Velocity Reconstruction from 3D Post-Stack Data in Frequency Domain Enrico Pieroni, Domenico Lahaye, Ernesto Bonomi Imaging & Numerical Geophysics area CRS4, Italy

Austin, March Zero-Offset Inversion 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

Austin, March Zero-Offset Inversion 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 SG1G1 G2G2 2 way travel path vith vel. v 1 way, v/2

Austin, March Zero-Offset Inversion  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 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

Austin, March Zero-Offset Inversion Upward propagator & FFT = Fourier matrix (x,y) -> (k x,k y ) V (n) = medium velocity at depth n Scalar wave equation  UPWARD + DOWNWARD separation Upward propagate data from reflectors to surface with halved velocity: one way wave equation Laterally invariant velocities: & for laterally variable velocities: PS exact vel. normal prop.correction

Austin, March Zero-Offset Inversion Build reflectivity from v Due to orthogonal incidence: velocity isosurfaces  reflectors  gradient filtering Edge detection for IP

Austin, March Zero-Offset Inversion  S = Z.O. “known” data  P (0) = Z.O. simulated data ( = field surface) Minimization Problem: Optimal Control Approach  Constrained minimization: Lagrange multipliers method  Find velocity v minimizing the misfit function j

Austin, March Zero-Offset Inversion  (n) = adjoint wave field at depth z (n) = (n-1)  z  D = adjoint operator, ~ downward propagator D* Adjoint Problem in the  Domain  L  P (n) = 0 Migration mapping: initial value problem (if misfit = 0 then =0)

Austin, March Zero-Offset Inversion Building the gradient Constraining (n) and P (n) to satisfy the direct and the adjoint equation: & from the first variation of the Lagrange function: = adjoint downward propagated Diff[D] direct upward prop. field = 0 if 

Austin, March Zero-Offset Inversion Optimization strategy  Number of parameters p = N x N y N z ~ 10 8  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

Austin, March Zero-Offset Inversion “scissors” ambiguity: v(z)  v’(z) =  v(z/  )  Good 1 st guess + adaptive in freq. 1D Test cases 1D is fully analytical both in the discrete and in the continuum z v

Austin, March Zero-Offset Inversion Test 1: discontinuous velocity nf=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

Austin, March Zero-Offset Inversion Test 1 [0,37.5 Hz] + 5 ord. mag. [0,50 Hz] + 3 ord. mag.

Austin, March Zero-Offset Inversion Test 2: continuous velocity nf=100 fmx=50 Hz [0,25 Hz] 50 itns showed every orders of magnitude final [0,12.5 Hz] 3 orders of magnitude final linear 1 st guess Parabolic exact

Austin, March Zero-Offset Inversion Test 3: disc. velocity + inversion nf=100 fmx=50 Hz exact [0,12.5 Hz] 100 itns Linear 1 st guess Handmade 1 st guess model, based on previous & 50 itns with all  assessing first 10 layers [0,25 Hz] 100 itns

Austin, March Zero-Offset Inversion Test 3 [0,37.5 Hz] 100 itns [0,50 Hz] 200 itns

Austin, March Zero-Offset Inversion  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

Austin, March Zero-Offset Inversion THE END

Austin, March Zero-Offset Inversion Pragmatic inversion: Migration & FFT = Fourier matrix (x,y) -> (k x,k y ) v j (n) = reference velocities  = shape functions & Downward propagate data at surface with halved velocity; & One way wave equation: for laterally invariant velocities & for laterally variable velocities

Austin, March Zero-Offset Inversion

Austin, March Zero-Offset Inversion Goal: subsoil imaging from post-stack data in frequency domain determing the velocity model of the subsoil in such a way that the simulated (modeled) and measured (given) pressure field at the surface (stacked sections) agree simulation 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 gradient algorithm: projected CG (PCG) optimization for the velocity model updating implementation: Fortran90 + MPI