seismic envelope inversion: reduction of local minima and ...wrs/publication/journal/... · seismic...

Geophysical Prospecting, 2015, 63, 597–614 doi: 10.1111/1365-2478.12208

Seismic envelope inversion: reduction of local minima and noiseresistance

Jingrui Luo1,2∗ and Ru-Shan Wu2

1Institute of Wave and Information, Xi’an Jiaotong University, Xi’an, 710049, China, and 2Modeling and Imaging Laboratory, Earth &Planetary Sciences, Univ. of California, Santa Cruz, CA 95064

Received September 2013, revision accepted July 2014

ABSTRACTWaveform inversion met severe challenge in retrieving long-wavelength backgroundstructure. We have proposed to use envelope inversion to recover the large-scalecomponent of the model. Using the large-scale background recovered by envelopeinversion as new starting model, we can get much better result than the conventionalfull waveform inversion. By comparing the configurations of the misfit functional be-tween the envelope inversion and the conventional waveform inversion, we show thatenvelope inversion can greatly reduce the local minimum problem. The combinationof envelope inversion and waveform inversion can deliver more faithful and accuratefinal result with almost no extra computation cost compared to the conventional fullwaveform inversion. We also tested the noise resistance ability of envelope inversionto Gaussian noise and seismic interference noise. The results showed that envelopeinversion is insensitive to Gaussian noise and, to a certain extent, insensitive to seismicinterference noise. This indicates the robustness of this method and its potential usefor noisy data.

Key words: Envelope inversion, Waveform inversion, misfit function, convergencerate, noise resistance.

INTRODUCTION

Full waveform inversion (FWI) was introduced in the earlyeighties in the time domain by Lailly (1983) and Tarantola(1984). The gradient of the misfit function was calculatedusing back propagation method, thus avoided the explicitlycomputing of the partial derivatives (Virieux and Operto,2009). Mora (1987, 1988) extended this technique to elasticproblems. Crase et al. (1990) applied this method to real data.Pratt (1999) and Pratt and Shipp (1999) applied the same ideato frequency domain full waveform inversion. Recently, fullwaveform inversion has been developed to 3D (Warner et al.,2013), and high performance computation has been applied(Mao et al., 2012). However, in full waveform inversion, wealways need a good initial model that is not far from the truemodel to get correct inversion results.

∗E-mail: [email protected]

In order to overcome this problem, some people usedglobal optimization methods (Boschetti, 1996; Mallick, 1999;Padhi et al., 2010). However, applications of these methodsare restricted because of the tremendous computation cost.Several other approaches have been developed in order toovercome this problem. Bunks et al. (1995) introduced mul-tiscale full waveform inversion method, where the scale de-composition is performed and allows us to do the inversionfrom low frequency to high frequency. Frequency domain fullwaveform inversion is an intrinsic multiscale approach (Pratt,1999; Pratt and Shipp, 1999). Brenders and Pratt (2007) usedcomplex-valued frequencies in frequency domain full wave-form tomography to ease the local minima problem. How-ever, the recovery of long wavelength component of the modeldepends on the availability of low-frequency signal in seis-mic source. To generate low-frequency signal below 5Hz isvery expensive, even not realistic below 1-2 Hz. Therefore,effort has been focused to the recovery of long wavelength

597C© 2014 European Association of Geoscientists & Engineers

598 J. Luo and R.-S. Wu

50 100 150 200

3

2

1

0

Receiver locations

Tim

e (s

)

3

2

1

0

Tim

e (s

)50 100 150 200

30

20

10

0

Receiver locations

Fre

quen

cy (

Hz)

30

20

10

0

Fre

quen

cy (

Hz)

(b)(a)

Figure 1 Data traces and their spectra. (a) A shot profile (common shot records) of the seismic data (top) and the envelope (bottom). (b) Seismicdata spectra (top) and envelope spectra (bottom). The data are from the synthetic data set of the Marmousi model.

0 0.5 1 1.5 2 2.5 3−100

0

100

200

(a) (b)

Time (s)

Am

plitu

de

Trace

Original trace

Trace envelope

0 10 20 30 40 500

1

2

3x 10

4 Trace spectrum

Frequency (Hz)

Am

plitu

de

Original trace

Trace envelope

0 0.5 1 1.5 2 2.5 3−100

0

100

200

Time (s)

Am

plitu

de

Trace

Original trace

Trace envelope

0 10 20 30 40 500

1

2

3x 10

4 Trace spectrum

Frequency (Hz)

Am

plitu

de

Original trace

Trace envelope

Figure 2 A trace from the shot profile in Fig. 1. (a) The top panel shows the time-domain trace and its envelope; the bottom panel shows theamplitude spectra of the trace and its envelope; (b) same as (a) except that the low-frequency component below 5 Hz was removed from thesource wavelet.

