Expressions tabulated for the effective antenna noise figure Fa usually assume an electric field antenna, since most measurements of radio noise are made on the electric field of the noise. Furthermore, the International Radio Consultative Committee (CCIR) noise model predictions for Fa are made only for electrically short grounded vertical monopoles over a perfect ground. However, at frequencies lower than those traditionally used for communications, i.e., at extremely low frequencies (ELF; frequencies in the range 3 Hz to 3 kHz) and very low frequencies (VLF; frequencies in the range 3–30 kHz), it is common for magnetic field loop antennas to be used, and the tabulated expressions for Fa do not apply. This communication reports the derivation of an expression for Fa as measured by a small vertical magnetic loop antenna and its subsequent application to ELF/VLF radio noise measurements made at a variety of locations around the world. There is good agreement between the measured Fa values and estimates of maximum and minimum Fa values for the ELF/VLF range published by Spaulding and Hagn in 1978, but an improved fit to the measurements can be obtained by making moderate adjustments to the maximum and minimum values at both the low (10–100 Hz) and high (8–32 kHz) frequency limits.
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 The effective antenna noise figure Fa is one of several parameters that are used to characterize radio noise. It is a measure of the noise power received by an antenna from sources external to the antenna [International Radio Consultative Committee (CCIR), 1964, 1982, 1988; Spaulding and Hagn, 1978; Spaulding and Washburn, 1985], and it is the ratio measured either in dB(kT0) or dB(kT0b) of the received noise power available from an equivalent lossless antenna (i.e., the power available after corrections for the antenna losses [CCIR, 1964, 1988; Lauber and Bertrand, 1994]) to the thermal, or Johnson, noise power available in the same bandwidth from a resistor at room temperature. As suggested by the wording of this definition, Fa values tend to be closely linked to the antennas used for their measurement, and this linking sets Fa apart from other radio noise parameters, which usually have little or no dependence on the antennas used for their measurement. In practice, the dependence means that an antenna-dependent numerical factor appears in the expressions relating Fa to the amplitudes of the electric or magnetic field strengths of the noise. This is not a fundamental problem, but the antennas traditionally used for the measurement of Fa have been electric field antennas, and the expressions that have been derived to convert the electric field measurements to Fa values have all been specific to electric field antennas. As a result, the survey of extremely low frequency (ELF)/very low frequency (VLF) radio noise being made by my Stanford University research group [Fraser-Smith and Helliwell, 1985; Fraser-Smith et al., 1988, 1991; Fraser-Smith, 1995], which involves measurements of the magnetic field of the noise by means of loop antennas, has until now been limited to the derivation of noise parameters other than Fa. In addition, since magnetic field loop antennas are conventionally used to make measurements on ELF/VLF radio signals and noise, and not electric field antennas, there have been few actual measurements of Fa for ELF/VLF radio noise. To remedy this situation, an expression for Fa as measured with a small vertical loop antenna has been derived, and the twofold purpose of this paper is (1) to document this derivation and (2) to provide representative measurements of Fa for the ELF/VLF radio noise occurring at a number of locations around the world and compare them with CCIR predictions.
Fa, the effective antenna noise figure, is derived from the effective antenna noise power factor fa, which, for any antenna, is given by
where pn is the mean noise power available from an equivalent loss-free antenna (W), k is the Boltzmann constant (1.38 × 10−23 J/K), T0 is a reference temperature (K), and b is the effective receiver noise bandwidth (Hz). The antenna noise figure is then defined by Fa = 10log10fa, with units of dB(kT0) or dB(kT0b). If we define Pn = 10log10pn (dB), we can further write Fa in the form
for a temperature T0 that is conventionally taken to be 288 K (15°C).
