A comparison of low‐frequency radio noise amplitude probability distribution models

Abstract

Abstract. One of the most commonly modeled statistics in atmospheric radio noise studies is the noise envelope voltage amplitude probability distribution (APD). Although a number of models have been introduced to characterize atmospheric noise envelope APDs, the quantity of real data that exist to verify their accuracy is somewhat limited, especially in the ELF and VLF bands. This paper presents the results of a statistical analysis in which thousands of hours of ELF/VLF noise are processed to derive APDs, which are then compared with various APD models to determine which of the models is most accurate. The error criterion used to find the optimal parameters of each APD model, as well as to compare the models against each other, is the expected value of the log error squared (where the log error is the difference in decibels between the data histogram and the model histogram). This criterion provides a means by which the models may be evaluated and compared numerically. The most accurate model is found to depend on geographic location, time of year and day, bandwidth, and center frequency, but two of the simplest models (i.e., each with only two parameters) are found to give extremely good performance in general. These are the Hall and alpha-stable (or a-stable) models, both of which approximate the Rayleigh distribution for low-amplitude values but decay with an inverse power law for high-amplitude values. This paper concludes that the Hall model is the optimal choice in terms of accuracy and simplicity for locations exposed to heavy sferic activity (e.g., lower latitudes) and the a-stable model is best for locations relatively distant from heavy sferic activity (e.g., the polar regions). IntroductionNaturally occurring radio noise above approximately 100 MHz is well modeled for most applications as a Gaussian random process; however, radio noise below 100 MHz (denoted atmospheric noise) is impulsive in nature and is not well modeled as Gaussian. Individual atmospheric events (mainly sferics, the electromagnetic emissions from lightning) produce large impulses in the noise waveform, so atmospheric noise consists of high-amplitude impulses superimposed on a background of low-level

Initial Results from SQUID Sensor: Analysis and Modeling for the ELF/VLF Atmospheric Noise

“…It has been demonstrated that the bit error rate of some digital modulation can be well estimated by APD [30]. Other widely used statistical definitions that characterize the envelope of the noise $A\left(t\right)$ are (1) the cumulative distribution function (CDF) $F_A\left(a\right)$, which is one minus the APD, (2) pdf $f_A\left(a\right)$ is the derivative of the CDF, and (3) the voltage deviation $V_d$ [5]. Traditional APD methods are based on the narrowband signal model.…”

“…It is known for a long time that the phase $\theta \left(t\right)$ of the atmospheric noise follows the uniform distribution over the angles $-\pi$ to π [5]. As for the envelope $A\left(t\right)$, there are several models used in recent literature to describe it, such as Hall model, Field and Lewenstein (F-L) model, Middleton’s Class A and Class B model, and α -stable distribution [32,33].…”

“…As for the envelope $A\left(t\right)$, there are several models used in recent literature to describe it, such as Hall model, Field and Lewenstein (F-L) model, Middleton’s Class A and Class B model, and α -stable distribution [32,33]. Considering the preliminary statistical results in [5], Hall model and S α S distribution are more suitable to describe the narrowband ELF/VLF noise. Consequently, only Rayleigh distribution, Hall model, and S α S distribution are considered in this paper.…”

“…The Hall model is presented by Hall in 1966 and the envelope pdf is described as a two-parameter distribution [5] $f_{\mathrm{Hall}}\left(r\right)=\left(m-1\right){\gamma }^{m-1}\frac{r}{{\left(r^2+{\gamma }^2\right)}^{\left(m+1\right)/2}},r⩾0,$ where m determines the impulsiveness of the noise and γ denotes the the scaling factor. It should be noted that the envelope has infinite variance for $m⩽3$, which is not physically possible.…”

A Global Survey of ELF/VLF Radio Noise

“…Our most recent work specifically describes improvements to low-frequency radio communications that should result from our analyses and modeling [Chrissan, 1998;Chrissan and Fraser-Smith, 2000;Chrissan and Fraser-Smith, 2002] and it provides what is undoubtedly the most comprehensive description of the operationally-important APD statistics for ELF/VLF noise as well as a detailed model of the clustering that occurs in ELF/VLF sferic occurrences.…”