seismic modulation model and envelope inversion with ...wrs/publication/journal/journal2018... ·...

Journal of Geophysics and Engineering

PAPER

Seismic modulation model and envelope inversionwith smoothed apparent polarityTo cite this article: Yuqing Wang et al 2018 J. Geophys. Eng. 15 2278

View the article online for updates and enhancements.

Related contentTime-domain full waveform inversion usinginstantaneous phase information withdampingJingrui Luo, Ru-Shan Wu and Fuchun Gao

-

Improving the gradient in least-squaresreverse time migrationQiancheng Liu

-

Time domain full waveform inversion withlow frequency wavefield decompressionLiang Zhanyuan, Wu Guochen and ZhangXiaoyu

-

This content was downloaded from IP address 128.114.69.29 on 23/08/2018 at 02:02

https://doi.org/10.1088/1742-2140/aac54dhttp://iopscience.iop.org/article/10.1088/1742-2140/aaa984http://iopscience.iop.org/article/10.1088/1742-2140/aaa984http://iopscience.iop.org/article/10.1088/1742-2140/aaa984http://iopscience.iop.org/article/10.1088/1742-2132/13/2/172http://iopscience.iop.org/article/10.1088/1742-2132/13/2/172http://iopscience.iop.org/article/10.1088/1742-2140/aac62fhttp://iopscience.iop.org/article/10.1088/1742-2140/aac62fhttp://oas.iop.org/5c/iopscience.iop.org/446723988/Middle/IOPP/IOPs-Mid-JGE-pdf/IOPs-Mid-JGE-pdf.jpg/1?

Seismic modulation model and envelopeinversion with smoothed apparent polarity

Yuqing Wang1,2,3 , Ru-Shan Wu3, Guoxin Chen3,4 and Zhenming Peng1,2

1 School of Optoelectronic Information, University of Electronic Science and Technology of China,Chengdu 610054, People’s Republic of China2 Center for Information Geoscience, University of Electronic Science and Technology of China, Chengdu610054, People’s Republic of China3University of California, Earth and Planetary Sciences, Modeling and Imaging Laboratory, Santa Cruz,California, United States of America4 School of Earth Sciences, Zhejiang University, Hangzhou 310027, People’s Republic of China

E-mail: [email protected]

Received 11 February 2018, revised 26 April 2018Accepted for publication 16 May 2018Published 10 July 2018

AbstractThe seismic modulation model analyzes a seismogram from the low and high-frequencyinformation, which is different from the traditional convolution model. In the modulation model,a seismogram is regarded as a modulated signal and its envelope is the amplitude-modulationcomponent containing the low-frequency information. On this foundation, the envelope ofseismograms can be used to recover a very smooth background structure. However, amplitudedemodulation methods can only obtain the absolute value of the envelope, which cannot reflectthe polarity changes of the amplitude information in seismograms. To solve this problem, weconsider the low-frequency modulation from both the amplitude and polarity points of view. Weextract a modulator signal with smoothed apparent polarity, which contains the amplitude andpolarity information in seismograms. The new approach can broaden the modulation modeltheory for seismic signals. Good results from examples for application to envelope inversiondemonstrate the good performance and potential of the proposed method.

Keywords: seismic modulation model, low-frequency, Hilbert envelope, smoothed apparentpolarity, envelope inversion

(Some figures may appear in colour only in the online journal)

Introduction

Historically, most signal analysis is based on linear transforms,which are additive decompositions. It assumes that a signal canbe represented by a superposition of independent components.This is the principle of linear superposition, such as inthe traditional linear data transforms, e.g., Fourier transform,Hilbert transform, or other transforms using basis and frames.