 The noise power (in W) available from a loss-free antenna must equate to the power flux density (in W/m2) of the electromagnetic radiation illuminating the antenna multiplied by the antenna's effective aperture. Thus we can write
where an impedance of 120π (ohms) is assumed for free space and where e is the root mean square (RMS) electric field strength (in V/m) of the noise (for a bandwidth b), g is the antenna gain relative to an isotropic radiator, and λ is the wavelength (in m) at the center of the bandwidth of the receiver. Typically, the ratio of this center frequency to the bandwidth is large; for the measurements we report below, the ratio is 20. Considering that one of the goals of this work is to derive an expression for Fa in terms of measurements made on the magnetic field of the radio noise illuminating an electrically small loop antenna, we could as well have written the power flux density (i.e., Poynting flux) in equation (3) in terms of the magnetic field strength of the radiation instead of the electric field strength e. To facilitate comparison with earlier work, we have, at this stage, retained the traditional electric field approach, but it is important to note that no additional assumptions or simplifications are involved when we finally convert from e to the corresponding magnetic field. If we now write G = 10log10g, E = 10log10e2, and the wavelength λ = 300/F (in m), where e is measured in μV/m and F is the center frequency measured in MHz, we have
where Pn is measured in dBW (dB above 1 W). Then, from equation (2)
The antenna dependence of the above two expressions linking Fa and E comes from the presence of the antenna gain factor G. Given a particular antenna, the gain factor is usually combined with the constant 96.79 to give an antenna-dependent constant. For example, for a short vertical monopole above a perfectly conducting ground plane, the gain G is −1.25 dB and (−G − 96.79) in equation (4b) is replaced by the new constant 1.25 − 96.79 = −95.54. A list of these constants for a variety of electric field antennas is provided in the study by Hagn and Shepherd , but no constants are available for magnetic field antennas.
2. Fa for a Small Vertical Loop Antenna
 In this section, we derive the gain factor G for a small vertical loop antenna of the kind frequently used to measure the magnetic field of low-frequency electromagnetic signals and noise. Small in this case means small relative to the free-space wavelength of the radiation, which is very large at ELF/VLF frequencies. Thus the actual physical size of the vertical loop antennas is not an issue. It is appropriate for us to assume a high efficiency factor, since we are considering the power available from a loss-free antenna, in which case the gain of the antenna is equal to its directivity, i.e., the ratio between its maximum radiated power density and its average power density over a sphere. Since we are only concerned with power densities measured in the far field, the directivity is proportional to the inverse of the integrated square of the far field of the antenna. Thus to obtain the gain, we need to work with the far field of the antenna.
 The far fields produced by electric dipoles and small loop antennas (magnetic dipoles) are given by the following expressions [Kraus, 1988, 1991]:(1) Electric dipole antenna
(2) Magnetic loop antenna
where I is the current (A), r is the distance from the antenna (m), θ is the angle measured from the vertical (radians), L is the length of the dipole antenna (m), A is the area of the loop (m2), λ is the wavelength (m), and the field quantities E and H are measured in V/m and A/m, respectively. Note that the square brackets in the above equations denote retarded current, i.e., [I] = I0sin2πf(t − r/c), and that both antennas have a gain of 3/2, or 1.76 dB.
 Now consider a small circular loop antenna above a perfectly conducting ground plane (Figure 1). First, we ignore the mutual impedance between the antenna and its image. Next, we note that the current in the image loop will have the same direction of circulation as the current in the loop above the ground plane. Replacing the circular loops by their equivalent horizontal magnetic dipole equivalents, we have two horizontally oriented dipole antennas varying in phase and separated by a distance 2h, where h ≪ λ. Thus the fields should effectively be double those produced by a single horizontal dipole antenna, which has a gain of 3/2. Thus for two such antennas, we expect the gain to be 3.
 To confirm this supposition, we now formally derive the gain of the pair of loop antennas (i.e., dipoles) shown in Figure 1. First consider the azimuthal field pattern for the dipole located at the origin of the geometrical construction in Figure 2. We know that the fields produced by both electric and magnetic dipoles depend on sin , where is the angle between the axis of the dipole and the line connecting the origin to the field point P. Converting sin θ to the equivalent angular term for the coordinate system in Figure 2, we obtain sin θ = . If we now move the dipole at the origin in Figure 2 a distance h up the z axis, and place an additional dipole with the same orientation along the x axis a distance h below the origin, an additional multiplicative angular term cos(βh sin α), where β = 2π/λ is the conventional phase constant, is introduced that takes account of the conversion of the source to a dipole pair. The total field pattern is therefore , and the gain is given by
where F(α, ϕ) = (f(α, ϕ))2 is the angular power pattern and dΩ = sin αdαdϕ is an element of solid angle. By making the approximations h ≪ λ and cos(βh sin α) ≈ 1 − (β2h2 sin2α)/2, we have
Then, to a first order,
where I have written x = cos α.
