PPT3_Refra-VA-modeling

Geophysical Signal AnalysisSlide 1 Ray tracing basics: • Snell’s Law • Fermat’s Principle • Huygens’ Principle: Ray width? • Ray dependency on velocity gradient • Circular ray for linear velocity gradient • Ray-associated wave energy • Multi-path rays Geophysical Signal AnalysisSlide 2 Ray tracing: HighRay tracing: High--frequency approximationfrequency approximation Geophysical Signal AnalysisSlide 3 velocity models (Image from John Lahr, USGS Open-File Report 99-132) Fig. RVA-2 Major layer elements of the planet Earth. Geophysical Signal AnalysisSlide 4 3.1 1D velocity models30 Traveltim e [sec] 0 0 Distance [degree] 180 30 Traveltim e [sec] 0 (b)(b) (a)(a) Fig. RVA-3 Traveltime versus distance graphs of: (a) Predictions from the IASP91 1D P-wave velocity model; and (b) Observed waveforms of some earthquakes. Geophysical Signal AnalysisSlide 5 Geophysical Signal AnalysisSlide 6 Geophysical Signal AnalysisSlide 7 Optics disk – an illustration of Snell’s Law Seismic modeling Geophysical Signal AnalysisSlide 8 Snell’s Law: A conceptual example. Seismic modeling Geophysical Signal AnalysisSlide 9 Fermat's principle states (in it's simplest form) that light waves of a given frequency traverse the path between two points which takes the least time. Or, traveltime is stationary w.r.t. perturbation of raypath. Fermat’s Principle One can derive Snell’s law from Fermat's principle, as shown in this sketch. Seismic modeling Geophysical Signal AnalysisSlide 10 Ray tracing basics: • Snell’s Law • Fermat’s Principle • Huygens’ Principle: Ray width? • Ray dependency on velocity gradient • Circular ray for linear velocity gradient • Ray-associated wave energy • Multi-path rays Seismic modeling Geophysical Signal AnalysisSlide 11 ‰ Rapid ray tracing in v(z) model using a reference table ReflectionsReflections z xx tt Surface Surface waves waves Refractions Refractions xx zz Seismic modeling Geophysical Signal AnalysisSlide 12 3.1 1D velocity models The left panel is an 1D velocity profile, and the right panels are raypaths from a surface source. Geophysical Signal AnalysisSlide 13 xx zz Depth gatherDepth gather xx tt Time gatherTime gather Surface waves Surface waves Refractions Refractions ReflectionsReflections Numerical modeling: Ray theory Ray method allows seismic rays and phases being connected and easily comprehensible! Seismic modeling Geophysical Signal AnalysisSlide 14 Q: What could be the raypaths of a first break arrival?Q: What could be the raypaths of a first break arrival? First break refers to the event that arrives the earliest on a seismic record panel, such as a common shot gather. The corresponding raypaths, however, depend on the velocity model. The near-offset is usually direct wave. The far-offset would be head wave for a layer-cake model, but would be turning waves in a gradient model. The situation will be more complex in the presence of lateral velocity variations. Geophysical Signal AnalysisSlide 15 (TOP) First arrival raypaths of a common shot gather in a four-layer velocity model (velocities are 1.0, 1.5, 2.0, and 2.5 km/s from the top layer down), with flat interfaces and constant receiver elevations. (LOWER) Reduced traveltimes using a 2.5 km/s reduction velocity. Ray tracing Geophysical Signal AnalysisSlide 16 (TOP) First arrival raypaths of a common shot gather in a four-layer velocity model (velocities are 1.0, 1.5, 2.0, and 2.5 km/s from the top layer down), with flat interfaces and varying receiver elevations. (LOWER) Reduced traveltimes using a 2.5 km/s reduction velocity. Ray tracing Geophysical Signal AnalysisSlide 17 (TOP) First arrival raypaths of a common shot gather in a four-layer velocity model (velocities are 1.0, 1.5, 2.0, and 2.5 km/s from the top layer down), with depth-varying interfaces and receiver elevations. (LOWER) Reduced traveltimes using a 2.5 km/s reduction velocity. Ray tracing Geophysical Signal AnalysisSlide 18 Kinematic ray tracing for traveltimes and