Asymptotic wave solutions

Jean VirieuxProfesseur UJF

Translucid Earth Source

Receiver

Same shape !

T(x)

Too diffracting medium : wavefront coherence lost !

Wavefront preserved

Wavefront : T(x)=T0

Travel-time T(x)

Amplitude A(x)

S(t)

( )

( , ) ( ) ( ( ))

( , ) ( ) ( ) i T x

u x t A x S t T x

u x A x S e

Eikonal equation

)(1)(

)(1)(

xcxT

xcLT

TLxc x

Two simple interpretation of wavefront evolutionOrthogonal trajectories are rays

T+T

T=cte

Velocity c(x)

L

Grad(T) orthogonal to wavefront

Direction ? : abs or square )(1))(( 2

2

xcxTx

Transport Equation

0).2(

0)()(0

...0

..0

..

2

22

211

'1

21222

22

1112

12222

2

2222

21112

1

222

2112

1

TATAA

TAdivdTAdiv

dSnTAdSnTAdSnTA

dSnTAdSnTA

dSnTAdSnTA

TdSATdSAd

cccc

Tracing neighboring rays defines a ray tube : variation of amplitude depends on section variation

0)()()().(2 2 xTxAxTxA

Ray tracing equations

1 tdsxdt

dsxd

)(sx

dsTd

)1(2

)(2 2

2

ccTc

dsTd

))(

1())(

1(xcds

xdxcds

d

Ray

T

T=cte

s curvilinear abscisse

)()(// xTxcdsxdT

dsxd

evolution of x

Evolution of is given by butT

.)(. Txctdsd )(. TTc

dsTd

therefore

)1(cds

Td

evolution of T

Curvature equation known as the ray equation

)(xTp )(sxq

We often note the slowness vector and the position

Evolution of

System of ray equations

1

dq cpdsdpds c

1 1

dq pddpd c c

2

1

dq c pdTdp cdT c

ddTc

dscdTcdsd

2

with

which ODE to select for numerical solving ? Either T or sampling

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

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

Main discontinuous boundaries Crustal discontinuity (Mohorovicic (moho – 30 km) Mantle discontinuity (Gutenberg – 2900 km) D’’ Core discontinuity (Lehman – 5100 km)

Shadow zone at the Gutenberg discontinuity: thickness of few kms

Minor discontinuities Interface at 100-200 km Interface at 670-700 km Interface at 15 km (discontinuity of Conrad)

These discontinuities are related to lithospheric, mesospheric and sismogenic structures.

They are not expected to be deployed over the entire globe

System of ray equations

cdspd

pcdsxd

1

ccdpd

pd

xd

11

cc

dTpd

pcdT

xd

1

2

ddTc

dscdTcdsd

2

with

which ODE to select for numerical solving ? Either T or samplingMany analytical solutions (gradient of velocity; gradient of slowness square)

( )dy A yd

Non-linear ODE !

Time integration of ray equations

Initial conditions EASY

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

source

receiver

Runge-Kutta second-order integrationPredictor-Corrector integration stiffness

source

receiver Boundary conditions VERY DIFFICULT

?

?

Shooting dp ?Bending dx ?Continuing dc ?

AND FROM TIME TO TIME IT FAILS !

But we need 2 points ray tracing because we have a source and a receiver to be connected !

Save p conditions if possible !

Eikonal Solvers Fast marching method (FMM)

22

222

)(1

1)()(

xT

czT

czT

xT

zT

xT

(tracking interface/wavefront motion : curve and surface evolution)Layered

medium

Let us assume T is known at a level z=cte

z=cte

z

z + dz

Compute along z=cte by a finite difference approximation

zT

Deduce and extend T estimation at depth z+dz

xT

Assuming 1D plane wave propagation, we have been able to estimate T at a depth z+dz

From Sethian, 1999

EIKONAL SOLVER

Fast marching method (FMM) 2 2

2

22

1( ) ( )( , )

1 ( )( , )

T Tx z c x z

T Tz c x z x

(tracking interface/wavefront motion : curve and surface evolution)

2D case

Strategy available in 2D and 3D BUT only for the minimum time in each node in the spatial domain (x,y,z).Possible extension in the phase domain (for multiphases) ?Sharp interfaces are always difficult to handle in this discrete formulation

From Sethian, 1999

Other techniques as fast sweeping methods3D case

Wavefront trackingBack-raytracing strategy

Once traveltime T is computed over the grid for one source, we may backtrace using the gradient of T from any point of the medium towards the source (should be applied from each receiver)

The surface {MIN TIME} is convex as time increases from the source : one solution !

