Basic Ray Theory
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Basic Ray Theory

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Brief heuristic, description of the application of ray theory to seismology and seismic exploration

Brief heuristic, description of the application of ray theory to seismology and seismic exploration

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Basic Ray Theory Basic Ray Theory Presentation Transcript

  • Teoría de Rayos Básica por Carlos César Piedrahita
  • Overview
    • Introduction
    • Wave propagation
    • Ray method
      • Ray path
      • Traveltimes
      • Amplitudes
  • Ray method Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes
  • Ray method Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes
  • Ray method Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes
  • Ray method Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes
  • Ray method Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes
  • Wave propagation Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Acoustic equation Elastodynamic equation Ray formulation
  • Wave propagation Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Acoustic equation Elastodynamic equation Ray formulation
  • Wave propagation Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Acoustic equation Elastodynamic equation Ray formulation
  • Wave propagation Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Acoustic equation Elastodynamic equation Ray formulation
  • Wave propagation Acoustic equation Elastodynamic equation Ray formulation
  • Wave propagation Acoustic equation Elastodynamic equation Ray formulation
  • Wave propagation Acoustic equation Elastodynamic equation Ray formulation
  • Wave propagation Acoustic equation Elastodynamic equation Ray formulation
  • Wave propagation Observed wavefield: superposition of isolated events Acoustic equation Elastodynamic equation Ray formulation
  • Wave propagation Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Acoustic equation Elastodynamic equation Ray formulation   
  • Ray method Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes 
  • Ray Ansatz  Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Reflections Reflection event Ray formulation
  • Ray Ansatz Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Reflections Reflection event Ray formulation 
  • Ray Ansatz Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Reflections Reflection event Ray formulation 
  • Ray Ansatz Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Reflections Reflection event Ray formulation 
  • Ray Ansatz Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Reflections Reflection event Ray formulation 
  • Ray Ansatz Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Reflections Reflection event Ray formulation 
  • Ray Ansatz Reflections Reflection event Ray formulation
  • Ray Ansatz  Reflections Reflection event Ray formulation t A
  • Ray Ansatz Reflections Reflection event Ray formulation t A 
  • Ray Ansatz Reflections Reflection event Ray formulation t A 
  • Ray Ansatz Reflections Reflection event Ray formulation t A 
  • Ray Ansatz  Reflections Reflection event Ray formulation t A
  • Ray Ansatz  Reflections Reflection event Ray formulation t A
  • Ray Ansatz  Reflections Reflection event Ray formulation t A
  • Ray Ansatz  Reflections Reflection event Ray formulation t A
  • Ray Ansatz  Reflections Reflection event Ray formulation t A
  • Ray Ansatz Reflections Reflection event Ray formulation Structure of reflected event A ( x ) = amplitude   ( x ) = traveltime Unknowns:  t A
  • Ray Ansatz Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Reflections Reflection event Ray formulation    
  • Ray method Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes  
  • Ray path and traveltimes Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Eikonal equation Ray equations Ray path   Traveltimes
  • Ray path and traveltimes Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Eikonal equation Ray equations Ray path   Traveltimes
  • Ray path and traveltimes Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Eikonal equation Ray equations Ray path   Traveltimes
  • Ray path and traveltimes Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Eikonal equation Ray equations Ray path   Traveltimes
  • Ray path and traveltimes Eikonal equation Ray equations Ray path   Traveltimes
  • Ray path and traveltimes Eikonal equation Ray equations Ray path Traveltimes
  • Ray path and traveltimes
    • Traveltimes (Ray Tracing)
    • Partial differential equation(PDE) of first order
    • Nonlinear PDE
    • Hamilton-Jacobi type
    Eikonal equation Ray equations Ray path Traveltimes
  • Ray path and traveltimes Eikonal equation Ray equations Ray path Traveltimes Hamiltonian Method of characteristics Ray equations Ray jacobian
  • Ray path and traveltimes Eikonal equation Ray equations Ray path Traveltimes Hamiltonian Method of characteristics Ray equations Ray jacobian
