2024/10/19 by Peter Kuchment, Kuchment, Peter, Leonid Kunyansky +1
Earth and Planetary Sciences · Physics and Astronomy · #Analysis of PDEs (math.AP) #Electromagnetic Scattering and Analysis #FOS: Mathematics #Seismic Imaging and Inversion Techniques #Underwater Acoustics Research
paper · pdf · doi:10.48550/arxiv.2410.14999
openalex publication_date 2024/10/19 · openalex created_date 2024/11/06 · openalex updated_date 2026/07/28
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].