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

Transcription

1 Radio Science, Volume 35, Number 1, Pages , January-February 2 A comparison of low-frequency radio noise amplitude probability distribution models D. A. Chrissan and A. C. Fraser-Smith Space, Telecommunications and Radioscience Laboratory, Stanford University, Stanford, California 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). 1. Introduction Naturally occurring radio noise above approximately MHz is well modeled for most applications as a Gaussian random process; however, radio noise below 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 im- pulses superimposed on a background of low-level Gaussian noise. The most commonly used and modeled statistic of atmospheric radio noise, other than the absolute power level, is the amplitude probability distribution (APD) of the noise envelope voltage. A number of models have been introduced to characterize the APD of atmospheric noise; however, the quantity of real data that exist to verify their accuracy is quite Copyright 2 by the American Geophysical Union. Paper number 1999RS //1999RS limited. The lack of data is especially pronounced for radio noise in the ELF (3 Hz to 3 khz) and VLF (3-3 khz) bands, for which only a few intervals of data with relatively few sample points are commonly referred to in the literature [e.g., Evans and Griffiths, 1974; International Radio Consultative Committee (CCIR), 1964, 1988; Middleton, 1977; Nakai, 1983; Watt and Maxwell, 1957a, b]. This paper presents the results of a statistical analysis in which thousands of hours of ELF/VLF noise recorded at a number of different sites around the world are processed and the resulting APDs are 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 model, as well as to compare the models against each other, is the expected value of the log error squared (between the model and data histograms). This criterion provides a means by which the models may be evaluated and compared numerically. The three noise models used for data comparison in this paper are the Hall model [Hall, 1966], the Field-Lewenstein model [FieM and Lewenstein, 1978], and the alpha-stable (or a-stable) model [Samorodnitsky and Taqqu, 1994];

2 196 CHRISSAN AND FRASER-SMITH: A COMPARISON OF RADIO NOISE MODELS they were chosen over a number of other tested noise models because of their accuracy and relative simplicity. Note that the widely known class A and B models of Middleton [1977] are not explicitly included be- American storm season) and at UTC, the peak of the diurnal cycle. The term heavy sferic activity is used in this paper to refer to the condition of numerous (and often overlapping) sferics as seen in cause (1) they to not fulfill the relative simplicity Figure 1; at locations relatively distant from thundercriterion, and (2) they are represented indirectly in that class B noise is a generalization of both the Hall storm centers, much less sferic activity is seen in the data spectrogram. and a-stable models. The data used for the analysis are from the Stan- For a given data histogram, each model's parameters are adjusted to minimize the error between the histogram and the model's estimate of it. After optimizing the parameters of each model, the minimum ford ELF/VLF Radio Noise Survey [Fraser-Smith and Helliwell, 1985; Fraser-Smith et al., 1988]. During the years a noise survey system of eight ELF/ VLF ( Hz to 32 khz) radio noise measurement errors achieved for the individual models are com- stations (or radiometers) was installed at a variety of high- and middle-latitude sites, in an effort to fill large gaps in the information available on radio noise in the ELF/VLF frequency range. A number of other ELF/ VLF measurement systems have been implemented, but this is the only system of its kind in terms of geographic coverage and continuity of simultaneous data collection. pared and the best model is determined. The best model depends on geographic location, time of year and day, frequency, and bandwidth, but the Hall and a-stable models are found to give extremely good performance in general. Both models have Rayleigh characteristics for low-amplitude values but decay with an inverse power law for high-amplitude values. The paper concludes that the Hall model is the optimal choice in terms of accuracy and simplicity for locations near 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). 2. ELF/VLF Noise ELF/VLF radio noise comprises both man-made and natural electromagnetic signals. Examples of man-made signals are power line harmonics, communication signals, and interference from electrically powered machinery; naturally occurring noise includes sferics, whistlers, polar chorus, and auroral hiss [Helliwell, 1965]. Sferics are typically the dominant source of naturally occurring low-frequency radio noise [Volland, 1995]. Even though lightning activity occurs mainly at lower latitudes, sferics can propagate for thousands of miles with little attenuation, so they are seen in noise data worldwide. The amount of sferic activity in a given noise sample depends on the worldwide source distribution of lightning relative to the receiver location, with nearby storms contributing a great deal and distant storms contributing less. A sample VLF spectrogram for 8 s of Grafton, New Hampshire, data is shown in Figure 1; the many vertical lines are sferics. A great deal of sferic activity is seen because Grafton is close to thunderstorm activity, especially in July (the peak of the North The radiometers are located at Arrival Heights, Antarctica (AH; 78øS, 167øE); Dunedin, New Zealand (DU; 46øS, 17øE); Grafton, New Hampshire (GN; 44øN, 72øW); Kochi, Japan (KO; 33øN, 133øE); L'Aquila, Italy (AQ; 42øN, 13øE); Sndrestromfjrd, Greenland (SS; 67øN, 51øW); Stanford, California (SU; 37øN, 122øW); and Thule, Greenland (TH; 77øN, 69øW). Most of the stations operated much longer than program expectations, and the systems at Stanford and Arrival Heights are still operating. A complete technical description of the radiometers is given by Fraser-Smith and Helliwell [1985]. Thousands of hours of ELF/VLF time-series data were recorded by the radiometers, and some additional LF data were recorded at Thule in the range 3-6 khz (i.e., in the lower part of the LF range, 3-3 khz). These ELF/VLF/LF time-series data are used to derive the APDs against which the models are tested. (In the remainder of this paper the terms ELF, VLF, and LF refer to the individual frequency bands, while the term low-frequency refers to them collectively.) The radiometers' time-capture interval for one data segment is 1 min, and the data from multiple time-capture intervals were used to obtain APDs with larger sample sizes. 3. Amplitude Probability Distributions As mentioned above, one of the most measured and modeled statistics of low-frequency radio noise is the APD. Since wideband time-series data contain a