A VERY GOOD TOOL FOR First Arrival Time

Tomography (FATT)

Time over the grid Ray

Smooth medium : simple case

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

Ray tracing by wavefronts

Rays and wavefronts in a constant gradient of the velocity. (Lambaré et al., 1996)

(Lambaré et al., 1996)

Rays and wave fronts in a complex medium. (Lambaré et al., 1996)

(Lambaré et al., 1996)

(Lambaré et al., 1996)

(Lambaré et al., 1996)

0 400 800 1200 1600 2000Z in m

Ray tracing by wavefronts Evolution over time :

folding of the wavefront is allowedDynamic sampling :

undersampling of ray fansoversampling of ray fans

Keep sampling of the medium by rays « uniform »

Heavy task 2D & 3D !

Example of wavefront evolution in Marmousi model

Hamilton Formulation

ppqq

dd

0

0 d q

d p

Information around the ray

Ray

2

121

cdpd

pd

qd

)1(21),( 2

2

cppqH

Hd

pd

Hd

qd

x

p

mechanics : ray tracing is a balistic problem

sympletic structure (FUN!)

Meaning of the neighbooring zone – Fresnel zone for example but also anything you wish

Paraxial Ray theory

yydd dd

)( yd

Estimation of ray tube : amplitude

Estimation of taking-off angles : shooting strategy…

does not depend on : LINEAR PROBLEM (SIMPLE) !

qpqHppqHd

qd

ppqqHd

qqd

qppp

pp

ddd

ddd

d

),(),(

),()(

0000

000

0000

0

0 0

0 0pq pp

qq qp

H Hq qdH Hp pd

d dd d

Practical issues

2 2, ,2 2, ,

0 0 1 00 0 0 1

0.5 0.5 0 00.5 0.5 0 0

x x

z z

xx xzx x

zx zzz z

q qq qd

u up pdu up p

d dd dd dd d

Four elementary paraxial trajectories

dy1t(0)=(1,0,0,0)

dy2t(0)=(0,1,0,0)

dy3t(0)=(0,0,1,0)

dy4t(0)=(0,0,0,1)

, , ,tx z x zy q q p pd d d d d

NOT A RAY !

Practical issues 0Hd

From paraxial trajectories, one can combine them for paraxial rays as long as the perturbation of the Hamiltonian is zero.For a point source, a small slowness perturbation (a=10-4) gives a good approximate ray depending on the velocity variation in the medium: it is a kind of derivative ….

(0) (0)(0) (0)

x z

z x

p pp p

d ad a

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

0x x z zp p p pd d

2 2, ,

1 1( , ) ( , ) 02 2x x z z x x z zp p p p u x z q u x z qd d d d

Point source conditions dqx(0)=dqz(0)=0

( ) (0) 3( ) (0) 4( )z xy p y p yd a d a d

This is enough to verify this condition initially

Solution

Practical issues

Two independent paraxial rays exist in 2D (while we have four elementary paraxial trajectories)

Point paraxial ray

Plane paraxial ray

2,

2,

(0) ( , ) (0)

(0) ( , ) (0)x x

z z

q u x z

q u x z

d a

d a

Plane wave conditions dpx(0)=dpz(0)=0

2 2, ,( ) ( , ) (0) 1( ) ( , ) (0) 2( )x zy u x z y u x z yd a d a d

This is enough to verify this condition initially

Solution

2 2, ,( , ) ( , ) 0x x z zu x z q u x z qd d

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 !

Time error over the grid (0)

NOT THE SAME COLOR SCALE (factor 100)

Coarser grid for computation

Errors through FMM times Errors through rays deduced after FMM times

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

Runge-Kutta integration

1/ 2 0 0 0

1 0 1/ 2 1/ 2

( )

. ( )2. ( )

df A fd

f f A f

f f A f

Second-order integration

Non-linear ray tracing

Second-order euler integration for paraxial ray tracing

1 0 0

.

