Fighting ambiguity of inverse problems in seismic imaging Jrg - - PowerPoint PPT Presentation

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions

Fighting ambiguity of inverse problems in seismic imaging

Jörg Schleicher

University of Campinas & INCT-GP

Colóquio Brasileiro de Matemática Rio de Janeiro, 30/07/2013

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions

Contents

1

Introduction

2

Reflector-oriented regularization in slope tomography

3

Decomposition of sensitivity kernels in full-waveform inversion

4

Conclusions

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions Motivation Objective Inversion

Contents

1

Introduction

2

Reflector-oriented regularization in slope tomography

3

Decomposition of sensitivity kernels in full-waveform inversion

4

Conclusions

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions Motivation Objective Inversion

Motivation

km/s 1.5 2 2.5 3 3.5 4 4.5 Distance (km) Depth (km) Velocity Model 3 4 5 6 7 8 9 0.5 1 1.5 2 2.5

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions Motivation Objective Inversion

Motivation

km/s 1.5 2 2.5 3 3.5 4 4.5 Distance (km) Depth (km) Velocity Model 3 4 5 6 7 8 9 0.5 1 1.5 2 2.5

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions Motivation Objective Inversion

Motivation

km/s 1.5 2 2.5 3 3.5 4 4.5 Distance (km) Depth (km) Velocity Model 3 4 5 6 7 8 9 0.5 1 1.5 2 2.5

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions Motivation Objective Inversion

Objectives

1

Slope tomography:

Introduce geologically meaningful constraints Improve velocity model building for depth migration Better recover large scale structural features Improve convergence of layer and grid-based tomography

2

Full-waveform inversion:

Decompose sensitivity kernels Understand contributions Invert only important ones

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions Motivation Objective Inversion

Objectives

1

Slope tomography:

Introduce geologically meaningful constraints Improve velocity model building for depth migration Better recover large scale structural features Improve convergence of layer and grid-based tomography

2

Full-waveform inversion:

Decompose sensitivity kernels Understand contributions Invert only important ones

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions Motivation Objective Inversion

Inversion

Problem: invert Nonlinear relationship between data and parameters d = F(m) m ≡ model parameters d ≡ data parameters F ≡ nonlinear functional (wave propagation)

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions Motivation Objective Inversion

Frechèt derivatives

Solution: Linear iterations: the Frechét derivative δd = DF(m0)δm m0 ≡ reference model parameters δd ≡ data perturbation δm ≡ model parameters perturbations around m0 DF ≡ Frechét derivative of F

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is slope tomography? Smoothness constraints Numerical Experiments Discussion

Contents

1

Introduction

2

Reflector-oriented regularization in slope tomography

3

Decomposition of sensitivity kernels in full-waveform inversion

4

Conclusions

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is slope tomography? Smoothness constraints Numerical Experiments Discussion

What is slope tomography?

Data space: d = {(xs, xr, T sr, ss, sr)n} for n = 1, . . . , N

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is slope tomography? Smoothness constraints Numerical Experiments Discussion

What is slope tomography?

Model space: m = {p, (X, τ s, τ r, θs, θr)n} for n = 1, . . . , N

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is slope tomography? Smoothness constraints Numerical Experiments Discussion

Frechét derivatives computation: dynamic ray tracing

The Hamiltonian (eikonal equation): H(x, s) = 1 2 (p(x)s · s − 1) = 0 x - position along the ray s - slowness vector along the ray p(x) - velocity square field

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is slope tomography? Smoothness constraints Numerical Experiments Discussion

Frechét derivatives computation: dynamic ray tracing

d dτ   x s   =   ∇xH −∇sH   d dτ   δx δs   =   ∇s∇T

x H

∇s∇T

s H

−∇x∇T

x H

−∇x∇T

s H

    δx δs   +   ∇s∇T

pHδp

−∇x(∇T

pHδp)

  .

