2) and v´eronique farra 1) ivan pˇsenˇc´ık

Approximate traveltimes of waves propagatingin laterally varying layered anisotropi media

Ivan Psencık1) and Veronique Farra2)

1) Institute of Geophysics, Acad. Sci. Praha, Czech Republic

2) Institut de Physique du Globe, Paris, France

14th IWSA, Perth 2010

Tuesday, April 13, 2010

Outline

OutlineIntrodu tion

OutlineIntrodu tionFirst-order ommon S-wave ray tra ing

OutlineIntrodu tionFirst-order ommon S-wave ray tra ingApproximate S-wave traveltime formulae

OutlineIntrodu tionFirst-order ommon S-wave ray tra ingApproximate S-wave traveltime formulaeTransformation of rst-order slowness ve torsat an interfa e (Snell's law)

OutlineIntrodu tionFirst-order ommon S-wave ray tra ingApproximate S-wave traveltime formulaeTransformation of rst-order slowness ve torsat an interfa e (Snell's law)Numeri al examples

OutlineIntrodu tionFirst-order ommon S-wave ray tra ingApproximate S-wave traveltime formulaeTransformation of rst-order slowness ve torsat an interfa e (Snell's law)Numeri al examplesCon lusions

Introdu tiontraveltimes important in migration and tomography


standard ”anisotropic” ray tracers for S waves



- break down in weakly anisotropic media

and in vicinities of singularities





- do not work in isotropic media





- do not work in isotropic media

- generate number of multiple reflections,

which may exceed acceptable limit

Introdu tionSolution: use of common S-wave ray concept

(Bakker, 2002; Klimes, 2006; Farra & Psencık, 2008)



- common ray - artificial trajectory approximating rays

of S1 and S2 waves




of S1 and S2 waves

- traveltimes of S1 and S2 waves evaluated by quadratures

along the common ray




of S1 and S2 waves

- traveltimes of S1 and S2 waves evaluated by quadratures

along the common ray

- at interfaces, slowness vectors of generated waves determined

by solving 4th-degree polynomial equation

First-order ommon S-wave ray tra ing

dxi/dτ = 12∂G[M]/∂pi , dpi/dτ = −1

2∂G[M]/∂xi

xi - coordinates of the first-order ray Ω

pi - components of the first-order slowness vector p

τ - first-order traveltime

G[M](x,p) - S-wave first-order mean eigenvalue

G[M](x,p) = 12[G

(1)S1(x,p) + G

(1)S2(x,p)]

G(1)SI (x,p) - S-wave first-order eigenvalues of Christoffel matrix Γ

Eikonal equation: G[M](x,p) = 1

Approximate S-wave traveltime formulae

τS1,S2(τ, τ0) = τ [M](τ, τ0) + ∆τ [M](τ, τ0) + ∆τS1,S2(τ, τ0)

τ [M](τ, τ0) - first-order traveltime between τ0 and τ on common ray

∆τ [M](τ, τ0) - ”averaging” correction of first-order traveltime

∆τ [M] = 14

∫ ττ0

(B213 + B2

23)/(B33 − 1)dτ

∆τS1,S2(τ, τ0) - ”separation” traveltime correction

∆τS1,S2 = ∓14

∫ ττ0

√

(M11 − M22)2 + 4M 212dτ

Approximate S-wave traveltime formulae

Bmn = Bmn(x,p) = Γik(x,p)e[m]i (x)e

[n]k (x)

Γik - elements of Christoffel matrix Γ

e[m] - triplet of orthonormal vectors

e[3] = c[M]p, e[1], e[2] arbitrarily in the plane ⊥ e[3]

c[M] - common S-wave phase velocity

MMN = MMN(x,p) = BMN − BM3BN3/(B33 − 1)

Transformation of rst-order slowness ve torsat an interfa e (Snell's law)

pGi = pi − (pkNk)Ni + ξGNi

p - first-order common S-wave slowness vector of incident wave

pG - first-order common S-wave slowness vector of generated wave

N - normal to the interface, ξG = pGk Nk

pGi − (pG

k Nk)Ni = pi − (pkNk)Ni - Snell’s law

Transformation of rst-order slowness ve torsat an interfa e (Snell's law)

pGi = pi − (pkNk)Ni + ξNi

G[M](x,pG) = 1 - eikonal equation

G[M](ξ) = 1 - polynomial equation of 4th degree for ξ

Iterative solution(Dehghan et al., 2007)

ξj = ξj−1 − [G[M](pj−1m ) − 1]/[Nk

∂G[M]

∂pk(pj−1

m )]

ξ0 - in reference isotropic medium

Numeri al examples

VSP CONFIGURATION

1.0 km

1.0 km

Numeri al examples (Klimes & Bulant, 2004)HTI rotated by 45 degrees0 . 2 0 . 4 0 . 6 0 . 8 0 .

2 . 4

2 . 2

2 . 0

QI (ANI 3%, SEPAR 1-4%)

z = 0 k m

S1

S2

P h a s e a n g l e ( d e g )

Ph

as

e v

elo

cit

y (

km

/s)

0 . 2 0 . 4 0 . 6 0 . 8 0 .

3 . 0

2 . 8

2 . 6

z = 1 k m


Numeri al examples0 . 0 0 . 2 0 . 4 0 . 6

0

- 0 . 0 5

S 1 A N D S 2 W A V E S

Q IR

el

tt

d

if.

