high resolution seismic imaging: asymptotic approachinduces a dramatic increasing complexity in ray...
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High resolution seismic imaging:
Asymptotic approach By Jean Virieux
Professeur
UGA
American Petroleum Institute, 1986
Salt Dome Fault
Unconformity
Pinchout
Anticline
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Time scales
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Source time
from 0.1 sec to 100 sec (rupture vel.)
Wave time
from secondes to hours
Window time
from few secondes to days
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Length scales
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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
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Examples
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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 !
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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)
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Rays Geometrical Optics Infinite Frequency Singularities topology
A link with the Catastrophe Theory (René Thom)
Do we need this complexity?
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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
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In fluid mechanism:
physics below a given small scale is parametrized (upscaled)
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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
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References (selection)
Červený, V., Seismic ray theory, Cambridge University Press, 2001
Chapman, C., Fundamentals of seismic wave propagation, Cambridge University
Press, 2004
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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
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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)
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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
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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
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(𝛻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
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Eikonal equation for anisotropic media
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(𝛻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
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Transport Equation Tracing neighboring rays defines a ray tube : variation of
amplitude depends on energy flux conservation through
sections.
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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
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Transport equation for anisotropic media
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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
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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
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Various non-linear ray equations
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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
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Polarization
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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
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How to solve these equations
ODE versus PDE !
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ODE
Lagrangian formulation
Wavefront complexity
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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
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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)
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In 3D: six or seven equations
In 2D: four or five equations
(1): rectilinear motion of a particle along this axis in mechanics
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Hamilton’s ray equations
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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"
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Ray equations for anisotropic media
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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
𝑐3(𝑞𝑖 , 𝑝𝑖)
𝜕𝑐(𝑞𝑖 , 𝑝𝑖)
𝜕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
𝑐3(𝑞𝑖 , 𝑝𝑖)
𝜕𝑐(𝑞𝑖 , 𝑝𝑖)
𝜕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
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Reduced Hamilton formulation
24/02/2016 HR seismig imaging 24
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)
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Reduced Hamilton formulation (2)
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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 …)
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Mechanical point of view
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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 !
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Mechanical point of view (2)
24/02/2016 HR seismig imaging 27
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.
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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
24/02/2016 HR seismig imaging 28
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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
24/02/2016 HR seismig imaging 29
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Velocity structure in the Earth
Radial Structure
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System of ray tracing equations
24/02/2016 HR seismig imaging 31
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
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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 !
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Ray tracing Constant angle step
Ray tracing by rays
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Two-point ray tracing
Ray tracing by rays
24/02/2016 HR seismig imaging 34
Source (0,0)
Receivers (2,-2),
(4,-2),(6,-2)
(depth is considered as
negative)
Please note the irregular
angle stepping.
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Hamilton’s ray equations
24/02/2016 HR seismig imaging 35
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"
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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
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Runge-Kutt a integration
Second-order RK integration
Non-linear ray tracing
Second-order euler integration for paraxial ray tracing is enough!
24/02/2016 HR seismig imaging 37
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
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Analytical solutions
24/02/2016 HR seismig imaging 38
Vertical gradient of
the velocity
Gradient of the
slowness square
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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
24/02/2016 HR seismig imaging 39
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Keeping complexity low
Solutions: moving from ODEs to PDEs
(Osher et al, 2002) for adequate spatial sampling of
the wavefront. Grids control the complexity!
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3D French’s model
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Interpolation challenge
24/02/2016 HR seismig imaging 41
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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 …
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Paraxial ray theory
similar to
Gauss optics
24/02/2016 HR seismig imaging 43
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Hamilton’s ray equations
24/02/2016 HR seismig imaging 44
Information around the ray 𝑦0
Ray
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 ))
𝑞 0 + 𝛿𝑞 𝑝 0 + 𝛿𝑝 𝛿𝑞 𝑎𝑛𝑑 𝛿𝑝 "𝑠𝑚𝑎𝑙𝑙"
𝑦 =𝑞
𝑝
𝛿𝑦 =𝛿𝑞
𝛿𝑝
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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) !
24/02/2016 HR seismig imaging 45
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
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Paraxial Ray theory
24/02/2016 HR seismig imaging 46
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)
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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 !
24/02/2016 HR seismig imaging 47
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
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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!
24/02/2016 HR seismig imaging 48
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
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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)
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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)
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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)
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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
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KMAH index tracking
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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
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Euler integration for paraxial ray tracing
First Euler integration for paraxial ray tracing is enough!