background without very low frequencies. Travel time in-version and migration velocity analysis are two traditionalmethods in this area. In travel time inversion, the travel timeinformation of the wavefield is used to invert the model pa-rameters. It can be classified to two categories. One is raybased travel time inversion (Dines and Lytle, 1979; Paulsson

et al., 1985; Justice et al., 1989) which needs the picking oftravel time, and the other is wave equation travel time in-version (Luo and Schuter, 1991; Woodward, 1992) which isbased on wave theory instead of ray theory. In migration ve-locity analysis, the residual moveout (Liu and Bleistein, 1995;Xie and Yang, 2008) or the image perturbation (Biondi and

C© 2014 European Association of Geoscientists & Engineers, Geophysical Prospecting, 63, 597–614

Seismic envelope inversion 599

0 0.5 1 1.5 2 2.5 3

−1

0

(a) (b)A

mpl

itude

0 0.5 1 1.5 2 2.5 3

−1

0

1

Am

plitu

de

0 0.5 1 1.5 2 2.5 3−3

−2

−1

0

1

Am

plitu

de

Time(s)

0 0.5 1 1.5 2 2.5 3

−1

0

1

Am

plitu

de

0 0.5 1 1.5 2 2.5 3

−1

0

1

Am

plitu

de

0 0.5 1 1.5 2 2.5 3

−1

0

1

Am

plitu

de

Time(s)

Figure 3 Effective data residual for back propagation in the time domain (the amplitudes have been normalized). (a) The first iteration. (b) The10th iteration. The top panels are the seismic data residuals for the conventional full-waveform inversion; the middle panels are the effectiveresiduals for the EI using envelope misfit; and the bottom panels are the effective residuals for EI using squared envelope misfit.

Sava, 1999; Sava and Biondi, 2004; Shen and Symes, 2008) isused, and the velocity perturbation is then obtained. In recentyears, Shin and Cha (2008) has developed Laplace domainfull waveform inversion which can give a smooth backgroundmodel from an inaccurate initial model. Later, they extendedthis method to the Laplace-Fourier domain full waveform in-version (Shin and Cha, 2009). There are also another kind ofmethods, which combines waveform inversion and some othertypes of inversion. For example, Zhou et al. (1995) and Zhouet al. (1997) combined traveltime and waveform inversion.Biondi and Almomin (2012) and Almomin and Biondi (2012)combined full waveform inversion with wave-equation migra-tion velocity analysis. Wang et al. (2012) used wave equationtomography and full waveform inversion. Liu et at. (2011)proposed the normalized integration method, in which theenvelope information of the data was used. In their method,they measure the misfit between the integral of the absolutevalue, or of the square, or of the envelope of the signal. The in-tegration is an increasing function, and the objective function

is convex. This method can provide an intermediate solutionbetween a very oscillating solution and a very smooth model.

We (Wu et al., 2013, 2014) have proposed an envelopeinversion (EI) method. In our method, we also use the en-velope information of the seismograms. However, instead ofmeasuring the integral of the envelope, we directly measurethe envelope or the squared envelope as a function of time.The low frequency information in the seismogram envelopecan be used, and a very smooth background structure canbe obtained. In this paper, we discuss some detailed imple-mentation and application issues of envelope inversion. Wefirst give the procedure about how to realize envelope inver-sion. Then we compare the behavior of the misfit functionsbetween envelope inversion and the conventional waveforminversion. From the numerical results we can see that enve-lope inversion can avoid or reduce the local minima problem.The combined inversion of EI and WI can deliver more faith-ful and accurate final result with almost no extra cost com-pared with conventional FWI. Also, we test the noise-resistant



property of envelope inversion. Numerical results showed thatthis method is insensitive to white Gaussian noise, and to acertain extent insensitive to seismic interference noise. Thisindicates the robustness of this method and its potential ap-plication in dealing with noisy data.

A B RIEF REVIEW OF THE C ONVENTIONALFULL WAVEFOR M I N V ER SI ON IN T H ET I M E D O M A I N

Full Waveform Inversion can retrieve information of subsur-face by fitting the observed data and the synthetic data. Theclassical least squares misfit functional in the time domain is,

σ (m) = 12

∑sr

∫ T

0[y(t) − u(t)]2dt, (1)

where u is the observed wavefield, y is the synthetic wavefield,m is the model parameter set. The summation is over all thesources and receivers.

We consider the acoustic situation with constant density,and consider velocity v as the model parameter, the gradientof the misfit function σ with respect to v can be obtained by,

∂σ

∂v=

∑sr

∫ T

0[y(t) − u(t)]

∂(t)∂v

dt. (2)

By introducing an operator J (Jacobian) and a vector (dataresidual) η, where

J = ∂y(t)∂v

, η = y(t) − u(t) (3)

equation (2) can be written as,

∂σ

∂v= JTη. (4)

The Jacobian J is also called the linear Frechet derivative.It is known that this gradient can be calculated by zero-lagcorrelation of the forward propagated source wavefields andthe backward propagated residual wavefields (Lailly, 1983;Tarantola, 1984; Bunks et al., 1995).

