high resolution seismic imaging: asymptotic approachinduces a dramatic increasing complexity in ray...

High resolution seismic imaging:

Asymptotic approach By Jean Virieux

Professeur

UGA

American Petroleum Institute, 1986

Salt Dome Fault

Unconformity

Pinchout

Anticline

Time scales

24/02/2016 HR seismig imaging 2

Source time

from 0.1 sec to 100 sec (rupture vel.)

Wave time

from secondes to hours

Window time

from few secondes to days

Length scales


Fault length

200 km for vr=2 km/s

Discontinuity distance

from few meters to few 100 kms

Volume sampling

from few kms to few 1000 kms

Examples


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 !

Motivation

Delayed travel-time tomography source:receiver

Fitting times => estimation of times

=> 2-pts rays: millions!

Full Waveform Inversion/Migration src/rec:focal point

Same challenge with an order increase in magnitude

Seismogram modeling

Handling multiple arrivals and filling gaps (shadows)


Rays Geometrical Optics Infinite Frequency Singularities topology

A link with the Catastrophe Theory (René Thom)

Do we need this complexity?

FWI: Ray+Born approach Ray tracing efficient only in smooth media

Updating the velocity structure through iterations

induces a dramatic increasing complexity in ray

tracing: people stays often (always) with the initial

ray tracing once done (Background model~Born)

Asymptotic theory (or high frequency

approximation) introduces a complexity in our

finite frequency seismic wave propagation : all

scales are there while they are not present in the

seismic data


In fluid mechanism:

physics below a given small scale is parametrized (upscaled)

References Červený, V., Seismic ray theory, Cambridge University Press, 2001

Chapman, C., Fundamentals of seismic wave propagation, Cambridge University

Press, 2004

Slawinski, M.A., Seismic waves and rays in elastic media, Handbook of geophysical

exploration, seismic, exploration, volume 34, Pergamon, 2003

Abdullaev, S.S., Chaos and dynamics of rays in waveguide media, Gordon and Breach

Science Publishers, 1993

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

Gilmore, R, Catastrophe theory for scientists and engineers, Wiley & Sons, Inc, 1981.

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 methods and

finite frequency effecs in Treatrise on Geophysics, Tome I by A. Diewonski and B.

Romanowicz, Elsevier, 2010


References (selection)

Červený, V., Seismic ray theory, Cambridge University Press, 2001

Chapman, C., Fundamentals of seismic wave propagation, Cambridge University

Press, 2004


Outline of this partial overview

Wavefronts and rays

Ray equation: non-linear ODE

Paraxial Ray equation: linear ODE

Hamilton-Jacobi equation: non-linear PDE

Conclusion

Perspectives


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)


𝑢 𝑥, 𝑡 = 𝐴 𝑥 𝑆 𝑡 − 𝑇 𝑥

𝑢 𝑥, 𝜔 = 𝐴 𝑥 𝑆(𝜔) 𝑒𝑖𝜔𝑇(𝑥)

𝜔𝑇 𝑥 is sometimes called the phase

Eikonal equation Two simple interpretations of

wavefront evolution

Orthogonal trajectories are rays in

an isotropic medium

Grad(T)=𝛻𝑥𝑇 orthogonal to wavefront

Direction ? : abs or square

The orientation of the wavefront could not

be guessed from the local information on a

specific wavefront

24/02/2016 HR seismig imaging

(𝛻𝑥𝑇(𝑥))2 =

1

𝑐2(𝑥)

T+DT

T=cte

Velocity c(x)

DL Ray 𝑐 𝑥 =Δ𝐿

Δ𝑇→Δ𝑇

Δ𝐿=

1

𝑐(𝑥)→ 𝛻𝑥𝑇 𝑥 =

1

𝑐(𝑥)

11

Eikonal equation for anisotropic media


(𝛻𝑥𝑇(𝑥))2 =

1

𝑐2(𝑥, 𝛻𝑥𝑇/ 𝛻𝑥𝑇 )

T=cte

Phase velocity c(x)

Ray (group) velocity v(x(t))

𝑛

Ray

𝑥

Seismic ray: a 1D curved line x(t)

Tangent 𝑑𝑥 𝑑𝑡 = 𝑥 = 𝑣(𝑡) defines the ray velocity

Phase velocity depends on the position

and on the orientation 𝑛.

12

Transport Equation Tracing neighboring rays defines a ray tube : variation of

amplitude depends on energy flux conservation through

sections.


2𝛻𝐴 𝑥 . 𝛻𝑇 𝑥 + 𝐴 𝑥 𝛻2𝑇 𝑥 = 0

∆ℇ1 = 𝐴12𝑑𝑆1∆𝑇1 = 𝐴2

2𝑑𝑆2∆𝑇2 = ∆ℇ2

→ 𝐴1

2𝛻𝑇1. 𝑛𝑑𝑆1 = 𝐴22𝛻𝑇2. 𝑛𝑑𝑆2

0 = −𝐴12𝛻𝑇1. 𝑛𝑑𝑆1 + 𝐴2

2𝛻𝑇2. 𝑛𝑑𝑆2

𝑅𝑎𝑦

𝑆𝑒𝑐𝑡𝑖𝑜𝑛

0 = 𝐴12𝛻𝑇1. 𝑛′𝑑𝑆1 + 𝐴2

2𝛻𝑇2. 𝑛𝑑𝑆2

𝑅𝑎𝑦

𝑆𝑒𝑐𝑡𝑖𝑜𝑛

+ 𝐴𝑐2𝛻𝑇𝑐 . 𝑛𝑐𝑑𝑆𝑐

𝑅𝑎𝑦

𝑇𝑢𝑏𝑒

0 = 𝑑𝑖𝑣(𝐴2𝑅𝑎𝑦

𝑉𝑜𝑙𝑢𝑚𝑒

𝛻𝑇)𝑑𝑉 ⟹ 𝑑𝑖𝑣 𝐴2𝛻𝑇 = 0

𝐴(𝑥)(2𝛻𝐴 𝑥 . 𝛻𝑇 𝑥 + 𝐴 𝑥 𝛻2𝑇 𝑥 ) = 0

Energy flux same at section one and at section two

net contribution=zero

Transport equation for anisotropic media


2𝛻𝐴 𝑥 . 𝛻𝑇 𝑥 + 𝐴 𝑥 𝛻2𝑇 𝑥 = 0

2𝜕𝐴

𝜕𝑥𝑖

𝜕𝑇

𝜕𝑥𝑖+ 𝐴

𝜕2𝑇

𝜕𝑥𝑖2 = 0

3

𝑖=1

∀𝑖 = 1,3 𝜕

𝜕𝑥𝑗𝑐𝑖𝑗𝑘𝑙𝐴𝑙

𝜕𝑇

𝜕𝑥𝑘+ 𝑐𝑖𝑗𝑘𝑙

𝜕𝐴𝑘𝜕𝑥𝑙

𝜕𝑇

𝜕𝑥𝑗

3

𝑙=1

3

𝑘=1

= 0

3

𝑗=1

u x, t = A x S t − T x

u x,ω = A x S(ω) eiωT(x)

𝑢 𝑥, 𝑡 = 𝐴 𝑥 𝑆 𝑡 − 𝑇 𝑥

u x,ω = A x S(ω) 𝑒𝑖𝜔𝑇(𝑥) Isotropic case: scalar functions

Anisotropic case: vectorial functions

Not easy to understand geometrically how energy flows

Slawinski, 2003, p167


Ray equation

Ray

T=cte

Evolution of 𝛻𝑇 is given by 𝑑𝛻𝑇

𝑑𝑠

but the operator 𝑑

𝑑𝑠= 𝑡 . 𝛻 = 𝑐𝛻𝑇. 𝛻

and, therefore, 𝑑𝛻𝑇

𝑑𝑠= 𝑐𝛻𝑇. 𝛻 𝛻𝑇

leading to 𝑑𝛻𝑇

𝑑𝑠=

𝑐

2𝛻 𝛻𝑇. 𝛻𝑇 =

𝑐

2𝛻(𝛻𝑇)2=

𝑐

2𝛻(

1

𝑐2)

Evolution of 𝑥 is given by 𝑑𝑥

𝑑𝑠

𝑥 𝑠 𝑤𝑖𝑡ℎ 𝑠 𝑎𝑠 𝑐𝑢𝑟𝑣𝑖𝑙𝑖𝑛𝑒𝑎𝑟 𝑎𝑏𝑠𝑐𝑖𝑠𝑠𝑒 𝑡 =𝑑𝑥

𝑑𝑠→ 𝑡 = 1

𝑡

Wavefront

𝑑𝑥

𝑑𝑠∥ 𝛻𝑇 →

𝑑𝑥

𝑑𝑠= 𝑐(𝑥)𝛻𝑇(𝑥)

𝑑𝛻𝑇

𝑑𝑠= 𝛻(

1

𝑐(𝑥))

Ray equations

𝑑

𝑑𝑠

1

𝑐(𝑥)

𝑑𝑥

𝑑𝑠= 𝛻(

1

𝑐 𝑥)

We define the slownss vector 𝑝 = 𝛻𝑇 𝑥 and the position 𝑞 = 𝑥 (𝑠) along the ray

Curvature equation

𝛻𝑇

Various non-linear ray equations


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

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

Curvilinear stepping

𝑑𝑞 (𝑠)

𝑑𝑠= 𝑐 𝑞 𝑝

𝑑𝑝 (𝑠)

𝑑𝑠= 𝛻𝑞

1

𝑐(𝑞 )

𝑑𝑇(𝑠)

𝑑𝑠=

1

𝑐(𝑞 )

Particule stepping

𝑑𝑞 (𝜉)

𝑑𝜉= 𝑝

𝑑𝑝 (𝜉)

𝑑𝜉=

1

𝑐(𝑞 )𝛻

1

𝑐(𝑞 )

𝑑𝑇(𝜉)

𝑑𝜉=

1

𝑐2(𝑞 )

Time stepping

𝑑𝑞 (𝑡)

