2D Laplace-Domain Waveform Inversion of Field Data Using a Power Objective Function

Abstract

The wavefield in the Laplace domain has a very small amplitude except only near the source point. In order to deal with this characteristic, the logarithmic objective function has been used in many Laplace domain inversion studies. The Laplace-domain waveform inversion using the logarithmic objective function has fewer local minima than the time- or frequency domain inversion. Recently, the power objective function was suggested as an alternative to the logarithmic objective function in the Laplace domain. Since amplitudes of wavefields are very small generally, a power <1 amplifies the wavefields especially at large offset. Therefore, the power objective function can enhance the Laplace-domain inversion results. In previous studies about synthetic datasets, it is confirmed that the inversion using a power objective function shows a similar result when compared with the inversion using a logarithmic objective function. In this paper, we apply an inversion algorithm using a power objective function to field datasets. We perform the waveform inversion using the power objective function and compare the result obtained by the logarithmic objective function. The Gulf of Mexico dataset is used for the comparison. When we use a power objective function in the inversion algorithm, it is important to choose the appropriate exponent. By testing the various exponents, we can select the range of the exponent from 5 × 10^sup -3^ to 5 × 10^sup -8^ in the Gulf of Mexico dataset. The results obtained from the power objective function with appropriate exponent are very similar to the results of the logarithmic objective function. Even though we do not get better results than the conventional method, we can confirm the possibility of applying the power objective function for field data. In addition, the power objective function shows good results in spite of little difference in the amplitude of the wavefield. Based on these results, we can expect that the power objective function will produce good results from the data with a small amplitude difference. Also, it can partially be utilized at the sections where the amplitude difference is very small.[PUBLICATION ABSTRACT]

Pure Appl. Geophys. 170 (2013), 20752085 2013 Springer Basel

DOI 10.1007/s00024-013-0651-4 Pure and Applied Geophysics

2D Laplace-Domain Waveform Inversion of Field Data Using a Power Objective Function

EUNJIN PARK,1 WANSOO HA,1 WOOKEEN CHUNG,2 CHANGSOO SHIN,1 and DONG-JOO MIN1

AbstractThe waveeld in the Laplace domain has a very small amplitude except only near the source point. In order to deal with this characteristic, the logarithmic objective function has been used in many Laplace domain inversion studies. The Laplace-domain waveform inversion using the logarithmic objective function has fewer local minima than the time- or frequency domain inversion. Recently, the power objective function was suggested as an alternative to the logarithmic objective function in the Laplace domain. Since amplitudes of waveelds are very small generally, a power \1 amplies the waveelds especially at large offset.

Therefore, the power objective function can enhance the Laplace-domain inversion results. In previous studies about synthetic datasets, it is conrmed that the inversion using a power objective function shows a similar result when compared with the inversion using a logarithmic objective function. In this paper, we apply an inversion algorithm using a power objective function to eld datasets. We perform the waveform inversion using the power objective function and compare the result obtained by the logarithmic objective function. The Gulf of Mexico dataset is used for the comparison. When we use a power objective function in the inversion algorithm, it is important to choose the appropriate exponent. By testing the various exponents, we can select the range of the exponent from 5 9 10-3 to 5 9 10-8 in the Gulf of Mexico dataset. The results obtained from the power objective function with appropriate exponent are very similar to the results of the logarithmic objective function. Even though we do not get better results than the conventional method, we can conrm the possibility of applying the power objective function for eld data. In addition, the power objective function shows good results in spite of little difference in the amplitude of the waveeld. Based on these results, we can expect that the power objective function will produce good results from the data with a small amplitude difference. Also, it can partially be utilized at the sections where the amplitude difference is very small.

Key words: Power objective function, eld data, inverse theory.

