Kirchhoff prestack depth migration in simple models of various anisotropy

Transcription

1 Kirchhoff prestack depth migration in simple models of various anisotropy Václav Bucha Department of Geophysics, Faculty of Mathematics and Physics, Charles University, Ke Karlovu, 6 Praha, Czech Republic, bucha@seis.karlov.mff.cuni.cz Summary We apply the Kirchhoff prestack depth migration to calculation of migrated sections in -D and -D simple anisotropic velocity models. Each velocity model is composed of two homogeneous layers separated by one curved interface. We use different types of anisotropy in the upper layer: isotropy, transversely isotropic media with a horizontal symmetry axis, triclinic anisotropy and monoclinic anisotropy. We test Kirchhoff prestack depth migration in two ways: a) the anisotropy used for computation of the recorded wave field is the same as the anisotropy used for migration, b) the anisotropy used for computation of the recorded wave field differs from the anisotropy used for migration. Keywords Kirchhoff prestack depth migration, anisotropic velocity model. Introduction We approximate the recorded wave field by ray-theory synthetic seismograms, and then apply ray-based Kirchhoff prestack depth migration to calculation of migrated sections. For the calculation of the recorded wave field, we use simple anisotropic velocity models composed of two homogeneous layers separated by one curved interface. The upper layer is anisotropic and velocity models differ by the anisotropy in the upper layer. The bottom layer is isotropic. The curved interface is the same in all models. Computation of the recorded wave field in the models is performed using the AN- RAY software package (Gajewski & Pšenčík, 99). Two-point rays are calculated for reflected P-wave in models with isotropy (ISO), transversely isotropic media with a horizontal symmetry axis (HTI), triclinic anisotropy (TA) and special case of monoclinic anisotropy (MA) in the upper layer. We use MODEL, CRT, FORMS and DATA packages for the Kirchhoff prestack depth migration (Červený, Klimeš & Pšenčík, 988; Bulant, 996). The velocity models for migration are homogeneous. In the first test, the anisotropy in the upper layer of the velocity model used for computation of the recorded wave field is the same as the anisotropy in the velocity model used for each migration. The bottom layer is isotropic in all models. We also test behaviour of -D Kirchhoff prestack depth migration in -D models in which the anisotropy in the upper layer of the velocity model used for computation of the recorded wave field is different from the anisotropy in the velocity model used for migration. We compare 4% HTI anisotropy used for the recorded wave field with 5% HTI and.5% HTI anisotropy used for the migration. Another test is a comparison In: Seismic Waves in Complex -D Structures, Report (Department of Geophysics, Charles University, Prague, ), pp

2 between 4% HTI anisotropy used for the recorded wave field and two values of isotropic velocity.5 km/s and 4.5 km/s used for the migration. Monoclinic anisotropy used for the recorded wave field is compared with isotropic velocity.45 km/s used for the migration. We use -D Kirchhoff prestack depth migration in a simple -D models with triclinic anisotropy (TA) and monoclinic anisotropy (MA). The anisotropy in the upper layer of the velocity model used for computation of the recorded wave field is the same as the anisotropy in the velocity model used for migration. Shot-receiver configuration consists of 8 parallel profile lines. We also test -D migration with profile line step twice enlarged (4 parallel profile lines). Programs used for the Kirchhoff prestack depth migration were introduced on compact disk SWD-CD-9 (Bucha & Bulant, 5). The GOCAD program has been used for visualization of the -D model with curved interface, shot-receiver configurations and computed two-point rays. The corresponding -D figures are screen snapshots of limited resolution.. Anisotropic velocity models The dimensions of the velocity models and measurement configurations are derived from the Marmousi model and dataset (Versteeg & Grau, 99). The horizontal dimension of the model is 9. km and the depth is km. Each velocity model is composed of two homogeneous layers separated by one curved interface (see Figures, ). Velocity models differ by the anisotropy of the upper layer. The bottom layer is isotropic in all models and has P-wave velocity V p =.6 km/s Figure. -D model with a curved interface. We also perform -D calculations in the model simply derived from the -D model by extension in the perpendicular direction (see Figure ). We compute recorded wave field in models with isotropy and various types of anisotropy in the upper layer: a) Isotropic (ISO) upper layer with P-wave velocity V p =.5 km/s. b) Transversely isotropic media with a horizontal symmetry axis (HTI) representing a medium with aligned thin dry cracks (Shearer & Chapman, 989). The axis of symmetry is parallel with the -D velocity model. Anisotropy of P waves is 4%. 6

3 Figure. -D model with a curved interface, top and bottom model planes. The horizontal dimensions of the model are 9. km x km, the depth is km. Reflected two-point rays are situated in the -D plane. Matrix of elastic parameters is defined by a table c) Triclinic anisotropy (TA) representing dry Vosges sandstone (Mensch & Rasolofosaon, 997). Reflected two-point rays propagate in -D volume. Matrix of elastic parameters is defined by a table d) Monoclinic anisotropy (MA). Reflected two-point rays are situated in -D plane x x. Matrix of elastic parameters is defined by a table