𝑑𝑡= 𝑐2 𝑞 𝑝

𝑑𝑝 (𝑡)

𝑑𝑡= 𝑐(𝑞 )𝛻

1

𝑐(𝑞 )

𝑑𝑇 =1

𝑐(𝑞 )𝑑𝑠 =

1

𝑐(𝑞 )2𝑑𝜉

under the condition of the eikonal 𝑝2 = 1/𝑐2(𝑞 )

The s

imple

st

set

Polarization


Chapman, 2004, p180

𝑔 1

𝑔2

𝑔3

Acoustic case: the unitary vector 𝑔 1 = 𝑐 𝑞 𝑝 supports the P wave vibration

Elastic case: the independent shear

vibration will be along two unitary vectors 𝑔 2 and 𝑔 3 such that

𝑑𝑔 2𝑑𝑡

= 𝑔 2. 𝛻𝑞𝑐 𝑔 1

𝑑𝑔 3𝑑𝑡

= 𝑔 3. 𝛻𝑞𝑐 𝑔 1

Isotropic case: shear

vibrations are orthogonal to

compression vibrations

𝑑𝑔 2𝑑𝜉

=𝑔 2. 𝛻𝑞𝑐

𝑝2𝑔 1

𝑑𝑔 3𝑑𝜉

=𝑔 3. 𝛻𝑞𝑐

𝑝2𝑔 1

Time stepping

Particule stepping

It is enough to follow the

evolution of the projection of

elastic unitary vectors on

one Cartesian coordinate:

𝑒 𝑧. 𝑔 2 and 𝑒 𝑧. 𝑔 3(Psencik,

perso. Comm.)

One additional equation for polarization

How to solve these equations

ODE versus PDE !


ODE

Lagrangian formulation

Wavefront complexity


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 where

we compute (ray) quantities at fixed positions


Properties of these ODEs

Intrinsic solutions independent of the

coordinate sytem used to solve it

If dummy variable for velocity, use it as the

variable stepping (often x coordinate)

𝛻𝑥1

𝑐(𝑞𝑧)= 0 ⇒ 𝑝𝑥 = 𝑐𝑡𝑒 ⇒ 𝑞𝑥 = 𝑞𝑥

0 + 𝜉𝑝𝑥 (1)

Eikonal equation: a good proxy for testing

the accuracy of the ray tracing (not enough

used)


In 3D: six or seven equations

In 2D: four or five equations

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

Hamilton’s ray equations


Information around the ray Ray

Mechanics : ray tracing as a particular balistic problem

sympletic structure (FUN!)

Meaning of the neighborhood zone

Fresnel zone if finite frequency

Any zone depending on your problem GBS

𝑑𝑞 (𝜉)

𝑑𝜉= 𝑝

𝑑𝑝 (𝜉)

𝑑𝜉=

1

𝑐(𝑞 )𝛻𝑞

1

𝑐(𝑞 )

𝑑𝑞 (𝜉)

𝑑𝜉= 𝛻𝑝 ℋ

𝑑𝑝 (𝜉)

𝑑𝜉= −𝛻𝑞ℋ

𝑑𝑇

𝑑𝜉= 𝑝 . 𝛻𝑝 ℋ

ℋ 𝑞 , 𝑝 =1

2(𝑝2 −

1

𝑐2(𝑞 ))

Hamilton approach suitable for perturbation

(Henri Poincaré en 1907 « Mécanique céleste »,

Richard Feymann Prix Nobel 1965) 𝑞 0 + 𝛿𝑞 𝑝 0 + 𝛿𝑝

𝛿𝑞 𝑎𝑛𝑑 𝛿𝑝 "𝑠𝑚𝑎𝑙𝑙"

Ray equations for anisotropic media


Eikonal equation

𝑝2 =1

𝑐2(𝑞 , 𝑝 )

Particule stepping

𝑑𝑞𝑖 (𝜉)

𝑑𝜉= 𝑝𝑖 +

1

𝑐3(𝑞𝑖 , 𝑝𝑖)

𝜕𝑐(𝑞𝑖 , 𝑝𝑖)

𝜕𝑝𝑖

𝑑𝑝𝑖 (𝜉)

𝑑𝜉=

1

𝑐(𝑞𝑖 , 𝑝𝑖)𝛻𝑞𝑖

1

𝑐(𝑞𝑖 , 𝑝𝑖)

𝑑𝑇(𝜉)

𝑑𝜉= 𝑝𝑖(𝑝𝑖 +

1



𝜕𝑝𝑖)

𝑖

Please, remember that the phase velocity

depends only on the orientation of slowness

vector and not on its length

Time stepping

𝑑𝑞𝑖 (𝑡)

𝑑𝑡= (𝑝𝑖+

1



𝜕𝑝𝑖)

1

𝑝𝑘(𝑝𝑘 +1

𝑐3(𝑞𝑘 , 𝑝𝑘)𝜕𝑐(𝑞𝑘 , 𝑝𝑘)

𝜕𝑝𝑘)𝑘

𝑑𝑝𝑖 (𝑡)

𝑑𝑡= (

1

𝑐 𝑞𝑖 , 𝑝𝑖𝛻𝑞𝑖

1

𝑐 𝑞𝑖 , 𝑝𝑖)

1

𝑝𝑖(𝑝𝑘 +1

𝑐3(𝑞𝑘 , 𝑝𝑘)𝜕𝑐(𝑞𝑘 , 𝑝𝑘)

𝜕𝑝𝑘)𝑘

Particule

formulation

does not show

simpler

structure than

time

formulation for

numerical point

of view

Reduced Hamilton formulation


In 3D, five equations

In 2D, three equations

including travel-time integration equation

Reduction of ray equations from six to four

Solving the eikonal equation for obtaining 𝑝3 reducing from one unknow

gives

𝑝3 = −ℋ𝑅 𝑥1, 𝑥2 , 𝑥3, 𝑝1, 𝑝2

which is a non-linear partial differential equation of first order known as

static Hamilton-Jacobi equations.

We may select 𝑥3 as the stepping variable, reducing once more from one

unknown, removing two unknows and therefore two equations are

cancelled.

Ray system (𝑥1, 𝑥2, 𝑝1, 𝑝2) with the evolution 𝑥3 with a variable ℋ𝑅

The evolution parameter 𝑥3 could not be monotonic (turning rays with an

extremum in variable 𝑥3).

(Cerveny, 2001, p107)

Reduced Hamilton formulation (2)


Turning rays leads us to consider centered ray coordinate system

and trigonometric functions ! (to be avoided … as slow crunching)

This is not intrinsic to ray equations which can be written in any coordinate

system (as well as the related paraxial/dynamic ray equations)

- Computational complexity for number crunching

- No evaluation in the vicinity of the hyper-surface of the eikonal

conservation as we live in lower-dimension space

What is best!

If you are afraid of curvilinear non-orthogonal coordinate system

intrinsically linked with the ray geometry, you may forget about that.

Cartesian coordinate system leads to higher dimensions but with simpler

expressions (somehow faster …)

Mechanical point of view


ℋ 𝑞 , 𝑝 =1

2

𝑝2

𝑚+ 𝑉(𝑞 )

ℋ 𝑞 , 𝑝 =1

2𝑝2 −

1

2

1

𝑐2(𝑞 )

𝑐 𝑧 = 𝑐0 1 + 𝛾𝑧 ray is a circle

Potential energy Kinetic energy

z

−1

2

1

𝑐2(𝑧)

Turning point

−1

𝛾 0

I should draw a 3D curve with the x coordinate :Help !

Just FUN?

A conservative system

ℋ 𝑞 , 𝑝 = 0 ℋ = ℰ

Chapman’s egg

S

R

Ray lives there (and not outside)

A non-conservative system

ℋ 𝑞 , 𝑝 = ℰ 𝑞 , 𝑝 ⇒ ℋ 𝑞 , 𝑝 − ℰ 𝑞 , 𝑝 = ℋ′ 𝑞,𝑝 = 0

Embedding it into an isolated system !

Mechanical point of view (2)


Just FUN: NO!

If perturbation of the velocity structure,

should we reset ray tracing?

ℋ 𝑞 , 𝑝 = ℋ0 𝑞 , 𝑝 + ∆ℋ 𝑞 , 𝑝

ℋ0 S

R

ℋ Rays on the Egg0 used for the

estimation of rays on the new Egg

Shifts in the phase space of both

sources and receivers: application to

extended image analysis as promoted

by different people: W. Symes, P. Sava

among others.

We may feel the extra-dimensionality around a given Chapman’s egg

in this full Hamiltonian formulation: this is not the case when we

consider the reduced Hamiltonian.

Velocity variation v(z)

dz

zduzu

d

dp

d

dp

d

dp

pd

dqp

d

dqp

d

dq

zyx

z

z

y

y

x

x

)()(;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.

dz

zduzu

d

dp

d

dp

pd

dqp

d

dq

zx

z

z

x

x

)()(;0

;

22)(

x

x

z

x

z

x

pzu

p

p

p

dq

dq

Where px is a constante

1

0

220011

)(

),(),(

z

zx

x

xxxxdz

pzu

ppzqpzq

For a ray towards the depth


Velocity variation v(z)

)(222

pxzupp

pp

pp

z

zx

z

zx

x

z

zx

x

z

zx

x

xxx

dz

pzu

zudz

pzu

zuTpzT

dz

pzu

pdz

pzu

pqpzq

10

10

22

2

22

2

011

2222011

)(

)(

)(

)(),(

)()(

),(

p

p

z

z

dz

pzu

zupT

dz

pzu

ppX

0

22

2

0

22

)(

)(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

( )

s in

p

p

a

r

a

r

p d r

rr u r p

r u r d rT

rr u r p

w i th p r u i

D

In Cartesian frame In Spherical frame

with p = usini


Velocity structure in the Earth

Radial Structure


System of ray tracing equations


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

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

Curvilinear stepping

𝑑𝑞 (𝑠)

𝑑𝑠= 𝑐 𝑞 𝑝

𝑑𝑝 (𝑠)

𝑑𝑠= 𝛻𝑞

1

𝑐(𝑞 )

𝑑𝑇(𝑠)

𝑑𝑠=

1

𝑐(𝑞 )

Particule stepping

𝑑𝑞 (𝜉)

𝑑𝜉= 𝑝

𝑑𝑝 (𝜉)

𝑑𝜉=

1

𝑐(𝑞 )𝛻

1

𝑐(𝑞 )

𝑑𝑇(𝜉)

𝑑𝜉=

1

𝑐2(𝑞 )

Time stepping

𝑑𝑞 (𝑡)

𝑑𝑡= 𝑐2 𝑞 𝑝

𝑑𝑝 (𝑡)

𝑑𝑡= 𝑐(𝑞 )𝛻

1

𝑐(𝑞 )

𝑑𝑇 =1

𝑐(𝑞 )𝑑𝑠 =

1

𝑐(𝑞 )2𝑑𝜉

under the condition of the eikonal 𝑝2 = 1/𝑐2(𝑞 )

The s

imple

st

set

Time integration of ray equations

Initial conditions EASY

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

source

receiver

Runge-Kutta second-order integration

Predictor-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 in seismology!

