Minimum Phase Wavelet Research Articles

English

The effect of wavelet phase rotation on post inversion in X-field Niger Delta has been investigated. This paper demonstrates the influence of wavelet phase employed in model based seismic inversion. It presents the problem associated with variation in wavelet phases employed in post-stack model based seismic inversion which affect the discrimination of litho-fluid contrast from inverted seismic volume. The study was carried out using 3D post stack seismic data and well logs from three (3) exploratory wells in the study area. The data was inverted into acoustic impedance (P–impedance) using five different wavelets with 0, 15, 30, 45 and 60° phases for each inversion process. The inverted P-impedance slices result for the different wavelet phases were compared and the effect of each wavelet on the inverted section was deciphered. Small decrease in impedance error was observed for zero degree (0°) and minimum (15 to 30°) wavelet phase. Alternatively, inverted P–impedance section using 45 and 60° wavelets phase depict high error in the impedance and less correlation coefficient with increase in the synthetic error. This study thus recommends zero degree (0°) or minimum phase (15 to 30°) wavelets for interpretation in the study area as the degree of variation in input wavelet greatly affects the inverted acoustic impedance results. Key words: Niger Delta, wavelet, well logs, seismic data, stack inversion

Combining CEEMD and recursive least square for the extraction of time-varying seismic wavelets

The vertical resolution of seismic data is greatly affected by time-varying seismic wavelets. To break through the shape restriction in amplitude estimation and the dependency upon the constant or piecewise stationary assumptions in phase estimation in the existing methods, we propose a time-varying wavelet extraction method combining CEEMD (Complementary Ensemble Empirical Mode Decomposition) and RLS (Recursive Least Square). According to the differences that the amplitude spectra of wavelets and reflection coefficient are smooth and oscillating respectively, the logarithm amplitude spectra of the seismogram at different times are decomposed into multi-layer components with different oscillation scales by CEEMD, and the amplitude spectra of time-varying wavelets can be estimated by filtering the oscillating components and reconstructing the smooth components. Hilbert transform is applied to the estimated amplitude spectra to build time-varying minimum-phase wavelets, and the system functions and pole-zeros in Z domain of minimum-phase wavelets can be estimated using RLS. Finally, time-varying mixed-phase wavelets are extracted by reconfiguring the pole-zeros and determining the optimal combinations under the constraint of local similarity. This method does not need to calculate the Q value. The numerical simulation and real seismic data processing results demonstrate that the proposed method can improve the accuracy of time-varying wavelet extraction compared to the conventional method.

Application of minimum phase wavelet for generation of synthetic seismic signals

Seismic signals are analyzed by using different wavelet transforms to evaluate the magnitude of the signal. Through experimentation, it is reported that surface wave magnitude is greater than equals to three, it would affect the environment. In this paper, an attempt is made to apply minimum phase wavelet technique to analyze the seismic signals through minimum phase wavelets using CREWES data.

Design of Time–Frequency-Localized Two-Band Orthogonal Wavelet Filter Banks

In this paper, we design time–frequency-localized two-band orthogonal wavelet filter banks using convex semidefinite programming (SDP). The sum of the time variance and frequency variance of the filter is used to formulate a real symmetric positive definite matrix for joint time–frequency localization of filters. Time–frequency-localized orthogonal low-pass filter with specified length and regularity order is designed. For nonmaximally regular two-band filter banks of length twenty, it is found that, as we increase the regularity order, the solution of the SDP converges to the filters with time–frequency product (TFP) almost same as the Daubechies maximally regular filter of length twenty. Unlike the class of Daubechies maximally regular minimum phase wavelet filter banks, a rank minimization algorithm in a SDP is employed to obtain mixed-phase low-pass filters with TFP of the filters as well as the scaling and wavelet function better than the equivalent two-band Daubechies filter bank.

