Reflection Seismology: Theory, Data Processing and Interpretation

By Yang Wencai

Authored by a geophysicist with more than 50 years of experience in research and instruction, Reflection Seismology: Theory, Data Processing and Interpretation provides a single source of foundational knowledge in reflection seismology principles and theory.Reflection seismology has a broad range of applications and is used primarily by the oil and gas industry to provide high-resolution maps and build a coherent geological story from maps of processed seismic reflections. Combined with seismic attribute analysis and other exploration geophysics tools, it aids geologists and geo-engineers in creating geological models of areas of exploration and extraction interest. Yet as important as reflection seismology is to the hydrocarbon industry, it’s difficult to find a single source that synthesizes the topic without having to wade through numerous journal articles from a range of different publishers. This book is a one-stop source of reflection seismology theory, helping scientists navigates through the wealth of new data processing techniques that have emerged in recent years.

• Provides geoscientists and geo-engineers with a theoretical framework for navigating the rapid emergence of new data processing techniques
• Presents a single source of reflection seismology content instead of a scattering of disparate journal articles
• Features more than 100 figures, illustrations, and working examples to aid the reader in retaining key concepts
• Arms geophysicists and geo-engineers with a solid foundation in seismic wave equation analysis and interpretation

• Language: English
• Publisher: Elsevier Science
• Release date: Sep 18, 2013
• ISBN: 9780124096004

Book preview

Preface

Reflection seismology belongs to a small branch of solid Earth physics. Does it have its own theoretical systems? This is the question that I keep considering. If one copies word by word the formula from elastic or continuum mechanics to form the theory of reflection seismology, one looks upon reflection seismology as a branch of applied technology, but not as a branch of applied science. Even today, many geoscientists still regard reflection seismology as engineering technology that surveys mineral exploration. As reflection seismologists, we cannot blame their ignorance, but can only blame ourselves for not having built the theoretical system perfectly and completely.

It is true that reflection seismology was an oil/gas exploration technique at the beginning. An oil field was found according to the seismic reflection data in the continent of Oklahoma in the USA in 1926, which proves that seismic reflection is really a good practical technique having enormous economic worth. After the 1930s, based on decades of effort, especially after the invention of computers, the theory of reflection seismology has been developed rapidly. Some monographs on reflection seismology have been published in the 1970s. However, because the renewal speed of seismic exploration methods has been too fast, these monographs cannot include enough newly developed content on the theory of reflection seismology. On the other hand, textbooks used in universities mostly quote elastic mechanics formulas, and have not absorbed enough newly developed theories on reflection seismology. From 1983 to 2013, I have written a few monographs on geophysical inversion theory and methods that have been my main research area. As the foundation of performing seismic inversion is built on solving the forward problems in reflection seismology, I have devoted my time for summarizing the theory of reflection seismology, merging forward and inverse problems into a textbook, and bringing out the best in each other.

This book is designed as a textbook for professional graduate students majoring in applied geophysics. It is impossible to cover all developments in reflection seismology theories. I have outlined just the equation systems of reflection seismology based on the theory of continuum mechanics. The equation family constructs the theoretical framework of reflection seismology. I do not want to emphasize the abstract beauty of these equations, but in this book, I try to put up systematically a skeleton that is formed with equations and mathematical formulae, showing the spirit and values of classical physics. Equations and mathematical formulae are the essential language for communication between human beings and computers, applicable to all. One can be an excellent geologist who may not be good at mastering this kind of language, but one cannot be an excellent seismologist who is not good at mastering this kind of language.

The corresponding teaching of how to use this textbook takes about 50 h. As teaching materials, the authors must consider the following three points and guard against three misleading factors, namely first, write teaching materials as research reports, discuss a lot of details, and mislead students to ignore the backbone. Second, write the textbook as a collection of practical techniques that will be updated soon, and do not probe into their theoretical bases deeply. Students tend to listen excitedly, think that a lot of useful knowledge would be obtained, but ignore the academic ability and marrow. Third, write the teaching materials as an encyclopedia that includes every aspect of the subject, but not fully systematically. Books become very voluminous when they provide too much of knowledge for students, but become unfavorable for training graduate students who have a systematic thinking ability. To avoid misleading the readers, this book adheres to the main idea, and does discuss the main branches in detail; it only explains the theory and not already existing market-based technology. I am not afraid of writing little, but only afraid of writing too much. The backbone must explain clearly, the minor branches may only be clicked. It not only lets students assimilate information but also confers the ability to combine bits of knowledge to solve practical problems in application.