The modulation–demodulation transform is a nonlinearmultiplicative decomposition which can reveal the structure ofthe process, such as the trends, slow evolution and variation,secular force for dynamics, etc. On the other hand, lineardecompositions, such as the Fourier transform, assume eachcomponent of the process as independent from the other com-ponents. In the time domain, the modulated signal equals to the

product of the modulation signal and the carrier signal. When aFourier transform is performed to a modulated signal, the Fourierspectrum of the modulated signal is convolved with the Fourierspectrum of the carrier signal. It means that the low-frequencyband of the modulation signal moves towards the high-frequencyends of the carrier frequency, so one will have a frequency shiftto high frequencies, and a frequency split of the carrier frequency.In the end, the low-frequency modulation signal cannot be foundby using the Fourier decomposition method. The linear decom-position and processing can only change the relative magnitudeof the spectral components of the signal without generating newfrequency components. So the linear decomposition cannotrecover the missing low-frequency information in the signal. Thatis why waveform inversion based on a linear signal model cannotsee the ultra-low-frequency information that is lower than the

Journal of Geophysics and Engineering

J. Geophys. Eng. 15 (2018) 2278–2286 (9pp) https://doi.org/10.1088/1742-2140/aac54d

1742-2132/18/0502278+09$33.00 © 2018 Sinopec Geophysical Research Institute Printed in the UK2278

https://orcid.org/0000-0003-1100-6005https://orcid.org/0000-0003-1100-6005mailto:[email protected]://doi.org/10.1088/1742-2140/aac54dhttp://crossmark.crossref.org/dialog/?doi=10.1088/1742-2140/aac54d&domain=pdf&date_stamp=2018-07-10http://crossmark.crossref.org/dialog/?doi=10.1088/1742-2140/aac54d&domain=pdf&date_stamp=2018-07-10

lowest frequency contained in the source wavelet. From themodulation model, the ultra-low-frequency information is in theenvelope which can be extracted nonlinearly by a demodulationoperator.

Wu et al (2014) proposed an envelope inversion (EI)method to extract the ultra-low-frequency information con-tained in seismograms, and considered the signal model forseismograms as a seismic modulation–convolution model.This new signal model can fully represent the low-frequencyand high-frequency information. Historically, the observedseismic records are commonly modeled as the convolution ofthe source wavelet and the Earth’s reflectivity series. Thetraditional convolution signal model (Robinson et al 1986) isa linear model suitable for seismic imaging. However, seis-mograms contain other information that the convolutionmodel does not include. The modulation signal model pro-vides a novel perspective for seismic exploration (Luo andWu 2015, Hu et al 2017, Wang and Wu et al 2017, Wang andYuan et al 2017, Wu and Chen 2017a, 2017b).

The modulation modeling technique has been successfullyapplied to some research fields, such as speech signalanalysis (Potamianos and Maragos 1999) and image processing(Havlicek et al 1996), and thus propelled the development ofmany demodulation methods (Loughlin and Tacer 1996, Turnerand Sahani 2011, Xu et al 2014). However, most amplitudedemodulation methods can only obtain the positive envelope,which cannot reflect the polarity changes in seismograms.Polarity plays an important role in the seismic instantaneousattributes, and many researches have been done on its definitionand criterion (Taner et al 1979, Robertson and Nogami 1984,White 1991, Brown et al 2002, Sheriff 2012, Wang et al 2016).Polarity can be defined as the sign of reflection coefficient thatindicates an increase or decrease in the interfacial wave impe-dance. So the EI without polarity information can not char-acterize the situation of wave impedance reduction. Xu et al(2014) proposed a signed demodulation method for amplitudeand frequency modulation signal. But this signed demodulationtheory is mainly applicable to the sine and cosine modulationsignal, not suitable for seismograms. Taner et al (1979) proposedthe concept of apparent polarity, which can represent the polarityinformation of seismogram. Inspired by the research above, wepropose including polarity information in the demodulation ofseismograms and in the application of EI.

The seismic modulation model

Using the modulation signal model, Wu et al (2014) proposedthat the reflection seismogram ( )p t x x, ,r sbsc formed by back-scattered waves with geophone xr and source xs is modeled as

g=( ) ( ) ( ) ( )p t G t tx x x x, , , , , 1r s sr r s wbsc

åg g g d t= =( ) ( )⁎ ( ) ( )⁎ ( ‐ ) ( )t s t t s t t , 2wi

i i

where g ( )tw serves as the carrier formed by the convolutionof the source wavelet ( )s t and the reflection coefficient functiong ( )t , d ( )t is a Dirac delta function, and ( )G t x x, ,sr r s isa smoothly varying Green’s function pair (background

propagator) which serves as the modulator and only modifies thetraveltimes and amplitudes of the reflection events.

