CIPR Seminar Lisbeth Engell Sorensen FinalToWeb

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Applied Seismic Simulation :  Applied Seismic Simulation by Lisbeth Engell-Sørensen Centre of Integrated Petroleum Research (CIPR) URL: http://www.ii.uib.no/~lisbeth e-mail: Lisbeth.Engell-Sorensen@cipr.uib.no From Reservoir to Geophysics: Seminar on Integrated Modelling CIPR, May 9, 2003 Introduction:  Introduction The main purpose of this talk is to resume the seismic methods and to help all CIPR employees and students to choose and understand the available codes for their purpose. Some definitions used in geophysics: Waveform modelling (inversion/trial and error/forward) Waveform simulation (forward) Asymptotic waveform modelling (phases) Full waveform modelling (seismograms) Contents: Theory, Applications, Areas of Interest for::  Contents: Theory, Applications, Areas of Interest for: Finite-difference simulation of equation of motion (10 slides) Full-waveform modelling with surface waves (3 slides) Asymptotic wave propagation with ray-tracing (10 slides) 1D simulation using reflection coefficients (3 slides) Other methods for surface waves and body waves: reflectivity method, spectral and pseudo spectral methods, finite element and spectral element methods (1 slide) Applied Seismic Simulation Codes (by L. Engell-Sørensen):  Applied Seismic Simulation Codes (by L. Engell-Sørensen) nc=non commercial=academic use Method, Finite Difference:  Method, Finite Difference Wave propagation in a general anisotropic, inhomogeneous elastic solid initiated by a body force is given by the equation of motion (Ben-Menahem et al, 1991): Einstein’s summation convention for repeated indices is used. fj is body force density, ui are displacement components, ij is stress tensor, Cijkl is the tensor of elastic moduli, k are spatial partial derivatives, t is temporal derivative, Ti = ijnj is the traction on a surface with normal n, Cijkl = Cjikl = Cijlk = Cklij (21 independent Cijkl coefficients). Representation Result:  Representation Result The displacement, u, due to body forces, f, throughout V and to boundary conditions, u and T, on S is (Aki and Richards, 1980) where G is Green’s tensor. The first term is the contribution from the body forces in V and the surface integral gives the contributions from the boundary conditions on the boundary S of V. General TI Medium:  General TI Medium The general anisotropic code uses 21 elastic coefficients. Eight independent coefficients are needed as input parameters for the general TI medium: density, P-, S-velocity, three Thomson parameters: , , , and two angles for the TI symmetry axis. (for the following FD modelling described here see: Hokstad et al., 2002; Holberg, 1987; Mittet et al., 1994; Petersen, 1999; Engell-Sørensen and Koster, 2002, Engell-Sørensen, 2002) FD Scheme: Higher Order Diff and Shift Operators (Staggered Grid):  FD Scheme: Higher Order Diff and Shift Operators (Staggered Grid) Newton’s law: Hooke’s law: Parallel Method:  Parallel Method The parallel code uses multiple processors in order to manipulate subsets of large amounts of data simultaneously 1. Model large problems in limited time 2. Larger memory problems on more processors Message Passing paradigm MPI (’nested’ parallelism) Data Exchange:  Data Exchange To compute a higher-order difference or shift operation several points are needed on each side of every grid point: Partition with overlapping sub-domains We define a common boundary with eight points for eight’s-order FD operator (half-length = eight) Since every grid point needs contributions from eight neighbouring grid points in forward and backward directions, the local data in the boundary that is shared by neighbouring sub-domains must be exchanged and added to the neighbouring sub-domain data 3D case: at most 26 neighbouring sub-domains with which information has to be exchanged Applied Finite Difference: Basalt Model in 3D Grid:  Applied Finite Difference: Basalt Model in 3D Grid The main purpose of the modelling is to study waveform propagation and generate realistic data for testing of processing and migration tools applied in basaltic regions. The work is based on a three-dimensional finite difference (FD) code, TIGER, made by SINTEF. The code computes wave propagation