Practical Migration, deMigration and Velocity Modeling Raytrace - PowerPoint PPT Presentation

Practical Migration, deMigration and Velocity Modeling Raytrace Methods Bee Bednar Panorama Technologies, Inc. 14811 St Marys Lane, Suite 150 Houston TX 77079 September 22, 2013 Bee Bednar (Panorama Technologies) Practical Migration, deMigration and Velocity Modeling September 22, 2013 1 / 20

Outline Raytrace Methods 1 Snell’s Law Raytrace modeling with Huygens’ Principle Dynamic Raytracing Anisotropic Raytracing Gaussian Beams Summary Bee Bednar (Panorama Technologies) Practical Migration, deMigration and Velocity Modeling September 22, 2013 2 / 20

Raytrace Methods Outline Raytrace Methods 1 Snell’s Law Raytrace modeling with Huygens’ Principle Dynamic Raytracing Anisotropic Raytracing Gaussian Beams Summary Bee Bednar (Panorama Technologies) Practical Migration, deMigration and Velocity Modeling September 22, 2013 3 / 20

Raytrace Methods Snell’s Law Simple Raytracing by Snell’s Law The surface slowness vector is p 0 = 1 ( cos θ 0 , sin θ 0 ) . v 1 At each step Snell’s law provides „ v ( z + ∆ z ) sin θ 1 « θ 2 = arcsin v ( z ) (1) and the new slowness vector 1 p 0 = v ( z + ∆ z )( cos θ 2 , sin θ 2 ) . to continue tracing. Figure: Raytracing in a v ( z ) medium. Bee Bednar (Panorama Technologies) Practical Migration, deMigration and Velocity Modeling September 22, 2013 4 / 20

Raytrace Methods Snell’s Law Simple Raytracing by Snell’s Law The ray is uniquely defined By its slowness vector By the normal angle at termination Decay along the ray is affected By the distance traveled By the local velocity By the local angle The initial slowness vector p 0 Represents apparent dip Ray fans when shooting Snell’s law From the wave equation Figure: Raytracing in a v ( z ) medium. Bee Bednar (Panorama Technologies) Practical Migration, deMigration and Velocity Modeling September 22, 2013 5 / 20

Raytrace Methods Snell’s Law Simple Raytracing by Snell’s Law In theory, rays are thin They have no thickness Amplitudes Approximate at best On ray is easy Off ray is hard Used to compute Wavefronts Shot records Common-offset sections Traveltime volumes Figure: Raytracing in a v ( z ) medium. Bee Bednar (Panorama Technologies) Practical Migration, deMigration and Velocity Modeling September 22, 2013 6 / 20

Raytrace Methods Snell’s Law Raytracing in Three Dimensions Given initial azimuth θ , declination φ , and position x 0 , set 1 p 0 = v ( x 0 ) ( sin θ cos φ, sin θ sin φ, cos θ ) , then calculate the next ray positions x n x n = x n − 1 + p n − 1 ∆ d and and slowness vectors p n p n = p n − 1 + P ∆ d (2) until the ray reaches its termination. Here ∆ d is the distance traveled at each step, x n is the position of the ray at the nth step, s = 1 / 2 v and and „ s ( x n + h , y , z ) − s ( x n , y , z ) , s ( x , y n + h , z ) − s ( x , y n , z ) , s ( x , y , z n + h ) − s ( x , y , z n ) « P = h h h for a sufficiently small h . Bee Bednar (Panorama Technologies) Practical Migration, deMigration and Velocity Modeling September 22, 2013 7 / 20

Raytrace Methods Raytrace modeling with Huygens’ Principle Exploding a Point Huygens’ Principle Only need point responses To produce the total response Up and Down tracing Down to each point Up to each receiver Up tracing Sources are receivers! Down tracing Receivers are sources! Very efficient Multiple arrivals Complex bookkeeping Bee Bednar (Panorama Technologies) Practical Migration, deMigration and Velocity Modeling September 22, 2013 8 / 20

Raytrace Methods Raytrace modeling with Huygens’ Principle Up and Down Rays (a) Downward rays (b) Upward rays Figure: Raytracing in complex salt. Source in (a) is on the surface at the intersection of the two planes. Source in (b) is in the center of the model below the salt. Note the sparsity of rays emerging on the left and the turning rays on the right. Bee Bednar (Panorama Technologies) Practical Migration, deMigration and Velocity Modeling September 22, 2013 9 / 20

Raytrace Methods Raytrace modeling with Huygens’ Principle Up and Down Rays (a) Downward rays (b) Upward rays Figure: Raytracing in complex salt. The number of rays that penetrate below the salt is quite small. Even when tracing huge numbers of rays, it is not obvious that one should expect the source on the left to explode the point on the right. Bee Bednar (Panorama Technologies) Practical Migration, deMigration and Velocity Modeling September 22, 2013 10 / 20