In the modulation signal model, g ( )tw still contains usefullow-frequency information of the medium velocity structure.In order to model the information contained in the seismogram,we decompose the seismogram ( )p t x x, ,r sbsc into twomultiplicative components: a low-frequency component

( )p t x x, ,r sbsc and a high-frequency component ˜ ( )p t x x, , ,r sbscthat is

=( ) ( ) · ˜ ( ) ( )p t p t p tx x x x x x, , , , , , . 3r s r s r sbsc bsc bsc

The low-frequency component ( )p t x x, ,r sbsc of the seis-mogram can be demodulated as Hilbert envelope, that is

g gg

== += +=

( ) { ( )}∣ ( ) [ ( )]∣∣ ( ) ( ) ( ) [ ( )]∣∣ ( )∣ { ( )} ( )

p t E p t

p t iH p tG t t iG t H tG t E t

x x x x

x x x xx x x xx x

, , , ,

, , , ,, , , ,, , , 4

r s r s

r s r s

sr r s w sr r s w

sr r s w

bsc bsc

bsc bsc

where [·]H denotes Hilbert transform operator, and [·]E isenvelope operator. In the demodulation process, the Green’sfunction ( )G t x x, ,sr r s is pulled out of the Hilbert transformbecause it is a smooth function satisfying the Bedrosian–Brown’s condition for Hilbert transform (Bedrosian 1962,Brown 1986). However, the low-frequency component inequation (4) contains only positive amplitude without polarityinformation.

In fact, the multiplicative decomposition 3 is non-uniqueand therefore has many possible solutions. We reconsider ademodulation method to obtain a low-frequency componentwith polarity information.

We assume that the low-frequency component containstwo parts: a positive Hilbert envelope and a smooth polaritycurve. Then the low-frequency component ( )p t x x, ,r sbsc ofseismogram can be reformulated as

=( ) { ( )} · ( ) ( )p t E p t sap tx x x x, , , , , 5r s r sbsc bsc

where ( )sap t represents the smooth polarity curve. In thisexpression, the envelope { ( )}E p t x x, ,r sbsc determines themagnitude of the low-frequency component, and ( )sap t is asmoothing curve with polarity. The low-frequency comp-onent ( )p t x x, ,r sbsc contains the amplitude and polarityinformation of the seismogram, representing the informationon the smoothed medium velocity structure. When the seis-mogram contains noise, its Hilbert envelope will have thehigh-frequency components caused by noise, and the ( )sap tcurve will also be affected. Correspondingly, there are high-frequency components of noise in the ( )p t x x, , .r sbsc In thiscase, ( )p t x x, ,r sbsc should be the data after low-pass filteringto reduce the influence of noise.

Smoothed apparent polarity

The apparent polarity is proposed by Taner et al (1979),defined as the sign of signal when Hilbert envelope has a localmaximum. Assuming a single reflection, a zero-phase waveletand no phase reverse, a positive or negative sign can be used torefer to positive or negative reflection coefficients. However,

2279

J. Geophys. Eng. 15 (2018) 2278 Y Wang et al

that is the ideal situation. Generally, most reflection eventsconsist of multiple reflectors. The interference between thereflectors may cause the location of the local maximum of theenvelope to be inconsistent with the position of the reflectioncoefficient, so that there is a lack of clear correlation betweenthe sign and the reflection coefficient. Hence, the sign isregarded as apparent polarity, showing a rough polarity of theseismogram.

From the definition, the apparent polarity has a valueonly at the local maximum of the envelope. If the apparentpolarity is considered as a portion of the low-frequencycomponent, we need to generate a smooth apparent polaritycurve by using these points. In this paper, we consider a zero-phase Ricker wavelet as the source wavelet, and get asmoothed apparent polarity (SAP) by interpolation to repre-sent the low-frequency polarity information of theseismogram.

