Retrieving the leaked signals from noise using a fast dictionary learning method

Most traditional seismic denoising algorithms will cause damages to useful signals, which are visible from the removed noise profiles and are known as signal leakage. The local signal-and-noise orthogonalization method is an effective method for retrieving the leaked signals from the removed noise. Retrieving leaked signals while rejecting the noise is compromised by the smoothing radius parameter in the local orthogonalization method. It is not convenient to adjust the smoothing radius because it is a global parameter while the seismic data is highly variable locally. To retrieve the leaked signals adaptively, we propose a new dictionary learning method. Because of the patch-based nature of the dictionary learning method, it can adapt to the local feature of seismic data. We train a dictionary of atoms that represent the features of the useful signals from the initially denoised data. Based on the learned features, we retrieve the weak leaked signals from the noise via a sparse co ding step. Considering the large computational cost when training a dictionary from high-dimensional seismic data, we leverage a fast dictionary up dating algorithm, where the singular value decomposition (SVD) is replaced via the algebraic mean to update the dictionary atom. We test the performance of the proposed method on several synthetic and field data examples, and compare it with that from the state-of-the-art local orthogonalization method.

Non-stationary local signal-and-noise orthogonalization

The local signal-and-noise orthogonalization method has been widely used in the seismic processing and imaging community. In the local signal-and-noise orthogonalization method, a fixed triangle smoother is used for regularizing the local orthogonalization weight, which is based on the assumption that the energy is homogeneously distributed across the whole seismic profile. The fixed triangle smoother limits the performance of the local orthogonalization method in processing complicated seismic datasets. Here, we propose a new local orthogonalization method that uses a variable triangle smoother. The non-stationary smoothing radius is obtained by solving an optimization problem, where the low-pass filtered seismic data are matched by the smoothed data in terms of the local frequency attribute. The new local orthogonalization method with non-stationary model smoothness constraint is called the non-stationary local orthogonalization method. We use several synthetic and field data examples to demonstrate the successful performance of the new method.

Gradient iterative algorithm for rational models based on Gram-Schmidt orthogonalization method

An improved gradient iterative algorithm, termed as Gram-Schmidt orthogonalization based gradient iterative algorithm, is proposed for rational models in this paper. The algorithm can obtain the optimal parameter estimates in one iteration for the reason that the information vectors obtained by using the Gram-Schmidt orthogonalization method are independent of each other. Compared to the least squares algorithm and the traditional gradient iterative algorithm, the proposed algorithm does not require the matrix inversion and eigenvalue calculation, thus it can be applied to nonlinear systems with complex structures or large-scale systems. Since the information vector of the rational models contains the latest output that is correlated with the noise, a biased compensation Gram-Schmidt orthogonalization based gradient iterative algorithm is introduced, by which the unbiased parameter estimates can be obtained. Two simulated examples are applied to demonstrate the efficiency of the proposed algorithm.