Save p conditions

if possible !


Ray tracing Constant angle step

Ray tracing by rays


Two-point ray tracing

Ray tracing by rays


Source (0,0)

Receivers (2,-2),

(4,-2),(6,-2)

(depth is considered as

negative)

Please note the irregular

angle stepping.



Information around the ray Ray

Mechanics : ray tracing as a particular balistic problem

sympletic structure (FUN!)

Meaning of the neighborhood zone

Fresnel zone if finite frequency

Any zone depending on your problem GBS

𝑑𝑞 (𝜉)

𝑑𝜉= 𝑝

𝑑𝑝 (𝜉)

𝑑𝜉=

1

𝑐(𝑞 )𝛻𝑞

1

𝑐(𝑞 )

𝑑𝑞 (𝜉)


𝑑𝑝 (𝜉)


𝑑𝑇


ℋ 𝑞 , 𝑝 =1

2(𝑝2 −

1

𝑐2(𝑞 ))

Hamilton approach suitable for perturbation

(Henri Poincaré en 1907 « Mécanique céleste »,

Richard Feymann Prix Nobel 1965) 𝑞 0 + 𝛿𝑞 𝑝 0 + 𝛿𝑝

𝛿𝑞 𝑎𝑛𝑑 𝛿𝑝 "𝑠𝑚𝑎𝑙𝑙"

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-Kutt a integration

Second-order RK integration

Non-linear ray tracing

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


𝑑𝑓

𝑑𝜉= 𝐴 𝑓

𝑓12 = 𝑓0 +

Δ𝜉

2 𝐴 𝑓0

𝑓1 = 𝑓0 + Δ𝜉 𝐴 𝑓12

𝑑𝛿𝑓

𝑑𝜉= 𝐴 𝑓 𝛿𝑓

𝛿𝑓1 = 𝛿𝑓0 + Δ𝜉 𝐴 𝑓0 𝛿𝑓0

Linear paraxial ray tracing

Propagator technique

Optical Lens technique

Analytical solutions


Vertical gradient of

the velocity

Gradient of the

slowness square

Keeping complexity low

Ray tracing is a fast 1D integration in 2D/3D

Ray tracing equations as ODEs may sample the

medium quite evenly

They are lagrangian formulation: we follow a

point while tracing rays without regarding the

density of rays


Keeping complexity low

Solutions: moving from ODEs to PDEs

(Osher et al, 2002) for adequate spatial sampling of

the wavefront. Grids control the complexity!


3D French’s model

Interpolation challenge


Mitigating the interpolation issue

We cannot guarantee that two rays will

stay nearby for small shooting angles

changes

Differential approach will be an answer:

the perturbed ray will stay nearby the

reference ray …


Paraxial ray theory

similar to

Gauss optics




Information around the ray 𝑦0

Ray

𝑑𝑞 (𝜉)

𝑑𝜉= 𝑝

𝑑𝑝 (𝜉)

𝑑𝜉=

1

𝑐(𝑞 )𝛻𝑞

1

𝑐(𝑞 )

𝑑𝑞 (𝜉)


𝑑𝑝 (𝜉)


𝑑𝑇


ℋ 𝑞 , 𝑝 =1

2(𝑝2 −

1

𝑐2(𝑞 ))

𝑞 0 + 𝛿𝑞 𝑝 0 + 𝛿𝑝 𝛿𝑞 𝑎𝑛𝑑 𝛿𝑝 "𝑠𝑚𝑎𝑙𝑙"

𝑦 =𝑞

𝑝

𝛿𝑦 =𝛿𝑞

𝛿𝑝

Paraxial Ray theory

Estimation of ray tube: KMAH

index tracking and amplitude

evaluation

Estimation of taking-off angles:

shooting strategy

The matrix 𝐴 does not depend on quantities from

𝛿𝑦 but only on quantities from 𝑦0: LINEAR PROBLEM

(SIMPLE) !


𝑑

𝑑𝜉𝛿𝑦 = 𝐴(𝑦0)𝛿𝑦

𝑑

𝑑𝜉

𝛿𝑞

𝛿𝑝=

𝛻𝑝𝑞ℋ0 𝛻𝑝𝑝ℋ

0

−𝛻𝑞𝑞ℋ0 −𝛻𝑞𝑝ℋ

0

𝛿𝑞

𝛿𝑝

𝑑(𝑞0 + 𝛿𝑞)

𝑑𝜉= 𝛻

𝑝0+𝛿𝑝ℋ 𝑞0 + 𝛿𝑞, 𝑝0 + 𝛿𝑝

𝑑𝛿𝑞

𝑑𝜉= 𝛻𝑝0𝛻𝑝0ℋ 𝑞0, 𝑝0 𝛿𝑝 + 𝛻𝑝0𝛻𝑞0ℋ 𝑞0, 𝑝0 𝛿𝑞

seismogram computation

Paraxial Ray theory


Solutions are coordinate dependent (differential

computation)

Not restricted to the so-called ray-centered

coordinate system (Cerveny, 2001)

Cartesian formulation is much simpler to handle

(Virieux & Farra, 1991)

2D simple linear system

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) NOT A paraxial RAY !


𝛿𝑦𝑡 = 𝛿𝑞𝑥, 𝛿𝑞𝑧 , 𝛿𝑝𝑥 , 𝛿𝑝𝑧

𝑑

𝑑𝜉

𝛿𝑞𝑥𝛿𝑞𝑧𝛿𝑝𝑥𝛿𝑝𝑧

=

0 0 1 00 0 0 1

0.5𝜕2 1 𝑐2(𝑞 0)

𝜕𝑥20.5

𝜕2 1 𝑐2(𝑞 0)

𝜕𝑥𝜕𝑧0 0

0.5𝜕2 1 𝑐2(𝑞 0)

𝜕𝑧𝜕𝑥0.5

𝜕2 1 𝑐2(𝑞 0)

𝜕𝑧20 0

𝛿𝑞𝑥𝛿𝑞𝑧𝛿𝑝𝑥𝛿𝑝𝑧

More complex anisotropic structure but still straightforward

Linear

system

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!


𝛿ℋ 𝜉 = 𝛿ℋ 0 = 0

𝑝𝑥𝛿𝑝𝑥 + 𝑝𝑧𝛿𝑝𝑧 −1

2

𝜕1 𝑐2 𝑥,𝑧

𝜕𝑥𝛿𝑞𝑥 −

1

2

𝜕1 𝑐2 𝑥,𝑧

𝜕𝑧𝛿𝑞𝑧 = 0

𝜕ℋ

𝜕𝑝𝑥𝛿𝑝𝑥 +

𝜕ℋ

𝜕𝑝𝑧𝛿𝑝𝑧 +

𝜕ℋ

𝜕𝑞𝑥𝛿𝑞𝑥 +

𝜕ℋ

𝜕𝑞𝑧𝛿𝑞𝑧 = 0

Or in the isotropic case

Similar conditions in 3D

Readily deduced for anisotropy Two 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 𝛼 could be set to an arbitrary small value: this is

a derivative or plan tangent computation (Gauss optics)

This is enough to verify this condition initially

Paraxial solution 𝛿𝑦𝑎 𝜉 = 𝛼𝑝𝑧 0 𝛿𝑦3 𝜉 − 𝛼𝑝𝑥 0 𝛿𝑦4(𝜉)


𝛿𝑞𝑥 𝑂 = 𝛿𝑞𝑧 𝑂 = 0 ⇒ 𝑝𝑥 0 𝛿𝑝𝑥 0 + 𝑝𝑧 0 𝛿𝑝𝑧 0 =0

𝛿𝑝𝑥(0) = 𝛼𝑝𝑧(0)

𝛿𝑝𝑧 0 = −𝛼𝑝𝑥(0)

𝛼 𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑦 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 (𝑙𝑖𝑛𝑒𝑎𝑟 𝑠𝑦𝑠𝑡𝑒𝑚)

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 𝛿𝑦𝑏 𝜉 = 𝛼𝜕1 𝑐2 𝑥,𝑧

𝜕𝑧(0)𝛿𝑦1 𝜉 − 𝛼

𝜕1 𝑐2 𝑥,𝑧

𝜕𝑥(0)𝛿𝑦2(𝜉)


𝛿𝑝𝑥 𝑂 = 𝛿𝑝𝑧 𝑂 = 0 ⇒

1

2

𝜕 1 𝑐2(𝑥, 𝑧)

𝜕𝑥0 𝛿𝑞𝑥 0 +

1

2

𝜕 1 𝑐2(𝑥, 𝑧)

𝜕𝑧0 𝛿𝑞𝑧 0 = 0

𝛿𝑞𝑥(0) = 𝛼𝜕 1 𝑐2(𝑥, 𝑧)

𝜕𝑧(0)

𝛿𝑞𝑧 0 = −𝛼𝜕 1 𝑐2(𝑥, 𝑧)

𝜕𝑥(0)

