Introduction to Inverse Scattering Theory Anthony J Devaney

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 A. J. Devaney Stanford Lectures-Lecture I 1

Problems Addressed by Inverse Scattering and DT Geophysical Electromagnetic Acoustic Medical Ultrasonic Optical Industrial Electromagnetic Ultrasonic Optical August, 1999 x x x x x x Off-set VSP/ cross-well tomography GPR surface imaging induction imaging Ultrasound tomography optical microscopy photon imaging x x x Ultrasound tomography optical microscopy induction imaging x A. J. Devaney Stanford Lectures-Lecture I 2

Time-dependent Fields Work entirely in frequency domain • Allows 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 August, 1999 A. J. Devaney Stanford Lectures-Lecture I 3

Canonical Inverse Scattering Configuration Sensor system Incident wave Scattered wave Inverse scattering problem: Given set of scattered field measurements determine object function August, 1999 A. J. Devaney Stanford Lectures-Lecture I 4

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 August, 1999 A. J. Devaney Stanford Lectures-Lecture I 5

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 6

Rayleigh Sommerfeld Formula Suppress frequency dependence S z Boundary Conditions Sommerfeld Radiation Condition in r. h. s. + Dirichlet or Neumann on bounding surface S Plane surface: August, 1999 A. J. Devaney Stanford Lectures-Lecture I 7

Angular Spectrum Expansion Weyl Expansion Homogeneous waves Evanescent waves Plane Wave Expansion August, 1999 A. J. Devaney Stanford Lectures-Lecture I 8

Angular Spectrum Representation of Free Fields Rayleigh Sommerfeld Formula August, 1999 A. J. Devaney Stanford Lectures-Lecture I 9

Propagation in Fourier Space Homogeneous waves Evanescent waves Free space propagation (z 1> z 0) corresponds to low pass filtering of the field data Backpropagation (z 1< z 0) requires high pass filtering and is unstable (not well posed) August, 1999 A. J. Devaney Stanford Lectures-Lecture I 10

Backpropagation of Bandlimited Fields Using A. S. E. z z 0 zmin Boundary value of field (or of normal derivative) on any plane z=z 0 zmin uniquely determines field throughout half-space z zmin August, 1999 A. J. Devaney Stanford Lectures-Lecture I 11

Backpropagation Using Conjugate Green Function Forward propagation=boundary value problem Backpropagation=inverse problem Forward Propagation S S 1 Boundary Conditions Incoming Wave Condition in l. h. s. + Dirichlet or Neumann on bounding surface S 1 Backpropagation Plane surface: AJD, Inverse Problems 2, p 161 (1986) August, 1999 A. J. Devaney Stanford Lectures-Lecture I 12

Approximation Equivalence of Two Forms of Backpropagation Homogeneous waves Evanescent waves August, 1999 A. J. Devaney Stanford Lectures-Lecture I 13

Potential Scattering Theory Lippmann Schwinger Equation August, 1999 A. J. Devaney Stanford Lectures-Lecture I 14

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 August, 1999 A. J. Devaney Stanford Lectures-Lecture I 15

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 August, 1999 A. J. Devaney Stanford Lectures-Lecture I 16

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 August, 1999 A. J. Devaney Stanford Lectures-Lecture I 17

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 August, 1999 A. J. Devaney Stanford Lectures-Lecture I 18

Born Inverse Scattering Ewald Spheres Back scatter data Forward scatter data k z Ewald Sphere August, 1999 A. J. Devaney Stanford Lectures-Lecture I 2 k Limiting Ewald Sphere 19

Using Multiple Frequencies Forward scatter data Back scatter data Multiple frequencies effective for backscatter but ineffective forward scatter August, 1999 A. J. Devaney Stanford Lectures-Lecture I 20

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 August, 1999 A. J. Devaney Stanford Lectures-Lecture I 21

Near Field Data Weyl Expansion August, 1999 A. J. Devaney Stanford Lectures-Lecture I 22

Spherical Incident Waves Lippmann Schwinger Equation Double slant-stack August, 1999 A. J. Devaney Stanford Lectures-Lecture I 23

Frequency Domain Slant Stacking Determine the plane wave response from the point source response Single slant-stack operation August, 1999 A. J. Devaney Stanford Lectures-Lecture I 24

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 August, 1999 A. J. Devaney Stanford Lectures-Lecture I 25

Slant-stacking Scattered Field Data Stack w. r. t. source coordinates z August, 1999 A. J. Devaney Stanford Lectures-Lecture I 26

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 27

Slant stack w. r. t. Receiver Coordinates Slant stack w. r. t. source coordinates z Slant stack w. r. t. receiver coordinates z August, 1999 A. J. Devaney Stanford Lectures-Lecture I 28

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 29

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 August, 1999 A. J. Devaney Stanford Lectures-Lecture I 30