GNGTS 2014 - Atti del 33° Convegno Nazionale

10 GNGTS 2014 S essione 3.1 The parameters which are required by our interpolation algorithm are the spatial window size and the maximum rank of the tensor. The spatial window size is a crucial parameter that can influence significantly the computational time since it affects the dimensions of the tensors to be completed. There are no known methods in the literature to estimate the optimal rank of each tensor. However, this fact does not influence significantly the reconstruction quality as long as the maximum rank is set to a value comparable to the largest fraction of the energy recalled by the approximating tensor. In all our experiments, it was enough to choose the maximum rank of the approximating tensor equal to half of the maximum rank of the tensor of observations. It is important to notice that the curvature of the events has not affected the quality of the reconstruction, thus demonstrating empirically that the proposed method turns out to be independent to the shape of the events. This appears to be an important advantage with respect to Spitz-like methods or algorithms based on local transform like seislet or curvelet, requiring proper geometrical assumptions . Similarly to all low-rank tensor completion methods, the interpolation algorithm proposed in this paper cannot handle spatially aliased data, but only dense data in a regular grid with irregularly missing traces. Conclusions. In this work we have extended the greedy Orthogonal Matching Pursuit algorithm to the multidimensional case, making making it applicable to low-rank tensor completion problem. We have introduced a new method for reconstructing and interpolating missing traces in 3D datasets. The algorithm operates in frequency-space domain, solving, for each temporal frequency, instances of the low-rank Hankel completion problem by the proposed OR1TPgreedy algorithm. At each iteration this algorithm searches for a rank-one tensor approximation of the residual, making the method efficient in terms of time and solution quality. Synthetic data extracted from the synthetic SEG Advanced Modeling Program (SEAM) dataset, was used to prove the ability of the proposed method to recover missing traces in a 3D irregularly sampled dataset. Since there are no assumptions about the data, the algorithm turns out to be robust with respect to the shape of the events, obtaining good results both in case of linear and curved events. Acknowledgements. The authors would like to thank Nicola Bienati (eni E&P) for fruitful discussions and valuable suggestions. References G. Bergqvist: The Higher-Order Singular Value Decomposition: Theory and an Application // Signal Processing Magazine, IEEE, vol. 27, iss.3, pp. 151-154, 2010 E. J. Candes, B. Recth: Exact Matrix Completion via Convex Optimization // Foundations of Computational Mathematics archive, vol. 9, iss. 9, pp. 717-772, 2009 E. J. Candes, and M. B. Wakin: An Introduction To Compressive Sampling // IEEE Signal Processing Magazine, vol.21, march 2008. C. Da Silva , F. J. Herrmann�: Hierarchical Tucker Tensor Optimization - Applications to Tensor Completion // SAMPTA, 2013 C. Da Silva, F. J. Herrmann: Low-rank promoting transformations and tensor interpolation: applications to seismic data denoising // Abstract, European Association of Geoscientists and Engineers, EAGE 2014. G. Davis, S. Mallat, Z. Zhang: Adaptive time-frequency decompositions with matching pursuits // Optical Engineering, 1994 L. De Lathauwer, B. De Moor and J. Vandewalle: A multilinear Singular Value Decomposition // SIAM 21, 2000 D. L. Donoho: For Most Large Underdetermined Systems of Linear Equations the Minimal l 1-norm Solution is also the Sparsest Solution // Comm. Pure Appl. Math, vol. 59, pp. 797—829, 2004 S. Fomel and Y. Liu: Seislet transform and seislet frame : Geophysics, vol. 75, pp. V25–V38, 2010 G. Hennenfent, L. Fenelon, and F. J. Herrmann: Nonequispaced curvelet transform for seismic data reconstruction: A sparsity-promoting approach // Geophysics, vol. 75, 2010 T. Kolda, Bader, W. Brett: Tensor Decompositions and Applications // SIAM Rev. 51: 455–500, 2009 B. Liu: Multi-dimensional reconstruction of seismic data // PhD thesis, University of Alberta, 2004.