Basic Ray Theory
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Transcript of Basic Ray Theory
![Page 1: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/1.jpg)
Teoría de Rayos BásicaTeoría de Rayos Básicaporpor
Carlos César PiedrahitaCarlos César Piedrahita
![Page 2: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/2.jpg)
OverviewOverview
• IntroductionIntroduction
• Wave propagationWave propagation
• Ray methodRay method– Ray pathRay path
– TraveltimesTraveltimes
– AmplitudesAmplitudes
![Page 3: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/3.jpg)
Ray methodRay methodWave Wave
propagationpropagation
Ray AnsatzRay Ansatz
Ray path andRay path andTraveltimesTraveltimes
AmplitudesAmplitudes
![Page 4: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/4.jpg)
Ray methodRay methodWave Wave
propagationpropagation
Ray AnsatzRay Ansatz
Ray path andRay path andTraveltimesTraveltimes
AmplitudesAmplitudes
![Page 5: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/5.jpg)
Ray methodRay methodWave Wave
propagationpropagation
Ray AnsatzRay Ansatz
Ray path andRay path andTraveltimesTraveltimes
AmplitudesAmplitudes
![Page 6: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/6.jpg)
Ray methodRay methodWave Wave
propagationpropagation
Ray AnsatzRay Ansatz
Ray path andRay path andTraveltimesTraveltimes
AmplitudesAmplitudes
![Page 7: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/7.jpg)
Ray methodRay methodWave Wave
propagationpropagation
Ray AnsatzRay Ansatz
Ray path andRay path andTraveltimesTraveltimes
AmplitudesAmplitudes
![Page 8: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/8.jpg)
Wave propagation Wave propagation Wave Wave
propagationpropagation
Ray AnsatzRay Ansatz
Ray path andRay path andTraveltimesTraveltimes
AmplitudesAmplitudes
Acoustic Acoustic equationequation
Elastodynamic Elastodynamic equationequation
Ray formulationRay formulation
![Page 9: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/9.jpg)
Wave propagation Wave propagation Wave Wave
propagationpropagation
Ray AnsatzRay Ansatz
Ray path andRay path andTraveltimesTraveltimes
AmplitudesAmplitudes
Acoustic Acoustic equationequation
Elastodynamic Elastodynamic equationequation
Ray formulationRay formulation
![Page 10: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/10.jpg)
Wave propagation Wave propagation Wave Wave
propagationpropagation
Ray AnsatzRay Ansatz
Ray path andRay path andTraveltimesTraveltimes
AmplitudesAmplitudes
Acoustic Acoustic equationequation
Elastodynamic Elastodynamic equationequation
Ray formulationRay formulation
![Page 11: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/11.jpg)
Wave propagation Wave propagation Wave Wave
propagationpropagation
Ray AnsatzRay Ansatz
Ray path andRay path andTraveltimesTraveltimes
AmplitudesAmplitudes
Acoustic Acoustic equationequation
Elastodynamic Elastodynamic equationequation
Ray formulationRay formulation
![Page 12: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/12.jpg)
Wave propagation Wave propagation Acoustic Acoustic equationequation
Elastodynamic Elastodynamic equationequation
Ray formulationRay formulation
![Page 13: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/13.jpg)
Wave propagation Wave propagation Acoustic Acoustic equationequation
Elastodynamic Elastodynamic equationequation
Ray formulationRay formulation
![Page 14: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/14.jpg)
Wave propagation Wave propagation
2
2
22 ),(
)(1),(
ttu
ctu
x
xx
Acoustic Acoustic equationequation
Elastodynamic Elastodynamic equationequation
Ray formulationRay formulation
![Page 15: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/15.jpg)
Wave propagation Wave propagation
ijijjiijj uuuu ,,,,, )]([)(
Acoustic Acoustic equationequation
Elastodynamic Elastodynamic equationequation
Ray formulationRay formulation
![Page 16: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/16.jpg)
Wave propagation Wave propagation
Observed wavefield:
superposition of isolated events
Acoustic Acoustic equationequation
Elastodynamic Elastodynamic equationequation
Ray formulationRay formulation
![Page 17: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/17.jpg)
Wave propagation Wave propagation Wave Wave
propagationpropagation
Ray AnsatzRay Ansatz
Ray path andRay path andTraveltimesTraveltimes
AmplitudesAmplitudes
Acoustic Acoustic equationequation
Elastodynamic Elastodynamic equationequation
Ray formulationRay formulation
![Page 18: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/18.jpg)
Ray methodRay methodWave Wave
propagationpropagation
Ray AnsatzRay Ansatz
Ray path andRay path andTraveltimesTraveltimes
AmplitudesAmplitudes
![Page 19: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/19.jpg)
Ray AnsatzRay AnsatzWave Wave
propagationpropagation
Ray AnsatzRay Ansatz
Ray path andRay path andTraveltimesTraveltimes
AmplitudesAmplitudes
ReflectionsReflections
ReflectionReflectioneventevent
Ray formulationRay formulation
