August, 1999A.J. Devaney Stanford Lectures-- Lecture I 1 Introduction to Inverse Scattering Theory...
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Transcript of August, 1999A.J. Devaney Stanford Lectures-- Lecture I 1 Introduction to Inverse Scattering Theory...
August, 1999 A.J. Devaney Stanford Lectures--Lecture I
1
Introduction to Inverse Scattering Theory
Anthony J. DevaneyDepartment of Electrical and Computer Engineering
Northeastern UniversityBoston, MA 02115
email: [email protected]
• Examples of inverse scattering problems• Free space propagation and backpropagation• Elementary potential scattering theory
• Lippmann Schwinger integral equation• Born series• Born approximation
• Born inversion from plane wave scattering data• far field data• near field data
• Born inversion from spherical wave scattering data• Slant stack w.r.t . source and receiver coordinates
August, 1999 A.J. Devaney Stanford Lectures--Lecture I
2
Problems Addressed by Inverse Scattering and DT
Geophysical
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x x x x x x x
Medical
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Industrial
ElectromagneticAcoustic
UltrasonicOptical
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ElectromagneticUltrasonic
Optical
Off-set VSP/ cross-well tomographyGPR surface imaging
induction imaging
Ultrasound tomographyoptical microscopy
photon imaging
Ultrasound tomographyoptical microscopyinduction imaging
August, 1999 A.J. Devaney Stanford Lectures--Lecture I
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Time-dependent Fields
Work entirely in frequency domain• Allows the theory to be applied to dispersive media problems
• Is ideally suited to incorporating LTI filters to scattered field data• Many applications employ narrow band sources
Wave equation becomesHelmholtz equation
Causal Fields
August, 1999 A.J. Devaney Stanford Lectures--Lecture I
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Canonical Inverse Scattering Configuration
Incident wave
Scattered wave
Sensor system
( ) ( , ) ( , ) ( , )
( , ) [ ( , )]
2 2
2 21
k O
O k n
r r r
r r
Inverse scattering problem: Given set of scattered field measurements determine object function
August, 1999 A.J. Devaney Stanford Lectures--Lecture I
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Mathematical Structure of Inverse Scattering
j=Zdrj±LO=d
Non-linear operator (Lippmann Schwinger equation) Object function
Scattered field data
Use physics to derive model and linearize mapping
Linear operator (Born approximation)
Form normal equations for least squares solution
Wavefield Backpropagation
Compute pseudo-inverse
Filtered backpropagation algorithm
Successful procedure require coupling of mathematicsphysics and signal processing
August, 1999 A.J. Devaney Stanford Lectures--Lecture I
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Ingredients of Inverse Scattering Theory
• Forward propagation (solution of boundary value problems)• Inverse propagation (computing boundary value from field measurements)• Devising workable scattering models for the inverse problem• Generating inversion algorithms for approximate scattering models• Test and evaluation
• Free space propagation and backpropagation• Elementary potential scattering theory
• Lippmann Schwinger integral equation• Born series• Born approximation
• Born inversion from plane wave scattering data• far field data• near field data
• Born inversion from spherical wave scattering data• Slant stack w.r.t . source and receiver coordinates
August, 1999 A.J. Devaney Stanford Lectures--Lecture I
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Rayleigh Sommerfeld Formula
=eikso¢r
Boundary Conditions
Sommerfeld Radiation Condition in r.h.s.
+Dirichlet or Neumannon bounding surface S
S
Plane surface:
z
Suppress frequencydependence
August, 1999 A.J. Devaney Stanford Lectures--Lecture I
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Angular Spectrum Expansion
Homogeneous waves
Evanescent waves
Weyl Expansion
Plane Wave Expansion
August, 1999 A.J. Devaney Stanford Lectures--Lecture I
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Angular Spectrum Representation of Free Fields
Rayleigh Sommerfeld Formula
August, 1999 A.J. Devaney Stanford Lectures--Lecture I
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Propagation in Fourier Space
Homogeneous waves
Evanescent waves
Free space propagation (z1> z0) corresponds to low pass filtering of the field dataBackpropagation (z1< z0) requires high pass filtering and is unstable (not well posed)
August, 1999 A.J. Devaney Stanford Lectures--Lecture I
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Backpropagation of Bandlimited Fields Using A.S.E.
Boundary value of field (or of normal derivative) on any planez=z0 zmin uniquely determines field throughout half-space z zmin
z
zmin
z0
August, 1999 A.J. Devaney Stanford Lectures--Lecture I
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Backpropagation Using Conjugate Green Function
Forward Propagation
Backpropagation
Forward propagation=boundary value problemBackpropagation=inverse problem
Incoming Wave Condition in l.h.s.