Reference ray

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is slope tomography? Smoothness constraints Numerical Experiments Discussion

Frechét derivatives computation: dynamic ray tracing

d dτ   x s   =   ∇xH −∇sH   d dτ   δx δs   =   ∇s∇T

x H

∇s∇T

s H

−∇x∇T

x H

−∇x∇T

s H

    δx δs   +   ∇s∇T

pHδp

−∇x(∇T

pHδp)

  .

Paraxial rays

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is slope tomography? Smoothness constraints Numerical Experiments Discussion

Initial Conditions

Slowness Direction: δx = 0 and δs = s

• I − n ∇T

s H

∇T

s H n

dn dθ δθ

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is slope tomography? Smoothness constraints Numerical Experiments Discussion

Initial Conditions

Scattering point position: δx = δX and δs = − ∇sH ∇sH ∇T

x HδX

∇sH

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is slope tomography? Smoothness constraints Numerical Experiments Discussion

Initial Conditions

Velocity model parameters: δx = 0 and δs = − ∇sH ∇sH ∇T

pHδp

∇sH

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is slope tomography? Smoothness constraints Numerical Experiments Discussion

Linear iterations

1- reference model m0 2- ray tracing is perfomed to calculate synthetic data (dc), δd = dOBS − dc 3- compute Frechét derivatives DF(m0) 4- solve for model perturbations δm 5- Update reference model m0 ← m0 + δm 6- If updated model fits the data within a specified tolerance stop; otherwise, iterate

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is slope tomography? Smoothness constraints Numerical Experiments Discussion

Linearized inversion

Estimation of a model consistent with the data: min

m

δd − DF(m0)δm2

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is slope tomography? Smoothness constraints Numerical Experiments Discussion

Linearized inversion

Estimation of a model consistent with the data: min

m

δd − DF(m0)δm2 Problem: There is no unique solution!

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is slope tomography? Smoothness constraints Numerical Experiments Discussion

Linearized inversion

Estimation of a model consistent with the data: min

m

δd − DF(m0)δm2 Problem: There is no unique solution! Remedy: Constrain the solution with additional properties. Regularization: smoothness of the velocity field.

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is slope tomography? Smoothness constraints Numerical Experiments Discussion

Smoothness constraints

Minimum curvature constraints

Minimize Laplacian Minimize second derivatives independently

Minimum inhomogeneity constraints

Minimize first derivatives independently Minimize directional derivatives along potential reflectors

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is slope tomography? Smoothness constraints Numerical Experiments Discussion

Slope tomography objective function

Φ(m; λi) = d − F(m)2

2 + λ2 0m − m02 2

+λ2

1D1p2 2 + λ2 2D3p2 2

+λ2

3D2 1p2 2 + λ2 4D2 3p2 2

+λ2

5(D2 1 + D2 3)p2 2

+λ2

6Drp2 2

Do not get too far from a prior (previous or initial) model

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is slope tomography? Smoothness constraints Numerical Experiments Discussion

Slope tomography objective function

Φ(m; λi) = d − F(m)2

2 + λ2 0m − m02 2

+λ2

1D1p2 2 + λ2 2D3p2 2

+λ2

3D2 1p2 2 + λ2 4D2 3p2 2

+λ2

5(D2 1 + D2 3)p2 2

+λ2

6Drp2 2

Gradient smoothness

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is slope tomography? Smoothness constraints Numerical Experiments Discussion

Slope tomography objective function

Φ(m; λi) = d − F(m)2

2 + λ2 0m − m02 2

+λ2

1D1p2 2 + λ2 2D3p2 2

+λ2

3D2 1p2 2 + λ2 4D2 3p2 2

+λ2

5(D2 1 + D2 3)p2 2

+λ2

6Drp2 2

Curvature smoothness

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is slope tomography? Smoothness constraints Numerical Experiments Discussion

Slope tomography objective function

Φ(m; λi) = d − F(m)2

2 + λ2 0m − m02 2

+λ2

1D1p2 2 + λ2 2D3p2 2

+λ2

3D2 1p2 2 + λ2 4D2 3p2 2

+λ2

5(D2 1 + D2 3)p2 2

+λ2

6Drp2 2

Laplacian isotropic smoothness

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is slope tomography? Smoothness constraints Numerical Experiments Discussion