![Page 20: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/20.jpg)
Ray AnsatzRay AnsatzWave Wave
propagationpropagation
Ray AnsatzRay Ansatz
Ray path andRay path andTraveltimesTraveltimes
AmplitudesAmplitudes
ReflectionsReflections
ReflectionReflectioneventevent
Ray formulationRay formulation
![Page 21: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/21.jpg)
Ray AnsatzRay AnsatzWave Wave
propagationpropagation
Ray AnsatzRay Ansatz
Ray path andRay path andTraveltimesTraveltimes
AmplitudesAmplitudes
ReflectionsReflections
ReflectionReflectioneventevent
Ray formulationRay formulation
![Page 22: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/22.jpg)
Ray AnsatzRay AnsatzWave Wave
propagationpropagation
Ray AnsatzRay Ansatz
Ray path andRay path andTraveltimesTraveltimes
AmplitudesAmplitudes
ReflectionsReflections
ReflectionReflectioneventevent
Ray formulationRay formulation
![Page 23: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/23.jpg)
Ray AnsatzRay AnsatzReflectionsReflections
ReflectionReflectioneventevent
Ray formulationRay formulation
![Page 24: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/24.jpg)
Ray AnsatzRay AnsatzReflectionsReflections
ReflectionReflectioneventevent
Ray formulationRay formulation
![Page 25: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/25.jpg)
Ray AnsatzRay AnsatzReflectionsReflections
ReflectionReflectioneventevent
Ray formulationRay formulation
![Page 26: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/26.jpg)
Ray AnsatzRay Ansatz
ReflectionsReflections
ReflectionReflectioneventevent
Ray formulationRay formulation
![Page 27: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/27.jpg)
Ray AnsatzRay Ansatz
t
ReflectionsReflections
ReflectionReflectioneventevent
Ray formulationRay formulation
![Page 28: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/28.jpg)
Ray AnsatzRay Ansatz
t
A
ReflectionsReflections
ReflectionReflectioneventevent
Ray formulationRay formulation
![Page 29: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/29.jpg)
Ray AnsatzRay Ansatz
t
A
ReflectionsReflections
ReflectionReflectioneventevent
Ray formulationRay formulation
![Page 30: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/30.jpg)
Ray AnsatzRay Ansatz
t
A
![Page 31: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/31.jpg)
Ray AnsatzRay Ansatz
t
A
ReflectionsReflections
ReflectionReflectioneventevent
Ray formulationRay formulation
![Page 32: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/32.jpg)
Ray AnsatzRay Ansatz
t
A
ReflectionsReflections
ReflectionReflectioneventevent
Ray formulationRay formulation
![Page 33: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/33.jpg)
Ray AnsatzRay Ansatz
t
A
ReflectionsReflections
ReflectionReflectioneventevent
Ray formulationRay formulation
![Page 34: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/34.jpg)
Ray AnsatzRay Ansatz
t
A
ReflectionsReflections
ReflectionReflectioneventevent
Ray formulationRay formulation
![Page 35: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/35.jpg)
Ray AnsatzRay Ansatz
t
A
))(()(),( xxx tFAtu
Structure of reflected event
A(x) = amplitude
(x) = traveltime
Unknowns:
![Page 36: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/36.jpg)
Ray AnsatzRay AnsatzWave Wave
propagationpropagation
Ray AnsatzRay Ansatz
Ray path andRay path andTraveltimesTraveltimes
AmplitudesAmplitudes
ReflectionsReflections
ReflectionReflectioneventevent
Ray formulationRay formulation
![Page 37: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/37.jpg)
Ray methodRay methodWave Wave
propagationpropagation
Ray AnsatzRay Ansatz
Ray path andRay path andTraveltimesTraveltimes
AmplitudesAmplitudes
![Page 38: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/38.jpg)
Ray path and traveltimesRay path and traveltimesWave Wave
propagationpropagation
Ray AnsatzRay Ansatz
Ray path andRay path andTraveltimesTraveltimes
AmplitudesAmplitudes
Eikonal Eikonal equationequation
Ray equationsRay equations
Ray pathRay path
TraveltimesTraveltimes
![Page 39: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/39.jpg)
Ray path and traveltimesRay path and traveltimesWave Wave
propagationpropagation
Ray AnsatzRay Ansatz
Ray path andRay path andTraveltimesTraveltimes
AmplitudesAmplitudes
Eikonal Eikonal equationequation
Ray equationsRay equations
Ray pathRay path
TraveltimesTraveltimes
![Page 40: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/40.jpg)
Ray path and traveltimesRay path and traveltimesWave Wave
propagationpropagation
Ray AnsatzRay Ansatz
Ray path andRay path andTraveltimesTraveltimes
AmplitudesAmplitudes
Eikonal Eikonal equationequation
Ray equationsRay equations
Ray pathRay path
TraveltimesTraveltimes
![Page 41: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/41.jpg)
Ray path and traveltimesRay path and traveltimesWave Wave
propagationpropagation
Ray AnsatzRay Ansatz
Ray path andRay path andTraveltimesTraveltimes
AmplitudesAmplitudes
Eikonal Eikonal equationequation
Ray equationsRay equations
Ray pathRay path
TraveltimesTraveltimes
