Fast Multisource Least Squares Migration of 3D Marine Data with Frequency Encoding Yunsong Huang and Gerard Schuster KAUST vs

RTM Problem & Possible Soln. Problem: RTM computationally costly; IO high Solution: Multisource LSM RTM Preconditioning speeds up by factor 2-3 Encoded LSM reduces crosstalk. Reduced comp. cost+memory 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. 2

Outline Phase Encoded Multisource LSM Problem with Marine Data Phase Encoded Multisource LSM with Frequency Selection Numerical results Conclusions 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. 3

Multisource Data vs mmig =[L +L ](d + d ) Time Shift Encoding 1 d1+d2 = [L1+L2]m migrate m ~ [L1+L2](d1+d2) T = L1d1+L2d2+L1d2+L2d1 T standard mig. crosstalk vs Benefit: Only one migration Problem: Low SNR

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

Phase Encoded Multisrce Least Squares Migration { L { d +d =[L +L ]m 1 2 Forward Model: Multisource Migration: mmig=LTd mmig T T =[L +L ](d + d ) 1 2 T T T T m = m + (k+1) (k) = L d +L d + 1 2 L d +L d 2 1 Crosstalk noise Standard migration K=20 K=1

Land SEG/EAGE Salt Reflectivity Model Z (km) 0 1.4 6 X (km) Use constant velocity model with c = 2.67 km/s Center frequency of source wavelet f = 20 Hz 320 shot gathers, Born approximation Encoding: Dynamic time, polarity statics + wavelet shaping Center frequency of source wavelet f = 20 Hz 320 shot gathers, Born approximation

Standard Land Phase Shift Migration vs MLSM Standard Phase Shift Migration (320 CSGs) 1 x Z k(m) 1.4 X (km) 6 Multisource PLSM (320 blended CSGs, 7 iterations) Z (km) 1 x 44 1.4 X (km) 6

Single-source Land PSLSM (Yunsong Huang) 1.0 Conventional encoding: Polarity+Time Shifts Model Error 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. Unconventional encoding 0.3 Iteration Number 50

Outline Phase Encoded Multisource LSM Problem with Marine Data Phase Encoded Multisource LSM with Frequency Selection Numerical results Conclusions 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. 10

the acquisition geometry Land Multisource FWI Fixed spread Simulation geometry must be consistent with the acquisition geometry

Marine Multisource FWI Mismatch solution with marine data Observed marine data Simulated land data Freq. encoding Decode & mute purify 4 Hz 8 Hz 4 Hz 8 Hz 4 Hz 8 Hz F.T., freq. selec. wrong misfit Blend Let’s focus on the white receiver…, how the frequency encoding could save the day. 8 Hz 4 Hz

Multisource FWI Freq. Sel. Workflow m(k+1) = m(k) + a L Dd ~T For k=1:K end Select unique frequency for each src d d  Filter and blend observed data: dd dpred dpred  Purify predicted data: dpreddpred Let’s focus on the white receiver…, how the frequency encoding could save the day. Data residual: Dd=dpred-d

Outline Aim of the study Multisource Low-discrepancy frequency coding Mismatch solution with marine data Low-discrepancy frequency coding Numerical results Conclusions 14

Low-discrepancy Frequency Encoding Standard Source index 1 60 Low-discrepancy encoding Frequency index 1 60 60 60 Frequency index Frequency index 1 1 Source index 1 60

Outline Aim of the study Multisource Low-discrepancy frequency coding Mismatch solution with marine data Low-discrepancy frequency coding Numerical results Conclusions 16

Multisource Least Squares Migration Marine Geometry m’ = m - LT[Lm - d] f ~ [LTL]-1 f Steepest Descent Preconditioned z (km) 4 8 x (km)

Multisource Least-squares vs Standard Migration a) Original b) Standard Migration Z (km) 1.48 c) 4 groups 50 iterations d) 1 group 50 iterations Z (km) 1.48 X (km) 6.75 X (km) 6.75

Conventional Migration 64 shots/gather 128 shots/gather 32 shots/gather 256 shots/gather Conventional Migration ‘14%’ is chosen to equate the computational complexity of this migration (with fewer # of sources) with that of Least Squares multisource migration. The idea is, less computation can be achieved simply by using fewer CSGs. The point made in this plot is, this naïve approach yields poor result, as indicated by the dashed yellow line, of relatively large model error. N_{shots}/ gather: # of shots contained in a super-gather SNR: the observed data (which forms the objective function of LS) has been intentionally added with random band-limited noise, such that the SNR=10dB. This is to study the robustness of our approach. Red dots signify where the LS multisource migration results are perceptually comparable to that of the standard migration. The computational cost of the LS multisource migration is computed based on those numbers.

