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.

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.

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.

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].

Selection of random vibration theory procedures for the NGA-East project and ground-motion modeling

2021 ◽  
Vol 37 (1_suppl) ◽  
pp. 1420-1439
Author(s):  
Albert R Kottke ◽  
Norman A Abrahamson ◽  
David M Boore ◽  
Yousef Bozorgnia ◽  
Christine A Goulet ◽  
...  
Keyword(s):  
Ground Motion ◽  
Time Domain ◽  
Random Vibration ◽  
Amplitude Spectrum ◽  
Motion Modeling ◽  
Peak Response ◽  
Vibration Theory ◽  
Random Vibration Theory ◽  
The Time Domain ◽  
The Impact

Traditional ground-motion models (GMMs) are used to compute pseudo-spectral acceleration (PSA) from future earthquakes and are generally developed by regression of PSA using a physics-based functional form. PSA is a relatively simple metric that correlates well with the response of several engineering systems and is a metric commonly used in engineering evaluations; however, characteristics of the PSA calculation make application of scaling factors dependent on the frequency content of the input motion, complicating the development and adaptability of GMMs. By comparison, Fourier amplitude spectrum (FAS) represents ground-motion amplitudes that are completely independent from the amplitudes at other frequencies, making them an attractive alternative for GMM development. Random vibration theory (RVT) predicts the peak response of motion in the time domain based on the FAS and a duration, and thus can be used to relate FAS to PSA. Using RVT to compute the expected peak response in the time domain for given FAS therefore presents a significant advantage that is gaining traction in the GMM field. This article provides recommended RVT procedures relevant to GMM development, which were developed for the Next Generation Attenuation (NGA)-East project. In addition, an orientation-independent FAS metric—called the effective amplitude spectrum (EAS)—is developed for use in conjunction with RVT to preserve the mean power of the corresponding two horizontal components considered in traditional PSA-based modeling (i.e., RotD50). The EAS uses a standardized smoothing approach to provide a practical representation of the FAS for ground-motion modeling, while minimizing the impact on the four RVT properties ( zeroth moment, [Formula: see text]; bandwidth parameter, [Formula: see text]; frequency of zero crossings, [Formula: see text]; and frequency of extrema, [Formula: see text]). Although the recommendations were originally developed for NGA-East, they and the methodology they are based on can be adapted to become portable to other GMM and engineering problems requiring the computation of PSA from FAS.

DESIGN OF RECURSIVE FILTERS

Geophysics ◽  
1970 ◽  
Vol 35 (2) ◽  
pp. 247-253 ◽  
Author(s):  
Raymundo Aguilera ◽  
J. CL. Debremaecker ◽  
Salvador Hernandez
Keyword(s):  
Frequency Domain ◽  
Time Domain ◽  
Minimum Phase ◽  
Recursive Filter ◽  
Recursive Filters ◽  
The Time Domain

Recursive filters are inherently more efficient than purely transverse or purely regressive ones. They can be computed in the frequency domain by a series of simple operations. The roots of the denominator must be computed and the moduli less than unity replaced by their inverses. If such an operation is also performed on the numerator, the resultant recursive filter is minimum phase. The same method can be used to construct a deconvolution operator in the time domain, starting with the autocorrelation. Two examples are given which show the efficiency of the method.

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.

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