![Page 42: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/42.jpg)
Ray path and traveltimesRay path and traveltimesEikonal Eikonal equationequation
Ray equationsRay equations
Ray pathRay path
TraveltimesTraveltimes
![Page 43: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/43.jpg)
Ray path and traveltimesRay path and traveltimesEikonal Eikonal
equationequation
Ray equationsRay equations
Ray pathRay path
TraveltimesTraveltimes
![Page 44: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/44.jpg)
Ray path and traveltimesRay path and traveltimes
• Traveltimes (Ray Tracing)Traveltimes (Ray Tracing)
• Partial differential equation(PDE) of Partial differential equation(PDE) of
first orderfirst order
• Nonlinear PDENonlinear PDE
• Hamilton-Jacobi typeHamilton-Jacobi type
Eikonal Eikonal equationequation
Ray equationsRay equations
Ray pathRay path
TraveltimesTraveltimes
22)()(
xx c
![Page 45: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/45.jpg)
Ray path and traveltimesRay path and traveltimesEikonal Eikonal
equationequation
Ray equationsRay equations
Ray pathRay path
TraveltimesTraveltimes
HamiltonianHamiltonian
Method of Method of characteristicscharacteristics
Ray equationsRay equations
Ray jacobianRay jacobian
![Page 46: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/46.jpg)
Ray path and traveltimesRay path and traveltimesEikonal Eikonal
equationequation
Ray equationsRay equations
Ray pathRay path
TraveltimesTraveltimes
HamiltonianHamiltonian
Method of Method of characteristicscharacteristics
Ray equationsRay equations
Ray jacobianRay jacobian
![Page 47: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/47.jpg)
Ray path and traveltimesRay path and traveltimesEikonal Eikonal
equationequation
Ray equationsRay equations
Ray pathRay path
TraveltimesTraveltimes
HamiltonianHamiltonian
Method of Method of characteristicscharacteristics
Ray equationsRay equations
Ray jacobianRay jacobian
![Page 48: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/48.jpg)
Ray path and traveltimesRay path and traveltimesHamiltonianHamiltonian
Method of Method of characteristicscharacteristics
Ray equationsRay equations
Ray jacobianRay jacobian
![Page 49: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/49.jpg)
Ray path and traveltimesRay path and traveltimesHamiltonianHamiltonian
Method of Method of characteristicscharacteristics
Ray equationsRay equations
Ray jacobianRay jacobian
![Page 50: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/50.jpg)
Ray path and traveltimesRay path and traveltimesHamiltonianHamiltonian
Method of Method of characteristicscharacteristics
Ray equationsRay equations
)(
)(21),,( 22
xp
xppx
cH
• Eikonal as a HamiltonianEikonal as a Hamiltonian
Ray jacobianRay jacobian
![Page 51: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/51.jpg)
Ray path and traveltimesRay path and traveltimesHamiltonianHamiltonian
Method of Method of characteristicscharacteristics
Ray equationsRay equations0Hd
• A special change of variablesA special change of variables
),,(),,( 21321 sxxx
where, along curves where, along curves ,
• is called characteristicRay jacobianRay jacobian
![Page 52: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/52.jpg)
Ray path and traveltimesRay path and traveltimesHamiltonianHamiltonian
Method of Method of characteristicscharacteristics
Ray equationsRay equations
0Hd• Along characteristics curves Along characteristics curves
Ray jacobianRay jacobian
![Page 53: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/53.jpg)
Ray path and traveltimesRay path and traveltimesHamiltonianHamiltonian
Method of Method of characteristicscharacteristics
Ray equationsRay equations
Ray jacobianRay jacobian
• System of 7 ordinary differential System of 7 ordinary differential equations (ODE)equations (ODE)
)(1
)(1
)(
x
xp
pxx
cdsd
cdsd
cdsd
![Page 54: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/54.jpg)
Ray path and traveltimesRay path and traveltimesHamiltonianHamiltonian
Method of Method of characteristicscharacteristics
Ray equationsRay equations
Ray jacobianRay jacobian
• Point source initial conditionPoint source initial condition
o
s
s
c
)0(
)cos,sinsin,sin(cos)0(
)0(
221211p
xx
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Ray path and traveltimesRay path and traveltimesHamiltonianHamiltonian
Method of Method of characteristicscharacteristics
Ray equationsRay equations
• Ray coordinatesRay coordinates
2
1
p0
(1, 2, s)
Ray jacobianRay jacobian
![Page 56: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/56.jpg)
Ray path and traveltimesRay path and traveltimesHamiltonianHamiltonian
Method of Method of characteristicscharacteristics
Ray equationsRay equations
Ray jacobianRay jacobians
x
xxxsJ
j
i
3
321
321
det
),,(),,()(
• Ray jacobianRay jacobian
![Page 57: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/57.jpg)
Ray path and traveltimesRay path and traveltimesHamiltonianHamiltonian
Method of Method of characteristicscharacteristics
