Efficient Gaussian Packets representation and seismic imaging Yu Geng*1, Ru-Shan Wu and Jinghuai Gao2

Modeling and Imaging Laboratory, IGPP, University of California, Santa Cruz, CA 95064

1Visiting from institute of Wave and Information, Xi’an Jiaotong University, Xi’an 710049, China 2Institute of Wave and Information, Xi’an Jiaotong University, Xi’an 710049, China

Summary In this paper we discuss an efficient way to apply Gaussian Packet method to data representation and seismic imaging. Similar to Gaussian Beam method, wavefield radiating from a seismic source as a set of Gaussian Packets can represent synthetic seismic data, and recorded wavefield at the surface can be expressed and downward continued by a set of Gaussian Packets at surface as well. The evolution of Gaussian Packet is determined by parameter of central frequency, local time, local space and ray emergent angle, and the shape of Gaussian Packet is determined by its initial value. Each Gaussian Packet is directly related to the ray emergent angle, and propagates along a central ray. Therefore, representation of seismic data using Gaussian Packets provides the local time slope and location information at certain central frequency, while summation of Gaussian Packets’ evolutions constructs the corresponding propagated wavefield. These properties also make seismic imaging using Gaussian Packets easily be understood and implemented. The method becomes efficient because 1) to represent seismic data, Gaussian Packets with given initial value can be used and inner product can be applied to obtain the useful information of seismic data; 2) with proper initial value, only Gaussian Packets with few central frequencies are needed for representing propagated seismic wavefield. Numerical examples on impulse responses and a 4layer zero-offset data are calculated to demonstrate the valid of the method. Introduction Previous study has shown how to use Gaussian Beams summation in computation of high frequency seismic wavefields in smoothly varying inhomogeneous media and successfully used to depth migration (Hill 1990, Cerveny 1985, Babich and Popov 1989, Popov 1982, Cerveny, Popov and Psencik 1982, Nowack and Aki 1984, Cerveny, Klimes and Psencik 2007, Nowack 2003, Hill 2001). This Beam form solutions are usually formulated in the frequency domain and it can decompose the wavefield into beam fields that are localized both in position and direction. Recently, studies of represent seismic data and imaging directly in time-space domain have shown up. Time localization for seismic imaging has been discussed by Wu (2008, 2009) using a Dreamlet decomposition and wave-equation based one-way propagating method. Leading-

order seismic imaging using Curvelets has been discussed (Douma and de Hoop 2007); Anton et al (2010) also discuss a sparse Gaussian wave Packet representation and seismic imaging. As a high-frequency asymptotic space-time particle-like solutions of the wave equation, Gaussian Packets were introduced (Babich and Ulin 1984). Decomposition of the data using optimized Gaussian Packets (Zacek 2006a, Klimes 1989b, Zacek 2006b) and related migration in common-shot domain was discussed by Zacek (Zacek 2005, Bucha 2008, Bucha 2009, Zacek 2004). However, in their method the dependence on parameter causes the decomposition into optimized Gaussian Packets complicated. The Gaussian Packets’ shapes are varying from different parameters, and the corresponding initial values have to be calculated for each group of parameter in their method. This causes the decomposition become time consuming. In this study, we discuss the representation of data using Gaussian Packets with no dependence on the parameter. The property of time localization determines that the frequency response of a Gaussian Packet is also centered at the central frequency. With proper initial value, only the Gaussian Packets with corresponding central frequency are needed to approximately represent the seismic data. This property helps to improve the efficiency. Impulse responses and migration result are shown to demonstrate the validity of this method. Data representation using Gaussian Packet In two dimensions case, a Gaussian Packet (Klimes 1989a, Klimes 2004) can be written as ( , , ) exp( )gpu x z t i A (1) is arbitrary positive parameter, and it can be understood as the center frequency of the wave packet. Both the phase-function and amplitude are complex-valued function of space coordinates ( , )x z and time t . Using Tayler expansion, the phase function can be further expanded to its second order as follow around point ( , , )r r rx z t on the central ray

2 2 2

( ) ( ) ( )

1 1 1 + ( ) ( ) ( )

2 2 2 + ( )( ) ( )( )

( )( )

x r z r t r

xx r zz r tt r

xz r r xt r r

zt r r

p x x p z z p t t

N x x N z z N t t

N x x z z N x x t t

N z z t t

(2)