ENVELOPE INVERSION M ETHOD

Extraction of a trace envelope

In order to realize envelope inversion, we should first get theenvelope of a signal (seismic trace). We extract envelope bytaking the amplitude after the analytical signal transform us-ing Hilbert transform. A signal without negative-frequency

components is called an analytic signal f (t) which can beconstructed from a real signal f (t) and its Hilbert transformH{ f (t)},f (t) = f (t) + i H{ f (t)}. (5)

The Hilbert transform is defined as,

H{ f (t)} = − 1π

P∫ +∞

−∞

f (τ )t − τ

, (6)

where P is the Cauchy principal value.The envelope of f (t) can then be derived by,

e(t) =√

f 2(t) + H{ f (t)}2. (7)

So we can easily get the envelope of this signal by using Hilberttransform.

Figure 1 gives an example of the seismic data and itsenvelope (Figure 1(a)) of a shot profile from the Marmousimodel and their spectra (Figure 1(b)). The data was generatedusing time domain finite difference method. The source is theRicker wavelet with a dominant frequency of 10Hz. We seethat the envelope traces are much smoother than the seismicdata traces, and the corresponding envelope spectra are muchricher in low frequency components. For a better look, inFigure 2 we zoom in one of the traces from the shot profile.Figure 2(a) shows the original trace and it’s envelope in bothtime domain and frequency domain. To better demonstratethe ability to extract low frequency information from the en-velope, we removed the low frequency component below 5Hzfrom the data trace (Figure 2(b)). We see that in this case, theenvelope is still very rich in low frequency components, whichcan be used to recover the long wavelength background that isbeyond the reach of the conventional full waveform inversion.For detailed signal model related to envelope inversion, pleasesee Wu et al. (2013b). In Wu et al. (2013b), we have shownthe inversion results for both the cases with and without lowfrequency source component below 5Hz (for the readers’ con-venience, we will show the results again in later sections ofthis paper).

Misfit function

In envelope inversion, we define the misfit function as theenvelope misfit (Wu et al. 2013a, b),

σ (m) = 14

∑sr

∫ T

0

∣∣e2syn − e2

obs

∣∣2dt. (8)

where m is the model parameter, esyn and eobs are theenvelopes of the synthetic trace and the observed trace



0 10 20 30 40 50 600

0.2

0.4

0.6

(a) (b)

1

Am

plitu

de

0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

1

Am

plitu

de

0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

1

Am

plitu

de

Frequency (Hz)

0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

1

Am

plitu

de

0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

1

Am

plitu

de

0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

1

Am

plitu

de

Frequency (Hz)

Figure 4 Same as Fig. 3 but in the frequency domain (the amplitudes have been normalized).

Distance (km)

Dep

th (

km)

2 4 6 8

1

2

1500

2000

2500

3000

3500

4000

4500

5000

5500

Figure 5 Test model for the misfit function configuration test. Thetest model is from the top left corner of the Marmousi model.

respectively. Using the Hilbert transform which has been in-troduced previously, the above equation can then be writtenas,

σ (m) = 14

∑sr

∫ T

0{[y2(t) + y2

H] − [u2(t) + u2H(t)]}2, (9)

= 14

∑sr

∫ T

0E2dt,

where y and u are the synthetic traces and the observed tracesrespectively, yH and uH are the corresponding Hilbert trans-forms. E is the instant envelope data residual. The summa-tion is over all the sources and receivers. Here we applieda square to the envelope, because the squared envelope hasbetter performance in long-wavelength background recovery.The difference of the envelope and its power in the misfitfunction has been discussed in Wu et al. (2013b). Here wealso show some simple numerical examples to briefly demon-strate this, but from the view of data residuals. We can see thedifference between the envelope and squared envelope fromFigure 3 and Figure 4, which show the effective data residualfor back-propagation (the amplitudes have been normalized).We see that the residuals for envelope inversion are muchricher in low frequency components compared to the conven-tional residual. Meanwhile the curves for the case of squaredenvelope are much smoother than that for the case of enve-lope, and the spectra for the case of squared envelope arericher in low frequencies than the case of envelope.



(a)

(b)

(c)

Figu

re6

Mis

fit

func

tion

conf

igur

atio

nsof

(a)

conv

enti

onal

wav

efor

min

vers

ion;

(b)

the

enla

rged

part

insi

deth

ere

dbo

rder

in(a

);an

d(c

)E

I.



Distance (km)

Dep

th (

km)

2 4 6 8

1

2

1500

2000

2500

3000

3500

4000

4500

5000

5500

line A line B

Figure 7 The Marmousi velocity model.

Distance (km)

Dep

th (

km)

2 4 6 8

1

2

2000

3000

4000

5000

Figure 8 Linear initial model.