  • Ray path and traveltimes Eikonal equation Ray equations Ray path Traveltimes Hamiltonian Method of characteristics Ray equations Ray jacobian
  • Ray path and traveltimes Eikonal equation Ray equations Ray path Traveltimes Hamiltonian Method of characteristics Ray equations Ray jacobian
  • Ray path and traveltimes Hamiltonian Method of characteristics Ray equations Ray jacobian
  • Ray path and traveltimes Hamiltonian Method of characteristics Ray equations
    • Eikonal as a Hamiltonian
    Ray jacobian
  • Ray path and traveltimes Hamiltonian Method of characteristics Ray equations
    • A special change of variables
    where, along curves  ,
    •   is called characteristic
    Ray jacobian
  • Ray path and traveltimes Hamiltonian Method of characteristics Ray equations
    • Along characteristics curves 
      Ray jacobian
  • Ray path and traveltimes Hamiltonian Method of characteristics Ray equations Ray jacobian
    • System of 7 ordinary differential equations (ODE)
  • Ray path and traveltimes Hamiltonian Method of characteristics Ray equations Ray jacobian
    • Point source initial condition
  • Ray path and traveltimes Hamiltonian Method of characteristics Ray equations
    • Ray coordinates
     2  1 p 0 (  1 ,  2 , s ) Ray jacobian
  • Ray path and traveltimes Hamiltonian Method of characteristics Ray equations Ray jacobian
    • Ray jacobian
  • Ray path and traveltimes Hamiltonian Method of characteristics Ray equations Ray jacobian
    • Example of ray jacobian: Point source and constant velocity
  • Ray path and traveltimes
    • System of 7 ordinary differential equations (ODE)
    Eikonal equation Ray equations Ray path Traveltimes
  • Ray path and traveltimes Eikonal equation Ray equations Ray path Traveltimes Without interfaces With interfaces
  • Ray path and traveltimes Without interfaces With interfaces
    • Requires a smooth velocity model and one initial condition
    • 4 th order Runge-Kutta can be used
    x S x p 0
  • Ray path and traveltimes Without interfaces With interfaces
    • Requires new initial conditions for each interface
    • Snell’s law must be used
    First initial condition Second initial condition Third initial condition Incident ray reflected ray transmitted ray
  • Ray path and traveltimes Without interfaces With interfaces
    • Snell’s law
     j  j+1 c j c j+!
  • Ray path and traveltimes Without interfaces With interfaces
    • Example: Six initial conditions
  • Ray path and traveltimes Without interfaces With interfaces
    • Example: Eight initial conditions (multiple)
  • Ray path and traveltimes Eikonal equation Ray equations Ray path Traveltimes
    • Traveltime is computed along the ray 
  • Ray path and traveltimes Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Eikonal equation Ray equations Ray path   Traveltimes    
  • Ray method Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes   
  • Amplitudes Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Transport equation Ray equations Geometric spreading   Amplitude 
  • Amplitudes Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Transport equation Ray equations Geometric spreading   Amplitude 
  • Amplitudes Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Transport equation Ray equations Geometric spreading   Amplitude 
  • Amplitudes Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Transport equation Ray equations Geometric spreading   Amplitude 
  • Amplitudes Transport equation Ray equations Geometric spreading   Amplitude 
  • Amplitudes Transport equation Ray equations Geometric spreading Amplitude
  • Amplitudes Transport equation Ray equations Geometric spreading Amplitude
    • Uses the ray solution
    • Partial Differential Equation (PDE) of First Order
  • Amplitudes Transport equation Ray equations Geometric spreading Amplitude
    • Alternative expression:
    • Applying Gauss’ theorem:
  • Amplitudes Transport equation Ray equations Geometric spreading Amplitude dS 0 dS 1
    • Ray tube
    D
  • Amplitudes Transport equation Ray equations Geometric spreading Amplitude
    • Ray tube
  • Amplitudes Transport equation Ray equations Geometric spreading Amplitude
    • Ray tube
  • Amplitudes Transport equation Ray equations Geometric spreading Amplitude
    • Ray tube
  • Amplitudes Transport equation Ray equations Geometric spreading Amplitude
    • Ray tube
    • Geometrical spreading
  • Amplitudes Transport equation Ray equations Geometric spreading Amplitude Without interfaces With interfaces Basic problems General expression
  • Amplitudes Without interfaces With interfaces A ( s ) A ( s 0 ) Basic problems General expression
  • Amplitudes Without interfaces With interfaces
    • Point source initial condition:
    Basic problems General expression
  • Amplitudes Without interfaces With interfaces
    • Requires new initial conditions for each interface
    • Snell’s law must be used
    • Reflection coefficient must be used (boundary conditions)
    Basic problems General expression
  • Amplitudes Without interfaces With interfaces
