Solar variations in extremely low frequency propagation parameters: 2. Observations of Schumann resonances and computation of the ELF attenuation parameter

Authors


Abstract

[1] Observations of resonant electromagnetic fields caused by global lightning activity are employed in determining the averaged parameters of the lower ionosphere. Using the two-dimensional telegraph equation (TDTE) transmission line model described by Kuequation imageak et al. [2003], we have computed the attenuation rate of the Earth-ionosphere waveguide from diurnal observations of the N-S magnetic component of the ELF field performed irregularly for 6 years in the East Carpathian mountains. As the measurements were carried out during both the minimum and the maximum of the solar cycle 23 we present how solar activity influence the first Schumann resonance frequency and the attenuation rate. The analysis of all the data indicates that the first Schumann resonance frequency increases from 7.75 Hz at solar minimum to about 7.95 Hz at solar maximum while the global mean attenuation rate α at 8 Hz varies from 0.31 dB/Mm at minimum to about 0.26 dB/Mm at maximum.

1. Introduction

[2] The ELF radio wave fields generated in thunderstorm regions propagate to large distances in the Earth-ionosphere waveguide. In the 3–3000 Hz frequency range the characteristics of these fields depend on the physical properties of the lower ionosphere and atmosphere, which affects the fundamental parameters of Schumann resonances (peak frequencies and amplitudes as well as quality factors). Study of the Schumann resonance measurements was used by some authors [e.g., Chapman et al., 1966; Berstein et al., 1974; Burrows and Niessen, 1972; Bannister, 1975; Burke and Jones, 1992; Jones, 1967, 1999] for determination of various ionospheric parameters. Besides, in several papers the influence of solar activity on Schumann resonances and thus on ionospheric parameters was also discussed [Greifinger and Greifinger, 1979; Tran and Polk, 1979; Schlegel and Füllekrug, 1999; Satori et al., 2000].

[3] In our first paper [Kuequation imageak et al., 2003, hereinafter referred to as paper 1] the two-dimensional telegraph equation (TDTE) transmission line model was presented together with spectral methods used for determining the signals generated at distances close to θ ≈ 90° (measured as the angle on the great circle between a source and a receiver) and those coming from very large distances (θ ≈ 130°–150°), when Schumann resonance measurements were done only at one station. In this paper we first describe our monitoring system and then we present results of our observations performed irregularly in the years 1995–2001. They indicate the influence of solar activity on daily mean values of the first Schumann resonance frequency and on the attenuation parameter χ. The discussion presented in section 5 shows that the increase of solar activity causes not only some systematic effects inside the Earth-ionosphere cavity but also produces a large scatter in values of various ionospheric parameters.

2. Instrumentation

[4] In 1995 we began irregular but long-term observations of the horizontal magnetic component of the ELF field in order to gather some statistically significant material used to study the influence of solar activity on the natural ELF fields present in the Earth-ionosphere cavity. Monitoring of the ELF field has been performed near the village of Zatwarnica on the outskirts of the Bieszczady National Park in the East Carpathian mountains. The site, located at ϕ = 49°20′ N, λ = 22°40′ E, is practically free from ELF electromagnetic disturbances of human origin. The monitoring station designed for the work reported here has one signal channel as illustrated schematically in Figure 1 [Kuequation imageak et al., 1996]. It is equipped with a solenoidal magnetic antenna which consists of 240,000 turns of copper wire distributed over five coils in order to reduce the antenna capacitance and thus provides a sufficiently wide transmission band (0.03–55 Hz) for the Schumann resonance measurements. The coil bobbins were threaded onto a silicon ferromagnetic core of length 100 cm and with a cross section 25 cm2. The antenna signal is passed through a low-noise preamplifier, which limits the low-pass frequency to 0.03 Hz. The preamplifier includes an integration stage so that the output voltage is proportional to the B-field received by the antennae. The total gain in this channel is 10/(signal frequency in Hz). The signal channel is sampled by a 12 bit analogue-to-digital converter working with a sampling frequency of 178 Hz, which is controlled by a 80C32 microcontroller. The analogue and digital parts are shielded separately to reduce possible electrical disturbances. The data are transferred to the computer via a 100 m long cable and a RS232 transmission system and then stored on the computer's hard disk. The monitoring station has an intrinsic noise level of 0.02 pT/Hz−1/2 at 10 Hz which allow signals to be recorded with maximum amplitudes 400 pTpp. It is built using low power technique with an inner power supply module.