1. Introduction

Since LAILLY (1983) and TARANTOLA (1984) sug

gested the back-propagation technique for waveform inversion, many studies on waveform inversion have been performed in the time domain (MORA 1987;

BUNKS et al. 1995; SHIPP and SINGH 2002) and in the

frequency domain (GELLER and HARA 1993; PRATT

et al. 1998; OPERTO et al. 2004; SHIN and MIN 2006;

SHIN et al. 2007).

However, both domains of waveform inversion have problems such as the existence of local minima and the absence of low-frequency components. In particular, it is difcult to generate or measure low-frequency components in seismic exploration. Therefore, conventional waveform inversion fails in recovering long-wavelength velocity models (SYMES

2008).SHIN and CHA (2008) proposed an acoustic

waveform inversion technique in the Laplace domain. The objective function in the Laplace domain has fewer local minima than in the frequency domain (SHIN and HA 2008), and it can construct the long-

wavelength velocity model reecting the characteristics of the subsurface medium. In particular, when we apply the method to the eld data that has small low-frequency components or high noise levels, it shows better results than other waveform inversions (SHIN and CHA 2008). Based on these advantages, the

Laplace-domain waveform inversion has been expanded to 3D acoustic mediums (PYUN et al. 2011),

elastic mediums (CHUNG et al., 2010) and acoustic-

elastic coupled mediums (BAE et al. 2010). To

increase the resolution of inversion results, the LaplaceFourier domain waveform inversion has also been studied (SHIN and CHA 2009).

For a waveform inversion based on the gradient method, the difference between the observed and

1 Department of Energy Systems Engineering, Seoul National University, Gwanak-ro 1, Gwanak-gu, 151-744 Seoul, South Korea. E-mail: [email protected]

2 Department of Energy and Resources Engineering, Korea Maritime University, 727 Taejong-ro, Yeongdo-gu, 606-791 Busan, South Korea.

2076 E. Park et al. Pure Appl. Geophys.

modeled data is dened by using an objective function. We obtain the velocity model by iteratively minimizing the objective function. Therefore, the choice of the objective function has a great deal of inuence on the inversion results (SHIN and HA 2008).

Waveform inversions in the time or frequency domain generally use an l2-norm objective function (TARANTOLA 1984; PRATT et al. 1998; SHIPP and SINGH

2002). However, in the Laplace domain, it is unsuitable to use the l2-norm as an objective function because the Laplace-transformed waveeld has very large amplitude only near the source. For this reason, most of the Laplace-domain waveform inversion algorithms use the logarithmic objective function suggested by SHIN and MIN (2006).

SHIN and HA (2008) performed the Laplace-

domain waveform inversion using various objective functions. The power objective function is suitable for nding the global minimum, when the exponent has a small value. SHIN and HA (2008) applied the

power objective function to a synthetic dataset in the Laplace domain, and they obtained very similar results to the logarithmic objective function.

In this paper, we focus on the power objective function and conduct the in-depth study for the Laplace domain waveform inversion. We develop the Laplace-domain waveform inversion algorithm using the power objective function, and the algorithm is applied to the eld dataset in acoustic media. From this research result, we can discuss the applicability of the power objective function in the Laplace domain waveform inversion.

2. Theory

2.1. Wave equation in the Laplace domain

The 2D acoustic wave equation in the time domain for a homogeneous and isotropic medium can be expressed as

r2u x; t

1 c2

The Laplace transform of Eq. (1) is given by

r2 x; s

s2c2 x; s

~f x; s

; 2

with

x; s

Z

u x; t

1 e stdt; 3

and

1 e stdt; 4

where s is a Laplace damping constant. In Eq. (3), u x; t

e st is the damped waveeld at a given s. This

equation can be written as

x; s

lim

x!0

Z1

0

~f x; s

Z

f x; t

u x; t

e s tdt

lim

x!0

Z1

1

u x; t

e ste ixtdt; 5

where x is the angular frequency. We can know that the Laplace-transformed waveeld is the zero-frequency component of the damped waveeld. Because of this characteristic, a long-wavelength velocity model is recovered and smooth inversion results are provided.