4 We perform the migration in homogeneous models (without curved interface) with anisotropy specified in items a) - d) and also in the following isotropic and anisotropic models: e) Isotropic (ISO) models with P-wave velocities V p =.45 km/s and V p = 4.5 km/s. f) Transversely isotropic media with a horizontal symmetry axis (HTI) representing a medium with aligned thin water-filled cracks (Shearer & Chapman, 989). The axis of symmetry is parallel with the -D velocity model. Anisotropy of P waves is 5%. Reflected two-point rays are situated in the -D plane. Matrix of elastic parameters is defined by a table g) Transversely isotropic media with a horizontal symmetry axis (HTI) representing again a medium with aligned thin water-filled cracks (Shearer & Chapman, 989). The axis of symmetry is parallel with the -D velocity model. Anisotropy of P waves is.5%. Reflected two-point rays are situated in the -D plane. Matrix of elastic parameters is defined by a table Shots and receivers The -D measurement configuration is derived from the Marmousi model and dataset (Versteeg & Grau, 99). One profile line is used for -D calculations. The first shot is at km, the last shot is at km, distance between shots is.5 km, the depth of shots is.8 km. The total number of shots is 4. The number of receivers per shot is 96, the first receiver is at offset.575 km left of shot location, the last receiver is at offset. km left of shot location, the distance between receivers is.5 km, the depth of receivers is km. The -D measurement configuration consists of 8 parallel profile lines, see Figures and 4. The measurement configuration along each profile line is the same as for the -D measurement. The distance between parallel profile lines is.5 km. We also test measurement configuration with twice greater line step.5 km (4 profile lines). 8

5 Figure. -D model with 8 parallel profile lines, curved interface and bottom model plane. The horizontal dimensions of the model are 9. km x km, the depth is km. We compute and stack migrated sections in -D plane (blue) located at the middle of the shot-receiver configuration (at horizontal coordinate of.5 km). Figure 4. Top view of 8 parallel profile lines. 4. Recorded wave field Computation of the recorded wave field in the models has been performed using the ANRAY software package (Gajewski & Pšenčík, 99). -D ray tracing (program anray.for) is used for calculation of two-point rays of reflected P-wave. Program fresan.for computes the frequency response at a system of receivers (vertical component, explosive source). Program synfan.for calculates ray-theory seismograms. The sampling interval is 4 ms and the number of samples per trace is 4. Seismograms are converted into GSE format and gridded. History files summarize parameters and execute programs for all 4 shot-receiver configurations in one profile line. In view of the fact that the layers are homogeneous and the curved interface is independent of the distance perpendicular to the profile lines, the recorded 9

6 wave field is the same for all parallel profile lines. Figures 5-9 show examples of two-point rays and seismograms for three selected shot-receiver configurations. Shots, and 4 are at horizontal coordinates of km, km and km. Rays and seismograms are computed in models with a) isotropy (ISO), b) transversely isotropic media with a horizontal symmetry axis (HTI), c) triclinic anisotropy (TA) and d) monoclinic anisotropy (MA) in the upper layer. Note the different illumination of the curved interface by two-point rays for the selected shots and receivers. shot shot shot 4 traces 6 9 traces 6 9 traces 6 9 shot shot shot 4 Figure 5: Isotropy (ISO) in the upper layer. Two-point rays and seismograms computed in isotropic model with P-wave velocity in the upper layer.5 km/s. Two-point rays of reflected P-wave propagate in -D plane. Shots, and 4 are located at horizontal coordinates of km, km and km. 4

7 shot shot shot 4 traces 6 9 traces 6 9 traces 6 9 shot shot shot 4 Figure 6: Transversely isotropic media with a horizontal symmetry axis (HTI) 4% in the upper layer. Two-point rays and seismograms computed in model with HTI 4% for P-waves in the upper layer. Two-point rays of reflected P-wave propagate in -D plane. Shots, and 4 are located at horizontal coordinates of km, km and km. 4

8 shot shot shot 4 traces 6 9 traces 6 9 traces 6 9 shot shot shot 4 Figure 7: Triclinic anisotropy (TA) in the upper layer. Two-point rays and seismograms computed in model with triclinic anisotropy (TA) in the upper layer. Two-point rays of reflected P- wave propagate in -D volume. Shots, and 4 are located at horizontal coordinates of km, km and km. 4

9 shot 4 shot shot Figure 8: Triclinic anisotropy (TA) in the upper layer. Detailed views of two-point rays computed for shots, and 4 in model with triclinic anisotropy (TA) in the upper layer and displayed in Figure 7. Note the curved path of reflections at the interface. 4