The gradient method in envelope inversion

Consider wave propagation velocity v as the model parameter.Calculating the derivative of the misfit function σ with respectto v, we get,

∂σ

∂v= 1

2

∑sr

∫ T

0E

∂[y2(t) + y2H(t)]

∂vdt,

= 12

∑sr

∫ T

0E[2y(t)

∂y(t)∂v

+ 2yH(t)∂yH(t)

∂v]dt,

=∑

sr

∫ T

0[Ey(t)

∂y(t)∂v

− H{EyH(t)}∂y(t)∂v

]dt,

=∑

sr

∫ T

0[Ey(t) − H{EyH(t)}]∂y(t)

∂vdt. (10)

We introduce the Frechet derivative operator J and the effec-tive envelope residual vector η, where

J = ∂y(t)∂v

, η = Ey(t) − H{EyH(t)} (11)

So equation (10) can be expressed as,

∂σ

∂v= JTη. (12)

From equation (12) we see that the derivative of theenvelope misfit function has the same form as that of theconventional waveform inversion (equation (4)) but with adifferent data residual vector. Therefore, envelope inversioncan also be realized by using back-propagation method. Wesee the forward modeling is done with the same frequencycontent as for the input seismic data. However, because η isrelated to the effective envelope residual not the seismic dataresidual, the low frequency information in signal envelope canbe lower than the lowest frequency in the source wavelet. Thislow frequency information can be used to construct the largescale background structure. The low-frequency informationcoded in the envelope is not directly from the source wavelet.

ANALYSIS AND T ESTS OF MISF ITFUNCTION CONFIGURATION

We first compare the configuration of the misfit functionsfrom envelope inversion and the conventional waveform in-version. We see from Figure 1 that the envelope fluctuatesalong time much smoothly and have much stronger low-frequency components than the seismic data, so we can expectthat the misfit function of the envelope varies also smootherthan that of the seismic data.

Figure 5 shows the model we used for this test. We usethe top left corner (inside the red border) of the Marmousimodel as the test model. As defined in equations (1) and (8),misfit function can be parameterized as a function of veloc-ity model, which varies in the multi-dimensional parameterspace. In order to show the behaviors of the misfit function,we simplify the multi-dimensional parameter space into a 2-dimensional space. We decompose the test model vtrue(x) intoa background model and a perturbation to the background.The background is a linear gradient background varies from1.5 km/s (Vmin) to 5.5 km/s (Vmax). We call this back-ground as the true background vtrue

0 (x). The true perturba-tion δvtrue(x) is obtained from the subtraction of vtrue(x) andvtrue

0 (x). We set a strength parameter α. By varing α, we get aseries perturbation functions δv(x) = αδvtrue(x). For the truemodel, α = 100%. The two parameters are Vmax and the



Distance (km)(a) (b)

(c)

Dep

th (

km)

2 4 6 8

1

2

2000

2500

3000

3500

4000

4500

5000

Distance (km)

Dep

th (

km)

2 4 6 8

1

2

2000

2500

3000

3500

4000

4500

5000

Distance (km)

Dep

th (

km)

2 4 6 8

1

2

2000

2500

3000

3500

4000

4500

5000

Figure 9 EI results after (a) one iteration; (b) five iterations; and (c) ten iterations.

0 50 100 150 2000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Iterations

Nor

mal

ized

mis

fit

Envelope inversionFWI using EI initial

Figure 10 Reduction of seismic data residuals. The misfit has beennormalized with the norm of the original seismic data. The solid lineis the seismic data misfit for the EI, and the dashed line is for EI+WI.

perturbation strength α. We keep Vmin as 1.5 km/s, andchange Vmax from 2.5km/s to 7.5km/s. We also change theperturbation strength α from 10% to 300%. By summing thebackground and the perturbation, we get a series of trial mod-els. Then we calculate the waveform and envelope data fromthe true model and from the trial models. The misfit for eachtrial model is obtained from the two data sets.

Distance (km)

Dep

th (

km)

2 4 6 8

1

2

2000

3000

4000

5000

Figure 11 EI+WI inversion result. This is the result of full-waveforminversion using the EI result in Fig. 9(c) as the initial model.

Figure 6 shows the configuration of the misfit functions(the variation of the misfit function with respect to the twoparameters). The horizontal axis is Vmax, and the verticalaxis is the perturbation strength. The amplitudes have beennormalized. (a) is for the conventional waveform inversion.To see the details, we enlarge the part inside the red border(Figure 6(b)). We see that besides the global minimum, thereare some local minima, which will lead the inversion to wrong



Distance (km)

Dep

th (

km)

2 4 6 8

1

2

2000

3000

4000

5000

Figure 12 Conventional FWI result. The starting model is the linearinitial model in Fig. 8.

results. (c) shows the configuration of the envelope misfit, wesee that the misfit function is very smooth, and there is onlya global minimum and no local minima. So using envelopeinversion can avoid or reduce the local minimum problem,and we will show this in the next section.

NUMERICAL EXAMPLES OF ENVELOPEINVERS ION

In this part, we will show some envelope inversion results. Weuse Marmousi velocity model and plotted it again in Figure 7,

and a 1-D linear gradient initial model (Figure 8) for the tests.The source wavelet is the Ricker wavelet with a dominant fre-quency of 10Hz. There are 50 shots and 228 receivers equallyspaced on the surface.

