RAYS AS ODESReferences Červený, V., Seismic ray theory, Cambridge University Press, 2001 Chapman,...

Seismic numerical forward modeling: Rays as ODEs

By Jean Virieux

Professeur

UJF-Grenoble I

28/09/2012Numerical methods : ODE integration

1

American Petroleum Institute, 1986

Salt DomeFault

Unconformity

Pinchout

Anticline

1

Time scales

28/09/2012 Numerical methods : ODE integration 2

Source timefrom 0.1 sec to 100 sec (rupture vel.)

Wave timefrom secondes to hours

Window timefrom few secondes to days

Length scales

28/09/2012 Numerical methods : ODE integration 3

Fault length200 km for vr=2 km/s

Discontinuity distancefrom few meters to few 100 kms

Volume samplingfrom few kms to few 1000 kms

Examples

28/09/2012 Numerical methods : ODE integration 4

Record of a far earthquake (Müller and Kind, 1976)

Traces from a oil reservoir (Thierry, 1997)

Records on the Moon (meteoritic impact) (Latham et al., 1971)

Seismic imaging is a tough problem on the Moon !

References Červený, V., Seismic ray theory, Cambridge University Press, 2001 Chapman, C., Fundamentals of seismic wave propagation, Cambridge University

Press, 2004 Goldstein, H, Classical mechanics, Addison-Wesley Publishing company, second

edition,1980 Glassner, A.S. (Editor), An introduction to ray tracing, Academic Press, second

edition, 1991 Sethian, J.A., Level Set Methods and Fast Marching Methods: Evolving Interfaces

in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science, Cambridge University Press, 1999.

Virieux, J. and Lambaré, G., Theory and Observations-Body waves: Ray methodsand finite frequency effecs in Treatrise on Geophysics, Tome I by A. Diewonskiand B. Romanowicz, Elsevier, 2010

28/09/2012 Numerical methods : ODE integration 5

Translucid Earth

Diffracting medium:

wavefront coherence lost !

Wavefront preserved

Wavefront : T(x)=T0

Travel-time T(x) and Amplitude A(x)

Source

Receiver

Same shape !

T(x)

S(t)

28/09/2012 Numerical methods : ODE integration 6

is sometimes called the phase

Eikonal equationTwo simple interpretations of wavefront evolution

Orthogonal trajectories are rays in an isotropic medium

Grad(T)= orthogonal to wavefront

Direction ? : abs or squareThe orientation of the wavefront could not be guessed from the local information on a specific wavefront

28/09/2012 Numerical methods : ODE integration

T+T

T=cte

Velocity c(x)

L RayΔΔ →

ΔΔ

1→

1

7

Transport Equation Tracing neighboring rays defines a ray tube : variation of amplitude depends on energy flux conservation throughsections.

28/09/2012 Numerical methods : ODE integration 8

2 . 0

∆ ∆ ∆ ∆

→ . .

0 . .

0 . ′ . .

0 ⟹ 0

2 . 0

Energy flux same at section one and at section two

net contribution=zero

Methods of characteristics

Differential geometry (Courant & Hilbert, 1966)

Non-linear ordinary differential equations Lagrangian formulation as we integrate

along rays

In opposition to Eulerian formulation wherewe compute (ray) quantities at fixed positions

28/09/2012 Numerical methods : ODE integration 9

28/09/2012 Numerical methods : ODE integration 10

Ray equation

Ray

T=cte

Evolution of is given by

but the operator . .

and, therefore, .

leading to .

Evolution of is given by

→ 1Wavefront

∥ →

1

Ray equations

1 1

We define the slownss vector and the position along the ray

Curvature equation

Various non-linear ray equations

28/09/2012 Numerical methods : ODE integration 11

Which ODE to select for numerical solving ? Either t or sampling.