10 shot shot shot 4 traces 6 9 traces 6 9 traces 6 9 shot shot shot 4 Figure 9: Monoclinic anisotropy (MA) in the upper layer. Two-point rays and seismograms computed in model with monoclinic anisotropy (MA) in the upper layer. Two-point rays of reflected P-wave propagate in -D plane. Shots, and 4 are located at horizontal coordinates of km, km and km. 44

11 5. Kirchhoff prestack depth migration We use MODEL, CRT, FORMS and DATA packages for the Kirchhoff prestack depth migration (Červený, Klimeš & Pšenčík, 988; Bulant, 996). The migration consists of one or two-parametric ray tracing from individual surface points (program crt.for), calculating grid values of travel time and amplitude (program mtt.for), common-shot migration (program grdmigr.for) and stacking of migrated images. History files summarize parameters and execute programs for all 4 shot-receiver configurations in profile lines. 5. -D migration Figure shows stacked migrated sections calculated in models with isotropy (ISO), transversely isotropic media with a horizontal symmetry axis (HTI) 4%, triclinic anisotropy (TA) and monoclinic anisotropy (MA). The anisotropy in the upper layer of the velocity model used for computation of the recorded wave field is the same as the anisotropy in the velocity model used for migration. The crosses denote the interface in the velocity model used for computation of the recorded wave field. Figures - show comparison of stacked migrated sections when the anisotropy in the upper layer of the velocity model used for computation of the recorded wave field is different from the anisotropy in the velocity model used for migration. We compare 4% HTI anisotropy used for the recorded wave field with 5% HTI and.5% HTI anisotropy used for the migration (see Figure ). Another test is a comparison between 4% HTI anisotropy used for the recorded wave field and two values of isotropic velocity.5 km/s and 4.5 km/s used for the migration (see Figure ). Figure shows comparison of monoclinic anisotropy (MA) used for the recorded wave field with isotropic velocity.45 km/s used for the migration. These tests should simulate situations when we made a bad guess of the velocity model for migration. The computing time of the Kirchhoff migration, for one -D stacked section computed on a grid of cells 6 6 metres and interpolated to grid of cells 4 4 metres, is approximately hours. The time of the computation corresponds to an Athlon one core. GHz processor. 5. -D migration Figures 4, 5 show stacked migrated sections calculated in models with triclinic anisotropy (TA) and monoclinic anisotropy (MA). The anisotropy in the upper layer of the velocity model used for computation of the recorded wave field is the same as the anisotropy in the velocity model used for migration. Shot-receiver configuration consists of 8 parallel profile lines with step.5 km (see Figures, 4). We also tested -D migration with profile line step twice enlarged (from.5 km to.5 km, 4 parallel profile lines). First profile line starts at horizontal coordinate of.5 km and the last profile line ends at horizontal coordinate of.5 km. We compute and stack 8 or 4 migrated sections in the -D plane located in the middle of the shot-receiver configuration (at horizontal coordinate of.5 km, see Figure ). Note that the migrated sections computed with profile line step.5 km (4 lines) for monoclinic anisotropy (MA, Figure 5) contain greater residua of individual migrated sections used for stacking than for triclinic anisotropy (TA, Figure 4). 45

12 4 6 8 ISO % HTI TA MA Figure. Stacked migrated sections calculated in models with isotropy (ISO), 4% HTI anisotropy, triclinic anisotropy (TA) and monoclinic anisotropy (MA). The anisotropy in the upper layer of the velocity model used for computation of the recorded wave field is the same as the anisotropy in the velocity model used for migration. The crosses denote the interface in the velocity model used for computation of the recorded wave field. 46

13 % HTI Recorded wave field - 4% HTI anisotropy Migration - 4% HTI anisotropy % HTI x 5% HTI Recorded wave field - 4% HTI anisotropy Migration - 5% HTI anisotropy % HTI x.5% HTI Recorded wave field - 4% HTI anisotropy Migration -.5% HTI anisotropy Figure. Comparison of stacked migrated sections when the percentage of HTI in the upper layer of the velocity model used for computation of the recorded wave field is different from the percentage of HTI in the velocity model used for migration. The crosses denote the interface in the velocity model used for computation of the recorded wave field. 47

14 % HTI Recorded wave field - 4% HTI anisotropy Migration - 4% HTI anisotropy % HTI x.5km/s ISO Recorded wave field - 4% HTI anisotropy Migration - isotropy (ISO), P-wave velocity is.5 km/s % HTI x 4.5km/s ISO Recorded wave field - 4% HTI anisotropy Migration - isotropy (ISO), P-wave velocity is 4.5 km/s Figure. Comparison of stacked migrated sections when the upper layer of the velocity model used for computation of the recorded wave field is anisotropic (4% HTI) and the velocity model used for migration is isotropic (ISO). The crosses denote the interface in the velocity model used for computation of the recorded wave field. 48