This book is divided into 7 chapters as follows: The first chapter reviews the basic wave theory of classical physics, and the second summarizes elastic wave equations that build up the foundation of reflection seismology. Particle dynamics used to be improperly applied to explain vibration and wave propagation in the past, and I have tried my best to correct improper concepts by using continuum mechanics. Chapter three summarizes seismic wave propagation in the solid Earth, especially some special characteristics different from elastic waves in common solid media. Chapter four deals with wave equation changes along with seismic data processing, together with changes on the relevant boundary conditions. Chapter five discusses the integral solutions of wave propagation problems and Green's function methods. Chapter six discusses the decomposition and continuation of seismic wave fields, with emphasis on the properties of wave equations with variable coefficients and the operator expansion method for the problems under study. The last chapter briefly introduces inverse problems involved in seismic exploration and typical solving numerical methods.

I would like to thank professors Fu Chengyi and W. Telford for guiding me on solid Earth geophysics. I also appreciate the help of Drs. Yang Wuyang, Li Lin, Xie Chunhui, Wang Enli, Wang Wanli, and Zhu Xiaoshan in the translation of this book to English. The author is much indebted to Mohanapriyan Rajendran for improving the English presentation of this book and grateful to all colleagues and readers who inform errors left in this book.

Wencai Yang

National Lab on Tectonics and Dynamics, Institute of Geology, CAGS.PRC

Chapter 1

Introduction to the Wave Theory

Abstract

Physics is the study of the motion of matter. It originates from Newtonian mechanics theory. In classical mechanics, which is established by Newton in the seventeenth century, particle dynamics was first employed to describe the motion of macroscopic objects. However, we must deal with continuously constructed media such as the earth, and the application of particle dynamics has some limitations. Thus, scientists developed continuum mechanics in the early twentieth century, including fluid mechanics and solid mechanics. Quantum mechanics has been developed in the same period to describe the motion of microscopic particles. The theory in this book is based on continuum mechanics and the methods of mathematical physics.

Keywords

Dynamics; Mechanics; Wave theory

Outline

1.1. Wave Motion in Continuous Media

1.2. Vibration

1.3. Propagation and Diffusion

1.4. Acoustic Wave Equation

1.5. Acoustic Wave Equation with Complex Coefficients

1.5.1. Complex Elastic Modulus and the Complex Wave Velocity

1.5.2. Damping Wave Equations in Viscoelastic Media

1.5.3. Viscoelastic Models

1.6. Acoustic Wave Equation with Variant Density or Velocity

1.7. Summary

Physics is the study of the motion of matter. It originates from Newtonian mechanics theory. In classical mechanics, which is established by Newton in the seventeenth century, particle dynamics was first employed to describe the motion of macroscopic objects. However, we must deal with continuously constructed media such as the earth; the application of particle dynamics has some limitations. Thus, scientists developed continuum mechanics in the early twentieth century, including fluid mechanics and solid mechanics. Quantum mechanics has been developed in the same period to describe the motion of microscopic particles. The theory in this book is based on continuum mechanics and the methods of mathematical physics.

The motion of macroscopic objects in general can be divided into three types: The first is displacement, such as linear movement, rotation, flight, and flow. The second is vibration and wave motion, such as periodic motion, water wave, acoustic wave, and seismic wave. The third is chaotic movement, such as intermittent motion, turbulence, and nonlinear wave motion. This book only discusses classical vibration and wave theory. As a kind of physical movement, vibration can be described using an initial value problem of ordinary differential equations for a closed system, in which energy and information do not get exchanged between the system and the outside world. For an open system, in which energy and information get exchanged between the system and the outside world, the initial and boundary value problems of partial differential equations must be applied. This chapter discusses the basic wave theory, focusing on the acoustic wave equation and related wave behaviors.

To make mathematical formulas clear, we use bold English letters for vectors or matrices, and normal Greek letters for scalars in this book.

1.1 Wave Motion in Continuous Media

Wave motion refers to the propagation of vibration in continuous media, which can be described by using the following formulations:

Wave motion = vibration + propagation (in continuous media);

Vibration = periodic motion that an object moves around its equilibrium point;

Propagation = interaction between the vibration and adjoining mass grains and diffusion of the vibration energy.

The above concepts are limited to the situation that the vibration equilibrium point of mass grains is fixed and is stationary during wave propagation; they are accurate for the elastic waves propagating in solid media. Generalized waves involve situations in which the vibration equilibrium point is not fixed and movable. For example, water waves are a kind of generalized waves with moving equilibrium points. We do not discuss generalized waves in this book.

Three assumptions are usually accepted in the study of continuum mechanics and are as follows (Fung, 1977; Spencer, 1980; Du Xun, 1985):

1. Mass motion follows the law of conservation of mass, that is the first derivative of mass M with respect to time t is zero:

(1.1)

2. The equilibrium point of mass–grain vibration in a wave field is fixed and stable.

3. Vibration occurs in continuous media, and the interaction between adjacent mass grains follows the continuity equation, which defines the continuous media.

What kind of media can be defined as continuous media? In other words, what kind of mechanical laws are followed by continuous media?

