Seismic Inversion

Theory and Applications

Yanghua Wang

• Publisher: Wiley-Blackwell
• Year: 2016
• Print ISBN: 9781119257981
• eBook ISBN: 9781119258025
• Edition: 1
• Topic: Physical Sciences
• Subtopic: Geology & Earth Sciences

Seismic inversion aims to reconstruct a quantitative model of the Earth subsurface, by solving an inverse problem based on seismic measurements. There are at least three fundamental issues to be solved simultaneously: non-linearity, non-uniqueness, and instability. This book covers the basic theory and techniques used in seismic inversion, corresponding to these three issues, emphasising the physical interpretation of theoretical concepts and practical solutions.

This book is written for master and doctoral students who need to understand the mathematical tools and the engineering aspects of the inverse problem needed to obtain geophysically meaningful solutions. Building on the basic theory of linear inverse problems, the methodologies of seismic inversion are explained in detail, including ray-impedance inversion and waveform tomography etc. The application methodologies are categorised into convolutional and wave-equation based groups. This systematic presentation simplifies the subject and enables an in-depth understanding of seismic inversion.

This book also provides a practical guide to reservoir geophysicists who are attempting quantitative reservoir characterisation based on seismic data. Philosophically, the seismic inverse problem allows for a range of possible solutions, but the techniques described herein enable geophysicists to exclude models that cannot satisfy the available data. This book summarises the author's extensive experience in both industry and academia and includes innovative techniques not previously published.

Chapter 1
Basics of seismic inversion

Seismic inversion attempts to extract spatially variable physical parameters from measured seismic data. These physical parameters may be representative of the Earth’s subsurface media, and have physical and geological meanings, and thus seismic inversion is a quantitative interpretation of seismic measurement. The inversion procedure is generally nonlinear, as the entire inversion engine to solve the inverse problem, at least partly, depends upon the solution. In practice, the inverse problem is often linearised, and the final nonlinear solution is obtained through the iterative application of linearised solvers. Therefore, this book will focus on the linear inverse problem.

1.1 The linear inverse problem

Linear seismic inversion may include at least three basic steps such as the following:

1. 1. Setting up an objective function, which describes how well a model estimate represents the seismic observation and meets our human expectation;
2. 2. Optimising the objective function based on a minimal variation principle, which leads to a linear system of equations, if the objective function is defined as a quadratic function;
3. 3. Solving this linear system, to obtain a quantitative solution.

Data fitting is a principal part of the objective function. Seismic inversion uses forward modelling to generate synthetic seismic data that will match the observed seismic data. Forward modelling can be presented in a linear form:

1.1

where G is a geophysical operator (a matrix), m is the ‘model’ vector, and d is the ‘data’ vector. Both vectors m and d are defined in the Hilbert space in which the structure (the length and angle) of an inner product of vectors can be measured. Row vectors and column vectors of matrix G are also defined in the Hilbert space.

For example, for a two-dimensional (2-D) velocity model defined in the x − z domain, we cannot straightforwardly include a fracture or a fault in the model parameterisation. Instead, we shall use an equivalent velocity model, which takes into account the effect of the fracture or the fault, so that the model vec...

Table of contents

• Cover
• Dedication
• Title Page
• Copyright
• Preface
• Chapter 1: Basics of seismic inversion
• Chapter 2: Linear systems for inversion
• Chapter 3: Least-squares solutions
• Chapter 4: Singular value analysis
• Chapter 5: Gradient-based methods
• Chapter 6: Regularisation
• Chapter 7: Localised average solutions
• Chapter 8: Seismic wavelet estimation
• Chapter 9: Seismic reflectivity inversion
• Chapter 10: Seismic ray-impedance inversion
• Chapter 11: Seismic tomography based on ray theory
• Chapter 12: Waveform tomography for the velocity model
• Chapter 13: Waveform tomography with irregular topography
• Chapter 14: Waveform tomography for seismic impedance
• Appendices
• Exercises and solutions
• References
• Author index
• Subject index
• End User License Agreement