When Can Poisson Random Variables Be Approximated as Gaussian?

In photon counting LiDAR, photon detection is generally well described as a Poisson point process as long as photon arrival rates are sufficiently low to avoid significant nonlinear effects in the detection chain (e.g., deadtime). In general, it is straightforward to employ the Poisson-negative log-likelihood as a noise model in order to obtain a maximum likelihood estimate of photon counting LiDAR data. However, data assimilation methods employing Kalman filters (KF) and some processing methods such as optimal estimation method (OEM) were originally designed for observational data with Gaussian distributed noise, thus allowing for closed-form solutions. A common assumption when applying these methods to photon counting LiDAR data is that, by the central limit theorem, the Poisson distributed photon counts follow Gaussian distributions. While this approximation is technically correct at high photon counts (assuming linear detection), the accuracy trade-off of the approximation is not always clear and is likely to depend on the particular estimation problem. In this chapter, we investigate a few simple cases and suggest areas for further research.