Acoustic waveform inversion of two-dimension Gulf of Mexico data

Acoustic waveform tomography can provide a velocity model with higher accuracy than traveltime tomography because the forward modeling is based on the complete wave equation rather than the high-frequency approximation of ray tracing. Moreover, each trace contains both shallow and deep reflection arrivals and so is richer in information content than the first arrival traveltime. As a result, waveform tomography provides a more detailed estimate of the velocity medium than does traveltime tomography. However, acoustic waveform tomography is a highly nonlinear inverse solution that requires a good starting velocity model compared to the smooth starting model that is sufficient for traveltime tomography. To overcome the nonlinear problems in waveform tomography, I develop a novel multiscale waveform tomography method and apply it to both synthetic and marine seismic data from the Gulf of Mexico. The inversion process is carried out using a multiscale acoustic method with a dynamic early-arrival muting window to mitigate the local minima problem of waveform tomography and elastic effects in the data. This multiscale approach is denoted as multiscale waveform tomography (MWT), and is first validated with synthetic acoustic data from the 2D SEG/EAGE salt model. Using the traveltime velocity tomogram as an initial model, MWT fails to converge to the global minimum. I found that the flooding technique normally used in subsalt migration can be used to significantly improve the convergence of MWT. The velocity model recovered by MWT using the flooding technique is more accurate and highly resolved than that obtained using conventional MWT. For the marine data, MWT can provide a more accurate velocity model than the initial model from traveltime tomography. The accuracy of the waveform velocity model is verified by comparing migration images and common image gathers. I also present ray-based spatial resolution formulas for migration and inversion with numerical tests on both homogeneous and heterogeneous media. Spatial resolution formulas are validated for both homogeneous and heterogeneous velocity models. For both models, the resolution of reverse-time migration images are consistent with the estimated resolution limits with respect to the Rayleigh's resolution criterion. A long wavelength corresponds to a high velocity value and therefore both vertical and horizontal resolution limits tend to degrade with depth in the heterogeneous model â€" deeper regions have higher velocity values.

WAVEFORM of in ACOUSTIC WAVEFO.RM INVERSION OF TWO-DIMENSIONAL GULF OF MEXICO DATA by Chaiwoot Boonyasiriwat A thesis submitted to the faculty of The University of Utah in partial fulfillment of the requirements for the degree of Master of Science In Geophysics Department of Geology and Geophysics The University of Utah May 2009 © Copyright © Chaiwoot Boonyasiriwat 2009 All Rights Reserved of a thesis submitted by Chaiwoot Boonyasiriwat satisfactory. THE UNIVERSITY OF UTAH GRADUATE SCHOOL SUPERVISORY COMMITTEE APPROVAL This thesis has been read by each member of the following supervisory committee and by majority vote has been found to be satisfactory. ~~ ~~ Ronald L. Bruhn Chaiwoot Boonyasiriwat in its final form and have format, the * . • ^ Dat/ ^ /^Gerard T. Schuster / Approved for the Major Department Marjorie Chair/ Dean Approved for the Graduate Council Dean of The Graduate School . THE UNIVERSITY ·OF UTAH GRADUATE SCHOOL FINAL READING APPROVAL To the Graduate Council of the University of Utah: I have read the thesis of found that (1) its format , citations, and bibliographic style are consistent and acceptable; (2) its illustrative materials including figures, tables, and charts are in place; and (3) the final manuscript is satisfactory to t he Supervisory Committee and is ready for submission to The Graduate School. ~~4T~ /Gerard Chair, Supervisory Committee ~~ a:A-Marjorie A. Chan Dean ~~~.CQ-f--' David S. Chapman tracing. Moreover, each trace contains both shallow and deep reflection arrivals and so is richer in information content than the first arrival traveltime. As a result, waveform tomography provides a more detailed estimate of the velocity medium than does traveltime tomography. However, acoustic waveform tomography is a highly nonlinear inverse solution that requires a good starting velocity model compared to the smooth starting model that is sufficient for traveltime tomography. multiscale acoustic method with a dynamic early-arrival muting window to mitigate the local minima problem of waveform tomography and elastic effects in the data. This multiscale approach is denoted as multiscale waveform tomography (MWT), and is first validated with synthetic acoustic data from the 2D SEG/EAGE salt model. Using the traveltime velocity tomogram as an initial model, MWT fails to converge to the global minimum. I found that the flooding technique normally used in subsalt migration can be used to significantly improve the convergence of MWT. The velocity model recovered by MWT using the flooding technique is more accurate and highly resolved than that obtained using conventional MWT. For the marine data, MWT can provide a more accurate velocity model than the initial model from traveltime tomography. The accuracy of the waveform velocity model is verified by comparing migration images and common image gathers. \ ABSTRACT Acoustic waveform tomography can provide a velocity model with higher ac­curacy than traveltime tomography because the forward modeling is based on the complete wave equation rather than the high-frequency approximation of ray To overcome the nonlinear problems in waveform tomography, I develop a novel multiscale waveform tomography method and apply it to both synthetic and marine seismic data from the Gulf of Mexico. The inversion process is carried out using a mitigate SEG /EAG E subsalt. model from traveltime tomography. The accuracy of the waveform velocity model is verified by comparing migration images and common image gathers. I also present ray-based spatial resolution formulas for migration and inversion .. with numerical tests on both homogeneous and heterogeneous media. Spatial resolution formulas are validated for both homogeneous and heterogeneous velocity models. For both models, the resolution of reverse-time migration images are consis­tent with the estimated resolution limits with respect to the Rayleigh's resolution criterion. A long wavelength corresponds to a high velocity value and therefore both vertical and horizontal resolution limits tend to degrade with depth in the heterogeneous model - deeper regions have higher velocity values. v numeriCal . ,reverse:-time l(mits resp~ct _- v TToo mmyy ppaarreennttss aanndd mmyy wwiiffee CONTENTS iv viii x 1 Introduction Theory Tomography Tomography Model Mexico Introduction Results Model 2.3.2 REFERENCES CONTENTS ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. IV LIST OF FIGURES . .......... . . . . ..................... . . . .... Vlll ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. CHAPTERS 1. ACOUSTIC WAVEFORM TOMOGRAPHY . . . . . . . . . . . . . . . .. 