Reflection Coefficient Sequence Research Articles

A general method for extracting the amplitude spectrum of the seismic wavelet from the seismic traces

When solving geophysical problems using reflection seismology, extracting the amplitude spectrum of the seismic wavelet (ASSW) from a seismic trace is the basis for many downstream works, such as deconvolution and sparse reflection coefficient inversion. This paper pertains to statistical methods for estimating ASSW, that is, separating the scattering effect of the formation from the amplitude spectrum of a seismic trace to obtain a band-limited ASSW. Common methods assume the reflection coefficient sequence (RCS) is white, making them unsatisfactory in many cases. Gao et al (2017 Inverse Problems 33 085005) proposed an ASSW extraction method based on contraction operator mapping (COM method) and experimentally proved it effective for non-white RCSs. However, this method is only valid for unimodal ASSW, which makes it unsuitable for most sources. We propose a general method for estimating ASSW which does not require the whiteness assumption of the RCS and does not restrict the specific form of ASSW. We show that the COM method has function representation limitations and redefines a new operator, proving that it is also a contractive mapping. Our method can be regarded as a general form of the COM method, filling the theoretical gap that existing methods cannot be applied to non-white RCS and non-unimodal ASSW. It provides a general method for the estimation of ASSW. We thoroughly verify this generality by simulating different source types while considering wavelet attenuation. Compared with the COM method, the proposed method has absolute advantages.

Prestack seismic inversion for elastic parameters using model-data-driven generative adversarial networks

Model-data-driven (MDD) generative adversarial networks (GANs) using prestack seismic data to estimate elastic parameters are proposed. First, by traverse sampling of elastic parameter model space and Gaussian sampling, a complete elastic parameter data set is generated. Second, several reflection coefficient sequences are constructed to produce synthetic prestack amplitude-versus-angle (AVA) gathers using the Zoeppritz equations. Third, conditional GANs (CGANs) are trained using synthetic data sets to establish a relationship between the synthetic prestack AVA gathers and elastic parameters. By using low-frequency constraints, the absolute values of the elastic parameters from the prestack AVA gathers are computed. To apply the networks trained with synthetic to field seismic data, Gaussian noise is added to the synthetic prestack AVA gathers of the training data set. Synthetic prestack AVA gathers with different signal-to-noise ratios are used to verify the robustness of the proposed method and explore its generalization to field seismic data. By testing MDD-CGANs applied to a 2D elastic parameter model, the inversion results by MDD-CGANs are closer to the true values than those by the limited-data-driven CGANs and fully convolutional networks. By applying MDD-CGANs to field prestack AVA gathers in the Tarim Basin, West China, more accurate and high-resolution results are obtained than those obtained using the conventional prestack amplitude-variation-with-offset elastic parameters inversion approach.

A Sparse Spike Deconvolution Algorithm Based on a Recurrent Neural Network and the Iterative Shrinkage-Thresholding Algorithm

Conventional sparse spike deconvolution algorithms that are based on the iterative shrinkage-thresholding algorithm (ISTA) are widely used. The aim of this type of algorithm is to obtain accurate seismic wavelets. When this is not fulfilled, the processing stops being optimum. Using a recurrent neural network (RNN) as deep learning method and applying backpropagation to ISTA, we have developed an RNN-like ISTA as an alternative sparse spike deconvolution algorithm. The algorithm is tested with both synthetic and real seismic data. The algorithm first builds a training dataset from existing well-logs seismic data and then extracts wavelets from those seismic data for further processing. Based on the extracted wavelets, the new method uses ISTA to calculate the reflection coefficients. Next, inspired by the backpropagation through time (BPTT) algorithm, backward error correction is performed on the wavelets while using the errors between the calculated reflection coefficients and the reflection coefficients corresponding to the training dataset. Finally, after performing backward correction over multiple iterations, a set of acceptable seismic wavelets is obtained, which is then used to deduce the sequence of reflection coefficients of the real data. The new algorithm improves the accuracy of the deconvolution results by reducing the effect of wrong seismic wavelets that are given by conventional ISTA. In this study, we account for the mechanism and the derivation of the proposed algorithm, and verify its effectiveness through experimentation using theoretical and real data.

