basic ray theory

Teoría de Rayos BásicaTeoría de Rayos Básicaporpor

Carlos César PiedrahitaCarlos César Piedrahita

OverviewOverview

• IntroductionIntroduction

• Wave propagationWave propagation

• Ray methodRay method– Ray pathRay path

– TraveltimesTraveltimes

– AmplitudesAmplitudes

Ray methodRay methodWave Wave

propagationpropagation

Ray AnsatzRay Ansatz

Ray path andRay path andTraveltimesTraveltimes

AmplitudesAmplitudes

Wave propagation Wave propagation Wave Wave





Acoustic Acoustic equationequation

Elastodynamic Elastodynamic equationequation

Ray formulationRay formulation

Wave propagation Wave propagation Acoustic Acoustic equationequation



Wave propagation Wave propagation

2

2

22 ),(

)(1),(

ttu

ctu

x

xx





ijijjiijj uuuu ,,,,, )]([)(





Observed wavefield:

superposition of isolated events




Wave propagation Wave propagation Wave Wave








Ray AnsatzRay AnsatzWave Wave





ReflectionsReflections

ReflectionReflectioneventevent


Ray AnsatzRay AnsatzReflectionsReflections




t





t

A





t

A

))(()(),( xxx tFAtu

Structure of reflected event

A(x) = amplitude

(x) = traveltime

Unknowns:

Ray AnsatzRay AnsatzWave Wave








Ray path and traveltimesRay path and traveltimesWave Wave





Eikonal Eikonal equationequation

Ray equationsRay equations

Ray pathRay path

TraveltimesTraveltimes

Ray path and traveltimesRay path and traveltimesEikonal Eikonal equationequation


Ray pathRay path


Ray path and traveltimesRay path and traveltimesEikonal Eikonal

equationequation


Ray pathRay path


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



Ray pathRay path


22)()(

xx c


equationequation


Ray pathRay path


HamiltonianHamiltonian

Method of Method of characteristicscharacteristics


Ray jacobianRay jacobian

Ray path and traveltimesRay path and traveltimesHamiltonianHamiltonian







)(

)(21),,( 22

xp

xppx

cH

• Eikonal as a HamiltonianEikonal as a Hamiltonian




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




0Hd• Along characteristics curves Along characteristics curves






• System of 7 ordinary differential System of 7 ordinary differential equations (ODE)equations (ODE)

)(1

)(1

)(

x

xp

pxx

cdsd

cdsd

cdsd





• Point source initial conditionPoint source initial condition

o

s

s

c

)0(

)cos,sinsin,sin(cos)0(

)0(

221211p

xx




• Ray coordinatesRay coordinates

2

1

p0

(1, 2, s)





Ray jacobianRay jacobians

x

xxxsJ

j

i

3

321

321

det

),,(),,()(

• Ray jacobianRay jacobian





22sin)( ssJ

• Example of ray jacobian: Example of ray jacobian:

Point source and constant velocity Point source and constant velocity

Ray path and traveltimesRay path and traveltimes• System of 7 ordinary differential System of 7 ordinary differential

equations (ODE)equations (ODE)



Ray pathRay path

TraveltimesTraveltimes )(1

)(1

)(

x

xp

pxx

cdsd

cdsd

cdsd


equationequation


Ray pathRay path


Without Without interfacesinterfaces

With interfacesWith interfaces

Ray path and traveltimesRay path and traveltimesWithout Without

interfacesinterfaces


• 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




• 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




• Snell’s lawSnell’s law

j

j

j

j

cc)(sin)(sin

1

1

j

j+1

cj

cj+!




