Efficient Multiscale Waveform Tomography and Flooding Method * * * C. Boonyasiriwat, P. Valasek, P. Routh, B. Macy, W. Cao, and G. T. Schuster * ConocoPhillips

Outline Goal Introduction Theory of Acoustic Waveform Tomography Multiscale Waveform Tomography: Efficiency Results Conclusions 1

Introduction: Waveform Tomography No high frequency approximation. Frequency domain: Pratt et al. (1998), etc. Time domain: Mora (1987, 1989); Zhou et al. (1995), Sheng et al. (2006), etc. Pratt and Brenders (2004) and Sheng et al. (2006) used early-arrival wavefields. Bunks et al. (1995) and Pratt et al. (1998) used multiscale approaches. 9

Outline Goal Introduction Theory of Acoustic Waveform Tomography Multiscale Waveform Tomography: Efficiency Results Conclusions

Why Acoustic? Elastic wave equation is expensive. Waveform inversion is also expensive. Previous research shows acoustics is adequate. Use acoustics and mute unpredicted wavefields. 11

Theory of Waveform Tomography An acoustic wave equation: The waveform misfit function is 12

Theory of Waveform Tomography The waveform residual is defined by The steepest descent method can be used to minimize the misfit function: 13

Theory of Waveform Tomography The gradient is calculated by where 14

Outline Goal Introduction Theory of Acoustic Waveform Tomography Multiscale Waveform Tomography: Efficient Smoothing Results Conclusions

Why Use Multiscale? Low Frequency Coarse Scale High Frequency Model parameter (m) Misfit function ( f ) Low Frequency Coarse Scale High Frequency Fine Scale Image from Bunks et al. (1995) 16

Our Multiscale Approach Combine Early-arrival Waveform Tomography (Sheng et al., 2006) and a time-domain multiscale approach (Bunks et al., 1995). Use a Wiener filter for low-pass filtering the data. Use a window function to mute all energy except early arrivals. Use multiscale V-cycles. 17

Why a Wiener Low-Pass Filter? Original Wavelet Target Wavelet Hamming Blackman Wiener Wavelet: Hamming Window Wavelet: Wiener Filter Lower frequencies, therefore dx, dt coarser 18

Multiscale V-Cycle High Frequency Fine Grid Low Frequency Coarse Grid 19

Outline Goal Introduction Theory of Acoustic Waveform Tomography Multiscale Waveform Tomography: Efficient Smoothing & Frequency Band Selection Multiscale Waveform Tomography: Efficient Smoothing Results Conclusions

Reflection Wavenumber Resolution Notation G(x|s)G(x|g)= exp(i )=exp(i )exp(i ) s g xo Parallel to t D and length = w/c gx kg ks Parallel to t D and magnitude = w/c sx txg= tx0g + dx txog D Expand txs around xo txs= tx0s + dx txos D kz = kg + ks txs+ txg = dx txos D) +txos+ txog + txog ( D ks + kg w kx= 0 Multiply by w w(txs+ txg) w(txs+ txg ) = dx (ks+kg) +w(txos+ txog ) dx (ks+kg) w(txos+ txog )

Reflection Wavenumber Resolution d(g|s) = ∫G(x|s)G(x|g)m(x)dx = ∫exp(iw[txs+txg])m(x)dx = exp(iw[txos +txog ]) ∫exp(i[ks +kg ] dx) m(x)dx (ignore geometrical spreading) s s g g This is in form of a Fourier transform of m(x). Therefore, d(g|s) ~ model spectrum M(k), i.e., after dividing d(g|s) by phase factor we get kg=k(sina,cosa) g=(xg,0); xo=(xo,zo) sina=(xo-xg)/√(xo-xg)2+zo2 M((ks+kg)w/c)=F(m(x))=d(g|s) ^ xo kg ks The data are Fourier transform of model, so inverse Fourier transform of data is model. ks=k(sina,cosa) s=(xs,0); xo=(xo,zo) sina=(xo-xs)/√(xo-xs)2+zo2 kz = kg + ks kg + ks Inverse transform of model spectrum/data=model m(x)= ∫G(x|s)*G(x|g)*d(x|g)dg A specified w and g -s pair will reconstruct part of wavenumber spectrum in model = ∫exp(-i[ks +kg ] dx) d(g|s)dg

Wavenumber vector k determines spectrum M(kx,kz) m(x)= ∫exp(-i[ks +kg ] dx) d(g|s)dg rs rg r Source Geophone k

Spatial Resolution Limit Formula g s Dx = minimum separation of two points so that they are distinguishable in image

Spatial Traveltime Resolution Limit Formula source-receiver pairs where the wavepath visits rx

Reflection Wavenumber Resolution M((ks+kg)w/c)=F(m(x))=d(g|s) ^ s g kz c = 2w kg ( |kg|=w/c) ks ( |ks|=w/c) w kz kz = kg + ks

Reflection Wavenumber Resolution M((ks+kg)w/c)=F(m(x))=d(g|s) ^ s g kz c = 2w ( |kg|=w/c) kg ks ( |ks|=w/c) kz kz = kg + ks w

Reflection Wavenumber Resolution M((ks+kg)w/c)=F(m(x))=d(g|s) ^ s g kz c = 2w ( |kg|=w/c) kg ks ( |ks|=w/c) kz kz = kg + ks w

Reflection Wavenumber Resolution M((ks+kg)w/c)=F(m(x))=d(g|s) ^ a cos(a)=L/( L +z ) 2 L s g z kg ks 2|k|cos(a) kz c = 2w kz = 2|w/c|cos(a) kz = 2|w/c|cos(a) kz w

