Fig. 5

Probability density function of the simulated trace of the power spectral density matrix obtained from a windowed noise waveform. (a) for n_l = 2 averaged spectral estimates; (b) for n_l = 6; (c) for n_l = 32; (c) for n_l = 256, relevant for the Van Allen probes EMFISIS Survey data. Red line: probability density estimated by a histogram from a Monte Carlo simulation of the sensor noise. Blue line: scaled \({\chi }^{2}\) distribution core model from the first derivative of the cumulative probability distribution according to Eq. 4. The correction factors \({C}_{c\nu }\) and \({C}_{{cs}}\) for this model are determined by a nonlinear fit to the simulated histogram. The model is valid for noise power spectral densities, for which the cumulative distribution function is below 0.99, shown by the blue vertical dashed line. At larger power spectral densities, this core model significantly deviates from the simulation results. Black line: scaled \({\chi }^{2}\) distribution tail model according to Eq. 5, valid for noise power spectral densities, for which the cumulative distribution function is above 0.99. Black vertical dashed line: power spectral density for which the count in the simulated histogram decreases below 100, and random errors of the simulation start to be larger than 10%. The two vertical dashed lines also show the interval for the nonlinear fit of the correction factors \({C}_{t\nu }\) and \({C}_{{ts}}\) for the tail distribution model to the logarithm of the simulated histogram. Vertical black dotted line: detection threshold \({S}_{0}\) (Eq. 6) for the trace of the power spectral density matrix to reach above \({S}_{0}\) with a probability P₀ = 10⁻⁷, according to the tail model from Eq. 5.