A Velocity Model Inference from the Inversion of 3D Zero-Offset Seismic
Traces in the Space-Frequency Domain
Domenico Lahaye
Enrico Pieroni - Ernesto Bonomi
Numerical Geophysics and Imaging Group - CRS4 Italy
Structure of the Presentation
./ Problem Formulation
./ Non-Linear Inversion Framework ./ Gradient Based Optimization
./ 1D Numerical Examples ./ Future Plans
./ Conclusions
Problem formulation
./ problem context
- non-invasive subsoil imaging for hydrocarbon prospecting
./ problem statement
- reconstruction of scalar velocity field (v) of the subsoil - given: . field measurements of acoustic traces, and
. a wave progation code
- find v such that k S∗ − S(v) k is minimized
simulated data measured data
Problem formulation
./ data
industrial context: inversion of large data sets (500 MB - 1GB) data compression . coincident source-receivers
(post-stack data)
. temporal frequency domain:
F F T {S(t), t → ω} = bS(ω)
⇒ inversion of 3D data sets feasable ./ wave propagation
one-way wave equation in frequency domain solved by phase shifting
⇒ highly innovative approach ⇐
Structure of the Presentation
./ Problem Formulation
./ Non-Linear Inversion Framework
∗ Wave Extrapolation
∗ Direct Field Equation
∗ Adjoint Field Equation
∗ Gradient
./ Gradient Based Optimization ./ 1D Numerical Examples
./ Future Plans ./ Conclusions
Wave Extrapolation
./ p acoustic wave field: 4(x,z) p = v12 ptt
./ bp(kx, z, ω) = F F T [p(x, z, t), x → kx, t → ω]
./ no lateral velocity variations: v = v(z) b
p(z± 4 z) = exp(±j kz 4 z) bp(z) kz = [(ω/v)2) − kx2]1/2 (+)downward/(-) upward propagation
⇒ phase-shift (PS) algorithm (Gazdag ’78)
./ with lateral velocity variations: v = v(x, z)
intepolation procedure on v = v(x, z) at each depth
⇒ phase-shift plus interpolation (PSPI) algorithm (Gadzag-Sguazzero ’83)
Direct Field Equation
./ Upward wavefield extrapolation - Demigration
PN = qN N # layers in depth
Pn = G(vn+1) Pn+1 + qn n = N − 1, . . . , 1 with G(vn+1) the propagation filter
./ P1 simulated surface data ( S(v) = P1 )
./ reflectivity qn ∗ isosurfaces of discontinuity of the velocity
∗ computed by edge-detection filter ./ no lateral velocity variations
G(vn+1) = exp[−j kz(vn+1) 4 z] and qn = vn+1 − vn vn+1 + vn
Direct Field Equation
Example in 1D
Given velocity as of function of depth
0 100 200 300 400 500 600
1000 1500 2000 2500 3000 3500
depth [m]
velocity [m/s]
N (#layers) = 64 dz = 10m
zmax = 640m
Direct Field Equation
Computed reflectivity
0 200 400 600 800
0 0.05 0.1 0.15 0.2 0.25
depth [m]
reflectivity
Direct Field Equation
Acoustic field PN at z = 640m Initial condition (time domain)
0 0.5 1 1.5 2
−1
−0.5 0 0.5 1
n = 64
Direct Field Equation
Acoustic field Pn at z = 350m Hit first velocity discontinuity
0 0.5 1 1.5 2
−0.03
−0.02
−0.01 0 0.01 0.02 0.03
n = 35
Direct Field Equation
Acoustic field Pn at z = 250m
First discontinuity further propagated
0 0.5 1 1.5 2
−0.03
−0.02
−0.01 0 0.01 0.02 0.03
n = 25
Direct Field Equation
Acoustic field Pn at z = 150m Hit second velocity discontinuity
0 0.5 1 1.5 2
−0.03
−0.02
−0.01 0 0.01 0.02 0.03
n = 15
Direct Field Equation
Acoustic field at the surface (z = 0) Simulated data S(v)
0 0.5 1 1.5 2
−0.03
−0.02
−0.01 0 0.01 0.02 0.03
n = 1
Adjoint Field Equation
./ L(v) =k S∗ − S(v) k + P