. .

df A fdf f A f

2 2, ,2 2, ,

0 0 1 00 0 0 1

0.5 0.5 0 00.5 0.5 0 0

x x

z z

xx xzx x

zx zzz z

q qq qd

u up pdu up p

d dd dd dd d

Four elementary paraxial trajectories

dy1t(0)=(1,0,0,0)

dy2t(0)=(0,1,0,0)

dy3t(0)=(0,0,1,0)

dy4t(0)=(0,0,0,1)

, , ,tx z x zy q q p pd d d d d

Remember !

3 is for point solution along x

4 is for point solution along z

Two points ray tracing: the paraxial shooting method

. where the shooting angle is .xi

i

d qxdd

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

0 0 0

0 0 0

3 4

0

x zx x i x i

x zi i i i iz z z

z xx x xi ix z

i i iz z z

z xxi i

i z

d q d q dp d q dpd dp d dp d

d q d q d qp p

d dp dp

d qqx p qx p

d

d d d

d d d

dd d

We can compute an estimated and, therefore, a new shooting angle

The estimation of the derivative is through paraxial computation

Amplitude estimation

2 2

3 4 3 4

2 2

( ) ( )

( ) ( )

( ) ( ) ( ) ( )

( ) ( )

x z z x

x z

z x z xx i x i z z i z i x

x z

q p r q p rL

p r p r

q p q p p r q p q p p rL

p r p r

d d

d d d d

Consider L the distance between the exit point of a ray and the paraxial ray running point.

L

Thanks to the point paraxial ray estimation dq3 and dq4, we may estimate the geometrical spreading L/and, therefore, the amplitude A(r)

( ) LA r

Stop and move to the numerical exercise

How to reconstruct the velocity structure ?

Forward problem (easy)from a known velocity structure, it is possible to compute

travel times, emergent distance and amplitudes.

Inverse problem (difficult)from travel times (or similarly emergent distances), it is

possible to deduce the velocity structure : this is the time tomographyeven more difficult is the diffraction tomography related to the

waveform and/or the preserved/true amplitude.

Tomographic approach

Very general problemmedicine; oceanography, climatology

Difficult problem when unknown a priori medium (travel time tomography)

Easier problem if a first medium could be constructed: perturbation techniques can be used for improving the reconstruction (delayed travel time tomography)

Travel Time tomography

We must « invert » the travel time or the emergent distance for getting z(u): we select the distance.

Abel problem (1826)

2

22

2

022

2

20

/2

)(;)(

2)( dupu

dudzppXdz

pzu

ppX

p

u

z p

Determination of the shape of a hill from travel times of a ball launched at the bottom of the hill with various initial velocity and coming back at the initial position.

ABEL PROBLEM

A point of mass ONE and initial velocity v0 reaches a maximal height x given by 2

021 vgx

We shall take as the zero value for the potential energy: this gives us the following equations and its integration:

2

0

/2 ( ) ; ( )2 ( )

xds ds dg x t x ddt g x

We may transform it into the so-called Abel integral where t(x) is known and f() is the shape of the hill to be found: this is an integrale equation.

dxfxt

x

0

)()(

dsd

y

x

The exact inverse solution

0

0

00

00

000

)(1)(

)()(

)()(

)()(

)()(

xdxxt

ddf

fx

dxxtdd

dfdxx

xt

xxdxdfdx

xxt

dxf

xdxdx

xxt x

We multiply and we integrate

We inverse the order of integration

We change variable of integration

We differentiate and write it down the final expression

22 sincos x

THE ABEL SOLUTION

a

x

a

xdxxt

ddf

dx

fxt

)(1)(

)()(By changing variable in a- and x in a-x, we get the standard formulae

We must havet(x) should be continuous,

t(0)=0 t(x) should have a finit derivative with a finite number of discontinuities.The most restrictive assumption is the continuity of the function t(x).

