Mathematics of Multidimensional Seismic Imaging, Migration, and Inversion

by Bleistein, Norman; Cohen, Jack K.; Stockwell, J. W., Jr.

• ISBN13: 9780387950617
• ISBN10: 0387950613
• Format: Hardcover
• Copyright: 2001-01-01
• Publisher: Springer Verlag

Summary

In the last 40 years geophysicists have found that it is possible to construct images and even determine important physical characteristics of rocks that can yield information about oil and gas bearing structures in the earth. To make these images and extract this information requires the application of an advanced understanding of the mathematical physics of wave propagation. The oil and gas industry labels a major collection of the necessary seismic data processing methods by the name seismic "migration". This text ist the first to treat many kinds of migration in a unified mahtematical way. The audience is mathematically oriented geophysicists or applied mathematicians working in the field of "inverse scattering imaging". The text can serve as a bridge between the applied math and geophysics community by presenting geophysicists with a practical introduction to advanced engineering mathematics, while presenting mathematicians with a window into the world of the mathematically sophistiated geophysicist.

Table of Contents

Preface	vii
List of Figures	xxiii
Multidimensional Seismic Inversion	1	(23)
Inverse Problems and Imaging	2	(2)
The Nonlinearity of the Seismic Inverse Problem	4	(1)
High Frequency	5	(2)
Migration Versus Inversion	7	(5)
Source-Receiver Configurations	12	(6)
Band and Aperture Limiting of Data	18	(2)
Dimensions: 2D Versus 2.5D Versus 3D	20	(1)
Acoustic Versus Elastic Inversion	20	(2)
A Mathematical Perspective on the Geometry of Migration	22	(2)
The One-Dimensional Inverse Problem	24	(64)
Problem Formulation in One Spatial Dimension	25	(3)
The 1D Model in a Geophysical Context	25	(2)
The 1D Model as a Mathematical Testground	27	(1)
Mathematical Tools for Forward Modeling	28	(7)
The Governing Equation and Radiation Condition	28	(1)
Fourier Transform Conventions	29	(3)
Green's Functions	32	(1)
Green's Theorem	33	(2)
The Forward Scattering Problem	35	(7)
The Forward Scattering Problem in ID	36	(4)
The Born Approximation and Its Consequences	40	(1)
The Inverse Scattering Integral Equation	41	(1)
Constant-Background, Zero-Offset Inversion	42	(18)
Constant-Background, Single Layer	43	(13)
More Layers, Accumulated Error	56	(2)
A Numerical Example	58	(1)
Summary	59	(1)
Inversion in a Variable-Background Medium	60	(17)
Modern Mathematical Issues	64	(2)
Summary	66	(4)
Implementation of the Variable-Wavespeed Theory	70	(6)
Summary	76	(1)
Reevaluation of the Small-Perturbation Assumption	77	(1)
Computer Implementation	78	(3)
Sampling	79	(2)
Variable Density	81	(7)
Inversion in Higher Dimensions	88	(73)
The Scattering Problem in Unbounded Media	90	(4)
The Born Approximation	94	(12)
The Born Approximation and High Frequency	99	(4)
The Constant-Background Zero-Offset Equation	103	(1)
One Experiment, One Degree of freedom in α	103	(3)
Zero-Offset Constant-Background Inversion in 3D	106	(7)
Restrictions on the Choice of k3	111	(2)
High Frequency, Again	113	(10)
Reflection from a Single Tilted Plane	117	(2)
The Reflectivity Function	119	(2)
Alternative Representations of the Reflectivity Function	121	(2)
Two-and-One-Half Dimensions	123	(4)
Zero-Offset Two-and-One-Half Dimensional Inversion	125	(2)
Kirchhoff Inversion	127	(17)
Stationary Phase Computations	127	(9)
Two-and-One-Half-Dimensional Kirchhoff Inversion	136	(2)
2D Modeling and Inversion	138	(6)
Testing the Inversion Formula with Kirchhoff Data	144	(12)
The Kirchhoff Approximation	144	(2)
Asymptotic Inversion of Kirchhoff Data	146	(6)
Summary	152	(4)
