Efficient Gaussian Packets representation and seismic imaging

Yu Geng, Ru-Shan Wu and Jinghuai Gao

WTOPI, Modeling & Imaging Lab., IGPP, UCSC Sanya 2011

OUTLINE

Introduction

Data representation using Gaussian Packets

• Impulse response in different media• Migration for a zero-offset data

Numerical examples

Conclusions

INTRODUCTION• Gaussian packets (GP), also called (space-time) Gaussian beams

(Raslton 1983) or quasiphotons (Babich and Ulin 1981), are high-frequency asymptotic space-time particle-like solutions of the wave equation(Klimes 1989; Klimes 2004). They are also waves whose envelops at a given time are nearly Gaussian functions.

• A Gaussian packet is concentrated to a real-valued space-time ray (in a stationary medium, a Gaussian packet propagates along its-real-valued spatial central ray), as a Gaussian beam to a spatial ray.

• Decomposition of the data using optimized Gaussian Packets and related migration in common-shot domain (Zacek 2004) are time-consuming.

• We discussed an efficient way to use Gaussian Packets to data decomposition and migration .

OUTLINE

Introduction

Data representation using Gaussian Packets

• Impulse response in different media• Migration for a zero-offset data

Numerical examples

Conclusions

Gaussian Packet• In the two-dimensional case, a Gaussian Packet can be written

as

is arbitrary positive parameter, and it can be understood as the center frequency of the wave packet.

• Using Tayler expansion, the phase function can be further expanded to its second order

( , , ) exp( )gpu x z t iA

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

Propagating a Gaussian Packet• While propagating, a Gaussian Packet can nearly keep its

Gaussian shape in space in any given time. To propagate a Gaussian Packet, we need to calculate

is the ray-theory slowness vector which can be determined by standard ray tracing at the central point and

is complex-valued and determines the shape of the Gaussian Packet, and its evolution along the spatial central ray is determined by quantities calculated by dynamic ray tracing. Amplitude can also be calculated during dynamic ray tracing.

, ,N N and AiN

1tN N

00

01 2det( )VA A

V

Q Q M

Propagating a Gaussian Packet

Profiles of Gaussian Packets

• When the Gaussian Packet can be written as

• Initial Parameters of Gaussian Packet: Beam width and pulse duration.

rt t

2

2

( , ) exp{ [ ( )1 ( ) ( )2

1 ( )2

( )( )]}

gps x r

z r xx r

zz r

xz r r

u x t i p x x

p z z N x x

N z z

N x x z z

A

Propagating a Gaussian Packet

Snapshot at for a Gaussian Packet with different initial beam curvature. White lines stand for corresponding central ray.

3 0.3 , 1.0st s

Snapshot for Gaussian Packets at with different pulse duration3 0.7t s

Data representation

• When the Gaussian Packet can be written as

• Parameters of Gaussian Packet

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

A0z

Space center location

Time center location

Central frequency

Ray parameter

Gaussian window width along space direction

Gaussian window width along time direction

rx

rt

xp

0w0ttN

Data representation

• The inner product between data and Gaussian Packet 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

• Cross term between time and space

2 2

( , ) exp{ [ ( ) ( )1 1 + ( ) ( ) ( )(+ }

2)]

2

gps x

xt r

r r

xx r tt r r

u x t i p x x t t

N x x N t N x x tt t

A

Data representation

• Because of the cross term between time and space , decomposition of the data into optimized Gaussian Packets becomes intricate.

0sinxt tt x ttN N p N

V

0o 79o

Data representation

• It is obvious that when is small enough, the cross term can be ignored. Thus, the Gaussian Packets are reduced into tensor product of two Gabor function.

• To fully cover the phase space, the time and frequency interval should satisfy

• The space and ray parameter interval should satisfy

2rt

2 /r rx p

( )( )xt r rN x x t t xtN

Data representation

• When the beam width is given as (Hill,1990) and time duration as the intervals can be written as

• The field contributed by Gaussian Packets at the point (x, z) and time t

0 2 a rw V 01 r ttN

0

2r

r tt

tN

2 rt 02rx w

04 r

pw

, , ,, , ,

( , , ) ( , , )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

OUTLINE

Introduction

Data representation using Gaussian Packets

• Impulse response in different media• Migration for a zero-offset data

Numerical examples

Conclusions

Numerical Examples

• Impulse response in different media• A ricker wavelet with dominant frequency 15Hz and time

delay 0.3s as the source time function

Impulse response (LCB method)impulse responses in a constant velocity media v=2km/s

with at time 0.8s

Impulse response (GP method)0 3ttN i

Impulse response (GP method)0 10ttN i

Impulse response (GP method)0 20ttN i

Impulse response (GP method)0 50ttN i

Impulse response (comparison)Different GP

GP compared with One-way method

One-way method

0 20ttN i

Impulse response (LCB method)The impulse responses with in a vertically linearly varying velocity media, minimum velocity 2km/s , and linear varying parameter dv/dz=0.5s-1 , t=0.5s

Impulse response (GP method) t=0.3s0 20ttN i

Impulse response (GP method) t=0.5s0 20ttN i

Impulse response (GP method) t=0.8s0 20ttN i

Impulse response (GP method)Minimum velocity 2km/s , and linear varying parameter dv/dx=0.5s-1 , dv/dz=0.5s-1, t=0.1s

Impulse response (GP method)Minimum velocity 2km/s , and linear varying parameter dv/dx=0.5s-1 , dv/dz=0.5s-1, t=0.3s

Impulse response (GP method)Minimum velocity 2km/s , and linear varying parameter dv/dx=0.5s-1 , dv/dz=0.5s-1, t=0.5s

Impulse response (GP method) t=0.4s0 20ttN i

Impulse response (GP method) t=0.5s0 20ttN i

Impulse response (GP method) t=0.8s0 20ttN i

Numerical Examples

• 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.

, , ,, , ,

( , , 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

Migration for a zero-offset data

Migration for a zero-offset data

Migration for a zero-offset data

Conclusions• 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 few central frequencies 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.

Acknowledgments

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