PS, SSP, PSPI, FFD KM SSP PSPI FFD

P(x,z,w) = P(x,0 ,w) e k = k 1 – k ~ k (1 – k + ..) k k k k 2 z ik(x)

PS, SSP, PSPI, FFD k = k 1 – k k k ~ k(1 – k ) ~ k (1 – .43k ) 1 -.5 z 2 k x P(x,z,w) = P(x,0 ,w) e z ik(x) -1 1 .2 k z ~ k(1 – k ) 2 x ~ k (1 – .43k ) 1 -.5 ; k = k k z k x

SSP Migration k = k(x) 1 – k k(x) = k 1 – k k - Dk P(x,z,w) = P(x,0 ,w) e z ik(x) k = k(x) 1 – k z 2 k(x) x = k 1 – k 2 k x - Dk Thin lens

FFD Migration k = k(x) 1 – k k(x) = k 1 – k k - Dk P(x,z,w) = P(x,0 ,w) e z ik(x) k = k(x) 1 – k z 2 k(x) x = k 1 – k 2 k x - Dk Thin lens

FFD Migration k = k(x) 1 – k k(x) = k 1 – k k - Dk P(x,z,w) = P(x,0 ,w) e z ik(x) k = k(x) 1 – k z 2 k(x) x = k 1 – k 2 k x - Dk Thin lens

FFD Migration P(x,z,w) = P(x,0 ,w) e z ik(x)

FFD Migration P(x,z,w) = P(x,0 ,w) e z ik(x) other term

FFD Migration P(x,z,w) = P(x,0 ,w) e other term PDE associated with ik(x) other term PDE associated with other term Rearrange PDE

Substitute FD approximations into above FFD Migration P(x,z,w) = P(x,0 ,w) e z ik(x) Substitute FD approximations into above

Substitute FD approximations into above FFD Migration P(x,z,w) = P(x,0 ,w) e z ik(x) Substitute FD approximations into above

FFD Migration k = k(x) 1 – k k(x) = k 1 – k k - Dk P(x,z,w) = P(x,0 ,w) e z ik(x) k = k(x) 1 – k z 2 k(x) x = k 1 – k 2 k x - Dk Thin lens

PS, SSP, PSPI, FFD

PS, SSP, PSPI, FFD

Summary Cost: Accuracy: KM SSP PSPI FFD

Course Summary m(x)= a(g,s,x) G(g|x)d(g|x)G(x|s)dgds g,s,w G(g|x) = G(g|x) + G(g|x) d(g|x) = d(g|x) + d(x|g) Filter G(g|x) = G(g|x) d(g|x) = d(g|x) RTM Asymptotic G + Fresnel Zone Asymptotic G 1-way G KM Phase Shift Beam

Multisource Seismic Imaging vs CPU Speed vs Year 100000 10000 copper 1000 Aluminum Speed VLIW 100 Superscalar 10 RISC 1 1970 1980 1980 1990 2000 2010 2020 Year

OUTLINE Theory I Numerical Results Theory II

RTM Problem & Possible Soln. Problem: RTM computationally costly Solution: Multisource LSM RTM Preconditioning speeds up by factor 2-3 LSM reduces crosstalk My talk is organized in the following way: 1. The first part is motivation. I will talk about a least squares migration (LSM ) advantages and challenges. 2. The second part is theory for a deblurring filter, which is an alternative method to LSM. 3. In the third part, I will show a numerical result of a deblurring filter. 4. The fourth is the main part of my talk. Deblurred LSM (DLSM) is a fast LSM with a deblurring filter. I will explain how to use the filter in LSM algorithm. 5. Then I will show numerical results of the DLSM. 6. Then I will conclude my presentation. Each figure has a slide number is shown at the footer. 19 5 19

Multisource Migration: Multisource Least Squares Migration d { L { d +d =[L +L ]m 1 2 Forward Model: Multisource Migration: mmig=LTd T T =[L +L ](d + d ) 1 2 Standard migration T T T T = L d +L d + 1 2 L d +L d 2 1 Crosstalk noise

Multisource Least Squares Phase-encoded Migration Orthogonal phase encoding s.t. <N* N >=0 1 2 =[N L +N L ](N d + N d ) 1 2 * T * mmig T Crosstalk noise =N*N L d +N*N L d + N*N L d + N*N L d 1 2 T T T T If <N N > = d(i-j) i j = L d + L d 1 2 Standard migration

Key Assumption + ~ ~ ~ [S(t) +N(t) ] d(t) = N(t ) <N(t)> ~ Zero-mean white noise: <N(t)>=0; <N(t) N(t’) >=0 k=1 M [S(t) +N(t) ] d(t) = M M vs M M= Stack Number Amplitude + k=1 M N(t ) (k) <N(t)> ~ 1 M k=1 M [ S(t) ] 2 (k) [ S(t) ] M 2 2 [ S(t) ] M 2 2 SNR ~ ~ ~ k=1 M [ N(t) ] 2 k=1 M [ N(t) ] 2 (k) (k) M s

