The effects of noise on minimum‐phase Vibroseis deconvolution

Geophysics ◽  
1982 ◽  
Vol 47 (8) ◽  
pp. 1174-1184 ◽  
Author(s):  
Samuel H. Bickel
Keyword(s):  
Phase Error ◽  
Amplitude Spectrum ◽  
Environmental Noise ◽  
Center Frequency ◽  
Earth Model ◽  
Minimum Phase ◽  
Low Frequencies ◽  
Residual Phase ◽  
Levinson Algorithm ◽  
Band Limited

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.

Air‐gun signatures and the minimum‐phase assumption

Geophysics ◽  
1992 ◽  
Vol 57 (2) ◽  
pp. 263-271 ◽  
Author(s):  
Neil D. Hargreaves
Keyword(s):  
Phase Error ◽  
Amplitude Spectrum ◽  
Phase Spectrum ◽  
Time Shift ◽  
Minimum Phase ◽  
Limited Bandwidth ◽  
Phase Errors ◽  
Phase Rotation ◽  
Air Gun ◽  
The Hilbert Transform

The air‐gun array signature is close to minimum‐phase as a function of continuous time, in the sense that for processing purposes its phase spectrum can be derived from the Hilbert transform of the logarithm of its amplitude spectrum. This phase spectrum is different, however, from the minimum‐phase spectrum that is estimated by spiking deconvolution for a sampled and time‐windowed version of the signature. As a consequence, there can be large phase errors when spiking deconvolution is applied to an air‐gun signature or to a recording instrument response. The errors can be shown to consist primarily of a time shift and, at least visually over a limited bandwidth, a phase rotation of the output wavelet. The time shift is introduced by time sampling, while the phase rotation is caused by the spectral smoothing generated by time windowing. If the seismic wavelet as a whole, and not just the air‐gun signature, is minimum‐phase, then the total residual phase error after spiking deconvolution, including also the error due to data noise, can also be shown to be close to a time shift and a phase rotation. This may be physical justification for the phase rotation schemes that are often successful in matching seismic data and well‐log synthetics. The minimum‐phase assumption can be used for statistical air‐gun array signature deconvolution, providing that a limited amount of deterministic information (the instrument slopes and the source and receiver depths in the approach used here) is available to guide the process in those areas of the spectrum that are critical to the phase computation. Date examples show that, with care, almost identical results can then be obtained from either purely statistical deconvolution or deterministic deconvolution plus statistical deconvolution of multiples and ghosting.

Q-factor estimation using bisection algorithm with power spectrum

Geophysics ◽  
2020 ◽  
Vol 85 (3) ◽  
pp. V233-V248
Author(s):  
Dengfeng Yang ◽  
Jun Liu ◽  
Jingnan Li ◽  
Daoli Liu
Keyword(s):  
Power Spectrum ◽  
Frequency Shift ◽  
Random Noise ◽  
Amplitude Spectrum ◽  
Seismic Profile ◽  
Estimation Accuracy ◽  
Low Frequencies ◽  
Bisection Algorithm ◽  
Centroid Frequency ◽  
Band Limited

The centroid frequency shift (CFS) method is a widely used [Formula: see text] estimation approach. However, the CFS approach assumes that the amplitude spectrum of a source wavelet has a particular shape, which can cause systematic error in [Formula: see text] estimation. Moreover, the amplitude spectrum at high and low frequencies is susceptible to random noise, which can reduce the robustness of [Formula: see text] estimation using the CFS method. To improve the accuracy and robustness of [Formula: see text] estimation, we have developed a [Formula: see text] extraction method using the bisection algorithm based on the centroid frequency shift of power spectrum (BPCFS). In the BPCFS approach, we first obtain the source and the received wavelet. Then, we calculate the centroid frequency of the attenuated wavelet and that of the received wavelet from the power spectrum. Based on the obtained centroid frequencies, we establish an equation containing only one variable — the [Formula: see text] factor. Introducing the Jeffrey divergence to measure the attenuation of the power spectrum, we prove that this equation has only one root when [Formula: see text] is greater than zero. The root of this equation, which is the desired [Formula: see text] factor, is obtained through the bisection algorithm; we do not make any assumption about the shape of the amplitude spectrum. The noise-free numerical tests indicate that the BPCFS gives more accurate results than the CFS, which demonstrates that the shape of the wavelet spectrum has little effect on the [Formula: see text] estimation accuracy for BPCFS. Gaussian random and blue noise tests also show that the stability of BPCFS is better than that of CFS. The frequency band selection for BPCFS is also more flexible when the amplitude spectrum of the seismic wavelet is band limited. The application to real vertical seismic profile data further demonstrate the effectiveness and feasibility of the new method.