Given a signal ( )x t with N sample points, if it containsnoise, a low-pass filter is applied to the signal first. Then itsenvelope with smoothed apparent polarity (E-SAP) can beobtained as follows with the outline in figure 1:

(1) Take the Hilbert transform of the signal ( )x t and obtainthe envelope ( )a t . Identify all local maxima of ( )a t ,denoted as ( )a IndMax (IndMax represents the coordi-nate location of local maxima);

(2) Get the apparent polarity of all local maxima points= ¹( ) ( ( ))ap IndMax sign x IndMax 0 (if =( ( ))sign x IndMax

0, =( )ap IndMax 1), and set the apparent polarityat beginning and end to zero, that is, =( )ap 1 0,

=( )ap N 0;

(3) Add an adjacent point to both sides of the local maximaIndMax, that is, =( ‐ ) ( ( ))ap IndMax sign x IndMax1 and

+ =( ) ( ( ))ap IndMax sign x IndMax1 . Connect all theapparent polarity points by a cubic spline line as thesmoothed apparent polarity ( )sap t ;

(4) The E-SAP of signal can be obtained by ( ) · ( )a t sap t .

In step (2), we make =( ( ))sign x IndMax 0 at=( )ap IndMax 1 to prevent the situation that apparent

polarity equals 0 at the local maxima due to interferences. Inaddition, setting the apparent polarity at the beginning andend to zero is to ensure that the interpolation curve does nothave large derivations at the beginning and end. In step (3),adding adjacent points is to stabilize the interpolation curve,which can be proved by figure 2. It is evident that addingadjacent points makes the interpolation curve more stable andprevents excessive amplitude in the SAP result. The inter-polation curve without adding adjacent points (figure 2(c)) iseasy to deflect, so that the E-SAP result (figure 2(d)) would bebadly affected.

In addition, the performance of E-SAP depends on thelocal maxima of Hilbert envelope. As shown in figure 3, wegive the synthetic seismogram of a wedge model that isgenerated by a wavelet convolved with two reflections ofequal amplitude and opposite polarity, and the layer thicknessbetween two reflections is gradually decreasing. The layerthickness has an influence on the detection of the localmaxima of the envelope. When the local maxima of the upperand lower boundaries can be separated, the E-SAP cancharacterize the polarity reversal. However, as the layerthickness decreases, the interference between the two reflec-tions becomes stronger, so that the two local maxima are

Figure 1. The outline of E-SAP. The black line represents the signal; the pink line represents the reflection series; the blue dotted linerepresents the Hilbert envelope of signal; and the blue triangles are the local maxima of the Hilbert envelope; the black circles are apparentpolarity points at the local maxima of the Hilbert envelope; the green line represents the smoothed apparent polarity obtained byinterpolation; and the red line represents the E-SAP result.

2280


confused. In figure 3, for a signal with layer thickness lessthan 26 ms, the E-SAP gives the same result as the Hilbertenvelope without polarity reversal. Therefore, if all the localmaxima of envelope are not detected due to the presence ofthin beds or other interferences, the E-SAP can not reflect the

polarity reversal between layers. In this case, the E-SAP resultis almost identical to the Hilbert envelope. So, even if thelocal maxima of the envelope is biased, the SAP is a slow-varying curve of polarity without amplitude gain, which doesnot cause serious damage to the E-SAP result.

Figure 2. SAP and E-SAP result (a)–(b) with adding adjacent points and (c)–(d) without adding adjacent points.

Figure 3. Wedge model that a wavelet convolved with two spikes of equal amplitude and opposite polarity.

2281


Experiments and analysis

Single-trace

A single-trace without thin layer is generated by a series ofreflection coefficients convolved with a 20 Hz Ricker waveletas shown in figure 4. The ten reflectors are located at 140,200, 320, 360, 460, 520, 666, 700, 750 and 900 ms. TheHilbert envelope (figure 4(a)) is always positive, which can-not reflect the polarity changes of the amplitude. All the local

maxima of envelope around the reflection series are identified.So the E-SAP result in figure 4(c) shows a rough smoothfitting curve of the reflection series, which contains thepolarity information of trace. Figures 4(b) and (d) give thefrequency spectrum distribution of trace, Hilbert envelope andE-SAP. We can see that the Hilbert envelope and E-SAPcontains the low frequencies below the frequency range oftrace. Figure 5 shows another single-trace with a thin bedproduced by multiple reflection coefficients convolved with a20 Hz Ricker wavelet. The ten reflectors are located at 140,

Figure 4. A synthetic trace response of ten reflectors convolved with a 20 Hz Ricker wavelet. Hilbert envelope in the time domain (a) andfrequency domain (b); E-SAP in the time domain (c) and frequency domain (d).