Slope tomography objective function

Φ(m; λi) = d − F(m)2

2 + λ2 0m − m02 2

+λ2

1D1p2 2 + λ2 2D3p2 2

+λ2

3D2 1p2 2 + λ2 4D2 3p2 2

+λ2

5(D2 1 + D2 3)p2 2

+λ2

6Drp2 2

Smoothness along reflectors

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is slope tomography? Smoothness constraints Numerical Experiments Discussion

Regularization along the reflectors

α α potential reflector source ray receiver ray source receiver scattering point

Drp operator α = θs + θr 2 n(α; X) × ∇p(X) = 0

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is slope tomography? Smoothness constraints Numerical Experiments Discussion

Slope tomography linear iterations

            DF(m0) λ0I λ1D1 λ2D3 λ3D2

1

λ4D2

3

λ5(D2

1 + D2 3)

λ6Dr             δm =             δd            

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is slope tomography? Smoothness constraints Numerical Experiments Discussion

Regularization

km/s 1.5 2 2.5 3 3.5 4 4.5 Distance (km) Depth (km) Velocity Model 3 4 5 6 7 8 9 0.5 1 1.5 2 2.5

Exact model

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is slope tomography? Smoothness constraints Numerical Experiments Discussion

Regularization

km/s 1.5 2 2.5 3 3.5 4 4.5 Distance (km) Depth (km) Velocity Model 3 4 5 6 7 8 9 0.5 1 1.5 2 2.5

Minimization of Laplacian

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is slope tomography? Smoothness constraints Numerical Experiments Discussion

Pre-Stack depth migration

1000 2000 Depth (m) 3000 4000 5000 6000 7000 8000 9000 Distance (m)

Exact model

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is slope tomography? Smoothness constraints Numerical Experiments Discussion

Pre-Stack depth migration

1000 2000 Depth (m) 3000 4000 5000 6000 7000 8000 9000 Distance (m)

Minimization of Laplacian

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is slope tomography? Smoothness constraints Numerical Experiments Discussion

Regularization

km/s 1.5 2 2.5 3 3.5 4 4.5 Distance (km) Depth (km) Velocity Model 3 4 5 6 7 8 9 0.5 1 1.5 2 2.5

Exact model

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is slope tomography? Smoothness constraints Numerical Experiments Discussion

Regularization

km/s 1.5 2 2.5 3 3.5 4 4.5 Distance (km) Depth (km) Velocity Model 3 4 5 6 7 8 9 0.5 1 1.5 2 2.5

Minimization of curvature

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is slope tomography? Smoothness constraints Numerical Experiments Discussion

Pre-Stack depth migration

1000 2000 Depth (m) 3000 4000 5000 6000 7000 8000 9000 Distance (m)

Exact model

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is slope tomography? Smoothness constraints Numerical Experiments Discussion

Pre-Stack depth migration

1000 2000 Depth (m) 3000 4000 5000 6000 7000 8000 9000 Distance (m)

Minimization of curvature

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is slope tomography? Smoothness constraints Numerical Experiments Discussion

Regularization

km/s 1.5 2 2.5 3 3.5 4 4.5 Distance (km) Depth (km) Velocity Model 3 4 5 6 7 8 9 0.5 1 1.5 2 2.5

Exact model

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is slope tomography? Smoothness constraints Numerical Experiments Discussion

Regularization

km/s 1.5 2 2.5 3 3.5 4 4.5 Distance (km) Depth (km) Velocity Model 3 4 5 6 7 8 9 0.5 1 1.5 2 2.5

Minimization of gradient

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is slope tomography? Smoothness constraints Numerical Experiments Discussion

Pre-Stack depth migration

1000 2000 Depth (m) 3000 4000 5000 6000 7000 8000 9000 Distance (m)

Exact model

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is slope tomography? Smoothness constraints Numerical Experiments Discussion

