Velocity Reconstruction from 3D Post-Stack Data in Frequency Domain Enrico Pieroni, Domenico Lahaye, Ernesto Bonomi Imaging & Numerical Geophysics area.

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