Figure 5. A synthetic trace with a thin bed response of ten reflectors convolved with a 20 Hz Ricker wavelet. Hilbert envelope in the timedomain (a) and frequency domain (b); E-SAP in the time domain (c) and frequency domain (d).

2282


200, 320, 335, 460, 520, 666, 700, 750 and 900 ms. Thewavelength of the Ricker wavelet can be estimated byl =

f

4

2.6 pwhere =f 20p is the peak frequency (Kallweit and

Wood 1982). The thickness of the thin bed at 320–335 ms isbelow the quarter wavelength. The Hilbert envelope(figure 5(a)) at the thin bed (dotted ellipse) only has one localmaxima so that the E-SAP (figure 5(c)) does not reverse thepolarity. However, the E-SAP result gives the polarity chan-ges at other locations (dotted box). The corresponding spec-tral range is still below the frequency band of trace.

The proposed method is also a test on a noisy single-traceas shown in figure 6. The noisy single-trace is generated byadding white Gaussian noise into the single-trace in figure 5,giving a signal-to-noise ratio of 5 dB. The Hilbert envelope(figure 6(a)) removes the high-frequency components ofnoise, and the E-SAP result (figure 6(c)) still give the polaritychanges at 200, 520 and 700 ms (dotted box). And their fre-quency spectral removes the high-frequency range of noisytrace.

Application in EI

The full-waveform inversion (FWI) requires a good initialmodel and performs poorly when the source wavelet missesthe low-frequency information (Virieux and Operto 2009).To overcome these deficiencies, the EI and its relatedmethods are developed. The misfit function of conventional

EI is defined as

òå ås = == -

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

⁎ ⁎r r t r t dt r t r t

r t e t e t

1

2

1

2,

, 6sr o

T

e esrT

e e

e obs

where ( )e t is the envelope of a synthetic wavefield, and ( )e tobsis the envelope of an observed waveform data. The summa-tion is over all the sources xs and receviers x .r We use the newdirect Envelope Fréchet derivative ¶

¶( )e tv

to get the sensitivityoperator for EI (Wu and Chen 2017a, 2017b) and the gradientoperator for EI is

å ås¶¶

=¶¶

=( ) ( ) · ( ) ( )( ) ⁎v

r te t

vQ r tG , 7

srTe

srT

ee

where summation of srT is over the source, receiver and time,v is the velocity model parameter, Q is virtual source operator,

( )G e is the envelope Green’s operator representing the forwardpropagation process, and ( ) ⁎G e is the adjoint operator of ( )G ,e

representing the reverse-time propagation of the envelope resi-dual ( )r t .e (For detailed formulation see Chen et al 2017, Wuand Chen 2017b and Chen et al 2018.) In the EI, the Hilbertenvelope is defined as envelope operator. In this paper, theE-SAP is used as the new envelope operator instead of theHilbert envelope. The EI result obtained by E-SAP is denotedas SAP-EI. We also compare the performance of the Hilbertenvelope and E-SAP in the combined EI+FWI inversion (Wuet al 2014). In the combined inversion, the FWI result isobtained by using the smooth background from EI as the initialmodel. In addition, the source wavelet used in the inversion is alow-cut Ricker wavelet (cut from 4Hz below).

Figure 6. A synthetic noisy trace with a thin bed response of ten reflectors convolved with a 20 Hz Ricker wavelet. The noisy trace is addedwith white Gaussian noise, giving a signal-to-noise ratio of 5 dB. Hilbert envelope in the time domain (a) and frequency domain (b); E-SAPin the time domain (c) and frequency domain (d).