© 2011 SEGSEG San Antonio 2011 Annual Meeting 33983398

Efficient Gaussian Packets representation and seismic imaging

where xp , zp are the components of slowness vector along

the raypath, 1tp , ij ijN N is the second space-time

derivatives of the phase function, and Im[ ( , , )] 0x z t . From equation (2), we can see that Gaussian Packets are particle-like solution of the wave equation, and they have both time and space localization. Suppose a Gaussian Packet travels along a central ray from point ( , , )r r rx z t to 1 1 1( , , )r r rx z t , Figure 1 shows the profiles

of the Gaussian Packet centered at point ( , , )r r rx z t on ( , )x t

plane with rz z and the Gaussian Packet centered at

1 1 1( , , )r r rx z t on ( , )x z plane with 1rt t . The profiles of

Gaussian Packets on ( , )x t plane are used to represent the seismic source or recorded data, while with the evolution of each Gaussian Packet, the profiles of Gaussian Packets on ( , )x z plane are used to synthetic the propagated wavefield. Unlike Curvelet or other wave-packet method, the representation using Gaussian Packet is directly related to the ray emergent angle.

Figure 1: Profiles of Gaussian Packets: dashed gray circle stands for the Gaussian Packet on z=zr plane, and solid gray circle stands for the Gaussian Packet on t=tr1 plane. In this paper, we will suppose that all the travel time along the central ray 0 occurs at 0z and the ray is shooting downward. Thus, we can see from equation(2), when 0z the Gaussian Packet can be written as

2 2

( , ) exp{ [ ( ) ( )

1 1 + ( ) ( ) + ( )( )]}

2 2

gps x r r

xx r tt r xt r r

u x t i p x x t t

N x x N t t N x x t t

A(3)

The profiles of Gaussian Packets at ( , )x t plane are

controlled by parameter rx , rt , and xp , together with

initial value of 0w , 0ttN , which controls the initial beam

width on the plane perpendicular to the central ray and the

pulse duration along the central ray. In this study, we will choose the initial value as discussed before (Geng, Wu and Gao 2010). To get parameter independent Gaussian Packet,

we suppose that initial value 0ttN is constant for the whole

Gaussian Packet set. Because of the cross term between time and space ( )( )xt r rN x x t t , decomposition of the data into

optimized Gaussian Packets becomes intricate (Zacek 2004) . To be able to correctly reconstrut the data using coeffcients calculated using optimized Gaussian Packets,

‘dual windows’ should be calculated, and initial value 0ttN is

dependent on the Gaussian Packet’s parameter and xp .

However, like discussed in fast beam migration (Gao et al. 2006), the local time slope and position information is what we need to represent the data. Fortunately, the inner product between data ( , )u x t and ( , )gp

su x t can directly

provides us the information of the local slope at certain central frequency in the local time and space area.

, , , ( , ), ( , )x r r

gp gpp t x sC u x t u x t (4)

At the surface, xtN can be written as

0sinxt tt x ttN N p N

V

(5)

which means that xtN is a pure imaginary number which

will affect the envelope of the Gaussian Packet. Figure 2 shows the envelope of Gaussian Packet at different ray emergent angle. The time localization property becomes really bad when the angle is large, and weighted window should be introduced to restrict the effecting area to prevent wrong information occurs.

a) b) Figure 2: Envelope of Gaussian Packet with different ray emergent

angle, 0 20ttN i , 30 /rad s : a) 0o , b) 79o

If we approximately function ( , )u x t using the inner product coefficients and Gaussian Packets, as follow

© 2011 SEGSEG San Antonio 2011 Annual Meeting 33993399

Efficient Gaussian Packets representation and seismic imaging

, , ,, , ,

( , )x r r

x r r

gp gpp t x s

p t x

u x t C u

(6)

The result ( , )u x t becomes incorrect when larger ray emergent angle is involved if shorter duration packet is used (imaginary part of 0

ttN is big).

