Anton A. Duchkov

Discrete Almost-Symmetric Wave Packets and Multiscale Geometrical Representation of (Seismic) Waves

We discuss a multiscale geometrical representation of (seismic) waves through a decomposition into wave packets. Wave packets can be thought of as certain localized “fat” plane waves. Here, we construct discrete almost-symmetric 3-D wave packets by using the unequally spaced fast Fourier transform. The resulting discrete transform is unitary, implying that the reconstruction operator is simply the adjoint of the decomposition operator. Another relevant aspect of the discretization scheme is the appearance of parameters that control the tiling of the phase space that corresponds with the dyadic parabolic decomposition, preserving the relative parabolic scaling while adapting to the physical problem at hand. We consider applications in exploration and global seismology, in particular for higher dimensional data regularization, seismic map migration, denoising, directional regularity analysis, and feature extraction.

Extended isochron rays in prestack depth (map) migration

Many processes in seismic data analysis and seismic imaging can be identified with solution operators of evolution equations. These include data downward continuation and velocity continuation. We have addressed the question of whether isochrons defined by imaging operators can be identified with wavefronts of solutions to an evolution equation. Rays associated with this equation then would provide a natural way of implementing prestack map migration. Assuming absence of caustics, we have developed constructive proof of the existence of a Hamiltonian describing propagation of isochrons in the context of common-offset depth migration. In the presence of caustics, one should recast to a sinking-survey migration framework. By manipulating the double-square-root operator, we obtain an evolution equation that describes sinking-survey migration as a propagation in two-way time with surface data being a source function. This formulation can be viewed as an extension of the exploding reflector concept from zero-o...

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.

Sparse wave-packet representations and seismic imaging

In this paper we introduce a new algorithm for seismic imaging based on the flow out of Gaussian wave packets. We follow the standard strategy of decomposing data into wave packets and flowing them out along rays to approximate the downward wavefield extrapolation, and finally applying an imaging condition. We revisit each computational step to gain efficiency. Furthermore, we develop procedure for seismic data decomposition in order to obtain highly sparse representations with Gaussian wave packets. As a result we get fast algorithm heavily exploiting sparse data representation and analytic description of Gaussian wave packets. We tests our algorithm on synthetic example of migrating common-shot gather.

Processing microseismic monitoring data, considering seismic anisotropy of rocks

Using the model information and in situ data on hydrofracturing in an oil and gas reservoir of the Bakken Formation (USA), potential of locating hypocenters of microseismic events concurrently with determining parameters of velocity anisotropy of seismic waves in rock mass is analyzed. It is shown that inclusion of anisotropy in the analysis improves accuracy of spatial location of microseismic event hypocenters and increases validity of estimation of the fracturing direction.

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.

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.

Seismic Wave Field in the Vicinity of Caustics and Higher-Order Travel Time Derivatives

In this paper we consider time-domain asymptotic solutions to the elastodynamic equation. We briefly describe the technique of constructing and analyzing the integral formulas describing seismic wave fields. The formulas are valid at regular points and in the vicinity of caustics.The analysis is performed locally, along one ray. Near a caustic we start with the Kirchhoff-type integral representation. The incident wave and Greens tensor (their discontinuous part or time-domain asymptotic) should be known at some previous regular point of the ray. Finally, we arrive at the integral description valid in the vicinity of caustics (time-domain equivalent of the oscillatory integral). This approach requires calculating higher-order derivatives of the travel time and ray amplitude along the ray. These derivatives may be found by solving differential equations. The equations are given in explicit form and can be used for calculations in isotropic media.

Analysis of seismic wave dynamics by means of integral representations and the method of discontinuities

We analyze the dynamics (amplitudes and phase distortions) of seismic waves as they propagate along the ray. Our analysis is performed via a ray series approximation in the time domain. That is, we concentrate on characterizing the sharp changes (discontinuities) of the signal that are localized near the wavefront. After convolution of the terms of such a series with a proper temporally short (high-frequency) wavelet, one obtains a synthetic seismic signal at a given point of interest. We present an outline of the proposed technique that yields integrals describing the wavefield. These integrals are similar to oscillatory integrals in the frequency domain. This description is uniformly valid near caustics, allowing the calculation of higher order terms of the ray series approximation. Practical use of the technique is illustrated by several examples which show two possible uses of the technique: general understanding of what is happening during wave propagation and practical calculations. First, we show how the structure of the ray decomposition changes near the simple caustic, and then we calculate a synthetic signal near the cusp caustic. The advantage of the technique is that the problem of seismic wave calculation is technically reduced to a problem of double integration of a Dirac δ-function; thus, it is computationally effective.

Equipment for microseismic monitoring of geodynamic processes in underground hard mineral mining

The article describes engineering decisions on equipment for acquisition of microseismicity data, that improve information content of microseismic monitoring of geodynamic processes in underground hard mineral mining.