We start envelope inversion (EI) directly from the lin-ear initial model. Figure 9 shows the results after 1 iteration,5 iterations and 10 iterations respectively. We can see thatthe large-scale structures have gradually shown up in the in-version results. After 10 iterations, we can already see thelarge-scale structures clearly. We plot the curve of the seismicdata residual (the residual between the observed seismic dataand the calculated seismic data) reduction with iterations ofenvelope inversion in Figure 10 as the solid line. The misfitvalue has been normalized with the norm of the observationdata. We see that the result has actually converged after about10 iterations, showing that envelope inversion converges fast.

To test the validity of the recovered large-scale structureby envelope inversion, we use the result in Figure 9(c) as thenew initial model and do a subsequent waveform inversion(WI). Figure 11 shows the final result of EI+WI. This result isobtained after 200 iterations. The convergence curve is plottedin Figure 10 as the dashed line. We implement the inversionon GPU, and the computation time of each iteration for EIand for WI are almost the same, which is about 2 minutes.So the computation time for envelope inversion is less than

2000 3000 4000 5000 6000

2

1

Amplitude

Dep

th (

km)

True modelEI+FWIFWI only

Amplitude

Dep

th (

km)

2000 3000 4000 5000 6000

2

1

True modelEI+FWIFWI only

(a) (b)

Figure 13 Comparison of inversion results. The comparison is along two vertical profiles from the velocity model (marked in Fig. 7). (a) TraceA. (b) Trace B. The solid lines are from the true model; dashed lines are from EI+WI; and dotted lines are from FWI alone.



2000

2500

3000

3500

4000

4500

5000


(c)

Dep

th (

km)

2 4 6 8

1

2

2000

2500

3000

3500

4000

4500

5000

Distance (km)

Dep

th (

km)

2 4 6 8

1

2

2000

2500

3000

3500

4000

4500

5000

Distance (km)

Dep

th (

km)

2 4 6 8

1

2

Figure 14 Inversion tests with low-cut (cut from 5 Hz below) source wavelet: (a) Smooth background obtained from EI; (b) waveform inversionusing (a) as the starting model; and (c) conventional full-waveform inversion starting from the linear initial model.

(a) (b) (c) (d)

Figure 15 A shot profile data set with Gaussian noise of (a) 0 dBW (without noise); (b) 10 dBW; (c) 20 dBW; and (d) 30 dBW.

5 percent of the whole EI+WI procedure. There is very littleextra cost to include EI into FWI.

To compare the inversion results between EI+WI andthe conventional FWI, we give the result of the conventionalFWI in Figure 12, which starts directly from the linear initial

model. We see that FWI result converged to a local minimum,while the EI+WI result is more faithful to the true model. Wepicked two traces from the model (the red lines in Figure 7)to see the inversion accuracy (Figure 13). The solid lines arethe true model, the dashed lines are the EI+WI results, and



00.

51

1.5

22.

53

−10

00

100

200

Amplitude

0dB

W G

auss

ian

nois

e

00.

51

1.5

22.

53

−10

00

100

200

Amplitude

10dB

W G

auss

ian

nois

e

00.

51

1.5

22.

53

−10

00

100

200

Amplitude

20dB

W G

auss

ian

nois

e

00.

51

1.5

22.

53

−10

00

100

200

Tim

e (s

)

Amplitude

30dB

W G

auss

ian

nois

e

Figu

re16

Ase

lect

edtr

ace

from

the

shot

prof

iles

inFi

g.15

.



00.

51

1.5

22.

53

−10

00

100

200

Amplitude

0dB

W G

auss

ian

nois

e

00.

51

1.5

22.

53

−10

00

100

200

Amplitude

10dB

W G

auss

ian

nois

e

00.

51

1.5

22.

53

−10

00

100

200

Amplitude

20dB

W G

auss

ian

nois

e

00.

51

1.5

22.

53

−10

00

100

200

Tim

e (s

)

Amplitude

30dB

W G

auss

ian

nois

e

Figu

re17

Cor

resp

ondi

ngen

velo

pes

ofth

etr

aces

inFi

g.16

.



Dis

tanc

e (k

m)

(a)

(b)

(c)

(d)

Depth (km)2

46

8

1 2

2000

2500

3000

3500

4000

4500

5000

Dis

tanc

e (k

m)

Depth (km)

24

68

1 2

2000

2500

3000

3500

4000

4500

5000

Dis

tanc

e (k

m)

Depth (km)

24

68

1 2

2000

2500

3000

3500

4000

4500

5000

Dis

tanc

e (k

m)

Depth (km)

24

68

1 2

2000

2500

3000

3500

4000

4500

5000

Figu

re18

EI

resu

lts

(aft

er10

iter

atio

ns)

usin

gda

taw

ith

Gau

ssia

nno

ise

of(a

)0

dBW

(wit

hout

nois

e);(

b)10

dBW

;(c)

20dB

W;a

nd(d

)30

dBW

.



Figure 19 Amplitude spectra of the traces in Fig. 16.