By using the nite element method (FEM), Eq. (2) can be written as

S

~f; 6

with

S K s2M; 7

where M is a mass matrix, K is a stiffness matrix, u is the waveeld, and f is the source vector.

2.2. Objective function and gradient direction

In this paper, we use the power objective function suggested by SHIN and HA (2008). The objective

function for a damping constant is expressed as

E m

1

2

o2u x; t

ot2 f x; t

; 1 where c is the velocity, u x; t

is the waveeld in the

time domain, and f x; t

is the source function in the

time domain.

XNsr j1

XNrck1pjk ~dpjk

2 ; 8

Vol. 170, (2013) Power Objective Function in the Laplace Domain 2077

The gradient direction of the logarithmic objective function is same as Eq. (13), but r is given by

where m is the model parameter vector representing characteristics of a medium, is the modeled data, ~d is the eld data, Nsr and Nrc are the number of sources and receivers. Because the characteristics of the waveeld depend on the exponent, it is important to select an appropriate exponent (SHIN and HA 2008).

After determining the objective function, we focus on minimizing it. We rst have to calculate the steepest-descent direction of the objective function. The derivative of Eq. (8) with respect to the l-th model parameter ml can be expressed as

oEoml X

Nsr

j1

1ij2 lnij2~d

ij2

...

1ijNr lnijNr~d

ijNr

0 0

... 0

r

2

666666666666664

3

777777777777775

1ij1 lnij1~d

ij1

: 15

In Eq. (13), S 1r is the back-propagated waveeld vector. We can obtain the steepest-descent direction by calculating the zero-lag convolution between the virtual source vector and the back-propagated wave-eld. The back-propagation algorithm helps the gradient direction to be calculated efciently because the partial derivative waveeld doesnt have to be calculated directly (TARANTOLA 1984; PRATT et al.

1998; PRATT 1999).

2.3. Normalization and inversion ow

In order to minimize the objective function, we iteratively update the model parameter ml. The model parameter is updated by the preconditioned gradient method (PRATT et al. 1998). It is expressed as

mn1l mnl dm 16

dm H 1rmE; 17

where H is a Hessian matrix. Although optimizing techniques have been developed for efciently accounting for the Hessian matrix in recent studies (ABUBAKAR et al. 2011; MTIVIER et al. 2012), the

Hessian matrix commonly requires large amount of computations. Therefore, we use the diagonal elements of the pseudo-Hessian matrix suggested by SHIN et al. (2001) instead of the Hessian.

Equation (16) can be rewritten as

mn1l mnl dm mnl angn; 18

where an is the step length at n-th iteration and gn is the steepest-descent direction normalized by the

" #: 9

Likewise, differentiating Eq. (6) with respect to a model parameter ml, we obtain

oSoml S

XNrck1ojkoml p p 1jkpjk ~dpjk

ooml 0; l 1; 2; . . .; Np; 10

where Np is the number of parameters. Equation (10) is simply expressed as

ooml S 1vl; l 1; 2; . . .; Np 11

with

vl

oSoml ; 12

where vl is the virtual source vector with respect to the l-th model parameter ml (SHIN and MIN 2006).

By substituting Eq. (11) by Eq. (9), we obtain

oEoml X

Nsr

j1

TS 1r

vl

13

and

~dpj1

p p 1j2pj2

~dpj2

...

p p 1jNrpjNr

p p 1j1pj1

~dpjNr

0 0

... 0

r

2

666666666666664

3

777777777777775

: 14

2078 E. Park et al. Pure Appl. Geophys.

Figure 1a The rst common shot gather and b its frequency spectrum obtained from the Gulf of Mexico dataset

Figure 2a The waveeld, b the powered waveeld (p = 0.005), and c the logarithmic waveeld of eld data in the Laplace-domain

where Nsr is the number of the damping constant, n is the iteration number, NRM is the normalizing operation and k is a stabilizing factor in the inversion. The

NRM (values) is to divide each value by the largest absolute value. The stabilizing factor k is used to avoid singular values of the pseudo-Hessian (MAR

QUARDT 1963; LEVENBERG 1994).