Pre-Stack depth migration

1000 2000 Depth (m) 3000 4000 5000 6000 7000 8000 9000 Distance (m)

Minimization of gradient

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is slope tomography? Smoothness constraints Numerical Experiments Discussion

Regularization

km/s 1.5 2 2.5 3 3.5 4 4.5 Distance (km) Depth (km) Velocity Model 3 4 5 6 7 8 9 0.5 1 1.5 2 2.5

Exact model

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is slope tomography? Smoothness constraints Numerical Experiments Discussion

Regularization

km/s 1.5 2 2.5 3 3.5 4 4.5 Distance (km) Depth (km) Velocity Model 3 4 5 6 7 8 9 0.5 1 1.5 2 2.5

Minimization of derivative along reflectors

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is slope tomography? Smoothness constraints Numerical Experiments Discussion

Pre-Stack depth migration

1000 2000 Depth (m) 3000 4000 5000 6000 7000 8000 9000 Distance (m)

Exact model

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is slope tomography? Smoothness constraints Numerical Experiments Discussion

Pre-Stack depth migration

1000 2000 Depth (m) 3000 4000 5000 6000 7000 8000 9000 Distance (m)

Minimization of derivative along reflectors

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is slope tomography? Smoothness constraints Numerical Experiments Discussion

Regularization

km/s 1.5 2 2.5 3 3.5 4 4.5 Distance (km) Depth (km) Velocity Model 3 4 5 6 7 8 9 0.5 1 1.5 2 2.5

Exact model

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is slope tomography? Smoothness constraints Numerical Experiments Discussion

Regularization

km/s 1.5 2 2.5 3 3.5 4 4.5 Distance (km) Depth (km) Velocity Model 3 4 5 6 7 8 9 0.5 1 1.5 2 2.5

Minimization of gradient and derivative along reflectors

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is slope tomography? Smoothness constraints Numerical Experiments Discussion

Pre-Stack depth migration

1000 2000 Depth (m) 3000 4000 5000 6000 7000 8000 9000 Distance (m)

Exact model

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is slope tomography? Smoothness constraints Numerical Experiments Discussion

Pre-Stack depth migration

1000 2000 Depth (m) 3000 4000 5000 6000 7000 8000 9000 Distance (m)

Minimization of gradient and derivative along reflectors

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is slope tomography? Smoothness constraints Numerical Experiments Discussion

Angle domain image gathers: x=4.0 km

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is slope tomography? Smoothness constraints Numerical Experiments Discussion

Angle domain image gathers: x=4.0 km

0.5 1.0 1.5 2.0 2.5 Depth(km)

• 50

50 Angle(degrees) 0.5 1.0 1.5 2.0 2.5 Depth(km)

• 50

50 Angle(degrees) 0.5 1.0 1.5 2.0 2.5 Depth(km)

• 50

50 Angle(degrees) 0.5 1.0 1.5 2.0 2.5 Depth(km)

• 50

50 Angle(degrees) 0.5 1.0 1.5 2.0 2.5 Depth(km)

• 50

50 Angle(degrees) 0.5 1.0 1.5 2.0 2.5 Depth(km)

• 50

50 Angle(degrees)

Exact Laplac Curv Grad Reflec Reflec-Grad Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is slope tomography? Smoothness constraints Numerical Experiments Discussion

Angle domain image gathers: x=6.5 km

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is slope tomography? Smoothness constraints Numerical Experiments Discussion

Angle domain image gathers: x=6.5 km

0.5 1.0 1.5 2.0 2.5 Depth(km)

• 50

50 Angle(degrees) 0.5 1.0 1.5 2.0 2.5 Depth(km)

• 50

50 Angle(degrees) 0.5 1.0 1.5 2.0 2.5 Depth(km)

• 50

50 Angle(degrees) 0.5 1.0 1.5 2.0 2.5 Depth(km)

• 50

50 Angle(degrees) 0.5 1.0 1.5 2.0 2.5 Depth(km)

• 50

50 Angle(degrees) 0.5 1.0 1.5 2.0 2.5 Depth(km)

• 50

50 Angle(degrees)

Exact Laplac Curv Grad Reflec Reflec-Grad Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is slope tomography? Smoothness constraints Numerical Experiments Discussion