n hλn, (Pn+1 − G(vn+1)Pn − qn)i Lagrange multiplier
./ Downward wave extrapolation
Migration for λn λ1 = S∗ − S(v) λn+1 = G+(vn+1) λn ./ no lateral velocity variations
G+(vn+1) = G∗(vn+1) = exp[j kz(vn+1) 4 z]
./ with lateral velocity variations
using interpolation (cfr. demigration)
Gradient
./ ∂L∂v = PN −1
n=1
λn, ∂G∂v (vn+1) Pn+1
+ PN
n=1
Dλn, ∂q∂vn E
./ direct field Pn : z → z + 4z adjoint field λn : z → z − 4z
⇒ synchronization required ./ no lateral velocity variations
∂G
∂v (vn+1) and ∂q∂vn can be computed analytically ./ with lateral velocity variations
∂G∂v (vn+1) requires derivative of the PSPI interpolation operator
Structure of the Presentation
./ Problem Formulation
./ Non-Linear Inversion Framework ./ Gradient Based Optimization
./ 1D Numerical Examples ./ Future Plans
./ Conclusions
Gradient based optimization
./ conjugate gradient (CG) method for optimization 1. determine search direction
2. perform line-search along established direction ./ projected conjugate gradient (PCG) method
vmin ≤ v ≤ vmax
project iterate onto set of feasable solutions ./ local optimization method
susceptible to local minina
enhancements of PCG required
Structure of the Presentation
./ Problem Formulation
./ Non-Linear Inversion Framework ./ Gradient Based Optimization
./ 1D Numerical Examples ./ Future Plans
./ Conclusions
1D Numerical examples
Test case scenario
1. Generate measured stack (surface data)
exact velocity + computed reflectivity DEMIG
⇒ measured surface stack : S(x, y, ω)
2. Generate simulated stack perturb velocity
initial guess vel. + comp. reflec. DEMIG
⇒ simulated surface stack : P (x, y, ω)
3. Reconstruct velocity
iterative updating of the velocity ( v ) by PCG algorithm search v such that k S − P (v) k is minimized
First Numerical example
Approximate and exact solution
start solution
0 100 200 300 400 500 600
1500 2000 2500 3000
depth [m]
velocity [m/s]
final solution (200 PCG it.)
0 100 200 300 400 500 600
1500 2000 2500 3000
number of iterations
velocity [m/s]
First Numerical example
Convergence history
0 50 100 150 200
10−6 10−4 10−2 100
number of iterations
norm cost function
Second Numerical example
Approximate and exact solution
start solution
0 100 200 300 400 500 600
1500 2000 2500 3000
depth [m]
velocity [m/s]
final solution (400 PCG it.)
0 100 200 300 400 500 600
1500 2000 2500 3000
depth [m]
velocity [m/s]
Second Numerical example
Convergence history
one-level approach - four-level approach
0 100 200 300 400
10−6 10−4 10−2 100 102
number of iterations
norm of cost function
Future Work
./ Further algorithmic developments in 1D
∗ Accelerate speed of convergence
∗ Avoid local minina
∗ Automate multilevel approaches
./ Extension to 2D and 3D
∗ Computation of ∂G
∂v in case of PSPI
∗ Computation of reflectivity
∗ Alignment of direct and adjoint fields in gradient computation
∗ ...
Conclusions
./ We proposed a new algorithm for the reconstruction of the scalar velocity field of a subsoil from post-stack
seismic data.
./ The algorithm is based on a non-linear Lagrangian framework in which direct and adjoint equations are solved by demigrating and migrating in the frequency domain respectively.
./ We presented encouraging preliminary results for 1D synthetic test cases.
./ Future work is required to extend this work in 2D and 3D.