Ray equationsRay equations
Ray jacobianRay jacobian
22sin)( ssJ
• Example of ray jacobian: Example of ray jacobian:
Point source and constant velocity Point source and constant velocity
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Ray path and traveltimesRay path and traveltimes• System of 7 ordinary differential System of 7 ordinary differential
equations (ODE)equations (ODE)
Eikonal Eikonal equationequation
Ray equationsRay equations
Ray pathRay path
TraveltimesTraveltimes )(1
)(1
)(
x
xp
pxx
cdsd
cdsd
cdsd
![Page 59: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/59.jpg)
Ray path and traveltimesRay path and traveltimesEikonal Eikonal
equationequation
Ray equationsRay equations
Ray pathRay path
TraveltimesTraveltimes
Without Without interfacesinterfaces
With interfacesWith interfaces
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Ray path and traveltimesRay path and traveltimesWithout Without
interfacesinterfaces
With interfacesWith interfaces
• Requires a smooth velocity model Requires a smooth velocity model and one initial conditionand one initial condition
• 44thth order Runge-Kutta can be used order Runge-Kutta can be used
xS
x
p0
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Ray path and traveltimesRay path and traveltimesWithout Without
interfacesinterfaces
With interfacesWith interfaces
• Requires new initial conditions for Requires new initial conditions for each interfaceeach interface
• Snell’s law must be usedSnell’s law must be used
First initialcondition
Second initialcondition
Third initialcondition
Incident ray
reflected ray
transmitted ray
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Ray path and traveltimesRay path and traveltimesWithout Without
interfacesinterfaces
With interfacesWith interfaces
• Snell’s lawSnell’s law
j
j
j
j
cc)(sin)(sin
1
1
j
j+1
cj
cj+!
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Ray path and traveltimesRay path and traveltimesWithout Without
interfacesinterfaces
With interfacesWith interfaces
• Example: Six initial conditionsExample: Six initial conditions
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Ray path and traveltimesRay path and traveltimesWithout Without
interfacesinterfaces
With interfacesWith interfaces
• Example: Eight initial conditions Example: Eight initial conditions (multiple)(multiple)
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Ray path and traveltimesRay path and traveltimesEikonal Eikonal
equationequation
Ray equationsRay equations
Ray pathRay path
TraveltimesTraveltimes
ray
))(()(
scdssx
• Traveltime is computed along the Traveltime is computed along the ray ray
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Ray path and traveltimesRay path and traveltimesWave Wave
propagationpropagation
Ray AnsatzRay Ansatz
Ray path andRay path andTraveltimesTraveltimes
AmplitudesAmplitudes
Eikonal Eikonal equationequation
Ray equationsRay equations
Ray pathRay path
TraveltimesTraveltimes
![Page 67: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/67.jpg)
Ray methodRay methodWave Wave
propagationpropagation
Ray AnsatzRay Ansatz
Ray path andRay path andTraveltimesTraveltimes
AmplitudesAmplitudes
![Page 68: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/68.jpg)
AmplitudesAmplitudesWave Wave
propagationpropagation
Ray AnsatzRay Ansatz
Ray path andRay path andTraveltimesTraveltimes
AmplitudesAmplitudes
TransportTransport equationequation
Ray equationsRay equations
GeometricGeometricspreadingspreading
AmplitudeAmplitude
![Page 69: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/69.jpg)
AmplitudesAmplitudesWave Wave
propagationpropagation
Ray AnsatzRay Ansatz
Ray path andRay path andTraveltimesTraveltimes
AmplitudesAmplitudes
TransportTransport equationequation
Ray equationsRay equations
GeometricGeometricspreadingspreading
AmplitudeAmplitude
![Page 70: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/70.jpg)
AmplitudesAmplitudesWave Wave
propagationpropagation
Ray AnsatzRay Ansatz
Ray path andRay path andTraveltimesTraveltimes
AmplitudesAmplitudes
TransportTransport equationequation
Ray equationsRay equations
GeometricGeometricspreadingspreading
AmplitudeAmplitude
![Page 71: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/71.jpg)
AmplitudesAmplitudesWave Wave
propagationpropagation
Ray AnsatzRay Ansatz
Ray path andRay path andTraveltimesTraveltimes
AmplitudesAmplitudes
TransportTransportequationequation
Ray equationsRay equations
GeometricGeometricspreadingspreading
AmplitudeAmplitude
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AmplitudesAmplitudesTransportTransportequationequation
Ray equationsRay equations
GeometricGeometricspreadingspreading
AmplitudeAmplitude
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AmplitudesAmplitudesTransportTransportequationequation
Ray equationsRay equations
GeometricGeometricspreadingspreading
AmplitudeAmplitude
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AmplitudesAmplitudesTransportTransport equationequation
Ray equationsRay equations