(

%)

D e p t h ( k m )0 . 0 0 . 2 0 . 4 0 . 6

Numeri al examples (Klimes & Bulant, 2004)HTI rotated by 45 degrees0 . 2 0 . 4 0 . 6 0 . 8 0 .

2 . 4

2 . 2

2 . 0

QI4 (ANI 6%, SEPAR 11-13%)

z = 0 k m

S1

S2


Ph

as

e v

elo

cit

y (

km

/s)

0 . 2 0 . 4 0 . 6 0 . 8 0 .

3 . 0

2 . 8

2 . 6

z = 1 k m


Numeri al examples0 . 0 0 . 2 0 . 4 0 . 6

0

- 0 . 4

- 0 . 8

S 1 A N D S 2 W A V E S

Q I 4R

el

tt

d

if.

(

%)

D e p t h ( k m )0 . 0 0 . 2 0 . 4 0 . 6

Numeri al examples (Shearer & Chapman, 1989)VTI: MODEL SC1 (ANI 11%, SEPAR 0-11%)

2.2

2.25

2.3

2.35

2.4

2.45

2.5

2.55

2.6

0 10 20 30 40 50 60 70 80 90

Pha

se v

eloc

ity (

km/s

)

Phase angle (deg)

SVSH

Numeri al examples0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0

0 . 6

0 . 5

0 . 4

0 . 3

S C 1 - H O M . , E X A C T ,

D E P T H I N K M

TR

AV

EL

TIM

E I

N S

EC

0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0

0 . 6

0 . 5

0 . 4

0 . 3

D E P T H I N K M

TR

AV

EL

TIM

E I

N S

EC


0 . 6

0 . 5

0 . 4

0 . 3

S C 1 - H O M . , E X A C T , A P P R .

D E P T H I N K M

TR

AV

EL

TIM

E I

N S

EC

0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0

0 . 6

0 . 5

0 . 4

0 . 3

D E P T H I N K M

TR

AV

EL

TIM

E I

N S

EC

0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0

0 . 6

0 . 5

0 . 4

0 . 3

D E P T H I N K M

TR

AV

EL

TIM

E I

N S

EC

0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0

0 . 6

0 . 5

0 . 4

0 . 3

D E P T H I N K M

TR

AV

EL

TIM

E I

N S

EC


0 . 5

0 . 4

0 . 3

S C G 1 - I N H O M . , E X A C T ,

D E P T H I N K M

TR

AV

EL

TIM

E I

N S

EC

0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0

0 . 5

0 . 4

0 . 3

D E P T H I N K M

TR

AV

EL

TIM

E I

N S

EC


0 . 5

0 . 4

0 . 3

S C G 1 - I N H O M . , E X A C T , A P P R .

D E P T H I N K M

TR

AV

EL

TIM

E I

N S

EC

0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0

0 . 5

0 . 4

0 . 3

D E P T H I N K M

TR

AV

EL

TIM

E I

N S

EC

0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0

0 . 5

0 . 4

0 . 3

D E P T H I N K M

TR

AV

EL

TIM

E I

N S

EC

0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0

0 . 5

0 . 4

0 . 3

D E P T H I N K M

TR

AV

EL

TIM

E I

N S

EC

Numeri al examples (Shearer & Chapman, 1989)VTI: MODEL SC4 (ANI 30%, SEPAR 0-30%)

1.8

1.9

2

2.1

2.2

2.3

2.4

2.5

2.6

2.7

0 10 20 30 40 50 60 70 80 90

Pha

se v

eloc

ity (

km/s

)

Phase angle (deg)

SVSH


1 . 2

1 . 0

0 . 8

0 . 6

0 . 4

SC4

D E P T H I N K M

TR

AV

EL

TIM

E I

N S

EC

0 . 0 0 . 4 0 . 8 1 . 2 1 . 6 2 . 0

1 . 2

1 . 0

0 . 8

0 . 6

0 . 4

D E P T H I N K M

TR

AV

EL

TIM

E I

N S

EC


1 . 2

1 . 0

0 . 8

0 . 6

0 . 4

SC4

D E P T H I N K M

TR

AV

EL

TIM

E I

N S

EC

0 . 0 0 . 4 0 . 8 1 . 2 1 . 6 2 . 0

1 . 2

1 . 0

0 . 8

0 . 6

0 . 4

D E P T H I N K M

TR

AV

EL

TIM

E I

N S

EC

0 . 0 0 . 4 0 . 8 1 . 2 1 . 6 2 . 0

1 . 2

1 . 0

0 . 8

0 . 6

0 . 4

D E P T H I N K M

TR

AV

EL

TIM

E I

N S

EC

0 . 0 0 . 4 0 . 8 1 . 2 1 . 6 2 . 0

1 . 2

1 . 0

0 . 8

0 . 6

0 . 4

D E P T H I N K M

TR

AV

EL

TIM

E I

N S

EC

Con lusions- appli able to S waves in inhomogeneous isotropi ,weakly anisotropi and moderately anisotropi media- in isotropi media exa t, in anisotropi media approximate- single ommon S-wave ray ne essary for omputation oftraveltimes of S1 and S2 waves- ommon S-wave ray tra ing stable, does not ollapse anywhere- omputer time savings in omputing traveltimesof re e ted/transmitted waves- performs better in inhomogeneous media; smoothes loopsin traveltime urves

SE

ISM

I CWAV ES I N

COM PLEX 3 − DS

TR

UC

TU

RES

2) and v´eronique farra 1) ivan pˇsenˇc´ık

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