August, 1999A.J. Devaney Stanford Lectures-- Lecture I 1 Introduction to Inverse Scattering Theory Anthony J. Devaney Department of Electrical and Computer

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


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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


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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


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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


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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


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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


7

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


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Angular Spectrum Expansion

Homogeneous waves

Evanescent waves

Weyl Expansion

Plane Wave Expansion


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Angular Spectrum Representation of Free Fields



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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)


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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


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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:


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Approximation Equivalence of Two Forms of Backpropagation

Homogeneous waves

Evanescent waves


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Potential Scattering Theory

Lippmann Schwinger Equation


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Born Series

• Linear mapping between incident and scattered field• Non-linear mapping between object profile and scattered field


Object functionNon-linear operator

Scattered field data


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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


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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


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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


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Born Inverse Scattering

Ewald Spheres

Forward scatter dataBack scatter data

z

Limiting Ewald SphereEwald Sphere

k2k


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Using Multiple Frequencies

Back scatter dataForward scatter data

Multiple frequencies effective for backscatterbut ineffective for forward scatter


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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


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Near Field Data

Weyl Expansion


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Spherical Incident Waves


Double slant-stack


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Determine the plane wave response from the point source response

Single slant-stack operation

Frequency Domain Slant Stacking


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Slant-stacking in Free-Space


Transform a set of spherical waves into a plane wave

Fourier transform w.r.t. source points

z


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Slant-stacking Scattered Field Data

Stack w.r.t. source coordinatesz


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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


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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


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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


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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

Documents

August, 1999A.J. Devaney Stanford Lectures-- Lecture I 1 Introduction to Inverse Scattering Theory Anthony J. Devaney Department of Electrical and Computer