The noise distribution in a shear wave speed image computed using arrival times at fixed spatial positions

In a number of applications, for example biomechanical imaging of tissue and geophysics, the goal is to recover wave speed from arrival times, the times when a wave front, initiated at a source, arrives at certain points in space. The mathematical model that relates the arrival times to the wave speed is assumed to be the Eikonal equation. Typically the measured arrival time data is noisy so that in turn the recovered wave speed is noisy. Here we assume the noise in the measured arrival times is Gaussian and show that then the recovered wave speed has noise that has an inverse Rician distribution. This latter distribution has infinite variance. Nevertheless, we show using the Kantorovich metric that for small variances in the Gaussian noise (corresponding to a small value of the shape parameter in the Rician random variable), the inverse Rician distribution can be approximated by a Gaussian distribution whose mean is the true wave speed.