Reconstruction of a Function From Parabolic Integrals With Random Exponential Attenuation

This paper considers the reconstruction of a deterministic function in a strip from independent repeated measurements on a family of parabolas. The exponential attenuation coefficient follows a known two-point distribution, and its realizations are unobserved. The averaged problem is reduced to a Volterra equation of the second kind; a representation of the solution in terms of the resolvent is obtained, and its uniqueness is proved. The exact covariance structure of the averaging error and the mean-square convergence of Tikhonov regularization are established. An estimate of the systematic error arising when the random coefficient is replaced by its mathematical expectation is derived. The theoretical relations are verified using a synthetic discrete model.