Angle domain image gathers: x=7.5 km

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is slope tomography? Smoothness constraints Numerical Experiments Discussion

Angle domain image gathers: x=7.5 km

0.5 1.0 1.5 2.0 2.5 Depth(km)

• 50

50 Angle(degrees) 0.5 1.0 1.5 2.0 2.5 Depth(km)

• 50

50 Angle(degrees) 0.5 1.0 1.5 2.0 2.5 Depth(km)

• 50

50 Angle(degrees) 0.5 1.0 1.5 2.0 2.5 Depth(km)

• 50

50 Angle(degrees) 0.5 1.0 1.5 2.0 2.5 Depth(km)

• 50

50 Angle(degrees) 0.5 1.0 1.5 2.0 2.5 Depth(km)

• 50

50 Angle(degrees)

Exact Laplac Curv Grad Reflec Reflec-Grad Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is slope tomography? Smoothness constraints Numerical Experiments Discussion

Discussion

Inverted velocity models depend strongly on regularization

Pure curvature constraints produced worst results Migrations results are less sensitive to regularization than velocity models

Regularization along the dip of possible reflectors

Implements in a natural way in slope tomography Reduces the differences between layer based and grid based velocity model parameterizations Highlights structural features in the velocity model Improves the velocity model in areas of poor ray coverage in a geologically plausible way

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is slope tomography? Smoothness constraints Numerical Experiments Discussion

Discussion

Inverted velocity models depend strongly on regularization

Pure curvature constraints produced worst results Migrations results are less sensitive to regularization than velocity models

Regularization along the dip of possible reflectors

Implements in a natural way in slope tomography Reduces the differences between layer based and grid based velocity model parameterizations Highlights structural features in the velocity model Improves the velocity model in areas of poor ray coverage in a geologically plausible way

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is full-waveform inversion? Secondary sources and sensitivity kernels Kernel decomposition Numerical experiment

Contents

1

Introduction

2

Reflector-oriented regularization in slope tomography

3

Decomposition of sensitivity kernels in full-waveform inversion

4

Conclusions

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is full-waveform inversion? Secondary sources and sensitivity kernels Kernel decomposition Numerical experiment

Forward problem

Non-linear problem, p = F(m). Small pertubations in the model parameters allow linearization, δp = Φδm. Frechèt derivatives for the acoustic wave equation δp = Φδm =

• Uf

Vf δK δρ

• .

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is full-waveform inversion? Secondary sources and sensitivity kernels Kernel decomposition Numerical experiment

Inverse problem

Adjoint Frechèt derivatives → back-project pertubations in the wavefield (data residual) onto model domain. δmk = δK k δρk

• =

Uf

∗

Vf

∗

• δp = Φ∗δp.

What a back-projection is needed for? mk+1 = mk + α Φ∗δpk

δmk

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is full-waveform inversion? Secondary sources and sensitivity kernels Kernel decomposition Numerical experiment

Secondary or adjoint sources

Want to know the Frechèt derivatives? Look for the secondary sources. Secondary sources Sources that will give rise to data residuals due to pertubations in the model parameters. Secondary sources are derived from the wave equation.

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is full-waveform inversion? Secondary sources and sensitivity kernels Kernel decomposition Numerical experiment

Sensitivity kernels from secondary sources

For the acoustic impulse response L [p(x, t; xs)] = δ(x − xs)S(t), the secondary sources are (Tarantola, 1984, Geophysics, 48) L [δp(x, t; xs)] = −δL [p(x, t; xs)]

• secondary sources

Wavefield perturbation δp(x, t; xs) = −

• V

d3x′ G(x, t; x′) ∗ δL

• p(x′, t; xs)
• .

