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Seismic numerical forward modeling: Rays as ODEs
By Jean Virieux
Professeur
UJF-Grenoble I
28/09/2012Numerical methods : ODE integration
1
American Petroleum Institute, 1986
Salt DomeFault
Unconformity
Pinchout
Anticline
1
Time scales
28/09/2012 Numerical methods : ODE integration 2
Source timefrom 0.1 sec to 100 sec (rupture vel.)
Wave timefrom secondes to hours
Window timefrom few secondes to days
Length scales
28/09/2012 Numerical methods : ODE integration 3
Fault length200 km for vr=2 km/s
Discontinuity distancefrom few meters to few 100 kms
Volume samplingfrom few kms to few 1000 kms
Examples
28/09/2012 Numerical methods : ODE integration 4
Record of a far earthquake (Müller and Kind, 1976)
Traces from a oil reservoir (Thierry, 1997)
Records on the Moon (meteoritic impact) (Latham et al., 1971)
Seismic imaging is a tough problem on the Moon !
References Červený, V., Seismic ray theory, Cambridge University Press, 2001 Chapman, C., Fundamentals of seismic wave propagation, Cambridge University
Press, 2004 Goldstein, H, Classical mechanics, Addison-Wesley Publishing company, second
edition,1980 Glassner, A.S. (Editor), An introduction to ray tracing, Academic Press, second
edition, 1991 Sethian, J.A., Level Set Methods and Fast Marching Methods: Evolving Interfaces
in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science, Cambridge University Press, 1999.
Virieux, J. and Lambaré, G., Theory and Observations-Body waves: Ray methodsand finite frequency effecs in Treatrise on Geophysics, Tome I by A. Diewonskiand B. Romanowicz, Elsevier, 2010
28/09/2012 Numerical methods : ODE integration 5
Translucid Earth
Diffracting medium:
wavefront coherence lost !
Wavefront preserved
Wavefront : T(x)=T0
Travel-time T(x) and Amplitude A(x)
Source
Receiver
Same shape !
T(x)
S(t)
28/09/2012 Numerical methods : ODE integration 6
is sometimes called the phase
Eikonal equationTwo simple interpretations of wavefront evolution
Orthogonal trajectories are rays in an isotropic medium
Grad(T)= orthogonal to wavefront
Direction ? : abs or squareThe orientation of the wavefront could not be guessed from the local information on a specific wavefront
28/09/2012 Numerical methods : ODE integration
T+T
T=cte
Velocity c(x)
L RayΔΔ →
ΔΔ
1→
1
7
Transport Equation Tracing neighboring rays defines a ray tube : variation of amplitude depends on energy flux conservation throughsections.
28/09/2012 Numerical methods : ODE integration 8
2 . 0
∆ ∆ ∆ ∆
→ . .
0 . .
0 . ′ . .
0 ⟹ 0
2 . 0
Energy flux same at section one and at section two
net contribution=zero
Methods of characteristics
Differential geometry (Courant & Hilbert, 1966)
Non-linear ordinary differential equations Lagrangian formulation as we integrate
along rays
In opposition to Eulerian formulation wherewe compute (ray) quantities at fixed positions
28/09/2012 Numerical methods : ODE integration 9
28/09/2012 Numerical methods : ODE integration 10
Ray equation
Ray
T=cte
Evolution of is given by
but the operator . .
and, therefore, .
leading to .
Evolution of is given by
→ 1Wavefront
∥ →
1
Ray equations
1 1
We define the slownss vector and the position along the ray
Curvature equation
Various non-linear ray equations
28/09/2012 Numerical methods : ODE integration 11
Which ODE to select for numerical solving ? Either t or sampling.