Figure 20 Amplitude spectra of the envelopes in Fig. 17.

the dotted lines are the conventional FWI results. We cansee that the results from EI+WI are close to the true model,and are much more accurate than that of the conventionalFWI.

To further demonstrate the ability of envelope inver-sion to recover the long-wavelength structure without lowfrequency in the source, we now show the results whenthe low frequency (below 5Hz) is removed from the sourcewavelet (this result has been shown in Wu et al. (2013b),here we show it again just for the readers’ convenience). FromFigure 2 we have seen that when the low frequency is removed

from the source wavelet, the envelope is still very rich in lowfrequencies. Figure 14(a) shows the envelope inversion result(starting from the linear initial model) after 10 iterations inthis situation. We can still see the large-scale structures fromthis result. Figure 14(b) is the final waveform inversion resultusing (a) as the new initial model. We can see that this result isvery similar to Figure 11. For comparison, Figure 14(c) showsthe conventional full waveform inversion result starting di-rectly from the linear initial model. We see that because of thelack of low frequencies in the data, this result is even worsethan Figure 12.




Dep

th (

km)

2 4 6 8

1

2

2000

2500

3000

3500

4000

4500

5000

Distance (km)

Dep

th (

km)

2 4 6 8

1

2

2000

2500

3000

3500

4000

4500

5000

Figure 21 Inversion results with Gaussian noise of 10 dBW. (a) EI+WI result. (b) FWI result.


Dep

th (

km)

2 4 6 8

1

2

2000

2500

3000

3500

4000

4500

5000

Distance (km)

Dep

th (

km)

2 4 6 8

1

2

2000

2500

3000

3500

4000

4500

5000

Figure 22 Inversion results with Gaussian noise of 30 dBW. (a) EI+WI result. (b) FWI result.

(a) (b) (c) (d)

Figure 23 A shot profile data set with SI noise. (a) is the original data set without noise; the SNR in (b), (c), and (d) are 2, 1, 0 respectively.

NOISE RES IST A N T PR OPER T Y OFENVELOPE INVERSION

Sensitivity to Gaussian noise

We first test the sensitivity of envelope inversion to Gaus-sian noise. Figure 15 shows one of the shot profiles (wave-form data) from the Marmousi model. (a) is the original

profile without noise. We gradually added Gaussian noise(with a white spectrum) with the power of 10dBW, 20dBWand 30dBW (the SNR (signal to noise ratio) are 1.43, 0.43and −0.57; SNR=log10 (signal power/noise power)), and theresulting profiles are shown in (b), (c) and (d) respectively.In order to show the influence of the noise to the data moreclearly, in Figure 16, we plot a trace from the shot profile.



Distance (km)

Dep

th (

km)

2 4 6 8

1

2

2000

2500

3000

3500

4000

4500

5000

Distance (km)

Dep

th (

km)

2 4 6 8

1

2

2000

2500

3000

3500

4000

4500

5000

Distance (km)

Dep

th (

km)

2 4 6 8

1

2

2000

2500

3000

3500

4000

4500

5000

(c) (d)

(a)

Figure 24 EI results (after 10 iterations) using data with SI noise. (a) The SNR is 2. (b) The SNR is 1. (c) The SNR is 0.

Figure 17 shows the corresponding envelopes. We see thatwith the increase of the noise power, the data is buried inthe noise gradually. Figure 18 shows the envelope inversionresults. Again, (a) is the result when no noise is added, (b), (c)and (d) are the results using the data contaminated by Gaus-sian noise with noise power of 10dBW, 20dBW and 30dBWrespectively. Comparing with the result without noise, theenvelope inversion results with different noise powers lookalmost the same as the one without Gaussian noise, showingthat envelope inversion is insensitive to white Gaussian noise.

Since EI+WI is a two-scale inversion method, and EI canrecover the large-scale structures using only the low frequencyinformation contained in the envelope without any interactionwith the high frequency seismic data, therefore it is muchless sensitive to high-frequency rich white Gaussian noise.Figure 19 shows the amplitude spectra of the traces inFigure 16, and Figure 20 shows the amplitude spectra of thetrace envelopes in Figure 17 (the amplitudes have been nor-malized for better look). We see that with the increase of thenoise power, the high frequency component of the seismic datais contaminated by the noise gradually, however, the envelopeis still dominant with low frequency information. We can useenvelope inversion to provide a good initial model even in the

noisy environment. Even though full waveform inversion iswhite noise sensitive and may be strongly affected by the exist-ing of noises, however, based on the above analysis we expectthat using EI+WI method will work better and more robustlyfor noisy data. We tried the cases when the powers of Gaussiannoise are 10dBW and 30dBW respectively. Figure 21 showsthe inversion results with Gaussian noise of 10dBW. (a) isEI+WI result using Figure 18(b) as the initial model; (b) isFWI result starting from the linear initial model. In this case,the noise is not very strong, and these two results look justlike the results without noise as shown in Figure 11 and 12.On the other hand, Figure 22 shows the inversion results withstrong Gaussian noise (30dBW). (a) is EI+WI result usingFigure 18(d) as the initial model; (b) is FWI result startingfrom the linear initial model. We see that in the case of strongnoise, although these two results are both affected, the EI+WIresult has less distortions. In Figure 22(a) we can still see themain structures of the model, especially the overthrust struc-ture. However, in Figure 22(b), the main structures are almostall buried into the noise. Based on this test we can say thatEI+WI can somewhat reduce the influence of Gaussian noiseto the inversion and therefore becomes a Gaussian noise re-sistant inversion method.