    • Acoustic reflection coefficient
    • Acoustic transmission coefficient
    Basic problems General expression
  • Amplitudes Without interfaces With interfaces S M G Basic problems General expression
    • Reflected ray
    • Known
  • Amplitudes Without interfaces With interfaces
    • Reflected ray
    S M G Basic problems General expression
  • Amplitudes Without interfaces With interfaces Basic problems General expression S M G
    • Reflected ray
  • Amplitudes Without interfaces With interfaces Basic problems General expression S M G
    • Reflected ray
  • Amplitudes Without interfaces With interfaces Basic problems General expression S M G
    • Reflected ray
    • Boundary condition
  • Amplitudes Without interfaces With interfaces Basic problems General expression S M G
    • Reflected ray
  • Amplitudes Without interfaces With interfaces
    • Transmitted ray
    Basic problems General expression S M G
    • Known
  • Amplitudes Without interfaces With interfaces
    • Transmitted ray
    Basic problems General expression S M G
  • Amplitudes Without interfaces With interfaces
    • Transmitted ray
    Basic problems General expression S M G
  • Amplitudes Without interfaces With interfaces
    • Transmitted ray
    Basic problems General expression S M G
  • Amplitudes Without interfaces With interfaces
    • Transmitted ray
    Basic problems General expression S M G
    • Boundary condition
  • Amplitudes Without interfaces With interfaces
    • Transmitted ray
    Basic problems General expression S M G
  • Amplitudes Without interfaces With interfaces Basic problems General expression 1 2 3 S G 5 4
    • To obtain a general amplitude expression, the two basic problems can be combined
  • Amplitudes Without interfaces With interfaces Basic problems General expression
  • Amplitudes Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes Ray equations Transport equation Geometric spreading   Amplitude     
  • Ray method Wave propagation Ray Ansatz Ray path and Traveltimes Amplitudes     Detalles de la teoría de rayos:
  • T H E E N D
  • Briefing
  • Appendix
  • Method of Characteristics  
  • PDE of First Order
  • Elastic Medium
  • Elastodynamic Wave Equation
    • Vector variables :
    • Wave Equation in Isotropic Media :
  • Acoustic Medium
  • Fourier Analysis
    • Before stating Zero-Order Ray hypothesis lets briefly review some
    • Fourier concepts
  • Fourier Transform Pair
    • Time to Frequency Fourier Transform :
    • Frequency to Time Transform :
  • Analytic Signal
    • Real and Causal signal:
    • Fourier representation:
  • Analytic Signal
    • Extend the definition to the Complex domain and define an Analytic function:
    • Hilbert Transform Pairs:
  • Analytic Signals
    • Hilbert Transform pairs and Signals :
  • Plane Wave
  • Plane Wave
    • Source Signal:
    • Fourier Transform :
  • Zero-Order Ray Theory(ZORT)
    • Elastodynamic case:
    • Acoustic case:
  • Transient Analytic ZORT
    • Elastodynamic case:
    • Acoustic case:
  • Hemlholtz Equation Fourier Transforming the Wave Equation we obtain Hemlholtz Equation
  • Asymptotic Aproximation
    • High frequency assumption:
    • Asymptotic Series, ZORT is the first term of the series:
  • Acoustic Case
  • Elastodynamics : Eikonal Equation
  • Elastodynamics : Transport Equation
  • Elastodynamics : Transport Equation
  • General Ray Equations
  • Eikonal Equation
    • Slowness Vector:
  • Eikonal Solution (Travel Times)
    • Characteristic Curve :
    • Eiconal equation in Hamiltonian form :
    • Traveltimes( Ray Tracing).
    • Partial Differential Equation(PDE) of First Order.
    • Nonlinear PDE.
    • Hamilton-Jacobi type.
    Eikonal Equation
  • Method of Characteristics
    • Solution Surface
     Initial Curve  , Characteristic Curves
  • PDE of First Order
  • Characteristic Equations Ordinary Differential Equation(ODE) System :
  • Ray Equations
  • Special Choices of  :
  • First Transport Equation
    • Integrating along a tube of rays :
  • Tube of Rays
  • Ray Coordinates
  • Jacobian of the Transformation :
  • Amplitude Formula
    • Integrating along the Ray Tube:
  • Initial Conditions
    • Point Source :
  • Amplitude due to a Point Source
  • Normalized Geometrical Spreading Factor
  • Line Caustic
  • Focus Caustic Point
  • Caustic Points
    • Line Caustic Points:
    • Focus Caustic Point:
  • Maslov Theory
    • Explain Ray theory in a Caustic Neighborhoods
  • R/T Coefficients(Snell´s Law)
    • Reflection
    • Condition :
    • Transmission
    • Condition :
  • Reflected Ray S G x z
  • Acoustic Reflection Coefficient
  • Transmitted Ray S x z
  • Acoustic Transmission Coefficient
  • Elastic Layered Medium Interface 1 Interface i Interface j Interface N-1 S = O 0 O 1  i +  i - O i G = O N  j +  j - O j
  • Amplitude at the Geophone
  • Target Reflector Amplitude
  • Differential Scalar Operators
    • Gradient of u:
    • Laplacian of u :
  • Polynomial Equation in 
    • Equating coefficients of  :