3 CHRISSAN AND FRASER-SMITH: A COMPARISON OF RADIO NOISE MODELS 197 o o Time (seconds) Figure 1. Grafton, New Hampshire, VLF spectrogram, July 2, 1988, at 5 UTC. The vertical lines are sferics; the lightning that causes them may be thousands of kilometers away. great deal of man-made interference, the noise APD is usually analyzed within a relatively narrow frequency range. In such narrowband analysis the broadband time-series data are digitally downconverted (i.e., frequency translated) using various center frequencies and low-pass filtered using various cutoff Using this latter notation, the signal n(t) can also be written frequencies. Statistics are then derived from the resulting low-pass equivalent signals. n(t) = (A(t)eJ(2*rœt+O(t))), A narrowband signal n(t) can be written where ( ) indicates the real part of the argument. n(t) = n/(t) cos (2,rft) - nq(t) sin (2,rft), The noise envelope A(t) is a random process, and where n/(t) is the inphase component, nq(t) is the its first-order statistics over a given time interval are quadrature component, and f is the center frequency. specified by its APD. The APD is defined as the The signals n/(t) and nq(t) are then low-passignals probability that the noise envelope A takes on a value with bandwidth much smaller than f. The complex- larger than some given value a: P(A > a). Other analytic representation is written commonly used statistical definitions that characterize A(t) are (1) the cumulative distribution function n(t) = A (t)e j(2*rœt+ø(t)), (CDF) FA(a), which is 1 minus the APD, (2) the where the magnitude (or envelope)a(t) is probability density function (pdf)fa (a), which is the derivative of the CDF, and (3) the voltage deviation Vd, defined as log(e[a 2]/E2[A]), which serves as (t) = + and the phase (t) is (t) = arctan no(t)] ni(t) ]'

