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.

A Dispersive Model for the Propagation of Ultrasound in Soft Tissue

1982 ◽  
Vol 4 (4) ◽  
pp. 355-377 ◽  
Author(s):  
K. V. Gurumurthy ◽  
R. Martin Arthur
Keyword(s):  
Soft Tissue ◽  
Phase Velocity ◽  
Hilbert Transform ◽  
Phase Function ◽  
Path Length ◽  
Minimum Phase ◽  
Velocity Measurements ◽  
Tissue Model ◽  
Dispersive Model ◽  
The Hilbert Transform

Although the dispersion of tissue is small and difficult to measure, it can be calculated from a knowledge of the tissue's attenuation. A minimum-phase function, which characterizes tissue dispersion was derived using the Hilbert transform. This function was incorporated into a tissue model which has a causal impulse response and from which accurate estimates of the slope of attenuation times path length can be extracted. Predictions of phase velocity closely match available dispersion measurements. The model suggests that phase velocity measurements must be much more accurate than attenuation measurements for a comparable description of tissue.

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.

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.

scholarly journals Enhancing the quality of interpretation magnetic data at low latitudes

2017 ◽  
Vol 1 (T4) ◽  
pp. 105-114
Author(s):  
Hai Hong Nguyen ◽  
Nhan Thanh Nhan ◽  
Liet Van Dang ◽  
Thu Ngoc Nguyen
Keyword(s):  
Magnetic Anomaly ◽  
Hilbert Transform ◽  
Amplitude Spectrum ◽  
Data Interpretation ◽  
Analytic Signal ◽  
Magnetic Data ◽  
Low Latitudes ◽  
Reduction To The Pole ◽  
Signal Method

Magnetic anomalies are antisymmetrical and often skewed to the location of the sources, because both of the magnetization and ambient field are not directed vertically, so it’s difficult to interpret. For reducing the magnetic anomaly to a symmetrical one – this located on the source of the anomaly – people often use the reduction to the pole (RTP) where the magnetization and ambient field are both directed vertically. However, at low latitudes (an absolute inclination less than 16o30’), the amplitude spectrum of the RTP’s operator was amplified at higher frequencies (short wavelengths) can form a narrow pie-shaped; so it produces artifacts elongated along the direction of the magnetic declination. Therefore, many methods of RTP at low latitudes are given to solve this problem, but most of them are not efficiency. In this paper, we performed enhancing the quality of interpretation of magnetic data at low latitudes by some RTP methods for magnetic data at low latitudes and the analytic signal method using gradient operator and Hilbert transform. This method is applied to a model and to a real magnetic anomaly to find out the best method. Then this method was applied to enhance the quality of magnetic data interpretation in the Southern Vietnam. The result showed that the analytic signal method using Hilbert transform allowed enhancing the quality of interpretatio of magnetic data n at low latitudes is the best.

Determination of the seismograph phase response from the amplitude response

1972 ◽  
Vol 62 (6) ◽  
pp. 1665-1672 ◽  
Author(s):  
P. M. Bolduc ◽  
R. M. Ellis ◽  
R. D. Russell
Keyword(s):  
Hilbert Transform ◽  
Rapid Determination ◽  
Phase Response ◽  
Minimum Phase ◽  
Amplitude Response ◽  
Amplitude Data ◽  
The Hilbert Transform ◽  
Seismic System

abstract Determination of the phase response of a minimum-phase seismic system directly from the amplitude response by the Hilbert transform is investigated. The error by this technique is found to be less than 2° with the major source of inaccuracy being the uncertainties in the amplitude data. This technique is useful for rapid determination of the phase response if the amplitude response is known.

Model of multicomponent narrow-band periodical non-stationary random signal

2020 ◽  
Vol 2020 (48) ◽  
pp. 17-24
Author(s):  
I.M. Javorskyj ◽  
◽  
R.M. Yuzefovych ◽  
P.R. Kurapov ◽  
◽  
...  
Keyword(s):  
Time Series ◽  
Real Time ◽  
Hilbert Transform ◽  
Spectral Properties ◽  
Narrow Band ◽  
Analytic Signal ◽  
Random Signal ◽  
Hilbert Transformation ◽  
The Hilbert Transform

The correlation and spectral properties of a multicomponent narrowband periodical non-stationary random signal (PNRS) and its Hilbert transformation are considered. It is shown that multicomponent narrowband PNRS differ from the monocomponent signal. This difference is caused by correlation of the quadratures for the different carrier harmonics. Such features of the analytic signal must be taken into account when we use the Hilbert transform for the analysis of real time series.

A Bayesian View on the Hilbert Transform and the Kramers-Kronig Transform of Electrochemical Impedance Data: Probabilistic Estimates and Quality Scores

2020 ◽  
Author(s):  
Jiapeng Liu ◽  
Ting Hei Wan ◽  
Francesco Ciucci
Keyword(s):  
Power Generation ◽  
Electrochemical Impedance Spectroscopy ◽  
Hilbert Transform ◽  
Electrochemical Impedance ◽  
The Self ◽  
Impedance Data ◽  
The Hilbert Transform ◽  
Validation Tests ◽  
Self Consistency ◽  
Mathematical Relations

<p>Electrochemical impedance spectroscopy (EIS) is one of the most widely used experimental tools in electrochemistry and has applications ranging from energy storage and power generation to medicine. Considering the broad applicability of the EIS technique, it is critical to validate the EIS data against the Hilbert transform (HT) or, equivalently, the Kramers–Kronig relations. These mathematical relations allow one to assess the self-consistency of obtained spectra. However, the use of validation tests is still uncommon. In the present article, we aim at bridging this gap by reformulating the HT under a Bayesian framework. In particular, we developed the Bayesian Hilbert transform (BHT) method that interprets the HT probabilistic. Leveraging the BHT, we proposed several scores that provide quick metrics for the evaluation of the EIS data quality.<br></p>

scholarly journals A Radial Basis Function Finite Difference Scheme for the Benjamin–Ono Equation

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 65
Author(s):  
Benjamin Akers ◽  
Tony Liu ◽  
Jonah Reeger
Keyword(s):  
Radial Basis Function ◽  
Hilbert Transform ◽  
Basis Function ◽  
Differential Operators ◽  
Convergence Rates ◽  
Traveling Wave Solutions ◽  
Algebraic Decay ◽  
Pseudo Differential Operators ◽  
Radial Basis ◽  
The Hilbert Transform

A radial basis function-finite differencing (RBF-FD) scheme was applied to the initial value problem of the Benjamin–Ono equation. The Benjamin–Ono equation has traveling wave solutions with algebraic decay and a nonlocal pseudo-differential operator, the Hilbert transform. When posed on R, the former makes Fourier collocation a poor discretization choice; the latter is challenging for any local method. We develop an RBF-FD approximation of the Hilbert transform, and discuss the challenges of implementing this and other pseudo-differential operators on unstructured grids. Numerical examples, simulation costs, convergence rates, and generalizations of this method are all discussed.

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