in a 3D anisotropic elastic media for micro seismic and traditional seismic sources. The geological model is a basalt model, which covers from 24 km to 37 km of a real shot line in horizontal direction and from the water surface to 3.5 km depth. The vertical parameter distribution is obtained from observations in two wells. At the depth of between 1100 m to 1500 m, a basalt horizon covers the whole sub surface layers. The 2½D model has been constructed using the compound modelling software from Norsk Hydro. The model is interpolated to a 3D grid. Each shot includes a subset of the global 3D grid in order to minimize computations. P Velocity:  P Velocity Shot Geometry and Source Signal:  Shot Geometry and Source Signal Performance:  Performance The computations were done on the IBM, p690 Regatta Turbo system at Parallab, University of Bergen (1.3 GHz Power4 processors), which consists of three 32 processor nodes with 64 Gbyte memory each (a total of 192 Gbyte memory). The system has a peak performance of 500 Gflops/s. The models applied here use about 12 Gbyte of memory. Each shot-model has 551 x 121 x 701 grid points. Temporal spacing in FD modelling is .25 ms and total recording time is 3 s. The wall-clock times have been measured for all 80 runs of the 3D wave propagation. Total time is smallest for 8 processors. Common Offset Gathers of Traction (z) for 80 shots of Displacement Source (z):  Common Offset Gathers of Traction (z) for 80 shots of Displacement Source (z) The seismic sections show clearly the wave propagation within and near the basalt layer. Diffractions are observed near the fault, possibly due to the fault geometry Slide16:  1025 m Slide17:  1275 m 1775 m:  1775 m Finite Difference, Areas of Interest:  Finite Difference, Areas of Interest To study waveform propagation and generate realistic data for testing of processing and migration tools applied in e.g. basaltic regions The parallel code enables us to model large-scale realistic geological models in reasonable time FD does not identify phases Method, Surface Waves 1:  Method, Surface Waves 1 Rayleigh waves are P-SV-type waves, that exist in a half space (Aki and Richards, 1980; Stein, 1991), and may be written as: Two conditions must be fulfilled for Rayleigh waves: The resulting linear system of equations have non-trivial solutions for four values of the apparent (=phase) velocity cx, where only one can be used Method, Surface Waves 2:  Method, Surface Waves 2 Love waves are SH-type waves, that exist in a layer over a half space (Aki and Richards, 1980; Stein, 1991), and may be written as: Four conditions must be fulfilled for Love waves: This gives several (multi mode) solutions for the phase velocity cx, and each mode is a function of kx=/cx (dispersion): Normal Mode Summation:  Normal Mode Summation Displacement due to P-SV and SH waves is where Mij and Gij are the moment tensor and the Green’s tensor, respectively, and an index after comma means spatial derivative at the source. For symmetric moment tensor and multi-layered, anelastic, isotropic medium we have (see Haskell, 1953; Ben-Menahem and Harkrider, 1964; Harkrider, 1970; Panza, 1985; Panza and Suhadolc, 1987; Florsch et al., 1991; Engell-Sørensen, 1993; Engell-Sørensen and Panza, 2000) Outward Radial P-SV Component :  Outward Radial P-SV Component Upward Vertical P-SV Component:  Upward Vertical P-SV Component Counter-Clockwise SH Component:  Counter-Clockwise SH Component Rayleigh Wave Eigenfunctions:  Rayleigh Wave Eigenfunctions Rayleigh Wave Parameters:  Rayleigh Wave Parameters Slide28:  Love Wave Eigenfunctions Love Wave Parameters:  Love Wave Parameters Applied Surface Waves:  Applied Surface Waves With mode summation of dispersed modes for P-SV and SH-type waves all P-SV type waves with phase velocity less than the P-wave velocity in the half space are modelled. Strategy for computing ground displacement : The dispersion relations for P-SV- and SH-type waves are found from the system of linear equations obtained by the boundary conditions at every interface and solved for the phase velocities (the eigenvalues) for all modes The associated eigenfunctions as function of depth, u(z), w(z), R(z), (z), v(z), L(z), are determined by solving the systems of linear equations obtained by the boundary