Seismic envelope inversion and modulation signal model

Ru-Shan Wu1, Jingrui Luo2, and Bangyu Wu3

ABSTRACT

We recognized that the envelope fluctuation and decay of seis-mic records carries ultra low-frequency (ULF, i.e., the frequencybelow the lowest frequency in the source spectrum) signals thatcan be used to estimate the long-wavelength velocity structure.We then developed envelope inversion for the recovery of low-wavenumber components of media (smooth background), so thatthe initial model dependence of waveform inversion can be re-duced. We derived the misfit function and the corresponding gra-dient operator for envelope inversion. To understand the long-wavelength recovery by the envelope inversion, we developeda nonlinear seismic signal model, the modulation signal model,as the basis for retrieving the ULF data and studied the nonlinearscale separation by the envelope operator. To separate the

envelope data from the wavefield data (envelope extraction), ademodulation operator (envelope operator) was applied to thewaveform data. Numerical tests using synthetic data for the Mar-mousi model proved the validity and feasibility of the proposedapproach. The final results of combined EIþWI (envelope-inversion for smooth background plus waveform-inversion forhigh-resolution velocity structure) indicated that it can delivermuch improved results compared with regular full-waveform in-version (FWI) alone. Furthermore, to test the independence ofthe envelope to the source frequency band, we used a low-cutsource wavelet (cut from 5 Hz below) to generate the syntheticdata. The envelope inversion and the combined EIþWI showedno appreciable difference from the full-band source results. Theproposed envelope inversion is also an efficient method withvery little extra work compared with conventional FWI.

INTRODUCTION

It is known that the lack of ultra low-frequency (ULF) data leadsto difficulty in recovering long-wavelength background structureand therefore to the starting-model dependence of full-waveform in-version (FWI) (see, e.g., Virieux and Operto, 2009). Traditionally,the starting model for FWI is provided by some other methods suchas traveltime tomography (including ray-based or wave-based, first-arrival traveltime tomography and reflection traveltime tomography)and velocity analysis. Within FWI, long-offset and multiscaleinversion has been developed to reduce the starting model depend-ence (Bunks et al., 1995; Pratt et al., 1996, 1998; Ravaut et al., 2004;Sirgue and Pratt, 2004; Plessix et al., 2010; Sirgue et al., 2010; Ficht-ner and Trampert, 2011; Vigh et al., 2011; Baeten et al., 2013; Brit-tan et al., 2013). Multiscale inversion is a cascade inversion startingfrom the lowest frequency available for the recovery of the largest

scale possible. Recent development of low-frequency land source(down to 1.5 Hz) has allowed multiscale FWI to use a 1D smoothstarting model (Baeten et al., 2013). They showed that the lowest-frequency band (1.5–2.0 Hz) is crucial in recovering the correctlong-wavelength background-velocity structure. However, in gen-eral, the ULF sources are not available and the standard seismic re-cords can go down only to approximately 5 Hz. Therefore, thestarting model is still a pressing problem for FWI. Shin and Cha(2009) perform inversion in the Laplace-Fourier domain using seis-mic data below 5 Hz to estimate the smooth, large-scale velocitystructure. A recent trend is that by introducing some extra termsin the misfit functional, FWI can incorporate into its frameworksome other inversion methods, such as traveltime tomography or in-version (Mora, 1989; Clément et al., 2001; Xu et al., 2012; Ma andHale, 2013; Wang et al., 2013) and velocity analysis (Almomin andBiondi, 2012; Biondi and Almomin, 2012, 2013; Tang et al., 2013).

Manuscript received by the Editor 4 August 2013; revised manuscript received 21 December 2013; published online 19 May 2014.1University of California, Earth and Planetary Sciences, Modeling and Imaging Laboratory, Santa Cruz, California, USA. E-mail: rwu@ucsc.edu.2University of California, Earth and Planetary Sciences, Modeling and Imaging Laboratory, Santa Cruz, California, USA and Xi’an Jiaotong University,

Xi’an, China. E-mail: bluebirdjingrui@gmail.com.3Formerly University of California, Earth and Planetary Sciences, Modeling and Imaging Laboratory, Santa Cruz, California, USA; presently Statoil, R&D

Center, Beijing, China. E-mail: bangyuwu@gmail.com.© 2014 Society of Exploration Geophysicists. All rights reserved.

WA13

GEOPHYSICS, VOL. 79, NO. 3 (MAY-JUNE 2014); P. WA13–WA24, 10 FIGS.10.1190/GEO2013-0294.1

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In this paper, we extract the ULF signals contained in seismictrace envelopes and apply them to the recovery of long-wavelengthbackground structure without the use of low-frequency sources thatare expensive or unavailable. The normalized integration method(Chauris et al., 2012; Donno et al., 2013) has used the normalizedtime integration of the squared waveform data as a new type of datafor least-squares (LS) minimization. However, much information ofthe envelope function has been lost due to the integration operator.We demonstrate that envelope fluctuation and decay of seismic re-cords carries ULF signals, but they are inaccessible from the con-ventional linear convolution signal model. To extract the envelopedata from the wavefield data and keep the useful information, weneed to perform a nonlinear operation using a demodulation oper-ator. For inversion, we derive the gradient operator for envelope dataand perform iterative inversion to obtain a smooth backgroundmodel. Then, we use the smooth model from envelope inversionas the starting model of waveform inversion to recover the high-wavenumber components of the model. Comparing with the linearmultiscale inversion of FWI, this is a nonlinear two-scale inversionin which the nonlinear scale separation is done by the envelopeoperator. The other advantage of the envelope inversion is its inde-pendence to the source wavelet. We use a low-cut source wavelet(cut from 5 Hz below) to generate the synthetic data. The envelopeinversion and the combined envelope inversion for a smoothbackground plus waveform inversion for high-resolution velocitystructure (EIþWI) show no appreciable difference from thefull-band-source results. Numerical tests using the Marmousi modeldemonstrated the validity and special features of the approach.