2283


We use two models to verify the effectiveness of theE-SAP method in the EI. Figure 7(a) shows the test modelwhich has three low-speed horizontal layers (T1, T3, T5),three high-speed horizontal layers (T2, T4, T6), and a high-speed block B. The high-speed (red) is 5 km s−1 and low-speed (blue) is 3 km s−1. There are 80 shots distributed alongthe model surface at intervals of 20 m and 240 receivers withintervals of 20 m for each shot. And the initial model is alinear gradient model given in figure 7(b). The source waveletis a Ricker wavelet with the dominant frequency 9 Hz, and thefrequency component below 4 Hz is removed.

For comparison, in figure 7(c), we show the FWIinversion result directly using the linear initial model. Thetraditional FWI result has large deviations from the S2 layer,which fails to give the whole structure of the model. Theinversion results by Hilbert Envelope EI and SAP-EI aregiven in figures 7(d) and (f), respectively. The HilbertEnvelope EI fills the velocity well in the T2 layer, but per-forms poorly for the structure below T2 layer. A betterinversion result of the structure is obtained by SAP-EI. TheSAP-EI result recovers the six horizontal layers, but deviatesfrom 1.3 km due to the influence of high-speed block B.Although SAP-EI does not reconstruct the high-speed blockB well, it clearly outlines the upper and lower boundaries. Inthe inversion result of SAP-EI, the velocity filling of T2 layer

is not as good as EI, but the boundaries of layers are bettercharacterized. In addition, we also give the combined EI plusFWI inversion results. Figures 7(e) and (g) show the finalresults of combined EI+FWI inversion using the recovered

Figure 7. Test model and inversion results using the low-cut source. (a) A velocity model with six horizontal layers (three low-speed layers(T1, T3, T5) and three high-speed layers (T2, T4, T6)) and a high-speed block B; (b) linear gradient initial model; (c) FWI inversion result;(d) Hilbert envelope EI inversion result; (e) Hilbert envelope EI+FWI inversion result; (f) SAP-EI inversion result; (g) SAP-EI+FWIinversion result.

Figure 8. One trace selected from the 60th shot data generated fromthe model and its envelopes in the time domain (a) and frequencydomain (b).

2284


smooth background by EI. It can be seen that there are sig-nificant improvements in the inversion results by combinedinversion EI+FWI. The EI+FWI using Hilbert envelopeperforms better than the EI alone, but it still fail to obtain anaccurate estimate of the target model, especially in the low-speed layer T5. The SAP-EI+FWI inversion result is closerto the true velocity model.

We select one trace from the 60th shot data generated usingthe model with the low-cut source wavelet as shown in figure 8.In figure 8(a), the Hilbert envelope produces a positive smoothenvelope for the trace, while the SAP produces a smooth curvewith positive and negative polarity. For convenience of displayand comparison, the amplitude unit of the frequency spectrum isconverted to dB. In figure 8(b), the original seismic trace lacksthe low-frequency component below 4 Hz, while the Hilbertenvelope and E-SAP contain rich low-frequency information.

Considering the effect of noise, we add white Gaussiannoise with signal-to-noise ratio of 5 dB into the shot data totest the anti-noise capability of the proposed method. We pre-filter the shot data with a low-pass filter and then obtain theSAP-EI+FWI inversion result shown in figure 9 (left),which still give an accurate estimate of the true velocitymodel under the noise interference. Figure 9 (right) representsthe Hilbert envelope and E-SAP result of one trace from the60th noisy shot data. These results are obtained after low-passfiltering on the noisy trace. Similar to the no-noise condition,the Hilbert envelope gives a positive smooth envelope, andthe E-SAP gives a smooth curve with positive and negativepolarity. The robustness of E-SAP is verified by the noisydata testing. The experiments indicate that the E-SAP canobtain the low-frequency information with polarity of seis-mogram, and perform well even in a noisy situation.