Comparison between deterministic and statistical wavelet estimation methods through predictive deconvolution: Seismic to well tie example from the North Sea

Wavelet estimation as well as seismic-to-well tie procedures are at the core of every seismic interpretation workflow. In this paper we perform a comparative study of wavelet estimation methods for seismic-to-well tie. Two approaches to wavelet estimation are discussed: a deterministic estimation, based on both seismic and well log data, and a statistical estimation, based on predictive deconvolution and the classical assumptions of the convolutional model, which provides a minimum-phase wavelet. Our algorithms, for both wavelet estimation methods introduce a semi-automatic approach to determine the optimum parameters of deterministic wavelet estimation and statistical wavelet estimation and, further, to estimate the optimum seismic wavelets by searching for the highest correlation coefficient between the recorded trace and the synthetic trace, when the time–depth relationship is accurate. Tests with numerical data show some qualitative conclusions, which are probably useful for seismic inversion and interpretation of field data, by comparing deterministic wavelet estimation and statistical wavelet estimation in detail, especially for field data example. The feasibility of this approach is verified on real seismic and well data from Viking Graben field, North Sea, Norway. Our results also show the influence of the washout zones on well log data on the quality of the well to seismic tie.

Nonminimum phase deconvolution in the log domain: A sparse inversion approach

Being disturbed by the discrepancy between the Ricker wavelet and minimum phase wavelets, we wondered if a sparseness criterion could get us deconvolved data with the event polarity being more clearly evident. Five data sets found it does. The sparseness criterion we used is a hyperbolic penalty function. It ranged from [Formula: see text] at small residuals to [Formula: see text] at large residuals. The main pitfall was that introducing negative filter lags introduced a null space (obviously so for Gaussian data). The null space demanded a regularization. We found a formulation in the domain of the Fourier transform of a log spectrum, in which a Ricker-style regularization appeared. Curiously, this regularization eliminated the leg jumps. A quasi-Newton solver was faster than that of our earlier work, a combination of conjugate directions with a Newton solver.

Application of Gabor deconvolution to the correction of nonstationarity for reasonable reflectivity estimations

Statistical white reflectivity, minimum-phase wavelets, and stationarity are regarded to be three major shortcomings in dealing with the Robinson convolution model. Modern reflectivity inversion methods (e.g., constrained reflectivity inversion) mainly attempt to reduce side effects of the first two assumptions, but most of them overlook the important fact that seismic signals are typically nonstationary. Numerical tests and practical applications have verified that nonstationarity does have great influence on reflectivity estimation and probably would result in misplaced positions or wrongly restored amplitudes for estimated reflectivity. Nonstationarity can be tackled from a perspective of hyperbolic smoothing in Gabor deconvolution. Specially, the energy relationship of hyperbolic strips in a log spectrum can be taken as a quantitative indicator in balancing nonstationarity and conditioning seismic traces to the assumption of unchanging wavelets. Applications to marine seismic data show that balancing nonstationarity finally helps to recover subtle reflectivity information and promotes better displays of geologic features of the subsurface.

Nonstationary sparsity-constrained seismic deconvolution

The Robinson convolution model is mainly restricted by three inappropriate assumptions, i.e., statistically white reflectivity, minimum-phase wavelet, and stationarity. Modern reflectivity inversion methods (e.g., sparsity-constrained deconvolution) generally attempt to suppress the problems associated with the first two assumptions but often ignore that seismic traces are nonstationary signals, which undermines the basic assumption of unchanging wavelet in reflectivity inversion. Through tests on reflectivity series, we confirm the effects of nonstationarity on reflectivity estimation and the loss of significant information, especially in deep layers. To overcome the problems caused by nonstationarity, we propose a nonstationary convolutional model, and then use the attenuation curve in log spectra to detect and correct the influences of nonstationarity. We use Gabor deconvolution to handle nonstationarity and sparsity-constrained deconvolution to separating reflectivity and wavelet. The combination of the two deconvolution methods effectively handles nonstationarity and greatly reduces the problems associated with the unreasonable assumptions regarding reflectivity and wavelet. Using marine seismic data, we show that correcting nonstationarity helps recover subtle reflectivity information and enhances the characterization of details with respect to the geological record.

Automatic detection of interfering seismic wavelets using fractal methods

Interference of reflections from upper and lower interfaces of geological units is a common source of interpretational ambiguity. This paper utilizes the fractal dimension of instantaneous seismic attributes to study this interfering phenomenon. Using a 1D synthetic seismic trace, we examine the divider, Hurst, spectral and variance fractal methods. Since each method utilizes a moving window technique as part of its analysis, a series of optimum window lengths needs to be determined. The four methods are applied to the instantaneous amplitude, phase and frequency attributes of a series of 1D synthetic seismic traces. To study wavelet interference on 2D seismic sections, we generate four 2D synthetic seismic sections by convolving a zero phase and a minimum phase wavelet with two different wedge models. For the first model, the reflection coefficient of the upper wedge interface is positive and the lower interface negative, whereas for the second model, the polarities are positive. Our results suggest that fractal analysis of the instantaneous phase, using the Hurst method, can separate two opposite polarity wavelets within a distance of λ/16 or greater and of λ/4 or greater when they have the same polarity. Our results are not sensitive to random noise when the signal-to-noise-ratio is greater than 10.