4 198 CHRISSAN AND FRASER-SMITH: A COMPARISON OF RADIO NOISE MODELS 1.8 data pdf 1.6 / x Rayleigh / \ / \ I \ 1.4! \ I.2 "" a Figure 2. Thule envelope probability density function (pdf) compared with a Rayleigh distribution with the same initial slope. The Thule data are normalized such that E[A] = 1, and the Rayleigh parameter is.26. an indicator of the impulsiveness of the noise. (The notation E[ ] represents the expected value.) The phase (t) of atmospheric noise has long been known to have a distribution that is uniform over the angles -rr to rr [Middleton, 1977]; several checks on the noise survey data provide additional evidence that this is true. Since low-frequency noise can be viewed as Gaussian background noise plus impulsive noise, its pdf looks like a bell curve but with a heavier low- probability tail (i.e., there is a higher probability of extreme values occurring). Likewise, since the envelope distribution of Gaussian noise is Rayleigh, the envelope distribution of low-frequency noise appears Rayleigh but with a heavier low-probability tail. An example of the difference between a data pdf and the Rayleigh distribution is shown in Figure 2, for July 1986 data from Thule, Greenland, in the to 37.6-kHz range. Data pdf's are found to decay with an inverse power law for large values (out to some limit set by the dynamic range of the system, as discussed by Chrissan [1998], and Fraser-Smith et al. [1988]), so the Rayleigh pdf has a tail which rolls off too quickly to accurately represent the data pdf. All data analyzed in this paper are normalized such that their average value E[A] is 1; however, it should be noted that the relative APD (with amplitude values expressed in decibels) is frequently normalized so that the RMS value is APD Models Extensive research has been conducted for over 4 years to model the APDs of atmospheric radio noise; the models that have been developed vary from general to specific in application and from simple to numerically intractable (see Spaulding [1995] for a complete overview). Some are based entirely on intuitive reasoning and/or fitting the data to mathe- matical functions (these are called empirical models); others start with assumptions on noise source distributions and the propagation of noise impulses to the receiver (statistical-physical models). Empirical models, in general, have been based on ad hoc curve fittings to limited data, but nonetheless a number of good models have been developed, including the Hall [1966] and Field and Lewenstein [1978] models. They are typically simpler and more mathematically tractable than statistical-physical models, but their parameters are often unrelated to the physical processes

5 CHRISSAN AND FRASER-SMITH: A COMPARISON OF RADIO NOISE MODELS 199 that create the noise. Statistical-physical models take into account the underlying physical processes of impulsive noise but are usually complicated and difficult to work with mathematically. In addition, working them into a tractable form often requires making approximations that are known not to be true for atmospheric noise, such as assuming that the impulsive sources are distributed independently and uniformly in space and time. (See Watt and Maxwell [1957b] and Chrissan [1998] for discussions on the clustering of sferics.) The most widely known statistical-physical models are the Class A and B noise models developed by Middleton [1977]. These are not considered explicitly in this paper due to their complexity, but the class B model proves to be very accurate since it is a generalization of both the Hall and a-stable models. It should be noted that class A noise is defined for cases in which the bandwidth of the input noise is comparable to or less than the detection bandwidth, and class B noise is defined for cases where the noise bandwidth is greater than the bandwidth of the detector (i.e., impulsive noise inputs produce transients in the receiver). It is the latter case which applies in this study, since the sferics in the noise data have much wider bandwidths than those used to analyze the noise. Another widely known model, the CCIR report APD model, is not included in the analysis because it cannot be represented in a simple form and has a somewhat ad hoc nature [CCIR, 1988; SpauMing and Washburn, 1985]. The complete set of noise model distributions examined for this paper includes power-rayleigh (or Weibull), Laplace, lognormal, hyperbolic, and any promising mixture processes or piecewise combinations of these, both with and without the Gaussian or Rayleigh distribution included. This encompasses most, if not all, of the commonly known APD models [e.g., Furutsu and Ishida, 1961; Giordano and Haber, 1972; Miller and Thomas, 1977]. It should be noted that the CCIR model mentioned above roughly falls into this category, since it is a piecewise combination of a Rayleigh distribution, a power-rayleigh distribution, and a curved region between the two [Spaulding and Washburn [1985]. The Hall model is presented and explained extensively by Hall [1966], but only the first-order pdf of the narrowband envelope.4 is needed here. The Hall model specifies this envelope pdf, f.4(a), as the two-parameter distribution f.4(a) = (m- 1)T m-1 [a 2 + T2](m+l)/2, a -->, called the Hall pdf from this point on. The term 3' is a scaling factor, and the term m determines the impulsiveness of the noise. Note that.4 has infinite variance for m _< 3; this implies infinite noise power, which is not physically possible. The Field and Lewenstein (F-L) model is an empirical model developed from the assumption that atmospheric noise is composed of impulsive noise superimposed on a background of low-level Gaussian noise. The envelope.4 is approximated as the sum of a Rayleigh-distributed random variable (for the Gaussian component) and a power-rayleigh-distributed random variable (for the impulsive component), so the envelope pdf is the convolution of the two densities: 2a 2 aa a-1 = -- e -a2/rø* -- e -(a/r) f.4 (a) Rø 2 R" ' a _> O. This distribution is referred to as the F-L model in the remainder of this paper. Note that this is a three-parameter distribution with no closed form, so it is cumbersome to work with mathematically. In addition, its parameters are not easily determined for a given data set. The a-stable pdf is directly specified in the charac- teristic function domain. The characteristic function (I)x(tO) of a random variable X is essentially a Fourier transform of the pdf: (I)x(co) = E[e jo'x] : f fx(x)e j dx, and the a-stable characteristic function is defined as ti)x(co) = e - lo, I, a two-parameter distribution. (The general form of the distribution includes two more parameters, defining an absolute shift and a skew, but these may be eliminated by assuming the noise is distributed symmetrically about zero. This the case for atmospheric noise time-series data.) For a = 2 the characteristic function defines a Gaussian distribution with mean zero and variance 23'; for a = 1 it defines a Cauchy distribution with parameter % Thus the Cauchy and Gaussian distributions are forms of the a-stable distribution. For a