Many analytical solutions (gradient of velocity; gradient of slowness square)

Curvilinearstepping

1

1

Particulestepping

1 1

1

Timestepping

1

1 1

undertheconditionoftheeikonal1/

The

sim

ples

tset

Properties of these ODEs Intrinsic solutions independent of the

coordinate sytem used to solve it If dummy variable for velocity, use itas the variable stepping (often x coordinate)

(1)

Eikonal equation: a good proxy for testingthe accuracy of the ray tracing (not enoughused)

28/09/2012 Numerical methods : ODE integration 12

In 3D: six or seven equationsIn 2D: four or five equations

(1): rectilinear motion of a particle along this axis in mechanics

Particulestepping

1 1

Velocity variation v(z)

dzzduzu

ddp

ddp

ddp

pddqp

ddq

pddq

zyx

zz

yy

xx

)()(;0;0

;;

Ray equations are

The horizontal component of the slowness vector is constant: the trajectory is inside a plan which is called the plan of propagation. We may define the frame (xoz) as this plane.

dzzduzu

ddp

ddp

pddqp

ddq

zx

zz

xx

)()(;0

;

22 )( x

x

z

x

z

x

pzu

ppp

dqdq

Where px is a constante

1

0

220011)(

),(),(z

z x

xxxxx dz

pzu

ppzqpzq

For a ray towards the depth

Velocity variation v(z)

)(222px zupp

pp

pp

z

z x

z

z x

x

z

z x

xz

z x

xxxx

dzpzu

zudzpzu

zuTpzT

dzpzu

pdzpzu

pqpzq

10

10

22

2

22

2

011

2222011

)(

)(

)(

)(),(

)()(),(

p

p

z

z

dzpzu

zupT

dzpzu

ppX

022

2

022

)()(2)(

)(2)(

At a given maximum depth zp, the slowness vector is horizontal following the equation

zp

If we consider a source at the free surface as well as the receiver, we get

2 2 2

2 2

2 2 2

2( )

( )2( )sin

p

p

a

r

a

r

p drrr u r p

r u r drTrr u r p

with p ru i

In Cartesian frame In Spherical framewith p = usini

Velocity structure in the Earth

Radial Structure

Integration of ray equations

28/09/2012 Numerical methods : ODE integration 16

Initial conditions EASY

1D sampling of 2D/3D medium : FAST

source

receiver

Runge-Kutta second-order integration

Predictor-Corrector integration

source

receiver Boundary conditions VERY DIFFICULT

?

?

Shooting p ?

Bending x ?

Continuing c ?

AND FROM TIME TO TIME IT FAILS ! (inherent to geometrical optics)

Save slownessconditions if possible !

But we need 2-points ray tracing because we have a source and a receiver to connect ! We even need more: branch identification (triplication for example)

A very good QC: the eikonal must be equal to zero !

Initial ray tracingRay tracing by rays

Two-point ray tracing

Ray tracing by rays

Hamilton’s ray equations

28/09/2012 Numerical methods : ODE integration 19

Information around the rayRay

Mechanics : ray tracing as a particular balistic problemsympletic structure (FUN!)

Meaning of the neighborhood zoneFresnel zone if finite frequencyAny zone depending on your problem GBS

1 1.,

12

1

Hamilton approach suitable for perturbation(Henri Poincaré en 1907 « Mécanique céleste »,Richard Feymann Prix Nobel 1965)

" "

Paraxial ray theorysimilar to

Gauss optics

28/09/2012 Numerical methods : ODE integration 20

Hamilton’s ray equations

28/09/2012 Numerical methods : ODE integration 21

Information around the ray

Ray

1 1.

,12

1

" "

Paraxial Ray theory

Estimation of ray tube: KMAHindex tracking and amplitudeevaluation

Estimation of taking-off angles: shooting strategy

The matrix does not depend on quantities frombut only on quantities from : LINEAR PROBLEM

(SIMPLE) !

28/09/2012 Numerical methods : ODE integration 22

,

, ,

seismogram computation

Paraxial Ray theory

28/09/2012 Numerical methods : ODE integration 23

Solutions are coordinate dependent (differentialcomputation)

Not restricted to the so-called ray-centeredcoordinate system (Cerveny, 2001)

Cartesian formulation is much simpler to handle(Virieux & Farra, 1991)

2D simple linear system

Four elementary paraxial trajectories

y1t(0)=(1,0,0,0)

y2t(0)=(0,1,0,0)

y3t(0)=(0,0,1,0)

y4t(0)=(0,0,0,1) NOT A paraxial RAY !

