Seismic multicomponent deconvolution and wavelet estimation by means of quaternions

Multicomponent seismic data are acquired by orthogonal geophones that record a vectorial wavefield. Since the single components are not independent, the processing should be performed jointly for all the components. In this thesis, hypercomplex numbers, specifically quaternions, are used to implement a fundamental step of seismic data processing applicable to multicomponent records: the Wiener deconvolution. This new approach directly derives from the complex Wiener filter theory, but special care must be taken in the algorithm implementation due to the peculiar properties of quaternion algebra. Synthetic and real data examples show that quaternion deconvolution, either spiking or predictive, generally performs superiorly to the standard (scalar) deconvolution because it takes advantage of the signal that is simultaneously present onto all the components. This provides a better wavelet estimation and thus an improved deconvolution performance. Besides the main research, a technique for computing the eigenvalue decomposition of a quaternion polynomial matrix is also proposed. The new algorithm is a generalisation of the second-order Sequential Best Rotation (SBR2) algorithm applicable to convolutive mixture of polarized signals recorded by multicomponent sensors. The application to seismic data is explained in terms of separation between signal and the uncorrelated noise and wavelet estimation. The simulations shows that not only does the quaternion SBR2 algorithm perform better with respect to the conventional scalar approach, but also to the alternative long-vector approach because it takes into account the possibly non-linear relations between the components. The results achieved by both quaternion deconvolution and quaternion SBR2 algorithm mean that the quaternion formulation is the optimal way to represent the vectorial nature of data.