Fast hyperbolic Radon transform represented as convolutions in log-polar coordinates

The hyperbolic Radon transform is a commonly used tool in seismic processing, for instance in seismic velocity analysis, data interpolation and for multiple removal. A direct implementation by summation of traces with different moveouts is computationally expensive for large data sets. In this paper we present a new method for fast computation of the hyperbolic Radon transforms. It is based on using a log-polar sampling with which the main computational parts reduce to computing convolutions. This allows for fast implementations by means of FFT. In addition to the FFT operations, interpolation procedures are required for switching between coordinates in the time-offset; Radon; and log-polar domains. Graphical Processor Units (GPUs) are suitable to use as a computational platform for this purpose, due to the hardware supported interpolation routines as well as optimized routines for FFT. Performance tests show large speed-ups of the proposed algorithm. Hence, it is suitable to use in iterative methods, and we provide examples for data interpolation and multiple removal using this approach.Comment: 21 pages, 10 figures, 2 table

Neural Eikonal Solver: improving accuracy of physics-informed neural networks for solving eikonal equation in case of caustics

The concept of physics-informed neural networks has become a useful tool for solving differential equations due to its flexibility. There are a few approaches using this concept to solve the eikonal equation which describes the first-arrival traveltimes of acoustic and elastic waves in smooth heterogeneous velocity models. However, the challenge of the eikonal is exacerbated by the velocity models producing caustics, resulting in instabilities and deterioration of accuracy due to the non-smooth solution behaviour. In this paper, we revisit the problem of solving the eikonal equation using neural networks to tackle the caustic pathologies. We introduce the novel Neural Eikonal Solver (NES) for solving the isotropic eikonal equation in two formulations: the one-point problem is for a fixed source location; the two-point problem is for an arbitrary source-receiver pair. We present several techniques which provide stability in velocity models producing caustics: improved factorization; non-symmetric loss function based on Hamiltonian; gaussian activation; symmetrization. In our tests, NES showed the relative-mean-absolute error of about 0.2-0.4% from the second-order factored Fast Marching Method, and outperformed existing neural-network solvers giving 10-60 times lower errors and 2-30 times faster training. The inference time of NES is comparable with the Fast Marching. The one-point NES provides the most accurate solution, whereas the two-point NES provides slightly lower accuracy but gives an extremely compact representation. It can be useful in various seismic applications where massive computations are required (millions of source-receiver pairs): ray modeling, traveltime tomography, hypocenter localization, and Kirchhoff migration.Comment: The paper has 14 pages and 6 figures. Source code is available at https://github.com/sgrubas/NE

Hybrid Kinematic-Dynamic Approach to Seismic Wave-Equation Modeling, Imaging, and Tomography

Estimation of the structure response to seismic motion is an important part of structural analysis related to mitigation of seismic risk caused by earthquakes. Many methods of computing structure response require knowledge of mechanical properties of the ground which could be derived from near-surface seismic studies. In this paper we address computationally efficient implementation of the wave-equation tomography. This method allows inverting first-arrival seismic waveforms for updating seismic velocity model which can be further used for estimating mechanical properties. We present computationally efficient hybrid kinematic-dynamic method for finite-difference (FD) modeling of the first-arrival seismic waveforms. At every time step the FD computations are performed only in a moving narrowband following the first-arrival wavefront. In terms of computations we get two advantages from this approach: computation speedup and memory savings when storing computed first-arrival waveforms (it is not necessary to make calculations or store the complete numerical grid). Proposed approach appears to be specifically useful for constructing the so-called sensitivity kernels widely used for tomographic velocity update from seismic data. We then apply the proposed approach for efficient implementation of the wave-equation tomography of the first-arrival seismic waveforms

Extended structure tensors for multiple directionality estimation

Standard structure tensors provide a robust way of directionality estimation of waves (or edges) but only for the case when they do not intersect. In this work, a structure tensor extension using a one-way wave equation is proposed as a tool for estimating directionality in seismic data and images in the presence of conflicting dips. Detection of two intersecting waves is possible in a two-dimensional case. In three dimensions both two and three intersecting waves can be detected. Moreover, a method for directionality filtering using the estimated directions is proposed. This method makes use of the ideas of a one-way wave equation but can be applied to generic images not related to wave propagation

Parallel algorithm of 3D wave-packet decomposition of seismic data: implementation and optimization for GPU

In this paper, we consider 3D wave-packet transform that is useful in 3D data processing. This transform is computationally intensive even though it has a computational complexity of O(N3 log N). Here we present its implementation on GPUs using NVIDIA CUDA technology. The code was tested on different types of graphical processors achieving the average speedup up to 46 times on Tesla M2050 compared to CPU sequential code. Also, we analyzed its scalability for several GPUs. The code was tested for processing synthetic seismic data set: data compression, de-noising, and interpolation

Prestack shot-gather depth migration by a `rigid' flow of Gaussian wave packets

Abstract in UndeterminedWe introduce a new method for prestack depth migration of seismic common-shot gathers.The computational procedure follows standard steps of the reverse-time migration, i.e.,downward continuation of the `source' and the `receiver' waveelds, followed by applicationof an imaging condition (e.g. zero-lag cross-correlation of these elds). In our method werst nd a sparse data representation with a small number of Gaussian wave packets. Wethen approximate the downward waveeld propagation (for the `source' and the `receiver'elds) by a `rigid'ow of these wave packets along seismic rays. In this case, the wavepackets are simply translated and rotated according to the ray geometry. One advantagewith using Gaussian wave packets is that analytic formulas can be used for translation, rotation, and the application of the cross-correlation imaging condition. Moreover, theyallow more sparse representations than competing methods. In combination, this yields acomputationally and memory ecient migration procedure, as only few rays have to betraced, and since it is cheap to compute the cross-correlation for the intersecting rays

The structure-tensor analysis for optimal microseismic data partial stack

Microseismic monitoring of hydrofrac is an actively developing technology utilizing various acquizition arrays. In this paper we consider processing of microseismic data recorded by specific surface network geometry-patch arrays (far separated local receiver groups). The project aim is to produce an optimal partial stacking of the data within patches for improving a signal to noise ratio for microseismic events detection and location. We propose to use a structure-tensor analysis for estimating directions of coherency in the data, which can be used for data stacking for each patch. Unlike to the standard slantstacking method, we do not scan all possible directions, but receive them as eigenvectors of the structure tensor. We used the synthetic data for testing our approach in presence of random and coherent noise, in the case of interfering events. The testing showed that the structure-tensor analysis provides robust coherent summation results. We also discuss the usefulness of the structure-tensor attributes for detecting (triggering) the arriving wave and separating body wave from surface waves based on the apparent velocity