A seismic section is a matrix m of dimensions Nt • Nx. A row is a trace function of time, whereas a column represents amplitude variations at a given time in function of the distance x. Filtering by singular value decomposition (SVD) decomposes the matrix m into Nx matrices of the same dimension called "eigensections". These matrices are arranged in descending order according to energy (Order 1 to Nx). These energies are the eigenvalues attributed to each section. The eigenvector u of an eigensection describes a wavelet in function of time. The eigenvector v describes the amplitude variations of the wavelet in function of the distance. The wave to be selected must first be flattened. Where the form of the dominant arrival changes little from one trace to another, the SVD method is a good tool for selecting the dominant wave which will be localized on the first eigensection.

A seismic reflection example, made up of a unique faulted horizontal reflector, is simulated, the presence of noise is also simulated. We would like to isolate the reflector on each side of the fault and from the noise.

The example shows:
Above, from left to right:

• the decay of the eigenvalues,
• the eigenvectors u (wavelets relative to each eigensection).

Below, from left to right:

• the initial section,
• the 1st eigensection,
• the 2nd eigensection,
• the amplitude variation (vectors v) of the 3 first eigensections (in blue, in red and then black),
• either: the signal on its own or additional noise or noise estimated as the difference between the initial section and the sum of the first 2 eigensections.

It is possible to study:
1) In the absence of noise (N/S=0), the dip of the reflector has an adverse influence on the separation or extraction of a wave. The dip is defined by the slope parameter expressed in number of samples between two traces, which gives the arrival time difference of the reflected wave.
2) The limiting effect of noise on the separation.