Single station estimation of seismic source time function from coda waves: The Kursk disaster

[1] We propose a high-resolution technique for estimating the source time function of a seismic event from only one record. This technique is based on the spectral factorization of the minimum phase wavelet from the most random part of a seismogram: its coda. As the coda non-stationarity is inconsistent with the classical spectral factorization theory, we develop a two-step algorithm: first, the diffuse coda field is whitened to remove the non-stationary attenuation effect; second, the minimum phase wavelet equivalent of the seismic source time function is estimated. Applied to the recordings of the Kursk's wreck, this method gives a source wavelet strikingly similar to the general shape of an underwater explosion, allowing us to infer its depth and yield. Based on the fundamental “random” character of diffusive waves, this approach opens up promising applications for new blind deconvolution methods.

Interpretive advantages of 90°-phase wavelets: Part 1 — Modeling

We discuss, in a two-part article, the benefits of 90°-phase wavelets in stratigraphic and lithologic interpretation of seismically thin beds. In Part 1, seismic models of Ricker wavelets with selected phases are constructed to assess interpretability of composite waveforms in increasingly complex geologic settings. Although superior for single surface and thick-layer interpretation, zero-phase seismic data are not optimal for interpreting beds thinner than a wavelength because their antisymmetric thin-bed responses tie to the reflectivity series rather than to impedance logs. Nonsymmetrical wavelets (e.g., minimum-phase wavelets) are generally not recommended for interpretation because their asymmetric composite waveforms have large side lobes. Integrated zero-phase traces are also less desirable because they lose high-frequency components in the integration process. However, the application of 90°-phase data consistently improves seismic interpretability. The unique symmetry of 90°-phase thin-bed response eliminates the dual polarity of thin-bed responses, resulting in better imagery of thin-bed geometry, impedance profiles, lithology, and stratigraphy. Less amplitude distortion and less stratigraphy-independent, thin-bed interference lead to more accurate acoustic impedance estimation from amplitude data and a better tie of seismic traces to lithology-indicative wireline logs. Field data applications are presented in part 2 of this article.

Zero-phasing the zero-phase source

This paradoxical title is in fact a tribute to the classic paper by Gibson and Larner, “Predictive deconvolution and the zero-phase source” (Geophysics, 1984). Their work was originally presented at the 1982 SEG Annual Meeting, which means that it has been almost 20 years since they exposed the problems associated with controlling the phase of vibroseis records, and we are still struggling with them. The main issue is not related to the zero-phase Klauder wavelet itself but rather with the rest of the recording chain which contains a series of minimum-phase filters (such as the geophone response and possibly the coupling conditions). The resulting seismic wavelet is therefore mixed phase, thus making it virtually impossible to process correctly and obtain final zero-phase records with standard deconvolution processing. To circumvent this problem, Gib-son and Larner recommended transforming the vibroseis data into minimum-phase records by applying a deterministic filter based on the known Klauder wavelet. The resulting records can then be processed within a controlled-phase sequence similar to that used for dynamite data. However, there is no clear minimum-phase equivalent for the vibroseis wavelet, and Gibson and Larner suggested that such a “beast” might not even exist. Figure 1, taken from their paper, shows four different minimum-phase equivalents of a Klauder wavelet, corresponding to four values of an obscure parameter in the minimum-phase transform. If the minimum-phase equivalent of the vibroseis source is nonunique, how can we guarantee a final zero-phase result? Figure 1. Klauder wavelet (top) and four minimum-phase equivalent wavelets generated using the indicated amounts of white-noise. (From Gibson and Larner.) This paper follows up the aforementioned classic. I will show that although a unique minimum-phase equivalent exists, it is neither practical nor necessary to compute. In-stead, I will show a simple and effective method to process vibroseis data …

The influence of reflection coefficient statistics on the seismic method: scattering and the minimum detectable reflection coefficient