Definition: for a small deformation, continuous media are defined by the continuity equation as follows:

(1.2)

In this equation, ρ denotes the density of a mass grain in the media with respect to time tindicates the divergence of the displacement vector. Equation (1.2) could be derived from Eqn (1.1), which describes the conservation of mass. The product of the grain volume and density is called the mass element.

Set V as the volume of continuous media and the mass in Eqn (1.1) can be expressed as follows:

In Cartesian coordinates, according to the law of the conservation of mass, mass can be expressed as follows:

(1.3)

, denoting a determinant as

(1.4)

We have

Substituting Eqn (1.4) into Eqn (1.3) yields

(1.5)

or

(1.6)

Hence, the integral kernel should be equal to zero as

(1.7)

Suppose a small deformation is being generated during the wave motion. Then,

If one ignores the second-order terms, Eqn (1.3) can be transformed as

(1.8)

If we substitute Eqn (1.8) into Eqn (1.7), the continuous Eqn (1.2) can be obtained.

It is true that wave motion involves only small deformations, but explosions involve large deformations. In seismic explorations, explosions are used as the vibration sources to produce seismic waves. Both vibration and movement of mass grains occur around the shots, where one has to use the theory of explosion that we do not discuss in this book. We will explain the movement in the far field area, where the equilibrium points are fixed and stable and small deformations are accepted.

The continuity Eqn (1.2) implies that the density of the mass elements may vary during wave motion and that the variation amplitude is proportional to the divergence of the displacement vector.

The movements of the mass grains in nature can be classified into several kinds, and vibration and wave motion are two of them. There are many physical branches that describe the movements of the mass grains, including classical mechanics, continuum mechanics, and nonlinear dynamics. Classical mechanics is the study of mass motion in free space, usually without additional constraints, and is helpful for the study of gas motion in a vacuum and the Brownian motion of molecules. Mass movement follows Newton's laws of motion and universal gravitation in an enclosed dynamic system and can be expressed by some ordinary differential equations with initial values. Continuum mechanics studies constrained motion in continuum space, it is based on Newton's equation of motion and specific constitutive equation, and it can be described by partial differential equations with initial and boundary values. The constitutive equation in elastic mechanics is called the generalized Hooke's law. Continuity Eqn (1.2) is the starting point of continuum mechanics, which used to be divided into statics and dynamics. Wave motion is the result of some forces, and belongs to dynamics.

1.2 Vibration

Vibration can be described as the motion of a mass constraint to an equilibrium point with a limited distance. No matter how the mass oscillates, it will always return to the equilibrium point, due to the direction of the force acting on the oscillator always pointing to the equilibrium point. Only when the direction of force (i.e. elastic force in a solid) is opposite that of the movement, the vibration can be always around the equilibrium point. Therefore, vibration is a kind of mass motion whose working force and displacement are in reverse directions.

In the case of one-dimensional (1D) motion, we denote the elastic force as F, and the displacement of movement is indicated as u; then, Hooke's law is

(1.9)

where k indicates the elastic coefficient, the negative sign indicates that the direction of the force is opposite to the displacement. Denoting the mass of the oscillator as m, we can apply the equation using Newton's second law as (Budak et al., 1964)

(1.10)

Setting the circular frequency as ωinto Eqn (1.10) and obtain

(1.11)

The above equation is the vibration equation without damping, and its general solution is

(1.12)

where A indicates the vibration phase.

as the proportional coefficient, then

(1.13)

Therefore, Eqn (1.10) can be rewritten as follows:

(1.14)

is the damping coefficient of the vibration. The damping vibration equation can be written as

(1.15)

Equation , its general solution is

(1.16)

is related to the damping coefficient.

The vibration of rock grains in reflection seismology mostly belongs to damping oscillations. Equation (1.16) shows that there are two outstanding features in real oscillations. The first is that the magnitude of the amplitude decreases according to the exponential law with respect to increasing time. The second is that the angular frequency ω is, the lower the frequency ω turns out to be.

1.3 Propagation and Diffusion

Vibration refers to the motion around an equilibrium point, while propagation refers the motion with energy spreading from a source in space. The existence of any object affects the surrounding mass, no matter whether these objects connect with each other or not. A physicist sitting outside a house would feel at least four kinds of forces acting toward him: the gravity from the earth, the pressure from the air, the thermal force from the sun, and the electric or magnetic forces from an antenna or transmission wires. The universe is made of all kinds of materials, and the effect between materials sustains the movements of the universe. Thus, there exists forces everywhere, and the space filled with forces is physically called the field. Forces change the movement of objects, so we can predict the magnitude and direction of a force by measuring the movement of objects (i.e. distance and velocity). Energy exists everywhere in the universe, and it can be embodied in matter, antimatter, or in the field. How is the field of forces generated? It is generated by the approaching or the entering of some kind of force sources. Explosions and vibrators are the force sources that produce reflective waves, and the vibration of a vibrator propagates through the interaction of molecules in the