Many analytical solutions (gradient of velocity; gradient of slowness square)
Curvilinearstepping
1
1
Particulestepping
1 1
1
Timestepping
1
1 1
undertheconditionoftheeikonal1/
The
sim
ples
tset
Properties of these ODEs Intrinsic solutions independent of the
coordinate sytem used to solve it If dummy variable for velocity, use itas the variable stepping (often x coordinate)
(1)
Eikonal equation: a good proxy for testingthe accuracy of the ray tracing (not enoughused)
28/09/2012 Numerical methods : ODE integration 12
In 3D: six or seven equationsIn 2D: four or five equations
(1): rectilinear motion of a particle along this axis in mechanics
Particulestepping
1 1
Velocity variation v(z)
dzzduzu
ddp
ddp
ddp
pddqp
ddq
pddq
zyx
zz
yy
xx
)()(;0;0
;;
Ray equations are
The horizontal component of the slowness vector is constant: the trajectory is inside a plan which is called the plan of propagation. We may define the frame (xoz) as this plane.
dzzduzu
ddp
ddp
pddqp
ddq
zx
zz
xx
)()(;0
;
22 )( x
x
z
x
z
x
pzu
ppp
dqdq
Where px is a constante
1
0
220011)(
),(),(z
z x
xxxxx dz
pzu
ppzqpzq
For a ray towards the depth
Velocity variation v(z)
)(222px zupp
pp
pp
z
z x
z
z x
x
z
z x
xz
z x
xxxx
dzpzu
zudzpzu
zuTpzT
dzpzu
pdzpzu
pqpzq
10
10
22
2
22
2
011
2222011
)(
)(
)(
)(),(
)()(),(
p
p
z
z
dzpzu
zupT
dzpzu
ppX
022
2
022
)()(2)(
)(2)(
At a given maximum depth zp, the slowness vector is horizontal following the equation
zp
If we consider a source at the free surface as well as the receiver, we get
2 2 2
2 2
2 2 2
2( )
( )2( )sin
p
p
a
r
a
r
p drrr u r p
r u r drTrr u r p
with p ru i
In Cartesian frame In Spherical framewith p = usini
Velocity structure in the Earth
Radial Structure
Integration of ray equations
28/09/2012 Numerical methods : ODE integration 16
Initial conditions EASY
1D sampling of 2D/3D medium : FAST
source
receiver
Runge-Kutta second-order integration
Predictor-Corrector integration
source
receiver Boundary conditions VERY DIFFICULT
?
?
Shooting p ?
Bending x ?
Continuing c ?
AND FROM TIME TO TIME IT FAILS ! (inherent to geometrical optics)
Save slownessconditions if possible !
But we need 2-points ray tracing because we have a source and a receiver to connect ! We even need more: branch identification (triplication for example)
A very good QC: the eikonal must be equal to zero !
Initial ray tracingRay tracing by rays
Two-point ray tracing
Ray tracing by rays
Hamilton’s ray equations
28/09/2012 Numerical methods : ODE integration 19
Information around the rayRay
Mechanics : ray tracing as a particular balistic problemsympletic structure (FUN!)
Meaning of the neighborhood zoneFresnel zone if finite frequencyAny zone depending on your problem GBS
1 1.,
12
1
Hamilton approach suitable for perturbation(Henri Poincaré en 1907 « Mécanique céleste »,Richard Feymann Prix Nobel 1965)
" "
Paraxial ray theorysimilar to
Gauss optics
28/09/2012 Numerical methods : ODE integration 20
Hamilton’s ray equations
28/09/2012 Numerical methods : ODE integration 21
Information around the ray
Ray
1 1.
,12
1
" "
Paraxial Ray theory
Estimation of ray tube: KMAHindex tracking and amplitudeevaluation
Estimation of taking-off angles: shooting strategy
The matrix does not depend on quantities frombut only on quantities from : LINEAR PROBLEM
(SIMPLE) !
28/09/2012 Numerical methods : ODE integration 22
,
, ,
seismogram computation
Paraxial Ray theory
28/09/2012 Numerical methods : ODE integration 23
Solutions are coordinate dependent (differentialcomputation)
Not restricted to the so-called ray-centeredcoordinate system (Cerveny, 2001)
Cartesian formulation is much simpler to handle(Virieux & Farra, 1991)
2D simple linear system
Four elementary paraxial trajectories
y1t(0)=(1,0,0,0)
y2t(0)=(0,1,0,0)
y3t(0)=(0,0,1,0)
y4t(0)=(0,0,0,1) NOT A paraxial RAY !