 The antenna gain factor g in equation (7a) is therefore given by g = 4π/(4π/3) = 3. Converting g = 3 to the corresponding value G = 4.77 and using equation (4a), we can finally write the expression for Fa for a small vertical loop antenna at an altitude h (where h ≪ λ) above a perfectly conducting ground plane in the following form:
 The electric field term E in equation (8) is no longer a directly measured field quantity. Converting to the directly measured magnetic field or, more specifically in our case, the directly measured amplitude spectral density of the noise magnetic induction Bn, we obtain our final expression for Fa as measured by a single small loop antenna:
where the units of Bn are fT/.
 A single small vertical loop antenna is not sensitive to ELF/VLF radio noise illuminating the antenna from directions that are predominantly perpendicular to the plane of the antenna, since the magnetic fields of the waves are both perpendicular to the direction of propagation and either largely or totally horizontally directed, which means that they induce only very small or negligible signals in the loop. This lack of sensitivity of a loop antenna to noise signals arriving from certain directions is not taken into account by the gain factor derived above, and equations (8) and (9) for Fa apply only for signals arriving in the plane of the loop. Since naturally occurring ELF/VLF noise can come from all directions, Fa values derived from measurements with a single loop are unlikely to be fully representative. This limitation of the measurements made with a single vertical loop antenna is well recognized, and it has become common practice to make measurements on low-frequency radio noise with crossed-loop antennas, i.e., two identical vertical loop antennas oriented at right angles to each other. If the square root of the sum of the squares of the outputs of the loops is computed, as is done in the Stanford University ELF/VLF noise survey, the magnetic field measurements become essentially omni-directional in the horizontal plane. For such a crossed-loop antenna, the average antenna gain over 360° of azimuth is approximately = 4.77, and Fa for such a system is given by
where the units of n are once again but where it is now understood that n is the square root of the sum of the squares of the measurements made simultaneously by each of the crossed loops. In conclusion, for the same noise fields, Fa for a pair of crossed loop antennas is 6 dB higher than the CCIR's Fa for a short monopole.
3. Measurements of Fa With Small Crossed-Loop Arrays
 As I have just described, the Stanford University ELF/VLF radio noise measurements are made with pairs of mutually perpendicular loop antennas, which are usually oriented in north-south (N-S) and east-west (E-W) directions [Fraser-Smith and Helliwell, 1985; Fraser-Smith et al., 1988, 1991; Fraser-Smith, 1995]. The loop outputs, after amplification, filtering, and analog-to-digital conversion, are used to compute the total RMS signal in each of 16 narrow frequency bands (5% bandwidth; the 16 frequencies cover the range 10 Hz to 32 kHz and are listed in Table 1) by taking the square root of the sum of the squares of the digitized N-S and E-W narrowband signals. This procedure has the effect of making the measurements omni-directional insofar as the noise sources are concerned. The measurements are stored on magnetic tape in a variety of forms; the data used in this communication were stored as 1-min average amplitudes of an original 600 measurements made at a rate of 10 s−1 on each of the 16 frequencies. An automatic calibration procedure is used to maintain the accuracy and consistency of the measurements over time.
Table 1. A Tabulation of Modified Minimum and Maximum Values of Fa for Frequencies in the Range 10 Hz to 32 kHz
Figures 3–5 show illustrative values of Fa that were calculated from the magnetic field measurements made by several Stanford measurement systems (or ELF/VLF “radiometers”), located at different locations around the world. The units used for the magnetic field measurements were , and the values of Fa were computed by using equation (10).
 The Fa values shown in Figure 3 were computed from the average magnetic field amplitudes measured during June 1986 at Arrival Heights, Antarctica. To search for possible diurnal changes, the averages and the corresponding Fa values were first computed for each 3-hour interval of the 24-hour day, and then overall average values were computed. As shown by the plots in the figure, there is little diurnal change in the average Fa values over the course of a day.
Figure 4 shows monthly average Fa values for the ELF/VLF radio noise measured at L'Aquila, Italy, during January and July 1987. The general decline of the values with frequency is the same as that for the Antarctic measurements, but the Italian values are roughly 10–15 dB higher than those at Arrival Heights.
