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Velocity Reconstruction from 3D Post-Stack Data
in Frequency Domain
Enrico Pieroni, Domenico Lahaye, Ernesto Bonomi
Imaging & Numerical Geophysics areaCRS4, Italy
Zero-Offset InversionAustin, March 2003 2
Non-linear inversion of post stack data for velocity analysis
Subsoil imaging by inverting Zero-Offset seismic data in space-frequency domain Optimal control of the error norm between real and simulated data
– direct problem: modeling by demigration – adjoint problem: error residual migration– minimization: line search along the error gradient in the velocity space
A sequence of nested non-linear inversions, from the lowest to the highest frequency Algorithm embarassingly parallel in frequencies
Zero-Offset InversionAustin, March 2003 3
What are post-stack data?
Offset acquisition:Hundreds of shots & Thousands of receivers
Stacking = compression of data to virtually zero-offset traces (S=G).Model: exploding reflectors with halved velocities
S G1 G2
2 way travel path vith vel. v
1 way, v/2
Zero-Offset InversionAustin, March 2003 4
P(n) = acoustic wave field at depth z(n) = (n-1)z D(n) = upward propagator v(n)(x) = v(x, y, z(n)) = velocity field q(n) = normal-incidence reflectivity
Direct Problem in the Domain
,, ,, NNfor nnnnn
NN
qPvDP
qP
1 221 )()( )())(,(),(
)()(),()()1()1()(
)()(
xxxx
xx
Demigration mapping: q(n) P(0) final value problem in which the zero offset data are modeled from medium reflectivity
Eqns decoupled in frequencies: embarassing data parallelism
Zero-Offset InversionAustin, March 2003 5
Upward propagator
exp , :PS 22
)(*)( FFTkFFT
nn
vzjvD
& FFT = Fourier matrix (x,y) -> (kx,ky)
V(n) = medium velocity at depth n
Scalar wave equation UPWARD + DOWNWARD separationUpward propagate data from reflectors to surface with halved velocity: one way wave equationLaterally invariant velocities:
& for laterally variable velocities:
* 1D D
)( exp exp )(, :PSPC
)()(*)(
xFFTEXPFFTx
nnn
vzj
VzjvD
)()( )()()( xx nnn cVv
EXP
PS exact vel. normal prop.correction
Zero-Offset InversionAustin, March 2003 6
Build reflectivity from v
nz
vN ˆsgnˆ
vz
v
vvn
vn
xv
xvvv
vvxq
nn
nn
sgn2
1ˆ
2
1)(
)(2
1)(
,1,2
,1,2
N̂vn ˆ
m/s1000
m/s2000
m/s3000
x
z
Due to orthogonal incidence:velocity isosurfaces reflectors
gradient filtering
Edge detection for IP
Zero-Offset InversionAustin, March 2003 7
S = Z.O. “known” data P(0) = Z.O. simulated data ( = field P @ surface)
Minimization Problem: Optimal Control
Approach
.. ,, )(*)( )()1()1()(*)(
1:1 ,
2 ccqqPDPdPjvPL NNnnnnn
Nn
x
2)0(2 ),(),( xxx SPdPj
Constrained minimization: Lagrange multipliers method
Find velocity v minimizing the misfit function j
Zero-Offset InversionAustin, March 2003 8
(n) = adjoint wave field at depth z(n) = (n-1)z D = adjoint operator, ~ downward propagator D*
Adjoint Problem in the Domain
, N,nnnn
n
vD
PS
2for ),())(,(),(
),(),(),()()1()1(
)()1(
xxx
xxx
LP(n) = 0 Migration mapping: initial value problem (if misfit = 0 then =0)
*DD
Zero-Offset InversionAustin, March 2003 9
Building the gradient
Constraining (n) and P(n) to satisfy the direct and the adjoint equation:
& from the first variation of the Lagrange function:
..)()(
)( ,
)()1(
)(
)1(*)(2
)(cc
v
qP
v
Dd
v
j
ni
nn
i
nn
i
ξξx
ξ
vPLPj ,,
= adjoint downward propagated Diff[D] direct upward prop. field
= 0 if
Zero-Offset InversionAustin, March 2003 10
Optimization strategy
Number of parameters p = NxNyNz ~ 108
huge search space: no Hessian or Montecarlo work lot of local minima Hessian evaluation requires running p direct problems
Conjugate gradient to reduce computation, storage and search time
- Gradient evaluated by automatic differentiation - CG + orthogonal projection Vmin .LE. v(x,y,z) .LE. Vmax - conjugate directions build with Fletcher-Reeves updating
Line search by Golden bracket + Polynomial - search interval bounded when the at least 1 velocity component reach the bound
Inversion adaptive in time-frequency to stabilize solution
Zero-Offset InversionAustin, March 2003 11
“scissors” ambiguity: v(z) v’(z) = v(z/) Good 1st guess + adaptive in freq.