1.1 Introduction. . ............. . .......... . .. . . .. .... . . . .... 1 1.2 Theory....... . ................... . .......... . . . . ... .. . 3 1.2.1 Acoustic Waveform Tomography. . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.2 Multiscale Early-Arrival Waveform Tomography. . . . . . . . . . .. 5 1.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 6 1.3.1 2D SEG/EAGE Salt Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3.2 Gulf of Mexico. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2. SPATIAL RESOLUTION ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . 20 2.1 Introduction.. . .. . ..... .. .. .. .. .. . .. ...... . ... .. ... ..... 20 2.2 Spatial Resolution Formulas . . .. .. .. . ........ . . .. .. ..... ... 20 2.3 Numerical Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 22 2.3.1 Homogeneous Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 22 2.3 .2 Smoothed SEG/EAGE Salt Model . . . . . . . . . . . . . . . . . . . . .. 25 3. CONCLUSIONS .. ... .. .. .... ......... .. . ....... . ......... 30 REFERENCES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 FIGURES models, model, v(z) model model, Original 20-data, data, data data, model, tomogram, v(z) technique process, model, flood, tomogram floods data Waveform data 1.7 data, x = filtered tomogram, gather obtained by using the waveform tomogram 14 data, tomography, tomogram tomography, tomogram data, tomogram, to­mogram the obtained tomogram model LIST OF FIGURES 1.1 The 2D SEG/EAGE salt model and initial models. a) The salt model. b) Traveltime velocity model. c) 'o(z) velocity model. ............. 7 1.2 Synthetic data from the 2D SEG/EAGE salt model. a) Original20-Hz data. b) Filtered 2.5-Hz data. c) Filtered 5-Hz data. . . . . . . . . . . . . .. 8 1.3 Waveform inversion results using 2.5-Hz data. a) True model. b) Waveform tomogram using the traveltime tomogram. c) Waveform tomogram using the 'o(z) model and the flooding technique. . . . . . . .. 9 1.4 Flooding process. a) Waveform tomogram using only the v(z) model. b) Waveform tomogram after a salt flood. c) Waveform tomogram after salt and sediment floods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 11 1.5 Waveform residual plot using 2.5-Hz data. . . . . . . . . . . . . . . . . . . . . .. 12 1.6 \V"aveform residual plot using 5-Hz data. . . . . . . . . . . . . . . . . . . . . . .. 13 1. 7 Marine data. (a) An original CSG from a source at = 0 km. The white line is the picked first-arrival traveltimes. (b) A filtered shot gather with a passband of 0-25 Hz. (c) A predicted shot gather obtained by using the traveltime tomogram. (d) A predicted shot tomogram. .. . . . . . . . . . . . .. 1.8 Inversion results from the marine data. (a) The initial velocity model obtained from traveltime tomography. (b) The velocity tomogram obtained from waveform tomography. (c) The vertical derivative of the waveform tomogram. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 16 1.9 Migration images from the marine data. (a) The Kirchhoff migration image obtained using the original data and the traveltime tomogram. (b) The Kirchhoff migration image obtained using the waveform to-mogram ................................................. 17 1.10 Zoomed views of migration images from the marine data. Using the traveltime tomogram, the Kirchhoff migration images in a) the solid box and b) the dashed box are obtained. Using the waveform tomogram, the Kirchhoff migration image in c) the solid box and d) the dashed box are obtained. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 18 1.11 Common image gathers (CIGs) obtained from the marine data mi­grated with the (a) traveltime tomogram and (b) waveform tomogram as the velocity model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 19 T8/rg The ks k9 . 22 Wavenumber nine model estimate loca­tions model SEG/scatterers SEG/model scat­terers = scatterers z = scatterers ix 2.1 The illumination of a diffractor at x by source/ receiver pair r s/rg in a heterogeneous medium. The. wavenumber vector k is composed of the . source-side wavenumber vector ks and the receiver-side wavenumber vector kg. . . .... . ..... 00 • • ••• • ' •••••••••••• •••••• ' . • • • • • • • • •• .22 2.2 \;Vavenumber illumination patterns at the n.ine scatterer locations in the homogeneous model. .. . ......... ...... ... .. ... . . .... . . . 23 2.3 The homogeneous velocity model with resolution bars that estimate the limits of horizontal and vertical resolution at nine different loca-tions in the model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 24 2.4 The reverse-time migration image for the homogeneous velocity model. 25 2.5 Smoothed SEG /EAGE salt model with scatterers. . ... . . . . ... .. . . 26 2.6 Wavenumber illumination patterns at the nine scatterer locations in the smoothed SEG /EAGE salt model. The traveltimes associated with the computed wavenumbers were computed by tracing rays through the salt model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.7 Reverse-time migration image for the smoothed salt model with scat-terers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 28 2.8 The reverse-time migration image line at x 8 km in the 2D SEG/EAGE salt model. The error bars indicate the estimated vertical resolution limits at the scatterers. ................ ..... ........ .. . . .. . 29 2.9 The reverse-time migration image line at = 2 km in the 2D SEG/ EAGE salt model. The error bars indicate the estimated horizontal resolution limits at the scatterers. ... . . . ..... . .......... . . ... . . . .... . . 