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𝑑𝛿𝑓
𝑑𝜉= 𝐴 𝑓0 𝛿𝑓
𝛿𝑓1 = 𝛿𝑓0 + Δ𝜉 𝐴 𝑓0 𝛿𝑓0
Linear paraxial ray tracing
Propagator technique
Optical Lens technique
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Paraxial trajectories and rays
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Paraxial rays have to be
understood as a kind of derivative
Paraxial solutions are coordinate-
dependent …
A differential form on
the Chapman’s egg
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Paraxial trajectories and rays
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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)
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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
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Solve iteratively Δx =𝑑𝑞𝑥
𝑑𝜃 Δθ
Δ𝑥
𝜃 + Δ𝜃
𝑑𝑞𝑥𝑑𝜃
=𝜕𝑞𝑥
𝜕𝑝𝑥(0) 𝑑𝑝𝑥𝑑𝜃
(0) +𝜕𝑞𝑥
𝜕𝑝𝑧(0) 𝑑𝑝𝑧𝑑𝜃
(0)
𝑑𝑞𝑥𝑑𝜃
=1
𝑝𝑥(0)2 + 𝑝𝑧(0)
2
𝜕𝑞𝑥𝜕𝑝𝑥(0)
𝑝𝑧 0 −𝜕𝑞𝑥
𝜕𝑝𝑧 0𝑝𝑥 0
𝑑𝑞𝑥𝑑𝜃
=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
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Efficient 2-pts-ray tracing
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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
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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
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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
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Polarization
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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.)
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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 !
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(Lambaré et al., 1996)
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STEP ONE: ray tracing
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STEP TWO: times and amplitudes (Lambaré et al., 1996)
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0 400 800 1200 1600 2000
Z in m
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STEP THREE: seismograms
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Stop and move to the
numerical exercise
by your own …
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Local ODE
Semi-Lagrangian approach
Sampling control
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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
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Lambaré et al (1996)
Vinje et al (1993) widely used in
NORSAR software
How to
compute
multiple
arrivals?
(Ray tracing by rays)
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Rays and wavefronts in an homogeneous medium.
(Lambaré et al., 1996)
Ray tracing by wavefronts
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An ODE is solved at
each point of the
wavefront while it is
spanned
Keeping the sampling
of the medium more
or less uniform
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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 !
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How to
compute
multiple
arrivals?
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(Runbord, 2007)
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Geometrical optics resolution
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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!
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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.
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PDE
Eulerian approach
Grid control
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Level-set functions
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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 )
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24/02/2016 HR seismig imaging 73
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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!
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𝜕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
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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)
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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
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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)
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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
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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
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(𝜕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
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Fast sweeping method (FSM) 2D case
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(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!
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Causality and viscosity 2D case
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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
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Fast sweeping method (FSM) 2D case
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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
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Number of iterations
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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
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2D examples
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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
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3D example
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Zhang, Zhao and Qian (2005)
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Drawback: only first-arrival times!
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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?
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(Runbord, 2007)
X X
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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
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Time-dependent Hamilton-Jacobi equation
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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.
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Static H-J equation versus dynamic one
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𝛻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
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Time-dependent Hamilton-Jacobi equation (2)
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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
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The paraxial strategy
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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)
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Application: Marmousi case
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(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
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The true 2D approach
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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
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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!)
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Strategies to be tuned in the future: interfaces can be
included (Cheng & Shu, 2007) … room for progress
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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)
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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).
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Interfaces in short !
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Grab expertise from synthetic image software (Pixar)!
But you need to curve your rays!
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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
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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 !
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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.
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Stop and move to the
numerical exercise
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http://seiscope.oca.eu
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Drawback: only first-arrival times!
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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
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106
Inverse problem
Jean Virieux
Year 2015-2016
1
Other inverse problems?
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From Brossier (2013,2014)
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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
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Few hints
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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?
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From Excel
It is an inversion ….
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Occam’s razor
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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)
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Approximations can’t be avoided …
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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)
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24/02/2016 HR seismig imaging 112 From Nissan-Meyer (Oxford)
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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.
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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)
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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.
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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
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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
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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).
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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
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Stratified medium
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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)
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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
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Velocity structure with depth
Velocity profile built without any a priori
An difficulty arises when
the velocity decreases.
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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 …
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The symmetrical radial EARTH
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Time tomography
Only for one dimension …
uptonow
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What can we do in 2D, 3D and 4D?
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A simple case : small perturbation
Initiale structure of
velocity
Search of small variation
of velocity or slowness
Linear approach
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Example: Massif Central
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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
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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
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« 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
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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
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Other shape functions are possible with two end members (nodal
versus global):
fourier functions (cos,sin), chebychev, spline … and so on
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Discrete Model Space
Discretization of the
medium fats the ray
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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
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Linear algebra
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𝛿𝑡 = 𝐽𝛿𝑢
𝑑 = 𝐺𝑚
𝑏 = 𝐴𝑥
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Least-squares solution
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•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
Least-squares solution
𝐽𝑡𝛿𝑡 = 𝐽𝑡𝐽𝛿𝑠 𝛿𝑠 = 𝐽𝑡(𝐽𝐽𝑡)−1𝛿𝑡
Least-norm solution
Remark
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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
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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 !
