Trazado de RayosTrazado de Rayos

porpor

Carlos César PiedrahitaCarlos César Piedrahita

Grupo de Geofísica, UIN, ICP-ECOPETROLGrupo de Geofísica, UIN, ICP-ECOPETROL

Numerical MethodsNumerical Methods

• Continuation Methods:Continuation Methods:

• Runge Kutta:Runge Kutta:

• Finite-Differences:Finite-Differences:

Continuation MethodContinuation Method

0XS

1X

jX

NX

1 NXR

Statement of the problemStatement of the problem

• Source and Receiver points given.Source and Receiver points given.

• N intersection points to be find.N intersection points to be find.

• Constant velocity layers.Constant velocity layers.

• Smooth interfaces between layers.Smooth interfaces between layers.

Mathematical problemMathematical problem

• The eikonal equation is satisfied by straight The eikonal equation is satisfied by straight

line segments in the layers.line segments in the layers.

• Snell´s law is satisfied at the intersection Snell´s law is satisfied at the intersection

points of the ray and the interfaces.points of the ray and the interfaces.

Model representationModel representation

)(0 xfz

)(1 xfz

)(2 xfz

)(xfz M

1211 ,,, vvR

2432 ,,, vvR

LLLL vvR ,,,

Snell´s law (Nonlinear Snell´s law (Nonlinear system)system)

1 11

1 1

, , .

Intersection point: , , ,

Tangent vector: 1, .

k

k

k k k kk k k k

k k k k

k k k k i k

k i k

X X X Xv v

X X X X

X x z x f x

f x

Nonlinear systemNonlinear system

).,())(,(),(

).,())(,(),(

.1,0,,,

11111

0

1

00000

21

RRiN

SSi

Nk

zxxfxzxX

zxxfxzxX

Nkxxx

NNNNN

Nonlinear systemNonlinear system

1 2 1 2

1 1 1

, , , , , , , .

, , , .

, , N equations and N unknowns.

T T

N N

T

x x x v v v

X V

X V

X V 0

Ray signaturesRay signatures

)(0 xfz

)(1 xfz

)(2 xfz

)(3 xfz

P

S

P

S

211 ,, vvR

432 ,, vvR

653 ,, vvR

Source Receiver

Propagation typesPropagation types

2 ,

2 1 ,

1 2 1 1

For 1,2, , define:

(compressive velocity),

(shear velocity).

velocity index of the kth region

crossed.

Example: 2, 3, 3, 2.

m P m

m S m

k

m M

v c

v c

j

j j j j

Ray signaturesRay signatures

)(0 xfz

)(1 xfz

)(2 xfz

)(3 xfz

P

S

P

S

211 ,, vvR

432 ,, vvR

653 ,, vvR

Source Receiver

P

PS

S

S

S

Ray signatureRay signature• The signature is defined as a finite sequence The signature is defined as a finite sequence

of positive integers which represent the order of positive integers which represent the order

of interfaces intersected and the propagation of interfaces intersected and the propagation

type of the ray in each layer. type of the ray in each layer.

• Propagation type :Propagation type :

• Intersection number:Intersection number:

.1, Nmjm

.1, Nkik

Ray signatureRay signature

12211 ;,;;,;, NNN jijijij

Class I ray: 2;1,3;2,3;1,2

Class II ray: 2;1,3;2,5;3,6;2,3;1,1

Solution ProceduresSolution Procedures

• Newton´s method is applied.Newton´s method is applied.

• It is a recursive algorithm.It is a recursive algorithm.

• It is a very fast method, but very dependent It is a very fast method, but very dependent

on the initial approximaion.on the initial approximaion.

Solution ProceduresSolution Procedures

.

,,

,,,,

,,,,)()()()(

)()(2

)(1

)(

X

VXVXA

VXAAVX

X

iiii

TiN

iii xxx

.

,)()()(

)()()1(

iii

iii

XA

XXX

Notation:

Properties of the systemProperties of the system

• Matriz A is an NXN tri-diadiagonal matrix.Matriz A is an NXN tri-diadiagonal matrix.