conditions for the two types of waves The frequency domain ground displacements ur, uz, and u are found by an integral expression over all wave numbers, and can be obtained by summing the residue contributions, which are all the modes Surface Waves, Areas of Interest:  Surface Waves, Areas of Interest Ocean-bottom registrations above an oil-field: Near-surface P- and S-velocities (anisotropy) Near-surface attenuation General seismic-source signature Method, Ray-Tracing:  Method, Ray-Tracing Rays and wave surface in anisotropic, inhomogeneous media: moving eigenvector trihedral on ray (g1,g2,g3) at M (quasi-longitudinal,quasi-shear,quasi-shear). Fixed reference Cartesian system at O. Wave-surface trihedral at P (e1,e2,e3=N). Slowness vector s, wave number k, phase velocity vector V, and τ are parallel to N. Group velocity vector W is tangent to the ray at P. Radiation Pattern of Point Sources:  Radiation Pattern of Point Sources The vector equation of motion for the spectral displacement, u, in general anisotropic and inhomogeneous elastic solid due to a point force F at r=r0 is (Ben-Menahem et al., 1991) where  is density, Cijkl(r) the tensor of elastic modules at r,  angular frequency, uj components of displacement vector. Green’s Function :  Green’s Function Green’s function is defined such that uj = Gjm Fm (Einstein summation convention). Substitution yields the wave equation of motion for Gjm Ray-approximation:  Ray-approximation Assume the ”ray approximation” of the high-frequency asymptotic-series solution where the sum is over all three types of waves propagating in the medium with unit polarization vectors, g, and the travel time between source and receiver is given by (). The polarization vectors are determined by an approximate equation obtained, when the ray-approximation Gjm defined here is inserted in the vector equation of motion (the Christoffel equation). Geometrical Spreading:  Geometrical Spreading The complex amplitude is given by (Červený et al., 1977, Aki and Richards, 1980, Hanyga et al.,1998) V()=(sisi)–1/2 = phase velocity R()(r0,r) = complete reciprocal reflection/transmission coefficient (r0,r) = complete phase shift due to caustics along the ray [J(r0)/J(r)]1/2 = geometrical spreading J(r) =det ((x1,x2,x3)/(,q1,q2))=|(r/q1r/q2)·(r/)ray| = Jacobian of the transformation from Cartesian coordinates of a point on the ray to its ray coordinates (r/)ray=group velocity (,q1,q2) = ray coordinates, = travel time along the ray (q1,q2) = curvilinear coordinates on the wave surface (=constant) Dipolar Point Source With Moment:  Dipolar Point Source With Moment Let us assume we have a dipolar point source with moment. The medium response is (Aki and Richards, 1980, Ben-Menahem and Singh, 1981) where the moment tensor M is a function of frequency. Ray Approximation Displacement:  Ray Approximation Displacement By taking the spatial derivative of the ray approximation at the source the displacement (4) becomes where s is slowness. By inserting Green’s tensor we finally get: Ray Approximation Displacement for Seismic Explosive Source:  Ray Approximation Displacement for Seismic Explosive Source For seismic explosive sources we have Mmk=M·δmk and obtain the displacement component Applied Ray-Tracing:  Applied Ray-Tracing Many ray-tracer codes: Recursive ray tracer (wave front tracing) (Hanyga et al., 1998, Moser and Pajchel, 1997;Vinje et al., 1993) Two-point ray tracing based on the eikonal equation that describes the kinematics of the wave field (e.g. Virieux, 1990, 1996; Hanyga and Pajchel, 1995; Engell-Sørensen, 1991) System of two coupled equations (the ray-equations). Can be solved by a Runge-Kutta scheme (Clarke, 1996). Recursive Ray Tracer:  Recursive Ray Tracer The recursive ray tracer (Hanyga et al., 1998, Moser and Pajchel, 1997) can be used. The geological model used is an n-layered 3D model with constant or linearly changing P- and S-velocities and density in the layers. The ray tracer finds the attributes in a 3D grid for rays from a source. Grid Locations West-East:  Grid Locations West-East Ray-Traced Attributes :  Ray-Traced Attributes Source (T1), Vertical Cross-Section along triangles, Light Blue: Structural Model Initial P-Polarity in X-Direction :  Initial P-Polarity in X-Direction Ekofisk Case Ea ------- Eb Eb ------- Ec Ec ------- Ed * P-Polarity in X-Direction:  P-Polarity in X-Direction Ekofisk Case Ea ------- Eb Eb ------- Ec Ec ------- Ed * Ray-Tracing, Areas of Interest:  Ray-Tracing, Areas of Interest Pre-stack and post-stack depth migration in complicated geological structures Tomography (i.e. inversion for geological structure) Ray tracing identify phases (i.e. remembers the history of individual rays) Applied 1D Methods:  Applied 1D Methods Apply impedance logs from a well Compute elastic impedance (EI) synthetic seismograms (e.g. 30 degrees) (Connolly, 1999) Stack synthetic EI seismograms (e.g. up till 30 degrees) Compare with migrated stacked CDP gathers (e.g. with offset up till 30 degrees) 1D Methods, Areas of Interest:  1D Methods, Areas of Interest Lithology and pore fluid identification Calibration of migration results: CDP-Gather Procedure for migration of data (GEOVECTEUR, SU, PROMAX, CREWES, e.t.c.): Decon (remove multiples) NMO-correction, stacking velocities, data Stack , one trace/CDP Migration with stacking velocities and stacked data -> migrated data Other Methods for Full Waveform Modelling:  Other Methods for Full Waveform Modelling Reflectivity method (Kind) Spectral methods (Bouchon, 1982) Pseudo Spectral Methods (Carcione) Finite element Spectral element methods (Caltech) Summary:  Summary All methods can be applied but for different purposes in connection with depletion of an oil reservoir Acknowledgements:  Acknowledgements The author would like to thank Norsk Hydro, Statoil, GEUS, and SINTEF for very helpful discussions and Parallab for being helpful with using the new IBM, p690 Regatta system Phillips Petroleum Company, Norway and Norsk Hydro ASA are thanked for financing the work presented here FD References:  FD References Aki K., and P. G. Richards (1980). Quantitative Seismology, Theory and Methods, vol. 1-2., W. H. Freeman, San Francisco, 931 pp. Ben-Menahem A., R. L. Gibson Jr and A. G. Sena (1991). Green's tensor and radiation patterns of point sources in general anisotropic inhomogeneous elastic media, Geophys. J. Int. 107, 297-308. Boore, D. M. (1972). Finite difference methods for seismic wave propagation in heterogeneous materials. In B. A. Bolt (editor), Seismology: Seismic Waves and Earth Oscillations (Methods in Computational Physics, Vol. 11). New York: Academic Press. CREWES MATLAB: Margrave, G. F. (2001). Numerical Methods of Exploration Seismology with algorithms in Matlab, Department of Geology and Geophysics, The University of Calgary, http://www.crewes.org/Samples/EduSoftware Engell-Sørensen, L. (2003). Optimized 3D Finite Difference Modelling of Basaltic Region, Extended Abstract, EAGE 65th Conference & Exhibition, Stavanger, 2003.  Engell-Sørensen, L. and J. Koster (2002). Optimization and Demonstration of 3D Finite Difference Tools for Modelling Seismic Acquisition: 64th EAGE Conference & Exhibition, Abstract. Hokstad, K., L. Engell-Sørensen, and F. Maaø (2002). 3-D elastic finite-difference modelling in tilted transversely isotropic media: 72nd Ann. Internat. Mtg: Soc. of Expl. Geophys. Holberg, O. (1987). Computational aspects of the choice of operator and sampling interval for numerical differentiation in large-scale simulation of wave phenomena, Geophys. Prosp. 35, 629-655. Mittet R., and L. Amundsen and B. Arntsen (1994). Iterative inversion/migration with complete boundary conditions for the residual misfit field, J. of Seismic Exploration, 1994, 3, p. 141-156. Petersen, S. A. (1999). Compound modelling – A geological approach to the construction of shared earth models: 61st EAGE Conference & Technical Exhibition, Abstract. SU: Stockwell, J. W. and J. K. Cohen (1998). The New SU User’s Manual, Version 2.5, Center for WavePhenomena, Colorado School of Mines, http://www.cwp.mines.edu/cwpcodes/ Surface Wave References:  Surface Wave References Aki K., and P. G. Richards (1980). Quantitative Seismology, Theory and Methods, vol. 1, W. H. Freeman, San Francisco, 557 pp. Ben-Menahem, A., Harkrider, D. G. (1964). Radiation patterns of seismic surface waves from buried dipolar point sources in a flat stratified Earth. J. Geophys. Res. 69, 2605-2620 Engell-Sørensen, L. and G. F. Panza (2000). Inversion for the Moment Tensor and Source Location Using Love- and Rayleigh-Type Waves, in