GeometricGeometricspreadingspreading
AmplitudeAmplitude
02 AA
• Uses the ray solutionUses the ray solution
• Partial Differential Equation (PDE) Partial Differential Equation (PDE)
of First Orderof First Order
![Page 75: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/75.jpg)
AmplitudesAmplitudesTransportTransport equationequation
Ray equationsRay equations
GeometricGeometricspreadingspreading
AmplitudeAmplitude
0div 2 pA
• Alternative expression:Alternative expression:
0ˆD
2
dSA np
• Applying Gauss’ theorem:Applying Gauss’ theorem:
![Page 76: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/76.jpg)
AmplitudesAmplitudesTransportTransport equationequation
Ray equationsRay equations
GeometricGeometricspreadingspreading
AmplitudeAmplitude
dS0
dS1
• Ray tubeRay tube
D
![Page 77: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/77.jpg)
AmplitudesAmplitudesTransportTransport equationequation
Ray equationsRay equations
GeometricGeometricspreadingspreading
AmplitudeAmplitude
• Ray tubeRay tube
00
20
11
21
)()(
)()( dS
scsAdS
scsA
![Page 78: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/78.jpg)
AmplitudesAmplitudesTransportTransport equationequation
Ray equationsRay equations
GeometricGeometricspreadingspreading
AmplitudeAmplitude
• Ray tubeRay tube
2100
20
2111
21
)()()(
)()()(
ddsJscsA
ddsJscsA
![Page 79: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/79.jpg)
AmplitudesAmplitudesTransportTransport equationequation
Ray equationsRay equations
GeometricGeometricspreadingspreading
AmplitudeAmplitude
• Ray tubeRay tube
)()(
)()()()(
1
0
0
120
21 sJ
sJscscsAsA
![Page 80: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/80.jpg)
AmplitudesAmplitudesTransportTransport equationequation
Ray equationsRay equations
GeometricGeometricspreadingspreading
AmplitudeAmplitude
• Ray tubeRay tube
)()(
)()()()(
1
0
0
120
21 sJ
sJscscsAsA
• Geometrical spreadingGeometrical spreading
)()()()(
10
121 scsc
sJs L
![Page 81: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/81.jpg)
AmplitudesAmplitudesTransportTransport equationequation
Ray equationsRay equations
GeometricGeometricspreadingspreading
AmplitudeAmplitude
Without Without interfacesinterfaces
With interfacesWith interfaces
Basic problemsBasic problems
GeneralGeneralexpressionexpression
![Page 82: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/82.jpg)
AmplitudesAmplitudesWithout Without
interfacesinterfaces
With interfacesWith interfaces
)()()(
)()()( 0
2/1
0
0
sAsJsJ
scscsA
A(s)
A(s0)Basic problemsBasic problems
GeneralGeneralexpressionexpression
![Page 83: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/83.jpg)
AmplitudesAmplitudesWithout Without
interfacesinterfaces
With interfacesWith interfaces
• Point source initial condition:Point source initial condition:
Basic problemsBasic problems
GeneralGeneralexpressionexpression
![Page 84: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/84.jpg)
AmplitudesAmplitudesWithout Without
interfacesinterfaces
With interfacesWith interfaces
• Requires new initial conditions for Requires new initial conditions for each interfaceeach interface
• Snell’s law must be usedSnell’s law must be used
• Reflection coefficient must be used Reflection coefficient must be used
(boundary conditions)(boundary conditions)Basic problemsBasic problems
GeneralGeneralexpressionexpression
![Page 85: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/85.jpg)
AmplitudesAmplitudesWithout Without
interfacesinterfaces
With interfacesWith interfaces
• Acoustic reflection coefficientAcoustic reflection coefficient
)(sin)cos(
)(sin)cos(22
1
221
jjjj
jjjjj
cc
ccR
• Acoustic transmission coefficientAcoustic transmission coefficient
jj RT 1
Basic problemsBasic problems
GeneralGeneralexpressionexpression
![Page 86: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/86.jpg)
SS JA ,
AmplitudesAmplitudesWithout Without
interfacesinterfaces
With interfacesWith interfacesS
M
G
Basic problemsBasic problems
GeneralGeneralexpressionexpression
• Reflected rayReflected ray
• KnownKnown
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AmplitudesAmplitudesWithout Without
interfacesinterfaces
With interfacesWith interfaces
• Reflected rayReflected ray
S
M
G
Basic problemsBasic problems
GeneralGeneralexpressionexpression
SincM
S
S
MM A
JJ
ccA
2/1
![Page 88: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/88.jpg)
AmplitudesAmplitudesWithout Without
interfacesinterfaces
With interfacesWith interfaces
Basic problemsBasic problems
GeneralGeneralexpressionexpression
S
M
G• Reflected rayReflected ray
MG
refM
M
GG A
JJ
ccA
![Page 89: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/89.jpg)
AmplitudesAmplitudesWithout Without
interfacesinterfaces
With interfacesWith interfaces
Basic problemsBasic problems
GeneralGeneralexpressionexpression
S
M
G• Reflected rayReflected ray
SincM
S
S
M
G
refM
M
GG A
JJ
cc
JJ
ccRA
2/1
![Page 90: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/90.jpg)
AmplitudesAmplitudesWithout Without