29 IX ACKNOWLEDGMENTS T. I also thank Paul Valasek and Partha Routh for their advice, and the Seismic Technology Group for their help during my internship at ConocoPhillips. I am grateful for the support from the members of the University of Utah Tomography and Modeling/ Migration Consortium and thank Amerada Hess for providing me with the Gulf of Mexico data set. I would like to thank the Development and Promotion of Sciences and Technology (DPST) of Thailand for financial support. Finally, I would like to thank UTAM colleagues especially Weiping Cao for useful discussion and help. I would like to thank Dr. Gerard T . Schuster and my committee members Dr. Ronald Bruhn and Dr. Richard Jarrard for their advice and constructive criticism. t hank financial support. Wei ping useful 1.1 Introduction accurate conventional prestack depth migration methods yield unreliable migration images unless the velocity model is accurate. traveltime and migration velocity analysis (MVA) (Stork, 1992; Tieman, 1995; Jiao et al., 2002; Sava et al., 2005; Sava and Vlad, 2008). Inverting only first-arrival traveltimes, traveltime tomography is computationally efficient but forward models the data by ray tracing, which is a high-frequency approximation that conflicts with the band-limited nature of seismic sources. The widely used MVA methods usually require intensive quality control and typically provide just a smooth velocity model. In contrast, waveform tomography directly inverts the seismic waveform data and can provide accurate and highly resolved velocity models (Bunks et al., 1995; Zhou et al, 1995, 1997; Sheng et al., 2006). et CHAPTER 1 ACOUSTIC WAVEFORM TOMOGRAPHY Accurate velocity models of subsurface structures are required to obtain reliable migration images of the subsurface. In areas with rugged surface topography and complex near-surface structures, conventional time-domain seismic imaging methods are likely to fail due to the severe statics problem. Using an accurate velocity model, a tomostatics method can improve the quality of the stacked section. The role of velocity models is more crucial in depth-domain seismic imaging where The standard tools for more accurate velocity estimation include traveltime tomography (Zhu and McMechan, 1989; Luo and Schuster, 1991; Pratt and Goulty, 1991; Schuster and Quintus-Bosz, 1993; Nemeth et al., 1997; Min and Shin, 2006) 1992: a1., traveltimcs, al., et Waveform tomography is a highly nonlinear inverse problem, and tends to con­verge to a local minimum if the starting model is not in the vicinity of the global minimum (Gauthier ct al., 1986). Therefore, a good initial velocity model is nonlinearity band-reconstructed than at high frequencies. This means that the multiscale inversion tends to converge to the global minimum (Sirgue and Pratt, 2004). At low frequencies, coarser grids can be used for computing numerical solutions of the wave equation than at high frequencies resulting in a greater computational efficiency. synthetic data from the 2D SEG/EAGE salt model and marine data from the Gulf of Mexico. There are two novel features in my procedure: first, I employ a multiscale approach in the space-time domain which departs from the conventional approach in the space-frequency domain. The possible advantage is that the 3D space-time approach can be more efficient than the 3D space-frequency method. Second, I use a flooding method to overcome the convergence problem when there is a large velocity contrast in the medium. The 2D synthetic data were generated using an acoustic finite-difference code, and results show that MWT with a flooding technique can successfully invert the salt-body data for an accurate and highly resolved velocity tomogram. In the case of marine data from the Gulf of Mexico, I reduce the elastic influences in the data by using MWT with a dynamic early-arrival muting window. Most seismic events in the data are muted except the early arrivals. My results suggest that the multiscale method can provide a velocity tomogram that is significantly more accurate than the initial model. 2 required by waveform tomography to partially overcome the local minima prob­lem. To further mitigate the nonlinearIty of waveform tomography, Bunks et al. (1995) introduced a multiscale method that sequentially inverts data band-passed from lower to higher frequencies. Once the smooth or low-wavenumber velocity structures are reconstructed, the higher-wavenumber structures are reconstructed using higher-frequency data. The multiscale approach mitigates the local minima problem and is computation­ally efficient. The nonlinearity of waveform tomography depends on the frequency content of seismic data so that the misfit function at low frequencies is more linear In this thesis, I apply multiscale waveform tomography (MWT) to both synthetic MvVT muting window. Most seismic events in the data are muted except the early arrivals. My results suggest that the multiscale method can provide a velocity tomogram that is significantly more accurate than the initial model. 1.2 Theory Tomography given = s(r,t|r5), 1.1) 1 <92p(r,t|rs) _ _ «(r) dt2 1 .P(r) •Vp(r,t|rs) p(r,t\rs) r rs; K(T) p(r) t|rs) t\rs) f t\r\ 0) * t\rs)dr', (1.2) JVQ t\r',0) VQ symbol * represents temporal convolution. In practice, the solution t\rs) is finite-4^-accuracy in space and 2n d-order accuracy in time (Levander, 1988). by Tarantola (1984). The data residual is defined as Sp(rg,t\rs) = [pobs(rg,t\rs) - pcaic(rg, t\rs)]m{rg, t\rs), (1.3) where rg is a receiver position vector, pobs(rg.t\rs) and pcaic{^g, t\rs) are, respec­tively, the observed and calculated data, and m(rg,t\rs) is an early-arrival window function. The velocity model is updated by minimizing the misfit function which is the Z/2 norm of the data residuals and is given by E=IY:^: f(6p(vs,t\va))2dt. i.4) 1 s g J A nonlinear preconditioned conjugate-gradient method (Sheng et al., 2006) is used to minimize the misfit function and obtain a velocity model r) - K(Y)/p{r). The gradient of the misfit function in equation 1.4 with respect to slowness perturbation is computed by the zero-lag correlation between the forward-propagated wavefields 3 1.2.1 Acoustic Waveform Tomography In this section, I present the theory of time-domain acoustic waveform tomog­raphy. The acoustic wave equation is used to describe wave propagation, glVen by 1 [J2p(r, tlrs) [ 1 l1:(r) 8t2 - \7. p(r) \7p(r, tlrs) = s(r, tlrs), (1.1 ) where p(r, tlrs) is a pressure field at position r at time t excited by a source at rs; K,(r) and p(r) are the bulk modulus and density distributions, respectively; and