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Half-time Range description for the free space wave operator and the spherical means transform

Peter Kuchment, Leonid Kunyansky

Abstract

The forward problem arising in several hybrid imaging modalities can be modeled by the Cauchy problem for the free space wave equation. Solution to this problems describes propagation of a pressure wave, generated by a source supported inside unit sphere $S$. The data $g$ represent the time-dependent values of the pressure on the observation surface $S$. Finding initial pressure $f$ from the known values of $g$ consitutes the inverse problem. The latter is also frequently formulated in terms of the spherical means of $f$ with centers on~$S$. Here we consider a problem of range description of the wave operator mapping $f$ into $g$. Such a problem was considered before, with data $g$ known on time interval at least $[0,2]$ (assuming the unit speed of sound). Range conditions were also found in terms of spherical means, with radii of integration spheres lying in the range $[0,2]$. However, such data are redundant. We present necessary and sufficient conditions for function $g$ to be in the range of the wave operator, for $g$ given on a half-time interval $[0,1]$. This also implies range conditions on spherical means measured for the radii in the range $[0,1]$.

Half-time Range description for the free space wave operator and the spherical means transform

Abstract

The forward problem arising in several hybrid imaging modalities can be modeled by the Cauchy problem for the free space wave equation. Solution to this problems describes propagation of a pressure wave, generated by a source supported inside unit sphere . The data represent the time-dependent values of the pressure on the observation surface . Finding initial pressure from the known values of consitutes the inverse problem. The latter is also frequently formulated in terms of the spherical means of with centers on~. Here we consider a problem of range description of the wave operator mapping into . Such a problem was considered before, with data known on time interval at least (assuming the unit speed of sound). Range conditions were also found in terms of spherical means, with radii of integration spheres lying in the range . However, such data are redundant. We present necessary and sufficient conditions for function to be in the range of the wave operator, for given on a half-time interval . This also implies range conditions on spherical means measured for the radii in the range .

Paper Structure

This paper contains 16 sections, 10 theorems, 74 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Formulation of the problem
  3. Some preliminary constructions
  4. The Radon transform and its range
  5. Exterior Radon transform
  6. Exterior initial/boundary value problem
  7. The implicit range characterization theorem
  8. A more explicit range characterization
  9. Range conditions for the Radon transform in terms of expansion in spherical harmonics
  10. Solving the exterior problem using separation of variables
  11. Fourier transform of $b(t,\theta)$ in time
  12. Solution obtained by expansion in spherical harmonics
  13. Range theorem in terms of spherical harmonics expansion
  14. Proof of Theorem \ref{['T:series']}
  15. Remarks and conclusions
  16. ...and 1 more sections

Key Result

Theorem 1

A function $F(\omega,p)$ defined on $S\times \mathbb{(}-1,1)$ can be represented as the Radon transform of a function $f\in C_{0}^{\infty}(B)$, if and only if the following conditions are satisfied:

Figures (1)

  • Figure 1: Geometry of the exterior problem: (a) Domain $(0,1)\times B^{c}$ where the problem is solved; (b) Support of the solution $v^{(b)}(t,x)$.

Theorems & Definitions (21)

  • Theorem 1
  • Remark 2
  • Theorem 3
  • Theorem 4
  • proof
  • Remark 5
  • Theorem 6
  • proof
  • Theorem 7
  • Remark 8
  • ...and 11 more