On double integrals over spheres

Formulae involving double integrals over spheres arise naturally in inverse scattering problems since the scattered data are measured in the space R x Sz x Sz. In this paper we derive a relation between differential forms on the space S"' x S"-' , and those on the space Z x S n-2 x S where Z is a real interval. Specifically, d( dq = sin"' 6 d6 d$ dv (t, 7) E S '-' x S '-' and (6, $, U) E Z x S"-2 x S"-'. This allows us to derive the results of John relating the iterated spherical mean of a function to its spherical mean in a simple way; to obtain new inversion formulae for the Fourier and Radon transforms; to extend formulae for linearised inverse quantum scattering and diffraction tomography to the multifrequency case; and also to establish a relation between multifrequency diffraction tomography and seismic migration algorithms.