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is full-waveform inversion? Secondary sources and sensitivity kernels Kernel decomposition Numerical experiment

Sensitivity kernel for the scattered field

Zhu et al, 2009, Geophysics, 74

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is full-waveform inversion? Secondary sources and sensitivity kernels Kernel decomposition Numerical experiment

Decomposition of sensitivity kernel

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is full-waveform inversion? Secondary sources and sensitivity kernels Kernel decomposition Numerical experiment

Decomposition of the model

= + Smooth part singular part (sharp contrasts)

Velocity model from velocity analysis Migrated image

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is full-waveform inversion? Secondary sources and sensitivity kernels Kernel decomposition Numerical experiment

Decomposition of the wavefield

L [p(x, t)] = δ(x − xs)S(t) LB [p0(x, t)] = δ(x − xs)S(t) L [ps(x, t)] = −V [p0(x, t)] + V = L − LB: Scattering potential

Conventionally: ps = δp is perturbation of p0 = p, V = δL is perturbation of L Here: Both contributions are perturbed − → δp0, δps, δLB, δV

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is full-waveform inversion? Secondary sources and sensitivity kernels Kernel decomposition Numerical experiment

Reparametrization

Conventionally: m = K ρ

• =

⇒ δm = δK δρ

• Here:

m =     KB ρB KS ρS     = ⇒ δm =     δKB δρB δKS δρS    

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is full-waveform inversion? Secondary sources and sensitivity kernels Kernel decomposition Numerical experiment

Reparametrization

Conventionally: δ p =

• Uf

Vf δK δρ

• Here:

δ p0 δ ps

• =

U V UB VB US VS

• 

   δKB δρB δKS δρS    

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is full-waveform inversion? Secondary sources and sensitivity kernels Kernel decomposition Numerical experiment

Reference wavefield residual and sensitivity kernel

Residual evaluated from reference secondary sources δ p0(x; xs) = −

• V

d3x′ G0(x; x′)δLB p0(x′; xs)

• .

Explicit bulk modulus contribution δ pK

0 (xg; xs) =

• V

d3x′

• −

ω2 K 2

B(x′)

• G0(x′; xg)

p0(x′; xs)

• δKB(x′).

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is full-waveform inversion? Secondary sources and sensitivity kernels Kernel decomposition Numerical experiment

Scattered wavefield residual

The residual evaluated from scattered secondary sources is given by

δ pS(x; xs) = −

• V

d3x′ GS(x′; x) V

• δ

p0(x′; xs)

• −
• V

d3x′ G0(x′; x) V

• δ

p0(x′; xs)

• −
• V

d3x′ GS(x′; x) δL

• p0(x′; xs)
• −
• V

d3x′ G0(x′; x) δL

• p0(x′; xs)
• −
• V

d3x′ GS(x′; x) δL

• pS(x′; xs)
• −
• V

d3x′ G0(x′; x) δL

• pS(x′; xs)
• +
• V

d3x′ GS(x′; x) δLB p0(x′; xs)

• +
• V

d3x′ G0(x′; x) δLB p0(x′; xs)

• Smooth part of δm

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is full-waveform inversion? Secondary sources and sensitivity kernels Kernel decomposition Numerical experiment

Scattered wavefield residual

The residual evaluated from scattered secondary sources is given by

δ pS(x; xs) = −

• V

d3x′ GS(x′; x) V

• δ

p0(x′; xs)

• −
• V

d3x′ G0(x′; x) V

• δ

p0(x′; xs)

• −
• V

d3x′ GS(x′; x) δL

• p0(x′; xs)
• −
• V

d3x′ G0(x′; x) δL

• p0(x′; xs)
• −
• V

d3x′ GS(x′; x) δL

• pS(x′; xs)
• −
• V

d3x′ G0(x′; x) δL

• pS(x′; xs)
• +
• V

d3x′ GS(x′; x) δLB p0(x′; xs)

• +
• V

d3x′ G0(x′; x) δLB p0(x′; xs)