15 4 6 8 MA Recorded wave field - monoclinic anisotropy (MA) Migration - monoclinic anisotropy (MA) MA x.45km/s ISO Recorded wave field - monoclinic anisotropy (MA) Migration - isotropy (ISO), P-wave velocity is.45 km/s Figure. Comparison of stacked migrated sections when the upper layer of the velocity model used for computation of the recorded wave field is anisotropic (MA) and the velocity model used for migration is isotropic (ISO). The crosses denote the interface in the velocity model used for computation of the recorded wave field. 49

16 4 6 8 TA -D, 8 profile lines Stacked migrated section for 8 parallel measurement lines TA -D, profile line at.5 km Migrated section for measurement line (coordinate of.5 km) TA -D, 4 profile lines Stacked migrated section for 4 parallel measurement lines Figure 4. Stacked migrated sections calculated in -D model with triclinic anisotropy (TA). The sections are stacked at the -D plane section located in middle of measurement configuration (coordinate of.5 km). The anisotropy in the upper layer of the velocity model used for computation of the recorded wave field is the same as the anisotropy in the velocity model used for migration. The crosses denote the interface in the velocity model used for computation of the recorded wave field. 5

17 4 6 8 MA -D, 8 profile lines Stacked migrated section for 8 parallel measurement lines MA -D, profile line at.5 km Migrated section for measurement line (coordinate of.5 km) MA -D, 4 profile lines Stacked migrated section for 4 parallel measurement lines Figure 5. Stacked migrated sections calculated in -D model with monoclinic anisotropy (MA). The sections are stacked at the -D plane section located in middle of measurement configuration (coordinate of.5 km). The anisotropy in the upper layer of the velocity model used for computation of the recorded wave field is the same as the anisotropy in the velocity model used for migration. The crosses denote the interface in the velocity model used for computation of the recorded wave field. 5

18 6. Conclusions We used Kirchhoff prestack depth migration for calculation of migrated sections in an anisotropic velocity model with one curved interface. The upper layer of the velocity model has several types of anisotropy. The bottom layer is isotropic. We computed -D stacked migrated sections when the anisotropy in the upper layer of the velocity model used for computation of the recorded wave field is the same as the anisotropy in the velocity model used for migration. In this case, migrated interface coincides nearly perfectly with the interface in the model used for computation of the recorded wave field. The length of the migrated interface is dependent on the illumination by rays that differs for different types of anisotropy. We compared -D stacked migrated sections when the anisotropy in the upper layer of the velocity model used for computation of the recorded wave field is different from the anisotropy in the velocity model used for migration. In this case migrated interface is shifted and deformed more by wrong guess of isotropic velocity than wrong guess of HTI anisotropy in the model used for migration. We tested -D Kirchhoff prestack depth migration in a simple -D model with success. We also tested -D migration with profile line step twice enlarged (from.5 km to.5 km). The migrated section computed with.5 km step for monoclinic anisotropy contains greater residua of individual sections used for stacking than migrated section computed for triclinic anisotropy. Acknowledgments The author thanks Ivan Pšenčík and Luděk Klimeš for great help throughout the work on this paper. The research has been supported by the Grant Agency of the Czech Republic under contracts 5/7/ and P//76, by the Ministry of Education of the Czech Republic within research project MSM686, and by the members of the consortium Seismic Waves in Complex -D Structures (see ). References Bulant, P. (996): Two-point ray tracing in -D. Pure appl. Geophys., 48, pp Bucha, V. & Bulant, P. (eds.) (5): SWD-CD-9 (DVD-ROM). In: Seismic Waves in Complex -D Structures, Report 5, pp , Dep. Geophys., Charles Univ., Prague, online at Červený, V., Klimeš, L. & Pšenčík, I. (988): Complete seismic-ray tracing in threedimensional structures. In: Doornbos, D.J.(ed.), Seismological Algorithms, Academic Press, New York, pp Gajewski, D. & Pšenčík, I. (99): Vertical seismic profile synthetics by dynamic ray tracing in laterally varying layered anisotropic structures. J. geophys. Res., 95B, pp. 5. Mensch, T. & Rasolofosaon, P. (997): Elastic-wave velocities in anisotropic media of arbitrary symmetry-generalization of Thomsens parameters ǫ, δ and γ. Geophys. J. Int., 8, pp Shearer, P.M. & C.H. Chapman (989): Ray tracing in azimuthally anisotropic media: I. Results for models of aligned cracks in the upper crust, Geophys. J., 96, pp Versteeg, R. J. & Grau, G. (eds.) (99): The Marmousi experience. Proc. EAGE workshop on Practical Aspects of Seismic Data Inversion (Copenhagen, 99), Eur. Assoc. Explor. Geophysicists, Zeist. 5