In order to obtain gn, we rst divide the steepest-descent direction by the pseudo-Hessian. Then we can normalize it using the maximum absolute value. Second, we repeatedly perform previous steps for each Laplace damping constant. Finally, we sum all values and normalize once more. In this manner, we can update the velocity by using the gradient direction. And by repeating the algorithm, we can identify subsurface structures.

pseudo-Hessian. The preconditioned gradient direction gn can be expressed as (HA et al. 2009)

gn NRM X

Ns

i1

( )

" #

" #;

NRM

PNsrj1 vl TS 1r

PNsrj1 vl T^vl

k

19

Vol. 170, (2013) Power Objective Function in the Laplace Domain 2079

3. Numerical results

3.1. Preprocessing and waveeld analysis

The eld datasets generally have a lot of noise, so the preprocessing processes are needed. We use a preprocessing technique for ltering and muting. We cut the low-frequency components using a low-cut lter and mute the upper part of the direct wave, because the signal is almost unclear but the noise is evident in these parts.

Figure 1 show the rst shot gather of the Gulf of Mexico dataset and its frequency spectrum after preprocessing. The frequency spectrum is expressed except for a 0.5 % value from the maximum and a0.5 % value from the minimum. We can note that the frequency components lower than 4 Hz are missing in

this data. When the data doesnt have low-frequency components, it is difcult to obtain correct subsurface structure. One way to improve this problem is applying the Laplace-domain waveform inversion, which generates the long-wavelength velocity model.

The Laplace-transformed waveeld obtained from the shot gather (Fig. 1a) is shown in Fig. 2. The Laplace damping constant is 7. As shown in Fig. 2a, the Laplace-transformed waveeld has values close to zero except for a very large amplitude near source. Figure 2b, c shows the waveeld to the power of0.005 and the logarithmic waveeld in Laplace-domain. Likewise, by taking the logarithm or the exponent to \1 at the waveeld, we can amplify the waveeld close to zero. Through this process, the difference of the modeled data and observed data is

Figure 3Laplace-transformed waveeld a with various damping constants (p = 0.05) and b with various exponent values (s = 7)

2080 E. Park et al. Pure Appl. Geophys.

Figure 4The comparison of the observed data and the inverted data of a rst shot gather a in the powered waveeld (p = 0.005) and b in the logarithmic waveeld

Figure 5The starting velocity model. (1st layer: 1.5 km/s, 2nd layer:3.3 km/s)

remarkably more clear. Then, each objective function is less likely to converge to local minima.

Figure 3 expresses the waveeld depending on various parameters in the Laplace domain. We rst

x the exponent at 0.05 and vary the damping constants (Fig. 3a). We can know that the lager damping constant includes a lot of relatively low wave-number components. Next, we x the damping constant at seven and use several exponents. When the exponent is smaller, the difference of the amplitude is smaller as shown in Fig. 3b.

3.2. Application on the eld dataset

We tested our waveform inversion algorithm on a eld dataset acquired from the Gulf of Mexico. There were 399 shots with a shot interval of 50 m and 408 receivers with a receiver interval of 25 m. The maximum recording length is 12 s and the time

Vol. 170, (2013) Power Objective Function in the Laplace Domain 2081

sampling interval is 4 ms. The offset ranges from 137 to 10,321 m and the water depth ranges from 420 to 910 m.

In order to conrm the applicability of the algorithm, we compare the observed data and the inverted data of a rst shot gather in the Laplace domain. Figure 4 is the result of each comparison in the powered waveeld and the logarithmic waveeld. The Laplace damping constant is ve and the exponent of the powered waveeld is 0.005. Because the inverted data has a similar result with the observed data, we verify that it is reasonable to apply this algorithm.