28/09/2012 Numerical methods : ODE integration 24
0 0 1 00 0 0 1
0.51⁄
0.51⁄
0 0
0.51⁄
0.51⁄
0 0
More complex anisotropic structure but still straightforward
Linearsystem
2D paraxial conditions
Paraxial rays require other conservative quantities : the perturbation of the Hamiltonian should be zero (or, in other words, the eikonal perturbation is zero)
If working with the reduced hamiltonian, this is implicitly set!
28/09/2012 Numerical methods : ODE integration 25
⁄ , ,⁄ 0
0
Or in the isotropic case
Similar conditions in 3DReadily deduced for anisotropyTwo independent solutions
Point source conditions
From paraxial trajectories, one can combine them for paraxial rays as long as the perturbation of the Hamiltonian is zero.For a point source, the parameter could be set to an arbitrary small value: this isa derivative or plan tangent computation (Gauss optics)
This is enough to verify this condition initially
Paraxial solution 0 3 0 4
28/09/2012 Numerical methods : ODE integration 26
0 ⇒ 0 0 0 0 =0
0 0
0 0
Plane source conditions
We combine the first two paraxial ray trajectories.
This is enough to verify this condition initially but gradient of velocity at the source could be quite arbitrariry
Cerveny’s condition
Chapman’s condition (only z variation)
Paraxial solution ′ ,⁄0 1 ′ ,⁄
0 2
28/09/2012 Numerical methods : ODE integration 27
0⇒
12
1 ,⁄0 0
12
1 ,⁄0 0 0
012
1 ,⁄0
012
1 ,⁄0
Two independent paraxial rays in 2D ( ): point (seismograms) and plane (beams) paraxial rays
KMAH index tracking
28/09/2012 Numerical methods : ODE integration 28
In 2D, the determinant 0 0 0
may change sign:
Increment by one the KMAH index as we have crossed a caustic
In 3D, the determinant
0 0 0 0
may change sign.
If minor determinants do not change sign, this is a plane caustic (add 1 to KMAH). If they change sign as well, this is a point caustic (add 2 to KMAH).
KMAH index key element for seismograms
ODE resolution Runge-Kutta of second order Write a computer program for an
analytical law for the velocity: take a gradient with a component along x and a component along z
Home work : redo the same thing with a Runge-Kutta of fourth order (look after its definition)
Consider a gradient of the square of slowness
28/09/2012 Numerical methods : ODE integration 29
Runge-Kutta integrationSecond-order RK integration
Non-linear ray tracing
Second-order euler integration for paraxial ray tracing is enough!
28/09/2012 Numerical methods : ODE integration 30
Linear paraxial ray tracing
Propagator technique
Optical Lens technique
Two points ray tracing: the paraxial shooting method
Consider x the distance between ray point atthe free surface and sensor position
The estimation of the derivative is through the point paraxial computation
28/09/2012 Numerical methods : ODE integration 31
Solve iteratively Δx Δθ
Δ
Δ
0 0 0 0
0 0 0 0
3 0 4 0Paraxial quantities for derivative
Derivative for shooting angle
Point paraxial condition
Amplitude estimationConsider L the distance between the exit point of a ray at the particule time and the paraxial ray running point.
Thanks to the point paraxial ray estimation dq3 and dq4, we mayestimate the geometrical spreadingΔ Δ⁄ and, therefore, the ray amplitude ∝ Δ /Δ
28/09/2012 Numerical methods : ODE integration 32
LΔ
From point paraxial ray
From point paraxial trajectories 3 and 4
ΔΔ
3 34 4
0 0 0
Using plane paraxial solutions 1 and 2, we can construct any beams as the Gaussian beams
Rays and wavefronts in an homogeneous medium. (Lambaré et al., 1996)
Ray tracing by wavefronts
Slow down the ray tracingefficiency as we samplethe entiremedium
(Lambaré et al., 1996)
28/09/2012 Numerical methods : ODE integration 34
STEP ONE: ray tracing
28/09/2012 Numerical methods : ODE integration 35
STEP TWO: times and amplitudes(Lambaré et al., 1996)
0 400 800 1200 1600 2000Z in m
28/09/2012 Numerical methods : ODE integration 36
STEP THREE: seismograms
GLOBAL Tomography Velocity variation at a
depth of 200 km : good correlation with superficial structures.