The solution HWB : HERGLOTZ-WIECHERT-BATEMAN

)(

0

22

2

22

2

22

2

)cosh(1)(

)(1)(

2/)(1)(

/2

)(

0

2

20

2

20

uX

u

u

u

u

p

u

pvdXvz

dpup

pXvz

dpup

ppXvz

dupu

dudzppX

In Cartesian frame In Spherical frame

)/(

0 )/cosh(1)

)(ln(

vr

rpvd

vrR

From the direct solution, we can deduce the inverson solutionAfter few manipulations, we can move from the Cartesian expression towards the Spherical expression

We find r(v) as a value of r/v

Stratified medium We may find the interface at a depth h

when considering all waves We may reconstruct an infinity of

structures with only direct and refracted waves.

We have a velocity jump when a velocity decrease

Velocity structure with depth Velocity profile built without any a priori

An difficulty arises when the velocity decreases.

An initial model through the HWB method

An initial model can be built The exact inverse formulae does not allow to introduce

additional information, F. Press in 1968 has preferred the exhaustive exploration of

possible profiles (5 millions !). The quality of the profile is appreciated using a misfit function as the sum of the square of delayed times as well as total volumic mass and inertial moments well constrained from celestial mechanics ... Exploration through grid search, Monte Carlo search,

simulated annealing, genetic algorithm, tabou method, hant search …

The symmetrical radial EARTH

A simple case : small perturbation Initiale structure of velocity Search of small variationof velocity or slowness Linear approach

Example: Massif Central

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 dld

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

d

d

d d

Consider small perturbations du(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 dt(s,r)=G(du)

DESCRIPTION OF THE VELOCITY PERTURBATION

The velocity perturbation field (or the slowness field) du(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 ,,,,),,( dd

Discrete Model Space

cube

kjikji huzyxu ,,,,),,( dd

m

m

m

nn

m

n

n

cubekji

cube rayonkjikji

rayon cubekjikji

uu

uu

ut

ut

ut

ut

tt

tt

uutrst

dlhudlhurst

dd

dd

dd

dd

dd

ddd

1

2

1

1

1

1

1

1

2

1

,,

,,,,,,,,

...

),(

),(00

Slowness perturbation description

0t G ud d

Matrice of sensitivity or of partial derivatives

Discretisation of the medium fats the raySensitivity matrice is a sparse matrice

LINEAR INVERSE PROBLEM

1 10 0

0 0

u G t m G dt G u d G md dd d

Updating slowness perturbation values from time residuals

Formally one can writewith the forward problem

Existence, Uniqueness, Stability, RobustnessDiscretisation

Identifiability

of the model

Small errors propagates

Outliers effects

NON-UNIQUENESS & NON-STABILITY : ILL-POSED PROBLEMREGULARISATION : ILL-POSED -> WELL-POSED

LEAST-SQUARES SOLUTIONS

AtDT=AtA DM

• The linear system can be recast into a least-square system, which means a system of normal equations. The resolution of this system gives the solution. DM=(AtA)-1AtDT

• The system is both under-determined and over-determined depending on the considered zone (and tne number of rays going through.

LEAST SQUARES METHOD

dGGGm

dGmGGmmE

mGdmGdmE

ttest

tt

t

0

1

00

000

00

0)()()()(

L2 norm

locates the minimum of E

normal equations

if exists 1

00

GG t

Least-squares estimation

Operator on data will derive a new model : this is called

the generalized inverse

tt GGG 01

00

gG0

G0 is a N by M matrice

is a M by M matrice 1

00

GG t

Under-determination M > NOver-determination N > M Mixed-determination – seismic tomography

Maximum Likelihood method One assume a gaussian distribution of dataJoint distribution could be written

)()(

21exp)( 0

10 mGdCmGddp d

Where G0m is the data mean and Cd is the data covariance matrice: this method is very similar to the least squares method

)()()()()()( 01

0100 mGdCmGdmEmGdmGdmE dt

)()()( 01

02 mGdWmGdmE d

Even without knowing the matrice Cd, we may consider data weight Wd through

SVD analysis for stability and uniqueness

SVD decomposition :

U : (N x N) orthogonal Ut=U-1

V : (M x M) orthogonal Vt=V-1

L : (N x M) diagonal matrice Null space for Li=0

tVUG L0

UtU=I and VtV=I (not the inverse !)

][

][

0

0

UUU

VVV

p

p

tpp

p VVUUG 000 000

L

Vp and V0 determine the uniqueness while Up and U0 determine the existence of the solution

tppp

tppp

UVG

VUG11

0

0

L

LUp and Vp have now inverses !