Figure 1.

Block schematic diagram of the magnetic ELF monitoring station.

[5] The data-acquisition computer has been programmed to carry out all of the analysis required for an interactive session assessing the quality of the data in real time. However, the station is fully automatic and allows for several days observations without an observer using only an additional external power supply for the computer. Our 24-hour observation routine gives 96 measurements each lasting 5 min of the north-south horizontal magnetic component Bx of the natural ELF field. Each 5-min series starts exactly at the beginning of the every quarter of hour, so we have 10-min gaps between successive observations. We have chosen the north-south component of the magnetic field because from our East Carpathian station both Asia and America thunderstorm centers are so far away that in our measurements we can await signals generated at large distances. The observation of these signals is needed if we want to use the spectral method described in paper 1 for the determination of the attenuation rate. If isotropic waveguide response is upheld, we expect the horizontal magnetic field from vertical lightning current in America and Asia to have a dominant north-south component in the Carpathians while the African lightning is expected to produce a dominant east-west magnetic field.

[6] Considerable care has been taken in the overall system design to provide adequate antialias filtering. This is achieved by a six-pole Chebyshev filter with cut off frequency equal to 55 Hz. We do not use a notch filter for the power line frequency of 50 Hz for two reasons. First, we wanted to search for correlations between every observed amplitude of Schumann resonances and disturbances in the power system and secondly as an additional calibration point on the frequency scale of signal spectra. As our Schumann resonances observations have been carried out far from the electrical power network the signal at 50 Hz does not interfere with our measurements. The measured absolute sensitivity of the magnetic channel is 12.6 V/T at 10 Hz. The amplitude and phase frequency-response of the system channel, relative to the fiduciary 10 Hz frequency, was measured using 2-m Helmholtz coils.

3. Numerical Results

[7] In order to consider how the general condition of the lower ionospheric layers affect the Schumann resonance parameters we have analyzed the data obtained in the years 1995–2001 during 35 observation sessions, each covering 24 hours. We usually begin our measurements at 1500 UT and stop at 1500 UT on the next day. As the observations span the minimum and the maximum of the solar cycle 23, we use them also to study solar activity influence on the ionosphere.

[8] There are many indices used for describing solar activity. We have chosen the following four fluxes: X-ray (SXF), electrons with energies E > 2 Mev (SEF), protons with E > 1 Mev (SPF), and radio (SRF) as the indices which measure the long-term changes of solar activity as well as unexpected solar events. The first three values are routinely measured with the space environment monitor system onboard of the 8-GOES satellite. The data were taken from the SPIDR database through the Internet. The X-ray fluxes are from 1–8 band. They represent the daily average fluxes interpolated for 0000 UT of the day following the day when our observations were began. The daily average solar radio flux at 2800 MHz was calculated from the Dominion Observatory (Penticton, Canada) measurements carried on every day at 1700 or 1800, 2000, and 2300 UT (series D).

[9] In Table 1 we present the observation log together with some characteristic Schumann resonances parameters calculated using our TDTE transmission line model (paper 1) as well as the solar data. First, the daily average values given in columns 8, 9, and 10 have been computed according to the following procedure:

Table 1. Daily Values of the Solar Activity Indices SXF, SEF, SPF, SRF, Values of the Characteristic Resonance Frequencies f1max, f1(90), 〈f1〉 and the ELF Attenuation Parameters χM., χ90, χ1, χ2 Calculated for the Days When our Observations Were Carried Outa
 DataDaySXF, 10−8, Wm−2SEF, ions, cm−2 s−1 sr−1SPF, ions, cm−2 s−1 sr−1SRF, sfuf1max, HzF1(90), Hzf1〉, HzχM, Hzχ90 10−6, Ω m−1 Hz−1/2χ1 10−6, Ω m−1 Hz−1/2χ210−6, Ω m−1 Hz−1/2
  • a

    The bold numeral numbers indicate active days when X-ray, electron or proton fluxes are evidently enhanced (see text).