Reflection Wavenumber Resolution Efficient Strategy for Choosing w (Sirgue&Pratt, 2004) (Single Frequency Strategy: Sirgue&Pratt, 2004) Just 2 frequencies & many offsets fill the kz line; and therefore reconstruct M(kx,kz) for wide range kz kz We get this range of kz’s for model, no need to be redundant In comparison, sloppy fine grid of ws will require lots of CPU time to cover same range of kz w Choose next w w Multiscale W(w) (Multiscale x-t Strategy: Boonyasiriwat et al., 2004) Wavelet Spectrum ½ power

Reflection Wavenumber Resolution Efficient Strategy for Choosing w (Sirgue&Pratt, 2004) (Single Frequency Strategy: Sirgue&Pratt, 2004) Just 2 frequencies & many offsets fill the kz line; and therefore reconstruct M(kx,kz) for wide range kz kz We get this range of kz’s for model, no need to be redundant In comparison, sloppy fine grid of ws will require lots of CPU time to cover same range of kz w Choose next w w Multiscale W(w) (Multiscale x-t Strategy: Boonyasiriwat et al., 2004) Wavelet Spectrum ½ power

Efficient Strategy for Choosing w (Boonyasiriwat et al., 2007)

Outline Goal Introduction Theory of Acoustic Waveform Tomography Efficient Multiscale Waveform Tomography Results: EWT vs MWT Conclusions 1

Layered Model with Scatterers 22

Initial Velocity Model 23

TRT Tomogram Gradient 24

EWT Tomogram using 15-Hz Data Gradient 25

MWT Tomogram using 2.5-Hz Data Gradient 26

MWT Tomogram using 5-Hz Data 27

MWT Tomogram using 10-Hz Data 28

MWT Tomogram using 15-Hz Data 29

Layered Model with Scatterers 30

Comparison of Misfit Function 15 Hz 15 Hz 5 Hz 10 Hz 2.5 Hz 31

Outline Goal Introduction Theory of Acoustic Waveform Tomography Efficient Multiscale Waveform Tomography Results with Flooding Conclusions 1

SEG Salt Velocity Model 32

Initial Velocity Models Depth (km) 4500 Traveltime Tomogram 4 Velocity (m/s) Depth (km) 1500 v(z) Model 4 X (km) 16 9

TRT Tomogram Gradient 33

MWT Tomogram (2.5,5 Hz) TRT 34

Introduction: Waveform Tomography 1 Introduction: Waveform Tomography 1.Not really good enough below salt 2. Flooding Method: Flood salt below top, invert for bottom, flood sediment below salt bottom. FWI. 7

Waveform Inversion Results Using Traveltime Tomogram Depth (km) 4500 4 Velocity (m/s) Using v(z) Model + Flooding Depth (km) 1500 4 X (km) 16 10

Waveform Inversion Results 4500 Velocity (m/s) True Model Depth (km) 1500 4 X (km) 16 10

Flooding Technique Using v(z) Model w/o Flooding Depth (km) 4500 4 Depth (km) 4500 4 Velocity (m/s) Waveform Tomogram after Salt Flood Depth (km) 1500 4 X (km) 16 11

Flooding Technique Waveform Tomogram after Sediment Flood Depth (km) Depth (km) 4500 4 Velocity (m/s) Waveform Tomogram using v(z) and Flooding Technique Depth (km) 1500 4 X (km) 16 12

SEG Salt Velocity Model 35

Outline Goal Introduction Theory of Acoustic Waveform Tomography Efficient Multiscale Waveform Tomography Results: Mapleton Conclusions 1

Mapleton Model 36

TRT Tomogram 37

MWT Tomogram (30, 50, 70 HZ) 38

Mapleton Model 39

Mapleton Land Seismic Data This is the zero-offset Mapleton land seismic data collected on an irregular surface in Mapleton, Utah.

Acquisition Geometry ~20 m ~78 m With the topographic phase-shift method, the data are firstly converted to the data at z=0 (the highest geophone elevation). In doing so, the exact geophone positions are used, and wavefield interpolation to a uniform grid is avoided. Geophone intervals are uniformly 0.5 m in distance, however, because of the relief, both dx, dz are nonuniform.

Tomography (Buddensiek and Sheng, 2004)

Wave-equation Migration Without correction

Wave-equation Migration With correction

Marine Data Results 40

Marine Data 480 Hydrophones 515 Shots 12.5 m dt = 2 ms Tmax = 10 s 41

Low-pass Filtering 42

Reconstructed Velocity 43

Observed Data vs Predicted Data 44

Waveform Residual vs Iteration Number 5 Hz 5 Hz 5 Hz 10 Hz 10 Hz 5 Hz 10 Hz 45

Common Image Gather 5 Hz 10 Hz 46

Outline Goal Introduction Theory of Acoustic Waveform Tomography Multiscale Waveform Tomography Results Conclusions 47

Conclusions MWT partly overcomes the local minima problem. MWT provides more accurate and highly resolved than TRT and EWT. MWT is much more expensive than TRT. Accuracy is more important than the cost. MWT provides very accurate tomograms for synthetic data and shows encouraging results for the marine data. 48

Future Work Use wider-window data and finally use all the data to obtain more accurate velocity distributions. Take into account the source radiation pattern. Apply MWT to land data. 49

Acknowledgment We are grateful for the support from the sponsors of UTAM consortium. Chaiwoot personally thanks ConocoPhillips for an internship and also appreciates the help from Seismic Technology Group at ConocoPhillips. 50