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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

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Structure of the Presentation

./ Problem Formulation

./ Non-Linear Inversion Framework ./ Gradient Based Optimization

./ 1D Numerical Examples ./ Future Plans

./ Conclusions

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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

(4)

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 ⇐

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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

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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)

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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

(8)

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

(9)

Direct Field Equation

Computed reflectivity

0 200 400 600 800

0 0.05 0.1 0.15 0.2 0.25

depth [m]

reflectivity

(10)

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

(11)

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

(12)

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

(13)

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

(14)

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

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Adjoint Field Equation

./ L(v) =k S − S(v) k + P

nn, (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)

(16)

Gradient

./ ∂L∂v = PN −1

n=1

λn, ∂G∂v (vn+1) Pn+1

+ PN

n=1

n, ∂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

(17)

Structure of the Presentation

./ Problem Formulation

./ Non-Linear Inversion Framework ./ Gradient Based Optimization

./ 1D Numerical Examples ./ Future Plans

./ Conclusions

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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

(19)

Structure of the Presentation

./ Problem Formulation

./ Non-Linear Inversion Framework ./ Gradient Based Optimization

./ 1D Numerical Examples ./ Future Plans

./ Conclusions

(20)

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

(21)

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]

(22)

First Numerical example

Convergence history

0 50 100 150 200

10−6 10−4 10−2 100

number of iterations

norm cost function

(23)

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]

(24)

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

(25)

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

∗ ...

(26)

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.

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