28/09/2012 Numerical methods : ODE integration 24

0 0 1 00 0 0 1

0.51⁄

0.51⁄

0 0

0.51⁄

0.51⁄

0 0

More complex anisotropic structure but still straightforward

Linearsystem

2D paraxial conditions

Paraxial rays require other conservative quantities : the perturbation of the Hamiltonian should be zero (or, in other words, the eikonal perturbation is zero)

If working with the reduced hamiltonian, this is implicitly set!

28/09/2012 Numerical methods : ODE integration 25

⁄ , ,⁄ 0

0

Or in the isotropic case

Similar conditions in 3DReadily deduced for anisotropyTwo independent solutions

Point source conditions

From paraxial trajectories, one can combine them for paraxial rays as long as the perturbation of the Hamiltonian is zero.For a point source, the parameter could be set to an arbitrary small value: this isa derivative or plan tangent computation (Gauss optics)

This is enough to verify this condition initially

Paraxial solution 0 3 0 4

28/09/2012 Numerical methods : ODE integration 26

0 ⇒ 0 0 0 0 =0

0 0

0 0

Plane source conditions

We combine the first two paraxial ray trajectories.

This is enough to verify this condition initially but gradient of velocity at the source could be quite arbitrariry

Cerveny’s condition

Chapman’s condition (only z variation)

Paraxial solution ′ ,⁄0 1 ′ ,⁄

0 2

28/09/2012 Numerical methods : ODE integration 27

0⇒

12

1 ,⁄0 0

12

1 ,⁄0 0 0

012

1 ,⁄0

012

1 ,⁄0

Two independent paraxial rays in 2D ( ): point (seismograms) and plane (beams) paraxial rays

KMAH index tracking

28/09/2012 Numerical methods : ODE integration 28

In 2D, the determinant 0 0 0

may change sign:

Increment by one the KMAH index as we have crossed a caustic

In 3D, the determinant

0 0 0 0

may change sign.

If minor determinants do not change sign, this is a plane caustic (add 1 to KMAH). If they change sign as well, this is a point caustic (add 2 to KMAH).

KMAH index key element for seismograms

ODE resolution Runge-Kutta of second order Write a computer program for an

analytical law for the velocity: take a gradient with a component along x and a component along z

Home work : redo the same thing with a Runge-Kutta of fourth order (look after its definition)

Consider a gradient of the square of slowness

28/09/2012 Numerical methods : ODE integration 29

Runge-Kutta integrationSecond-order RK integration

Non-linear ray tracing

Second-order euler integration for paraxial ray tracing is enough!

28/09/2012 Numerical methods : ODE integration 30

Linear paraxial ray tracing

Propagator technique

Optical Lens technique

Two points ray tracing: the paraxial shooting method

Consider x the distance between ray point atthe free surface and sensor position

The estimation of the derivative is through the point paraxial computation

28/09/2012 Numerical methods : ODE integration 31

Solve iteratively Δx Δθ

Δ

Δ

0 0 0 0

0 0 0 0

3 0 4 0Paraxial quantities for derivative

Derivative for shooting angle

Point paraxial condition

Amplitude estimationConsider L the distance between the exit point of a ray at the particule time and the paraxial ray running point.

Thanks to the point paraxial ray estimation dq3 and dq4, we mayestimate the geometrical spreadingΔ Δ⁄ and, therefore, the ray amplitude ∝ Δ /Δ

28/09/2012 Numerical methods : ODE integration 32

LΔ

From point paraxial ray

From point paraxial trajectories 3 and 4

ΔΔ

3 34 4

0 0 0

Using plane paraxial solutions 1 and 2, we can construct any beams as the Gaussian beams

Rays and wavefronts in an homogeneous medium. (Lambaré et al., 1996)

Ray tracing by wavefronts

Slow down the ray tracingefficiency as we samplethe entiremedium

(Lambaré et al., 1996)

28/09/2012 Numerical methods : ODE integration 34

STEP ONE: ray tracing

28/09/2012 Numerical methods : ODE integration 35

STEP TWO: times and amplitudes(Lambaré et al., 1996)

0 400 800 1200 1600 2000Z in m

28/09/2012 Numerical methods : ODE integration 36

STEP THREE: seismograms

GLOBAL Tomography Velocity variation at a

depth of 200 km : good correlation with superficial structures.