raypaths • One point: shooting • Two point: refraction • Three point: reflection Algorithms • Shooting and bending (Julian & Gubbins, 1977, Three-dimensional seismic ray tracing: J. Geophys., 43, 95-113) • eikonal equation - Vidale, 1990, Finite-difference calculation of traveltimes in three dimensions - Qin, et al., 1992, Finite-difference solution of the eikonal equation along expanding wavefronts - Sethian & Popovici, 1999, 3-D traveltime computation using the fast marching method - Mo & Harris, 2002, Finite-difference calculation of direct-arrival traveltimes using the eikonal equation • Shortest–time (Moser, 1991, Shortest path calculation of seismic rays) Seismic modeling Geophysical Signal AnalysisSlide 19 Traveltimes from a fast marching method (Sethian & Popovici, 1999) Seismic modeling Geophysical Signal AnalysisSlide 20 (Mo & Harris, 2002, Finite-difference calculation of direct-arrival traveltimes) Seismic modeling Geophysical Signal AnalysisSlide 21 (Mo & Harris, 2002, Finite-difference calculation of direct-arrival traveltimes) Seismic modeling Geophysical Signal AnalysisSlide 22 (Mo & Harris, 2002, Finite-difference calculation of direct-arrival traveltimes) Seismic modeling Geophysical Signal AnalysisSlide 23 Multi-path rays Seismic modeling Geophysical Signal AnalysisSlide 24 Multi-path wavefronts Seismic modeling Geophysical Signal AnalysisSlide 25 9.3 Regularized inversion of prestack migration Figure 9.23: Waveform and ray-tracing modeling of seismic propagation from a point source located in the poorly illuminated area of the subsurface. Notice that the energy corresponding to the middle of the aperture-angle range never reaches the surface. (Biondi, 2004) Geophysical Signal AnalysisSlide 26 Approximations • Shortest raypath to approximate max-energy • Use of reference table to explore symmetry of the model • Interpolation Dynamic ray tracing for • Traveltime • Raypath • Amplitude and phase • Frequency dependent • Similar to result of wave theory Future issues • Maximum-energy (dynamic ray) • Multiple arrival times • Model partition and interpolation Seismic modeling Geophysical Signal AnalysisSlide 27 Tracing of Reflection RaysTracing of Reflection Rays Seismic modeling Geophysical Signal AnalysisSlide 28 Raypaths (dashed curves) in a 3D VSP modelRaypaths (dashed curves) in a 3D VSP model Seismic modeling Geophysical Signal AnalysisSlide 29 Isochrons (wavefronts) in a 3D VSP modelIsochrons (wavefronts) in a 3D VSP model Seismic modeling Geophysical Signal AnalysisSlide 30 Waveform modeling Forward and inverse problems Forward and inverse problems are two opposite approaches of are two opposite approaches of derivation. Although the sense derivation. Although the sense of "of "forwardforward" and "" and "inverseinverse" is " is only relative in view of only relative in view of mathematics, geoscientists mathematics, geoscientists usually refer to a forward usually refer to a forward problem as a process that goes problem as a process that goes from intrinsic variables (model) from intrinsic variables (model) to observational parameters to observational parameters (data), and an inverse problem (data), and an inverse problem as how to find out intrinsic as how to find out intrinsic parameters (model) from parameters (model) from observation values (data).observation values (data). Waveform modeling Geophysical Signal AnalysisSlide 31 The Huygens Principle was developed in the 17th century by the Dutch physicist Huygens about the propagation of waves. He stated that when you have a wavefront, you could synthesize the next wavefront by imagining on the wavefront (being decomposed into) an infinite number of small sound sources, whose waves (being superposed) together would form the next wavefront. Decomposition & superposition Waveform modeling Geophysical Signal AnalysisSlide 32 A listener would then not be able to determine the difference between a situation where the wavefront is real, or when it is