Minimum
12.05.951220.9728110345.3363.08.027.807.783.643.933.993.78
26.06.965233.473.661.1966.88.007.837.803.723.833.933.78
322.07.965690.3813.021.4961.67.847.693.794.293.79
46.06.978883.2114.560.8768.77.807.733.934.163.93
515.08.979585.28193.111.2371.97.977.857.803.843.763.933.80
  Mean2.66  66.47.9977.8247.7603.7333.8484.0603.816
  Sigma1.99  4.2.025.023.048.101.079.159.064
 
RisingPhase
628.09.97100213.03656.152.4979.67.857.853.763.763.76
72.05.98121871.3012.68147.15106.27.987.907.903.803.593.593.70
826.06.98127367.2318.901.17102.07.987.813.803.893.80
911.08.98131951.2453.221.95138.57.867.843.733.793.73
101.05.99158286.523835.463.64111.18.047.887.883.563.663.663.61
1111.06.99162393.6759.022.23154.08.127.977.873.233.363.693.30
  Mean63.82  115.28.0307.8927.8583.5983.6203.7303.650
  Sigma29.00  26.8.066.048.032.270.160.106.183
 
Maximum
121.05.00194878.0093.722.74144.18.117.907.923.273.593.533.43
1330.05.001977117.891356.464.71135.58.097.927.923.363.533.533.44
149.06.001987229.62264.33115.67159.78.118.027.993.273.193.293.23
1525.06.002003116.1316.06165.92160.48.097.917.953.363.563.433.46
161.09.00207165.28495.535.77145.18.067.847.853.483.793.763.64
1716.09.002086155.6927.58746.08158.48.147.907.933.153.593.493.37
1810.10.00211073.3419.684.25126.28.127.998.003.233.293.263.26
1920.10.002120141.72881.465.80142.88.077.977.933.443.363.493.40
2015.11.00214696.302849.9222.03129.48.158.007.993.113.263.293.18
2128.11.002159127.7425.78225.47174.08.167.968.033.073.393.163.23
2211.12.002172118.901032.411.63124.28.107.907.983.313.593.333.45
231.04.0122834434.74145.2628.28223.88.137.987.983.193.333.333.26
2421.04.012303138.594.505.92173.18.138.007.993.193.263.293.22
2528.04.012310126.2312.5220.26173.28.027.957.943.643.433.463.54
262.05.012314120.9519.390.99159.28.077.907.923.443.593.533.52
274.05.012316102.573.190.35159.68.067.947.903.483.463.593.47
2821.05.01233350.68129.396.02137.87.947.883.463.663.46
297.06.012350181.3811.932.83152.78.057.927.903.523.533.593.52
3014.06.012357275.9431.2342.79182.18.057.927.883.523.533.663.52
3111.07.01238446.037.003.93123.08.017.897.883.683.633.663.66
3220.07.01239349.341327.911.74133.98.137.873.193.693.19
3311.08.01241587.2553.62109.44152.78.158.047.853.113.133.763.12
3416.08.01242050.9341.47325.45131.68.077.853.443.763.44
3528.08.012432230.3657.5446.98184.88.147.947.923.153.463.533.30
  Mean300.65  153.68.0967.9427.9273.3303.4523.5033.388
  Sigma882.61  23.7.043.049.053.176.162.175.150
  Totalmean    8.0777.9167.8913.4063.5403.6213.494
  Sigma    .056.063.077.229.209.256.219

[10] 1. Initially each of the 96 registered time series of the horizontal component Bx was Fourier transformed to the frequency domain using a FFT algorithm and 770 points Hamming window [Oppenheim and Schaffer, 1975].

[11] 2. Then, for all the 96 spectra the peak frequencies at the local maximum power f1, f2 and the amplitudes a1, a2 of the first and the second Schumann resonances were determined. Figure 2 shows some examples of 24 hours variation of f1, F = f2/f1, A = a2/a1.

Figure 2.

The 24 hour variation of the first Schumann resonance frequency f1, the ratio F = f2/f1 of the second Schumann resonance frequency to the first, the ratio A = a2/a1 of the amplitudes of the two first Schumann resonances.

[12] 3. Using the methods described in paper 1, section 3 we calculated: (1) the average value f1max (column 8) from the five largest values f1 satisfying the requirements for a distant source, i.e., the conditions 1.74 ≤ F ≤ 1.78 and A ≥ 0.8, and (2) the average value f1(90) (column 9) from these five values f1 for which the parameters A received the smallest values simultaneously with the condition F ≤ 1.72 or F ≥ 1.8 which is the requirement for the sources originating at distance θ ≈ 90° from the observer.

