Introduction to Inverse Scattering
Theory
Anthony J. Devaney
Department of Electrical and Computer Engineering
Northeastern University
Boston, MA 02115
email: devaney@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
August, 1999
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Problems Addressed by Inverse
Scattering and DT
Geophysical
Electromagnetic
Acoustic
Medical
Ultrasonic
Optical
Industrial
Electromagnetic
Ultrasonic
Optical
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Off-set VSP/ cross-well tomography
GPR surface imaging
induction imaging
Ultrasound tomography
optical microscopy
photon imaging
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Ultrasound tomography
optical microscopy
induction 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
Causal Fields
Wave equation becomes
Helmholtz equation
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Canonical Inverse Scattering
Configuration
Sensor system
Incident wave
Scattered wave
( 2  k 2 )  ( r ,  )  O( r ,  )  ( r ,  )
O ( r ,  )  k 2 [1  n 2 ( r ,  )]
Inverse scattering problem: Given set of scattered field measurements
determine object function
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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
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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
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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:
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Angular Spectrum Expansion
Weyl Expansion
Homogeneous waves
Evanescent waves
Plane Wave Expansion
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Angular Spectrum Representation of
Free Fields
Rayleigh Sommerfeld Formula
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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 data
Backpropagation (z1< z0) requires high pass filtering and is unstable (not well posed)
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Backpropagation of Bandlimited
Fields Using A.S.E.
z
z0
zmin
Boundary value of field (or of normal derivative) on any plane
z=z0  zmin uniquely determines field throughout half-space z  zmin
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Backpropagation Using Conjugate
Green Function
Forward propagation=boundary value problem
Backpropagation=inverse problem
Forward Propagation
S
S1
Boundary Conditions
Incoming Wave
Condition in l.h.s.
+
Dirichlet or Neumann
on bounding surface S1
Backpropagation
Plane surface:
AJD, Inverse Problems 2, p161 (1986)
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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
Lippmann Schwinger Equation
• Linear mapping between incident and scattered field
• Non-linear mapping between object profile and scattered field
Non-linear operator
Object function
Scattered field data
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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
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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
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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
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Born Inverse Scattering
Ewald Spheres
Back scatter data
Forward scatter data
k
2k
z
Ewald Sphere
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Limiting Ewald Sphere
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Using Multiple Frequencies
Forward scatter data
Back scatter data
Multiple frequencies effective for backscatter
but ineffective for forward scatter
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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
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Near Field Data
Weyl Expansion
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Spherical Incident Waves
Lippmann Schwinger Equation
Double slant-stack
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Frequency Domain Slant Stacking
Determine the plane wave response from the point source response
Single slant-stack operation
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Slant-stacking in Free-Space
Transform a set of spherical waves into a plane wave
Rayleigh Sommerfeld Formula
z
Fourier transform w.r.t. source points
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Slant-stacking Scattered Field Data
Stack w.r.t. source coordinates
z
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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
Slant stack w.r.t. source coordinates
z
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 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
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