• Smooth part of δm

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is full-waveform inversion? Secondary sources and sensitivity kernels Kernel decomposition Numerical experiment

Scattered wavefield residual

The residual evaluated from scattered secondary sources is given by

δ pS(x; xs) = −

• V

d3x′ GS(x′; x) V

• δ

p0(x′; xs)

• −
• V

d3x′ G0(x′; x) V

• δ

p0(x′; xs)

• −
• V

d3x′ GS(x′; x) δL

• p0(x′; xs)
• −
• V

d3x′ G0(x′; x) δL

• p0(x′; xs)
• −
• V

d3x′ GS(x′; x) δL

• pS(x′; xs)
• −
• V

d3x′ G0(x′; x) δL

• pS(x′; xs)
• +
• V

d3x′ GS(x′; x) δLB p0(x′; xs)

• +
• V

d3x′ G0(x′; x) δLB p0(x′; xs)

• Singular part of δm

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is full-waveform inversion? Secondary sources and sensitivity kernels Kernel decomposition Numerical experiment

Kernel decomposition

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is full-waveform inversion? Secondary sources and sensitivity kernels Kernel decomposition Numerical experiment

Different levels of non-linearity

δ pS(x; xs) = −

• V

d3x′ GS(x′; x) V

• δ

p0(x′; xs)

• −
• V

d3x′ G0(x′; x) V

• δ

p0(x′; xs)

• −
• V

d3x′ GS(x′; x) δL

• p0(x′; xs)
• −
• V

d3x′ G0(x′; x) δL

• p0(x′; xs)
• −
• V

d3x′ GS(x′; x) δL

• pS(x′; xs)
• −
• V

d3x′ G0(x′; x) δL

• pS(x′; xs)
• +
• V

d3x′ GS(x′; x) δLB p0(x′; xs)

• +
• V

d3x′ G0(x′; x) δLB p0(x′; xs)

• .

Scattering: single, multiple, strong mutiple

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is full-waveform inversion? Secondary sources and sensitivity kernels Kernel decomposition Numerical experiment

Different levels of non-linearity

Single scattering: −

• V

d3x′ G0(x′; x) δL

• p0(x′; xs)
• Jörg Schleicher

Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is full-waveform inversion? Secondary sources and sensitivity kernels Kernel decomposition Numerical experiment

Different levels of non-linearity

Mutiple scattering: −

• V

d3x′ GS(x′; x) δL

• p0(x′; xs)
• Jörg Schleicher

Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is full-waveform inversion? Secondary sources and sensitivity kernels Kernel decomposition Numerical experiment

Different levels of non-linearity

Strong multiple scattering: −

• V

d3x′ GS(x′; x) δL

• pS(x′; xs)
• Jörg Schleicher

Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is full-waveform inversion? Secondary sources and sensitivity kernels Kernel decomposition Numerical experiment

Forward and adjoint decomposition

Bulk modulus contribution: δ p0 δ ps

• =
• U

n=8

i=1 UB,i

n=6

i=3 US,i

δKB δKS

• The backprojection based on the above decomposition is

δKBest δKS

est

• =
• U†

n=8

i=1 UB,i †

n=6

i=3 US,i †

δ p0 δ ps

• Jörg Schleicher

Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is full-waveform inversion? Secondary sources and sensitivity kernels Kernel decomposition Numerical experiment

Numerical experiment

Residual-wavefield backprojection Perturbation of the singular part

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is full-waveform inversion? Secondary sources and sensitivity kernels Kernel decomposition Numerical experiment

Numerical experiment

Unperturbed model: Perturbation: Random change of the scatterer positions.