The starting model is a two-layer velocity model (Fig. 5). The rst layer, the seawater layer, is xed at1.5 km/s and the second layer has a homogeneous velocity of 3.3 km/s. The grid interval for inversion is 25 m. Nine Laplace damping constants are used, ranging from 1 to 13 with an interval of 1.5.

Our algorithm performed 100 iterations. The source wavelet and velocity are updated in each iteration. The source wavelet is estimated along the method suggested by SHIN et al. (2007). We inverted

the velocity model by changing the exponents in the power objective function. SHIN and HA (2008) found

from a test of a salt model that the power objective function was very smooth and desirable for local optimization methods when the exponent was equal to or smaller than 0.1. From this result, we choose the rst exponent of 0.05 and next values are reduced to a tenth of the time of the preceding value. We tested the exponent from 5 9 10-2 to 5 9 10-10. The result was good and almost the same with each other when the exponent was from 5 9 10-3 to 5 9 10-8 and the rest of the exponents were not good. Therefore, we selected three exponents for analysis of the power objective function in this test.

The velocity model using the logarithmic wave-eld is shown in Fig. 6d. When the exponent p is 5 9 10-2 (Fig. 6a), the result is poorer than in Fig. 6d. However, inverted velocity models are similar to Fig. 6d when the exponent p is 5 9 10-5

and 5 9 10-8. The reason why Fig. 6a shows a low-quality result is that it used too large an exponent value. As shown in Fig. 3b, too large an exponent cant amplify the waveeld enough in the power objective function. If the value of the waveeld is too

small, the objective function is more affected by variations in the background velocity than variations in the anomalous-body velocity, and then the objective function has more of a possibility to converge to the local minima. If the exponent is equal to one in the power objective function, it is same for the l2-norm objective function. As mentioned earlier, the l2-norm objective function is not recommended for the Laplace-domain waveform inversion because it is very vulnerable to the local minima problem.

In contrast, if the exponent is too small it is also difcult to obtain good inversion results, since most of the values of the waveeld converge to one. In this test, all values of the waveeld are calculated into one when the exponent is 5 9 10-10, so the algo

rithm failed to nd the velocity model of the

substructure.

We calculate the velocity error of the each power objective function based on the logarithmic objective

Figure 6The inverted velocity model using the power objective function with a p = 5 9 10-2, b p = 5 9 10-5 and c p = 5 9 10-8, and d using the logarithmic objective function after the 100th iteration

2082 E. Park et al. Pure Appl. Geophys.

Figure 7The velocity error of several power objective functions about the logarithmic objective function at 8 km (a and b) and 10 km (c and d) offset

function in order to know the similarity between two objective functions. The equation is as follows;

error %

vp vl

vl 100; 20

where vp is the velocity obtained by using the power objective function and vl is the velocity obtained by using the logarithmic objective function. We choose the velocity prole at 8 and 10 km from the left, extracted from Fig. 6ad. Figure 7a, c show that the power objective function with p = 5 9 10-2 has a relatively large difference about the logarithmic objective function. In contrast, the power objective function with p = 5 9 10-5 and with p = 5 9 10-8 have small errors, so we do not distinguish it with the naked eye. Therefore, we draw only these two power

objective functions in Fig. 7b, d. The error of the power objective function with p = 5 9 10-5 is \0.1 percent, so we can know that it has very similar results to the logarithmic objective function. The result obtained from the logarithmic objective function is not an exact solution, but its accuracy has been veried through many experiments about various synthetic data. Therefore, we can determine that the power objective function with suitable exponent gives good results in the Laplace-domain waveform inversion.