[13] 4. The average value 〈f1〉 (column 10) was calculated as the arithmetic mean from all the 96 measurements of f1.

[14] Having the daily average values f1max, f1(90), 〈f1〉 we computed the average values of the attenuation rate χ using the two linear functions χM = 36.2 − 4.06·f1max and χ90 = 29.9 − 3.33 · f1(90) derived in paper 1. The values χM and χ90 obtained are given in columns 11 and 12. For a few days (8 days altogether) we were not able to find values of f1max or f1(90). The reason for this was the lack of 5-min intervals that satisfied requirements for a distant source (5 days) or a source originating at angular distance θ ≈ 900 from the observer (3 days). In these two last cases a large electrical disturbances coming from heavy and prolonged thunderstorms were noticed at the time when the sources at θ ≈ 90° should be observed. The column 13 comprises values of χ1 which have been calculated from the second function above but using average frequency 〈f1〉 instead of the frequency f1(90). Values of χ2 given in column 11 stand for the mean of χM and χ90. In cases when one of values χM or χ90 were not obtained χ2 is equal to χ90 or χM.

[15] Table 1 is divided into three parts according to changes of solar activity observed in cycle 23 [Zięba et al., 2001]. The first part comprises 5 days from the minimum of the activity, the second comprises 6 days from the rising phase, and the third comprises 24 days from the beginning of the maximum.

[16] It is evident from Table 1 that in some days the electron or proton fluxes registered onboard the 8-GOES satellite are clearly enhanced. The numeral numbers of these days (column 1), called further as active days, are written in boldface. They were defined if X-ray, electron and proton fluxes exceeded 10−5 Wm−2, 103 ions cm−2s−1sr−1, and 102 ions cm−2s−1sr−1, respectively. One day (21.04.2001) is also treated as active because between 16 and 19 April 2001 large electrons and proton fluxes were observed. These events probably changed ionospheric parameters for a few days. The influence of high-energy particle precipitation events on Schumann resonance parameters was studied by Schlegel and Füllekrug [1999]. They showed that the first Schumann resonance frequency increases by about 0.04–0.14 Hz, depending on the intensity of the proton flux. The electron flux produces considerably smaller variations of the SR frequency.

4. Discussion

[17] From 6 years of our Schumann resonance observations it is evident that the characteristic frequencies of the first resonance clearly increase during the active phase of the solar cycle. Figure 3a shows how the mean frequency 〈f1〉 changes from the minimum of the solar cycle to the maximum. The increase of 〈f1〉 is evident. The same is observed for the two characteristic resonance frequencies f1(90), f1max from Figure 3b. Recently, the increase of SR frequency with solar activity was found by Satori et al. [2000]. The values of all the indices used by us for describing solar activity in the days of our SR measurements are presented in Figure 4. As only the X-ray and radio fluxes increase monotonically with time from the solar minimum to the maximum, we will use them further to study how SR frequencies 〈f1〉, f1(90), f1max change with solar activity. The fluxes of electrons and protons (Figure 4c) show an apparent fluctuation which evidently is enhanced at the maximum and so can be used as indicators of some solar events.

Figure 3.

(a) The variation of mean frequency 〈f1〉 during the analysed time period. The solid squares indicate the active days. The distance weighted least squares fit is also presented. (b) The mean, the standard error and the standard deviation of the resonance frequencies 〈f1〉, f1(90), f1max in three analyzed solar phases.

Figure 4.

The variation of the solar radio flux (SRF) (a), and of the X-ray flux (SXF) (b) during the analyzed time period. (c) The variation the electron flux (SEF) (triangles) and of the proton flux (SPF) (squares). The solid symbols indicate the active days. The distance weighted least squares fits are also presented.

[18] Figures 5, 6, and 7show how the resonance frequencies 〈f1, f1(90), f1max depend on the daily X-ray (SXF) and radio (SRF) fluxes. The fitted curves presented in Figures 5, 6, and 7 are described by formulas:

equation image

and

equation image

In the square brackets the percentage of the data variance explained by the fits are given. A close look at Figures 5, 6, and 7 shows that the majority of days with high values of SR frequency are the active days with enhanced proton or electron flux.

Figure 5.