Propagation of seismic reflection energy through layered media is discussed in terms of one-dimensional elastic scattering and the effect of a layered overburden on the detectability of the underlying target horizons is investigated. In a previous paper, using Walden & Hosken’s statistical models of real reflection series, Q -like attenuation laws were derived for the two-way transmission. Considerable use was made of the O’Doherty and Anstey relation between the amplitude spectrum of the two-way transmission and the energy spectrum of the reflection coefficients. With reference to the seismic reflection bandwidth, in general the equivalent Q is seen to increase with frequency, except over an intermediate band of frequencies where it often decreases with frequency. Also, the minimum phase wavelet predicted by the theory was shown to model adequately the first pulse of the two-way transmission waveform, carrying the greater part of the energy, and the lag of the first peak was approximately described in terms of the statistical parameters of the reflection coefficients in the overburden. The spectrum of the back-scattered energy can be determined from the conservation of energy into and out of the overburden and is seen to be complementary to the forward scattered signal and thus it can also be described in term s of the Walden and Hosken statistical parameters. The back-scattered energy can be divided into two components: (i) the primary reflections from within the overburden together with their associated short period multiples and (ii) the long period internal multiple noise which may arrive at the same time as the reflections from the underlying target horizons, obscuring them . The ratio of forward-scattered signal to back-scattered noise is a function of frequency and travel time through the overburden and it sets a fundamental signal-to-noise ratio for the section. An approximate expression is derived for the signal-to-noise peak power ratio which we use to determine both a natural cut-off frequency above which the noise dominates over signal from a given target strength, and α min , the minimum reflection coefficient detectable below the overburden. α min is seen to depend on the statistical properties of the overburden and can usually be decreased by decreasing the high and low-cut frequencies of the seismic bandwidth. There is thus a trade-off between detectability level and resolution.

The influence of reflection coefficient statistics on the seismic method: scattering attenuation and transmission wavelets

The propagation through layered media of seismic energy from reflection seismic surveys is discussed in terms of one dimensional elastic scattering. The effect of a layered overburden on the detectability of the underlying target horizons is investigated. The required signal from the target reflectors arises from the two-way forward-scattered component whereas the internal multiple noise (which tends to obscure the target reflections) arises from the back-scattered component. The starting point of the investigation is the O’Doherty-Anstey relation for the two-way transmission response. In this paper, using statistical models of real reflection series, we derive Q -like attenuation laws for the two-way transmission. Most real sequences of reflection coefficients have spectra which rise with frequency in the seismic band and this leads to signal attenuation which only approximates to that of a 'constant Q ’ type over small bands of frequency. The implications of the theory are checked for two very different types of overburden, one being a repetitive type of sedimentary sequence with a large mean square reflection coefficient and the other a non-repetitive sequence with a small mean square reflection coefficient, against synthetic seismograms derived from real sonic logs. The minimum phase wavelet predicted by the theory is shown to model adequately the first pulse of the two-way transmission waveform, carrying the greater part of the energy, and the lag of the first peak is given approximately in terms of the statistical parameters of the reflection coefficients in the overburden.

Minimum phase wavelet by ARMA factorization

An algorithm is presented for computing a minimum phase wavelet, given only the causal part of its autocorrelation function r/sup +/ (k),k>or=0. The algorithm falls in the category of spectral factorization techniques, with the difference that instead of factoring the symmetric autocorrelation, its ARMA model is factored, and the ARMA model for symmetric autocorrelation is obtained directly from that of r/sup +/ (k) via a simple identity. It is found that, at least in seismic context, this procedure works better than the conventional spectral factorization as it involves ARMA polynomials which are of much lower order than MA polynomials. The algorithm is supported by two theorems and a detailed numerical example. The treatment is essentially deterministic.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Optimal Identification of Non-minimum Phase ARMA (P, Q) Seismic Wavelet

This paper is dealing with the application of parametric methods to the classical seismic problem. A new maximum-likelihood algorithm is proposed, which gives a fast and globally optimal identification of non-minimum phase seismic wavelets. As ARMA(p, q) model is used, we can identify the MA part beyond the limit of the AH order so that the wavelet model can more easily represent a geophysical seismic wavelet. First, we find a minimum phase wavelet to estimate the AA part. Then, we search for the globally optimal MA parameters by using the zero-discrimination method the resulting algorithm is 2p times faster than Mendel's method. Moreover the solution is globally optimal and can avoid suboptimal solution that may result from other methods. Identification in synthetic data demonstrates the effectiveness and efficiency of the algorithm.