𝛼 𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑦 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 (𝑙𝑖𝑛𝑒𝑎𝑟 𝑠𝑦𝑠𝑡𝑒𝑚)

Two independent paraxial rays in 2D (𝛿𝑦𝑎 𝑎𝑛𝑑 𝛿𝑦𝑏): point

(seismograms) and plane (beams) paraxial rays

KMAH index tracking


In 2D, the determinant

𝑝𝑥(𝜉) 𝛿𝑞𝑥3 𝛿𝑞𝑧

3

𝑝𝑧(𝜉) 𝛿𝑞𝑥4 𝛿𝑞𝑧

4

0 𝑝𝑥(0) 𝑝𝑧(0)

may change sign:

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

𝑢 𝑞 , 𝑡 = 𝐴 𝑞 𝑒𝑖𝜔𝑇(𝑥)𝑒−𝑖12𝑠𝑔𝑛 𝜔 𝐾𝑀𝐴𝐻

In 3D, the determinant

𝑝𝑥(𝜉) 𝛿𝑞𝑥4 𝛿𝑞𝑦

4 𝛿𝑞𝑧4

𝑝𝑦(𝜉) 𝛿𝑞𝑥5 𝛿𝑞𝑦

5 𝛿𝑞𝑧5

𝑝𝑧(𝜉) 𝛿𝑞𝑥6 𝛿𝑞𝑦

6 𝛿𝑞𝑧6

0 𝑝𝑥(0) 𝑝𝑦(0) 𝑝𝑧(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

Euler integration for paraxial ray tracing

First Euler integration for paraxial ray tracing is enough!


𝑑𝛿𝑓

𝑑𝜉= 𝐴 𝑓0 𝛿𝑓

𝛿𝑓1 = 𝛿𝑓0 + Δ𝜉 𝐴 𝑓0 𝛿𝑓0

Linear paraxial ray tracing

Propagator technique

Optical Lens technique

Paraxial trajectories and rays


Paraxial rays have to be

understood as a kind of derivative

Paraxial solutions are coordinate-

dependent …

A differential form on

the Chapman’s egg

Paraxial trajectories and rays


Fine sampling:

True ray traced at an angle of 31° (blue)

Paraxial ray deduced at an angle of 31°

(green) from a ray traced at an angle of

30° (red)

Coarse sampling:

True ray traced at an angle of 35° (blue)

Paraxial ray deduced at an angle of 35°

(green) from a ray traced at an angle of

30° (red)

Two points ray tracing: the paraxial shooting method

Consider Dx the distance between ray point at

the free surface and sensor position

The estimation of the derivative 𝑑𝛿𝑞𝑥

𝑑𝜃 is through the point paraxial computation


Solve iteratively Δx =𝑑𝑞𝑥

𝑑𝜃 Δθ

Δ𝑥

𝜃 + Δ𝜃

𝑑𝑞𝑥𝑑𝜃

=𝜕𝑞𝑥

𝜕𝑝𝑥(0) 𝑑𝑝𝑥𝑑𝜃

(0) +𝜕𝑞𝑥

𝜕𝑝𝑧(0) 𝑑𝑝𝑧𝑑𝜃

(0)


=1

𝑝𝑥(0)2 + 𝑝𝑧(0)

2

𝜕𝑞𝑥𝜕𝑝𝑥(0)

𝑝𝑧 0 −𝜕𝑞𝑥

𝜕𝑝𝑧 0𝑝𝑥 0


=1

𝑝𝑥(0)2 + 𝑝𝑧(0)

2𝛿𝑞𝑥3 𝑝𝑧 0 − 𝛿𝑞𝑥4 𝑝𝑥 0

Paraxial quantities for derivative

Derivative for shooting angle

Point paraxial condition

(orthogonal perturbation to 𝑝 )

trajectory

Efficient 2-pts-ray tracing


An unevenly shooting sampling for reaching receivers at a

depth of 2 km with offsets 2, 4,6,8 km

Thanks to the paraxial information

Optimization approach

Amplitude estimation Consider DL the distance between the exit point of a ray at the

particule time 𝜉 and the paraxial ray running point.

Thanks to the point paraxial

trajectory estimation 𝛿𝑞3 and 𝛿𝑞4,

we may estimate the geometrical

spreading Δ𝐿 Δ𝜃 and, therefore, the

ray amplitude 𝐴 𝜉 ∝ Δ𝐿/Δ𝜃


DL

𝑝

Δ𝐿 Δ𝜃 ∝𝛿𝑞𝑥

𝑎 𝜉 𝑝𝑧 𝜉 − 𝛿𝑞𝑧𝑎 (𝜉)𝑝𝑥(𝜉)

𝑝𝑥(𝜉)2 + 𝑝𝑧(𝜉)

2 𝑝𝑥(0)2 + 𝑝𝑧(0)

2

Δ𝐿/Δ𝜃 ∝ 𝛿𝑞𝑥3 𝜉 𝑝𝑧 0 −𝛿𝑞𝑥4(𝜉)𝑝𝑥(0) 𝑝𝑧 𝜉 − 𝛿𝑞𝑧3 𝜉 𝑝𝑧 0 −𝛿𝑞𝑧4 𝜉 𝑝𝑥 0 𝑝𝑥(𝜉)

𝑝𝑥(𝜉)2+𝑝𝑧(𝜉)

2 𝑝𝑥(0)2+𝑝𝑧(0)

2

From point paraxial ray 𝛿𝑦𝑎

From point paraxial trajectories 𝛿𝑦𝑎3 and 𝛿𝑦𝑎4

Δ𝐿/Δ𝜃 ∝

𝑝𝑥(𝜉) 𝛿𝑞𝑥3(𝜉) 𝛿𝑞𝑧3(𝜉)𝑝𝑧(𝜉) 𝛿𝑞𝑥4(𝜉) 𝛿𝑞𝑧4(𝜉)

0 𝑝𝑥(0) 𝑝𝑧(0)

𝑝𝑥(𝜉)2 + 𝑝𝑧(𝜉)

2 𝑝𝑥(0)2 + 𝑝𝑧(0)

2

Using plane paraxial solutions 𝛿𝑦1 and 𝛿𝑦2, we can construct a beam

as the Gaussian beams

Polarization


Chapman, 2004, p180

𝑔 1

𝑔2

𝑔3

Acoustic case: the unitary vector 𝑔 1 = 𝑐 𝑞 𝑝 supports the P wave vibration

Elastic case: the independent shear

vibration will be along two unitary vectors 𝑔 2 and 𝑔 3 such that

𝑑𝑔 2𝑑𝑡

= 𝑔 2. 𝛻𝑞𝑐 𝑔 1

𝑑𝑔 3𝑑𝑡

= 𝑔 3. 𝛻𝑞𝑐 𝑔 1

Isotropic case

𝑑𝑔 2𝑑𝜉

=𝑔 2. 𝛻𝑞𝑐

𝑝2𝑔 1

𝑑𝑔 3𝑑𝜉𝑡

=𝑔 3. 𝛻𝑞𝑐

𝑝2𝑔 1

Time stepping

Particule stepping

It is enough to follow the

evolution of the projection of

elastic unitary vectors on

one Cartesian coordinate:

𝑒 𝑧. 𝑔 2 and 𝑒 𝑧. 𝑔 2

(Psencik, perso. Comm.)

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 !


(Lambaré et al., 1996)


STEP ONE: ray tracing


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

0 400 800 1200 1600 2000

Z in m


STEP THREE: seismograms

Stop and move to the

numerical exercise

by your own …


Local ODE

Semi-Lagrangian approach

Sampling control


Ray tracing by wavefronts Evolution over time:

folding of the wavefront is allowed

(still a significant curse of complexity!)

Dynamic sampling:

undersampling of ray fans

oversampling of ray fans

Keep an « uniform » sampling of the medium by rays

by tracking the surrounding density of rays

by estimating through paraxial approach the ray density


Lambaré et al (1996)

Vinje et al (1993) widely used in

NORSAR software

How to

compute

multiple

arrivals?

(Ray tracing by rays)

Rays and wavefronts in an homogeneous medium.

(Lambaré et al., 1996)

Ray tracing by wavefronts


An ODE is solved at

each point of the

wavefront while it is

spanned

Keeping the sampling

of the medium more

or less uniform

Ray tracing by wavefronts

Sampling the wavefront is an heavy task in

2D & 3D !

Still better than oversampling through ray

tracing by rays

Example of wavefront

evolution in a smooth version

of the Marmousi model

Smoothness is required !


How to

compute

multiple

arrivals?


(Runbord, 2007)

Geometrical optics resolution


No scale as we are at

infinite frequency !

Is it a fair assumption

while we have finite

frequency wave

content?

We must proceed down

to a given resolution

length under which we

do not want to decipher

the wavefront:

wavefront healing

related to so-called

viscous solution.

Do we need this complexity of tracking

seismic wavefronts!

Few attempts

The medium should be smooth enough

for avoiding the wavefront tracking …

Usually done once in the initial model

when performing imaging procedure …

A better strategy? Yes, for first arrivals;

maybe for multiple arrivals.


PDE

Eulerian approach

Grid control


Level-set functions


𝑝2 =1

𝑐2(𝑥 )

ℋ 𝑥 , 𝑝 =1

2𝑝2 −

1

𝑐2 𝑥 = 0

Eikonal equation

The eikonal equation is a first-order non-linear partial differential equation of static

Hamilton-Jacobi equations: Crandall & Lions (1983, 1984) have shown that

« viscous » solutions of such equation can be obtained.

