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.

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 Feature Preserving Autofocus Algorithm for Phase Error Correction of SAR Images

Sensors ◽  
2021 ◽  
Vol 21 (7) ◽  
pp. 2370
Author(s):  
Haemin Lee ◽  
Chang-Sik Jung ◽  
Ki-Wan Kim
Keyword(s):  
Phase Error ◽  
Minimum Entropy ◽  
Motion Error ◽  
Phase Gradient ◽  
Limited Bandwidth ◽  
Phase Errors ◽  
Feature Preserving ◽  
Autofocus Algorithm ◽  
Autofocus Algorithms ◽  
The Cost

Autofocus is an essential technique for airborne synthetic aperture radar (SAR) imaging to correct phase errors mainly due to unexpected motion error. There are several well-known conventional autofocus methods such as phase gradient autofocus (PGA) and minimum entropy (ME). Although these methods are still widely used for various SAR applications, each method has drawbacks such as limited bandwidth of estimation, low convergence rate, huge computation burden, etc. In this paper, feature preserving autofocus (FPA) algorithm is newly proposed. The algorithm is based on the minimization of the cost function containing a regularization term. The algorithm is designed for postprocessing purpose, which is different from the existing regularization-based algorithms such as sparsity-driven autofocus (SDA). This difference makes the proposed method far more straightforward and efficient than those existing algorithms. The experimental results show that the proposed algorithm achieves better performance, convergence, and robustness than the existing postprocessing autofocus algorithms.

ON THE MINIMUM‐LENGTH PROPERTY OF ONE‐SIDED SIGNALS

Geophysics ◽  
1973 ◽  
Vol 38 (4) ◽  
pp. 657-672 ◽  
Author(s):  
A. J. Berkhout
Keyword(s):  
Amplitude Spectrum ◽  
Phase Spectrum ◽  
Signal Duration ◽  
Complex Conjugate ◽  
Minimum Length ◽  
Time Interval ◽  
Minimum Phase ◽  
Inverse Filtering ◽  
Complex Signals ◽  
Phase Property

Apart from a visual inspection, the length property of a signal is mostly judged by its amplitude spectrum (e.g., bandwidth or “roughness” of the spectrum). However, due to the important influence of the phase spectrum. a sensible yardstick for signal length should consider both amplitude and phase. In addition, a complete investigation should also include evaluation of the length of the (partly) deconvolved signal. In this paper, signal duration and signal length are introduced as two different concepts. Unlike signal duration, giving the minimum time interval outside of which the signal equals zero, signal length gives some information as to how the signal energy is distributed within this interval. An objective yardstick for signal length is proposed, and it is shown how signal length depends on both the amplitude and phase spectrum. The minimum‐length signal is introduced and it is shown that for one‐sided signals the minimum signal‐length property implies the minimum‐phase property and vice‐versa. The efficiency of causal least‐square inverse filtering on minimum‐length and nonminimum‐length signals is evaluated. An investigation scheme is given which enables the user to determine whether a one‐sided signal has the minimum‐length (minimum‐phase) property. The theory is applicable to complex signals. For real signals the * (symbol for complex conjugate) and absolute value marks may be omitted: [Formula: see text] and [Formula: see text].

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.

THE APPLICATION OF HOMOMORPHIC DECONVOLUTION TO SHALLOW‐WATER MARINE SEISMOLOGY—PART I: MODELS

Geophysics ◽  
1974 ◽  
Vol 39 (4) ◽  
pp. 401-416 ◽  
Author(s):  
Paul L. Stoffa ◽  
Peter Buhl ◽  
George M. Bryan
Keyword(s):  
Shallow Water ◽  
Water Column ◽  
Amplitude Spectrum ◽  
Phase Spectrum ◽  
Phase Curve ◽  
Minimum Phase ◽  
Cepstrum Analysis ◽  
Infinite Extent ◽  
Complex Cepstrum ◽  
The Time Domain

The complex cepstrum is investigated mathematically and through models for functions of interest in shallow‐water marine seismology. Association of the slowly varying components of the phase spectrum with the source replaces the usual minimum‐phase assumption. This is analogous to the usual treatment of the amplitude spectrum. Complex cepstrum expressions are developed for an arbitrary (but minimum‐phase) reflector series water‐column multiple generator, and simplified bubble‐pulse oscillation. While the complex cepstrum of all functions is of infinite extent, removing only the first n nonzero complex‐cepstrum contributions of a decaying, impulsive, periodic time function (such as the water‐column multiple generator) serves to eliminate the first n multiples entirely in the time domain and reduces the remaining multiples to at most 1/(n+1) of their original value. A new method of computing the continuous, ramp‐free phase spectrum required for complex‐cepstrum analysis is developed on the basis of the derivative of the phase curve.