6 2 CHRISSAN AND FRASER-SMITH: A COMPARISON OF RADIO NOISE MODELS close to 2 the distribution is essentially Gaussian but Finally, the results are compared with the voltage with heavier tails. deviation parameter Vd. If atmospheric noise is a-stable distributed, its The data set that was analyzed spans various times envelope distribution is the Fourier-Bessel transform of the year and all times of the day at seven of the of e- P : stations. Space limitations preclude the presentation of the thousands of graphs and model parameter values resulting from the analysis; instead, we discuss f/i (a) = a fo 9e-YP Jo(a9) d9, a -> O. the qualitative features of the data while providing examples of numerical results. The APDs presented Note for a = 2 that this is the Rayleigh distribution. in this section are each determined using data taken For a close to 2 it is essentially a Rayleigh distribution from the same 4-hour diurnal period of four consecbut with a heavier tail. From this point on, this fa (a) will be referred to as the a-stable envelope pdf. utive days. The introduction mentions that the error criterion The a-stable pdf does not exist in closed form used to find the optimal parameters of each model as except for a = 1 or a = 2, but it can be approximated well as to compare the models against each other is numerically without a great deal of computational the expected value of the log error squared between complexity. In addition, the parameter 3/is a simple the model and data histograms, also called the meanscaling factor such that squared log error (MSLE). The MSLE is defined as 1 fx(x;., )=- fx,y and likewise for the envelope pdf, fa(a; a, 3')= 7-5f/i ;a, 1, so a lookup table must vary only over the one parameter a. Such a lookup table method is used in these analyses. The a-stable model is an empirical model, but like the Hall model it does have some physical justification. Nikias and Shao [1995] show that under certain assumptions on the underlying noise sources and the propagation characteristics between them and the receiver, atmospheric noise is expected to exhibit an a-stable pdf. These assumptions, such as sources independently distributed in space and time, are not true in practice; however, they are approached closely enough in some cases to explain the accuracy of the a-stable model. 5. Data Analysis This section presents the general results found when fitting all three models to all of the data, followed by specific results for each measurement location. This is followed by a discussion of each of the models, noting the range of parameters each model uses in fitting the data and how these parameters vary with season, time of day, and station. x MSLE: f f(x) [ log f(x)] f x)j dx, where f(x) is the data pdf and f(x) is the model's estimate of it. Note that this expression is similar to the relative entropy definition in information theory [Cover, 1991], except for squaring the term in brackets. After optimizing the parameters of each model, the minimum errors achieved for the individual mod- els may be compared in order to determine the best model. The data clearly reveal a pattern defining which model works best under which conditions, and the findings are as follows: The Hall model is found to be very accurate in modeling the amplitude pdf of VLF radio noise under the condition of heavy sferic activity in the noise; otherwise, the a-stable model is best. In addition, there is a fairly large transition region (as a function of time, location, frequency, etc.) where both models are equally accurate. The F-L model exhibits an MSLE roughly - times higher than the other two for most of the data samples examined, but it is still reasonably accurate. The sferic activity at a given location has both a seasonal and diurnal cycle [Chrissan and Fraser- Smith, 1996a, b; Chrissan and Fraser-Smith, 1997]. Seasonal variations peak during the local summer (i.e., Southern Hemisphere locations are most active from December to February), while diurnal variations are as follows: North America peaks at roughly UTC, South America peaks at roughly 2 UTC, Europe and Africa peak at roughly 16 UTC, and

