August, 1999A.J. Devaney Stanford Lectures-- Lecture I 1 Introduction to Inverse Scattering Theory Anthony J. Devaney Department of Electrical and Computer Engineering Northeastern University Boston, MA 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, 1999A.J. Devaney Stanford Lectures-- Lecture I 2 Problems Addressed by Inverse Scattering and DT Geophysical x x x x x x x x x x x x xxxxxxx Medical x x x x x x Industrial Electromagnetic Acoustic Ultrasonic Optical x x x x x x Electromagnetic Ultrasonic Optical Off-set VSP/ cross-well tomography GPR surface imaging induction imaging Ultrasound tomography optical microscopy photon imaging Ultrasound tomography optical microscopy induction imaging

August, 1999A.J. Devaney Stanford Lectures-- Lecture I 3 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 becomes Helmholtz equation Causal Fields

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

August, 1999A.J. Devaney Stanford Lectures-- Lecture I 5 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, 1999A.J. Devaney Stanford Lectures-- Lecture I 6 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, 1999A.J. Devaney Stanford Lectures-- Lecture I 7 Rayleigh Sommerfeld Formula Boundary Conditions Sommerfeld Radiation Condition in r.h.s. + Dirichlet or Neumann on bounding surface S S Plane surface: z Suppress frequency dependence

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

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

August, 1999A.J. Devaney Stanford Lectures-- Lecture I 10 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, 1999A.J. Devaney Stanford Lectures-- Lecture I 11 Backpropagation of Bandlimited Fields Using A.S.E. Boundary value of field (or of normal derivative) on any plane z=z 0  z min uniquely determines field throughout half-space z  z min z z min z0z0

August, 1999A.J. Devaney Stanford Lectures-- Lecture I 12 Backpropagation Using Conjugate Green Function Forward Propagation Backpropagation Forward propagation=boundary value problem Backpropagation=inverse problem Incoming Wave Condition in l.h.s. + Dirichlet or Neumann on bounding surface S 1 S S1S1 Boundary Conditions AJD, Inverse Problems 2, p161 (1986) Plane surface:

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

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

August, 1999A.J. Devaney Stanford Lectures-- Lecture I 15 Born Series Linear mapping between incident and scattered field Non-linear mapping between object profile and scattered field Lippmann Schwinger Equation Object function Non-linear operator Scattered field data

August, 1999A.J. Devaney Stanford Lectures-- Lecture I 16 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, 1999A.J. Devaney Stanford Lectures-- Lecture I 17 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, 1999A.J. Devaney Stanford Lectures-- Lecture I 18 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, 1999A.J. Devaney Stanford Lectures-- Lecture I 19 Born Inverse Scattering Ewald Spheres Forward scatter data Back scatter data z Limiting Ewald Sphere Ewald Sphere k 2k

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

August, 1999A.J. Devaney Stanford Lectures-- Lecture I 21 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, 1999A.J. Devaney Stanford Lectures-- Lecture I 22 Near Field Data Weyl Expansion

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

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

August, 1999A.J. Devaney Stanford Lectures-- Lecture I 25 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, 1999A.J. Devaney Stanford Lectures-- Lecture I 26 Slant-stacking Scattered Field Data Stack w.r.t. source coordinates z

August, 1999A.J. Devaney Stanford Lectures-- Lecture I 27 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, 1999A.J. Devaney Stanford Lectures-- Lecture I 28 Slant stack w.r.t. Receiver Coordinates z Slant stack w.r.t. source coordinates Slant stack w.r.t. receiver coordinates z

August, 1999A.J. Devaney Stanford Lectures-- Lecture I 29 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, 1999A.J. Devaney Stanford Lectures-- Lecture I 30 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