• Given a “good” initial approximation the Given a “good” initial approximation the

convergence is quadratic:convergence is quadratic:

,2,1,2)1()( iK ii XX

Continuation in velocitiesContinuation in velocities

Source Receiver

S

S

S

P

P

P

P

P

Continuation in velocitiesContinuation in velocities

).()()(

.10,ˆ)1()(

XXX

VVV

0

If we find the derivative, we can find an appropiate value for the initial approximation.

Continuation in receiversContinuation in receivers

Source R1 R2 R3

Continuation in receiversContinuation in receivers

).()()(

),()1(),(

.10,)1()()()()()(

)()(

XXX

xxx

0

1N

iR

iR

jR

jR

iR

jR

yxyx

If we find the derivative, we can find an appropiate value for the initial approximation.

First ray in a class of raysFirst ray in a class of rays

• A shooting procedure obtains the first ray of A shooting procedure obtains the first ray of

a class. An angle search is executed.a class. An angle search is executed.

• Obtained the first ray of a given class, the Obtained the first ray of a given class, the

rest of the rays of the class is obtained by rest of the rays of the class is obtained by

receiver and velocity continuation.receiver and velocity continuation.

TraveltimeTraveltime

kkk

N

k i

k

D

v

Dt

k

xx

1

1

1

,

Amplitude(2½D medium)Amplitude(2½D medium)

1X

2X

3X

Ray coordinates(polar angles)Ray coordinates(polar angles)

2X

3X

1X

,,

Cross-sectional areaCross-sectional area

.1

),,(

),,(,,

,1

321

xxx

xxx

p

xxxJ

ddp

dS

Tube of rays crossing the Tube of rays crossing the interfacesinterfaces

1j

j

1jdS

jdS

j

j

jn

1jn

Tube of rays crossing the Tube of rays crossing the interfacesinterfaces• R/T coefficients:R/T coefficients:

• Recursive expression:Recursive expression:

. jjj AKA

.)cos(

)cos(1

21

21

)1(2

jjjj

jjj dSAKdSA

Tube of raysTube of rays

Source Receiver

1iv

2iv

1Niv

Niv

Amplitude at the receiverAmplitude at the receiver

• Green´s function solution.Green´s function solution.

• Out-of-Plane spreading factor.Out-of-Plane spreading factor.

• In-plane spreading factor.In-plane spreading factor.

• Discontinuous change in cross-sectionalDiscontinuous change in cross-sectional

area at an interface.area at an interface.

Amplitude at the receiverAmplitude at the receiver

1

1/ 2 1/ 2 1/ 21

1 11 1

Green R/T coefficientsIn-Plane SpreadingOut-of-Plane Spreading

cos( )1cos( ) .

4 cos( )j

NNi j

N N jj ji

v D dA K

v d

Derivate of receiver position Derivate of receiver position respect shooting angle respect shooting angle

Source

Phase shiftsPhase shifts

• Reflection: Reflection:

• Caustic points: Caustic points:

• Critical or supercritical rays(Head waves):Critical or supercritical rays(Head waves):

..

2

., ie

Critical/Over-critical Critical/Over-critical angle(head waves)angle(head waves)

Source

Phase shifts Phase shifts

.

.

,

1

321

11

N

k

kkk

kk

k

k

:shift phase Total

:shift Phase kth

xx :segment Ray kth

SeismogramSeismogram

Earth ModelSource Trace

)(ts )(tf

SeismogramSeismogram

).()(

,)](exp[)(2

1)(

,]exp[)()(

1 traveltimeN TtwAtf

dtiStw

dtiSts

: Trace

: Wavelet

: Source

Initial value Problem(Shooting Initial value Problem(Shooting ))

.prh

parameter,

prh

hfh

Tooo

T

d

d

,

,,

,,

Runge-KuttaRunge-Kutta

2X

3X

1X

ih

1ih

2ih

0h

Runge-Kutta schemeRunge-Kutta scheme

.1

k

jji iji1i fhh

Eikonal solversEikonal solvers

• Global grid(finite differences) (Vidale)Global grid(finite differences) (Vidale)

• Fast Marching Methods(Sethian)Fast Marching Methods(Sethian)