A new approach for extracting the amplitude spectrum of the seismic wavelet from the seismic traces

In reflection seismology, knowing the seismic wavelet is important both for processing seismic data and for modeling the seismic response. There are two approaches to obtain the seismic wavelet. One approach is deterministic and the other is statistic. This work belongs to the second category. A seismic wavelet is determined by the product of amplitude spectrum and phase spectrum. A conventional method uses a two-step procedure to estimate the seismic wavelet. The first step is for the amplitude spectrum, and the second step is for the phase spectrum. So extracting the amplitude spectrum of a seismic wavelet (ASSW) from the amplitude spectrum of a seismic trace is a key step. The commonly used methods are correlation-based method, the log-spectrum-averaging method and spectrum-shaping method. All these methods assume the reflection coefficient sequence is white, which may not be valid under some conditions.In this paper, we propose a new approach to obtain ASSW without the whiteness assumption about reflectivity. We define an operator on a properly chosen function space and prove that the operator is contractive in this space. Then, we convert the problem of estimating the ASSW into one of finding the fixed point of the operator. We give the algorithm in detail based on the contraction operator mapping (COM method). We compare our method with the widely used methods by synthetic signals in which the reflectivity is not white. The results show that our method performs satisfactorily for nonwhite reflection series, on which other methods do not work well. Moreover, our method is robust for the noise and the frequency interval on which COM works.

Improving the resolution of seismic traces based on the secondary time–frequency spectrum

The resolution of seismic data is critical to seismic data processing and the subsequent interpretation of fine structures. In conventional resolution improvement methods, the seismic data is assumed stationary and the noise level not changes with space, whereas the actual situation does not satisfy this assumption, so that results after resolution improvement processing is not up to the expected effect. To solve these problems, we propose a seismic resolution improvement method based on the secondary time–frequency spectrum. First, we propose the secondary time-frequency spectrum based on S transform (ST) and discuss the reflection coefficient sequence and time-dependent wavelet in the secondary time–frequency spectrum. Second, using the secondary time–frequency spectrum, we design a twodimensional filter to extract the amplitude spectrum of the time-dependent wavelet. Then, we discuss the improvement of the resolution operator in noisy environments and propose a novel approach for determining the broad frequency range of the resolution operator in the time–frequency–space domain. Finally, we apply the proposed method to synthetic and real data and compare the results of the traditional spectrum-modeling deconvolution and Q compensation method. The results suggest that the proposed method does not need to estimate the Q value and the resolution is not limited by the bandwidth of the source. Thus, the resolution of the seismic data is improved sufficiently based on the signal-to-noise ratio (SNR).

An Application of the Frequency-Domain Experience Mode Decomposition to Enhance Deconvolution Results

To improve the resolution of the seismic section after deconvolution, a method based on frequency-domain experience mode decomposition was proposed. Empirical mode decomposition (EMD) method is usually used to analyze the time domain non-stationary signal, in order to better recover original reflection coefficient sequence, empirical mode decomposition was implemented for frequency-domain amplitude spectrum. Through the different characteristics between the equivalent filter amplitude after deconvolution and reflection coefficient sequence amplitude in frequency-domain, the real reflection coefficient sequence was recovered. Simulation results indicate that the method is effective and feasible.

Ultrasonic Inspection of Composite Materials Using Minimum Entropy Deconvolution

In this work, the Minimum Entropy Deconvolution (MED) method, developed for ultrasonic signals, is used to address the problem of delamination defect detection in Composite Materials. Standard deconvolution techniques suppose that the wavelet is minimum phase but generally make no assumptions about the amplitude distribution of the primary reflection coefficient sequence. For a white reflection sequence the assumption of a Gaussian distribution means that recovery of the true phase of the wavelet is impossible; however, a non-Gaussian distribution in theory allows recovery of the phase. It is generally recognized that primary reflection coefficients typically have a non-Gaussian amplitude distribution. The minimum entropy deconvolution (MED) method supposes whiteness but seek to exploit the non-Gaussianity. This method do not assume minimum phase. The deconvolution filter is defined by the maximization of a function called the objective. The algorithm is tested on simulated data and also tested on real ultrasonic data from multilayered composite materials.