Tracking Control of Non-Minimum Phase Systems Using Filtered Basis Functions: A NURBS-Based Approach

2015 ◽  
Author(s):  
Molong Duan ◽  
Keval S. Ramani ◽  
Chinedum E. Okwudire
Keyword(s):  
Tracking Control ◽  
Phase Error ◽  
Approximate Model ◽  
Basis Functions ◽  
Minimum Phase ◽  
Tracking Errors ◽  
Model Inversion ◽  
B Spline ◽  
Phase Systems ◽  
Zero Phase

This paper proposes an approach for minimizing tracking errors in systems with non-minimum phase (NMP) zeros by using filtered basis functions. The output of the tracking controller is represented as a linear combination of basis functions having unknown coefficients. The basis functions are forward filtered using the dynamics of the NMP system and their coefficients selected to minimize the errors in tracking a given trajectory. The control designer is free to choose any suitable set of basis functions but, in this paper, a set of basis functions derived from the widely-used non uniform rational B-spline (NURBS) curve is employed. Analyses and illustrative examples are presented to demonstrate the effectiveness of the proposed approach in comparison to popular approximate model inversion methods like zero phase error tracking control.

Prediction and suppression of long-period nonpropagating seismic noise

1973 ◽  
Vol 63 (3) ◽  
pp. 937-958
Author(s):  
Anton Ziolkowski
Keyword(s):  
Transfer Function ◽  
Pressure Distribution ◽  
Noise Level ◽  
Seismic Noise ◽  
Wave Number ◽  
Earth Model ◽  
Noise Power ◽  
Long Period ◽  
Band Limited ◽  
Time Invariant

abstract Approximately half the noise observed by long-period seismometers at LASA is nonpropagating; that is, it is incoherent over distances greater than a few kilometers. However, because it is often strongly coherent with microbarograph data recorded at the same site, a large proportion of it can be predicted by convolving the microbarogram with some transfer function. The reduction in noise level using this technique can be as high as 5 db on the vertical seismometer and higher still on the horizontals. If the source of this noise on the vertical seismogram were predominantly buoyancy, the transfer function would be time-invariant. It is not. Buoyancy on the LASA long-period instruments is quite negligible. The noise is caused by atmospheric deformation of the ground and, since so much of it can be predicted from the output of a single nearby microbarograph, it must be of very local origin. The loading process may be adequately described by the static deformation of a flat-earth model; however, for the expectation of the noise to be finite, it is shown that the wave number spectrum of the pressure distribution must be band-limited. An expression for the expected noise power is derived which agrees very well with observations and predicts the correct attenuation with depth. It is apparent from the form of this expression why it is impossible to obtain a stable transfer function to predict the noise without an array of microbarographs and excessive data processing. The most effective way to suppress this kind of noise is to bury the seismometer: at 150 m the reduction in noise level would be about 10 db.

CEPSTRUM ALIASING AND THE CALCULATION OF THE HILBERT TRANSFORM

Geophysics ◽  
1974 ◽  
Vol 39 (4) ◽  
pp. 543-544 ◽  
Author(s):  
Paul L. Stoffa ◽  
Peter Buhl ◽  
George M. Bryan
Keyword(s):  
Hilbert Transform ◽  
Phase Function ◽  
Amplitude Spectrum ◽  
Magnetic Data ◽  
Minimum Phase ◽  
Original Function ◽  
Cepstrum Analysis ◽  
Complex Cepstrum ◽  
The Hilbert Transform ◽  
Special Case

Schafer (1969) has pointed out that the Hilbert transform approach used in computing the minimum‐phase spectrum of a given amplitude spectrum corresponds to a special case of complex‐cepstrum analysis in which the phase information of the original function is ignored. The resulting complex cepstrum is an even function. Since a minimum‐phase function has no complex‐cepstrum contributions for T<0, its even part must exactly cancel the odd part for T<0. Thus, by setting all complex‐cepstrum contributions for T<0 equal to zero and doubling all contributions for T>0, we obtain the complex cepstrum of the minimum‐phase function corresponding to the original function. However, the DFT-calculated complex cepstrum is an aliased function (Stoffa et al., 1974). Thus some negative periods will appear at positive locations and vice versa. Appending the original function with zeros will reduce the aliasing. Shuey (1972), in computing the Hilbert transform for magnetic data, indicates that the computation breaks down near the end of the profile, or at long cepstrum periods. This is precisely the point in the even cepstrum where aliasing will have its greatest effect.

scholarly journals Response Analysis of Asymmetric Monostable Harvesters Driven by Color Noise and Band-Limited Noise

2021 ◽  
Vol 11 (19) ◽  
pp. 9227
Author(s):  
Shuangyan Liu ◽  
Wei Wang
Keyword(s):  
Output Power ◽  
Harmonic Excitation ◽  
Center Frequency ◽  
Response Analysis ◽  
Color Noise ◽  
Energy Harvesters ◽  
Output Performance ◽  
Response Bandwidth ◽  
Band Limited ◽  
The Moment

