FGDP -- TABLE OF CONTENTS
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`Fundamentals of Geophysical Data Processing
`(FGDP)
`by Jon Claerbout
`
`TABLE OF CONTENTS
`
`Preface (ps 15K)
`Transforms (ps 106K) (src 30K)
`
`• SAMPLED DATA AND Z TRANSFORMS
`• Z-TRANSFORM TO FOURIER TRANSFORM
`• THE FAST FOURIER TRANSFORM
`• Phase delay and group delay
`• Correlation and spectra
`• Hilbert transform
`
`One-sided functions (ps 306K) (src 61K)
`
`• INVERSE FILTERS
`• MINIMUM PHASE
`• FILTERS IN PARALLEL
`• POSITIVE REAL FUNCTIONS
`• NARROW-BAND FILTERS
`• ALL-PASS FILTERS
`• NOTCH FILTER AND POLE ON PEDESTAL
`• THE BILINEAR TRANSFORM
`
`Spectral factorization (ps 142K) (src 52K)
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`• ROOT METHOD
`• ROBINSON'S ENERGY DELAY THEOREM
`• THE TOEPLITZ METHOD
`• WHITTLE'S EXP-LOG METHOD
`• THE KOLMOGOROFF METHOD
`• CAUSALITY AND WAVE PROPAGATION
`
`Resolution (ps 435K) (src 65K)
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`• TIME-FREQUENCY RESOLUTION
`• TIME-STATISTICAL RESOLUTION
`• FREQUENCY-STATISTICAL RESOLUTION
`• TIME-FREQUENCY-STATISTICAL RESOLUTION
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`9/23/2015
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`FGDP -- TABLE OF CONTENTS
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`Page 2 of 3
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`• THE CENTRAL-LIMIT THEOREM
`• CONFIDENCE INTERVALS
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`Matrices and multichannel time series (ps 130K) (src 52K)
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`• REVIEW OF MATRICES
`• SYLVESTER'S MATRIX THEOREM
`• MATRIX FILTERS, SPECTRA, AND FACTORING
`• MARKOV PROCESSES
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`Data modeling by least squares (ps 164K) (src 84K)
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`• MORE EQUATIONS THAN UNKNOWNS
`• WEIGHTS AND CONSTRAINTS
`• FEWER EQUATIONS THAN UNKNOWNS
`• HOUSEHOLDER TRANSFORMATIONS AND GOLUB'S METHOD
`• CHOICE OF A MODEL NORM
`• ROBUST MODELING
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`Waveform applications of least squares (ps 105K) (src 47K)
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`• PREDICTION AND SHAPING FILTERS
`• BURG SPECTRAL ESTIMATION
`• ADAPTIVE FILTERS
`• DESIGN OF MULTICHANNEL FILTERS
`• LEVINSON RECURSION
`• CONSTRAINED FILTERS
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`Layers revealed by scattered wave filtering (ps 125K) (src 56K)
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`• REFLECTION AND TRANSMISSION COEFFICIENTS
`• ENERGY FLUX IN LAYERED MEDIA
`• GETTING THE WAVES FROM THE REFLECTION COEFFICIENTS
`• GETTING THE REFLECTION COEFFICIENTS FROM THE WAVES
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`Mathematical physics in stratified media (ps 263K) (src 117K)
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`• FROM PHYSICS TO MATHEMATICS
`• NUMERICAL MATRIZANTS
`• UP- AND DOWNGOING WAVES
`• SOURCE-RECEIVER RECIPROCITY
`• CONSERVATION PRINCIPLES AND MODE ORTHOGONALITY
`• ELASTIC WAVES
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`• About this document ...