Reflection coefficient (schur parameter) representation for convex compact sets in the plane

We obtain a one-to-one relation between the shape and orientation of a convex compact planar set and a complex-valued reflection coefficient (Schur (1917) parameter) sequence, such that (1) the reflection coefficient magnitudes are less than or equal to one, (2) if any reflection coefficient has a magnitude equal to one, then all subsequent reflection coefficients are equal to zero, and (3) the first reflection coefficient is equal to zero. Three additional independent parameters specify the position of the set in the plane, and the size of the set (specifically its circumference). For a finite duration reflection coefficient sequence, if the last nonzero reflection coefficient has a magnitude that is less than one, then the boundary of the set is an infinitely differentiable convex curve. The boundary is a convex polygon if and only if the magnitude of the last reflection coefficient is equal to one; the number of sides of the polygon is equal to the index of the last reflection coefficient. Almost all planar convex compact sets have reflection coefficient sequences of infinite duration. Such sets can be accurately approximated with convex compact sets that are generated from relatively small numbers of reflection coefficients.

Ratio Asymptotics for Polynomials Orthogonal on Arcs of the Unit Circle

Ratio asymptotics for orthogonal polynomials on the unit circle is characterized in terms of the existence of limn |Φn(0)| and {limn [Φn+1(0)/Φn(0)] , where \( \{\Phi_n(0)\}_{n \geq 0} \) denotes the sequence of reflection coefficients. The limit periodic case, that is, when these limits exist for n = j mod k , j = 1, . . ., k , is also considered.

Optimal seismic deconvolution: distributed algorithms

Deconvolution is one of the most important aspects of seismic signal processing. The objective of the deconvolution procedure is to remove the obscuring effect of the wavelet's replica making up the seismic trace and therefore obtain an estimate of the reflection coefficient sequence. This paper introduces a new deconvolution algorithm. Optimal distributed estimators and smoothers are utilized in the proposed solution. The new distributed methodology, perfectly suitable for a multisensor environment, such as the seismic signal processing, is compared to the centralized approach, with respect to computational complexity and architectural efficiency. It is shown that the distributed approach greatly outperforms the currently used centralized methodology offering flexibility in the design of the data fusion network.

Spectral factorization technique for estimation of an ARMA operator for multichannel deconvolution of seismic data

A new spectral factorization method is presented for the estimation of a causal as well as a causally invertible ARMA operator from the correlation sequence of seismic traces. The method has been implemented for multichannel deconvolution of seismic traces with the aim of exploiting the trace‐to‐trace correlation that exists within seismograms. A layered earth model with a small reflectivity sequence has been considered, and the seismic traces have been considered as the output of a linear system driven by white noise reflection coefficient sequences. The present method is the concatenation of three algorithms, namely Kung's method for state variable (F,G,H) realization using a singular value decomposition (SVD) algorithm, Faurre's technique for computation of the strong spectral factor and Leverrier's algorithm for ARMA representation of the spectral factor. The inverted ARMA operator is used as a recursive filter for deconvolution of seismic traces. In the example shown, two traces with a covariance sequence of 160 ms length have been considered for multichannel deconvolution of stacked seismic traces. The results presented, when compared with those obtained from a conventional deconvolution algorithm, have shown encouraging prospects.

Asymptotic Behaviour of Orthogonal Polynomials on the Unit Circle with Asymptotically Periodic Reflection Coefficients

Let {an}n∈N0withan∈C,an+N=anand |an|<1 for alln∈N0, be a periodic sequence of reflection coefficients and let {Pn}n∈N0be the associated sequence of orthogonal polynomials generated byPn+1=zPn−ānP*n. Furthermore let {bn}n∈N0be an asymptotically periodic sequence of reflection coefficients which arises by a perturbation of the sequence {an}n∈N0and thus satisfies the conditions limν→∞bj+νN=ajforj=0, …, N−1, and |bn|<1 for alln∈N0. Let {Pn}n∈N0generated byPn+1=zPn−b&nP*nbe the disturbed orthogonal polynomials. Using the “periodic” polynomials {Pn}n∈N0as a comparison system we derive so-called comparative asymptotics for the disturbed polynomials on and off the support of the disturbed orthogonality measure, which consists essentially of several arcs of the unit circle. As a by-product of these results we obtain asymptotically a description of the location of the zeros of {Pn}n∈N0. Finally, a representation for the absolutely continuous part of the disturbed orthogonality measure is derived, and it is shown that there are at most finitely many point measures if thebn's converge geometrically fast to thean's.

