...Diego Takahashi; Vanderlei C. Oliveira Jr.; Valéria C. F. Barbosa ABSTRACT We have developed a fast equivalent-layer method for processing large-scale magnetic data. We demonstrate that the sensitivity matrix associated with an equivalent layer of dipoles can be arranged to a block-Toeplitz...

... for the LS solution cannot be stored within the memory of a single computer. A new technique is described for parallel computation of the LS operator that is based on a partitioned-matrix algorithm. The classical LS method for solution of block-Toeplitz systems of normal equation (NE) to the general case...

... the straightforward least-squares error solution without simplifying to a Toeplitz matrix. Also we show that the conjugate-gradient algorithm used in conjunction with the least-squares problem leads to a satisfactory simplification; that in the computation of the operators, the square matrix involved in the normal...

... wavefield. The two transforms are band-limited inverse operations. The transforms can be implemented by using time-step independent, noncausal time-varying digital filters that can be precomputed exactly from sums over Bessel functions. Their product becomes the symmetric Toeplitz matrix with the elements...

.... By exploiting the special structure of the level-one Toeplitz matrix T ( 1 ) , one can adopt the FFT algorithm to compute these products in O ( M 1 log 2 M 1 ) operations with M 1 defined in next section. Figure 4. Denoising and reconstruction using...

..., which involves an SVD. In the appendices, we will describe how this step can be carried out efficiently in O ( N   log   N ) floating point operations with O ( N ) storage thanks to the low-rank requirements and the Toeplitz (BTTB) matrix structure, enabling fast matrix-vector...

... advantages are obtained. The Toeplitz operator becomes independent of frequency and is calculated only once. Furthermore, the nonuniform FFT is initialized only once. The computations for each frequency slice are now Since the initialization of the nonuniform FFT costs approximately as much...

...Lingling Wang; Qian Zhao; Jinghuai Gao; Zongben Xu; Michael Fehler; Xiudi Jiang ABSTRACT We have developed a new sparse-spike deconvolution (SSD) method based on Toeplitz-sparse matrix factorization (TSMF), a bilinear decomposition of a matrix into the product of a Toeplitz matrix and a sparse...

... the inherent regularizing properties of CG iterations, and by using a block-diagonal structure of Toeplitz operators for fast matrix-vector products. Furthermore, performing plane-wave decompositions with respect to the source and receiver components separately, as in Algorithm  1 , minimizes the size...

... at depths 20, 25, and 30 km. First, we imitate the approach of an experienced operator of traditional (pre‐2024) ISOLA. We select by trial and error a relatively low‐noise range of 0.03–0.07 Hz (Fig.  1b ). Here, the operator would consider two options—to use all three components or just the less noisy Z...

.... Taking advantage of the symmetric block-Toeplitz Toeplitz-block (BTTB) structure of the sensitivity matrix that arises when regular grids of observation points and equivalent sources (point masses) are used to set up a fictitious equivalent layer, we develop an algorithm that greatly reduces...

... that the operator A has a Toeplitz structure, and thus is easily diagonalizable in the frequency domain. Hence, equation  12 can be efficiently solved by least-squares estimators in the frequency domain. Steps (1 and 3) have cost functions with a general form of x = argmin x ‖ b − Ax ‖ 2 2...

.... (4) A H WA is Hermitian. The operator ( A H WA + k 2 I ) has a so-called block-Toeplitz-Toeplitz-block (BTTB) structure and is independent of the temporal frequency, apart from k 2 I . Recalling that the BTTB operator is data independent and needs to be computed only once...

... spacings leads to fast acquisition but also, obviously, to sparse sampling. Both irregular and sparse sampling are often not optimally handled in conventional multitrace processing methods Irregular sampling leads, for example, to a leaking stack operator (see Vermeer, 1990 ), and to problems with f - k...

... to remove it. However, at a given frequency such a filter would also attenuate the signal. This is because a bandpass filter operates on the summed amplitude of the signal and the noise and cannot determine how much of a given amplitude is signal and how much is noise. For this reason bandpass filters...

... seismic wavelet. To obtain reflectivity coefficients with more accurate relative amplitudes, we should compute a nonstationary deconvolution of this seismogram, which might be difficult to solve. We have extended sparse spike deconvolution via Toeplitz-sparse matrix factorization to a nonstationary sparse...

...Figure 2. Illustration of nonstationary convolution as described by equation 9 . Like stationary convolution, the operation is a matrix-vector product, but here the matrix does not possess Toeplitz symmetry. Each column of the matrix contains the source waveform as modified by the attenuation...

... or nonsymmetric Toeplitz systems, tridiagonal, Hessenberg, Vandermonde, and Hankel systems can be solved recursively with O ( N 2 ) multiply and divide operations. In the general case of a simple symmetric matrix with no special structure, the application of Levinson's principle gives us...

... filtering and wavelet effects. Similarly, Sui and Ma (2019) extend the work of Wang et al. (2016) by introducing a nonstationary spike deconvolution method via Toeplitz-sparse matrix factorization. Chen et al. (2019) propose a multitrace semiblind deconvolution method, wherein “semiblind” refers...