Full Waveform Inversion in an Anisotropic World (EET 10)

Where are the parameters hiding?

image of Full Waveform Inversion in an Anisotropic World (EET 10)
  • By Tariq A. Alkhalifah
  • Format: EPUB
  • Publication Year: 2016
  • Number of Pages: 197
  • Language: English
  • Ebook ISBN: 9789462822023

Why full waveform inversion? Is not imaging (at a very reduced cost) enough? Well, a simple visual characterization depicting the difference between what the two approaches bring to the table can be realized using the common street artist pencil and brush. While imaging provides us with what we could describe as a pencil sketch of the Earth’s content, outline its complications, FWI fills in the colours, with a more in depth description and clarity to the Earth’s picture. While a pencil sketch may, depending on the pencil (or the frequency), provide a map of the major elements in the drawings, FWI provides colours, usually with definite crisp borders between the colours, reflecting the high resolution nature of the technique.

Table of Contents

      0.1 Why?
      0.2 Who should read this book?

1    Introduction

1.1 Defining full waveform inversion
1.2 What is available for velocity model building?
1.3 Why FWI?
1.4 FWI challenges
1.5 What are we covering in this book?
1.5.1 2D versus 3D

2    The wavefield
2.1 The components of wave propagation
2.2 The coordinate system
2.3 The medium
2.4 The acoustic (scalar) wavefield
2.5 The acoustic wave equation
2.5.1 The wave equation in the time domain
2.5.2 Derivation via the dispersion relation
2.5.3 The Helmholtz wave equation
2.6 Solving the wave equation
2.6.1 Analytical solutions of the acoustic wave equation
2.7 Finite-difference solutions of the wave equation
2.8 The Courant–Friedrichs–Lewy condition
2.8.1 The non-linear aspect

3    Anisotropy: the acoustic case
3.1 The case for anisotropy
3.1.1 The acoustic assumption for anisotropy
3.1.2 Anisotropic parameterization in the acoustic case
3.2 The dispersion relation
3.3 The VTI acoustic wave equation
3.4 Finite difference solutions of the wave equation
3.4.1 Eliminating the S-wave artefact
3.4.2 A practical subsurface model
3.4.3 Other forms of the equation
3.5 Summary

4    The fundamentals of FWI
4.1 Background and FWI issues
4.2 Modelling
4.2.1 The model
4.2.2 The data
4.2.3 The forward problem
4.3 The objective function
4.3.1 The measure of misfit
4.3.2 Global versus local minima
4.3.3 The basin of attraction
4.3.4 An example
4.4 The update
4.4.1 The gradient
4.4.2 Optimization
4.4.3 More on the Hessian
4.4.4 Practical application
4.5 The Born approximation
4.5.1 The Born approximation and sensitivity kernels
4.6 Sensitivity kernels
4.6.1 Reflections versus transmissions
4.6.2 As a function of offset and frequency
4.6.3 The background velocity and the perturbation
4.7 Model and data weights
4.7.1 Illumination and the Null space
4.7.2 The data quality
4.7.3 The weights
4.8 The algorithm
4.8.1 The initial model
4.8.2 The modelling
4.8.3 The residual
4.8.4 The update
4.8.5 The convergence
4.9 Summary

5    The FWI nonlinearity and potential solutions
5.1 What happens if low frequencies are not present?
5.2 The model wavenumber
5.3 The relation between model wavenumbers and data
5.3.1 The critical issue of nonlinearity
5.3.2 Addressing the nonlinearity
5.4 Phase inversion and the instantaneous traveltime
5.4.1 Multiple events
5.4.2 Model induced nonlinearity
5.4.3 Beyond the kinematics
5.5 Damping the data
5.5.1 The influence of damping
5.6 Why damp IT?
5.7 The new objective function
5.7.1 The gradient
5.7.2 In action
5.8 Summary

6    Anisotropic model building
6.1 The model
6.1.1 The SEG/EAGE salt-body model
6.1.2 The anisotropy parameter η
6.1.3 The symmetry axis direction
6.2 A DTI model
6.2.1 Dip-constrained TTI media
6.2.2 Extended imaging condition
6.2.3 Moveout analysis
6.2.4 Angle decomposition
6.2.5 Downward continuation
6.2.6 Domain of applicability
6.3 Velocity analysis
6.3.1 Putting the inhomogeneity in the background (back burner)
6.3.2 Using the eikonal
6.3.3 Using the wavefield
6.3.4 A practical implementation
6.4 Summary: linearized anisotropy

7    Practical FWI in anisotropic media
7.1 The choice of parameters
7.2 The Born approximation
7.2.1 The radiation pattern
7.2.2 The trade-off
7.2.3 Using the horizontal velocity
7.3 The model update wavenumber
7.4 The anisotropic sensitivity kernels
7.4.1 As a function of frequency and offset
7.5 Summary

8    Conditioning the full waveform inversion gradient to welcome anisotropy
8.1 Introduction
8.2 The model’s update characteristics
8.2.1 The model’s wavenumber update
8.2.2 The myth of transmitted and reflected waves in FWI
8.2.3 The radiation pattern
8.3 Anisotropic radiation patterns
8.4 Gradients for classical and reflection-based MVA
8.5 Filtering the gradient
8.6 Gradients under filtering
8.7 For inhomogeneous media
8.8 Gradient wavenumber distribution for anisotropic media
8.9 Filtering anisotropy
8.9.1 The vncombination
8.9.2 The vh combination
8.10 Summary

9    Insights into the data dependency on anisotropy: an inversion prospective
9.1 The concept
9.2 The angular sensitivity of waves
9.2.1 The kinematic transmission components
9.2.2 The scattering components
9.3 The wavenumber content from the kinematics and scattering
9.4 Maps of resolvability
9.5 A simple example of influence
9.6 The case for dimensionless parameters
9.7 The scattering potential in elastic media
9.8 An elastic VTI Marmousi — model
9.9 Discussions
9.10 Summary

10    Final remarks
10.1  What did we get from FWI?
10.2  Looking forward



This is a required field
Please enter a valid email address
Approval was a Success
Invalid data
An Error Occurred
Approval was partially successful, following selected items could not be processed due to error