Velocity variations at a depth of 1325 km : good correlation with the Geoid.
Courtesy of W. Spakman
Delayed Travel-time tomography
0( , ) ( , , ) ( , , ) ( , , )receiver receiver receiver
source source source
t s r u x y z dl u x y z dl u x y z dl
0 0
0 0
0
0
0
0
0
0
( , ) ( , , ) ( , , )
( , ) ( , ) ( , , )
( , ) ( , , )
receiver receiver
source source
receiver
source
receiver
source
t s r u x y z dl u x y z dl
t s r t s r u x y z dl
t s r u x y z dl
Consider small perturbations u(x) of the slowness field u(x)
station
source
dlzyxustationsourcet ),,(),( Finding the slowness u(x) from t(s,r) is a difficult problem: only techniques for one variable !
This a LINEAR PROBLEM t(s,r)=G(u)
DESCRIPTION OF THE VELOCITY PERTURBATION
The velocity perturbation field (or the slowness field) u(x,y,z) can be described into a meshed cube regularly spaced in the three directions.For each node, we specify a value ui,j,k. The interpolation will be performed with functions as step funcitons. We have found shape functions h,i,j,k=1 pour i,j,k, and zero for other indices.
cube
kjikji huzyxu ,,,,),,(
Discrete Model Spacecube
kjikji huzyxu ,,,,),,(
m
m
m
nn
m
n
n
cubekji
cube rayonkjikji
rayon cubekjikji
uu
uu
ut
ut
ut
ut
tt
tt
uutrst
dlhudlhurst
1
2
1
1
1
1
1
1
2
1
,,
,,,,,,,,
...
),(
),(00
Slowness perturbation description
0t G u
Matrice of sensitivity or of partial derivatives
Discretisation of the medium fats the ray
Sensitivity matrice is a sparse matrice
Corinth GulfAn extension zone where there is a deep drilling project.
How this rift is opening?
What are the physical mechanisms of extension (fractures, fluides, isostatic equilibrium)
Work of Diana Latorre and of Vadim Monteiller
Seismic experiment 1991 (and one in 2001)
MEDIUM 1D : HWB ANDRANDOM SELECTION
Velocity structure imageHorizontal sections
Velocity structure image Vertical sections
P S
Vp/Vs ratio:fluid existence ?
Recovered parameters might have diferent interpretation and the ratio Vp/Vs has a strong relation with the presence of fluids or the relation Vp*Vs may be related to porosity
Conclusion FATT
Selection of an enough fine grid Selection of the a priori model information Selection of an initial model 2PT-RT Time and derivatives estimation LSQR inversion Update the model Uncertainty analysis (Lanzos or numerical)
THANK YOU !
Many figures have come from people I have worked with: many thanks to them !
28/09/2012 Numerical methods : ODE integration 49
http://seiscope.oca.eu
Seismic attributes
Travel time evolution with the grid step : blue for FMM and black for recomputed time
One ray Log scale in time
Grid step
S
R
A ray
2 PT ray tracing non-linear problem solved, any attribute could be computed along this line :
-Time (for tomography)
-Amplitude (through paraxial ODE integration fast)
-Polarisation, anisotropy and so on
Moreover, we may bend the ray for a more accurate ray tracing less dependent of the grid step (FMM)
Keep values of p at source and receiver !
28/09/2012 Numerical methods : ODE integration 50
Polarization
28/09/2012 Numerical methods : ODE integration 51
Chapman, 2004, p180
Acoustic case: the unitary vector supports the P wave vibration
Elastic case: the independent shearvibration will be along two unitary vectors
and such that
.
.
Isotropic case
.
.
Time stepping
Particule stepping
It is enough to follow the evolution of the projection of elastic unitary vectors on one Cartesian coordinate: . and . (Psencik,
perso. Comm.)
Semi-lagrandian approach as we track the evolution of the wavefront and its complex folding (Lambaré et al, 1996)
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