7 CHRISSAN AND FRASER-SMITH: A COMPARISON OF RADIO NOISE MODELS 21 Southeast Asia peaks at roughly 8 UTC. Since the optimal model depends on sferic activity, it is related to season and time of day. The sferics themselves have a frequency response (as seen at the receiver) that peaks in the 8- to 14-kHz range and decreases with increasing frequency; therefore only the stronger sferics create large impulses at width for 23 khz and below is 4 Hz; at the higher frequencies tested it is 2 khz. It is found that the Hall the higher VLF frequencies (as seen in spectrograms similar to Figure 1). Because of this frequency response, narrowband time-series data appear to contain lower sferic activity as the center frequency is increased. The bandwidth of the receiver is known to affect The Arrival Heights VLF data are taken from the time period May 1995 through June 1996; the center frequencies analyzed span the range khz, and the noise bandwidth is chosen to be 6 Hz (in order to reject adjacent man-made signals). A fairly wide uncontaminated band from 25.5 to 27.5 khz is ana- lyzed as well. It is found that the a-stable model performs best over the whole data set, with an average MSLE of.8, compared with.6 for the Hall and.153 for the F-L. Average percentage error is computed as x V'MSLE percent, so on average, the a-stable model is within approximately 7% ( v'ø'øøø8) of the true pdf. The Hall model's error is 19%, and the F-L model's error is 33%. The a-stable model's accuracy is especially good above 2 khz, where the respective average errors are 5%, 22%, and 29%. The Dunedin VLF data cover the year 1989; the center frequencies analyzed span the range khz, and the noise bandwidth is 4 Hz. The band khz is analyzed as well. It is found that the Hall model performs best below approximately 23 khz; above this range the a-stable model is slightly better. The respective percentage errors for the a-stable, Hall, and F-L pdf's are 14%, %, and 33% for frequencies below 23 khz and 8%, %, and 31% for those above. The Thule data include June 1986 to February 1987, with center frequencies ranging from 15 to 43 khz (LF data were collected at Thule). The band- and a-stable models are comparable below 23 khz except during the peak of the seasonal and diurnal cycles, when the Hall model exhibits 1/2 to 1/3 the MSLE of the a-stable model. The average percentage errors for the a-stable, Hall, and F-L models below 22 khz are 12%, %, and 34%, respectively; at higher frequencies those errors are 6%, 15%, and 35%. Even at higher frequencies the seasonal and diurnal variations have an effect: At 36 khz the the impulsiveness of the noise because a narrow bandwidth spreads the sferics in time, causing them to overlap more and appear less impulsive (this lowers the V d as well; see section 5.5). Thus the a-stable model exhibits a larger error than the Hall model at a-stable error varies roughly from 5% to 11% seasonally and from 9% to 15% diurnally (during the low bandwidths since the impulses are less distinctive seasonal peak). against the background noise, but this is true only for The SondrestrOm data include September 1993 to very small bandwidths. As the bandwidth approaches June 1994, with frequencies from 17 to 26 khz and zero, the pdf distribution approaches Gaussian, for which the envelope is Rayleigh. We now present specific results related to the discussion above. bandwidths of 6 Hz. The accuracy of the models follows the same diurnal, seasonal, and frequency patterns as at the stations discussed above, as shown in Figure 3. The three rows of this figure correspond 5.1. Results by Location to center frequencies of 17.35, 22.55, and 26.5 khz, and the four columns cover four consecutive seasons from September 1993 to June The plots are of the average percentage error (as calculated using V'MSLE); the Hall model error is depicted by the dashed line, and the a-stable error is depicted by the solid one. Note that the a-stable model has considerably less error than the Hall model for the months of Decem- ber and March, when the Northern Hemisphere seasonal variation is at a minimum. However, in June and September (near the peak of storm season) and at 17 khz, the Hall model gives better performance. In addition, there is a diurnal variation of the a-stable error in September and June; notice that it increases with the passing of North American storms at approximately UTC. The Grafton, Stanford, and L'Aquila data cover the same general frequency and time ranges as the data presented above, with the same general results. Grafton is especially close to heavy storm activity in North America, which typically occurs in the Northern Hemisphere summer, so the accuracy of the a-stable model is noticeably worse during this season. This is depicted in Figures 4 and 5, which show the average percentage errors (averaged over all frequen-

