#### Transcript Efficient Gaussian Packets representation and seismic imaging

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 Numerical examples • Impulse response in different media • Migration for a zero-offset data Conclusions INTRODUCTION • Gaussian packets (GP), also called (space-time) Gaussian beams (Raslton 1983) or quasiphotons (Babich and Ulin 1981), are highfrequency 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-realvalued 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 timeconsuming. • We discussed an efficient way to use Gaussian Packets to data decomposition and migration . OUTLINE Introduction Data representation using Gaussian Packets Numerical examples • Impulse response in different media • Migration for a zero-offset data Conclusions Gaussian Packet • In the two-dimensional case, a Gaussian Packet can be written as gp u ( x, z, t ) A exp(i ) 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 px ( x xr ) pz ( z zr ) pt (t tr ) 1 1 1 2 2 + N xx ( x xr ) N zz ( z zr ) Ntt (t tr ) 2 2 2 2 +N xz ( x xr )( z zr ) N xt ( x xr )(t tr ) N zt ( z zr )(t tr ) 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 N , N , and A Ni is the ray-theory slowness vector which can be determined by standard ray tracing at the central point and Nt 1 N 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. A A 0 V0 V det(Q1 Q 2 M 0 ) Propagating a Gaussian Packet • When t tr the Gaussian Packet can be written as usgp ( x, t ) A exp{i[ px ( x xr ) pz ( z zr ) 1 N xx ( x xr ) 2 2 1 N zz ( z zr ) 2 2 N xz ( x xr )( z zr )]} • Initial Parameters of Gaussian Packet: Beam width and pulse duration. Profiles of Gaussian Packets Propagating a Gaussian Packet Snapshot at t 3 0.3s, 1.0s for a Gaussian Packet with different initial beam curvature. White lines stand for corresponding central ray. Snapshot for Gaussian Packets at t 3 0.7 s with different pulse duration Data representation • When z 0 the Gaussian Packet can be written as usgp ( x, t ) A exp{i[ px ( x xr ) (t tr ) 1 1 + N xx ( x xr ) 2 Ntt (t tr ) 2 +N xt ( x xr )(t tr )]} 2 2 • Parameters of Gaussian Packet xr Space center location tr Time center location Central frequency px Ray parameter w0 Gaussian window width along space direction Ntt0 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. C pgpx , ,tr , xr u ( x, t ), usgp ( x, t ) • Cross term between time and space usgp ( x, t ) A exp{i[ px ( x xr ) (t tr ) 1 1 2 + N xx ( x xr ) Ntt (t tr ) 2 +N xt ( x xr )(t tr )]} 2 2 Data representation • Because of the cross term between time and space , decomposition of the data into optimized Gaussian Packets becomes intricate. sin N xt N tt px N tt0 V 0o 79o Data representation • It is obvious that when N xt is small enough, the cross term Nxt ( x xr )(t tr ) 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 tr 2 • The space and ray parameter interval should satisfy xr p 2 / r Data representation • When the beam width is given as (Hill,1990) w0 2Va r and time duration as1 r Ntt0 the intervals can be written as 2 tr p x 2 w r 0 0 2tr r Ntt 4w0r • The field contributed by Gaussian Packets at the point (x, z) and time t u( x, z, t ) px , ,tr , xr C pgpx , ,tr , xr usgp ( x, z, t tr ) OUTLINE Introduction Data representation using Gaussian Packets Numerical examples • Impulse response in different media • Migration for a zero-offset data 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) N 3i 0 tt Impulse response (GP method) N 10i 0 tt Impulse response (GP method) N 20i 0 tt Impulse response (GP method) N 50i 0 tt Impulse response (comparison) Different GP One-way method Ntt0 20i GP compared with 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) Ntt0 20i t=0.3s Impulse response (GP method) Ntt0 20i t=0.5s Impulse response (GP method) Ntt0 20i 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) Ntt0 20i t=0.4s Impulse response (GP method) Ntt0 20i t=0.5s Impulse response (GP method) Ntt0 20i 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. u( x, z, t 0) px , ,tr , xr Cpgpx , ,tr , xr usgp ( x, z, t tr ) 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.