GNGTS 2014 - Atti del 33° Convegno Nazionale

GNGTS 2014 S essione 3.1 To recover the missing samples, each frequency slice of the data in frequency-space domain is rearranged in a 4D Hankel tensor. This tensor can be obtained with an appropriate transformation as described in Trickett (2010). Given a raw 2D frequency slice , to form the 4D Hankel tensor , we apply the following: where the four tensor directions have length, respectively, , , and , with , , and . Once the tensor T is obtained, the low-rank tensor completion is performed by the OR1TP algorithm and the interpolated frequency slice is obtained applying the inverse transform. The interpolation procedure described above is summarized in Tab. 2. • Take the Discrete Fourier Transform (DFT) of each trace in the grid • For each frequency within the signal band • Form a complex-value Hankel tensor T • Perform tensor completion on T by OR1TP algorithm • Recover the interpolated frequency slice from the completed tensor • Take the Inverse Discrete Fourier Transform (IDFT) of each trace. Tab. 2. Interpolation procedure by low-rank Hankel tensor completion in frequency domain performed by the OR1TP greedy algorithm. Examples. We chose to test the proposed approach with a portion of the synthetic SEG Advanced Modeling Program (SEAM) 3D dataset to prove the effectiveness of our method. We obtained an irregularly sampled dataset from the (regularly sampled) reference data by removing a randomly selected subset of traces. Two different parameter sets have been tested: in the first experiment (Fig. 1), each 2D frequency slice is of size 7 × 7 samples, that rearranged in a Hankel tensor becomes a 4D volume of size 4 × 4 × 4 × 4. Due to the fact that the maximum rank of our 4D Hankel tensor is , we recovered each tensor by approximating it with a tensor having maximum rank equal to    . In the second experiment, the frequency slice is a window of size 5 × 5 samples that, once mapped to a Hankel tensor, becomes a multidimensional array of order 4 and size 3 × 3 × 3 × 3. Similarly to the experiment previously described, the maximum rank was chosen equal to      . To better highlight the interpolation capability of the proposed approach, both experiments have being carried out by eliminating half of the traces in the original 3D dataset. The computation of the rank-one tensor that approximates the residual tensor at each iteration of the OR1TP is performed through the generalized iterative Rayleigh quotient method (Lathauwer, 2000), choosing as stopping criteria either maximum number of iterations MaxIter = 20 and tolerance tol =1e-3. Fig. 1 shows the results obtained by OR1TP greedy algorithm with frequency slices of size 7 × 7 samples over a portion of the data containing linear events. Fig. 2 shows the results obtained by filling the missing traces with OR1TP algorithm. In this case, the frequency slice size is equal to 5 × 5 samples. In this portion of data there is the presence of hyperbolic events with high curvature that does not influence the results of the reconstruction. To evaluate quantitatively the performance of our interpolation algorithm, we chose to vary the percentage of randomly zeroed traces from 20% to 60% for fixed instances (5×5 and