensions (f,k) amplitude spectrum associated to the selected section. The 2D spectra can be normalized in wavenumber k. Above the 2D spectrum, we present 4 eigenvalue curves according to the frequency. The black curve represents the sum of the eigenvalues. The blue curve is the first eigenvalue associated to the first eigenvector (first eigensection), the green curve is associated to the second eigenvalue associated to the second eigenvector (second eigensection). The red curve that is the difference between the black curve and the sum of the blue and green curves represents the sum of the other eigenvalues.