ENVELOPE INVERSION

Review of conventional FWI in the time domain

FWI can retrieve information of the subsurface by fitting thedifference between the recorded data dobs and the simulated datadsyn with the proposed model. The classical LS misfit functionalis given by

σðdÞ ¼ 1

2

Xsr

ZT

0

ðdsyn − dobsÞ2dt;

¼ 1

2

Xsr

ZT

0

½yðtÞ − uðtÞ�2dt; (1)

where u ¼ dobs is the observed wavefield and y ¼ dsyn is thesynthetic wavefield. The summation is over all the sources andreceivers.The goal is to obtain the model parameter m, which can be up-

dated as follows:

mnþ1 ¼ mn þ αnβn; (2)

where αn is the step length in the nth iteration and βn corresponds tothe updating direction, which can be obtained from the gradient ofthe misfit function, so the gradient of the misfit function with re-spect to the model m must be calculated first.Consider velocity v as the model parameter, the gradient of the

misfit function with respect to v can be obtained by

∂σ∂v

¼Xsr

ZT

0

½yðtÞ − uðtÞ� ∂yðtÞ∂v

dt. (3)

Introduce an operator J (Jacobian) and a vector (data residual) η,where

J ¼ ∂yðtÞ∂v

; η ¼ yðtÞ − uðtÞ; (4)

then, equation 3 can be written as

∂σ∂v

¼ JTη. (5)

The Jacobian J is also called the linear Fréchet derivative. It isknown that this gradient can be calculated by zero-lag correlationof the forward propagated source wavefields and the backward-propagated residual wavefields η (Lailly, 1983; Tarantola, 1984,1987; Pratt et al., 1998; Pratt, 1999).

Misfit functional and gradient operatorfor envelope inversion

Now, we define the misfit functional and derive the correspond-ing gradient operator for the envelope inversion. Here, the data to beused in the misfit functional are the trace envelopes, not directly thetraces (waveforms). Bozdag et al. (2011) discuss the envelope misfitfunctional and its use in the kernel sensitivity analysis of globalseismic tomography. However, their purpose is to use the instanta-neous amplitude and phase information of individual arrivals, suchas the first P or S arrivals, so they apply a short time window to thetraces for isolating the chosen arrivals. For our envelope inversion,we need to extract and treat the envelope of the whole trace.We adopt the power objective functional to minimize the

residual:

σðdÞ¼ 1

2

Xsr

ZT

0

����dpsyn−dpobs

����2

dt

¼ 1

2

Xsr

ZT

0

����epsyn−epobs

����2

dt;

¼ 1

2

Xsr

ZT

0

� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiy2ðtÞþy2HðtÞ

qp−

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2ðtÞþu2HðtÞ

qp�

2

dt;

¼ 1

2

Xsr

ZT

0

E2dt; (6)

where e is the envelope function; u and y are the observed and syn-thetic waveforms, respectively; and uH and yH are the correspond-ing Hilbert transforms. Here, E is the instant envelope data residualand the summation is over all the sources and receivers, p is thepower for the envelope data, it can be any positive number. Theextraction of the envelope function and the information containedin envelopes will be discussed in detail in the next section. The gra-dient operator can be derived from the partial derivative of the misfitwith respect to the model perturbations as follows (for a detailedderivation, see Appendix A):

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∂σ∂v

¼Xsr

ZT

0

E∂

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiy2ðtÞþ y2HðtÞ

pp

∂vdt;

¼ pXsr

ZT

0

E

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiy2ðtÞþ y2HðtÞ

qp−2

�yðtÞ ∂yðtÞ

∂vþ yHðtÞ

∂yHðtÞ∂v

�dt;

¼ pXsr

ZT

0

Eep−2syn

�yðtÞ ∂yðtÞ

∂vþ yHðtÞ

∂yHðtÞ∂v

�dt;

¼ pXsr

ZT

0

�EyðtÞep−2syn

∂yðtÞ∂v

−HfEyHðtÞep−2syn g ∂yðtÞ∂v

�dt;

¼ pXsr

ZT

0

½EyðtÞep−2syn −HfEyHðtÞep−2syn g� ∂yðtÞ∂vdt. (7)

We introduce the Fréchet derivative operator (matrix) J and theeffective residual vector η, defined as

J ¼ ∂yðtÞ∂v

; η ¼ EyðtÞep−2syn −HfEyHðtÞep−2syn g. (8)

So equation 7 can also be written as a matrix equation:

∂σ∂v

¼ JTη. (9)

