Attenuation of multiple in reflection seismic data using Kalman–Bucy filter Marcus P.C. Rocha a, * , Lourenildo W.B. Leite b , Mauro de L. Santos a , Valcir J.da C. Farias a a Department of Mathematics, Federal University of Para ´ , Brazil b Department of Geophysics, Graduate Course in Geophysics, Federal University of Para ´ , Brazil Abstract The main objective of this work is the study and the application of the Kalman–Bucy method in the deconvolution pro- cess with prediction, considering the observed data as non-stationary. We propose a new prediction deconvolution oper- ator (attenuation of multiple). The data used in this work are synthetic and, based on this, this work has characteristics of a numerical research. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Stochastic process; Kalman–Bucy filter; Deconvolution; State space 1. Introduction The main objective of this paper is the study and the application of the Kalman–Bucy method in the decon- volution process with prediction considering the observed data as non-stationary. The Kalman–Bucy method is considered an alternative process in time domain in addition to the Wiener–Hopf theory. The valorization and importance of the Kalman–Bucy filter derives from its applicability and versatility as well as the essence of its conceptualization, this is a basic condition to geophysical processes. The basic application references in seismic in this paper are Crump [1], Mendel et al. [6], Mendel [4,5] and Robinson [7]. The prediction operator (KBCP) is based on Crump [7] and Mendel et al. [6] theories. Its structure resem- bles that of Wiener–Hopf filter, where the coefficients of the operator (WHLP) are obtained through autocor- relation, in the case of (KBCP) are obtained from the function b i ðkÞ. The objective of the seismic section of reflection is to allow the interpretation of the registered data. The detailed representation of the seismic signal demands a relatively complex model, and its processing makes use of a set of techniques based on information stochastic properties. Non-stationary stochastic processes are the basic characteristics of geophysical data and it is necessary to the application of the method used here. 0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.11.186 * Corresponding author. E-mail address: mrocha@ufpa.br (M.P.C. Rocha). Applied Mathematics and Computation 189 (2007) 805–815 www.elsevier.com/locate/amc