« Computing first-arrival times is equivalent to tracking an interface advancing at a

local speed normal to itself » from Sethian & Popovici (1999)

- Level-set methods (Osher & Sethian, 1988)

- Fast marching methods (Sethian, 1996a; Sethian and Popovici, 1999) 𝒪(𝑁𝐿𝑜𝑔𝑁) - Fast sweeping methods (Zhao, 2005) 𝒪(𝑁) - Discontinuous Finite-Element methods (Li et al, 2008; Yan & Osher, 2011)

𝑝 =1

𝑐(𝑥 )= 𝛻𝑥 𝑇(𝑥 ) 𝑥 ∈ Ω

T 𝑥 = 𝑇0 𝑥 𝑥 ∈ Γ ⊂ 𝜕Ω

Using an upwind, monotone and

consistent discretization of 𝛻𝑥 𝑇(𝑥 )

Eikonal Solvers Fast marching method (FMM)

(tracking interface/wavefront motion :

curve and surface evolution)

In a 2D layered medium

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

z=cte

z

z + dz

Compute 𝜕𝑇 𝜕𝑥 along z=cte by a finite difference

approximation

Deduce 𝜕𝑇 𝜕𝑧 from eikonal

From T at a depth of z, we have been

able to estimate T at a depth of z+dz

EIKONAL SOLVER

ONLY FIRST-ARRIVAL!


𝜕𝑇

𝜕𝑥

2

+𝜕𝑇

𝜕𝑧

2

=1

𝑐(𝑧)2

𝜕𝑇

𝜕𝑧= ±

1

𝑐(𝑧)2−

𝜕𝑇

𝜕𝑥

2

a) 𝜕𝑇

𝜕𝑥

b) 𝜕𝑇

𝜕𝑧

c) down

Extend T estimation at depth z+dz

Fast marching method (FMM)

Computing T(x,z) is a stationay

boundary problem: discretize it on a

grid and find an efficient numerical

method to solve it.

2D case

The solution is updated by following the

causality in a sequential way: updated

pointwise in the order the solution is

strictly increasing (upwind difference

scheme and a heap-sort algorithm)

Sharp interfaces are difficult to describe

From Sethian & Podovici (1999)


𝒪(𝑁 𝐿𝑜𝑔𝑁) where N is the number of

grid points in a direction

𝜕𝑇

𝜕𝑥

2

+𝜕𝑇

𝜕𝑧

2

=1

𝑐(𝑥, 𝑧)2

𝜕𝑇

𝜕𝑧= ±

1

𝑐(𝑥, 𝑧)2−

𝜕𝑇

𝜕𝑥

2

The ENO or WENO stencil How to estimate the discrete 𝛻𝑇 ?

A first-order Godunov upwind difference scheme

High-order improved ENO or WENO stencils (Liu

et al, 1994; Jiang and Shu, 1996)

See Sethian’s book (1999)


𝑇𝑖,𝑗 − 𝑇𝑖,𝑗𝑥𝑚𝑖𝑛

ℎ

+ 2

+𝑇𝑖,𝑗 − 𝑇𝑖,𝑗

𝑦𝑚𝑖𝑛

ℎ

+ 2

=1

𝑐2(𝑥, 𝑧)

with 𝑇𝑖,𝑗𝑥𝑚𝑖𝑛 = min (𝑇𝑖−1,𝑗 , 𝑇𝑖+1,𝑗) and 𝑇𝑖,𝑗

𝑦𝑚𝑖𝑛 = min (𝑇𝑖,𝑗−1, 𝑇𝑖,𝑗+1)

and with 𝑥 + = 𝑥 𝑖𝑓 𝑥 > 0 or 𝑥 + = 0 𝑖𝑓 𝑥 ≤ 0

Fast sweeping method (FSM)

Gauss-Seidel iterations with alternating direction sweepings are incorporated

into same upwind finite difference stencil: complexity in 𝒪 𝑁 .

2D case


(𝜕𝑇

𝜕𝑥)2+(

𝜕𝑇

𝜕𝑧)2=

1

𝑐2(𝑥, 𝑧)

Velocity Travel time

In 2D at least four sweeps are

needed and six sweeps in 3D

Iterations are independent of the

grid size

Singularities at the boundary may induce

errors, especially for a point source (Luo &

Qian, 2010)

From Zhao (2005)

Any mesh could be used

Computing T(x,z) is a stationay

boundary problem

Fast sweeping method (FSM) 2D case


(𝑝𝑥)2+(𝑝𝑧 )

2=1

𝑐2(𝑥, 𝑧)

Any mesh could be used

𝑇𝐶 = 𝑇𝐴 +𝐴𝐶.𝑝

𝐴𝐶AC 𝑇𝐶 > 𝑇𝐴

𝑇𝐶 = 𝑇𝐵 +𝐵𝐶.𝑝

𝐵𝐶BC 𝑇𝐶 > 𝑇𝐵

Waves travel along AC or BC direction with an

apparent slowness which is the projected

slowness value from the true slowness vector.

𝑥𝐶 − 𝑥𝐴𝐴𝐶

𝑧𝐶 − 𝑧𝐴𝐴𝐶

𝑥𝐶 − 𝑥𝐵𝐵𝐶

𝑧𝐶 − 𝑧𝐵𝐵𝐶

𝑝𝑥𝑝𝑧

=

𝑇𝐶 − 𝑇𝐴𝐴𝐶

𝑇𝐶 − 𝑇𝐵𝐵𝐶

Han et al (2015)

𝑝𝑥𝑝𝑧

=𝑀𝑇𝐶 + 𝑁𝑃𝑇𝐶 + 𝑄

(𝑀𝑇𝐶 + 𝑁)2+(𝑃𝑇𝐶 + 𝑄)2=1

𝑐2(𝑥, 𝑧)

Quadratic equation: real solution 𝑇𝐶 needed!

Causality and viscosity 2D case


𝑇𝐴

𝑇𝐵

𝑇𝐶?

P

(𝑀𝑇𝐶 + 𝑁)2+(𝑃𝑇𝐶 + 𝑄)2=1

𝑐2(𝑥, 𝑧)

𝑀2 + 𝑃2 𝑇𝐶2 + 2 𝑀𝑁 + 𝑃𝑄 𝑇𝐶 +𝑁2 + 𝑄2 −

1

𝑐2 𝑥, 𝑧= 0

IF 𝑇𝐴 unset and IF 𝑇𝐵 unset, do nothing

IF 𝑇𝐴 set and IF 𝑇𝐵 unset, compute 𝑇𝐶 as if wave comes from A

IF 𝑇𝐴 unset and IF 𝑇𝐵 set, compute 𝑇𝐶 as if wave comes from B

IF 𝑇𝐴 set and IF 𝑇𝐵 set, do

compute roots of equation (1)

if real solution

check if the « backward ray » intersects the segment [AB]

if yes, update 𝑇𝐶 by this value if it is smaller

if no real solution, compute the viscous solution

as wave is coming from A or from B: select the smallest value

enddo

equation (1) A

B

C

We provide always

a value if A and/or

B have a value

# ray tracing

Fast sweeping method (FSM) 2D case


Eight points stencil:

From the possible eight values (if one is

set), take the smallest one.

Sweeping technique:

Four sweeping when applying the stencil

Sweep 1: 𝑖𝑖 = 1, 𝑛1, 1; 𝑖2 = 1, 𝑛2, 1

Sweep 2: 𝑖𝑖 = 𝑛1, 1, −1; 𝑖2 = 1, 𝑛2, 1

Sweep 3: 𝑖𝑖 = 𝑛1, 1, −1; 𝑖2 = 𝑛2, 1, −1

Sweep 4: 𝑖𝑖 = 1, 𝑛1, 1; 𝑖2 = 𝑛2, 1, −1

Iteration over sweeps until convergence

(no more updating of 𝑇𝐶)

We need to setup a set of values

which will be fixed:

at the source, 𝑇 = 0, for example

Number of iterations


The number of iterations depend on the

medium structure: one must be aware that

the characteristics of the hyperbolic system

should be sampled at least once by the

sweeping loop.

In seismics or seismology, we have less

dramatic configuration than the one shown

in this figure

Few iterations are necessary to achieve

convergence

2D examples


See Taillandier et al (1999) for an application to first-arrival time tomography

using the adjoint formulation

(Luo and Qian, 2008)

A smooth model constructed from Marmousi Wavefronts as deduced

from travel-time computation

Only first-arrival times

3D example


Zhang, Zhao and Qian (2005)

Drawback: only first-arrival times!


Time over the grid Ray

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 !

Back to inversion through rays

Still very usefuf to back-raytracing once we know times: getting the Jacobian matrix

Could we do better: multi-arrival times and amplitudes?


(Runbord, 2007)

X X

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


Time-dependent Hamilton-Jacobi equation


ℋ 𝑥 , 𝑝 =1

2𝑝2 −

1

𝑐2 𝑥 = ℋ 𝑥 , 𝛻𝑥 𝑇 = 0 for any 𝑥

with 𝑇 𝑥 = 𝑇𝑜𝑏𝑠(𝑥 ) for position 𝑥 on boundary

Two techniques for making a link between static HJ equations and time-dependent

HJ equations

- The level-set idea (Osher, 1993) by adding a dimension to the problem

Introducing the new function 𝜓 such that 𝜓𝑡 + ℋ 𝑥 , 𝛻𝑥 𝜓 = 0, one can find the

solution T(𝑥 ) as the zero-level 𝜓𝑡 𝑥 = 0.

- The so-called paraxial formulation (Leung et al, 2004) similar to what we have

done for the reduced Hamiltonian. One prefered spatial direction is assumed as

the evolution (« time ») direction: .

Often, a semi-lagrangian approach is used with such increase in the space

dimensions for computer efficiency (Fomel & Sethian, 2002).

Another way for

multiple arrivals,

thanks to PDE.

Static H-J equation versus dynamic one


𝛻𝑇(𝑥, 𝑦, 𝑧) =1

𝑐(𝑥,𝑦,𝑧) Eikonal or static Hamilton-Jacobi equation

𝜕𝜙

𝜕𝑡x, y, z, t + c x, y, z 𝛻𝜙 𝑥, 𝑦, 𝑧, 𝑡 = 0 Dynamic H-J equation

The solution should converge to the static one.

𝜙 x, y, z, t = T x, y, z − t

We may use numerical approaches which consider PDE of H-J equations

Time-dependent Hamilton-Jacobi equation (2)


We are back to PDE strategy on fixed grids!

This is not an artificial tool but a very deep insight in physics (Goldstein, 1980, chap10)

- Hamilton (1834) has shown the equivalence between Eikonal equations and H-J

equations.