s(r, tlrs) is the source function. The solution to equation 1.1 can be written as p(r, tlrs) = r G(r, tlr', * s(r', tlrs)dr', JVa where G(r, tlr', 0) is the Green's function, Vo denotes the model volume, and the * p(r, tlrs) computed using a staggered-grid, explicit finite-difference method with 4th-order 2nd-order The inversion scheme used in this work is based on the adjoint method proposed rg pobs(rg, tlrs) Pcalc(rg, tlrs) r g, tlrs) L2 (1.4) gradient v( r) = J 11:( r) / p( r). . , 4 et ah, 9(r) ^ 4 T E 7 * l r * ) p ' ( r , ^ d t , (1.5) p r, t|rs) forward-t\rs) p'(r, t\rs) = I G(r, -t\r', 0) 6s(v', t\rs)dr', (1.6) Ss(r\t\rs) = 528(r'-r9)8p(rg,t\rs). (1.7) 9 dfc = -Pfcgfc + Afcdfc-i, (1.8) = 1 , 2 , f c m a x , al., d0 = -go- ft g[ • (Pfcgfc - Pfc-igfc-i) gfc_i • Pfc-igfc- ft = (1.9) vk+1(r) = vk(r) + Af c4(r), (1.10) Xk 1999), c4(r) dfc back-projection Vo(r) is obtained from traveltime tomography with dynamic smoothing filters (Nemeth et al., 1997). and the back-projected wavefield residuals (Tarantola, 1984; Luo and Schuster, 1991; Zhou et al., 1995, 1997; Sheng 'et' aL, 2006) g(= _(2) 2: Jp(r , tlrs)p'(r, tlrs)dt, v r s . (1.5) where p denotes the time derivative of p, p(r, tlr s) represents the forward-propagated wavefields, and p'(r, tlrs) represents the back-projected wavefield residuals given by p'(r, tlrs) J G(r, -tlr', 0) * os(r', tlrs)dr', (1.6) and the perturbed slowness field is os(r', tlrs) = 2: o(r' - rg)op(rg , tlrs). (1.7) The velocity model is iteratively updated along the conjugate directions defined by (1.8) where iterations k 1,2, ... , kmax , gk is the gradient direction, and P is the conventional geometrical-spreading preconditioner (Causse et al. , 1999). At the first iteration do -go. The parameter {3k is obtained using the Polak-Ribiere formula (Nocedal and Wright, 1999) f3 - g[ . (Pkgk - Pk-1 g k - 1) k - T P g k - l· k-lgk-l (1.9) The velocity model is updated by (1.10) where Ak is the step length, which is determined by a quadratic line-search method (Nocedal and Wright, 1999) , and dk(r) is the component of the conjugate-direction vector dk at position r. At each iteration, one forward-propagation and one back­projection are needed for computing the gradient direction. Additional forward propagations are required for the line search. In this work, the starting model va (al. , Arrival Tomography domain MWT). al. al. ah, , . v WTARGET(u)WlIGINAL(L>) JWIENER{U) - -R- ' 5-, l1 - 1 1; \vvoriginai(u)\2 + e2 fWIENER Wiener WORIGINAI WTARGET 1S the u e dx , . Xmin ^ CM IN dx < - < - - , (1.12) ^ °JMAX where X m i n is the minimum wavelength, cm i n is the minimum velocity, and fmax is the maximum frequency of the band. At low frequencies, coarser grids can be used than at high frequencies. Therefore, low-frequency inversions will be fast and efficient compared to high-frequency inversions, and can afford to take a large I 5 1.2.2 Multiscale Early-Arrival Waveform Tomography In this section, I describe a time-do~ain implementation of multiscale waveform tomography (M\iVT). In the frequency domain, the data are decomposed into separate frequency components and it is straightforward to apply the multiscale method. In contrast, time-domain inversion simultaneously uses multiple frequency components of the data in a frequency band. Thus, the data must be band-pass fil­tered into multiple frequency bands with various peak frequencies, and the inversion process can proceed using low-frequency data and then high-frequency data. A Wiener filter is used for low-pass filtering the data, where Boonyasiriwat et a1. (2009) show that Wiener filtering is more accurate than the filtering method proposed by Bunks et a1. (1995). A low-pass Wiener filter (Boonyasiriwat et al., 2009) can be computed by f . () = Wta'rget (w ) WJriginal (w ) W tener W 1 W () 12 2' original W + E (1.11) where fWiener is the \iViener filter, l¥original is the original wavelet, vVtarget is the low-frequency target wavelet, W is an angular frequency, f is a damping factor to prevent numerical instability, and t denotes the complex conjugate. The Wiener filter is applied to the source wavelet and data in the frequency domain. Once the source and data are filtered to a low frequency band, the spacing of the finite-difference grid points can be determined by the maximum frequency of the band. The numerical dispersion condition for the finite-difference scheme used in this work requires at least 5 grid points per minimum wavelength (Levander, 1988). A square grid (dx = dz) is utilized in the finite-difference scheme so that the grid spacing d:c is determined by dx< Am' m<c~· - 5 - 5}' , max (1.12) where Amin is the minimum wavelength, Cmin is the minimum velocity, and fmax is the maximum frequency of the band. At low frequencies, coarser grids can be used than at high frequencies. Therefore, low-frequency inversions will be fast and efficient compared to high-frequency inversions, and can afford to take a large al., radiation by using a multiscale method with a dynamic early-arrival muting window, which partly corrects for attenuation and wavelet distortion effects. The inversion initially inverts data low-pass filtered and windowed about a short time-window. After some iterations, higher-frequency data are used in the inversion with the same muting window. Then, the inversion proceeds with longer time-windows. 1.3 Numerical Results SEG/EAGE receivers are located along the free surface and I employ two initial models. The first one is from traveltime tomography (Figure 1.1b) and the second one is the v(model (Figure 1.1c) obtained from a smooth ID sediment velocity profile of the true model. The original data are generated using a 20-Hz Ricker wavelet. A common shot gather from a source at the center of the free surface is shown in Figure 1.2a. To use the multiscale method, the original data are low-pass filtered to 2 frequency bands with peak frequencies of 2.5 Hz and 5 Hz; the filtered data are shown in Figures 1.2b and 1.2c, and the MWT results are shown in Figure 1.3. 