It is obvious that when xtN is small enough, the cross

term ( )( )xt r rN x x t t can be ignored. Thus, the Gaussian

Packets are reduced into tensor product of two Gabor function, and can be written as follow

2 2

( , ) exp{ [ ( ) ( )

1 1 + ( ) ( ) ]}

2 2

gps x r r

xx r tt r

u x t i p x x t t

N x x N t t

A (7)

Thus, to fully cover the phase space, the time and frequency interval should satisfy 2rt , and the

space and ray parameter interval should satisfy 2 /r rx p , r is reference frequency (Hill

1990). We will choose / 2r r rt x p . When the

beam width is given as 0

2 a

r

Vw

and time duration

as0

1

r ttN, the intervals can be written as

0

2r

r tt

tN

,2 rt

, 02rx w ,04 r

pw

(8)

The field contributed by Gaussian Packets at the point ( , )x z and time t is then can be approximated by the summation of each Gaussian Packet as follow

, , ,, , ,

( , , ) ( , , )x r r

x r r

gp gpp t x s r

p t x

u x z t C u x z t t

(9)

Numerical examples Impulse response in different media Due to the time localization, the oscillation in a given time window is controlled by the central frequency of each

Gaussian Packet. When the initial parameter 0ttN is chosen

properly, we can approximate the source time function just by its dominant frequency. In this case, the local time rt stands for the time delay of each Gaussian Packet.

Figure 3 shows a ricker wavelet with dominant frequency 15Hz and time delay 0.3s as the source time function. To approximate the source time function, different Gaussian

Packets are plotted with 0 3 ,10 ,20 ,50ttN i i i i , respectively,

only with central frequency equals to 15Hz.

Figure 3: Ricker wavelet and approximated source time function using Gaussian Packets.

When the initial value 0 20ttN i , the approximated source

function is very close to the original ricker wavelet. The corresponding impulse responses in a constant velocity media with 2 /v km s at time 0.8s is shown in Figure 4.

The impulse responses with 0 20ttN i in a vertically

linearly varying velocity media with minimum velocity 2 /v km s , and linear varying parameter

1/ 0.5dv dz s at time 0.5s is shown in Figure 5. To show the validity of the method, impulse responses calculated by one-way LCB propagator are also shown in Figure 4 and Figure 5. Migration for a zero-offset data The exploding reflector principle (Claerbout 1985) states that the seismic image is equal to the downward-continued, zero-offset data evaluated at time zero, if the seismic velocities are halved. In this case, the local time rt stands for the travel time of

each Gaussian Packet, and when we define the raypath upcoming and origin travel time equal to zero (raypath pointing to the opposite direction in Figure 1), the contributions of the Gaussian Packets in the image become

, , ,, , ,

( , , 0) ( , , )x r r

x r r

gp gpp t x s r

p t x

u x z t C u x z t t

(10)

A 4 layer velocity model and the corresponding zero-offset dataset synthetic by ricker wavelet with dominant frequency 15Hz and time delay 0.2s are shown in Figure 6. The imaging result using Gaussian Packet with central frequency 15Hz and 0 20ttN i is shown in Figure 7.

a)

© 2011 SEGSEG San Antonio 2011 Annual Meeting 34003400

Efficient Gaussian Packets representation and seismic imaging

b)

c)

d)

e)

Figure 4: Comparison of impulses responses in constant media. a),

b), c), d) are Gaussian Packet method with 0 3 ,10 , 20 ,50ttN i i i i ,

respectively; e) one-way LCB method.

a)

b)

Figure 5: Comparison of impulses responses in vertically linearly

varying media. a), Gaussian Packet method with 0 20ttN i ; b),

one-way LCB method. Conclusions In this paper, we have introduced and tested an efficient Gaussian Packet zero-offset migration method. The representation of the data using Gaussian Packets directly

provides the local time slope and position information. We also have shown that with proper choice of time duration parameter, only Gaussian Packets with one central frequency are needed to obtain propagated seismic data. Imaging for a 4 layer velocity model zero-offset dataset shows valid of the method. Although velocity has to be smoothed before ray tracing, this method can be competitive as a preliminary imaging method, and the localized time property is suitable for target-oriented imaging.

a)

b)

Figure 6. a), 4 layer velocity model. b), zero-offset data.

Figure 7: Imaging result using Gaussian Packets with central

frequency 15Hz and 0 20ttN i .

Acknowledgments The author would like to thank Prof. Ludek Klimes, Yingcai Zheng and Yaofeng He for useful information and fruitful discussion, for fruitful discussions. This work is supported by WTOPI (Wavelet Transform On Propagation and Imaging for seismic exploration) Project at University of California, Santa Cruz. The author will also thank China Scholarship Council.

© 2011 SEGSEG San Antonio 2011 Annual Meeting 34013401

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