interfacesinterfaces
With interfacesWith interfaces
Basic problemsBasic problems
GeneralGeneralexpressionexpression
S
M
G• Reflected rayReflected ray
1coscos
ref
incincM
refM
JJ
• Boundary conditionBoundary condition
![Page 91: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/91.jpg)
AmplitudesAmplitudesWithout Without
interfacesinterfaces
With interfacesWith interfaces
Basic problemsBasic problems
GeneralGeneralexpressionexpression
S
M
G• Reflected rayReflected ray
SG
S
S
GG A
JJ
ccRA
2/1
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AmplitudesAmplitudesWithout Without
interfacesinterfaces
With interfacesWith interfaces
• Transmitted rayTransmitted ray
Basic problemsBasic problems
GeneralGeneralexpressionexpression
S
M G
SS JA ,• KnownKnown
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AmplitudesAmplitudesWithout Without
interfacesinterfaces
With interfacesWith interfaces
• Transmitted rayTransmitted ray
Basic problemsBasic problems
GeneralGeneralexpressionexpression
S
M G
SincM
S
S
MM A
JJ
ccA
2/1
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AmplitudesAmplitudesWithout Without
interfacesinterfaces
With interfacesWith interfaces
• Transmitted rayTransmitted ray
Basic problemsBasic problems
GeneralGeneralexpressionexpression
S
M G
MG
traM
M
GG A
JJ
ccA
![Page 95: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/95.jpg)
AmplitudesAmplitudesWithout Without
interfacesinterfaces
With interfacesWith interfaces
• Transmitted rayTransmitted ray
Basic problemsBasic problems
GeneralGeneralexpressionexpression
S
M G
SincM
S
S
M
G
traM
M
GG A
JJ
cc
JJ
ccTA
2/1
![Page 96: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/96.jpg)
AmplitudesAmplitudesWithout Without
interfacesinterfaces
With interfacesWith interfaces
• Transmitted rayTransmitted ray
Basic problemsBasic problems
GeneralGeneralexpressionexpression
S
M G
• Boundary conditionBoundary condition
1coscos
tra
incincM
traM
JJ
![Page 97: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/97.jpg)
AmplitudesAmplitudesWithout Without
interfacesinterfaces
With interfacesWith interfaces
• Transmitted rayTransmitted ray
Basic problemsBasic problems
GeneralGeneralexpressionexpression
S
M G
SG
S
tra
inc
S
GG A
JJ
ccRA
2/1
coscos
![Page 98: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/98.jpg)
AmplitudesAmplitudesWithout Without
interfacesinterfaces
With interfacesWith interfaces
Basic problemsBasic problems
GeneralGeneralexpressionexpression
1
2
3
S G
5
4
• To obtain a general amplitude To obtain a general amplitude expression, the two basic problems expression, the two basic problems can be combinedcan be combined
![Page 99: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/99.jpg)
AmplitudesAmplitudesWithout Without
interfacesinterfaces
With interfacesWith interfaces
ii
iii
S
N
iii
G
S
G
SG
ccK
ARKccA
coscos
1LL
Basic problemsBasic problems
GeneralGeneralexpressionexpression
![Page 100: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/100.jpg)
AmplitudesAmplitudesWave Wave
propagationpropagation
Ray AnsatzRay Ansatz
Ray path andRay path andTraveltimesTraveltimes
AmplitudesAmplitudes
Ray equationsRay equations
TransportTransport equationequation
GeometricGeometricspreadingspreading
AmplitudeAmplitude
![Page 101: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/101.jpg)
Ray methodRay methodWave Wave
propagationpropagation
Ray AnsatzRay Ansatz
Ray path andRay path andTraveltimesTraveltimes
AmplitudesAmplitudes
Detalles de la teoría de rayos:
![Page 102: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/102.jpg)
T H E E N D
![Page 103: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/103.jpg)
BriefingBriefing
A cou s ticeq u a tion
E las tod yn am iceq u a tion
R ay fo rm u la tion
W aveP rop ag a tion
E ven ts
R eflec ted even ts
R ay fo rm u la tion
R ay A n sa tz
E ikon a leq u ation
R ay eq u ation s
R ay p a th
Trave lt im e
R ay p a th an dtrave lt im e
Tran sp ortE q u ation
R ay eq u a tion s
Jacob ian
A m p litu d e
A m p litu d e
R ay M eth od
![Page 104: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/104.jpg)
AppendixAppendix
![Page 105: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/105.jpg)
Method of CharacteristicsMethod of Characteristics
![Page 106: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/106.jpg)
PDE of First OrderPDE of First Order
)(
,,,;,, where
,,)()(,RI,0),,(
321321
00
3
xpp
xxxxpx
pppxxxx
gH
![Page 107: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/107.jpg)
Elastic MediumElastic Medium
1r
2r
3r pSHu
SVu
r
![Page 108: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/108.jpg)
Elastodynamic Wave Elastodynamic Wave EquationEquation• Vector variables :Vector variables :
• Wave Equation in Isotropic Media :Wave Equation in Isotropic Media :
vector ntdisplaceme
vector position
),,(
),,,( 321
t
rrrruu
r
.3,2,1,)]([)( ,,,,, iuuuu ijijjiijj
![Page 109: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/109.jpg)
Acoustic MediumAcoustic Medium
.)(,0 2
2
t
uu
ctp
cp
p
,1
,.