Spectrum estimation using maximum entropy and multiresolution considerations

Given a sample autocovariance sequence of finite length for some observed random process, the spectrum estimation problem involves the extension of this sequence for the required Fourier transformation. The maximum entropy approach which is based on the optimal use of information contents, leads to a dual sequence of reflection coefficients with reciprocal spectrum of the process. The estimation of the maximum entropy spectrum implies results identical to those using autoregressive modeling in one dimension under appropriate white noise assumptions. In cases of a non-white noise component, the approach is generalized to an autoregressive-moving-average model. Recent developments in multiresolution analysis with spectral domain decompositions also offer possibilities of subband spectrum estimation for specific applications. Using a simulated data sequence with two close frequencies, the estimated spectrum from a two-level decomposition with autoregressive modeling shows better resolution than with conventional processing. Geodetic and geophysical applications are briefly indicated.

On the distribution of seismic reflection coefficients and seismic amplitudes

Reflection coefficient sequences from 14 wells in Australia have a statistical character consistent with a non‐Gaussian scaling noise model based on the Lévy‐stable family of probability distributions. Experimental histograms of reflection coefficients are accurately approximated by symmetric Lévy‐stable probability density functions with Lévy index between 0.99 and 1.43. These distributions have the same canonical role in mathematical statistics as the Gaussian distribution, but they have slowly decaying tails and infinite moments. The distribution of reflection coefficients is independent of the spatial scale (statistically self‐similar), and the reflection coefficient sequences have long‐range dependence. These results suggest that the logarithm of seismic impedance can be modeled accurately using fractional Lévy motion, which is a generalization of fractional Brownian motion. Synthetic seismograms produced from our model for the reflection coefficients also have Lévy‐stable distributions. These simulations include transmission losses, the effects of reverberations, and the loss of resolution caused by band‐limited wavelets, and suggest that actual seismic amplitudes with sufficient signal‐to‐noise ratio should also have a Lévy‐stable distribution. This prediction is verified using poststack seismic data acquired in the Timor Sea and in the continental USA. However, prestack seismic amplitudes from the Timor Sea are nearly Gaussian. We attribute the difference between prestack and poststack data to the high level of measurement noise in the prestack data. Many of the basic statistical techniques upon which seismic deconvolution and wavelet estimation are based presume implicitly that the underlying probability distribution has finite variance. Sample variance, autocorrelation, and even sample means are not reliable for infinite‐variance Lévy‐distributed variables. Although the seismic amplitudes and reflection coefficients are bounded and cannot truly have a Lévy‐stable distribution, some of their statistical properties are similar to those of Lévy‐stable variables. In other words, they behave as if they were following a Lévy‐stable distribution, which suggests that methods developed for Gaussian or near‐Gaussian variables may not be the most appropriate. The highly non‐Gaussian nature of the seismic amplitudes and the reflection coefficients should be considered in the design of seismic deconvolution and inversion techniques.

Multiscale signal processing: isotropic random fields on homogeneous trees

In this paper we consider isotropic processes indexed by the nodes of a homogeneous tree of order q. An (hanging) version of the qth order tree appears naturally when we consider successive filtering-and-decimation (by a factor of q) operations, as in multirate filtering and wavelet transforms. We derive Levinson and Schur recursions which provide us with a parametrization of an isotropic process via its reflection (or PARCOR) coefficient sequence. This can be done in an elegant way in the non-oriented setting, by making use of some general prediction errors. We state the counterpart of these results for the case of the tree, where we have forward and backward prediction errors, defined according to a notion of causality from coarse to fine scales. These oriented results represent generalizations of the Levinson and Schur recursions derived in for the case of the dyadic tree (i.e., q=2), which were used in to develop modeling and whitening filters for isotropic processes. >

The Schur Algorithm for Matrix-Valued Meromorphic Functions

This article extends the classical Schur algorithm to matrix-valued functions that are bounded on the unit circle and have a finite number of Smith–McMillan poles inside the unit disc. With each such function this article associates two infinite sequences: one is the well-known sequence of reflection coefficients (all less than one in magnitude), whereas the other is a sequence of signs. Under certain assumptions, the number of negative signs equals the number of poles within the unit disc. This article shows how to solve tangential interpolation problems using the algorithm and gives a simple proof for the connection between the number of poles inside the unit disc of each solution to the inertia of a certain Pick matrix. Also described is a numerically efficient procedure for carrying out the algorithm that involves only scalar operations.