6 number of iterations in order to obtain an accurate estimate of low-wavenumber components in the velocity model. The multiscale approach has the ability to mitigate the local minima problem commonly encountered in waveform tomography (Bunks et al., 1995; Boonyasiriwat et a1., 2009). The velocity model with accurate low-wavenumber components is a good initial model for higher-frequency inversions while higher-frequency data progressively recover the higher-wavenumber parts of the model. In practice, my acoustic modeling does not account for elastic effects in the data, attenuation, unknown density, unknown source wavelet, and source radiation patterns, which can lead to a poor convergence. I try to partly overcome these prob­lems To demonstrate its effectiveness, I apply MWT to synthetic data from the 2D SEG/EAGE salt model and to marine data from the Gulf of Mexico. 1.3.1 2D SEGjEAGE Salt Model The 2D SEG/EAGE salt model (Figure 1.1a) has dimensions of 16 km x 3.7 km with a source spacing of 40 m and a receiver spacing of 20 m. Sources and 1.1 b) v (z) 1D . , X(km) X(km) c) v(z) Model F i g u r e models, model, model, v(z) model. 7 a) SEG Salt Model Velocity (m/s) 0 4500 4000 _1 E 3500 C. £2 3000 0aQ ). ----- - 2500 3 -. 2000 0 2 4 6 8 10 12 14 16 1500 X (km) b) Traveltime Tomogram Velocity (m/s) 4500 4000 -1 E 3500 C. 3000 2500 2000 0 2 4 6 8 10 12 14 16 1500 X (km) c) v(z) Model Velocity (m/s) 0 4500 4000 -1 E 3500 C. £2 3000 a. Q) 0 2500 3 2000 0 2 4 6 8 10 12 14 16 1500 X (km) Figure 1.1. The 2D SEG/EAGE salt model and initial models. a) The salt modeL b) Traveltime velocity modeL c) v(z) velocity modeL 8 Hz SEG/EAGE model, Original 20-Hz data, data, Hz o (l) E i=4 o -.2 -(/-) (l) E i=4 o -.2 -(/-) (l) E i=4 6 -8 -6 -4 -6 -4 -6 -4 a) Original 20-Hz Data -2 0 2 4 6 8 Offset (km) b) Filtered 2.5-Hz Data -2 0 2 4 6 8 Offset (km) c) Filtered 5-Hz data -2 0 2 4 6 8 Offset (km) Figure 1.2. Synthetic data from the 2D SEGjEAGE salt model. a) Original20-Hz data. b) Filtered 2.5-Hz data. c) Filtered 5-Hz data. a) True Velocity Model Velocity (m/s) 2 4 6 8 X(km) b) Waveform Tomogram using Traveltime Tomogram Velocity (m/s) 6 8 X(km) c) Waveform Tomogram using v(z) Model and Flooding Technique Velocity (m/s) .4500 Figure 1.3. 2.5-Hz data, model, b) Waveform tomogram using the traveltime tomogram, c) Waveform tomogram using the model and the flooding technique. -1 E C :E. 2 ---___ Q) --- o 3 -- o 10 12 14 X (km) o -1 E -.::.t! £a.2 Q) o 3 o 2 4 10 12 14 X (km) 9 4500 16 4000 3500 3000 2500 2000 1500 4500 16 4000 3500 3000 2500 2000 1500 -1 E -.::.t! £a.2 Q) 0 3 0 2 4 6 8 10 X (km) 12 14 16 4500 4000 3500 3000 2500 2000 1500 Waveform inversion results using data. a) True model. travelt ime tomogram. v(z) Using the traveltime tomogram (Figure 1.1b), MWT converges to a local minima and provides an inaccurate velocity tomogram (Figure 1.3b). Using the model and the flooding techniques, MWT can provide an accurate and highly resolved velocity tomogram (Figure 1.3c). yield of m/s below the salt top and the resulting velocity (Figure 1.4b) is used to migrate the data. The bottom of the salt body is picked from the migration image (not shown here) and the velocity model in Figure 1.4b is flooded with a sediment velocity of 3000 m/s below the salt bottom to obtain the velocity model shown in Figure 1.4c. Then the velocity model obtained after the flooding process is used in the inversion process to give the residual plot using 2.5-Hz data shown in Figure 1.5. The model is used in the first 100 iterations and the flooding model (Figure 1.4c) is used in the last 30 iterations. The waveform tomogram is then is used to invert 5-Hz data and the final velocity model is obtained as shown in Figure 1.3c. The residual plot using 5-Hz data is shown in Figure 1.6. 1.3.2 Gulf of Mexico A streamer data set from the Gulf of Mexico was acquired using 515 shots with a shot interval of 37.5 m, a time-sampling interval of 2 ms, a trace length of 10 s, and 480 active hydrophones per shot. The hydrophone interval is 12.5 m with a near offset of 198 m and a far offset of about 6 km. The data are transformed from 3D to 2D format by applying the filter Ji/u in the frequency domain to correct for 3D geometrical spreading (Zhou et al., 1995). The attenuation factor is estimated by the spectral ratio method (Maresh et al., 2006), and the attenuation effect is compensated by applying an inverse-Q filter (Wang, 2006) to the data. The source wavelet is estimated by stacking along the water-bottom reflection. Then, the data are low-pass filtered to 2 frequency bands with passbands of 0-15 Hz and 0-25 Hz. Figure 1.7a and 1.7b show the original and filtered shot gathers, respectively. 10 LIb), t'omogram v(z) Now I describe how the flooding technique can y iel-d an accurate tomogram. Using the v( z) model, MWT yields the tomogram in Figure 1.4a. After the top of the salt boundary is picked, the salt velocity of 4500 mls is used to flood the region l.4b of mls l.4c. t he v(z) t he l .4c) Ji lw al. , Q al. , water-bottom reflection. Then, the data are low-pass filtered to 2 frequency bands with passbands of 0-15 Hz and 0-25 Hz. Figure 1.7a and 1.7b show the original and filtered shot gathers, respectively. X(km) b) Waveform Tomogram after Salt Flood X(km) 4500 Velocity (m/s) 4500 Velocity 4500 F i g u r e 1.4. Flooding process, a) Waveform tomogram using only the model, b) Waveform tomogram after a salt flood, c) Waveform tomogram after salt and sediment floods. 11 a) Waveform Tomogram using v(z) Model Velocity (m/s) 0 -1 E ..l<: --- £2 0- w 0 3 0 2 4 6 8 10 12 14 16 X (km) 0 -1 E ~ £2 0- w 0 3 0 2 4 6 8 10 12 14 16 X (km) c) Waveform Tomogram after Sediment Flood (m/s) 00 00 -1 E ..l<: --- £2 0- w 0 3 0 2 4 6 8 10 12 14 16 X (km) Figure process. v(z) model. flood. Waveform Residual using 2.5-Hz Data 70 I 1 1 1 1 r 11 i i i i 1 1- 0 40 120 Iteration Number 70r------r------~----_.------~----~r_----_r__. ro ::l "0 60 50 .~ 40 0:: E.... -$ 30 >ro ~ 20 10 OL-----~------~------~----~-------L------~~ o 20 60 80 100 120 Figure 1.5. Waveform residual plot using 2.5-Hz data. 12 Waveform Residual using 5-Hz Data 351 1 1 1 - 0 10 20 50 Iteration Number 35.--------.--------.--------.--------~-------. 30 co 25 :::I -0 ·iii Q) 0:: .E-2- 20 Q) c>o ~ 15 10 5~------~--------~--------~--------L-------~ o 30 40 Figure 1.6. Waveform residual plot using 5-Hz data. 13 CSG 0 25 CSG 0 25 CSG 0 data, CSG x = arrival traveltimes. traveltime tomogram, 0 1 .- -C-/) QE) 2 i= 3 o 1 .- -C-/) Q) 2 E i= 3 a) Original CSG 2 4 Offset (km) c) 0-25 Hz TRT Predicted CSG 2 4 Offset (km) 0 1 .