The dependence of frequency 〈f1〉 upon X-ray (SXF) and radio (SRF) fluxes. The solid squares indicate the active days. The best logarithmic fits to all the data are also presented.

Figure 6.

The dependence of frequency f1(90) upon X-ray (SXF) and radio (SRF) fluxes. The solid squares indicate the active days. The best logarithmic fits to all the data are also presented.

Figure 7.

The dependence of frequency f1max upon X-ray (SXF) and radio (SRF) fluxes. The solid squares indicate the active days. The best logarithmic fits to all the data are also presented.

[19] Before coming to the discussion of the attenuation parameter χ we want to mention a seasonal effect seen in our characteristic frequencies f1(90) and f1max. When all the data are divided into two subsets, one consisting of observations from April to July and the second from August to December, we notice the evident difference in the deviations of f1(90) and f1max from the best-fitted logarithmic curves. In the case of frequency f1(90) (Figure 8a) a reduction of the variance, from 0.0118 in the August–December subset to 0.0064 in the April–July subset, is observed but without change of the curve shape. f1max shows opposite behavior (Figure 8b). The shape of the fitted curve changes and the variance 0.00685 of the April–July subset decreases to 0.00357 for the August–December one. We infer that these effects are connected with seasonally changing localizations of the world thunderstorms centers. Because for the determination of the characteristic frequency f1max, the presence of very distant sources (θ > 130°) is needed, we deduce that this condition is better fulfilled during the August–December months when the spring comes on the Southern Hemisphere. However, a more careful analysis of this effect will be possible in future, when more data are gathered.

Figure 8.

The dependence of frequencies f1(90) and f1max upon the SRF. The open squares indicate data taken in the April–July measurements, while the solid squares indicate those from the August–December measurements. The dashed curves fitted to the April–July data are given by f1(90) = 7.3 + 0.31 log (SRF), and f1max = 7.1 + 0.23 log (SRF); the solid curves fitted to the August–December data are given by f1(90) = 7.4 + 0.26 log (SRF), f1max = 7.2 + 0.42 log (SRF).

[20] From the above discussion it is evident that the daily average frequencies f1max, f1(90), and even 〈f1〉 can be accepted as representative observational parameters describing the physical conditions of the Earth-ionosphere cavity. These frequencies are strictly connected with the daily-averaged value of the attenuation parameter χ of the Earth-ionosphere waveguide which can be independently calculated from them using equation (10) given in paper 1: χ = 36.2 − 4.06 · f1max and χ = 29.9 − 3.33 · f1(90). Because the frequencies 〈f1〉 are similar to f1(90) we use for both these frequencies the same formula. Values of χM., χ90, χ1 obtained in this way are presented in columns 11, 12, and 13 of Table 1, respectively, while values of χ2 given in column 14 stand for the mean of χM and χ90. As in the case of the frequencies, we examinse how the attenuation parameter χ changes during the solar activity cycle. All the data have been gathered in three subsamples grouping the values from the minimum, rising phase, and maximum of solar cycle 23. The mean values, the standard errors, and the standard deviations obtained in the four different ways used for the calculation of the attenuation parameters are presented in Figure 9a for the three solar phases analyzed. The mean values indicate that all the daily average attenuation parameters χ decline in the active phase of solar cycle. However, in our simple model in which the attenuation parameter depends on the product of electric conductivity and the ionosphere height squared, we cannot distinguish which of these two factors is more important. The variation of daily values of χ2 is presented in Figure 9b. The evident decline of χ2 values occurs about one year after the beginning of the rising phase. Probably, this time interval is needed before a slow reconstruction of the general propagation conditions in the ionosphere begins. We find that during the maximum, the scattering of χ2 is quite large and the smallest χ2 values are observed in active days. The difference between the mean χ2 values in the minimum and in the maximum of the solar cycle 23 is equal to 0.428 × 10−6 Ω m−1 Hz−1/2. It is larger (0.565) when calculated from the means of the nine smallest (3.221 ± 0.051) and the nine largest (3.786 ± 0.064) parameters. This difference does not change too much if the other ways of the χ calculation are used (0.403, 0.396, and 0.557 for χM., χ90 and χ1, respectively). Therefore from all of these we conclude that a value of Δχ equal to 0.5 × 10−6 Ω m−2 Hz−1/2 can be taken as the difference between the solar minimum and maximum values of the attenuation parameter. Using the formula αdB/Mm = 1.15 × 104 χ · ω1/2 given in paper 1, it is possible to calculate the values of the global attenuation rate α at 8 Hz. We take for the minimum 0.31 dB/Mm and for the maximum 0.26 dB/Mm. The solar minimum value is larger than given by some authors [Burrows and Niessen, 1972; Berstein et al., 1974; Burke and Jones, 1992; Jones, 1999] but it is smaller than calculated from the measurements of Bannister [1975].

