Communications in Mathematical Sciences

Volume 16 (2018)

Number 4

Recovery of the sound speed for the acoustic wave equation from phaseless measurements

Pages: 1017 – 1041



Joonas Ilmavirta (Department of Mathematics and Statistics, University of Jyväskylä, Finland)

Alden Waters (Johann Bernoulli Instituut Rijksuniversiteit Groningen, University of Groningen, The Netherlands)


We recover the higher order terms for the acoustic wave equation from measurements of the modulus of the solution. The recovery of these coefficients is reduced to a question of stability for inverting a Hamiltonian flow transform, not the geodesic X-ray transform encountered in other inverse boundary problems like the determination of covector fields for the wave equation. Under some geometric assumptions, we reduce this to a question of boundary rigidity, which allows recovery of the sound speed for the acoustic wave equation. Previous techniques do not measure the full amplitude of the outgoing scattered wave, which is the main novelty in our approach.


phaseless measurements, Helmholtz equation, acoustic wave equation, Gaussian beams, inverse problems, integral geometry

2010 Mathematics Subject Classification

35C07, 35Q99, 35R30, 53C65, 58J37

J.I. was partially supported by an ERC Starting Grant (grant agreement no 307023). A.W. acknowledges support by EPSRC grant EP/L01937X/1 and ERC Advanced Grant MULTIMOD 26718.

Received 23 August 2016

Accepted 17 March 2018

Published 31 October 2018