Distance (km)

Dep

th (

km)

2 4 6 8

1

2

2000

2500

3000

3500

4000

4500

5000

(a)

(b) (c)Distance (km)

Dep

th (

km)

2 4 6 8

1

2

2000

2500

3000

3500

4000

4500

5000

Distance (km)

Dep

th (

km)

2 4 6 8

1

2

2000

2500

3000

3500

4000

4500

5000

Figure 25 EI+FWI results using data with SI noise. (a), (b), and (c) are the results using Fig. 24 (a), (b), and (c) as initial models, respectively.

Sensitivity to seismic interference noise

We also tested the sensitivity of envelope inversion to seismicinterference (SI) noise. SI usually comes from others sourcesoperating in the same area. As we mentioned previously, thereare 50 shots equally spaced on the surface. To simulate SI, weput another source behind the 50th shot, on the right side ofthe surface. Figure 23 shows a shot profile. (a) is the originaldata, (b), (c), and (d) are the data with SI noise and the SNR are2, 1, and 0 respectively. We see that the noise becomes strongerfrom (a) to (c), and the effective signal is affected by SI.

Figure 24 shows the envelope inversion results with SI inthe data. (a) is the result when the SNR is 2, (b) is the resultwhen the SNR is 1, and (c) is the result when the SNR is 0.We see that when the SNR is 2, the noise is not very strong,the inversion result looks just the same as the one withoutany noise (Figure 18(a)). The result is almost not affected bythe noise at all. We can see the final EI+FWI result shownin Figure 25(a) is as good as the one in Figure 11. Whenthe SNR is 1, the noise is a little stronger, and the envelopeinversion result (Figure 24(b)) is affected a little in this case.However we can still see the large-scale component in theresult. Figure 25(b) shows the final EI+FWI result in thiscase, we see that the final result is still as good as (a) with thelarge-scale component in the envelope inversion result. When

the SNR is 0, the noise is very strong (same level as the data)in this case. We see that the envelope inversion is affected verymuch (Figure 24(c)), and we can only see some structures inthe very shallow part. Figure 25(c) shows the final EI+FWIresults. We can see that in this case the result has convergedto a local minimum.

From the above test we can see that when the SI noise istoo strong, the envelope inversion will lose its effectiveness,however when the SI noise is not very strong, the envelopeinversion can be noise-resistant. So we can conclude that theenvelope inversion has potential to apply to noisy data.

CONCLUSION

Numerical examples have shown that envelope inversion ismore reliable and effective to retrieve the large-scale compo-nent in the model than the conventional full waveform in-version and can avoid or reduce the local minimum problem.The combined inversion EI+WI can deliver more faithful andaccurate final result with very little extra computation cost.Also, numerical examples showed that envelope inversion isinsensitive to white Gaussian noise and to some extent, toseismic interference noise, indicating the robustness of thismethod and its potential use for noisy data.



ACKNOWLEDG E ME N T S

We thank Xiao-bi Xie, Rui Yan, Jinghuai Gao, Jicheng Liu,Bo Chen and Benfeng Wang for helpful discussions. The workis supported by WTOPI (Wavelet Transform On Propagationand Imaging for seismic exploration) Project at University ofCalifornia, Santa Cruz.

REFERENCES

Almomin A. and Biondi B. 2012. Tomographic Full Waveform In-version: Practical and Computationally Feasible Approach. 82ndAnnual International Meeting, SEG, Expanded Abstracts, 1–5.

Biondi B. and Almomin A. 2012. Tomographic full waveform in-version (TFWI) by combining full waveform inversion with wave-equation migration velocity analysis. 82nd Annual InternationalMeeting, SEG, Expanded Abstracts, 1–5.

Biondi B. and Sava P. 1999. Wave-equation migration velocityanalysis. 69th Annual International Meeting, SEG, ExpandedAbstracts, 1723–1726.

Boschetti F., Dentith M.C. and List D. 1996. Inversion of seismicrefraction data using genetic algorithms. Geophysics 61, 1715–1727.

Brenders A.J. and Pratt R.G. 2007. Full waveform tomography forlithospheric imaging: Results from a blind test in a realistic crustalmodel. Geophysical Journal International, 168, 133–151.

Bunks C., Saleck F.M., Zaleski S. and Chavent G. 1995. Multiscaleseismic waveform Inversion. Geophysics 60, 1457–1473.

Crase E., Pica A., Noble M., McDonald J. and Tarantola A. 1990.Robust elastic nonlinear waveform inversion: Application to realdata. Geophysics 55, 527–538.