SEG/EAGE Model+Marine Data sources in total 40 m 100 m 16 swaths, 50% overlap 256 sources a swath 6 km 3.7 km 20 m 16 cables 13.4 km

Numerical Results 8 x True reflectivities Conventional migration 6.7 km True reflectivities Conventional migration 3.7 km 256 shots/super-gather, 16 iterations The multisource result is of slightly higher resolution than the conventional shot-gather migration; indistinguishable in this size on screen. 13.4 km 8 x

Frequency-selection FWI of 2D Marine Data Shots: 60 Source freq: 8 Hz Receivers/shot: 84 Cable length: 2.3 km Z (km) 1.5 4.5 (km/s) 1.5 6.8 X (km)

FWI images Actual model Starting model Standard FWI Multisource FWI Z (km) 1.5 Starting model Standard FWI (69 iterations) Z (km) 1.5 X (km) 6.8 Multisource FWI (262 iterations) X (km) 6.8

Conclusions Frequency selection is implemented in FDTD 2 x time steps per forward or backward modeling Low-discrepancy frequency encoding affects no asymptotic rate of convergence helps to reduce model error in the beginning of simulation 4x speedup for the multisource FWI on the synthetic marine model 24

Step 1: Frequency Selection/Shot Step 2: Pass only red to right of Pass only green between d = [L mk - d] Simulation result Observed data Still mismatch between d and L mk  erroneous big d combined by frequency selective multisource

Outline Phase Encoded Multisource LSM Problem with Marine Data Phase Encoded Multisource LSM with Frequency Selection Numerical results Conclusions 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. 26

Multisource Data mmig =[L +L ](d + d ) mmig =[L +L ](d + d ) Time Shift Encoding d1 +d2 mmig T T =[L +L ](d + d ) Time d1 d2 1 d1+d2 = [L1+L2]m migrate m ~ [L1+L2](d1+d2) T = L1d1+L2d2+L1d2+L2d1 T crosstalk Time Shift + Frequency Encoding S(w) Time 1 T d1 +d2 mmig T =[L +L ](d + d ) w d1 d2 2 d1+d2 = [L1+L2]m migrate m ~ [L1+L2](d1+d2) T = L1d1+L2d2+L1d2+L2d1 T

Frequency Selection per Shot Gather R(w) Frequency bands of source spectrum w 20 m 216 channels Group 1 76 sources/group The colors of every hydrophone denote its ‘pass frequency’. That is, non-colored frequency bands are filtered out. By this frequency-selection mechanism, we can winnow out the undesired contributions at each hydrophone from other sources when these sources are grouped together for simulation. 2.2 km

Outline Phase Encoded Multisource LSM Problem with Marine Data Phase Encoded Multisource LSM with Frequency Selection Numerical results: 2D Conclusions 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. 29

Sensitivity to Noise Level SNR=30dB SNR=10dB SNR=20dB LSFDMS stands for Least-squares Frequency Selection Multi-Source Different noise levels have been tried, with essentially the same results. Here, strict frequency selection is honored, that is, simultaneous sources have absolutely non-overlapping frequency channels. This restriction precludes more aggressive multi-sourcing. Thus the speedup factor is smaller than the eyebrow-raising 40ish. 256 frequency channels are assumed to be available. Thus each super-gather can meaningfully include at most about 256 sources.

Outline Phase Encoded Multisource LSM Problem with Marine Data Phase Encoded Multisource LSM with Frequency Selection Numerical results: 3D Conclusions 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. 31

Cost vs Quality: Can I<<S? Yes. What have we empirically learned? Stnd. Mig Multsrc. LSM IO 1 1/320 1 0.1-.01 Cost ~ 1 ~1 SNR~ Resolution dx 1 double Cost vs Quality: Can I<<S? Yes.

Conclusions Mig vs MLSM 1. SNR: VS GS GI # geo # srcs # iterations 2. Memory: 1/S Cost: O(1/10) 2. Cost: S vs I 3. Caveat: Mig. & Modeling were adjoints of one another. LSM sensitive starting model Second I compute reflectivity model within this offset range from the velocity and density models. I also created a source wavelet that mimics an air gun source signature. Fdom = 25 Hz. 4. Unconventional encoding: I << S Next Step: Sensitivity analysis to starting model 33

Acknowledgments Schlum-WesternGeco Aramco Total BP Tullow Chevron Thanks Sponsors of Center for Seimsic Imaging And Fluid Modeling (CSIM) at KAUST http://csim.kaust.edu.sa Schlum-WesternGeco Total Tullow Veritas Aramco BP Chevron Pemex Petrobras Second I compute reflectivity model within this offset range from the velocity and density models. I also created a source wavelet that mimics an air gun source signature. Fdom = 25 Hz. 34

Land Data: Src shootsall phones, so does FD modeler. This is good. Marine Data: Src shootslimited phones, but FD modeler shoots all. This is bad. d = [L mk - d] Simulation result Observed data combined by multisource Mismatch between d and L mk  erroneous big d

Outline Aim of the study Multisource Migration Mismatch solution with marine data Low-discrepancy frequency coding Numerical results Conclusions 36

low-discrepancy encoding Convergence Rates Waveform error Faster initial convergence rate of the white curve 1 1 supergather, standard encoding Same asymptotic convergence rate of the red and white curves Log normalized by individual sources 1 supergather, low-discrepancy encoding 3.8 x 0.025 1 69 262 Log iteration number

low-discrepancy encoding Convergence Rates Speedup 60 / 2 / 2 / 3.8 = 4 Velocity error 1 Gain 60: sources Overhead factors: 2 x FDTD steps 2 x domain size 3.8 x iteration number 1 supergather, standard encoding Log normalized by individual sources 1 supergather, low-discrepancy encoding 3.8 x 0.35 1 69 262 Log iteration number

Low-discrepancy encoding is 12% to 3x faster initially than Convergence Rates Velocity error (normalized) Low-discrepancy encoding is 12% to 3x faster initially than Standard encoding 1 standard encoding 10 0.75 1 iteration number