This envelope inversion can also be implemented using the back-propagation method, and the terms in the square brackets serve asthe effective residuals, which is the envelope residual riding on thecarrier signal (modulation). Note that operator J includes a back-propagation operator and a virtual source operator (see, e.g., Prattet al., 1998). From the above equations, we see that the LS residualis calculated using envelope data, but the back projection of theenvelope residual to the model space is done by back propagationusing the carrying signal with the envelope residual riding on itsshoulder. Based on the above equations, the envelope inversion al-gorithm can be developed following the algorithm of the time-domain waveform inversion.To see the effect of the parameter p, we rewrite the misfit function

in equation 6 as follows:

σðdÞ ¼ 1

2

Xsr

ZT

0

½epsyn − epobs�2ðtÞdt;

¼ 1

2

Xsr

ZW

0

½epsyn − epobs�2ðωÞdω;

¼ 1

2

Xsr

ZW

0

½epsynðωÞ − epobsðωÞ�2dω. (10)

Calculating the gradient operator, we get

∂σ∂v

¼Xsr

ZW

0

½epsynðωÞ − epobsðωÞ�∂epsynðωÞ

∂vdω. (11)

Because esyn is a function with respect with y and yH, using thechain rule we know that the above equation can be written in thefollowing form:

∂σ∂v

¼Xsr

ZW

0

½epsynðωÞ − epobsðωÞ��∂epsynðωÞ

∂y∂y∂v

þ ∂epsynðωÞ∂yH

∂yH∂v

�dω. (12)

Figure 1 shows two traces generated from the Marmousi model.In Figure 1a and 1b, the top panels show the original traces; themiddle panels show the trace envelope in the time domain witha different value of p (the amplitudes have been normalized);the bottom panels show the trace envelope in the frequency domainwith a different value of p (the amplitudes have been normalized).We see that different values of power p of the envelope data havethe effect of preconditioning the data in the time and frequency do-mains. In the time domain higher power p, we put heavier weight onthe energetic arrivals, in this case (acoustic wave) earlier envelopearrivals, especially the direct arrivals (including turning waves),whereas in the frequency domain, we put heavier weight on thelow-frequency envelope data. We tested three cases, p ¼ 1, 2,and 3. The parameter p ¼ 3 diminishes the later arrivals too much,and the inversion cannot penetrate to the depth. Therefore, in ap-plications we use only p ¼ 1 and 2. In next section, we show thatp ¼ 2 is a good choice for the Marmousi data set, which provides agood balance on windowing the data for stable inversion to recoverthe long-wavelength background velocity structure.

INFORMATION IN THE ENVELOPE AND THEMODULATION SIGNAL MODEL

Envelope operator (demodulation operator)

In equation 6, we have introduced the envelope function for themisfit functional. The envelope function is extracted from the wave-form data by an envelope operator (or demodulation operator). Theenvelope operator E½fðtÞ� consists of two steps:

1) Analytic transform of trace fðtÞ:

faðtÞ ¼ fðtÞ þ iHðfÞðtÞ ¼ fðtÞ þ ifHðtÞ; (13)

where HðfÞðtÞ is the Hilbert transform of a real function f andfa is the analytic signal (complex) corresponding to f.

2) Take the magnitude of the analytic signal:

E2fðtÞ ¼ E2

f½fðtÞ� ¼ jfaðtÞj2¼ Re½faðtÞ�2 þ Im½faðtÞ�2 ¼ fðtÞ2 þ fHðtÞ2;

EfðtÞ ¼ sqrtfE2fðtÞg. (14)

Now, we discuss the properties of the envelope operator and infor-mation contained in envelopes.Because the envelope operator contains a square-root operator, it

is a nonlinear operator (nonlinear filtering). From the modulationtheory in signal processing (e.g., Robinson et al., 1986), we see thatthe envelope operator is equivalent to a demodulation operator if weconsider a seismogram as a modulated signal: the low-frequencyenvelope is the modulation signal and the high-frequency reflectionwaveforms as the carrier signal. We will discuss the signal model inthe next subsection. But let us now concentrate on the properties of

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the extracted envelopes. From the modulation theory, we know thatamplitude modulation is a nonlinear operation, and the modulationfrequency is riding on the shoulders of the carrying frequency, asshown schematically in Figure 2. Although the modulation fre-quency may be very low, it does not show up in the correspondinglow-frequency range in the frequency-domain representation of themodulated signal. Figure 2a is the modulation signal (the envelope);the carrier signal is a sinusoidal signal with a carrier frequency of8 Hz. Figure 2b is the modulated signal, which is the product of themodulation signal and the carrier signal, and Figure 2c and 2d is thespectra of the modulation signal and the modulated signal, respec-tively (the amplitudes have been normalized). Only the nonlineardemodulation operator can extract the low-frequency informationcoded in the envelope.Figure 3a shows a shot profile of waveform data (top) and

envelope data (bottom) from the synthetic data set of the Marmousimodel (see the next section); the corresponding waveform spectra(top) and envelope spectra (bottom) are plotted in Figure 3b. Weclearly see that the envelope data have ULF spectra compared with

the waveform data. From Figures 1 and 3, we can also see thatenvelope shapes are different for different parts of the traces. Forthe early arrivals of near offsets, the envelopes look like individualenvelopes of the source wavelet; however, for later arrivals and evenearly arrivals of far offsets, the envelopes become continuous fluc-tuations due to the interference between many arrivals and the non-linearity of the envelope operator. The match of the envelope dataresidual may avoid the high-frequency cycle skipping, reduce localminima, and decrease the sensitivity to high-frequency noises.