In this paper, we investigate the response of asymmetric potential monostable energy harvesters (MEHs) excited by color noise and band-limited noise. The motivation for this study is that environmental vibrations always have the characteristic of randomness, and it is difficult to modulate a perfectly symmetric MEH. For the excitation of exponentially correlated color noise, the moment differential equation was applied to evaluate the output performance of the asymmetric potential MEHs. Numerical and theoretical analyses were carried out to investigate the influence of noise intensity and internal system parameters on the output power of the system. Our results demonstrate that the output performance of the asymmetric MEH decreases with the increase in the correlation time, which determines the character of the color noise. On the contrary, the increase in the asymmetric degree enhances the output power of the asymmetric MEH subjected to color noise. For the band-limited noise excitation, numerical simulation is undertaken to consider the response of the asymmetric MEHs, and outcomes indicate that the frequency bandwidth and center frequency have a significant influence on the output performance. Regarding the asymmetric potential, its appearance leads the MEHs to generate higher output power at lower frequencies and this phenomenon is more obvious with the increase in the degree of asymmetry. Finally, we observed that the characteristics of the response bandwidth of asymmetric MEHs subjected to band-limited noise excitation are similar to the response under harmonic excitation.

Use of autoconvolution to suppress first‐order, long‐period multiples

Geophysics ◽  
1985 ◽  
Vol 50 (9) ◽  
pp. 1410-1425 ◽  
Author(s):  
C. J. Tsai
Keyword(s):  
Single Channel ◽  
Primary Energy ◽  
Low Frequencies ◽  
First Order ◽  
Seismic Wavelet ◽  
Common Depth Point ◽  
Band Limited ◽  
Marine Seismic ◽  
Prediction Problems ◽  
Water Bottom

A common problem in interpreting marine seismic data is the interference of water‐bottom multiples with primary reflections containing the structural or stratigraphic information. In deep ‐water areas, where considerable primary energy arrives before the first simple water‐bottom multiple, weak and deep crustal reflections are often obscured by the first‐order water‐bottom multiples. In order to obtain a more interpretable section, a technique involving a two‐step process was developed to suppress the first‐order water‐bottom multiples. First, the relation between the zero‐order, water‐bottom primary and its first‐order, simple water‐bottom multiple is used to derive statistically an inverse of the seismic wavelet in order to remove its effect, i.e., to wavelet‐shape the data. This wavelet processing provides a band‐limited estimate of the subsurface impulse response. The second step consists of using the autoconvolution of the wavelet‐shaped primary energy to estimate deterministically and subtract the actual first‐order, water‐bottom multiples, The method was applied to field data from the deep Gulf of Mexico. Different incidence angles for the input primaries and multiples, as well as dipping reflecting interfaces, introduce uncompensated traveltime errors. These errors reduce the ability to suppress multiples, thus restricting the validity of the method to low frequencies where common‐depth‐point stacking is less effective. On the other hand, curved interfaces may also cause amplitude prediction problems. In spite of this, the first‐order, water‐bottom multiple energy is significantly reduced (by up to 18 dB) on dip‐filtered, single‐channel data.

SEISMIC WAVE PROPAGATION IN LAYERED MEDIA IN TERMS OF COMMUNICATION THEORY

Geophysics ◽  
1966 ◽  
Vol 31 (1) ◽  
pp. 17-32 ◽  
Author(s):  
S. Treitel ◽  
E. A. Robinson
Keyword(s):  
Impulse Response ◽  
Communication Theory ◽  
Amplitude Spectrum ◽  
P Wave ◽  
Earth Model ◽  
Phase Lag ◽  
Elastic Plates ◽  
Pass System ◽  
Maximum Delay ◽  
Minimum Delay

The problem of a normally incident plane P wave propagating in a system of horizontally layered homogeneous perfectly elastic plates is reformulated in terms of concepts drawn from communication theory. We show how both the reflected and transmitted responses of such a system can be expressed as a z transform which is the ratio of two polynomials in z. Since this response must be stable, the denominators of the z transforms describing the reflected and transmitted motion are minimum delay (i.e., minimum‐phase lag). If the layered medium is bounded at depth by a perfect reflector, then the reflected impulse response recorded at the surface is in the form of a dispersive all‐pass z transform. A dispersive all‐pass system is one whose z transform is the ratio of the z transform of a maximum‐delay wavelet to that of its corresponding minimum‐delay wavelet; hence, the amplitude spectrum of a dispersive all‐pass system is unity for all frequencies. This means that the amplitude spectrum of the reflected response is identical to the amplitude spectrum of the input wavelet used to excite the system. More specifically, all the energy put in is returned with the same frequency content, but is differentially delayed. The phase‐lag spectrum of the reflected response lies everywhere above the phase‐lag spectrum of the input wavelet. Thus, the all‐pass situation implies that the layered earth model considered here, while not able to alter the amplitude of the frequency components of the input wavelet, will introduce differential time delays with certain properties into each such component. Finally, since the reflected impulse response is an all‐pass wavelet, its autocorrelation is a spike of unit magnitude at τ=0, and zero for all other lags.

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