Reverse-Time Wave-Equation Migration Deduced from the Kirchhoff Approximation	156	(5)
Large-Wavenumber Fourier Imaging	161	(55)
The Concept of Aperture	163	(1)
The Relationship Between Aperture and Survey Parameters	164	(12)
Rays, Fourier Transforms, and Apertures	165	(1)
Aperture and Migration Dip	166	(2)
Migration Dip and Apertures	168	(8)
Summary	176	(1)
Examples of Aperture-Limited Fourier Inversion	176	(20)
Aperture Limited Inversion of a Diary Delta Function (A Point Scatterer)	177	(2)
Aperture-Limited Inversion of a Singular Function (a Reflecting Plane)	179	(5)
Generalization to Singular Functions of Other Types of Surfaces-Asymptotic Evaluation	184	(5)
Relevance to Inverse Scattering	189	(1)
Aperture-Limited Fourier Inversion of Smoother Functions	189	(1)
Aperture-Limited Fourier Inversion of Steplike Functions	189	(3)
Aperture-Limited Fourier Inversion of a Ramplike Function	192	(2)
Aperture-Limited Inversion of an Infinitely Differentiable Function	194	(2)
Summary	196	(1)
Aperture-Limited Fourier Identity Operators	196	(20)
The Significance of the Boundary Values in Dy	198	(3)
Stationary Phase Analysis for I0	201	(7)
The Near-Surface Condition	208	(1)
Extracting Information About f on Sy	208	(1)
Processing for a Scaled Singular Function of the Boundary Surface Sy	209	(2)
The Normal Direction	211	(1)
Integrands with Other Types of Singularities	212	(2)
Summary	214	(1)
Modern Mathematical Issues	215	(1)
Inversion in Heterogeneous Media	216	(66)
Asymptotic Inversion of the Born-Approximate Integral Equation--General Results	217	(25)
Recording Geometries	217	(3)
Formulation of the 3D, Variable-Background, Inverse-Scattering Problem	220	(5)
Inversion for a Reflectivity Function	225	(2)
Summary of Asymptotic Verification	227	(1)
Inversion in Two Dimension	227	(5)
General Inversion Results, Stationary Triples, and cosθs	232	(6)
An Alternative Derivation: Removing the Small-Perturbation Restriction at the Reflector	238	(3)
Discussion	241	(1)
The Beylkin Determinant h, and Special Cases of 3D Inversion	242	(8)
General Properties of the Beylkin Determinant	243	(2)
Common-Shot Inversion	245	(2)
Common-Offset Inversion	247	(2)
Zero-Offset Inversion	249	(1)
Beylkin determinants and Ray Jacobians in the Common-Shot and Common-Receiver Configurations	250	(7)
Asymptotic Inversion of Kirchhoff Data for a Single Reflector	257	(14)
Stationary Phase Analysis of the Inversion of Kirchhoff Data	258	(5)
Determination of cosθs and c+	263	(2)
Finding Stationary Points	265	(3)
Determination of the Matrix signature	268	(1)
The Quotient h/|det[Φξσ]|1/2	269	(2)
Verification Based on the Fourier Imaging Principle	271	(5)
Variable Density	276	(3)
Variable-Density Reflectivity Inversion Formulas	277	(1)
The Meaning of the Variable-Density Reflectivity Formulas	278	(1)
Discussion of Results and Limitations	279	(3)
Summary	281	(1)
Two-and-One-Half-Dimensional Inversion	282	(29)
2.5D Ray Theory and Modeling	283	(7)
Two-and-One-Half-Dimensional Ray Theory	283	(7)
2.5D Inversion and Ray Theory	290	(7)
The 2.5D Beylkin Determinant	292	(1)
The General 2.5D Inversion Formulas for Reflectivity	293	(4)
The Beylkin Determinant H and Special Cases of 2.5D Inversion	297	(14)
General Properties of the Beylkin Determinant	297	(2)
Common-Shot Inversion	299	(1)
A Numerical Example-Extraction of Reflectivity from a Common-Shot Inversion	299	(2)
Constant-Background Propagation Speed	301	(1)
Vertical Seismic Profiling	302	(2)
Well-to-Well Inversion	304	(1)
Invert for What?	304	(1)
Common-Offset Inversion	305	(2)
A Numerical Examples--Extraction of the Reflection Coefficient and cosθs from a Common-Offset Inversion	307	(1)
A Numerical Example-Imaging a Syncline with Common-offset Inversion	307	(2)
Constant Background Inversion	309	(1)
