GNGTS 2014 - Atti del 33° Convegno Nazionale

64 GNGTS 2014 S essione 3.1 dictionary. This alternative approach can be viewed as a mere application of the well-known Matching Pursuit (MP) algorithm first proposed by Mallat and Zhang (1993). We bring here the contribution of Blumensath and Davies (2009) to the MP approach for handling the data interpolation problem. We apply the so-called Conjugate Gradient Pursuit (CGP) with the Stagewise selection strategy instead of the simple MP for estimating the Fourier components. The usage of this engine in the seismic data processing context has been inspired by the work of Adamo et al. (2014). The CGP differs from MP for the fact that, at each iteration, all so far collected components are updated for being conjugate to components at previous step. Moreover, the Stagewise selection strategy suggests picking a number of new components per iteration instead of a single one. This leads to a reduction of the final number of iteration needed to get to convergence. On the other hand, the MP, in which the same component is selected again and again until its energy is exhausted, needs a lot of iterations. We make a comparison among the two approaches, and show that they produce comparable results in terms of accuracy but CGP converges much faster than MP as expected. Fourier representation of seismic data for interpolation. An interpolation algorithm, in this context, aims to estimate the Fourier spectrum from irregularly sampled seismic traces while avoiding wavenumber energy leakage. Since seismic data is usually well-sampled in the time direction, it is efficient and accurate to use the FFT for representing data in the f-x domain. Then, for each temporal frequency the algorithm estimates the spatial Fourier components (in the f-k domain), and uses the expansion for reconstructing data at any desired spatial location. For this reason, to illustrate the concept of the method, we consider the interpolation problem of a single frequency slice of a seismic data set. Let us consider the complex-valued function f ( x ) defined in the interval [0,1[ and sampled at a set of N x points x l . The sampled function f ( x l ) represents the temporarily transformed seismic data at a given frequency and spatial position x l . We compute Fourier transformation of irregularly sampled data by direct evaluation of trigonometric sums: (1) while the input data expansion is computed as: (2) In matrix notation, we write data samples f ( x l ) as the vector f and Fourier coefficients fˆ ( k ) as the vector fˆ . The exponential wave functions e 2 πikx l , are represented by the vectors Φ k which constitutes the columns of the matrix operator Φ of size N x × N k . Thus, Eq. (2) can be rewritten in the form f = Φ · fˆ (3) which represents an undetermined system when N k > N n . Interpolation via Matching Pursuit. The basic assumption which makes this kind of approaches possible (MP, ALFT, and similar ones), is that seismic data have a sparse representation in the Fourier domain, which means that the input data can be accurately approximated by few dominant wavenumbers. It is by this hypothesis that the usage of greedy algorithms is legitimate: Mallat and Zhang (1993) firstly suggest to handle sparse representation problems by using the MP algorithm. This simple approach has already proved to be effective when applied to seismic data regularization and interpolation problems, as Nguyen and Winnett (2011) demonstrated. We here recall first the MP approach, in order to underline afterward our upgrade by the application of CGP.