2
2
22
constant,
pressure,u
![Page 110: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/110.jpg)
Fourier AnalysisFourier Analysis
• Before stating Zero-Order Ray hypothesis Before stating Zero-Order Ray hypothesis lets briefly review somelets briefly review some
Fourier conceptsFourier concepts
![Page 111: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/111.jpg)
Fourier Transform PairFourier Transform Pair• Time to Frequency Fourier Transform :Time to Frequency Fourier Transform :
• Frequency to Time Transform : Frequency to Time Transform :
titfF exp
tiFtf
exp)(21)(
![Page 112: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/112.jpg)
Analytic SignalAnalytic Signal• Real and Causal signal:Real and Causal signal:
• Fourier representation:Fourier representation:
.0,0)( ttf
.)exp(][Re1
)(0
dtiftf
![Page 113: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/113.jpg)
Analytic SignalAnalytic Signal• Extend the definition to the Complex Extend the definition to the Complex
domain and define an Analytic function:domain and define an Analytic function:
• Hilbert Transform Pairs:Hilbert Transform Pairs:]].[[][],[][][ tfHtgtgitftF
.)exp(][1
][0
dtiftF
![Page 114: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/114.jpg)
Analytic SignalsAnalytic Signals• Hilbert Transform pairs and Signals :Hilbert Transform pairs and Signals :
.1
][][,1
]][[
].[]cos[][],[]][cos[
tittF
ttH
tsinittFtsintH
![Page 115: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/115.jpg)
2r
Plane WavePlane Wave
1r
3r
n
![Page 116: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/116.jpg)
Plane WavePlane Wave• Source Signal:Source Signal:
• Fourier Transform :Fourier Transform :
].[)( tFt Uu
][)exp()( fit Uu
![Page 117: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/117.jpg)
Zero-Order Ray Zero-Order Ray Theory(ZORT)Theory(ZORT)
• Elastodynamic case:Elastodynamic case:
• Acoustic case:Acoustic case:
].[)](exp[)(),( fi rrUru
].[)](exp[)(),( fiPp rrr
![Page 118: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/118.jpg)
Transient Analytic ZORTTransient Analytic ZORT• Elastodynamic case:Elastodynamic case:
• Acoustic case:Acoustic case:
)].([)(),( rrUru tFt
)].([)(),( rrr tFPtp
![Page 119: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/119.jpg)
Hemlholtz EquationHemlholtz Equation
medium. the of velocity
frequency,
vector, position ,
r
r
rx
c
ucrrr
rrr
321
2
2
,,
,,ˆ)(
0321
Fourier Transforming the Wave Equation we obtain Hemlholtz Equation
![Page 120: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/120.jpg)
Asymptotic AproximationAsymptotic Aproximation• High frequency assumption:High frequency assumption:
• Asymptotic Series, ZORT is the first term of Asymptotic Series, ZORT is the first term of the series:the series:
.1,/ cLo
![Page 121: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/121.jpg)
Acoustic CaseAcoustic Case
./1)(:
,021:
,0:
222
21
20
cT
TP
PP
PP
![Page 122: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/122.jpg)
Elastodynamics : Eikonal Elastodynamics : Eikonal EquationEquation
wave.-S
wave.-P
matrix. lChristoffe
0U
,,1)(
.,,1)(
ˆ
,)ˆˆ(
22
22
2
T
T
I
![Page 123: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/123.jpg)
Elastodynamics : Transport EquationElastodynamics : Transport Equation
.0
,
22
)(
TA
TAP
U
:waves-P
![Page 124: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/124.jpg)
Elastodynamics : Transport EquationElastodynamics : Transport Equation
.0)(
,
222
2)(
TCB
CBS
eeU
:waves-S
1
![Page 125: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/125.jpg)
General Ray EquationsGeneral Ray Equations
waves.S
waves.P
waves.Acoustic
Equation. Transport General
Equation. Eikonal General
,,
,,2
,1,1
,0
,
2
2
gf
gf
gf
TAf
fgT
![Page 126: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/126.jpg)
Eikonal EquationEikonal Equation• Slowness Vector:Slowness Vector:
.Tp
p
Wavefront
![Page 127: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/127.jpg)
Eikonal Solution (Travel Eikonal Solution (Travel Times)Times)• Characteristic Curve :Characteristic Curve :
• Eiconal equation in Hamiltonian form :Eiconal equation in Hamiltonian form :
,)(),(|)(),( 6 tttt pprrpr
.)(
121,, 2
2
rppr
cH
![Page 128: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/128.jpg)
• Traveltimes( Ray Tracing).Traveltimes( Ray Tracing).
• Partial Differential Equation(PDE) of First Partial Differential Equation(PDE) of First Order.Order.
• Nonlinear PDE.Nonlinear PDE.
• Hamilton-Jacobi type.Hamilton-Jacobi type.