Modulation signal model

To better understand the nonlinear nature of the envelope oper-ator and its effects on information extraction from seismograms, wefirst discuss the modulation signal model for surface reflection seis-mic data.In the surface reflection survey, the data set (wavefield records) is

composed of direct arrivals (including turning waves) and scatteredwaves from subsurface reflectors. For the formation of reflectiondata, the roles of forward scattering and backscattering are very dif-ferent from each other. Forward scattering only modifies Green’sfunction (propagator) of the background medium (smooth part),whereas backscattering of reflectors is actually responsible forthe generation of reflection signals.In the frequency domain, the wavefield measured on the surface

can be modeled as

pðω; xr; xsÞ ¼ p0ðω; xr; xsÞ þ pscðω; xr; xsÞ; (15)

where p0ðω; xr; xsÞ ¼ gðω; xr; xsÞ is the direct wavefield in thebackground medium (including turning waves) and pscðxr; xsÞ isthe scattered field received by the geophone at xr due to a sourceat xs.Now, we discuss the seismograms formed by backscattered

waves, which can be modeled as

pbscðω; xr; xsÞ ¼ sðωÞZSgMðx; xsÞgMðxr; xÞγðxÞdSðxÞ;

(16)

where S represents all the reflection surfaces in the model space, gMis the modeling Green’s function, which is a full-wave Green’sfunction, sðωÞ is the source spectrum, and γðxÞ is the scattering co-efficient of the reflector element. The full-wave Green’s functioncan be decomposed into a primary-wave part and a multiple-wavepart:

gM ¼ gF þ gmul; (17)

where gF is the forward-scattering Green’s function, which includesall the forward-scattering effects (the transmission mode), such asrefraction, diffraction, geometric spreading, and focusing effectsand therefore contains the average velocity information along thepropagation path, whereas gmul contains multiply scattered wavesfrom other scatterers. To simplify the treatment, we consider onlythe primary reflections, so gmul is neglected. The approximation issimilar to the De Wolf (1971, 1985) approximation (Wu, 1994,2003). We know that the scattering models for forward scatteringand backscattering are very different, and therefore the informationcontained in the reflection series γðxÞ and in the forward scattering

0 0.5 1 1.5 2 2.5 3−200

0

200Original trace

Am

plitu

de

Time (s)

0 0.5 1

Time domain trace envelope with different p

Time domain trace envelope with different p

Frequency domain trace envelope with different p

Frequency domain trace envelope with different p

1.5 2 2.5 30

0.5

1

Time (s)

Am

plitu

de p = 1p = 2

0

b)

a)

5 10 15 20 25 30 35 40 45 500

0.5

1

Frequency (Hz)

Am

plitu

de p = 1p = 2

0 0.5 1 1.5 2 2.5 3−100

0

100Original trace

Am

plitu

de

Time (s)

0 0.5 1 1.5 2 2.5 30

0.5

1

Time (s)

Am

plitu

de p = 1p = 2

0 5 10 15 20 25 30 35 40 45 500

0.5

1

Frequency (Hz)

Am

plitu

de p = 1p = 2

Figure 1. Data traces generated from Marmousi model and theirenvelopes with different powers (different values of p) in the timeand frequency domains.

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approximated Green’s function gF represents different properties ofmedium perturbations. In the case of direct arrivals, gF containsvelocity perturbations along the propagation path; whereas for re-flected arrivals, γðxÞ corresponds to local reflectivity due to imped-ance jump (velocity jump for acoustic media with constant density).Assuming there are totally L’s interfaces (including reflector sur-

faces) and each interface has Ml elements (after discretization), sothe total number of scattering elements is N ¼ P

Ll¼1 Ml. Then, the

primary backscattered field (discretized version of equation 16) canbe modeled as a reflection series:

pbscðω;xr;xsÞ¼sðωÞXNi¼1

gFðω;xi;xsÞgFðω;xr;xiÞγðxiÞdsðxiÞ.

(18)

For simplicity, we take the reflection coefficients as frequencyindependent. The frequency- and time-domain solutions for re-flected signals can be written as

pbscðω; xr; xsÞ ¼ sðωÞXi

gFðω; xi; xsÞgFðω; xr; xiÞγðxiÞ;

pbscðt; xr; xsÞ ¼ sðtÞ �Xi

gFðt; xi; xsÞ � gFðt; xr; xiÞγðxiÞ;

(19)

where * denotes the time convolution. Green’s functions can be ex-pressed in a form,

gFðω; xi; xsÞ ¼ jgsðωÞjðxiÞ exp½−iωτsðxiÞ�;gFðω; xi; xrÞ ¼ jgrðωÞjðxiÞ exp½−iωτrðxiÞ�; (20)

where τs and τr are the corresponding time delays due to forwardscattering along the source and receiver paths, respectively. We de-fine a propagator as follows

Gsrðω; xi; xs; xrÞ ¼ jgsðωÞjjgrðωÞj expf−iω½τsðxiÞþ τrðxiÞ�g; (21)

so the reflection seismograms can be modeled as a reflection timeseries as follows

pbscðt; xr; xsÞ ¼Xi

Rwðxr; xs; t − τiÞ

¼ sðtÞ �Xi

Gsrðt; xi; xs; xrÞγðt; xiÞ;