8 22 CHRISSAN AND FRASER-SMITH: A COMPARISON OF RADIO NOISE MODELS 2O Sep. 93 Dec. 93 Mar. 94 Jun ß \ /? \ / 2O 2 / "'" \ O /\ / / hour (UTC) hour (UTC) hour (UTC) hour (UTC) Figure 3. Errors of the Hall (dashed line) and a-stable (solid line) models as a function of frequency and season at SOndrestrOm. cies) of the three models as a function of time of day for the months of January and July, The a-stable model is clearly the best in January, but in July its performance degrades to a greater error than the Hall model. In addition, a fairly large diurnal error variation is seen in July, which is during the seasonal peak. Thule data at center frequencies of 36.6 and 43.4 khz and with bandwidths of 25, 5,, 2, 4, 8, and 16 Hz were processed in order to test the effect of increasing bandwidth. The Hall model exhibits a larger MSLE and the a-stable model exhibits a smaller one as bandwidth increases, but the effect is only seen during increased storm activity, and the max- imum error of either model is only approximately 13%. An additional sample of Arrival Heights May 1995 data was processed with a center frequency of 8 khz and bandwidth varying from 25 to 16 Hz. In this case the a-stable model is only 7.5% in error across the whole range of bandwidths, while the Hall mod- el's error increases from 9.5% to 21% with increasing bandwidth. The F-L error varies between 4% and 6%. For a detailed discussion of bandwidth effects on the APD, see SpauMing and Washburn [1985] Parameters of the Models In this section we discuss how well each model fits various portions of the dynamic range of the noise envelope. This information is not contained in the MSLE since it is an average over the entire dynamic range; therefore this section provides insight as to why a model may perform well or poorly. In addition, it is stated for each model the range of parameters exhibited in fitting the data histograms and whether or not these parameters are correlated with location, center frequency, bandwidth, time of day, or season of the year. Figures 6 (pdf) and 7 (APD) show the fit of the three models using a typical data sample for which the a-stable model is optimal. The data are from Arrival Heights during the 4-8 UTC diurnal time period in May (May 8-11, 1995), a time and month of relatively low sferic activity. The center frequency is 22.7 khz, and the bandwidth is 6 Hz. Parameters and errors for the three models are given in Table 1. The envelope pdf of Figure 6 is essentially Rayleigh except for a heavy tail due to occasional sferics, starting at db in the dynamic range. The Hall and F-L models fit the curve accurately in the high

9 CHRISSAN AND FRASER-SMITH: A COMPARISON OF RADIO NOISE MODELS 23 5 Jan hall F-L -- stable o O 6 I hour (UTC) Figure 4. Errors of the Hall and a-stable models as a function of frequency and season, Grafton, New Hampshire, January Jul. 88 5,, 45 4 hall F-L stable o O O hour (UTC) Figure 5. Errors of the Hall and a-stable models as a function of frequency and season, Grafton, New Hampshire, July 1988.

10 24 CHRISSAN AND FRASER-SMITH: A COMPARISON OF RADIO NOISE MODELS -1-2 Arrival Heights amplitude pdf, 22.7 khz, 6 Hz BW, May 1995, 6 UTC " \ ' - data.. -' xx. ' Hall x x ' stable -7-8,: -9 I I I ; amplitude (db) Figure 6. Fit of Hall, Feld and Lewenstein (F-L), and a-stable pdf models to a sample of Arrival Heights data Arrival Heights APD, 22.7 khz, 6 Hz BW, May 1995, 6 UTC \\ i i i i i! > xx x \ \ x x data Hall F-L x\ x\ x\ o x le-6% amplitude (db) Figure 7. Fit of Hall, F-L, and a-stable amplitude probability distribution (APD) models to a sample of Arrival Heights data.