- De Broglie & Schrödinger (1926) has made the connection with wave equation.

The action 𝒮(𝑥 , 𝑝 , 𝜉) = ℋ(𝑥 , 𝑝 ) − ℰ𝜉 where the evolution variable 𝜉 acts as a time for the

related particle and the energy ℰ is zero for rays. The action 𝒮 is proportional (at the

Planck constant limit) to the phase for a general dispersive wave

𝑢 = 𝐴 𝑒𝑖 𝒮(𝜉,𝑥 ,𝑝 ) and one has the « particle-time » dependent H-J equation,

𝜕𝒮(𝜉, 𝑥 , 𝑝 )

𝜕𝜉+ℋ 𝑥 ,

𝜕𝒮

𝜕𝜉= 0

The paraxial strategy


In a 2D medium, we can consider the eikonal 𝛻𝑥 𝑇(𝑥 ) =1

𝑐(𝑥 ) for sub-vertical

rays and, therefore, we may choose the variable 𝑧 as the evolution. We have

𝜕𝑇 𝑧, 𝑥, 𝑝𝑥

𝜕𝑧+ℋ 𝑧, 𝑥, 𝑝𝑥 = 0

A 2D level-set motion in the space 𝑥, 𝑝𝑥 as we treat the variable 𝑧 as an

artificial time variable and we consider the function 𝜑(𝑧, 𝑥, 𝑝𝑥). We have

𝐷𝜑

𝐷𝑧=𝜕𝜑

𝜕𝑧+𝜕𝜑

𝜕𝑥

𝑑𝑥

𝑑𝑧+𝜕𝜑

𝜕𝑝𝑥

𝑑𝑝𝑥𝑑𝑧

with initial values 𝜑 0, 𝑥, 𝑝𝑥 = 𝑥𝑠 as the position of the source. The zero

level-set intersection will provide 𝜑 𝑧, 𝑥 𝑧 , 𝑝𝑥 𝑧 =0, i.e rays intersections at

constant depth. Travel-times can be deduced by integration as well.

(Leung & Qian & Osher, 2004)

Application: Marmousi case


(Qian & Leung, 2004;

Qian & Leung, 2006)

Medium from 4.8 km to 7.2 km in x and from 0 km to 3 km in x

Grid (x,z) = 384 x 122 with a stepping 25 m x 25 m

The source is at x=6.0 km and z=2.8 km

(100 x 200 x 122)

for (𝑝𝑥, 𝑥, 𝑧)

𝑧 = 0

𝑝𝑥

𝑥

𝑝𝑥

𝑥

Superposition of ODE

& PDE solutions

Contours for various z

PD

E

Finer grid

The true 2D approach


In a 2D medium, we could consider time as the evolution variable and solve

the level-set problem for the function 𝜑 𝑡, 𝑥, 𝑧, 𝑝𝑥, 𝑝𝑧 defined by

𝜕𝜑

𝜕𝑡+𝑑𝑥

𝑑𝑡

𝜕𝜑

𝜕𝑥+𝑑𝑦

𝑑𝑡

𝜕𝜑

𝜕𝑧+𝑑𝑝𝑥𝑑𝑡

𝜕𝜑

𝜕𝑝𝑥+𝑑𝑝𝑧𝑑𝑡

𝜕𝜑

𝜕𝑝𝑧

with two initial conditions 𝜑1 0, 𝑥, 𝑧, 𝑝𝑥, 𝑝𝑧 = 𝑥 and 𝜑2 0, 𝑥, 𝑧, 𝑝𝑥, 𝑝𝑧 = z.

The intersection of the zero level-set values of these two functions gives the

phase space solution 𝑥, 𝑧, 𝑝𝑥, 𝑝𝑧 𝑡 for source (0,0). In fact, we can have all

solutions by selection another level-set values other than zero.

It is an eulerian approach in a 5-D space in 2D medium which could be 7-D

space for 3D medium. The approach is not restricted to paraxial formulation.

Strategies have been defined to reduce the cost: reduced phase space,

semi-lagrangian integration (Fomel & Sethian, 2002; Osher et al, 2002).

Work in progress

not limited to

paraxial approach

Osher’s strategy

Purpose of eulerian approach Very robust technique for multiple travel-

times estimation: grid controls the

complexity (challenging problem)

Estimating amplitudes has also been

setup through eulerian approach

Identification of branches at the grid

spacing accuracy (practical in 2D!)


Strategies to be tuned in the future: interfaces can be

included (Cheng & Shu, 2007) … room for progress

Interfaces in short ! Incompatible with high frequency solution as we request to have a smooth medium

If an interface which have a smooth pattern, reset the problem by computing the asymptotic solution on the interface and restart a new problem (reflection or transmission at your own choice) using the solution as the initial solution: this is defined by the signature of the ray (PPSP ray for example)


If abrupt interface (no first derivative), go to geometrical theory of diffraction (Keller, 1962) or (Klem-Musatov, 1994) … Maybe too complex to manipulate for what we need in seismology and seismics? Towards whisperring galleries (Babic, 1962; Thomson, 1995).

Interfaces in short !


Grab expertise from synthetic image software (Pixar)!

But you need to curve your rays!

Interfaces in short !

Lagrangian approach: resume solutions at the interface and restart the ODE integration (Snell-Descartes law and paraxial Snell Descartes law) (Farra et al, 1989)

Semi-lagrangian approach: procedure back to ODE (Rawlison & Sambdrige, 2003)

PDE: rely on discontinuous finite element methods (Cheng & Shu, 2007) or boundary formulation


Conclusion Rays and Waves Geometrical optics: ODE versus PDE

Choose PDE when possible !

ODE: tracing one (paraxial) ray is fast

Please trace paraxial rays as incremental cost

Keep complexity low (seismic waves are finite frequency waves)

Do not drown yourself into the no-scale optical singularities

Identification of rays is a key problem

PDE seems to solve it explicitly !


Perspectives Rays and Waves

Fast sweeping method O(N) for travel-times and for amplitudes

Amplitude equations have been designed

Finite element methods put into the scene

Stencils are moving to higher orders and h-adaptivity

Discontinuous Galerkin methods

This is the road to take for interface investigation in the frame of PDE.


Stop and move to the

numerical exercise



http://seiscope.oca.eu

Drawback: only first-arrival times!


Performing the forward problem and the adjoint

problem allows one to estimate efficiently the gradient

of the misfit function without rays

Only the gradient and not the Jacobian matrix

A VERY GOOD TOOL

for First Arrival Time Tomography (FATT)

Four

succesiv

e s

weeps

and o

ne ite

ration

(Taillandier et al, 2009)

An alternative to two tomographic techniques

- Inversion through rays

- Inversion through times without ray tracing

using simulated annealing (Asad et al, 1999)

Initial

106

Inverse problem

Jean Virieux

Year 2015-2016

1

Other inverse problems?


From Brossier (2013,2014)

Outline of this partial overview

Motivation and hints

Travel time tomography (ABEL/HWB)

Delayed travel time tomography

Least-squares solution

Quality control

Applications

Conclusion & Perspectives


Few hints


Least-square method

Sum of vertical distances between data points

and expected y values from the unknown line

y=ax+b should be minimum: find a and b?


From Excel

It is an inversion ….

Occam’s razor


Do not use more complicated maths than the data

deserves

Approximate the least constrained quantity

Given: data (observed and modeled)

Assumed: wave propagation

Unknown: Earth structure

Occam’s Razor: parcimonious principle

When you have many explanations for predicting exactly the

same quantities and that there is no way to distinguish them,

select the simplest one… until you end up with a

contradiction.

Ockham (~1295-~1349)

Approximations can’t be avoided …


Incomplete data

Limited understanding

Limited crunching capacity

Limited frequency range: translucid Earth

Limited dataset: direct access impossible (or difficult)

Data windowing (observables): extraction of robust information

Dimensionality (2D vs 3D …)

Structural complexity (multi-scale heterogeneities)

Physics approximation (elastic vs acoustic)

« An approximate solution to

a real problem is better than

an exact solution to an ideal

problem » (Tarantola, 2007)

24/02/2016 HR seismig imaging 112 From Nissan-Meyer (Oxford)

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 tomography

even more difficult is the diffraction tomography related to the

waveform and/or the preserved/true amplitude.


Tomographic approach

Very general problem

medicine; 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

0

22

2

2

0

/

2

)(;

)(

2)( du

pu

dudz

p

pXdz

pzu

ppX

p

u

zp

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

0

2

1vgx

We shall take as the zero value for the

potential energy: this gives us the

following equations and its integration: 2

0

/2 ( ) ; ( )

2 ( )

x

d s d s dg x t x d

d t g x

xx x

x

We may transform it into the so-called Abel integral

where t(x) is known and f(x) is the shape

of the hill to be found: this is an

integrale equation.

xx

xd

x

fxt

x

0

)()(

ds dx

x

y

x

The exact inverse solution

x

x

xxx

xx

xxx

xx

x

0

0

00

00

000

)(1)(

)()(

)()(

)()(

)()(

x

dxxt

d

df

f

x

dxxt

d

d

dfdx

x

xt

xx

dxdfdx

x

xt

d

x

f

x

dxdx

x

xtx

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

x22

sincos x

THE ABEL SOLUTION

a

x

a

x

dxxt

d

df

d

x

fxt

xxx

x

xx

x

)(1)(

)()(

By changing variable x in

a-x and x in a-x, we get the

standard formulae

We must have

t(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

2

0

2

2

0

uX

u

u

u

u

p

u

pv

dXvz

dp

up

pXvz

dp

up

ppXvz

du

pu

dudz

p

pX

In Cartesian frame In spherical frame

D

D

)/(

0)/cosh(

1)

)(ln(

vr

rpv

d

vr

R

From the direct solution, we

can deduce the inverson

solution

After 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


When considering discontinuities, ABEL/HWB method

based on first arrival times has not an unique solution.

More over there will be an ambiguity when the velocity

decreases (can only defined the velocity jump in this zone)

In fact, we avec an infinity of solutions when considering

only direct and refracted waves.