Deconvolution filter in seismic data processing.

Information of importance in reflection seismic method resides in the reflectivity series. In order to extract this information from obtained data, a deconvolution filter is the core of seismic data processing. Since whitening deconvolution was first introduced by Robinson1), it has been used in the oil industry on a routine basis. Whitening deconvolution is based on following assumptions: (1) A reflectivity series is an uncorrelated random series. (2) A seismic wavelet is minimum phase. Deconvolution outputs, however, ofter show that the minimum phase assumption on seismic wavelet is incorrect. In this paper we deeply investigate in depth this assumption and consider several types of deconvolution methods. Predictive deconvolution can be applied to both minimum and non-minimum phase wavelets and it is equivalent to phase correction method2), 3) for Vibroseis data, in a mathematical sense. In the case of whitening deconvolution, to obviate the minimum phase assumption, a statistical wavelet estimation approach, 4) using property of reflectivity series, is explained.

The effects of noise on minimum‐phase Vibroseis deconvolution

Ristow and Jurczyk (1975) proposed a mixed‐phase Vibroseis® inverse filter which is the usual minimum phase spiking deconvolution filter convolved with a Weiner‐Levinson minimum phase wavelet having the same amplitude spectrum as the Vibroseis wavelet. A problem exists since (for large time‐bandwidth products) the Vibroseis signal approximates a band‐limited signal and noise may have to be added to ensure convergence of the Wiener‐Levinson algorithm. This processing noise level can alter the resulting minimum phase wavelet. Since the deconvolution filter is influenced by the ambient or environmental noise as well as by the processing noise, the proposed correction to spiking deconvolution may not always yield meaningful results. It is shown that although the Vibroseis wavelet may span several octaves, it is not only band‐limited but can be approximated by a narrow‐band signal representation. In this formulation, the center frequency for the wavelet is considered to be the average of the high and low frequencies. The phase associated with this center frequency is independent of time but depends upon both the signal bandwidth and the deconvolution noise platform. Finally, this paper examines distortions in the deconvolved wavelet arising from both processing and environmental noise‐induced variations in the phase and envelope delay. The reflection sequence is assumed to be white, and a minimum phase, nearly constant Q, earth model is assumed. Curves are presented which show the residual phase error as a function of attenuation, processing noise, and the environmental signal‐to‐noise (S/N) ratio. It was found that although both types of noise will cause some residual phase error, phase compensation can correct for many of the phase distortions and polarity reversals that may be present in the deconvolved data only if the environmental noise is smaller than the processing noise.

THE PHASE PROPERTY OF A FINITE REALIZATION OF RANDOM NOISE AND ITS INFLUENCE ON DECONVOLUTION *

AbstractA finite realization of a discrete random noise process may be considered as a one‐sided energy signal. Its phase property can then be described by means of the center position. The samples of such a realization are the components of a random signal vector and the center position is therefore a random variable. A statistical analysis shows that the expected value of the center position equals half the time duration of the realization. This implies that the Z‐transform of the realization may be expected to have an equal number of poles and zeros inside and outside the unit circle. The standard deviation from the expected value of the center position is shown to depend on the time duration of the realization and on the autocorrelation of the process. It follows that, for processes that can be described by the convolution of a white series and a disturbance wavelet, the center position is independent of the phase property of the wavelet. A conclusion based on these results is that the homomorphic technique of wavelet estimation through cepstrum stacking must give questionable outcomes. Another conclusion is that the super‐position of a realization of random noise on a minimum phase wavelet will in general give a mixed phase resulting signal. It is pointed out that schemes for the derivation of deconvolution filters do not take account of this phenomenon.

Possible instability in the Burg maximum entropy method.

The maximum entropy method (MEM) suggested by Burg is claimed to give power spectral estimates with high resolution, especially for short records. However, we can produce examples for which Burg's algorithm gives unrealistic spectra. It is shown that the algorithm fails when applied to minimum phase wavelets. A new algorithm, one-sided Burg algorithm, is proposed to circumvent this difficulty. Both the new and original algorithms give reasonable spectral estimates when they are applied to stationary data.