γðt; xiÞ ¼ γiδðt − τiÞ. (22)

Here, the spatial distribution of reflectors in the 3D space is trans-formed into a time series by a distance-time correspondence in abackground medium (e.g., homogeneous medium). The aboveequation is the traditional convolution signal model (Robinson,1957; Robinson et al., 1986), in which the propagator (Green’sfunction pairs)Gsrðt; xi; xs; xrÞ only modify the traveltimes and am-plitudes of the reflection events, and the signal spectrum is mainlydetermined by the source wavelet. The amplitude and phase func-tions of the propagator Gsr are smoothly varying functions. Theneglect of gmul is equivalent to the drop of the reverberation wave-form in the convolution model (Robinson et al., 1986). We see thatthe propagator, although carrying smooth medium information, canonly modify the source spectrum. Based on the linear signal model,we are restricted to only access the information provided withinsource spectra. This convolution model is good for seismic imagingbecause it provides the mathematical model for resolution analysis

−3 −2 −1 0 1 2 3−1

−0.5

0

0.5

a)

b)

c)

d)

1

Time (s)

Am

plitu

de

−3 −2 −1 0 1 2 3−1

−0.5

0

0.5

1

Time (s)

Am

plitu

de

−12 −8 −4 0 4 8 120

0.2

0.4

0.6

0.8

1

Frequency (Hz)

Am

plitu

de

−12 −8 −4 0 4 8 120

0.2

0.4

0.6

0.8

1

Frequency (Hz)

Am

plitu

de

Figure 2. (a) The modulation signal, (b) themodulated signal, (c) the spectra of the modulationsignal (with a carrier frequency of 8 Hz), and(d) the spectra of the modulated signal.

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and resolution enhancement (such as deconvolution) in high-resolution imaging. However, depending on the information youwant to extract from the seismic records (seismograms), one candefine different signal models. In fact, seismograms have much richinformation than the convolution model predicts. For seismic inver-sion, we need the low-frequency information contained in seismo-grams for the recovery of long-wavelength background velocitystructure. We see from the above analysis that there is low-frequency information coded into the envelope fluctuation and de-cay of seismograms, but not accessible from linear convolutionmodel. The problem is how to make use of the information asso-

ciated with gF coded in the envelope for the estimate of long-wave-length velocity structure in waveform inversion. The demodulationoperator can peel off the envelope from the seismograms, and there-fore can in a certain way decode the low-frequency information notreachable by linear methods. To understand the demodulation op-erator and its effect on the signal spectra, we need to consider themodulation signal theory.We reformulate and redefine the signal model of equation 18 as a

modulation model. Because the propagator Gsrðt; xi; xs; xrÞ is asmooth function of time with respect to the reflection series, thesignal model 18 can be rewritten as

pbscðt; xr; xsÞ ¼Xi

Gsrðt; xs; xrÞγiðt; xiÞ � sðtÞ;

≃ Gsrðt; xs; xrÞγwðtÞ;γwðtÞ ¼

Xi

sðtÞ � γiðt; xiÞ; (23)

where * denotes convolution; γwðtÞ is the reflectivity series with thesource wavelet signature, which is considered as the carrier signal;whereas Gsr is the modulation signal. The high-frequency carriersignal is modulated by the earth medium though wave propagation(only forward-scattering involved). We can consider the reflectionseries 23 as a Modulation-convolution signal model, or as a modu-lation signal model for the reflection series if we are concernedmainly with the low-wavenumber background recovery. (This sig-nal model has been briefly discussed in Wu et al., 2012, 2013). Infact, we can say that the carrier and the modulation signals containquite different informations about the earth media. The carrier γwðtÞis formed by impedance discontinuities convolved with the sourcewavelet; whereas the modulator Gsr contains mainly the informa-tion on the smoothed medium velocity structure (under the forward-scattering approximation).Now, we include the direct arrivals, which can be written as

pdðω; xr; xsÞ ¼ p0ðω; xr; xsÞ þ pdscðω; xr; xsÞ;pdðt; xr; xsÞ ¼ gdðt; xr; xsÞ � sðtÞ; (24)

where

gdðt; xr; xsÞ ¼Z

dωe−iωtgdðω; xr; xsÞ; (25)

is Green’s function for the direct arrival and is assumed to be a muchslower time-varying function than the source wavelet. Because thedirect arrivals are single arrivals or sparse arrivals, so the convolu-tion can be an approximated byproduct. Including the direct arrivals24 and the reflection series 23, the modulation signal model for thewhole seismogram can be expressed as

pðt; xr; xsÞ ¼ gdðt; xr; xsÞsðtÞ þ Gsrðt; xs; xrÞγwðtÞ. (26)

For a seismogram including the reflection series, we can view itas a product of two functions: a carrier signal and a modulator. Forthe direct arrivals, the carrier is the source wavelet, and the modu-lator is the direct propagator (Green’s function); for the reflected

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Figure 3. (a) A shot profile of waveform data (top) and envelopedata (bottom) and (b) waveform spectra (top) and envelope spectra(bottom). The data are from the synthetic data set of the Marmousimodel.