Velocity variations at a depth of 1325 km : good correlation with the Geoid.

Courtesy of W. Spakman

Delayed Travel-time tomography

0( , ) ( , , ) ( , , ) ( , , )receiver receiver receiver

source source source

t s r u x y z dl u x y z dl u x y z dl

0 0

0 0

0

0

0

0

0

0

( , ) ( , , ) ( , , )

( , ) ( , ) ( , , )

( , ) ( , , )

receiver receiver

source source

receiver

source

receiver

source

t s r u x y z dl u x y z dl

t s r t s r u x y z dl

t s r u x y z dl

Consider small perturbations u(x) of the slowness field u(x)

station

source

dlzyxustationsourcet ),,(),( Finding the slowness u(x) from t(s,r) is a difficult problem: only techniques for one variable !

This a LINEAR PROBLEM t(s,r)=G(u)

DESCRIPTION OF THE VELOCITY PERTURBATION

The velocity perturbation field (or the slowness field) u(x,y,z) can be described into a meshed cube regularly spaced in the three directions.For each node, we specify a value ui,j,k. The interpolation will be performed with functions as step funcitons. We have found shape functions h,i,j,k=1 pour i,j,k, and zero for other indices.

cube

kjikji huzyxu ,,,,),,(

Discrete Model Spacecube

kjikji huzyxu ,,,,),,(

m

m

m

nn

m

n

n

cubekji

cube rayonkjikji

rayon cubekjikji

uu

uu

ut

ut

ut

ut

tt

tt

uutrst

dlhudlhurst

1

2

1

1

1

1

1

1

2

1

,,

,,,,,,,,

...

),(

),(00

Slowness perturbation description

0t G u

Matrice of sensitivity or of partial derivatives

Discretisation of the medium fats the ray

Sensitivity matrice is a sparse matrice

Corinth GulfAn extension zone where there is a deep drilling project.

How this rift is opening?

What are the physical mechanisms of extension (fractures, fluides, isostatic equilibrium)

Work of Diana Latorre and of Vadim Monteiller

Seismic experiment 1991 (and one in 2001)

MEDIUM 1D : HWB ANDRANDOM SELECTION

Velocity structure imageHorizontal sections

Velocity structure image Vertical sections

P S

Vp/Vs ratio:fluid existence ?

Recovered parameters might have diferent interpretation and the ratio Vp/Vs has a strong relation with the presence of fluids or the relation Vp*Vs may be related to porosity

Conclusion FATT

Selection of an enough fine grid Selection of the a priori model information Selection of an initial model 2PT-RT Time and derivatives estimation LSQR inversion Update the model Uncertainty analysis (Lanzos or numerical)

THANK YOU !

Many figures have come from people I have worked with: many thanks to them !

28/09/2012 Numerical methods : ODE integration 49

http://seiscope.oca.eu

Seismic attributes

Travel time evolution with the grid step : blue for FMM and black for recomputed time

One ray Log scale in time

Grid step

S

R

A ray

2 PT ray tracing non-linear problem solved, any attribute could be computed along this line :

-Time (for tomography)

-Amplitude (through paraxial ODE integration fast)

-Polarisation, anisotropy and so on

Moreover, we may bend the ray for a more accurate ray tracing less dependent of the grid step (FMM)

Keep values of p at source and receiver !

28/09/2012 Numerical methods : ODE integration 50

Polarization

28/09/2012 Numerical methods : ODE integration 51

Chapman, 2004, p180

Acoustic case: the unitary vector supports the P wave vibration

Elastic case: the independent shearvibration will be along two unitary vectors

and such that

.

.

Isotropic case

.

.

Time stepping

Particule stepping

It is enough to follow the evolution of the projection of elastic unitary vectors on one Cartesian coordinate: . and . (Psencik,

perso. Comm.)

Semi-lagrandian approach as we track the evolution of the wavefront and its complex folding (Lambaré et al, 1996)