We may find interfaces when considering all waves

(including reflection waves)

24/02/2016 121

Phase signature P : mantle P wave

S : mantle S wave

K : Outer core P wave

I : Inner core P wave

J : Inner core S wave

c : Reflected P wave

(outer core)

i : Reflected P wave

(inner core)

m : number of reflections

Please, describe the wave with signature

PKP, PKKP, SKKKS=S3KS

HR seismig imaging

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


Time tomography

Only for one dimension …

uptonow


What can we do in 2D, 3D and 4D?

A simple case : small perturbation

Initiale structure of

velocity

Search of small variation

of 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

Consider small perturbations 𝛿𝑠(𝑥, 𝑦, 𝑧) of the slowness field 𝑠0(𝑥, 𝑦, 𝑧)

Finding the slowness u(x,y,z) from t(s,r) is a

difficult problem: only techniques for one

variable u(z) (Abel) !

LINEARIZED PROBLEM 𝛿𝑡 𝑑 = 𝐽(𝑑,𝑚) 𝛿𝑠(𝑚) from the model domain to the data domain


« frozen » ray approximation

(ray connecting

source/receiver for the

known slowness 𝑠0)

𝑡 𝑠𝑜𝑢𝑟𝑐𝑒, 𝑟𝑒𝑐𝑒𝑖𝑣𝑒𝑟 = 𝑠 𝑥, 𝑦, 𝑧 𝑑𝑙𝑟

𝑠

𝑡 𝑠, 𝑟 = 𝑠 𝑥, 𝑦, 𝑧 𝑑𝑙 = 𝑠0 𝑥, 𝑦, 𝑧 𝑑𝑙 + 𝛿𝑠 𝑥, 𝑦, 𝑧 𝑑𝑙𝑟

𝑠

𝑟

𝑠

𝑟

𝑠

𝑡 𝑠, 𝑟 = 𝑠0 𝑥, 𝑦, 𝑧 𝑑𝑙𝑟0

𝑠0

+ 𝛿𝑠 𝑥, 𝑦, 𝑧 𝑑𝑙𝑟0

𝑠0

𝑡 𝑠, 𝑟 − 𝑡0 𝑠, 𝑟 = 𝛿𝑠 𝑥, 𝑦, 𝑧 𝑑𝑙𝑟0

𝑠0

𝛿𝑡 𝑠, 𝑟 = 𝛿𝑠 𝑥, 𝑦, 𝑧 𝑑𝑙𝑟0

𝑠0

Discretization of the slowness perturbation

The velocity perturbation field (or the slowness field) 𝛿𝑠(𝑥, 𝑦, 𝑧) can be

described into a meshed cube regularly spaced in the three directions.

For each node, we specify a value 𝛿𝑠𝑖,𝑗,𝑘. The interpolation will be performed

with functions as step functions. For each grid point (i,j,k), shape functions

ℎ𝑖,𝑗,𝑘 = 1 for i,j,k, and zero for other indices.

𝛿𝑠 𝑥, 𝑦, 𝑧 = 𝛿𝑠𝑖,𝑗,𝑘 ℎ𝑖,𝑗,𝑘𝑐𝑢𝑏𝑒

Nodal approach


Other shape functions are possible with two end members (nodal

versus global):

fourier functions (cos,sin), chebychev, spline … and so on

Discrete Model Space

Discretization of the

medium fats the ray


Weighted

ray

segment

𝛿𝑡 s, r = 𝛿𝑠𝑖,𝑗,𝑘 ℎ𝑖,𝑗,𝑘𝑐𝑢𝑏𝑒𝑟𝑎𝑦0

𝑑𝑙 = 𝛿𝑠𝑖,𝑗,𝑘 ℎ𝑖,𝑗,𝑘𝑑𝑙

𝑟𝑎𝑦0𝑐𝑢𝑏𝑒

𝛿𝑡 𝑠, 𝑟 = 𝛿𝑠𝑖,𝑗,𝑘Δ𝑙𝑖,𝑗,𝑘𝑖,𝑗,𝑘

= 𝜕𝑡

𝜕𝑠𝑖,𝑗,𝑘𝛿𝑠𝑖,𝑗,𝑘

𝑖,𝑗,𝑘

𝛿𝑡 𝑠, 𝑟 = 𝐽𝑖,𝑗,𝑘𝑖,𝑗,𝑘

δ𝑠𝑖,𝑗,𝑘

Sensitivity matrix J is a sparse matrix

also named Fréchet derivative or

Jacobian matrix …

𝛿𝑡 𝑑 = 𝐽(𝑑,𝑚) 𝛿𝑠(𝑚) 𝛿𝑡1𝛿𝑡2⋮

𝛿𝑡𝑛−1𝛿𝑡𝑛

=

𝜕𝑡1𝜕𝑠1

⋯𝜕𝑡1𝜕𝑠𝑚

𝜕𝑡2𝜕𝑠1

…𝜕𝑡2𝜕𝑠𝑚

⋮ ⋱ ⋮𝜕𝑡𝑛−1𝜕𝑠1

⋯𝜕𝑡𝑛−1𝜕𝑠𝑚

𝜕𝑡𝑛𝜕𝑠1

⋯𝜕𝑡𝑛𝜕𝑠𝑚

𝛿𝑠1𝛿𝑠2⋮

𝛿𝑠𝑚−1

𝛿𝑠𝑚

To be solved in least squares sense

Linear algebra


𝛿𝑡 = 𝐽𝛿𝑢

𝑑 = 𝐺𝑚

𝑏 = 𝐴𝑥



•The rectangular system can be

recast into a square system

(sometimes called normal

equations).

• Solving this square linear

system gives the so-called least-

squares solution.

•Another interesting solution

The system is both under-determined and over-determined depending on

the considered zone (and the number of rays going through.

𝐽𝑡𝛿𝑡 = 𝐽𝑡𝐽𝛿𝑠 𝛿𝑠 = 𝐽𝑡𝐽−1 𝐽𝑡𝛿𝑡


𝐽𝑡𝛿𝑡 = 𝐽𝑡𝐽𝛿𝑠 𝛿𝑠 = 𝐽𝑡(𝐽𝐽𝑡)−1𝛿𝑡

Least-norm solution

Remark

LEAST SQUARES METHOD

dGGGm

dGmGG

m

mE

mGdmGdmE

tt

est

tt

t

0

1

00

000

00

0)(

)()()(

L2 norm

locates the minimum of E

normal equations

if exists 1

00

GGt

Least-squares estimation

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

the generalized inverse

ttGGG

0

1

00

gG

0

G0 is a N by M matrice

is a M by M matrice 1

00

GGt

Under-determination M > N

Over-determination N > M Mixed-determination – seismic tomography


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 L

0

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

][

][

0

0

UUU

VVV

p

p

t

p

p

pVVUUG

000

00

0

L

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

the solution

t

ppp

t

ppp

UVG

VUG

11

0

0

L

L

Up and Vp have now inverses !


Solution, model & data resolution

RmmVVmVUUVmGGdGmt

p

t

ppp

t

pppestLL

1

0

1

0

1

0)(

The solution is

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

ppVVR

NddUUmGdt

ppestest

0

dUUNt

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

importance matrice

Goodness of resolution

SPREAD(R)=

SPREAD(N)=

Spreading functions

2

2

IN

IR

Good tools for quality estimation


LINEAR INVERSE PROBLEM

1 1

0 0

0 0

u G t m G d

t G u d G m

d d

d d

Updating slowness perturbation values from time residuals

Formally one can write

with the forward problem

Existence, Uniqueness, Stability, Robustness

Discretisation

Identifiability

of the model

Small errors

propagates Outliers effects

NON-UNIQUENESS & NON-STABILITY : ILL-POSED PROBLEM

REGULARISATION : ILL-POSED -> WELL-POSED


Linearized Inverse Problem


No explicit evaluation of the Hessian 𝐽𝑘𝑡𝐽𝑘:

only products « 𝐽 𝛿𝑠 » and « 𝐽𝑡𝛿𝑡 » are required in LSQR algorithm

Paige & Sanders (1982)

Misfit function

ℂ 𝑠 =1

2𝛿𝑡𝑡𝛿𝑡

First loop over models : for current model 𝑠𝑘 (iteration k)

Forward problem

A. Solve eikonal equation

B. Compute rays and synthetic travel-times

Build Fréchet derivative matrix 𝐽𝑘and delayed times 𝛿𝑡𝑘

Second loop over linear system: iteratively solve 𝐽𝑘𝛿𝑠𝑘 = 𝛿𝑡𝑘 , ie 𝐽𝑘𝑡𝐽𝑘𝛿𝑠𝑘 = 𝛿𝑡𝑘

using conjugate gradient (LSQR, for example)

Update the model 𝑠𝑘+1 = 𝑠𝑘 + 𝛿𝑠𝑘

Two-loops procedure



Misfit function

ℂ 𝑠 =1



Forward problem


B. Compute rays and synthetic travel-times

Build Fréchet derivative matrix 𝐽𝑘and delayed times 𝛿𝑡𝑘

Second loop over linear system: iteratively solve 𝐽𝑘𝛿𝑠𝑘 = 𝛿𝑡𝑘 , ie 𝐽𝑘𝑡𝐽𝑘𝛿𝑠𝑘 = 𝛿𝑡𝑘

using conjugate gradient (LSQR, for example)

Update the model 𝑠𝑘+1 = 𝑠𝑘 + 𝛿𝑠𝑘

Two-loops procedure

Complexity: ℴ 𝑁𝑠𝑟𝑐 ∗ 𝑁𝑟𝑒𝑐 for forward/adjoint formulation and

ℴ 𝑁𝑚 ∗ 𝑁𝑠𝑟𝑐 ∗ 𝑁𝑟𝑒𝑐 ∗ 𝑘 for gradient storage (𝑘: sparsity )