+Dirichlet or Neumann
on bounding surface S1
S S1
Boundary Conditions
AJD, Inverse Problems 2, p161 (1986)
Plane surface:
August, 1999 A.J. Devaney Stanford Lectures--Lecture I
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Approximation Equivalence of Two Forms of Backpropagation
Homogeneous waves
Evanescent waves
August, 1999 A.J. Devaney Stanford Lectures--Lecture I
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Potential Scattering Theory
Lippmann Schwinger Equation
August, 1999 A.J. Devaney Stanford Lectures--Lecture I
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Born Series
• Linear mapping between incident and scattered field• Non-linear mapping between object profile and scattered field
Lippmann Schwinger Equation
Object functionNon-linear operator
Scattered field data
August, 1999 A.J. Devaney Stanford Lectures--Lecture I
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Scattering Amplitude
Boundary value of the spatial Fourier transform of the induced source on a sphere of radius k (Ewald sphere)
Induced Source
Inverse Source Problem: Estimate source Inverse Scattering Problem: Estimate object profile
Non-linear functional of O
Linear functional of
August, 1999 A.J. Devaney Stanford Lectures--Lecture I
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Non-uniqueness--Non-radiating Sources
Inverse source problem does not possess a unique solutionInverse scattering problem for a single experiment does not possess a unique solution
Use multiple experiments to exclude NR sources
Difficulty: Each induced source depends on the (unknown)internal field--non-linear character of problem
August, 1999 A.J. Devaney Stanford Lectures--Lecture I
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Born Approximation
Boundary value of the spatial Fourier transform of the object function on a set of spheres of radius k (Ewald spheres)
Generalized Projection-Slice Theorem in DT
Linear functional of O
August, 1999 A.J. Devaney Stanford Lectures--Lecture I
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Born Inverse Scattering
Ewald Spheres
Forward scatter dataBack scatter data
z
Limiting Ewald SphereEwald Sphere
k2k
August, 1999 A.J. Devaney Stanford Lectures--Lecture I
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Using Multiple Frequencies
Back scatter dataForward scatter data
Multiple frequencies effective for backscatterbut ineffective for forward scatter
August, 1999 A.J. Devaney Stanford Lectures--Lecture I
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Born Inversion for Fixed Frequency
Inversion Algorithms: Fourier interpolation (classical X-ray crystallography)
Filtered backpropagation (diffraction tomography)
Problem: How to generate inversion from Fourier data on spherical surfaces
A.J.D. Opts Letts, 7, p.111 (1982)
Filtering of data followed by backpropagation: Filtered Backpropagation Algorithm
August, 1999 A.J. Devaney Stanford Lectures--Lecture I
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Near Field Data
Weyl Expansion
August, 1999 A.J. Devaney Stanford Lectures--Lecture I
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Spherical Incident Waves
Lippmann Schwinger Equation
Double slant-stack
August, 1999 A.J. Devaney Stanford Lectures--Lecture I
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Determine the plane wave response from the point source response
Single slant-stack operation
Frequency Domain Slant Stacking
August, 1999 A.J. Devaney Stanford Lectures--Lecture I
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Slant-stacking in Free-Space
Rayleigh Sommerfeld Formula
Transform a set of spherical waves into a plane wave
Fourier transform w.r.t. source points
z
August, 1999 A.J. Devaney Stanford Lectures--Lecture I
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Slant-stacking Scattered Field Data
Stack w.r.t. source coordinatesz
August, 1999 A.J. Devaney Stanford Lectures--Lecture I
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Born Inversion from Stacked Data
Use either far field data (scattering amplitude) or near field data
Far field data:
Near field data:
Near field data generated using double slant stack
August, 1999 A.J. Devaney Stanford Lectures--Lecture I
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Slant stack w.r.t. Receiver Coordinates
z
Slant stack w.r.t. source coordinates
Slant stack w.r.t. receiver coordinates
z
August, 1999 A.J. Devaney Stanford Lectures--Lecture I
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Born Inversion from Double Stacked Data
Fourier transform w.r.t source and receiver coordinates Use Fourier interpolation or filtered backpropagation to generate reconstruction
August, 1999 A.J. Devaney Stanford Lectures--Lecture I
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Next Lecture
Diffraction tomography=Re-packaged inverse scattering theory
Key ingredients of Diffraction Tomography (DT)Employs improved weak scattering model (Rytov approximation)
Is more appropriate to geophysical inverse problemsHas formal mathematical structure completely analogous to conventionaltomography (CT)
Inversion algorithms analogous to those of CTReconstruction algorithms also apply to the Born scattering model ofinverse scattering theory