Fixed point algorithms for detection of parabolic events

In this paper we show how to convert the problem of estimating delay, slope and curvature of a parabolic event into a frequency estimation problem. Two dimensional data (time and offset) is converted into samples on a two-dimensional manifold embedded in a three-dimensional spaced. To conduct frequency estimation on this manifold we design general domain Hankel matrices and make use of a fixed point algorithm that is designed to find minima of convex envelopes of functionals using a combination of a rank penalty and a misfit function, under the constraint of a certain matrix structure. We illustrate that the proposed method can successfully detect the parameters of the parabolic events also in the case of unequally spaced spatial sampling and in the presence of rather high levels of noise.