- -C-/) QE) 2 i= 3 o 1 Q) 2 E i= 3 14 b) 0-25 Hz Observed CSG 2 4 Offset (km) d) 0-25 Hz MWT Predicted CSG 2 4 Offset (km) Figure 1.7. Marine data. (a) An original eSG from a source at = 0 km. The white line is the picked first- arrival t raveltimes. (b) A filtered shot gather with a passband of 0-25 Hz. (c) A predicted shot gather obtained by using the traveltime tomogram. (d) A predicted shot gather obtained by using the waveform tomogram. wave­form resolution than the initial model (Figure 1.8a). To reveal the high-wavenumber details of the waveform tomogram, the derivative with respect to depth is applied to the waveform tomogram (Figure 1.8c). The predicted shot gathers obtained by using traveltime tomogram and waveform tomogram are shown in Figures 1.7c and 1.7d, respectively. traveltime and waveform tomograms are shown in Figures 1.9a and 1.9b, respectively. The zoomed views of the migration images are shown in Figure 1.10 for more detailed comparisons. Using the waveform tomogram as the migration velocity, the resulting migration image appears to be better focused than that obtained by using the traveltime tomogram as the migration velocity. Comparing the CIGs in Figure 1.11, the waveform tomogram is more accurate than the traveltime tomogram since the corresponding CIGs are flatter. Horizontal reflectors in a common image gather are an indication that the migration velocity model is accurate (Yilmaz, 2001). 15 Traveltime tomography is utilized to provide an initial velocity model for wave­fo~ m tomography (Figure 1.8a). The inversion process is composed of 3 parts and in each part a muting window with a different length is used for inversion of both the low-pass and high-pass data. In the first part, a muting window of length 1 s is applied to the filtered data, and the inversion sequentially proceeds using data with passbands of 0-15 Hz and 0-25 Hz. The reconstructed velocity from the first part is used as an initial model in the second part where a 2-second window is used, and, in the last part, a 3-second window is applied. Figure 1.8b shows the reconstructed velocity tomogram from waveform tomography which has a higher 1. 7 d, To verify that the reconstructed velocity tomogram is more accurate than the initial model, I compare the migration images and common image gathers (CIG) obtained by using the initial model and the final model. The original data were migrated using Kirchhoff migration, and the migration images using the traveltime resulting flatter. Velocity (m/s) -50 Figure 1.8. data, obtained from traveltime tomography, (b) The velocity tomogram obtained from waveform tomography, (c) The vertical derivative of the waveform tomogram. 16 a) Traveltime Velocity Tomogram 0 3000 0.5 - 2500 CE 1 -..r:: a. ~ 1.5 2000 2 2.5 1500 0 5 10 15 20 X (km) b) Waveform Velocity Tomogram Velocity (m/s) 0 2800 0.5 2600 E 1 2400 C -..r:: 2200 a. ~ 1.5 2000 2 1800 1600 2.5 0 5 10 15 20 X (km) c) Vertical Derivative of Waveform Velocity Tomogram 0 250 0.5 200 E 150 ~ - 1 -..r:: 100 a. ~ 1.5 50 2 2.5 -50 0 5 10 15 20 X (km) F igure Inversion results from the marine data. (a) The initial velocity model tomography. tomography. tomogram. a) Migration Image using Traveltime Tomogram 8 12 14 16 18 X(km) b) Migration Image using Waveform Tomogram X(km) Figure 1.9. Migration images from the marine data, (a) The KirchhorT migration image obtained using the original data and the traveltime tomogram, (b) The Kirchhoff migration image obtained using the waveform tomogram. 17 TraveHifne 0.5 ..--.. 1 E 6 J:: Q. 1.5 Q) 0 2 2.5 0 2 4 6 10 20 X (km) 0.5 ..--.. 1 E ~ --- J:: 0. 1.5 Q) 0 2 2.5 0 2 4 6 8 10 12 14 16 18 20 X (km) data. Kirchhoff tomogram. a) TRT Migration Image in Solid Box b) TRT Migration Image in Dashed Box (km) X (km) (km) X (km) 1.1 1.4 3 4 5 6 X c) MWT Migration Image in Solid Box 1.1~~m 1.4 1.5 2 3 4 5 6 X 18 3 4 5 6 d) MWT Migration Image in Dashed Box 1.9~~:;::;; 2 ~ 2.1 .c a. ~ 2.2 2.3 2.4 3 4 5 6 Figure 1.10. Zoomed views of migration images from the marine data. Using the traveltime tomogram, the Kirchhoff migration images in a) the solid box and b) the dashed box are obtained. Using the waveform tomogram, the Kirchhoff migration image in c) the solid box and d) the dashed box are obtained. 19 a) CIGs using Traveltime Tomogram X(km) b) CIGs using Waveform Tomogram X(km) Figure 1.11. Common image gathers (CIGs) obtained from the marine data migrated with the (a) traveltime tomogram and (b) waveform tomogram as the 19 · 0.5 .-... 1 E -~- .s:: a. 1.5 Q) 0 2 2.5 0 2 4 6 8 10 12 14 16 18 X (km) 0.5 .-... 1 E 6 .s:: a. 1.5 Q) 0 2 2.5 0 2 4 6 8 10 12 14 16 18 X (velocity model. 2.1 Introduction The theory of spatial resolution is well-established in seismic migration and inversion (Beylkin et al., 1985; Cohen et al., 1986; Bleistein, 1987; Vermeer, 1999; Chen and Schuster, 1999). Using the Born approximation, Beylkin et al. (1985) presented a formula that connects the source frequency and acquisition geometry to the spatial resolution of the migration image. This spatial resolution formula is quite simple and, in theory, applicable to heterogeneous media. In contrast, Chen and Schuster (1999) used the far-field approximation to derive spatial resolution limits for homogeneous media so that the resolution formula of Beylkin et al. (1985) is more general. Consequently, I choose to apply the formula of Beylkin et al. (1985) for resolution analysis tests in this work. 2.2 Spatial Resolution Formulas In this section I present some formulas that map acquisition geometry configu­ration to spatial resolution in both the wavenumber and the space domains. Since 2D surface-seismic-profile (SSP) data are only dealt with in this work, I will limit my derivation for 2D SSP acquisition geometry. The source/ receiver configuation can be described by coordinates £ so that for fixed X, rs (X, 0) and r9 (£,0) describe a 2D common-shot gather. Beylkin et al. (1985) derived a formula to compute the acoustic impedance con­trast as a function of position r from seismic measurements with limited aperture. In 2D the limited aperture is defined by the range of £. Their formula is a mapping from (£, a;) (the coordinates of the observed data; LJ is the angular frequency of CHAPTER 2 SPATIAL RESOLUTION ANALYSIS 2 .1 Introduction al. , al. , Schust er , t he resolut ion t o for resolution analysis t ests in this work. t he dat a t his receiver ~ X , rs = X ,O) rg = (C apert ure ~. ( ~ , w) dat a; w of the source signature) to (kx,kz) (the coordinates of the migration image). This mapping is given by r)=a;V0(r, O, (2.1) where k = (kX) kz) is the wavenumber vector in the reconstructed (migration) do­main, and (f)(r, {) is the traveltime surface of a