11 CHRISSAN AND FRASER-SMITH: A COMPARISON OF RADIO NOISE MODELS 25 Table 1. Parameter Values for the Hall, F-L, and Alpha-Stable Models in Figures 6 and 7 Mean-Squared Percent Model Parameters Log Error Error Hall m / F-L a = r =.77 r =.18 Alpha-stable a = /-.293 probability -- to -db range but are unable to accurately model the higher range. In fact, they are often off by orders of magnitude. Figures 8 (pdf) and 9 (APD) show the fit for data typical of when the Hall model is optimal. The data are from Grafton, in March (March 7-, 1988), and contain reasonably heavy sferic activity. The center frequency is 17.5 khz, and the bandwidth is 4 Hz. Parameters and errors for the three models are given in Table 2. The value of m for the Hall model and the value of a for the a-stable model are significantly less in Table 2 than in Table 1, and the curve of Figure 8 does not appear as much to be a Rayleigh distribution with a heavy tail. The average amplitude is higher relative to the background noise, and so more of the probability occurs at decibel values less than zero. In addition, the data curve has no convex bend at db, as in Figure 6, making the Hall pdf an almost perfect fit. The upward curve of the a-stable pdf near 4 db in Figure 8 is due to limits of numerical accuracy for the algorithm used to determine the pdf. These inaccuracies do not pose a problem because it is found that the data histograms contain zero probability where accuracy is degraded. It should be noted that although the F-L model does not in general perform as well as the other two models, it is still quite accurate in comparison with a number of other impulsive noise models that have been proposed, and it was the optimal third choice for these analyses based on the criteria given in the introduction. The F-L model's problem is primarily in its asymptotic behavior when approaching low or high values, since it is neither Rayleigh at low levels nor polynomial at high ones. At the high end of the dynamic range the F-L model decreases exponentially as a -, which is too rapid a decay to match the data. 1 Grafton amplitude pdf, 17.5 khz, 4 Hz BW, Mar 1988, 18 UTC data -1 i stabl e X XX\ -6 I I I I I \., / amplitude (db) Figure 8. Fit of Hall, F-L, and a-stable pdf models to a sample of Grafton data.

12 _ 26 CHRISSAN AND FRASER-SMITH: A COMPARISON OF RADIO NOISE MODELS 99.9 Grafton APD, 17.5 khz, 4 Hz BW, Mar 1988, 18 UTC i i i I i data Hall F-L stable O o x 6O 4O 2O e-6% -4 i i I -3O -2O amplitude (db) Figure 9. Fit of Hall, F-L, and a-stable APD models to a sample of Grafton data Typical Parameter Values Typical values of the F-L parameters are as follows: a is usually in the range.5-.6, although values as low as.34 and as high as 1.15 are found. The values of r and r both range from.1 to 1. but typically take values of.7 and.25, respectively. There is only a mild dependence of the three parameters on season and time of day at the high-latitude sites, but at lower latitudes there is a large dependence. At Grafton, for example, the value of a jumps from.5 in January to.9 in July, and there is an additional.2 diurnal variation about these values as well. The term r correspondingly drops from.8 to.2, and r increases from.2 to.8, from January to July. The Hall model has values of m that range from 2.2 to 11, with typical values between 3 and 4. Values of 7 range from.6 to 2.4 and are typically There is no strong relationship between the Hall parameters and time, location, etc., but the values of m and 7 do tend to rise as the center frequency is increased through the 15- to 27-kHz range. The a-stable model has values of 7 that are mostly near.3. The value of a is usually between 1.6 and 2. but can be as low as 1.1. A seasonal and diurnal dependence is seen in a, but it is strong only at the low-latitude sites. Table 2. Parameter Values for the Hall, F-L, and Alpha-Stable Models in Figures 8 and 9 Mean-Squared Percent Model Parameters Log Error Error Hall m = F-L a = r =.4 r =.4 Alpha-stable a = ELF Results The ELF data are severely contaminated by power line harmonics spaced either 5 or 6 Hz apart, and the need for total rejection of these signals allows only for very narrow analysis bandwidths. (The need for total rejection of power line harmonics is demonstrated by the fact that even small leakage causes a significant drop in Vd.) Fortunately, several data samples were found for which the 24-Hz harmonic did not exist at all for an entire month, so the usable bandwidth spanned the range 18-3 Hz. These