Figure 9.

(a) The mean, the standard error, and the standard deviation of the attenuation parameter calculated in four different ways in three solar phases. (b) The variation of the attenuation parameter χ2 in the analyzed time period. The solid squares indicate the active days. The distance weighted least squares fit is also presented.

[21] The dependence of χ2 upon X-ray flux and solar radio flux are shown in Figure 10. Table 2 gives the parameters of the best-fitted curves presented in this figure. Most of the smallest χ2 values are observed on active days. To obtain a more objective description of the χ2 dependence on X-ray and radio fluxes we analyze the subset of all the data, omitting the six active days with the smallest χ2. From Table 2 it is evident that the changes of the mean global attenuation parameters with the solar activity measured by solar radio flux (SRF)) can be quite well described by a linear or a logarithmic function with the dispersion of the points around the fitted curve equal to about 35% of the total data variance. The dependence of χ2 on X-ray flux is much better described by the logarithmic function.

Figure 10.

The dependence of the attenuation parameter χ2 upon X-ray (XRF) and solar radio (SRF) fluxes. The solid squares indicate the active days. The solid curves are the best fits to all the data while the dashed ones omit data from the six active days with the smallest χ2.

Table 2. Parameters of the Best Fitted Linear and Logarithmic Function to the χ2 Values
SampleAmount of the DataBest Linear Fit χ2 = a + b·(SRF)% of Variance Explained by the ModelBest Logarithmic Fit χ2 = a + b·log (SRF)% of Variance Explained by the Model
abab
All the data354.05 ± 0.10−0.0041 ± 0.000752.25.9 ± 0.4−1.13 ± 0.1855.3
All the data without six active days with the smallest χ2 values294.03 ± 0.07−0.0036 ± 0.000562.85.6 ± 0.3−0.98 ± 0.1464.5
  Best Linear Fit χ2 = a + b·(SXF) Best Logarithmic Fit χ2 = a + b·log (SXF) 
  ab ab 
All the data353.51 ± 0.04−0.0001 ± 0.000055.53.84 ± 0.07−0.20 ± 0.0444.0
All the data without six active days with the smallest χ2 values293.58 ± 0.04−0.0001 ± 0.0000412.53.85 ± 0.06−0.17 ± 0.0355.6

5. Conclusions

[22] The basic results obtained in this work can be summarized as follows:

[23] 1.We show that the everyday measurements of Schumann resonances carried on one station can be effectively used to study the influence of various factors on the observed diurnal changes of the first Schumann resonance frequency.

[24] 2. The past 6 years of Schumann resonance measurements have shown that there is a definite increase of the first Schumann resonance frequency with solar activity (from about 7.75 Hz at minimum to 7.95 Hz at maximum). As a result the daily average attenuation parameter χ decreases indicating also that some long-term changes have to take place in the Earth-ionosphere cavity during the solar cycle. The worldwide reaction of the Earth ionosphere to the long-term changes of solar activity begins probably after a 1-year delay.

[25] 3. The dependence of the attenuation parameter χ upon X-ray flux can be described by the logarithmic function χ = 3.85 − 0.17 · log(SXF) and upon solar radio flux at 2800 MHz (SRF) by the linear function χ = 4.03 − 0.0036 · (SRF) or the logarithmic one χ = 5.6 − 0.98 · log(SRF).

[26] 4. The difference between the solar phase minimum (0.31 dB/Mm) and maximum (0.26 dB/Mm) values of the global attenuation rate α at 8 Hz is approximately 0.05 dB/Mm.

Acknowledgments

[27] We would like to thank both referees for very valuable remarks and suggestions, which helped us to improve the current version of the paper significantly. This work was supported by the State Committee for Scientific Research, project 6 PO4D 043 17.

[28] Arthur Richmond thanks Valentin Roldugin and Earle Williams for their assistance in evaluating this paper.

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