Conclustion

The modulation signal model can provide low-frequency andhigh-frequency information of a seismogram, especially forthe ultra-low-frequency information which is lost in theconvolution model. A reasonable demodulation method is

required for extracting the low-frequency component. In theseismic modulation model, the low-frequency component isusually extracted as a positive envelope by most demodula-tion methods, losing the polarity information of the seismo-gram. Inclusion of polarity information is important tocharacterize the reflected energy of different interfaces. Onthis basis, we proposed an envelope with smoothed apparentpolarity (E-SAP) as a new demodulation method. The E-SAPcontains the smooth amplitude and polarity information of theseismograms. In addition, we applied the E-SAP to the EI asnew envelope operator instead of Hilbert envelope. Comparedwith convention EI, the EI using E-SAP can better reconstructthe velocity structure and characterize the boundary. Thenumerical examples demonstrated the effectiveness of theproposed method and its good performance in the EI andcombined EI+FWI.

Acknowledgments

This work is supported by the WTOPI Research Consortiumat University of California, Santa Cruz, USA. Y Wangacknowledges the financial support from China ScholarshipCouncil and the National Natural Science Foundation ofChina (Grant Nos. 61571096, 41274127 and 40874066).

ORCID iDs

Yuqing Wang https://orcid.org/0000-0003-1100-6005

References

Bedrosian E 1962 A Product Theorem for Hilbert Transforms RANDMemorandum (Research Memoranda) RM-3439-PR

Brown J L Jr 1986 A Hilbert transform product theorem Proc. IEEE74 520–1

Brown R J, Stewart R R and Lawton D C 2002 A proposed polaritystandard for multicomponent seismic data Geophysics 671028–37

Figure 9. The SAP-EI+FWI inversion result in the case of white Gaussian noise with signal-to-noise ratio of 1 dB (left), and its one traceselected from the 60th noisy shot data and its envelopes in the time domain (right).

2285


https://orcid.org/0000-0003-1100-6005https://orcid.org/0000-0003-1100-6005https://orcid.org/0000-0003-1100-6005https://orcid.org/0000-0003-1100-6005https://doi.org/10.1109/PROC.1986.13495https://doi.org/10.1109/PROC.1986.13495https://doi.org/10.1109/PROC.1986.13495https://doi.org/10.1190/1.1500363https://doi.org/10.1190/1.1500363https://doi.org/10.1190/1.1500363https://doi.org/10.1190/1.1500363

Chen G, Wu R S and Chen S 2017 The nonlinear data functional andmultiscale seismic envelope inversion: algorithm andmethodology for application to salt structure inversion 87thAnnual Int. Meeting, SEG Expanded Abstracts pp 1697–701

Chen G, Wu R S, Wang Y and Chen S 2018 Multi-scale signedenvelope inversion J. Appl. Geophys. 153 113–26

Havlicek J P, Harding D S and Bovik A C 1996 Themulticomponent AM-FM image representation IEEE Trans.Image Process. 5 1094–100

Hu Y, Han L, Xu Z, Zhang F and Zeng J 2017 Adaptive multi-stepfull waveform inversion based on waveform modedecomposition J. Appl. Geophys. 139 195–210

Kallweit R S and Wood L C 1982 The limits of resolution of zero-phase wavelets Geophysics 47 1035–46

Luo J and Wu R S 2015 Seismic envelope inversion: reduction oflocal minima and noise resistance Geophys. Prospect. 63597–614

Loughlin P J and Tacer B 1996 On the amplitude- and frequency-modulation decomposition of signals J. Acoust. Soc. Am. 1001594–601

Potamianos A and Maragos P 1999 Speech analysis and synthesisusing an AM-FM modulation model Speech Commun. 28195–209

Robertson J D and Nogami H H 1984 Complex seismic traceanalysis of thin beds Geophysics 49 344–52

Robinson E A, Durrani T S and Peardon L G 1986 GeophysicalSignal Processing (Englewood Cliffs, NJ: Prentice-Hall)

Sheriff R E 2012 Seismic Stratigraphy (Berlin: Springer)

Taner M T, Koehler F and Sheriff R E 1979 Complex seismic traceanalysis Geophysics 44 1041–63

Turner R E and Sahani M 2011 Demodulation as probabilistic inferenceIEEE Trans. Audio Speech Lang. Process. 19 2398–411