P. C. Hansen, B. H. Jacobsen & K. Mosegaard (Eds.), Methods and Applications of Inversion, Lecture Notes in Earth Sciences 92, Springer, Berlin, 2000. Engell-Sørensen, L. (1993). North Sea earthquake source parameters, Ph D. Thesis, Institute of Solid Earth Physics, University of Bergen, Norway. March 1993. Florsch, N., Fäh, D., Suhadolc, P., and Panza, G. F. (1991). Complete Synthetic Seismograms for High-Frequency Multimode SH-waves. Pageph 136, 4, 529-560. Herrman, R. (2002). Computer Programs in Seismology – Version 3.20, http://www.eas.slu.edu/People/RBHerrmann/CPS32.html Harkrider, D. G., (1970). Surface waves in multilayered elastic media. Part II. Higher mode spectra and spectral ratios from point sources in plane layered earth models, Bull. Seism. Soc. Am. 60, 6, 1937-1987. Haskell, N. A. (1953). Dispersion of surface waves on multilayered media, Bull. Seism. Soc. Am. 43,17-34. Panza, G. F. (1985). Synthetic seismograms: the Rayleigh waves modal summation. J. Geophys. 58, 125-145. Panza, G. F., Suhadolc, P. (1987). Complete strong motion synthetics. Ed. Bolt, A. B., Academic Press, Orlando, FL., pp. 135-204. Stein, S. and M. Wysession (2002). Introduction to Seismology, Earthquakes and Earth Structure, Blackwell Science Inc; 1st ed., 498 pp. Ray-Tracing References:  Ray-Tracing References Aki K. and P. Richards (1980). Quantitative Seismology, Theory and Methods, vol. 1, W. H. Freeman, San Francisco, 557 pp. Ben-Menahem A., R. L. Gibson Jr and A. G. Sena (1991). Green's tensor and radiation patterns of point sources in general anisotropic inhomogeneous elastic media, Geophys. J. Int. 107, 297-308. Ben-Menahem A. and S. J. Singh (1981). Seismic Waves and Sources, Springer-Verlag, New York, 1108 pp. Clarke R. A. (1996). Two point raytracing in 3D blocky models: KIM 1996 Annual Report, Istitut Français du Pétrole, Pau, France. Červený, V., I. A. Molotkov, and I. Pšenčik (1977). Ray Methods in Seismology, University Karlova, Praha. Engell-Sørensen, L. (1991). Inversion of arrival times of microearthquake sources in the North Sea using a 3-D velocity structure and prior information. Part I. Method. Bull. Seism. Soc. Am. 81 1183-1194. Hanyga A. and Pajchel J.(1995). Point-to-curve ray tracing in complicated geological models, Geophysical Prospecting 43, 859-872. Hanyga, A., A. B. Druzhinin, A. D. Dzhafarov and L. Frøyland (1998). 3-D Recursive Cell Ray Tracing in Inhomogeneous and Weakly Anisotropic Discontinuous Media. Norsk Hydro Report. Moser, T. J. and J. Pajchel (1997). Recursive seismic ray modelling application in inversion and VSP, Geophysical Prospecting, 45, 885-908. NORSAR-2D Ray Modelling (2003). http://www.norsar.no/Seismod/Products/N2D.html NORSAR-3D Ray Modelling (2003). http://www.norsar.no/Seismod/Products/N3D.html SEIS88 (2003). Seismic Waves in Complex 3-D Structures (SW3D), Department of Geophysics, Charles University Prague, http://seis.karlov.mff.cuni.cz/software/index.htm Vinje V., Iversen E. and Gjøystdal H. (1993). Traveltime and amplitude estimation using wavefront construction. Geophysics 58, 1157-1166. Virieux J. (1990). Propagation in inhomogeneous media: Ray theory workshop, Institut de Géodynamique, Université de Nice. Virieux J. (1996). Seismic Ray Tracing in Seismic Modelling of Earth Structure, revue OCEANIS, Institut Océanographique, Paris, vol. 23 (2), 153-205. 1D Method References:  1D Method References Aki K., and P. G. Richards (1980). Quantitative Seismology, Theory and Methods, vol. 1., W. H. Freeman, San Francisco, 557 pp. Conolly, P. (1999). Elastic impedance, The Leading Edge, April 1999. CREWES MATLAB: Margrave, G. F. (2001). Numerical Methods of Exploration Seismology with algorithms in Matlab, Department of Geology and Geophysics, The University of Calgary, http://www.crewes.org/Samples/EduSoftware Geovecteur®Plus 7.1. (2003). CGG Data Processing Software, http://www.cgg.com/proserv/software/products/geovecteur/GVT1.html PROMAX (2002) Seismic Processing Software, Landmark A Halliburton Company, http://www.lgc.com/news/pressreleases/20020819-landmark+releases+seisspace.htm SU: Stockwell, J. W. and J. K. Cohen (1998). The New SU User’s Manual, Version 2.5, Center for Wave Phenomena, Colorado School of Mines, http://www.cwp.mines.edu/cwpcodes/

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