`
`Next: About this document ... Up: Table of Contents
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`http://sepwww.stanford.edu/sep/prof/fgdp/toc_html/index.html
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`FGDP -- TABLE OF CONTENTS
`
`Page 3 of 3
`
`Stanford Exploration Project
`10/30/1997
`
`http://sepwww.stanford.edu/sep/prof/fgdp/toc_html/index.html
`
`9/23/2015
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`PGS Exhibit 2045, pg. 3
`WesternGeco v. PGS (IPR2015-00313)
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`

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`SOURCE-RECEIVER RECIPROCITY
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`Page 1 of 5
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`Next: CONSERVATION PRINCIPLES AND MODE Up: Mathematical physics in stratified
`Previous: UP- AND DOWNGOING WAVES
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`SOURCE-RECEIVER RECIPROCITY
`
`The principle of reciprocity states that a source and receiver may (under some conditions) be
`interchanged and the same waveform will be observed. This principle is often used to advantage in
`calculations and may also be used to simplify data collection. It is somewhat amazing that this
`principle applies t other earth with its complicated inhomogeneities. Intuitively, the main reason for
`validity of the reciprocal principle is that energy propagates equally well along a given ray in either
`direction. Either way, it goes at the same speed with the same attenuation. This is true for all common
`types of waves.
`
`Little more would need to be said if all waves were scalar phenomena with scalar sources and scalar
`receivers as, for example, acoustic pressure waves with explosive sources and pressure-sensitive
`receivers. The situation becomes more complicated when the sources or receivers are moving
`diaphragms, because then their orientations become important. The directional properties of the
`source and receiver are often referred to as radiation patterns. To apply the reciprocity principle it is
`necessary to regard the radiation patterns as attached to the medium, not as being attached to the
`source and receiver. Thus, when source and receiver are said to be interchanged, it is only a scalar
`magnitude which is interchanged; the radiation patterns stay fixed at the same place. These general
`ideas are made more precise in the following derivation. It will be seen that the notion of rays actually
`turns out to be irrelevant. Reciprocity also works in diffusion and potential problems.
`
`Theoretical treatments are often somewhat hard to read. They often begin by specifying that the
`differential operator along with suitable boundary conditions should constitute a self-adjoint problem.
`This means that when you reexpress the differential equations in difference form you discover that the
`matrix of coefficients is symmetric. Let us take the example of acoustic waves in one dimension.
`Newton's equation says that mass density times accelerations
`equals the negative of the pressure
` plus the external force Fx. Utilizing
`gradient
` time dependence we have
`
`which, defining F = - Fx, may be written
`
`The other important equation of acoustics says that the incompressibility K-1 multiplied by the
`pressure p plus the divergence of displacement
`equals the external (relative) volume injection V,
`that is
`
`(2)
`
`.In
`We will now combine (9-4-1) and (9-4-2) in a finite difference form with, for convenience,
`practice, one might like to use many grid points to approximate the behavior of continuous functions,
`but for the sake of illustration we only need use a few grid points. Luckily, in this case reciprocity will
`be exactly true despite the small number of grid points. We have
`
`(1)
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`SOURCE-RECEIVER RECIPROCITY
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`(3)
`
`The first and last rows of (9-4-3) requires some special comment. The quantities I0 and In are called
`impedances. If they vanish, we have zero pressure end conditions; if they are infinite, we have zero
`motion end conditions.
`
`Now with all this fuss we have gone through to obtain the matrix (9-4-3), the only thing we want from
`it is to observe that the matrix is indeed a symmetric matrix (even if and K-1 were functions of x). In
`the exercises it is shown that a symmetric matrix may also be attained in two dimensions. That the
`matrix is symmetric is partly a result of the physical nature of sound and partly a result of careful
`planning on the part of the author. To obtain the correct statement of reciprocity in other situations
`you may have to do some careful planning too. The essence of reciprocity is that since the matrix of
`(9-4-3) is symmetric then the inverse matrix will also be symmetric. Premultiplying (9-4-3) through
`by the inverse matrix we get the responses as a result of matrix multiplication on the external
`excitations.
`
`(4)
`
`The letters A, B, C, and D indicate the symmetry of the matrix of (9-4-4). Now if all external sources
`vanish except on one end where there is a unit strength volume source V0 = 1, then according to
`(9-4-4) the pressure in the middle p1 will equal A. If in a second experiment all the external sources
`vanish except the middle volume source V1 = 1, then according to (9-4-4) the pressure response p0 at
`the end will also equal A. This is the reciprocal principle. Note that with the letter D in (9-4-4) a like
`statement applies to the forces and the displacements. A mixed statement applies with the letters C
`and B.
`
`In a realistic experiment it may not be possible to have a pure volume source or a pure external force.