difTractor in r for shot/receiver pairs described by £. The traveltime surface 0(r,£) is computed from the background velocity model and V0(r, £) represents the derivative of 0(r, f) with respect to the point of reconstruction r. Equation 2.1 determines the region of coverage D in the spatial wavenumber domain which is the estimate of spatial resolution (Beylkin et al., 1985; Vermeer, 1999). Vermeer (1999) states, "the larger the region of coverage in k, the better the spatial resolution." Equation 2.1 makes resolution analysis quite simple since resolution can be obtained by analyzing the spatial gradients of the diffraction traveltime surface 0(r, f) in the given experiment configuration (Vermeer, 1999). The maximum wavenumber that corresponds to the maximum gradient of <f>(r, £) gives a fair indication of resolution. The diffraction traveltime 0(r, f) can be described as 0(r, 0 = r(r, r.) + r(r, rg) = rs + r9, (where r(x, y) is the traveltime from surface position y to subsurface position x. Similarly, k can be written as the vectorial sum k = k5 + k5 , where ks and kff are the contributions of shot and receiver, respectively, to the wavenumber vector k as illustrated in Figure 2.1. Each shot/receiver pair in the acquisition geometry corresponds to a point k in the wavenumber space. Taking all shot/receiver pairs of a configuration leads to a collection of points in wavenumber space. Using the Beylkin's formula, horizontal resolution Ax and vertical resolution Az can be written as 21 kx, kz) k(r) = w\7¢(~), (2.1) = kx, kz) ¢(r, 0 is the traveltime surface of a diffractor in r for shot/receiver pairs by~. ¢(r,O \7 ¢( r,~) ¢( 0 smce ¢(O ¢(r,O gives a fair ¢(r,~) (2.2) T(X, (2.3) ks kg ~x ~z . , Figure 2.1. The illumination of a diffractor at x by source/ receiver pair r s/r9 in a heterogeneous medium. The wavenumber vector k is composed of the source-side wavenumber vector ks and the receiver-side wavenumber vector k 9. and Ax = 77 (2.4) max \kx{r, £, u)\ Az= -4 TT. max \kz(r. £, u)\ where kx and kz are the horizontal and vertical components of the wavenumber vector k. 2.3 Numerical Results 2.3.1 Homogeneous Model I first validate the resolution formulas for a homogeneous model (velocity = 2000 m/ s) with a source spacing of 20 m and a receiver spacing of 10 m. Sources and receivers are located along the free surface, and there are nine scatterers in the model. The wavenumber illuminations for this model are shown Figure 2.2 and the estimated spatial resolutions are given in Table 2.1 and illustrated in Figure 2.3 as 22 s/ rg ks g. 7r 6.x = ---.,.----,- max Ikx(r,~, w)I ' (2.4) 7r 6.z = ---.,.---.,.---.,. max Ikz(r, ~ , w)I ' (2.5) kx kz Results a) x = 200 m, z = 200 m b) x = 500 m, z = 200 m c) x = 800 m, z = 200 m E 0.05 O CD ,N 0.1 E 0.05 T3 CD . N 0.1 0 0.2 k (rad/m) d) x = 200 m, z = 500 m E 0.05 "D CD , N 0.1 k x (rad/m) e) x = 500 m, z = 50 0 0 0.2 k (rad/m) f) x = 800 m, z = 500 m 0.1-50 .2 0 0.2 k (rad/m) g) x = 200 m, z = 800 m 0.15 0 0.2 k (rad/m) 0.15 h) x = 500 m, z = 800 m -0.2 0 0.2 k (rad/m) i) x = 800 m, z = 800 m E 0.05 •o CD E 0.05 -o CD 0.05 X5 CD -0.2 0 0.2 k (rad/m) 0.1-05 .2 0 0.2 k (rad/m) . N 0.1 -0.2 0 0.2 k (rad/m) Figure 2.2. Wavenumber illumination patterns at the nine scatterer locations in the homogeneous model. model for a Ricker source wavelet with a peak frequency of 20 Hz. The values in the parentheses are the horizontal resolution limit Ax and the vertical resolution limit Az, respectively. Scatterer Position x 2 km x 8 km x 14 km 2 1 km (26 m, 25 m) (27 m, 25 m) (26 m, 25 m) z 2 km (30 m, 25 m) (36 m, 25 m) (30 m, 25 m) z - 3 km (36 m, 25 m) (49 m, 25 m) (36 m, 25 m) --- Z 0,---...,..---, -E- 0.05 "0 <Il -=- ~N 0.1 0.15 '-----------' -0.2 x Z 0,..-----.--------, -E- 0.05 "0 -~ ~N 0.1 0.15 '----------' -0.2 0.2 x Z 0,..-----.--------, --- -E- 0.05 "0 ~ ~N 0.1 0.15 '----------' 0.2 x =: Z 0,-----...---, --- 0.05 =c ~ ~N 0.1 _ 0.15 '----------' -0.2 0 0.2 x Z 500 m o ,.-----y--------, --- E 0.05 =c -~ ~N 0.1 0.15 '----------' -0.2 x Z 0,----,----, -E- 0.05 "0 ~ ~N 0.1 0.15 '----------' -0.2 0.2 x --- 23 = Z = 0,-----..:----, 0.05 =c -~ ~N 0.1 0.15 '-----------' -0.2 x Z = 500 m o ,.-----y--------, --- E 0.05 =c <Il -=- ~N 0.1 --- 0.15 '-----------' -0.2 x = Z = 0,------.---, -E- "0 ~ ~ N 0.1 0.15 '-----------' 0.2 x 2.2 . Table 2.1. Estimated spatial resolutions at nine scatterer locations in the homo­geneous .6.x .6.z, = = = Z = Z = Z = CD Ci 10001= 0 300 X(m) ,..-.. E ------ £ ........ D... cu o 24 O~------~------~------~------~-------' 200 400 600 800 1000------~----~------~----~----~ o 200 400 600 800 1000 X (m) Figure 2.3. The homogeneous velocity model with resolution bars that estimate the limits of horizontal and vertical resolution at nine different locations in the model. image of this model is shown in Figure 2.4. Comparing the migration image with that - the vertical resolution limits are about the same and the horizontal resolution limits are decreasing with depth. this experiment I validate the resolution formulas for a heterogeneous model which is a smoothed version of the 2D SEG/EAGE salt model (Figure 2.5). The 1001 3001 4001 £ CL CD Q 600 0 "ifc";:'-.vs;j.:;v:! ::s/l l i t X(The reverse-time migration image for the homogeneous velocity model. 25 error bars. According to the wavenumber illumination patterns, it is expected . . that the horizontal resolution will degrade with depth. The reverse-time migration the estimated resolutions, it is clear that-the spatial resolutions of the migration image at nine scatterer locations are consistent with the estimated resolutions _ . 2.3.2 Smoothed SEG/EAGE Salt Model In salt model (Figure 2.5). The Migration Image for Homogeneous Model 0 100 200 300 -- 400 -E .