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.

scholarly journals Grey Spectrum Analysis of Air Quality Index and Housing Price in Handan

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-6 ◽  
Author(s):  
Kai Zhang ◽  
Yan Chen ◽  
Lifeng Wu
Keyword(s):  
Air Quality ◽  
Quality Index ◽  
Amplitude Spectrum ◽  
Housing Price ◽  
Lag Time ◽  
Phase Spectrum ◽  
Air Quality Index ◽  
The Relationship ◽  
Grey Phase ◽  
Main Change

To analyze the relationship between air quality index (AQI) and housing price, six relationship indexes between air quality index and housing price were calculated using grey spectrum theory, specifically grey association spectrum, grey cospectrum, grey amplitude spectrum, grey phase spectrum, grey lag time length, and grey condense spectrum. Three main change periods were extracted. There was a negative correction between the air quality and the housing price in Handan. The results provide a basis for the government’s measures to prevent haze.

scholarly journals Image Saliency Prediction in Transformed Domain: A Deep Complex Neural Network Method

2019 ◽  
Vol 33 ◽  
pp. 8521-8528 ◽  
Author(s):  
Lai Jiang ◽  
Zhe Wang ◽  
Mai Xu ◽  
Zulin Wang
Keyword(s):  
Neural Network ◽  
Amplitude Spectrum ◽  
Ground Truth ◽  
Phase Spectrum ◽  
Saliency Maps ◽  
Image Saliency ◽  
Saliency Prediction ◽  
Network Method ◽  
Salient Regions ◽  
Robust To Noise

The transformed domain fearures of images show effectiveness in distinguishing salient and non-salient regions. In this paper, we propose a novel deep complex neural network, named SalDCNN, to predict image saliency by learning features in both pixel and transformed domains. Before proposing Sal-DCNN, we analyze the saliency cues encoded in discrete Fourier transform (DFT) domain. Consequently, we have the following findings: 1) the phase spectrum encodes most saliency cues; 2) a certain pattern of the amplitude spectrum is important for saliency prediction; 3) the transformed domain spectrum is robust to noise and down-sampling for saliency prediction. According to these findings, we develop the structure of SalDCNN, including two main stages: the complex dense encoder and three-stream multi-domain decoder. Given the new SalDCNN structure, the saliency maps can be predicted under the supervision of ground-truth fixation maps in both pixel and transformed domains. Finally, the experimental results show that our Sal-DCNN method outperforms other 8 state-of-theart methods for image saliency prediction on 3 databases.

scholarly journals Strip-Mode Microwave Staring Correlated Imaging with Self-Calibration of Gain–Phase Errors

Sensors ◽  
2019 ◽  
Vol 19 (5) ◽  
pp. 1079 ◽  
Author(s):  
Rui Xia ◽  
Yuanyue Guo ◽  
Weidong Chen ◽  
Dongjin Wang
Keyword(s):  
Error Estimation ◽  
Phase Error ◽  
Super Resolution ◽  
Image Splicing ◽  
Initial Value ◽  
Self Calibration ◽  
Phase Errors ◽  
Imaging Model ◽  
Multiple Imaging ◽  
Imaging Performance

Microwave staring correlated imaging (MSCI) can realize super resolution imaging without the limit of relative motion with the target. However, gain–phase errors generally exist in the multi-transmitter array, which results in imaging model mismatch and degrades the imaging performance considerably. In order to solve the problem of MSCI with gain–phase error in a large scene, a method of MSCI with strip-mode self-calibration of gain–phase errors is proposed. The method divides the whole imaging scene into multiple imaging strips, then the strip target scattering coefficient and the gain–phase errors are combined into a multi-parameter optimization problem that can be solved by alternate iteration, and the error estimation results of the previous strip can be carried into the next strip as the initial value. All strips are processed in multiple rounds, and the gain–phase error estimation results of the last strip can be taken as the initial value and substituted into the first strip for the correlated processing of the next round. Finally, the whole imaging in a large scene can be achieved by multi-strip image splicing. Numerical simulations validate its potential advantages to shorten the imaging time dramatically and improve the imaging and gain–phase error estimation performance.

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