Dines K. and Lytle R. 1979. Computerized geophysical tomography.Proc. IEEE 67, 1065–1072.

Justice J., Vassiliou A., Singh S., Logel J., Hansen P., Hall B., HuttP. and Solankil J. 1989. Acoustic tomography for enhancing oilrecovery. The Leading Edge 8, 12–19.

Lailly P. 1983. The seismic inverse problem as a sequence of be-fore stack migration. Proceeding of SIAM Phiadelphia Conferenceon Inverse Scattering: Theory and Applications (eds E. Robinsonand A. Weglein), pp. 206–220. Society for Industrial and AppliedMathematics.

Liu J., Chauris H. and Calandra H. 2011. The normalized integrationmethod – an alternative to full waveform inversion? Near Sruface2011–17th European Meeting of Environmental and EngineeringGeophysics, B07.

Liu Z. and Bleistein N. 1995. Migration velocity analysis: Theory andan iterative algorithm. Geophysics. 60, 142–153.

Luo Y., and Schuster G.T. 1991. Wave-equation traveltime inversion.Geophysics 56, 645–653.

Mallick S. 1999. Some practical aspects of prestack waveform inver-sion using a genetic algorithm: An example from the east TexasWoodbine gas sand. Geophysics 64, 326–336.

Mao J., Wu R. and Wang B. 2012. Multiscale full waveform inversionusing GPU. 82nd Annual International Meeting, SEG, ExpandedAbstracts. 1–7.

Mora P. 1987. Nonlinear two-dimensional elastic inversion of multi-offset seismic data. Geophysics 52, 1211–1228.

Mora P. 1988. Elastic wave-field inversion of reflection and transmis-sion data. Geophysics 53, 750–759.

Padhi A., Mukhopadhyay P., Blacic T., Fortin W., Holbrook W.S.and Mallick S. 2010. Prestack waveform inversion for the water-column velocity structure- the present state and the road ahead.80th Annual International Meeting, SEG, Expanded Abstracts,2845–2849.

Paulssonet B., Cook N. and McEvilly T. 1985. Elastic wave velocitiesand attenuation in an underground repository of nuclear waste.Geophysics 50, 551–570.

Pratt R.G. 1999. Seismic waveform inversion in the frequency do-main, Part 1: Theory and verification in a physical scale model.Geophysics 64, 888–901.

Pratt R.G. and Shipp R.M. 1999. Seismic waveform inversion inthe frequency domain, Part 2: Fault delineation in sediments us-ing crosshole data. Geophysics 64, 902–914.

Sava P. and Biondi B. 2004. Wave-equation migration velocity anal-ysis. I. Theory. Geophysical Prospecting. 52, 593–606.

Shen P. and Symes W. 2008. Automatic velocity analysis via shotprofile migration. Geophysics 73, VE49–VE59.

Shin C. and Cha Y.H. 2008. Waveform inversion in the Laplacedomain. Geophys. J. Int. 173, 922–931.

Shin C. and Cha Y.H. 2009. Waveform inversion in the Laplace-Fourier domain. Geophys. J. Int. 177, 1067–1079.

Tarantola A. 1984. Inversion of seismic reflection data in the acousticapproximation. Geophysics 49, 1259–1266.

Virieux J. and Operto S. 2009. An overview of full-waveforminversion in exploration geophysics. Geophysics 74, WCC1–WCC26.

Wang H., Singh S.C., Jian H. and Calandra H. 2012. Integrated in-version of subsurface velocity structures using wave equation to-mography and full waveform inversion. 82nd Annual InternationalMeeting, SEG, Expanded Abstracts, 1–5.

Warner M., Ratcliffe A., Nangoo T., Morgan J., Umpleby A., ShahN., Vinje V., A tekl I., Guasch L., Win C., Conroy G. and BertrandA. 2013. Anisotropic 3D full-waveform inversion. Geophysics 78,R59–R80.

Woodward M.J. 1992. Wave-equation tomography. Geophysics 57,15–26.

Wu R.S., Luo J. and Wu B. 2013. Ultra-low-frequency informationin seismic data and envelope inversion. 83rd Annual InternationalMeeting, SEG, Expanded Abstracts, 3078–3082.

Wu R.S., Luo J. and Wu B. 2014. Seismic envelope inversion andmodulation signal model. Geophysics 79, WA13–WA24.

Xie X. and Yang H. 2008. The finite-frequency sensitivity kernelfor migration residual moveout and its applications in migrationvelocity analysis. Geophysics 73, S241–S249.

Zhou C., Cai W., Luo Y., Schuster G.T. and HassanzadehS. 1995. Acoustic wave-equation traveltime and waveforminversion of crosshole seismic data. Geophysics 60, 765–773.

Zhou C., Schuster G.T. and Hassanzadeh S. 1997. Elastic waveequation traveltime and waveform inversion of crosswell data.Geophysics 62, 853–868.


seismic envelope inversion: reduction of local minima and ...wrs/publication/journal/... · seismic...

Documents