Virieux J and Operto S 2009 An overview of full-waveforminversion in exploration geophysics Geophysics 74 WCC1–26

Wang G C, Yuan S Y, Wang S X, Yan B P and Luo C M 2017Multi-scale envelope inversion method based on scaleseparation 79th EAGE Conf. and Exhibition, 2017

Wang S X, Yuan S Y, Yan B P, He Y X and Sun W J 2016Directional complex-valued coherence attributes fordiscontinuous edge detection J. Appl. Geophys. 129 1–7

Wang Y Q, Wu R S, Peng Z M, Yang L F and You J C 2017Comparison of demodulation methods for seismic signals 87thAnnual Int. Meeting, SEG Expanded Abstracts pp 5506–10

White R E 1991 Properties of instantaneous seismic attributesLeading Edge 10 26–32

Wu R S and Chen G X 2017a Multi-scale seismic envelopeinversion for strong-nonlinear FWI Int. Geophysical Conf.(Qingdao, China, April 2017) pp 29–33

Wu R S and Chen G X 2017b New Fréchet derivative for envelopedata and multi-scale envelope inversion 79th EAGE Conf. andExhibition, 2017

Wu R S, Luo J and Wu B 2014 Seismic envelope inversion andmodulation signal model Geophysics 79 WA13–24

Xu G, Wang X, Xu X and Shao L 2014 Amplitude and phaseanalysis based on signed demodulation for AM-FM signalJ. Comput. Commun. 2 87

2286


https://doi.org/10.1016/j.jappgeo.2018.04.008https://doi.org/10.1016/j.jappgeo.2018.04.008https://doi.org/10.1016/j.jappgeo.2018.04.008https://doi.org/10.1109/83.503927https://doi.org/10.1109/83.503927https://doi.org/10.1109/83.503927https://doi.org/10.1016/j.jappgeo.2017.02.017https://doi.org/10.1016/j.jappgeo.2017.02.017https://doi.org/10.1016/j.jappgeo.2017.02.017https://doi.org/10.1190/1.1441367https://doi.org/10.1190/1.1441367https://doi.org/10.1190/1.1441367https://doi.org/10.1111/1365-2478.12208https://doi.org/10.1111/1365-2478.12208https://doi.org/10.1111/1365-2478.12208https://doi.org/10.1111/1365-2478.12208https://doi.org/10.1121/1.416061https://doi.org/10.1121/1.416061https://doi.org/10.1121/1.416061https://doi.org/10.1121/1.416061https://doi.org/10.1016/S0167-6393(99)00012-6https://doi.org/10.1016/S0167-6393(99)00012-6https://doi.org/10.1016/S0167-6393(99)00012-6https://doi.org/10.1016/S0167-6393(99)00012-6https://doi.org/10.1190/1.1441670https://doi.org/10.1190/1.1441670https://doi.org/10.1190/1.1441670https://doi.org/10.1190/1.1440994https://doi.org/10.1190/1.1440994https://doi.org/10.1190/1.1440994https://doi.org/10.1109/TASL.2011.2135852https://doi.org/10.1109/TASL.2011.2135852https://doi.org/10.1109/TASL.2011.2135852https://doi.org/10.1190/1.3238367https://doi.org/10.1190/1.3238367https://doi.org/10.1190/1.3238367https://doi.org/10.1016/j.jappgeo.2016.03.016https://doi.org/10.1016/j.jappgeo.2016.03.016https://doi.org/10.1016/j.jappgeo.2016.03.016https://doi.org/10.1190/1.1436827https://doi.org/10.1190/1.1436827https://doi.org/10.1190/1.1436827https://doi.org/10.1190/geo2013-0294.1https://doi.org/10.1190/geo2013-0294.1https://doi.org/10.1190/geo2013-0294.1https://doi.org/10.4236/jcc.2014.29012

IntroductionThe seismic modulation modelSmoothed apparent polarityExperiments and analysisSingle-trace

Application in EIConclustionAcknowledgmentsReferences

seismic modulation model and envelope inversion with ...wrs/publication/journal/journal2018... ·...

Documents