Reconstruction of the initial state from the data measured on a sphere for plasma-acoustic wave equations

Abstract

We establish a reconstruction formula for the initial information of a density (pressure) from the time records of data measured on a sphere for a certain class of wave-type equations. We first derive a reconstruction formula of integral form, and by applying the theory of Fourier Bessel series for this, we also obtain a formula of discrete version, which is more robust and numerically advantageous in the presence of noises. The reconstruction formulas we propose in this work are explicit and applicable to a wider class of wave-type equations including the plasma-acoustic wave equations.

Orthogonal function series formulas for inversion of the spherical Radon transform

“…An inversion formula for the spherical Radon transform Mf (θ, r)| (θ,r)∈S n−1 ×[0,∞) was studied in many works [13-15, 18, 19, 27, 28, 36, 44]. (Also, the inverting problem to recover photoacoustic source or the initial function from the acoustic pressure satisfying the wave equation with the constant coefficient restricted to the sphere was studied in [8,29,37,45,46].) Especially in [36], Norton found the relation between the Fourier coefficients of the given function f and the spherical Radon transform Mf for n = 2.…”