synthesized. Decomposition & superposition Waveform modeling Geophysical Signal AnalysisSlide 33 Decomposition & superposition Waveform modeling Geophysical Signal AnalysisSlide 34 Waveform modeling Geophysical Signal AnalysisSlide 35 Physical modeling BB CC AA DD receiversreceivers shotsshots Waveform modeling Geophysical Signal AnalysisSlide 36 Wave theory numerical modelingWave theory numerical modeling Waveform modeling Geophysical Signal AnalysisSlide 37 Animation of 2004 Indonesia tsunami Source: NOAA (http://www.noaanews.no aa.gov/video/tsunami- indonesia2004.mov) Animation provided by Vasily V. Titov, Associate Director, Tsunami Inundation Mapping Efforts (TIME), NOAA/PMEL - UW/JISAO, USA. Waveform modeling Geophysical Signal AnalysisSlide 38 Finite differencing wave equationFinite differencing wave equation Finite differencing is the easiest way to simulate wave motion in generally heterogeneous velocity and density models. Many challenges for the applications include: • Numerical dispersion • Boundary conditions • Book keeping of parameters Selected references: Claerbout, J. F., 1971, Toward a unified theory of reflector mapping: Geophysics, 36, 467-481. Clayton & Engquist, 1980, Absorbing side boundary conditions for wave- equation migration, Geophysics, 45, 895-904. Dablain, M. A., 1986, The application of high-order differencing to the scalar wave equation: Geophysics, 51, 54-63. Waveform modeling Geophysical Signal AnalysisSlide 39 (Yan et al., 2004) Waveform modeling Geophysical Signal AnalysisSlide 40 Synthetic waveforms by acoustic finite-difference method (Yan et al., 2004) Waveform modeling Geophysical Signal AnalysisSlide 41 Comparison between synthetic and observed first arrivals (Yan et al., 2004) Waveform modeling Geophysical Signal AnalysisSlide 42 Cross-line section of the interval velocity model for the SEG-EAGE salt data set. (Biondi, 2004) Waveform modeling Geophysical Signal AnalysisSlide 43 Snapshots of the wavefield at t=0 s and t=1 s, when the source is located below a salt body with a rugose top. (Biondi, 2004) Waveform modeling Geophysical Signal AnalysisSlide 44 (Biondi, 2004) Wavefield recorded at the surface, corresponding to the wave modeling shown in Figure 4.3. The trajectory superimposed onto the wavefield represents the time-delay function computed by numerically solving the Eikonal equation. The Eikonal solution is a poor representation of the full- wavefield Green function. Waveform modeling Geophysical Signal AnalysisSlide 45 Waveform modeling in a cross-well setup (Pratt, 2005). Yellow star is shot. Green boxes are geophones. (a) A checkerboard model with cell size 10.5mx10.5m and velocities at 2.3 and 3.0 km/s. (b) Shot gather from the setup of (a). 150 m 3 0 0 m (c) (d) 50 t[ms] 110150 m 3 0 0 m (a) (b) 50 t[ms] 110 (c) Another checkerboard model with cell size 21mx21m and velocities at 2.3 and 3.0 km/s. (d) Shot gather from the setup of (c). Waveform modeling Geophysical Signal AnalysisSlide 46 Figure 3.6: Simple model case of ambiguity in constant-offset migration. The velocity model is a constant background with a circular slow velocity anomaly. The positions of the diffractors are marked as X1 and X2. (Biondi, 2004) Waveform modeling Geophysical Signal AnalysisSlide 47 Figure 3.7: Diffraction curves corresponding to the velocity model and diffractor locations shown in Figure 3.3. Notice that the two curves are tangent to each other at the apex. (Courtesy of Marie Clapp) (Biondi, 2004) Waveform modeling Geophysical Signal AnalysisSlide 48 Figure 3.8: The zero-offset section (panel a) and the CMP gather located at the origin of the midpoint axis (panel b) extracted from the synthetic data set used to illustrate the problem of artifacts in ADCIGs. The artifacts are related to the curved event recorded at zero offset around 4.5 seconds and corresponding to a reflection with opening angle γ ≠ 0. (Biondi, 2004) Waveform modeling Geophysical Signal AnalysisSlide 49 Thank You Thank You !!