• Example: Six initial conditionsExample: Six initial conditions




• Example: Eight initial conditions Example: Eight initial conditions (multiple)(multiple)


equationequation


Ray pathRay path


ray

))(()(

scdssx

• Traveltime is computed along the Traveltime is computed along the ray ray

Ray path and traveltimesRay path and traveltimesWave Wave







Ray pathRay path


AmplitudesAmplitudesWave Wave





TransportTransport equationequation


GeometricGeometricspreadingspreading

AmplitudeAmplitude






TransportTransportequationequation



AmplitudeAmplitude

AmplitudesAmplitudesTransportTransportequationequation



AmplitudeAmplitude

AmplitudesAmplitudesTransportTransport equationequation



AmplitudeAmplitude

02 AA

• Uses the ray solutionUses the ray solution

• Partial Differential Equation (PDE) Partial Differential Equation (PDE)

of First Orderof First Order




AmplitudeAmplitude

0div 2 pA

• Alternative expression:Alternative expression:

0ˆD

2

dSA np

• Applying Gauss’ theorem:Applying Gauss’ theorem:




AmplitudeAmplitude

dS0

dS1

• Ray tubeRay tube

D




AmplitudeAmplitude


00

20

11

21

)()(

)()( dS

scsAdS

scsA




AmplitudeAmplitude


2100

20

2111

21

)()()(

)()()(

ddsJscsA

ddsJscsA




AmplitudeAmplitude


)()(

)()()()(

1

0

0

120

21 sJ

sJscscsAsA




AmplitudeAmplitude


)()(

)()()()(

1

0

0

120

21 sJ

sJscscsAsA

• Geometrical spreadingGeometrical spreading

)()()()(

10

121 scsc

sJs L




AmplitudeAmplitude

Without Without interfacesinterfaces


Basic problemsBasic problems

GeneralGeneralexpressionexpression

AmplitudesAmplitudesWithout Without



)()()(

)()()( 0

2/1

0

0

sAsJsJ

scscsA

A(s)

A(s0)Basic problemsBasic problems





• Point source initial condition:Point source initial condition:






• 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





• Acoustic reflection coefficientAcoustic reflection coefficient

)(sin)cos(

)(sin)cos(22

1

221

jjjj

jjjjj

cc

ccR

• Acoustic transmission coefficientAcoustic transmission coefficient

jj RT 1



SS JA ,



With interfacesWith interfacesS

M

G



• Reflected rayReflected ray

• KnownKnown




• Reflected rayReflected ray

S

M

G



SincM

S

S

MM A

JJ

ccA

2/1






S

M

G• Reflected rayReflected ray

MG

refM

M

GG A

JJ

ccA






S

M


SincM

S

S

M

G

refM

M

GG A

JJ

cc

JJ

ccRA

2/1






S

M


1coscos

ref

incincM

refM

JJ

• Boundary conditionBoundary condition






S

M


SG

S

S

GG A

JJ

ccRA

2/1




• Transmitted rayTransmitted ray



S

M G

SS JA ,• KnownKnown







S

M G

SincM

S

S

MM A

JJ

ccA

2/1







S

M G

MG

traM

M

GG A

JJ

ccA







S

M G

SincM

S

S

M

G

traM

M

GG A

JJ

cc

JJ

ccTA

2/1







S

M G

• Boundary conditionBoundary condition

1coscos

tra

incincM

traM

JJ







S

M G

SG

S

tra

inc

S

GG A

JJ

ccRA

2/1

coscos






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




ii

iii

S

N

iii

G

S

G

SG

ccK

ARKccA

coscos

1LL









TransportTransport equationequation


AmplitudeAmplitude






Detalles de la teoría de rayos:

T H E E N D

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

AppendixAppendix

Method of CharacteristicsMethod of Characteristics

PDE of First OrderPDE of First Order

)(

,,,;,, where

,,)()(,RI,0),,(

321321

00

3

xpp

xxxxpx

pppxxxx

gH

Elastic MediumElastic Medium

1r

2r

3r pSHu

SVu

r

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

Acoustic MediumAcoustic Medium

.)(,0 2

2

t

uu

ctp

cp

p

,1

,.

2

2

22

constant,

pressure,u

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

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)(

Analytic SignalAnalytic Signal• Real and Causal signal:Real and Causal signal:

• Fourier representation:Fourier representation:

.0,0)( ttf

.)exp(][Re1

)(0

dtiftf

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

Analytic SignalsAnalytic Signals• Hilbert Transform pairs and Signals :Hilbert Transform pairs and Signals :

.1

][][,1

]][[

].[]cos[][],[]][cos[

tittF

ttH

tsinittFtsintH

2r

Plane WavePlane Wave

1r

3r

n

Plane WavePlane Wave• Source Signal:Source Signal:

• Fourier Transform :Fourier Transform :