Misfit function

ℂ 𝑠 =1



Forward problem


B. Compute synthetic travel-times and delayed times 𝛿𝑡𝑘

Adjoint problem

A. Solve adjoint equation

B. Build gradient 𝛾𝑘

Descent method (one-step gradient or conjugate gradient or l-BFGS) 𝛿𝑠𝑘

Update the model 𝑠𝑘+1 = 𝑠𝑘 + 𝛼𝑘𝛿𝑠𝑘

(Sei & Symes, 1994; Leung & Qian, 2006; Taillandier et al, 2009)

One-loop procedure

No explicit evaluation of the Hessian

l-BFGS provides an evaluation of the H-1 from previous stored gradients



Misfit function

ℂ 𝑠 =1



Forward problem



Adjoint problem



Descent method (one-step gradient or conjugate gradient or l-BFGS) 𝛿𝑠𝑘

Update the model 𝑠𝑘+1 = 𝑠𝑘 + 𝛼𝑘𝛿𝑠𝑘

(Sei & Symes, 1994; Leung & Qian, 2006; Taillandier et al, 2009)

One-loop procedure

Complexity: ℴ 𝑁𝑠𝑟𝑐 for forward/adjoint formulation and ℴ 𝑁𝑚 for gradient



Misfit function

ℂ 𝑠 =1



Forward problem



Adjoint problem



Second loop over linear system : 𝐻𝑘 𝛿𝑠𝑘 = −𝛾𝑘

Truncated Newton method based on iterative conjugate gradient

(matrix free approach for product 𝐻𝑘 𝑣, thanks to second-order adjoint

formulation)

Update the model 𝑠𝑘+1 = 𝑠𝑘 + 𝛼𝑘𝛿𝑠𝑘 (Métivier et al, 2013)

Two-loop procedure

Full Hessian impact 𝐻 = 𝐽𝑡𝐽 +𝜕𝐽

𝜕𝑠𝛿𝑡 is evaluated

Maximum Likelihood method One assume a gaussian distribution of data

Data distribution could be written

where 𝐺0𝑚𝑒𝑠𝑡 is the data mean and Cd is the data covariance matrice.

This method is very similar to the least squares method where we define the

following misfit function 𝐸1.

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

through the misfit function


𝐸2 𝑚 = (𝑑 − 𝐺0𝑚)𝑡𝑊𝑑 𝑑 − 𝐺0𝑚

𝐸 𝑚 = 𝑑 −𝐺0 𝑚𝑡(𝑑 − 𝐺0𝑚) → 𝐸1 𝑚 = 𝑑 −𝐺0 𝑚

𝑡𝐶𝑑−1(𝑑 − 𝐺0𝑚)

𝑝(𝑑) ∝ 𝑒𝑥𝑝 −1

2𝑑 −𝐺0 𝑚

𝑡𝐶𝑑−1(𝑑 − 𝐺0𝑚)

The inferred model distribution will be gaussian with the main difficulty of

estimating the posterior model covariance 𝐶𝑚 connected with the curvature of

the misfit function 𝐸2.

PRIOR INFORMATION Hard bounds

Prior model

The parameter 휀 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 pm

t

pd

tmmWmmmGdWmGdmE

with Wd data weighting and Wm model weighting

t

md

tg

pm

t

pd

t

GCGCGG

mmCmmmGdCmGdmE

0

11

0

1

00

1

0

1

04)()()()()(

)()()()()(003 p

t

p

tmmmmmGdmGdmE

BmAi

Seismic velocity should be positive


L curve


Estimating the parameter 휀 is

always difficult

We can test different values

which provide a L curve

If a knee appears, this could

be an adequat value …

although this knee is often

difficult to identify in seismic

tomography.

Other techniques are available as LASSO approach but

they are intensive and sensitive to noise.

UNCERTAINTY ESTIMATION

Least squares solution

Model covariance : uncertainty in the data

curvature of the error function

Sampling the misfit function around the estimated model:

often this has to be done numerically

dGdGGGmgtt

est 00

1

00

1

2

2

2

1

00

2

2

0000

2

1cov

cov

covcov

estmm

dest

t

dest

dd

gt

d

ggtg

est

m

Em

GGm

IC

GCGGdGm

Uncorrelated data


A posteriori model covariance matrice True a posteriori distribution

Tangent gaussian distribution

S diagonal matrice eigenvalues

U orthogonal matrice eigenvectors

Error ellipsoidal could be estimated

WARNING : formal estimation related

to the gaussian distribution hypothesis

t

md

t

USUCGCG 1

0

1

0

If one can decompose this matrice


A priori & A posteriori information

What is the meaning of the « final » model we provide ?

acceptable 24/02/2016 HR seismig imaging 148

Steepest descent methods )()(

1 kkmEmE

kk

kk

kkk

k

k

kkk

DmE

mEmEd

dE

Ed

mEd

mEmEtmE

)(

)()(

)(

)())((

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

)(

2

1

mC

mgC

mC

dC

m

d

pm

dMinimisation of this vector

2/1

2/1

m

kd

k

C

GCAIf one computes

then

)(

))((

0

2/1

2/1

km

kdt

kk

t

k

mmC

dmgCAmAA d

Gaussian error distribution of data and of a posteriori model

Easy 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)


Flow chart

true ray tracing

data residual

sensitivity matrice

model update

new model

mmm

dGm

mgG

ddd

mgd

m

d

synobs

syn

obs

D

DD

D

1

0

0

)(

collected data

starting model loop

Calculate for formal uncertainty estimation

small model variation or small errors exit

2

2

m

E


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


Ray approximation


𝛿T s, r ≈ 𝑢 𝑥(𝑙) 𝑑𝑙 ≈ 𝑢 𝑥 𝛿 𝑥 − 𝑥 𝑙 𝐾(𝑠, 𝑟, 𝑥)𝑑𝑣

Line

Volume

𝐾(𝑥, 𝑦, 𝑧) is the sensitivity kernel

Ray approximation reasonnable approximation


Same gradient

once estimated

on the model

discretization grid

Gradient

𝐽𝑡𝛿𝑡

Delayed Travel-time Tomography


Still DTT provides

impressive images while we

do believe that DFT would

provide better images in the

future, thanks to finer

discretization coming with

the densification of the

available data.

(Li & van der Hilst, 2010)

𝛿T s, r = 𝑢 𝑥(𝑙) 𝑑𝑙 = 𝑢 𝑥 𝛿 𝑥 − 𝑥 𝑙 𝐾(𝑠, 𝑟, 𝑥)𝑑𝑣

Delayed Fresnel Tomography

𝛿T 𝑠, 𝑟 = 𝑢 𝑥 𝐾(𝑠, 𝑟, 𝑥)𝑑𝑣



𝛿T 𝑠, 𝑟 = 𝑢 𝑥 𝐾(𝑠, 𝑟, 𝑥)𝑑𝑣

No amplitude measurements

Only phase information

Valid also for dispersive waves

Improved resolution?

Windows definition


(LTA/STA;

Maggi et al, 2009)

(Wavelet Freq/time;

Lee and Chen, 2013)

Phases measurements


(Zaroli et al, 2010)

Workflow for DFT

Taking seismograms

Windowing phases

Picking phases (cross-correlation)

Definition of a misfit function (L2)


Model perturbation (velocity)


Phase differences


(Fichtner et al, 2009)

We do not pick phases as for DFT:

Full Phases Inversion


(Fichtner et al, 2009)

Do we improve the resolution compared to DFT?

If equivalent, very expensive; if not, smooth transition to FWI

Workflow for FPI

Taking seismograms

Windowing phases

Definition of a misfit function (time differences)

Full phases inversion

Model perturbation (velocity)


Applications


Corinth Gulf

An 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 24/02/2016 HR seismig imaging 167

Seismic experiment 1991 (and one in 2001)


MEDIUM 1D : HWB AND RANDOM SELECTION


Velocity structure image Horizontal sections


Basin structure

Fast variation

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)


Dramatic increase of

acquisition density both

in the academic world

(increasing density and

in the industrial world

(continuous recording)

General trend

for

seismic tomography


IRIS DATA CENTER



http://seiscope.oca.eu

THANK YOU !

Many figures have come from people I have worked with:

many thanks to them !


END HERE


THE Cm-1/2 MATRICE

)exp(2

ji

ij

xx

c

Shape independent of

Values depend on

SATURATION

The matrice Cm has a band diagonal shape

- is the standard error (same for all nodes)

is the correlation length

n=nx.ny.nz=104 Cm=USUt

(Lanzos decomposition)

t

mUUSC

2/12/1


Analysis of coefficients Values independent of n (n>5000)

Values are only related to and

2/12/1 ~0

m

n

m

nCC

Typical sizes 200x200x50

deduced from 20x20x5 (few minutes)

Strategy of libraries of Cm-1/2 for

various and =1

Other coefficients could be deduced

R: Cm-1/2 sparse matrice


An example

=0.8

v=100 km/s

x=100 km

t=100 s

=0.1

Ray imprints

Same numerical grid for all

simulations (either 100x100 or

400x400)

Same results at the limit of

numerical precision related to

the estimation of the sensitivity

matrice


Illustration of selection {,v}

= 5 km and v= 3 km/s

Error function analysis will give us optimal

values of a priori standard error and

correlation length (2D analysis)

v influence

influence


A posteriori information What is the meaning of the « final » model we provide ?

acceptable


Probability density

)()(),( mdmdMD

The probability density in the model space is related to the

Data space


Different informations


Perspectives

A priori model covariance : location dependent

wavelet decomposition allows local

analysis

(work of Matthieu Delost-Geoazur)

Fresnel tomography : introduction of the first

Fresnel zone influence in the forward problem

(work of Stéphanie Gautier-Montpellier)


Kirchhoff approximation

1) Representation theorem

2) Kirchhoff summation

3) Reciprocity

Born approximation

1) Single scattering approximation

2) Surface approximation

(Forgues et al., 1996) 24/02/2016 HR seismig imaging 195

(Forgues et al., 1996) 24/02/2016 HR seismig imaging 196


http://seiscope2.osug.fr

high resolution seismic imaging: asymptotic approachinduces a dramatic increasing complexity in ray...

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