Multisource S/N Ratio d , d , …. d +d +…. L [d + d +.. ] 1 2 1 2 1 2 1 2 # CSGs # geophones/CSG

MS S-1 M ~ MS MI MS vs vs Multisrc. Migration vs Standard Migration # geophones/CSG # CSGs MS S-1 M ~ vs MS Iterative Multisrc. Migration vs Standard Migration # iterations vs MI MS

Summary T T L d +L d 2 1 Time Statics 1. Multisource crosstalk term analyzed analytically 2. Crosstalk decreases with increasing w, randomness, dimension, iteration #, and decreasing depth Time+Amplitude Statics 3. Crosstalk decrease can now be tuned QM Statics 4. Some detailed analysis and testing needed to refine predictions.

OUTLINE Theory I Numerical Results Theory II

The Marmousi2 Model Z k(m) 3 X (km) 16 The area in the white box is used for S/N calculation.

Conventional Source: KM vs LSM (50 iterations) Z k(m) 3 X (km) 16 Z (km) 3 X (km) 16

200-source Supergather: KM vs LSM (300 its.) Z k(m) 3 X (km) 16 Z (km) 3 X (km) 16

I S/N = S/N Number of Iterations The S/N of MLSM image grows as the square root of the number of iterations. 7 S/N 1 Number of Iterations 300

Multisource Technology Fast Multisource Least Squares Phase Shift. Multisource Waveform Inversion (Ge Zhan) Theory of Crosstalk Noise (Schuster) 8

The True Model use constant velocity model with c = 2.67 km/s center frequency of source wavelet f = 20 Hz

Multi-source PSLSM 645 receivers and 100 sources, equally spaced 10 sets of sources, staggered; each set constitutes a supergather 50 iterations of steepest descent

Single-source PSLSM 645 receivers and 100 sources, equally spaced Jerry, The multi-source and single-source approaches have used different strategies for the step length. Therefore direct comparison of their misfit error is not applicable. Sorry about that. 645 receivers and 100 sources, equally spaced 100 individual shots 50 iterations of steepest descent

Multi-Source Waveform Inversion Strategy (Ge Zhan) Generate multisource field data with known time shift 144 shot gathers Initial velocity model Generate synthetic multisource data with known time shift from estimated velocity model Using multiscale, multisource CG to update the velocity model with regularization Multisource deblurring filter

3D SEG Overthrust Model (1089 CSGs) 15 km 3.5 km 15 km

Dynamic Polarity Tomogram Numerical Results 3.5 km Dynamic QMC Tomogram (99 CSGs/supergather) Static QMC Tomogram (99 CSGs/supergather) 15 km Dynamic Polarity Tomogram (1089 CSGs/supergather)

OUTLINE Theory I Numerical Results Theory II

Multisource Least Squares Migration Crosstalk term Time Statics Time+Amplitude Statics QM Statics 36

Summary 37 1. Multisource crosstalk term analyzed analytically Time Statics 1. Multisource crosstalk term analyzed analytically 2. Crosstalk decreases with increasing w, randomness, dimension, and decreasing depth Time+Amplitude Statics 3. Crosstalk decrease can now be tuned QM Statics 4. Some detailed analysis and testing needed to refine predictions. 37

Multisource Migration: Multisource Least Squares Migration d { L { d +d =[L +L ]m 1 2 Forward Model: Phase encoding Multisource Migration: mmig=LTd Kirchhoff kernel Standard migration Crosstalk term 34

Multisource Least Squares Migration Crosstalk term 35

Multisource Least Squares Migration Crosstalk term Time Statics Time+Amplitude Statics QM Statics 36

Crosstalk Term L d +L d T T Time Statics Time+Amplitude Statics 2 1 Time Statics Time+Amplitude Statics QM Statics

Summary 37 1. Multisource crosstalk term analyzed analytically Time Statics 1. Multisource crosstalk term analyzed analytically 2. Crosstalk decreases with increasing w, randomness, dimension, and decreasing depth Time+Amplitude Statics 3. Crosstalk decrease can now be tuned QM Statics 4. Some detailed analysis and testing needed to refine predictions. 37

Multisource FWI Summary (We need faster migration algorithms & better velocity models) Stnd. FWI Multsrc. FWI IO 1 vs 1/20 Cost 1 vs 1/20 or better Sig/MultsSig ? Resolution dx 1 vs 1

Key Assumption + n n <d(t)>= <S(t)> + <N(t)> N(t ) Zero-mean white noise: <N>=0; <N N >=0 i j <d(t)>= <S(t)> + <N(t)> n= Stack Number Amplitude + k=1 n N(t ) (k) <N(t)> ~ n n 1/n 2 2 2 <N(t) > ~ 2 k=1 n [ N(t ) ] (k) 1/n <N(t)> ~ <S(t)> ~ <N(t) > ~ 2 k=1 n [ N(t ) ] (k) 1/n