Efficient Gaussian Packets representation and seismic imaging Yu Geng, Ru-Shan Wu and Jinghuai Gao WTOPI, Modeling & Imaging Lab., IGPP, UCSC Sanya 2011

OUTLINE IntroductionData representation using Gaussian Packets Impulse response in different media Migration for a zero-offset data Numerical examplesConclusions

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 IntroductionData representation using Gaussian Packets Impulse response in different media Migration for a zero-offset data Numerical examplesConclusions

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

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.

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.

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

Data representation When the Gaussian Packet can be written as Parameters of Gaussian Packet Space center location Time center location Central frequency Ray parameter Gaussian window width along space direction Gaussian window width along time direction

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. Cross term between time and space

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

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

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

OUTLINE IntroductionData representation using Gaussian Packets Impulse response in different media Migration for a zero-offset data Numerical examplesConclusions

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)

Impulse response (comparison) Different GP GP compared with One-way method One-way method

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=0.3s

Impulse response (GP method) t=0.5s

Impulse response (GP method) t=0.8s

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=0.4s

Impulse response (GP method) t=0.5s

Impulse response (GP method) t=0.8s

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.

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.