Figure 5 compares plots of monthly average Fa values for Kochi, Japan (KO), Søndrestrømfjord, Greenland (SS), Arrival Heights (AH), and Thule, Greenland (TH), and it further compares the various plots with estimates of Fa published by the CCIR  and by Spaulding and Hagn  for the frequency range 10 Hz to 10 kHz. These latter estimates are the most complete of the limited data available for Fa at frequencies in the ELF and VLF ranges, and there is excellent agreement between the estimates and the measurements over most of the displayed frequency range. The Fa values at the high-latitude measurement locations are noticeably smaller than those at the middle-latitude locations, as would be expected considering the greater distance to high latitudes from the largely tropical (thunderstorm) source locations, but most of the measurements are well contained within the ranges provided by the estimated maximums and minimums. In addition, Spaulding and Hagn's  estimates decline with frequency very similarly to our measurements, and they similarly show the effect of increased attenuation for radio wave propagation in the earth-ionosphere waveguide for frequencies in the range 1–5 kHz, which straddle the earth-ionosphere waveguide cutoff frequency.
 There are some relatively minor discrepancies between the estimates and the measurements at the low (10–100 Hz) and high (8–32 kHz) ends of the frequency range covered by the data. I have adjusted Spaulding and Hagn's  estimates at these two extremes of the ELF/VLF frequency range to contain all our measurements (with the exception of the Thule measurements for f < 135 Hz, which are anomalous and appear to be contaminated) and extended the estimates from 10 to 32 kHz, as shown in Figure 6. The modified estimates are listed in Table 1 for the 16 frequencies at which the measurements of Fa were made.
 The measurements of Fa reported here differ from those normally encountered in that they were obtained by using magnetic loop antennas. Nevertheless, they agree well with estimates from the study by Spaulding and Hagn [1978; CCIR, 1982], with the exception of some relatively minor discrepancies at the upper and lower ends of the ELF/VLF frequency range. For f < 100 Hz, the estimates of maximum possible values of Fa can be too small by as much as 12 dB, and the estimated minimums also tend to be lower than the measured minimums, although by only a few decibels. Differences of this order are not surprising, considering the few actual measurements of radio noise at frequencies less than 100 Hz that must have been available before 1978. For frequencies in the range 8–10 kHz, the estimates of minimum and maximum Fa values are both high; for the worst case, at 10 kHz, the estimate of minimum Fa is too high by about 15 dB. The modifications to Spaulding and Hagn's estimates of minimum and maximum values of Fa, and their extension from 10 to 32 kHz, result in a new series of Fa maximum and minimum values which agree well with measurements over the entire 10 Hz to 32 kHz frequency range.
 Radio noise statistical quantities such as Fa often play an essential role in the design of communication systems, and they have been used mostly, if not entirely, for these design and other related engineering studies. However, it must be remembered that the predominant source of radio noise in the ELF/VLF range is the lightning occurring in thunderstorms. Furthermore, because ELF/VLF radio signals propagate for large distances over the Earth's surface with relatively small attenuation, measurements of ELF/VLF radio noise at any one location are influenced by the lightning emissions (i.e., sferics) from thunderstorms over large areas of the globe. At the lowest frequencies, where the wavelengths begin to become comparable to the Earth's circumference and the attenuation is the least, the noise statistics respond to lightning activity occurring all over the world. As a result, the statistics of ELF/VLF radio noise relate to the variation of thunderstorm activity on a global basis, which means that their variation over time must also relate to changes in global weather and climate. More recent studies by the author and his colleagues have shown that ELF radio noise measurements can provide new information about global lightning and climate variability [e.g., Füllekrug and Fraser-Smith, 1997], and it is possible that at some time in the future, ELF/VLF radio noise measurements could have an important application in studies of the variation of the world's climate.
 I appreciate the assistance of Frankie Y. Liu during the early stages of the preparation of this paper, and I am grateful to W. R. Lauber, the late G. H. Hagn, and the late A. D. Spaulding for many helpful discussions. The measurements at L'Aquila, Italy, were made in collaboration with A. Meloni and P. Palangio and those at Kochi, Japan, were made in cooperation with T. Ogawa; their important contributions are gratefully acknowledged. This research was sponsored by the Office of Naval Research through grants N00014-92-J-1576 and N00014-04-1-0748. Logistical support for the Søndrestrømfjord, Greenland, and Arrival Heights, Antarctica, measurements was provided by the National Science Foundation through grants ATM-9813556, ATM-0334122, and ANT-0138126.