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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
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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
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Linearized Inverse Problem
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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
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Linearized Inverse Problem
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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
Complexity: ℴ 𝑁𝑠𝑟𝑐 ∗ 𝑁𝑟𝑒𝑐 for forward/adjoint formulation and
ℴ 𝑁𝑚 ∗ 𝑁𝑠𝑟𝑐 ∗ 𝑁𝑟𝑒𝑐 ∗ 𝑘 for gradient storage (𝑘: sparsity )
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Linearized Inverse Problem
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Misfit function
ℂ 𝑠 =1
2𝛿𝑡𝑡𝛿𝑡
First loop over models : for current model 𝑠𝑘 (iteration k)
Forward problem
A. Solve eikonal equation
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
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Linearized Inverse Problem
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Misfit function
ℂ 𝑠 =1
2𝛿𝑡𝑡𝛿𝑡
First loop over models : for current model 𝑠𝑘 (iteration k)
Forward problem
A. Solve eikonal equation
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
Complexity: ℴ 𝑁𝑠𝑟𝑐 for forward/adjoint formulation and ℴ 𝑁𝑚 for gradient
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Linearized Inverse Problem
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Misfit function
ℂ 𝑠 =1
2𝛿𝑡𝑡𝛿𝑡
First loop over models : for current model 𝑠𝑘 (iteration k)
Forward problem
A. Solve eikonal equation
B. Compute synthetic travel-times and delayed times 𝛿𝑡𝑘
Adjoint problem
A. Solve adjoint equation
B. Build gradient 𝛾𝑘
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
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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
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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.
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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
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L curve
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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.
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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
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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
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A priori & A posteriori information
What is the meaning of the « final » model we provide ?
acceptable 24/02/2016 HR seismig imaging 148
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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
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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)
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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)
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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
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Sampling a posteriori distribution
Resolution estimation : spike test
Boot-Strapping
Jack-knifing
Natural Neighboring
Monte-Carlo
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Sampling a posteriori distribution
Uncertainty estimation for P and S velocities using boot-strapping techniques
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Ray approximation
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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
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Ray approximation reasonnable approximation
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Same gradient
once estimated
on the model
discretization grid
Gradient
𝐽𝑡𝛿𝑡
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Delayed Travel-time Tomography
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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
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Delayed Fresnel Tomography
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𝛿T 𝑠, 𝑟 = 𝑢 𝑥 𝐾(𝑠, 𝑟, 𝑥)𝑑𝑣
No amplitude measurements
Only phase information
Valid also for dispersive waves
Improved resolution?
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Windows definition
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(LTA/STA;
Maggi et al, 2009)
(Wavelet Freq/time;
Lee and Chen, 2013)
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Phases measurements
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(Zaroli et al, 2010)
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Workflow for DFT
Taking seismograms
Windowing phases
Picking phases (cross-correlation)
Definition of a misfit function (L2)
Delayed Fresnel Tomography
Model perturbation (velocity)
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Phase differences
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(Fichtner et al, 2009)
We do not pick phases as for DFT:
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Full Phases Inversion
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(Fichtner et al, 2009)
Do we improve the resolution compared to DFT?
If equivalent, very expensive; if not, smooth transition to FWI
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Workflow for FPI
Taking seismograms
Windowing phases
Definition of a misfit function (time differences)
Full phases inversion
Model perturbation (velocity)
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Applications
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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
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Seismic experiment 1991 (and one in 2001)
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MEDIUM 1D : HWB AND RANDOM SELECTION
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Velocity structure image Horizontal sections
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Basin structure
Fast variation
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Velocity structure image Vertical sections
P S
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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
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Other methods of exploration
Grid search
Monte-Carlo (ponctual or continuous)
Genetic algorithm
Simulated annealing and co
Tabou method
Natural Neighboring method
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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)
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Dramatic increase of
acquisition density both
in the academic world
(increasing density and
in the industrial world
(continuous recording)
General trend
for
seismic tomography
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IRIS DATA CENTER
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http://seiscope.oca.eu
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THANK YOU !
Many figures have come from people I have worked with:
many thanks to them !
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END HERE
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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
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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
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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
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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
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A posteriori information What is the meaning of the « final » model we provide ?
acceptable
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Probability density
)()(),( mdmdMD
The probability density in the model space is related to the
Data space
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Different informations
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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)
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Kirchhoff approximation
1) Representation theorem
2) Kirchhoff summation
3) Reciprocity
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Born approximation
1) Single scattering approximation
2) Surface approximation
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http://seiscope2.osug.fr