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is full-waveform inversion? Secondary sources and sensitivity kernels Kernel decomposition Numerical experiment

Perturbations on the singular part of the model

No background perturbation means δKB = 0. Then δp0 = 0 and δps = n=6

• i=3

US,i

• δKS

Backprojection of the scattered-wavefield residual yields δKS

est =

n=6

• i=3

US,i

†

• δps

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is full-waveform inversion? Secondary sources and sensitivity kernels Kernel decomposition Numerical experiment

Perturbations on the singular part of the model

δKS,4

est(x′

i ) = −

• dω

ω2 K 2(x′

i )

direct wavefield cross-correlation

• p∗

0 (x′ i , ω; xs) background extrapolator

• G∗

0(x′ i , ω; xg) δ

ps(xg, ω; xs)

• back-propagation of δ

pS

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is full-waveform inversion? Secondary sources and sensitivity kernels Kernel decomposition Numerical experiment

Perturbations on the singular part of the model

δKS,3

est(x′

i ) = −

• dω

ω2 K 2(x′

i )

direct wavefield cross-correlation

• p∗

0 (x′ i , ω; xs) scattered wave extrapolator

• G∗

S(x′ i , ω; xg) δ

ps(xg, ω; xs)

• back-propagation of δ

pS

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is full-waveform inversion? Secondary sources and sensitivity kernels Kernel decomposition Numerical experiment

Perturbations on the singular part of the model

δKS,6

est(x′

i ) = −

• dω

ω2 K 2(x′

i )

scattered wavefield cross-correlation

• p∗

S(x′ i , ω; xs) background extrapolator

• G∗

0(x′ i , ω; xg) δ

ps(xg, ω; xs)

• back-propagation of δ

pS

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is full-waveform inversion? Secondary sources and sensitivity kernels Kernel decomposition Numerical experiment

Perturbations on the singular part of the model

δKS,5

est(x′

i ) = −

• dω

ω2 K 2(x′

i )

scattered wavefield cross-correlation

• p∗

S(x′ i , ω; xs) scattered wave extrapolator

• G∗

S(x′ i , ω; xg) δ

ps(xg, ω; xs)

• back-propagation of δ

pS

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is full-waveform inversion? Secondary sources and sensitivity kernels Kernel decomposition Numerical experiment

Discussion

Successful kernel decomposition

Perturbation of background medium and singular part Based on model building and migration-type imaging

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is full-waveform inversion? Secondary sources and sensitivity kernels Kernel decomposition Numerical experiment

Discussion

Successful kernel decomposition

Perturbation of background medium and singular part Based on model building and migration-type imaging

Potential for better control over FWI optimization:

Contributions show different levels of non-linearity Multiple scattering carries important information

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions What is full-waveform inversion? Secondary sources and sensitivity kernels Kernel decomposition Numerical experiment

Discussion

Successful kernel decomposition

Perturbation of background medium and singular part Based on model building and migration-type imaging

Potential for better control over FWI optimization:

Contributions show different levels of non-linearity Multiple scattering carries important information

Pratical challenges on separation of the model/data components Potential use in 4D-inversion problems

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions

Contents

1

Introduction

2

Reflector-oriented regularization in slope tomography

3

Decomposition of sensitivity kernels in full-waveform inversion

4

Conclusions

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions

Conclusions

Seismic Inverse problems are (partly) underdetermined Something has to be done about ambiguity

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions

Conclusions

Seismic Inverse problems are (partly) underdetermined Something has to be done about ambiguity Slope tomography

Model-geometry-based regularization Led to more realistic velocity model

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions

Conclusions

Seismic Inverse problems are (partly) underdetermined Something has to be done about ambiguity Slope tomography

Model-geometry-based regularization Led to more realistic velocity model

Full-waveform inversion

Sensitivity kernel decomposition Led to better understanding of contributions

Jörg Schleicher Fighting ambiguity in seismic inverse problems

Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions

Acknowledgements

Contributors to these topics were J. C. Costa, F . J. C. da Silva,

• E. N. S. Gomes, A. Mello, D. Amazonas, D. L. Macedo, and
• I. Vasconcelos.

We thank Gilles Lambaré for provinding the Marmousoft data set and the picked events on this data set. This research was supported by CAPES, FINEP , CNPq, as well as Petrobras, Schlumberger, and the sponsors of the WIT consortium. Thank you for your attention

Jörg Schleicher Fighting ambiguity in seismic inverse problems