`In other words, the external source may have some finite, nonzero impedance. Then the first
`experiment we would perform would be with the excitation at the middle, getting for the end
`response:
`
`Interchanging source and receiver locations, we have
`
`(5)
`
`(6)
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`SOURCE-RECEIVER RECIPROCITY
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`Page 3 of 5
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`The notable feature of (9-4-5) and (9-4-6) is that the matrices are transposes of one another. This
`feature would not be lost if we were to consider a more elaborate experiment where the vectors in
`(9-4-5) and (9-4-6) contained more elements. For example, a vector in (9-4-5) or (9-4-6) could
`contain elements of an array of physically separated volume sources or pressure sensors. In fact, if the
`reader is able to frame elastic, electromagnetic, diffusion, or potential problems as symmetric
`algebraic equations like (9-4-3), then the matrices like (9-4-5) and (9-4-6) will still be transposes of
`one another. The setting up of symmetric equations like (9-4-3) is often not difficult, although it may
`get somewhat complicated in multidimensional noncartesian geometry.
`
`In such a more general case we may denote the right-hand vectors in (9-4-5) or (9-4-6) by
`to denote
`excitation and the left-hand vectors by
` to denote response. Using
` for the matrix of (9-4-5) and
` for the transposed matrix, (9-4-5) and (9-4-6) would be
`
`(7)
`(8)
`
`Now let us deduce a physical statement from (9-4-7) and (9-4-8). First take the inner product of
`(9-4-7) with
`
`The right-hand side, which is a scalar, may be transposed
`
`substituting from (9-4-8) we have
`
`(9)
`Equation (9-4-9) is the basic statement of reciprocity; the inner product of the excitation vector and
`the response vector at place 0 equals their inner product at place 1. Notice that the inner products are
`between vectors which occur in different experiments.
`
`9-2
`Figure 2 A reciprocity example. Reciprocity says that u0 = w0 + 2w1 + w2.
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`SOURCE-RECEIVER RECIPROCITY
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`An example of an elastic system with vector-directed displacement and force vectors is depicted in
`Figure 9-2. A laboratory example by J. E. White which combines electromagnetic, solid, liquid, and
`gaseous media is down in Figure 9-3. A geophone is a spring pendulum couped to an induction coil.
`The first geophone is mounted on a pipe which rests on the bottom of a glass desiccator. The second
`geophone is attached to the glass with a chunk of modeling clay, below the water level. The top pair
`of traces shows the (source) current into the first geophone and the (open circuit) voltage at the
`second; the bottom traces show the current in the second geophone and the voltage at the first.
`
`9-3
`Figure 3 An example of the reciprocal principle in a combined electromagnetic, solid, liquid, and
`gaseous system [J.E. White].
`
`EXERCISES:
`
`on five grid points where the boundary conditions are
`1. Consider Poisson's equation
`that the end points are zero. A unit excitation at the third grid point gives the solution
`.Find the solution with a unit excitation in the second grid point. Observe
`reciprocity if you do it right.
`2. Write an equation like (9-4-3) for the heat-flow equation. How will the introduction of
`imaginary numbers change the statement of the reciprocal principle?
`3. Write the three first-order partial differential equations of acoustics in two-dimensional
`cartesian geometry. Observe the gridding arrangement below.
`
`Write a set of
` equations for the vector
`it come out symmetric and in an obviously orderly form.
`
`.Make
`
`4. In Sec 8-3, Exercises 5 and 6 taken together illustrate the reciprocity theorem which states, ``If
`source and receiver are interchanged, the same waveform will be observed.'' Solve the problem
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`SOURCE-RECEIVER RECIPROCITY
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`Page 5 of 5
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`of a surface source with a receiver in the middle of the layers and solve the same problem with
`interchanged source and receiver to test the reciprocity theorems.
`
`Next: CONSERVATION PRINCIPLES AND MODE Up: Mathematical physics in stratified
`Previous: UP- AND DOWNGOING WAVES
`Stanford Exploration Project
`10/30/1997
`
`http://sepwww.stanford.edu/sep/prof/fgdp/c9/paper_html/node5.html
`
`9/23/2015
`
`PGS Exhibit 2045, pg. 8
`WesternGeco v. PGS (IPR2015-00313)