xxx SEISMIC DATA FILTERINGWITH HIERARCHICAL LAPPED TRANSFORMS AND HIDDEN MARKOV MODELS LAURENT DUVAL IFP, 95852 Rueil-Malmaison Cedex, France, laurent.duval@ifp.fr Abstract We propose a method for uncoherent noise removal in geophysical data. The Multiple Wavelet Stacking is based on a concurrent use of wavelet-based shrinkage and data and time-scale depen- dent threshold choice. Since one singular wavelet does not match all the time-varying properties of a signal, the simultaneous use of several wavelets is able to lower some shrinkage shortcomings, such as wavelet dependency, and to further reduce the residual noises. Motivation and related works Wavelet transforms have emerged as efficient tools for signal separation and noise filtering in several geophysical applications. In heavy noise conditions, wavelet-based techniques generally owe their robustness to their attractive time-scale properties. Wavelet based denoising, or wavelet shrinkage [Miao and Cheadle, 1998], arises from the work of D. Donoho, based on the energy compaction properties of these time-scale operators [Donoho, 1995]. With the most widely used signal model [Ulrych et al., 1999], let represent an actual geophysical record. It can be decom- posed in an underlying signal , and two noise components and , respectively coherent and random: In the remaining of this paper, we will focus on the random noise component . One useful prop- erty is that the orthogonal transform of a white noise remains a white noise. Meanwhile, a coherent signal is generally efficiently and sparsely represented after a wavelet transform. Donoho [1995] demonstrated that for additive noise, a simple thresholding procedure would discard ”mainly” noise coefficients. This result has been shown to be asymptotically near optimal for a wide class of signals corrupted by gaussian white noise. As stated in [Ioup and Ioup, 1998], ”the choice of the wavelet and its associated scaling function are very important to obtain the most useful wavelet transform”. As a result, a good wavelet should be matched to the data properties. Unfortunately, discrete wavelets with fast algorithms are relatively rare, and not trivialy related to geophysical signals. Moreover, we cannot expect optimal energy compaction from a single wavelet domain, since most signals are non-stationary and contain a variety of frequency contents. Wavelet packets represent an enrichment of the projecting vectors. This concept is generalized for instance in [Saito, 1994]. The signal is estimated from a noisy observation using a library of orthonormal bases, consisting in various wavelets, wavelet packets and local trigonometrics bases. A procedure, based the information-theoretic Minimum Length Description criterion, chooses the best base as a compromise between the fidelity (denoising) and the efficiency of the signal estimation (compression). Both goals may be attained simultaneously EAGE 65th Conference & Technical Exhibition — Stavanger, Norway,2 - 5 June 2003