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series, the carrier is γwðtÞ and the modulator is the propagatorGsrðt; xs; xrÞ. If we apply a long-wavelength velocity perturbationto a given model, but keep the impedance jumps unchanged, thenthe propagator will change, resulting in a modulation in amplitudeand phase to the original reflection series. After demodulatingthe traces, the envelope changes (residuals) contain the long-wavelength information of the velocity perturbation. This providesthe signal basis for long-wavelength background recovery.Now, we look at the consequence and effect of the demodulation

operator applied to the modulation model (equation 26). We applythe Hilbert transform product theorem, i.e., the Bedrosian-Browntheorem (Bedrosian, 1962; Brown, 1986), to trace model 26:

Hfpðt;xr;xsÞg ¼Hfgdðt;xr;xsÞsðtÞ þGsrðt;xs;xrÞγwðtÞg;¼ gdðt;xr;xsÞHfsðtÞgþGsrðt;xs;xrÞHfγwðtÞg.

(27)

Because the propagator functions gdðt; xr; xsÞ and Gsrðt; xs; xrÞare low-pass functions (slowly varying function) (under the for-ward-scattering approximation) and satisfies the Brown’s conditionfor demodulation, so it can be pulled out of the Hilbert transform.Therefore, the envelope function of the seismic traces can be ap-proximated by

E2ðtÞ ¼ jpaðtÞj2 ¼ p2ðtÞ þ p2HðtÞ;

≐ ðgdðtÞÞ2½s2ðtÞ þ s2HðtÞ� þ G2srðtÞ½γ2wðtÞ þ ðHfγwðtÞgÞ2�;

¼ ðgdðtÞÞ2E2½sðtÞ� þ G2srðtÞE2½γwðtÞ�; (28)

where E2½sðtÞ� is the squared envelope of the source wavelet. Inderiving the above equation, we neglected the crosstalk betweenthe direct arrival energy and the reflection series energy. Thisassumption is valid for near and medium offsets due to the travel-time difference. For far offsets, the approximation may not be validand needs further study. However, this approximation is not criticalfor envelope extraction. We see that taking an envelope of seismo-grams (demodulation) is equivalent to the application of a nonlin-ear signal filtering, resulting in the extraction of low-frequencyinformation coded in the seismograms. For direct arrivals (the firstterm in the right side), the resulting envelope-gram (envelogram)has the effect of using a low-resolution wavelet (a Gaussian pulse)to replace the original high-frequency wavelet (Ricker wavelet).For the reflection series (the second term in the right side),E2½γwðtÞ� will extract only the low-frequency part of the carriersignal γwðtÞ. We also noticed that the envelope operator keepsthe long-wavelength information riding on the propagator Gsr un-touched. In this way, the high-wavenumber reflection informationis peeled off and only the low-wavenumber background is kept inthe envelograms.

Nonlinear scale separation by envelope operator

It is well-known that seismic data have wavenumber gaps (orscale gaps) in terms of subsurface structure inversion. That is thelack of intermediate wavenumber (medium-scale) informationabout the subsurface media in the data (Claerbout, 1985; Jannaneet al., 1989; Mora, 1989; Ghosh, 2000; Latimer et al., 2000; Plessix

et al., 2012; Baeten et al., 2013). For 3D earth structure, this gap isthe combined results of limited acquisition frequency band and lim-ited observation aperture. For surface reflection survey with limitedacquisition aperture, the gap is mainly caused by the lack of low-frequencies in source spectra (e.g., Latimer et al., 2000; Baetenet al., 2013). Therefore, substantial efforts have been made to de-velop low-frequency sources. Big gun arrays and a deeper towingdepth can provide low frequencies down to as low as 2 Hz, but evenlower frequencies are still needed (Vigh et al., 2011; Baeten et al.,2013). Recently, low-frequency land source (vibrator with nonlinearfrequency sweep) has been reported (Plessix et al., 2012; Baetenet al., 2013), which can extend the low-frequency end to 1.5 Hz.FWI using the corresponding data showed improved low-wavenum-ber background recovery. Due to the availability of low-frequencydata, they could use a 1D smooth starting model in their multiscale

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inversion scheme. They showed that the lowest-frequency band(1.5–2.0 Hz) is crucial in recovering the correct low-wavenumberbackground velocity structure.As we mentioned above, the linear scale separation is usually

realized by a multiscale inversion: the large-scale recovery is done

using the low-frequency data and the smaller scales are recoveredfrom the high-frequency data subsequently. In this linear approach,the expensive low-frequency source is a necessary and critical arma-ment for success. In the following, we discuss the nonlinear scaleseparation through envelope operator.To simplify the scale analysis, in the following we mainly discuss

the scale response for vertical structural variations as in previousdiscussions (Jannane et al., 1989; Baeten et al., 2013). From equa-tion 28 and Figures 2 and 3, we see that the envelope operator canpeel off the low-frequency envelope from the waveform traces. Thelow-frequency nonlinear extraction in fact separates the large-scaleresponse that is coded in the envelope, from the waveform data thatare the band-limited responses to the medium perturbation. This isvery different from the concept of linear scale separation, whichdepends on the linear correspondence between the signal frequencyand medium scale. In Figure 4a, we show the comparison of wave-form spectrum (blue line) and envelope spectrum (red line) for theMarmousi data with a full Ricker source excitation. In the figure, wealso plot the average medium velocity spectrum (thin gray line) andthe average velocity perturbation (medium velocity subtracts thebackground velocity) spectrum (dotted line). To see the correspon-dence of medium wavenumber spectra to the data frequency spec-tra, we perform a z‐t transform using the known velocity structure.The medium spectrum shown in the figure is the overall average ofthe vertical profiles. The same has done for the perturbation spec-trum. From Figure 4a, we see that for the waveform data, the low-frequency spectral components are missing so the long-wavelengthpart of the perturbation spectrum is not covered. Compared with theenvelope spectrum, we see the complementary role of envelopedata, which has strong low-frequency components but has veryweak high-frequency components. This property of the envelopefunction can reduce the cycle skipping and local minima problemsof waveform data and can recover the long-wavelength components

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Figure 5. (a) True Marmousi model and (b) linear gradient initialmodel.

