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

1

Raytrace Methods 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

1

Raytrace Methods 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

Figure: Raytracing in a v(z) medium. The surface slowness vector is p0 = 1 v1 (cosθ0, sinθ0). At each step Snell’s law provides θ2 = arcsin „v(z + ∆z) sin θ1 v(z) « (1) and the new slowness vector p0 = 1 v(z + ∆z)(cosθ2, sinθ2). to continue tracing.

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

Figure: Raytracing in a v(z) medium. 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 p0 Represents apparent dip Ray fans when shooting Snell’s law From the wave equation

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

Figure: Raytracing in a v(z) medium. 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

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Raytrace Methods Snell’s Law

Raytracing in Three Dimensions

Given initial azimuth θ, declination φ, and position x0, set p0 = 1 v(x0) (sin θ cos φ, sin θ sin φ, cos θ) , then calculate the next ray positions xn xn = xn−1 + pn−1∆d and and slowness vectors pn pn = pn−1 + P∆d (2) until the ray reaches its termination. Here ∆d is the distance traveled at each step, xn is the position of the ray at the nth step, s = 1/2v and and

P = „ s(xn + h, y, z) − s(xn, y, z) h , s(x, yn + h, z) − s(x, yn, z) h , s(x, y, zn + h) − s(x, y, zn) h «

for a sufficiently small h.

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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

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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

• f 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

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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 ei. 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

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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

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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

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Raytrace Methods Summary

Questions?

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