Eikonal EquationEikonal Equation
![Page 129: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/129.jpg)
Method of CharacteristicsMethod of Characteristics• Solution SurfaceSolution Surface
InitialCurve
, Characteristic Curves
![Page 130: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/130.jpg)
PDE of First OrderPDE of First Order
.
,,,,,,
,,,,0,,H
321321
3
0
rppand r
where
rrrrpr
ppprrr
g
![Page 131: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/131.jpg)
Characteristic EquationsCharacteristic Equations
Parameter Ray
p
pp
r
Hdtd
HHdtd
Hdtd
p
r
p
Ordinary Differential Equation(ODE) System :
![Page 132: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/132.jpg)
Ray EquationsRay Equations
2)(1
)(1
)(1
r
rrp
pr
cdtd
ccdtddtd
![Page 133: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/133.jpg)
Special Choices of Special Choices of : :
time. travelx
. lenght arcx
physical. non ,
,,)(
,),(
,1
2
tc
stc
t
![Page 134: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/134.jpg)
First Transport EquationFirst Transport Equation• Integrating along a tube of rays : Integrating along a tube of rays :
SVdTAv
TAv
.0.
,0
220
22
Slim
:theorem Green´s using
![Page 135: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/135.jpg)
Tube of RaysTube of Rays
1dS
2dS
B
![Page 136: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/136.jpg)
Ray CoordinatesRay Coordinates
1r
2r
3r
0p
O
1 2
s),,( Point 21
![Page 137: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/137.jpg)
Jacobian of the Jacobian of the Transformation :Transformation :
.
,),,(
),,(
),,,(),,(
21
21
321
21321
dsddJV
sxxxJ
sxxx
![Page 138: Basic Ray Theory](https://reader036.fdocuments.in/reader036/viewer/2022062401/5403305c8d7f72703c8b48ec/html5/thumbnails/138.jpg)
Amplitude FormulaAmplitude Formula• Integrating along the Ray Tube:Integrating along the Ray Tube:
Factor Spreading lGeometrica
JJ
JvJvtAtA
o
oooo
)()(
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Initial ConditionsInitial Conditions• Point Source :Point Source :
.p
rr
o
s
s
sinsinsinc
)0(
),cos,,(cos)0(
,)0(
221211
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Amplitude due to a Point Amplitude due to a Point SourceSource
strenght. source,lim
,1
ooooSP
GGSS
AJvg
JvvgA
o
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Normalized Geometrical Normalized Geometrical Spreading FactorSpreading Factor
.1
,
22 Lvv
gA
vvJL
GGSS
GS
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Line CausticLine Caustic
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Focus Caustic PointFocus Caustic Point
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Caustic PointsCaustic Points• Line Caustic Points: Line Caustic Points:
• Focus Caustic Point:Focus Caustic Point:
0J
.
,2)(
2 shift phase
rank
J
. shift phaserank
,1)(J
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Maslov TheoryMaslov Theory• Explain Ray theory in a Caustic Explain Ray theory in a Caustic
NeighborhoodsNeighborhoods
caustic point2caustic line
Index KMAH1
,22/1
ieJJ
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R/T Coefficients(Snell´s Law)R/T Coefficients(Snell´s Law)• ReflectionReflection
Condition :Condition :
• Transmission Transmission
Condition :Condition :
j
j
j
j
csin
csin )()(
1
1
jj 1
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Reflected RayReflected Ray
x
z
S G
A Medium
B Medium
j Interfacej 1j
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Acoustic Reflection Acoustic Reflection CoefficientCoefficient
)()cos(
)()cos(
2
2
1
2
2
1
jj
jj
jj
jj
sincc
sincc
R
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Transmitted RayTransmitted Ray
x
z
S
A Medium
B Medium
j Interfacej
1j
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Acoustic Transmission Acoustic Transmission CoefficientCoefficient
)()cos(
)cos(2
2
2
1j
j
jj
j
sincc
T
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Elastic Layered MediumElastic Layered Medium
Interface 1
Interface i
Interface j
Interface N-1
S = O0
O1
i+
i-
Oi
G = ON
j+j
-Oj
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Amplitude at the GeophoneAmplitude at the Geophone
1
2/1
111
2/1
2 coscos A
JvJvc
vvA
GGGi
iii
iiiN
iG
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Target Reflector AmplitudeTarget Reflector Amplitude
.cos
cos~
,~
2/1
jjjj
jjjj
jG
cv
vc
cLA
A
.1
,~122
G
SG
i
N
jii
SSGG
Jvv
L
cvv
gA
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Differential Scalar OperatorsDifferential Scalar Operators• Gradient of u:Gradient of u:
• Laplacian of u :Laplacian of u :
uxuxux
xxx
uuuu
321
111
u xxu xxu xx
xxxxxx
uuuu
332211
332211
222
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Polynomial Equation in Polynomial Equation in • Equating coefficients of Equating coefficients of : :
,2,1,2
02
)(
122
22
kAAA
AA
c
kkk
: Equation Transport Order Higher
.: Equation Transport First
: Equation Eikonal
xxxx
xxxx
xx