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RickerwaveletEnvelope

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Rickerwavelet

Envelope

Figure 6. Ricker source wavelet (solid line) andits envelope (dashed line). (a) Full-band sourcein time domain, (b) full-band source in frequencydomain, (c) low-cut source in time domain, and(d) low-cut source in frequency domain.

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of the perturbation structure as shown in the next section of the Mar-mousi model. Of course, its limitation needs to be further studied.Because of the nonlinear nature of the envelope operator, the low-

frequency contents of the envelope data do not tightly depend on thesource spectrum, which is a sharp contrast to the conventional FWI.In Figure 4b, we plot a similar spectral comparison as in Figure 4a,but using a low-cut Ricker source. When generating synthetic seis-mograms, the source wavelet was filtered with a 5-Hz low-cut taper.From the spectrum of waveform data thus generated, we see verylittle energy exists below 5 Hz. Later in the FWI tests, we can seemuch worse results than the full-band source because of the lack oflow-frequency energy. However, the spectrum of the envelope data(average over all the traces) does not show too much difference fromthe full-band source case. This demonstrates the nonlinear nature ofthe envelope extraction, which does not have the linear correspon-dence between the source spectrum and the data spectrum. Later inthe inversion test section, we will show that the inversion results forthese two cases are very similar, showing the independence ofenvelope inversion to the source spectra.This kind of nonlinear scale separation is similar to the Laplace

domain inversion (Shin and Ha, 2008) and the normalized integra-tion method (Chauris et al., 2012; Donno et al., 2013). For linearscale separation in the multiscale inversion method, the inversiontheory is based on Born modeling. However, for nonlinear scaleseparation, the forward modeling theory for the long-wavelengthpart is not Born modeling. The forward modeling theory forenvelope inversion needs further investigation, which may providemore insight on the benefit and limitation of the envelope inversion.

INVERSION TESTS WITH THE MARMOUSIMODEL

The data were generated by an FD algorithm with an acquisitionsystem composed of 50 shots evenly distributed along the surface.We used a total of 228 receivers across the surface for each shot.The true model is shown in Figure 5a. A linear gradient model isused as the initial model (Figure 5b).To test the independence of envelope inversion to source spectra,

especially the existence of low frequency in the source wavelet, weuse two types of Ricker wavelets as the source wavelet: one is thefull-band wavelet, and the other is the low-cut wavelet. Figure 6shows the two types of Ricker wavelet and their envelopes inthe time and frequency domains. Figure 7 shows two traces selectedfrom the data generated from the Marmousi model when using thelow-cut source wavelet in Figure 6. The upper panels are the timedomain traces and their envelopes, and the lower panels are the cor-responding frequency-domain spectra. We can see the rich low-frequency information in the envelopes.Starting from the linear initial model, the smooth backgrounds

obtained from envelope inversion (after 10 iterations) are shownin Figure 8a and 8b for the two types of sources, respectively.We see the strong similarity between these two recovered back-ground structures. This shows the insensitivity of EI to the sourcefrequency band, and further demonstrates that the long-wavelengthrecovery of background is from the demodulated envelope curve,not from the low-frequency source excitation. Figure 8c and 8dshows the final results of combined inversion (EIþWI) by wave-form inversion using the recovered smooth background by EI. Thisstrategy is similar to that used in the combined inversion of Chauriset al. (2012) and Donno et al. (2013), where they use the estimated

background velocity model from the normalized integration methodas the starting model for the high-wavenumber recovery by FWI.For comparison, in Figure 8e and 8f, we show the conventional FWIresults starting directly from the linear initial model. We see thesignificant improvement in inversion fidelity by combined inversionEIþWI. This is mainly due to the scale separation of the envelopeoperator, so that the long-wavelength background can be recoveredcorrectly by EI. From Figure 8e and 8f, we also see that conven-tional FWI is sensitive to the source frequency band. The inversionresult using the low-cut source is noticeably worse than that of thefull-band source; whereas the corresponding two results by EIþWI

show no appreciable difference.Figure 9 shows the reduction of LS residuals with iterations.

Compared with FWI (dashed line), the convergence of EIþWI

(solid line) is faster and has avoided the false local minima.

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Trace envelope

Figure 7. Two traces (a and b) selected from the data generatedfrom the Marmousi model using the low-cut Ricker wavelet andtheir spectra. The upper panels are the time domain traces and theirenvelopes, and the lower panels are the corresponding spectra in thefrequency domain.