Raytrace Methods Dynamic Raytracing Raytrace Amplitude Correction Downward � = Upward amplitude Mostly ignored, or Use one direction for both Amplitude Computation Dynamic raytracer Multiple arrival phase changes Dynamic raytracer Reflection angle compensation Dynamic raytracer Bee Bednar (Panorama Technologies) Practical Migration, deMigration and Velocity Modeling September 22, 2013 11 / 20

Raytrace Methods Dynamic Raytracing Dynamic Rays Figure: The amplitude, A ( d ) , at any point, x ( d ) , on any ray is a function of the local ray coordinate system e i . Computing amplitudes on the ray requires propagation of the ray coordinate system during the ray tracing. This coordinate system propagation is similar to the raytracing itself and is relatively simple, but the mathematics is not. While beyond the scope of these notes efficient computation of the amplitudes along the ray is straightforward. The computed ray with amplitudes satisfies the wave equation kinematically and dynamically at each ray point. Bee Bednar (Panorama Technologies) Practical Migration, deMigration and Velocity Modeling September 22, 2013 12 / 20

Raytrace Methods Anisotropic Raytracing Anisotropic Raytracing Figure: Anisotropic model. Clockwise from top right: Vertical Velocity, η , symmetry axis theta , symmetry dip angle φ . Bee Bednar (Panorama Technologies) Practical Migration, deMigration and Velocity Modeling September 22, 2013 13 / 20

Raytrace Methods Anisotropic Raytracing Anisotropic Raytracing Figure: With a bit more work, one can extend raytracing to anisotropic models. The figures above show raytrace wavefronts based on the model in the previous slide. In this case the blue wavefronts are from the full anisotropic model and the magenta wavefronts are from the isotropic part of model. Bee Bednar (Panorama Technologies) Practical Migration, deMigration and Velocity Modeling September 22, 2013 14 / 20

Raytrace Methods Gaussian Beams Gaussian Beams Figure: Computing amplitudes off the central ray requires even more complicated mathematics. Getting accurate amplitudes in the vicinity of a ray necessitates computing complex traveltimes. At this point the mathematics is horrendous and way beyond what we wish to accomplish here. In this case the amplitude correction has a Gaussian decay off the ray but the actual beam satisfies the wave equation. Bee Bednar (Panorama Technologies) Practical Migration, deMigration and Velocity Modeling September 22, 2013 15 / 20

Raytrace Methods Gaussian Beams A single Gaussian Beam — After Hale 1993 Figure: A single Gaussian Beam or Fat Ray. Amplitudes off the central ray die off as a Gaussian Bell. The Gaussian decay is such that summing Gaussian Beams together in the proper manner produces extremely accurate approximations to the full wavefield. After Hale 1993 Bee Bednar (Panorama Technologies) Practical Migration, deMigration and Velocity Modeling September 22, 2013 16 / 20

Raytrace Methods Gaussian Beams Gaussian Beam Shot Record — After Hale 1993 Figure: Gaussian Beam Forward Modeled Shot. Multiple Gaussian Beams are summed together to produce a one-way forward propagated shot. The Gaussian decay is required to assure that the sum of a sufficient number of Gaussian Beams produces an accurate wavefield. After Hale 1993 Bee Bednar (Panorama Technologies) Practical Migration, deMigration and Velocity Modeling September 22, 2013 17 / 20

Raytrace Methods Summary Three Raytracing Methods Pure Snell’s law Compute the incidence and transmission angles Amplitudes generally not computed Accurate traveltimes Used in no-Amplitude Kirchhoff and Beam Dynamic Raytracing Accurate amplitudes on the ray But only on the ray Accurate traveltimes Used in Amplitude Kirchhoff and Beam Gaussian Beams Accurate amplitudes on and off the ray Accurate but complex traveltimes Used in Gaussian Beam Migration Bee Bednar (Panorama Technologies) Practical Migration, deMigration and Velocity Modeling September 22, 2013 18 / 20

Raytrace Methods Summary Summary The Hierarchy (Decreases from left to right in all) Accuracy hierarchy RTM → WEM → Gaussian Beams → Kirchhoff → Beam Ray based methods sensitive to sharp lateral velocity variations Gaussian Beam issues Almost as complex as RTM Kirchhoff and Beam issues Poor amplitude handling Beam issues Lack of resolution Velocity sensitivity Beam → Kirchhoff → Gaussian Beam → WEM → RTM Computational efficiency Beam → Kirchhoff → WEM → Gaussian Beam → RTM → GB Bee Bednar (Panorama Technologies) Practical Migration, deMigration and Velocity Modeling September 22, 2013 19 / 20

Raytrace Methods Summary Questions? Bee Bednar (Panorama Technologies) Practical Migration, deMigration and Velocity Modeling September 22, 2013 20 / 20