-c: 500 Cl. Q) 0 600 700 800 900 1000 0 200 400 600 800 1000 X (m) Figure 2.4. Smoothed SEG/EAGE Salt Model with Scatterers Velocity (m/s) X(km) Figure 2.5. Smoothed SEG/ EAGE salt model with scatterers. per shot with a source spacing of 80 m and a receiver spacing of 20 m. Sources and receivers are located along the free surface, and there are nine scatterers in the model. Using the Beylkin's spatial resolution analysis, the wavenumber illuminations at the nine locations are computed as shown in Figure 2.6, and the resolution limits are estimated as shown in Table 2.2 and illustrated in Figure 2.5 at the scatterers' locations as the error bars. By using the velocity model in Figure 2.5 as the migration velocity, the reverse-time migration image of this model is obtained as shown in Figure 2.7. Since it is difficult to compare the scatterers' responses in the migration image with the estimated resolution limits, I plot the migration image along a vertical line at x 8 km (Figure 2.8) and along a horizontal line at z 2 km (Figure 2.9). Along the vertical line the scatterer at z 2 km has the poorest vertical resolution and the scatterer at z 1 km has the best vertical resolution. This result is consistent with the vertical resolution limits estimated from the Beylkin's formula. Similarly, along the horizontal line, the scatterer at x = 8 km has the poorest resolution which is also consistent with the estimated horizontal resolution limit. The scatterer at the middle of the model (x 8 km 3 3.5 o 2 4 6 8 10 12 X (km) 14 26 Velodty (m/s) 6000 5500 5000 4500 4000 3500 3000 2500 2000 16 F igure scatterers. acquisition geometry for this experiment consists of 201 sources and 806 receivers = = 2.9) . = = = 8 km has the poorest resolution which is also consistent with the stimated horizontal resolution limit. The scatterer at the middle of the model = 8 km a) x = 2 km, z = 1 km b) x = 8 km, z = 1 km T3 GO 0 k x (rad/m) d) x = 2 km, z = 2 km CD T3 CD 0 k x (rad/m) g) x = 2 km, z = 3 km 0.2 0 k x (rad/m) h) x = 8 km, z = 3 km 0 k (rad/m) k x (rad/m) c) x = 14 km, z = 1 km 0 k x (rad/m) f) x = 14 km, z = 2 km x (rad/z = velocity with depth. This is also due to the increasing velocity with depth. = Z = km . -0.2 ~ N -0.1 o 0.1 0.2 l...-. __ ~ __ ----' -0.2 o 0.2 x = Z = -0.2 - -0.1 E =0 ~ o .:.::N 0.1 0.2 '-----~------' -0.2 o 0.2 x Z -0.2 -0.1 E =0 ro 0 -!::.. .:.::N 0.1 0.2 L-_____ ----' -0.2 o 0.2 x Z km -0.2 _ -0.1 -E- "0 -~ 0 0.2 L-_____ -----' -0.2 o 0.2 k (rad/m) x e) x = 8 km, Z = 2 km -0.2 _ -0.1 -E- "0 ~ 0 .:.::N 0.1 0.2 '------------' - 0.2 o 0.2 x = Z = -0.2 -0.1 E =0 ~ o .:.::N 0.1 0.2 l...-. _____ ----' -0.2 o 0.2 x 27 = Z = km -0.2 _ -0.1 -E- "0 ~ 0 .:.::N -E- "0 ~ .:.::N 0.1 0.2 '-----~------' -0.2 o 0.2 x = Z = km -0.2 -0.1 0.1 o • 0.2 L-_____ -----' -0.2 o 0.2 k (rad/m) x i) x = 14 km, Z = 3 km -0.2 _ -0.1 --Eg- 0 -!::.. 0.1 • 0.2 l...-. _____ -----' -0.2 o 0.2 k (rad/m) x Figure 2.6. Wavenumber illumination patterns at the nine scatterer locations in the smoothed SEG/EAGE salt model. The traveltimes associated with the computed wavenumbers were computed by tracing rays through the salt model. and 2 km) has the poorest resolution in both the horizontal and the vertical directions because it is located within the salt body which has a very high velocity value compared to the surrounding sediments. The wavelength of seismic wave is longer in regions with high velocity values than lower-velocity regions. For this heterogeneous model, both horizontal and vertical resolution limits tend to degrade SEG/Ax Az, x = x - = - m. = 2 3 km 0 2 4 6 8 10 12 14 16 X(km) 28 Table 2.2. Estimated spatial resolutions at nine scatterer locations in the smoothed SEG j EAGE salt model for a Ricker source wavelet with a peak frequency of 20 Hz. The values in the parentheses are the horizontal resolution limit 6 x and the vertical resolution limit 6 z, respectively. o 0.5 1 .- E 1.5 ~ ..r: a. 2 Q) o 2.5 3 3.5 o Scatterer Position = 2 km = 8 km x = 14 km z = 1 km (32 m, 33 m) (18 m, 28 m) (30 m, 31 m) z = 2 km (39 m, 40 m) (56 m, 55 m) (35 m, 32 m) z = (43 m, 41 m) (35 m, 32 m) (46 m, 35 m) Migration Image for SEG/EAGE Salt Model X (km) Figure 2.7. Reverse-time migration image for the smoothed salt model with scatterers. 29 Resolution at z = 1 km Resolution at z 2 km Resolution at z 3 km 0.9i : 1 i --. 1 i • Depth (km) Depth (km) Depth (km) x = SEG/vertical resolution limits at the scatterers. Resolution at x = 2 km Resolution at x = 8 km o1- 1.8 2 X (km) = km 0.025 0.02 •D E < 0.015 0.01 0.005 2.2 = 2D EAGE limits at the scatterers. 0.9 ,-------,--- -----, 0.8 0.7 0.6 Q) ~ 0.5 a. ~ 0.4 0.3 0.2 0.1 o L-----1......w...l.-l....I..-f\-><--J\->-.J 0.9 1 1.1 = 0.16 0.14 0.12 Q) 0.1 "0 :e ~0 .08 « 0.06 0.04 0.02 O~ 1.9 2 r 2.1 Q) "0 0.12 0.1 0.08 :E a. E 0.06 « 0.04 = km H r\1\ 0.02 (Vi ~.9 3 29 3.1 Figure 2.8. The reverse-time migration image line at = 8 km in the 2D SEG / EAGE salt model. The error bars indicate the estimated vert ical resolut ion scatterers. = = Resolution at x = 14 km 0.09 H H H 0.08 0.07 0.1 0.06 Q) Q) Q) "0 "0 ~ 0.05 ~ ~ a. a. a. «E «~ 0.04 0.05 0.03 0.02 0.01 0 ~ .8 1 ) ~ i 0 J. ~ _\ 1.8 8 13.8 14 14.2 (km) X(km) X (km) Figure 2.9. The reverse-time migration image line at z = 2 km in the 2D SEG/EAGE salt model. The error bars indicate the estimated horizontal resolution inversion as it causes artifacts at the tip of the salt body. Consequently, waveform inversion converged to a local minimum which was inaccurate compared to the true model. To overcome this problem, I discovered that the flooding technique, commonly used in subsalt imaging, can be used to improve the convergence of waveform inversion. Surprisingly, the velocity model, which is only a ID velocity profile, when combined with the flooding technique can provide an accurate velocity model by multiscale waveform tomography. a- early-arrival Since in this case, the true velocity structure is not known, the accuracy of the waveform tomogram is assessed by comparing the migration images and common image gathers. The results showed that acoustic waveform tomography can be used to invert these elastic field data. This success is attributed to the fact that marine data are simpler than land data which are usually corrupted by surface waves, strong random noise, and strong elastic effects. Beylkin's resolution analysis with respect to the Rayleigh's resolution criterion. A long wavelength corresponds to a high velocity value and therefore both vertical and CHAPTER 3 CONCLUSIONS Acoustic waveform tomography is used to invert both 2D synthetic and field data for the velocity models. In the case of 2D synthetic data from the SEG/ EAGE salt model, the traveltime velocity model is not a good starting model for waveform t rue t he v(z) lD In the marine data case, multiscale waveform tomogram with a dynamic early­arrival muting window successfully inverted the marine data set to obtain a velocity tomogram that is more accurate than the initial model from traveltime tomography. Beylkin 's spatial resolution formulas are validated for both homogeneous and heterogeneous velocity models. 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