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Introduction to Inverse Scattering Theory

Anthony J. Devaney Department of Electrical and

Computer Engineering Northeastern

University Boston, MA 02115 email

devaney_at_ece.neu.edu

- 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

Problems Addressed by Inverse Scattering and DT

Geophysical

x

x

x

x

x

x

x

x

Electromagnetic Acoustic

Off-set VSP/ cross-well tomography GPR surface

imaging induction imaging

x

x

x

x

x

Medical

Ultrasound tomography optical microscopy photon

imaging

Ultrasonic Optical

Industrial

Ultrasound tomography optical microscopy induction

imaging

Electromagnetic Ultrasonic Optical

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

Causal Fields

Wave equation becomes Helmholtz equation

Canonical Inverse Scattering Configuration

Inverse scattering problem Given set of

scattered field measurements

determine object

function

Mathematical Structure of Inverse Scattering

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

mathematics physics and signal processing

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

Rayleigh Sommerfeld Formula

Suppress frequency dependence

S

Boundary Conditions

z

Sommerfeld Radiation Condition in r.h.s.

Dirichlet or Neumann on bounding surface S

Plane surface

Angular Spectrum Expansion

Weyl Expansion

Plane Wave Expansion

Angular Spectrum Representation of Free Fields

Rayleigh Sommerfeld Formula

Propagation in Fourier Space

Homogeneous waves Evanescent waves

Free space propagation (z1gt z0) corresponds to

low pass filtering of the field

data Backpropagation (z1lt z0) requires high pass

filtering and is unstable (not well posed)

Backpropagation of Bandlimited Fields Using A.S.E.

Propagation

z

Backpropagation

z0

zmin

Boundary value of field (or of normal derivative)

on any plane zz0 ? zmin uniquely determines

field throughout half-space z ? zmin

Backpropagation Using Conjugate Green Function

Forward propagationboundary value

problem Backpropagationinverse problem

Boundary Conditions

S

S1

Incoming Wave Condition in l.h.s.

Dirichlet or Neumann on bounding surface S1

Plane surface

AJD, Inverse Problems 2, p161 (1986)

Approximation Equivalence of Two Forms of

Backpropagation

Homogeneous waves Evanescent waves

Potential Scattering Theory

Lippmann Schwinger Equation

Born Series

Lippmann Schwinger Equation

- Linear mapping between incident and scattered

field - Non-linear mapping between object profile and

scattered field

Object function

Scattering Amplitude

Non-linear functional of O

Induced Source

Linear functional of ?

Boundary value of the spatial Fourier transform

of the induced source on a sphere of radius k

(Ewald sphere)

Inverse Source Problem Estimate source Inverse

Scattering Problem Estimate object profile

Non-uniqueness--Non-radiating Sources

Inverse source problem does not possess a unique

solution Inverse 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

Born Approximation

Linear functional of O

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

Born Inverse Scattering

Ewald Spheres

k

2k

Ewald Sphere

Limiting Ewald Sphere

Using Multiple Frequencies

Back scatter data

Multiple frequencies effective for

backscatter but ineffective for forward scatter

Born Inversion for Fixed Frequency

Problem How to generate inversion from Fourier

data on spherical surfaces

Inversion Algorithms Fourier interpolation

(classical X-ray crystallography) Filtered

backpropagation (diffraction tomography)

A.J.D. Opts Letts, 7, p.111 (1982)

Filtering of data followed by backpropagation

Filtered Backpropagation Algorithm

Near Field Data

Weyl Expansion

Spherical Incident Waves

Lippmann Schwinger Equation

Double slant-stack

Frequency Domain Slant Stacking

Determine the plane wave response from the point

source response

Single slant-stack operation

Slant-stacking in Free-Space

Transform a set of spherical waves into a plane

wave

Rayleigh Sommerfeld Formula

Fourier transform w.r.t. source points

Slant-stacking Scattered Field Data

Stack w.r.t. source coordinates

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

Slant stack w.r.t. Receiver Coordinates

Slant stack w.r.t. source coordinates

Slant stack w.r.t. receiver coordinates

z

Born Inversion from Double Stacked Data

Fourier transform w.r.t source and receiver

coordinates Use Fourier interpolation or

filtered backpropagation to generate

reconstruction

Next Lecture

Diffraction tomographyRe-packaged inverse

scattering theory

Key ingredients of Diffraction

Tomography (DT) Employs improved weak scattering

model (Rytov approximation) Is more appropriate

to geophysical inverse problems Has formal

mathematical structure completely analogous to

conventional tomography (CT) Inversion algorithms

analogous to those of CT Reconstruction

algorithms also apply to the Born scattering

model of inverse scattering theory