13 CHRISSAN AND FRASER-SMITH: A COMPARISON OF RADIO NOISE MODELS 27 data samples are Arrival Heights for September 1994 and Thule and Sndrestrm for April 199. They are analyzed using a center frequency of 24 Hz and bandwidths ranging over the values 5,, 15, 2, 25, and 3 Hz. The ELF results do not differ significantly from the VLF results either qualitatively or quantitatively. The average percentage errors of the a-stable, Hall, and F-L models at Arrival Heights are 4%, 12%, and 36%, respectively; at Sndrestrm they are 9%, 6%, and 42%, respectively; and at Thule they are 8%, 7%, and 37%, respectively. These numbers are averaged across bandwidth; at the larger bandwidths the a-sta- ble model always outperforms the Hall. The parameter ranges of the models are the same as described above for VLF Discussion of Voltage Deviation The Va statistic is reintroduced at this point because it is a fundamental parameter in previous work on modeling the APDs of atmospheric noise [CCIR, 1988; SpauMing et al., 1962]. It is computed as V a = log(e[a2]/e2[a]), and as such it is the RMS value of the noise envelope divided by the average value, in decibels. Thus it can also be expressed as V a = 2 g(vrms/vavg), in decibels. V a is a mea- sure of the spikiness of the noise (since sharper impulses result in a higher RMS value relative to the mean value) and is greatly dependent on the noise bandwidth for a given noise environment; however, it is not necessarily an indicator of heavy sferic activity. For the data of Figures 6 and 7, the V a is 3.5 db, and for the data of Figures 8 and 9, it is 4.9 db, but there are many cases where the a-stable model is optimal for higher V a as well. The typical range of V a values is db, although values as high as 9- db can be seen at high-latitude sites. V a is also found to vary with season, location, and bandwidth. Seasonal and diurnal variations of V a are seen in the data, but they cannot be systematically correlated with known storm distributions. For instance, V a does not necessarily rise and fall with storm season: At Grafton (for one sample frequency band) it reaches a low of 2.5 db in both January and July, from approximately 4.5 db in spring and fall. Other data show that Grafton's Va varies out of phase with its diurnal variation, while Dunedin's Va variation is in phase with its diurnal variation. Va is also found to relate somewhato the parameters of the three pdf models. For one set of Arrival Heights data samples, the Hall parameters m and 3/ decrease from to 2.8 and 2. to.4, respectively, as V a increases from 1.5 to 9.. The F-L parameters a, r, and r change from.7 to.4, 1. to.4, and.1 to.2, respectively, as Va increases, and the a-stable parameters a and 3/decrease from to 1.2 and.3 to.2, respectively. The errors of the three models bear fairly little relation to Va, except that the Hall model tends to be more accurate than the a-stable model for lower values of m, which corresponds to higher values of Va. 6. Conclusions This paper presents many APD results derived from a statistical analysis of low-frequency radio noise. Three relatively simple models for the noise envelope voltage APD are described, and the parameters and accuracy of each model are determined as a function of location, time, and frequency. The parameters and errors of each model are found to vary with thunderstorm activity and the noise frequency and bandwidth. The main conclusion of this paper is that the Hall model is the best choice in terms of accuracy and simplicity for locations near heavy sferic activity, and the a-stable model is best for those removed from heavy sferic activity. A general rule based on the results would be to use the a-stable pdf in the polar regions except at the peak of the diurnal and seasonal storm cycle and to use the Hall model at low and middle latitudes except at the null of the seasonal and diurnal cycle. The crossover point is not critical; there are a broad range of conditions where both models have roughly the same accuracy. It is thus also concluded that the Middleton class B noise model could be used, at the expense of added complexity, as a unified APD noise model that is optimum over all conditions. Acknowledgments. This research was sponsored by the Office of Naval Research through grant N14-92-J Logistic support for the measurements at Sndrestromfjord, Greenland, and Arrival Heights, Antarctica, was provided by the National Science Foundation through NSF cooperative agreement ATM and NSF grants DPP and OPP , respectively. We also thank the reviewers for many useful comments. References Chrissan, D., Statistical analysis and modeling of lowfrequency radio noise and improvement of low-frequency

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