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To demonstrate the effect of data power p to the result ofenvelope inversion, in Figure 10, we show the result of EI with p ¼1 and the final result of EIþWI. As we discussed in the previoussection, the EI results in this case are much rougher than the case ofp ¼ 2 due to the heavier weight on the later arrivals (reflections)and on the high-frequency components of the envelope data. Weknow that the back propagation of envelope data is similar toenergy-pact imaging, so there is no destructive interference, result-ing in a noisier image than the FWI image. Note that even thoughthe EI results of p ¼ 1 are very different from those of p ¼ 2; thefinal result of EIþWI in this case is very similar to the case ofp ¼ 2. This indicates that long-wavelength background media re-covered in these two cases are fairly close to each other.

CONCLUSION

Envelope fluctuation and decay of seismic records carries ULFsignals, which can be used to estimate the long-wavelength velocitystructure. We proposed a nonlinear seismic signal model, a modu-lation model, for retrieving the ULF information in seismic wave-form data. Envelope inversion based on LS minimization ofenvelope data can recover the low-wavenumber components of un-known velocity structures (smooth backgrounds), so that the initialmodel dependence of waveform inversion can be reduced. This isdemonstrated by the Marmousi model tests, in which a 1D lineargradient starting model is used, and the combined EIþWI can

correctively recover the low- and high-wavenumber structure ofthe model. The other advantages of the envelope inversion areits independence of the source wavelet and its low cost (very littleextra cost beyond the regular FWI cost). Further study is needed forthe inversion dependence on reflector distribution, acquisition aper-ture, and other limitations of the envelope inversion. The other im-portant research topic for better understanding of the advantagesand limitations of envelope inversion is the forward modeling ofenvelope formation, which will be investigated in the future study.

0 50 100 150 2000

1

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4

5

6

7

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9

Iterations

Mis

fit

EI + FWIFWI

Figure 9. Reduction of LS residuals with iterations. Comparison ofEIþWI (solid line) and FWI (dashed line).

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Figure 8. Envelope inversion tests on the Mar-mousi model. (a) Smooth background obtainedfrom envelope inversion (EI) using the full-bandsource, (b) same as (a) but with the low-cut source,(c) combined EIþWI inversion result (waveforminversion using the smooth background fromenvelope inversion) using the full-band source,(d) combined EIþWI inversion result using thelow-cut source, (e) direct waveform inversion(FWI) from the linear initial model using thefull-band source, and (f) same as (e) but usingthe low-cut source.

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ACKNOWLEDGMENTS

We thank M. Dai, J. Mao, Y. Geng, L. Ye, R. Yan, X.-B. Xie fordiscussions and helps. The suggestions and comments of reviewersare appreciated. We thank also the WTOPI members who partici-pated in the discussions during our last year’s annual WTOPI meet-ing. This research is supported by the WTOPI Research Consortiumat the University of California, Santa Cruz, USA.

APPENDIX A

GRADIENT CALCULATION FOR ENVELOPE IN-VERSION

In this appendix, we show how to derive the gradient of theenvelope misfit as shown in equation 7.First, to calculate the gradient of the misfit as in equation 6 with

respect to velocity, we obtain

∂σ∂v

¼Xsr

ZT

0

E ×∂

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiy2ðtÞ þ y2HðtÞ

pp

∂vdt;

¼ pXsr

ZT

0

Effiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiy2ðtÞ þ y2HðtÞ

qp−2

�yðtÞ ∂yðtÞ

∂v

þ yHðtÞ∂yHðtÞ∂v

�dt;

¼ pXsr

ZT

0

Eep−2syn ½yðtÞ ∂yðtÞ∂vþ yHðtÞ

∂yHðtÞ∂v

�dt.

(A-1)

Let us define the Hilbert transform of signal fðtÞ as

HffðtÞg ¼ hðtÞ � fðtÞ; (A-2)

where * denotes convolution, and

hðtÞ ¼ 1

πt. (A-3)

Note that hð−tÞ ¼ −hðtÞ. We then have

ZsðtÞδHffðtÞgdt ¼

ZsðtÞ

Zhðt − t 0Þδfðt 0Þdt 0dt;

¼ZZ

hðt − t 0ÞsðtÞdtδfðt 0Þdt 0;

¼ZZ

h½−ðt 0 − tÞ�sðtÞdtδfðt 0Þdt 0;

¼ −ZZ

hðt 0 − tÞsðtÞdtδfðt 0Þdt 0;

¼ −Z

HfsðtÞgδðfÞdt; (A-4)

where δ means differentiation with respect to some kind of param-eter (other than t).Using the equation in above, equation A-1 becomes

∂σ∂v

¼pXsr

ZT

0

Eep−2syn

�yðtÞ ∂yðtÞ

∂vþyHðtÞ

∂yHðtÞ∂v

�dt;

¼pXsr

ZT

0

�EyðtÞep−2syn

∂yðtÞ∂v

þEyHðtÞep−2syn∂yHðtÞ∂v

�;

¼pXsr

ZT

0

�EyðtÞep−2syn ×

∂yðtÞ∂v

−HfEyHðtÞep−2syn g ∂yðtÞ∂v

�dt;

¼pXsr

ZT

0

�EyðtÞep−2syn −HfEyHðtÞep−2syn g

�∂yðtÞ∂v

dt.

(A-5)

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Figure 10. EI with p ¼ 1 (a) and the final result of EIþWI (b).Note that even the EI results of p ¼ 1 are very different from thoseof p ¼ 2, the final result of EIþWI in this case is very similar tothe case of p ¼ 2.

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