1D Test cases
1D is fully analytical both in the discrete and in the continuum
z
v
0
)()0( 0)(
)(
1
2
1]),0[),(,(
H
v
di
dzedz
zdv
zvHzvP
z
]),0[),/(,(]),0[),(,( )0()0( HzvPHzvP
Zero-Offset InversionAustin, March 2003 12
Test 1: discontinuous velocitynf=100 fmx=50 Hz
[0,25 Hz]200 itns
+ 4 ord. mag.[0,12.5 Hz]200 itns
5 ord. mag.
constant initial guess
Step exact vel
Zero-Offset InversionAustin, March 2003 13
Test 1
[0,37.5 Hz]
+ 5 ord. mag.
[0,50 Hz]
+ 3 ord. mag.
Zero-Offset InversionAustin, March 2003 14
Test 2: continuous velocity
nf=100 fmx=50 Hz
[0,25 Hz]
50 itns showed every 10
+ 5 orders of magnitude
final
[0,12.5 Hz]
3 orders of magnitude
final
linear 1st guess
Parabolic exact
Zero-Offset InversionAustin, March 2003 15
Test 3: disc. velocity + inversion
nf=100 fmx=50 Hz
exact
[0,12.5 Hz]100 itns
Linear 1st guess
Handmade 1st guess model, based on previous& 50 itns with all assessing first 10 layers
[0,25 Hz]100 itns
Zero-Offset InversionAustin, March 2003 16
Test 3
[0,37.5 Hz]
100 itns
[0,50 Hz]
200 itns
Zero-Offset InversionAustin, March 2003 17
Control of the error to decide how to proceed, to be done automatically Velocity estimate from the lowest to the highest frequency (other: sliding windows, back & forth, …) , to be done automatically Dev: 3D feasible thanks to parallelization in frequencies (3D should also remove a lot of ambiguities) Perspective: integrate with a multi-scale spatial approach from the lowest to the highest depth Dev: correct ZO for geometrical spreading & amplitude Open problem: optimal tuning of the reflectivity for real data
Conclusion & Further Developments
Zero-Offset InversionAustin, March 2003 18
THE END
Zero-Offset InversionAustin, March 2003 19
Pragmatic inversion: Migration
exp , :PS 222
)(*)( FFTFFT
yxn
n kkv
zjvM
& FFT = Fourier matrix (x,y) -> (kx,ky)
j
nj
nj
n vMvM ),()())(,( :PSPI )()( xx
otherwise 0 ; )(
)()(
if 1)( njv
nv
nj xx
vj(n) = reference velocities
= shape functions
),(),(
),())(,(),()1(
)()1()1( 2for
xx
xxx
SP
PvMP , N,nnnn
& Downward propagatedata at surface withhalved velocity;
& One way wave equation: for laterally invariant velocities
& for laterally variable velocities
* 1M M
Zero-Offset InversionAustin, March 2003 20
)( ))(,( 2121
)0( tjtj eqqezvP
2
122
1
11
23
232
12
121
v
zzt
v
zt
vv
vvq
vv
vvq
Zero-Offset InversionAustin, March 2003 21
Goal: subsoil imaging from post-stack data in frequency domaindeterming the velocity model of the subsoil in such a waythat the simulated (modeled) and measured (given) pressure field at the surface (stacked sections) agreesimulation code:- in frequency domain:* data compression and hence reduced computational cost* typical problem dimension: 500 MB - 1 GB* direct/adjoint propagation of data by phase shifting- in 3D- highly innovative approach- weak points:* reflectivity (isosurfaces of velocity discontinuities)* amplitude mathematical model: Lagrangian formulation- cost function: difference between simulated and measured stacks- constraints: one-way wave equation (in frequency domain)- direct field: demigration (upward) from reflectivity to recorded data- adjoint field: migration (downward) driven by source term (residual error) from surface data to adjoint field- migration operator and derivative- gradient:Integral of (direct field * OP * adjoint field) dx dy dz dw- weak points: computation of OP and gradientalgorithm: projected CG (PCG) optimization for the velocity model updatingimplementation: Fortran90 + MPI