In the previous Fig. 3b, we know that the smaller the exponent is, the smaller the difference of the amplitude is. Despite the power objective function with p = 5 9 10-8 having almost no difference for the amplitude, it shows a good inversion result. From

Vol. 170, (2013) Power Objective Function in the Laplace Domain 2083

Figure 7 continued

Figure 8The migrated images generated from the starting velocity models in

Fig. 5

this result, we can see that the power objective function has strengths for the data with small amplitude differences.

The migrated images generated from the velocity models in Fig. 5 and Fig. 6 are shown in Fig. 8 and Fig. 9. We use the simple reverse-time migration (RTM) algorithm. Figure 9d is obtained from the logarithmic objective function and shows overall improved results compared to Fig. 8 generated from the starting velocity model. Figure 9b, c obtained from the power objective function show similar results to Fig. 9d because of very small error in the velocity model.

Finally, in order to verify the accuracy of the velocity models and the migrated images, we make the common image gathers (CIGs). While the

migrated images are obtained by overlapping images of all shot points, CIGs are obtained by horizontally arraying images of a specic shot point. Therefore, theoretically, CIGs obtained by exact velocity should

2084 E. Park et al. Pure Appl. Geophys.

Figure 9The migrated images generated from the corresponding velocity models in Fig. 6

Figure 10The common-image gathers (CIGs) obtained by a using the initial velocity model and using the inverted velocity model with b p = 5 9 10-5 and c p = 5 9 10-8

have at structures. Figure 10 shows the CIGs obtained from the 225th shot. Figure 10a shows the CIGs using the initial velocity model, whereas Fig. 10b, c show the CIGs using the inverted velocity model with p = 5 9 10-5 and with p = 5 9 10-8. Most reection events in the CIGs using the inverted velocity model are attened. Therefore, we can conrm that the proposed algorithm of the power objective function can be applied to the eld dataset.

4. Conclusions

We performed the Laplace-domain waveform inversion on the eld dataset. The eld datasets generally have a lot of noise, so we applied the preprocessing step and analyzed the Laplace waveeld

before inversion. The Laplace waveelds generally have very small absolute values; therefore, they are amplied by taking the logarithm or the exponent as \1. Through the waveeld analysis, we can know that the logarithmic waveeld and the powered waveeld have less risk of falling into local minima than the Laplace waveeld. Also, we conrm the accuracy of this algorithm through comparing the inverted waveeld and the real waveeld.

We tested our algorithm on eld data by using several waveeld exponents and comparing the results to those of the logarithmic objective function. In the case of the Gulf of Mexico dataset, exponents from 5 9 10-3 to 5 9 10-8 in the power objective function yielded very similar results to that of the logarithmic objective function. The inversion result has low quality when the exponent is larger than 5 9 10-3 or smaller than 5 9 10-8, because too large an exponent cant amplify the waveeld enough and too small an exponent makes all values of the waveeld converge to one.

Vol. 170, (2013) Power Objective Function in the Laplace Domain 2085

Here, we can see that subsurface structures are well recovered by using the power objective function algorithm with an appropriate exponent. Although the power objective function does not show signicantly better results than the logarithmic objective function, we can conrm the applicability of the Laplace-domain waveform inversion using the power objective function on eld datasets. Especially, the power objective function shows good results despite the waveeld having almost no difference in the amplitude. Therefore, we can draw a conclusion that the power objective function has strengths on the data with small amplitude difference. Based on these results, we can expect that the proposed algorithm using the power objective function will produce good results from the data with small amplitude differences. Also, it can partially be utilized at the sections where the amplitude difference is very small.

Acknowledgments

This work was supported nancially by the Energy Efciency & Resources (No. 2010T100200376) and Human Resources Development program (No. 20124010203200) of the Korea Institute of Energy Technology Evaluation and Planning (KETEP) grant funded by the Korea government Ministry of Knowledge Economy. We thank GX Technology for providing us with the eld data.

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(Received February 17, 2012, revised January 9, 2013, accepted February 2, 2013, Published online March 15, 2013)