The calculation of moment uncertainties from velocity distribution functions with random errors

Instrumentation that detects individual plasma particles is susceptible to random counting errors. These errors propagate into the calculations of moments of measured particle velocity distribution functions. Although rules of thumb exist for the effects of random errors on the calculation of lower order moments (e.g., density, velocity, and temperature) of Maxwell-Boltzmann distributions, they do not generally apply to nonthermal distributions or to higher-order moments. To date, such errors have only been estimated using brute force Monte Carlo techniques, i.e., repeated (~50) samplings of distribution functions. Here we present a mathematical formalism for analytically obtaining uncertainty estimates of plasma moments due to random errors either measured in situ by instruments or synthesized by particle simulations. Our uncertainty estimates precisely match the statistical variation of simulated plasma moments and carry the computational cost equivalent of only ~15 Monte Carlo samplings. In addition, we provide the means to calculate a covariance matrix that can be reported along with typical plasma moments. This matrix enables the propagation of statistical errors into arbitrary coordinate systems or functions of plasma moments without the need to reanalyze full distribution functions. Our methodology, which is applied to electron data from Plasma Electron and Current Experiment on the Cluster spacecraft as an example, is relevant to both existing and future data sets and requires only instrument-measured counts and phase space densities reported for a set of calibrated energy-angle targets.