Prestack seismic data are multidimensional signals that can be described as a low-rank fourth-order tensor in the f requency − space domain. Tensor completion strategies can be used to recover unrecorded observations and to improve the signal-to-noise ratio of prestack volumes. Additionally, tensor completion can be posed as an inverse problem and solved using convex optimization algorithms. The objective function for this problem contains a data misfit term and a term that serves to minimize the rank of the tensor. The alternating direction method of multipliers offers automatic rank determination and it is used to obtain a reconstructed seismic volume. The proposed method converges to a good approximation of the rank of the tensor given the input data. We present synthetic examples to illustrate the behaviour of the algorithm in terms of trade-off parameters that control the quality of the reconstruction. We further illustrate the performance of the algorithm in a land data survey from Alberta, Canada.
Articles you may be interested inFast optimization of binary clusters using a novel dynamic lattice searching method We compare two implementations of a new algorithm called the pivot method for the location of the global minimum of a multiple minima problem. The pivot method uses a series of randomly placed probes in phase space, moving the worst probes to be near better probes iteratively until the system converges. The original implementation, called the ''lowest energy pivot method,'' chooses the pivot probes with a probability based on the energy of the probe. The second approach, called the ''nearest neighbor pivot method,'' chooses the pivot probes to be the nearest neighbor points in the phase space. We examine the choice of distribution by comparing the efficiency of the methods for Gaussian versus generalized q-distribution, based on the Tsallis entropy in the relocation of the probes. The two implementations of the method are tested with a series of test functions and with several Lennard-Jones clusters of various sizes. It appears that the nearest neighbor pivot method using the generalized q-distribution is superior to previous methods.
A pivot algorithm for the location of a global minimum of a multiple-minimum problem is presented. The pivot method uses a series of randomly placed probes in phase space, moving the worst probes to be near better probes iteratively until the system converges. The approach chooses nearest-neighbor pivot probes to search the entire phase space by using a nonlocal distribution for the placement of the relocated probes. To test the algorithm, a standard suite of functions is given, as well as the energies and geometric structures of LennardJones clusters, demonstrating the extreme efficiency of the method. Significant improvement over previous methods for high-dimensional systems is shown. ͓S1063-651X͑97͒08801-6͔
A reconstruction method known as Projection Onto Convex Sets (POCS) is an effective, uncomplicated and robust method for the recovery of irregularly missing seismic traces. However, slow convergence of the POCS reconstruction method could jeopardize its computational appeal. For this reason, we investigate the performance of the POCS reconstruction method in terms of different threshold schedules and present a new data driven threshold that leads to an efficient implementation of the POCS method. In particular, we show that high quality solutions can be obtained in a few iterations. In addition, we address an important issue with the classical implementations of POCS reconstruction in that they cannot interpolate regularly missing data. To solve this problem, we introduce a masking operator that is based on a dominant dip scanning method into the POCS iteration. At the end, we present a variant of the POCS method that permits de‐noising seismic volumes during the reconstruction stage. This is achieved by defining a weighted trace re‐insertion strategy that alleviates the influence of noisy traces in the final reconstruction of the seismic volume. We show the effectiveness of the proposed method using synthetic and field data.
Tensors, also called multilinear arrays, have been receiving attention from the seismic processing community. Tensors permit us to generalize processing methodologies to multidimensional structures that depend on more than 2D. Recent studies on seismic data reconstruction via tensor completion have led to new and interesting results. For instance, fully sampled noise-free multidimensional seismic data can be represented by a low-rank tensor. Missing traces and random noise increase the rank of the tensor. Hence, multidimensional prestack seismic data denoising and reconstruction can be tackled with tools that have been studied in the field of tensor completion. We have investigated and applied the recently proposed parallel matrix factorization (PMF) method to solve the 5D seismic data reconstruction problem. We have evaluated the efficiency of the PMF method in comparison with our previously reported algorithms that used singular value decomposition to solve the tensor completion problem for prestack seismic data. We examined the performance of PMF with synthetic data sets and with a field data set from a heavy oil survey in the Western Canadian Sedimentary Basin.
A new algorithm is presented for the location of the global minimum of a multiple minima problem. It begins with a series of randomly placed probes in phase space, and then uses an iterative Gaussian redistribution of the worst probes into better regions of phase space until all probes converge to a single point. The method quickly converges, does not require derivatives, and is resistant to becoming trapped in local minima. Comparison of this algorithm with others using a standard test suite demonstrates that the number of function calls has been decreased conservatively by a factor of about three with the same degree of accuracy. A sample problem of a system of seven Lennard᎐Jones particles is presented as a concrete example.
A multicomponent seismic record was transformed to the frequency-wavenumber domain using the quaternion Fourier transform. This transform was integrated into the projection onto convex sets algorithm to allow for the reconstruction of vector signals. The method took advantage of the spectral overlap of components in the frequency-wavenumber domain. The results of the method were compared with standard component-by-component reconstruction. Results were found for synthetic and real data including an example of 5D reconstruction of a converted-wave data set acquired over a heavy oil reservoir. We found an improvement in reconstruction quality producing fully sampled radial and transverse offset-azimuth gathers with a preserved vector relationship.
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