MULTICHANNEL LINEAR FILTERS FOR OPTIMAL REJECTION OF MULTIPLE REFLECTIONS

Abstract

In a CDP family after normal moveout correction, multiple reflections generally have a residual moveout hyperbolically increasing with the distance from the shotpoint. Simple stochastic models of primary signal, coherent noise and noncoherent noise can be generated, from which auto and mutual‐correlation functions can be derived. The set of linear operators corresponding to the multichannel filter for optimal rejection of coherent and noncoherent noise can be obtained by solving the proper Wiener‐Hopf equations. In the solutions already published some difficulties may arise, due either to the precision of the computations necessary to avoid numerical instabilities, or to the restrictive hypotheses on which the model is founded (multiples having precisely known moveout). In this case the difficulties are overcome with a proper model and by solving the Wiener‐Hopf equations in the frequency domain at different frequencies; the time operators are obtained by means of a discrete Fourier transformation. With n‐fold coverage, q linear systems of n equations in n unknowns have to be solved; the matrices of the coefficients are symmetric or Hermitian depending on the model chosen. With 48‐fold coverage, a set of 48 operators having a length of about 0.5 sec can be obtained in less than 40 sec of computing time on an IBM 360/44 with Array Processor. Several curves are shown comparing (for each residual moveout) the attenuation of the multiple obtained by multichannel filtering versus the one obtained by straight stacking. It can be seen that the increase of attenuation of the coherent noise may amount to 15–25 db in a broad range of moveouts. The filter can be designed to reject multiples having residual moveout in one or two disjoint intervals, or having fixed or varying amplitude.