].[)( tFt Uu

][)exp()( fit Uu

Zero-Order Ray Zero-Order Ray Theory(ZORT)Theory(ZORT)

• Elastodynamic case:Elastodynamic case:

• Acoustic case:Acoustic case:

].[)](exp[)(),( fi rrUru

].[)](exp[)(),( fiPp rrr

Transient Analytic ZORTTransient Analytic ZORT• Elastodynamic case:Elastodynamic case:

• Acoustic case:Acoustic case:

)].([)(),( rrUru tFt

)].([)(),( rrr tFPtp

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

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

Acoustic CaseAcoustic Case

./1)(:

,021:

,0:

222

21

20

cT

TP

PP

PP

Elastodynamics : Eikonal Elastodynamics : Eikonal EquationEquation

wave.-S

wave.-P

matrix. lChristoffe

0U

,,1)(

.,,1)(

ˆ

,)ˆˆ(

22

22

2

T

T

I

Elastodynamics : Transport EquationElastodynamics : Transport Equation

.0

,

22

)(

TA

TAP

U

:waves-P

Elastodynamics : Transport EquationElastodynamics : Transport Equation

.0)(

,

222

2)(

TCB

CBS

eeU

:waves-S

1

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

Eikonal EquationEikonal Equation• Slowness Vector:Slowness Vector:

.Tp

p

Wavefront

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

• 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

Method of CharacteristicsMethod of Characteristics• Solution SurfaceSolution Surface

InitialCurve

, Characteristic Curves

PDE of First OrderPDE of First Order

.

,,,,,,

,,,,0,,H

321321

3

0

rppand r

where

rrrrpr

ppprrr

g

Characteristic EquationsCharacteristic Equations

Parameter Ray

p

pp

r

Hdtd

HHdtd

Hdtd

p

r

p

Ordinary Differential Equation(ODE) System :

Ray EquationsRay Equations

2)(1

)(1

)(1

r

rrp

pr

cdtd

ccdtddtd

Special Choices of Special Choices of : :

time. travelx

. lenght arcx

physical. non ,

,,)(

,),(

,1

2

tc

stc

t

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

Tube of RaysTube of Rays

1dS

2dS

B

Ray CoordinatesRay Coordinates

1r

2r

3r

0p

O

1 2

s),,( Point 21

Jacobian of the Jacobian of the Transformation :Transformation :

.

,),,(

),,(

),,,(),,(

21

21

321

21321

dsddJV

sxxxJ

sxxx

Amplitude FormulaAmplitude Formula• Integrating along the Ray Tube:Integrating along the Ray Tube:

Factor Spreading lGeometrica

JJ

JvJvtAtA

o

oooo

)()(

Initial ConditionsInitial Conditions• Point Source :Point Source :

.p

rr

o

s

s

sinsinsinc

)0(

),cos,,(cos)0(

,)0(

221211

Amplitude due to a Point Amplitude due to a Point SourceSource

strenght. source,lim

,1

ooooSP

GGSS

AJvg

JvvgA

o

Normalized Geometrical Normalized Geometrical Spreading FactorSpreading Factor

.1

,

22 Lvv

gA

vvJL

GGSS

GS

Line CausticLine Caustic

Focus Caustic PointFocus Caustic Point

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

Maslov TheoryMaslov Theory• Explain Ray theory in a Caustic Explain Ray theory in a Caustic

NeighborhoodsNeighborhoods

caustic point2caustic line

Index KMAH1

,22/1

ieJJ

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

Reflected RayReflected Ray

x

z

S G

A Medium

B Medium

j Interfacej 1j

Acoustic Reflection Acoustic Reflection CoefficientCoefficient

)()cos(

)()cos(

2

2

1

2

2

1

jj

jj

jj

jj

sincc

sincc

R

Transmitted RayTransmitted Ray

x

z

S

A Medium

B Medium

j Interfacej

1j

Acoustic Transmission Acoustic Transmission CoefficientCoefficient

)()cos(

)cos(2

2

2

1j

j

jj

j

sincc

T

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

Amplitude at the GeophoneAmplitude at the Geophone

1

2/1

111

2/1

2 coscos A

JvJvc

vvA

GGGi

iii

iiiN

iG

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

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

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

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