THE KURTOSIS OF REFLECTION COEFFICIENTS IN A FRACTAL SEQUENCE OF SEDIMENTARY LAYERS

In the reflection seismic technique of hydrocarbon prospecting, the measured signal is the convolution of an unknown sequence of reflection coefficients by a statistically known source signature. A basic task of seismic processing is to estimate the sequence of reflection coefficients. The aim of this paper is to show that in case of a fractal sedimentation model, the conventional statistical descriptors (variance, skewness, kurtosis) of the reflection coefficient sequence are strongly dependent on, or are even divergent functions of, the resolution.

ARMA realization from the reflection coefficient sequence

The problem of ARMA modeling is focused in this paper. It is shown how to compute the ARMA parameters from a finite set of consecutive reflection coefficients in a completely recursive way. The algorithm consists of simple recursions for both the AR and MA parameters and it generalizes in a unifying framework other existing AR and MA algorithms.

Transmission through a Thue-Morse chain.

We study the reflection \ensuremath{\Vert}${\mathit{r}}_{\mathit{N}}$\ensuremath{\Vert} of a plane wave (with wave number k&gt;0) through a one-dimensional array of N \ensuremath{\delta}-function potentials with equal strengths v located on a Thue-Morse chain with distances ${\mathit{d}}_{1}$ and ${\mathit{d}}_{2}$. Our principal results are: (1) If k is an integer multiple of \ensuremath{\pi}/\ensuremath{\Vert}${\mathit{d}}_{1}$-${\mathit{d}}_{2}$\ensuremath{\Vert}, then there is a threshold value ${\mathit{v}}_{0}$ for v; if v\ensuremath{\ge}${\mathit{v}}_{0}$, then \ensuremath{\Vert}${\mathit{r}}_{\mathit{N}}$\ensuremath{\Vert}\ensuremath{\rightarrow}1 as N\ensuremath{\rightarrow}\ensuremath{\infty}, whereas if v${\mathit{v}}_{0}$, then \ensuremath{\Vert}${\mathit{r}}_{\mathit{N}}$\ensuremath{\Vert}?1. In other words, the system exhibits a metal-insulator transition at that energy. (2) For any k, if v is sufficiently large, the sequence of reflection coefficients \ensuremath{\Vert}${\mathit{r}}_{\mathit{N}}$\ensuremath{\Vert} has a subsequence \ensuremath{\Vert}${\mathit{r}}_{2}^{\mathit{N}}$\ensuremath{\Vert}, which tends exponentially to unity. (3) Theoretical considerations are presented giving some evidence to the conjecture that if k is not a multiple of \ensuremath{\pi}/\ensuremath{\Vert}${\mathit{d}}_{1}$-${\mathit{d}}_{2}$\ensuremath{\Vert}, actually \ensuremath{\Vert}${\mathit{r}}_{2}^{\mathit{N}}$\ensuremath{\Vert}\ensuremath{\rightarrow}1 for any v&gt;0 except for a ``small'' set (say, of measure 0). However, this exceptional set is in general nonempty. Numerical calculations we have carried out seem to hint that the behavior of the subsequence \ensuremath{\Vert}${\mathit{r}}_{2}^{\mathit{N}}$\ensuremath{\Vert} is not special, but rather typical of that of the whole sequence \ensuremath{\Vert}${\mathit{r}}_{\mathit{N}}$\ensuremath{\Vert}. (4) An instructive example shows that it is possible to have \ensuremath{\Vert}${\mathit{r}}_{\mathit{N}}$\ensuremath{\Vert}\ensuremath{\rightarrow}1 for some strength v while \ensuremath{\Vert}${\mathit{r}}_{\mathit{N}}$\ensuremath{\Vert}?1 for a larger value of v. It is also possible to have a diverging sequence of transfer matrices with a bounded sequence of traces.

Multiscale autoregressive processes. II. Lattice structures for whitening and modeling

For pt.I see ibid., vol.40, no.8, p.1915-34 (1992). Lattice structures for the whitening and modeling of isotropic processes on trees are developed, and a result relating the stability properties of these models to the reflection coefficient sequence introduced in pt.I is presented. This framework allows one to obtain a detailed analysis of the Wold decomposition of processes on trees. One interesting aspect of this is that there is a significantly larger class of singular processes on dyadic trees than on the integers. >