Solution, model & data resolution

RmmVVmVUUVmGGdGm tp

tppp

tpppest LL 1

01

01

0 )(The solution is

where Model resolution matrice : if V0=0 then R=VVt=I tppVVR

NddUUmGd tppestest 0

dUUN tppwhere Data resolution matrice : if U0=0 then N=UUt=I

importance matriceGoodness of resolution

SPREAD(R)=

SPREAD(N)=Spreading functions

2

2

IN

IR

Good tools for quality estimation

PRIOR INFORMATION Hard boundsPrior model

is the damping parameter controlling the importance of the model mp

Gaussian distribution

Model smoothness

Penalty approach add additional relations between model parameters (new lines)

)()()()()( 005 pmt

pdt mmWmmmGdWmGdmE

With Wd data weighting and Wm model weighting

tmd

tg

pmt

pdt

GCGCGG

mmCmmmGdCmGdmE

011

01

00

10

104 )()()()()(

)()()()()( 003 pt

pt mmmmmGdmGdmE

BmA i Seismic velocity should be positive

UNCERTAINTY ESTIMATION Least squares solution

Model covariance : uncertainty in the datacurvature of the error function

Sampling the error function around the estimated model often this has to be done numerically

dGdGGGm gttest 00

100

1

2

22

100

2

2

0000

21cov

cov

covcov

estmmdest

tdest

dd

gtd

ggtgest

mEm

GGm

IC

GCGGdGm

Uncorrelated data

A posteriori model covariance matrice True a posteriori distribution

Tangent gaussian distribution

S diagonal matrice eigenvaluesU orthogonal matrice eigenvectors

Error ellipsoidal could be estimatedWARNING : formal estimation related to the gaussian distribution hypothesis

tmd

t USUCGCG 10

10

If one can decompose this matrice

A priori & A posteriori informationWhat is the meaning of the « final » model we provide ?

acceptable

Flow chart

true ray tracing

data residual

sensitivity matrice

model update

new model

mmmdGm

mgG

ddd

mgdmd

synobs

syn

obs

10

0

)(

collected data

starting modelloop

Calculate for formal uncertainty estimation

small model variation or small errors exit

22

mE

Sampling a posteriori distribution

Resolution estimation : spike test

Boot-Strapping

Jack-knifing

Natural Neighboring

Monte-Carlo

Sampling a posteriori distribution

Uncertainty estimation for P and S velocities using boot-strapping techniques

Steepest descent methods )()( 1 kk mEmE

kk

kk

kkkk

k

kkk

DmE

mEmEd

dEE

d

mEdmEmEtmE

)(

)()(

)()())((

2

12

1

0

Gradient method

Conjugate gradient

Newton

Quasi-Newton

Gauss-Newton is Quasi-Newton for L2 norm

quadratic approximation of E

Tomographic descent 2

2/1

2/1

2/1

2/1 )(21

mCmgC

mCdC

m

d

pm

dMinimisation of this vector

2/1

2/1

m

kdk C

GCAIf one computes

then

)())((

02/1

2/1

km

kdtkk

tk mmC

dmgCAmAA d

Gaussian error distribution of data and of a posteriori modelEasy implementation once Gk has been computed

Extension using Sech transformation (reducing outliers effects while keeping L2 norm simplicity)

LSQR method The LSQR method is a conjugate gradient method developped by Paige & Saunders

Good numerical behaviour for ill-conditioned matrices

When compared to an SVD exact solution, LSQR gives main components of the solution while SVD requires the entire set of eigenvectors

Fast convergence and minimal norm solution (zero components in the null space if any)

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 AND RANDOM 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

Other methods of exploration Grid search Monte-Carlo (ponctual or

continuous) Genetic algorithm Simulated annealing and co Tabou method Natural Neighboring method

Conclusion FATT Selection of an enough fine grid Selection of the a priori model information Selection of an initial model FMM and BRT for 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 !

Kirchhoff approximation

1) Representation theorem2) Kirchhoff summation3) Reciprocity

Born approximation

1) Single scattering approximation2) Surface approximation

(Forgues et al., 1996)

(Forgues et al., 1996)