Zero-Offset Inversion	309	(2)
The General Theory of Data Mapping	311	(78)
Introduction to Data Mapping	312	(7)
Kirchhoff Data Mapping (KDM)	314	(1)
Amplitude Preservation	315	(1)
A Rough Sketch of the Formulation of the KDM Platform	315	(2)
Possible Kirchhoff Data Mappings	317	(2)
Derivation of a 3D Kirchhoff Data Mapping Formula	319	(9)
Spatial Structure of the KDM Operator	322	(1)
Frequency Structure of the Operator and Asymptotic Preliminaries	323	(4)
Determination of Incidence Angle	327	(1)
2.5D Kirchhoff Data Mapping	328	(3)
Determination of Incidence Angle	330	(1)
Application of KDM to Kirchhoff Data in 2.5D	331	(16)
Asymptotic Analysis of 2.5D KDM	339	(2)
Stationary Phase Analysis in &gama;	341	(3)
Validity of the Stationary Phase Analysis	344	(3)
Common-Shot Downward Continuation of Receivers (or Sources)	347	(7)
Time-Domain Data Mapping for Other Implementations	349	(2)
Stationary Phase in tI	351	(3)
2.5D Transformation to Zero-Offset (TZO)	354	(20)
TZO in the Frequency Domain	355	(6)
A Hale-Type TZO	361	(2)
Gardner/Forel-Type TZO	363	(2)
On the Simplification of the Second Derivatives of the Phase	365	(9)
3D Data Mapping	374	(13)
Stationary Phase in &gama;	375	(2)
Discussion of the Second Derivatives of the Phase	377	(3)
3D Constant-Background TZO	380	(1)
The &gama;2 Integral As a Bandlimited Delta Function	381	(3)
Space/Frequency TZO in Constant Background	384	(2)
A Hale-Type 3D TZO	386	(1)
Summary and Conclusions	387	(2)
Distribution Theory	389	(20)
Introduction	389	(1)
Localization via Dirac Delta functions	390	(7)
Fourier Transforms of Distributions	397	(2)
Rapidly Decreasing Functions	399	(1)
Temperate Distributions	400	(1)
The Support of Distributions	401	(1)
Step Functions	402	(3)
Hilbert Transforms	404	(1)
Bandlimited Distributions	405	(4)
The Fourier Transform of Causal Functions	409	(9)
Introduction	409	(6)
Example: the 1D Free-Space Green's Function	415	(3)
Dimensional Versus Dimensionless Variables	418	(12)
The Wave Equation	419	(3)
Mathematical Dimensional Analysis	419	(2)
Physical Dimensional Analysis	421	(1)
The Helmholtz Equation	422	(3)
Inversion Formulas	425	(5)
An Example of Ill-Posedness	430	(5)
Ill-posedness in Inversion	431	(4)
An Elementary Introduction to Ray Theory and the Kirchhoff Approximation	435	(54)
The Eikonal and Transport Equations	436	(2)
Solving the Eikonal Equations by the Method of Characteristics	438	(8)
Characteristic Equations for the Eikonal Equation	442	(1)
Choosing λ = 1/2: σ as the Running Parameter	443	(1)
Choosing λ = c2/2: τ Traveltime, as the Running Parameter	444	(1)
Choosing λ = c(x)/2: s, Arclength, as the Running Parameter	444	(2)
Ray Amplitude Theory	446	(7)
The ODE For of the Transport Equation	448	(1)
Differentiation of a Determinant	449	(3)
Verification of (E.3.12)	452	(1)
Higher-Order Transport Equations	453	(1)
Determining Initial Data for the Ray Equations	453	(10)
Initial Data for the 3D Green's Function	454	(3)
Initial Data for the 2D Green's Function	457	(2)
Initial Data for Reflected and Transmitted Rays	459	(4)
2.5D Ray Theory	463	(4)
2.5D Ray Equations	464	(1)
2.5D Amplitudes	465	(1)
The 2.5D Transport Equation	465	(2)
Raytracing in Variable-Density Media	467	(3)
Ray Amplitude Theory in Variable-Density Media	468	(1)
Reflected and Transmitted Rays in Variable Density Media	469	(1)
Dynamic Raytracing	470	(6)
A Simple Example, Retracing in Constant-Wavespeed Media	473	(1)
Dynamic Retracing in σ	474	(1)
Dynamic Retracing in τ	475	(1)
Two Dimensions	475	(1)
Conclusions	475	(1)
The Kirchhoff Approximation	476	(13)
Problem Formulation	478	(1)
Green's Theorem and the Wavefield Representation	479	(4)
The Kirchhoff Approximation	483	(3)
2.5D	486	(1)
Summary	487	(2)
References	489	(10)
Author Index	499	(4)
Subject Index	503