ACCOUNTING FOR THE INFLUENCE OF COSMIC NOISE ON THE NUMBER OF METEORS DETECTED BY RADAR

G. N. GLAZOV

(Presented by the scientific seminar of the Radio Engineering Faculty)

One of the most important tasks of meteor radar is to determine the variation of meteor activity with time. However, the recording capability of a meteor radar, unfortunately, is different under different conditions and is not constant in time. It is known that the hourly number of meteors \(N\) detected by radar depends on the magnitude of the threshold signal. The magnitude of the threshold signal varies with time as a result of temporal changes in the level of cosmic noise, which chiefly determines the threshold signal in the wavelength range usually used for meteor radar.

In [1] it is shown that the correction to the hourly number, caused by the nonconstancy of the level of cosmic interference when receiving on a weakly directional antenna, is small and, for a wavelength \(\lambda = 10\) m, for example, lies within \(\pm 15\%\).

The correction to the hourly number may prove significant when receiving on a directional antenna, in particular because of “bursts” of interference of cosmic or solar origin and because of different orientations of the antenna beam relative to “hot” and “cold” sources of cosmic noise at different times.

It is of interest to consider the question of the influence of the level and course of cosmic interference on the number of meteors detected over a sufficiently long interval of time. Since, generally speaking, the number of meteors detected by the radar during a time \(T\) is a random variable, it is of interest to find the dependence of the parameters of the distribution of this random variable on the behavior of the level of cosmic noise in the interval \(T\); it is clear in advance that the desired distribution parameters and, in particular, the mean number of registrations during the time \(T\), depend on the mean noise level in the interval \(T\) and on the degree of their variation relative to the mean (i.e., on the variance) in this interval.

Knowing such dependences, one can compare observational results obtained under different conditions—for example, the number of meteors recorded by a radar for different positions of the lobe of a directional antenna, or the results of observations with directional antennas at different points on the Earth. When considering the dependence of the number of detected meteors on wavelength, radar coordinates, antenna-lobe direction, etc., the factor of the influence of cosmic noise can be properly excluded. It is of interest to find-

the determination of the dependence of the variance of the number of meteors detected during time \(T\) (under the same observing conditions) on the characteristics of cosmic noise. This makes it possible to indicate the limits within which the number of meteors detected during time \(T\) may deviate for purely random reasons.

Following Kaiser [2], one may write the dependence of the mean hourly number of detected meteors \(N\) on the threshold-signal power \(\varepsilon\) as follows:

\[ N = k_1 \varepsilon^{-\frac{s-1}{2}}, \tag{1} \]

where \(k_1\) is a proportionality coefficient independent of the magnitude of the threshold signal; \(s\) is a parameter of the distribution of meteoroid bodies by mass [2]: \(p(m)\sim m^{-s}\).

The threshold-signal power may be represented in the form [3]

\[ \varepsilon = k T_o \Delta f \left(\frac{T_a}{T_o} - 1 + F_c\right)\rho, \tag{2} \]

where \(k\) is Boltzmann’s constant; \(T_o\) is room temperature; \(T_a\) is the effective antenna temperature characterizing the intensity of external (in the present case, cosmic) noise; \(\Delta f\) is the effective receiver noise band; \(F_c\) is the receiver noise factor proper, characterizing the intensity of the internal (fluctuation) noise of the receiver; \(\rho\) is the observability coefficient.

In the general case \(k_1 = k_1(t)\) and \(\varepsilon=\varepsilon(t)\) are realizations of a random function of time. In the particular case when the mean number of meteors detected per unit time, \(N\), is constant, the distribution of the number \(n\) of meteors detected during time \(T\) is given by the Poisson formula

\[ P_T(n)=\frac{(N\cdot T)^n e^{-NT}}{n!}. \tag{3} \]

In the general case, when \(N=N(t)\) is a function of time [4,5],

\[ P_T(n)=\frac{\mu^n e^{-\mu}}{n!}, \tag{4} \]

where

\[ \mu=\int_{t_o}^{t_o+T} N(t)\,dt, \tag{5} \]

\(t_o\) is the time of the beginning of the interval \(T\).

Thus, the process of meteor registration is specified as a nonstationary “Poisson” random process. The necessary conditions [5] under which (3), (4) are valid are fulfilled for the process of meteor registration if each registered meteor is made to correspond to a point on the time axis (corresponding, for example, to the beginning of the radio echo).

In particular, the theoretical distribution, following from (4), of the intervals between meteor radio echoes is in good agreement with experimental data [6,7].

The average number of meteors registered during time \(T\), according to (4), is equal to

\[ N_{T\mathrm{cp}}=\mu=\int_{t_0}^{t_0+T} k_1(t)\,\varepsilon(t)^{-\frac{s-1}{2}}\,dt, \tag{6} \]

and the root-mean-square deviation is

\[ \sigma(N_T)=\sqrt{N_{T\mathrm{cp}}}. \tag{7} \]

For sufficiently large \(T\), when \(\mu\gg 1\), the Poisson distribution (4) is approximated sufficiently well by the normal distribution [4]

\[ P_T(n)=\frac{1}{\sqrt{2\pi\mu}}\,e^{-\frac{(n-\mu)^2}{2\mu}}. \tag{8} \]

Introduce the quantity \(k_{1\mathrm{cp}}\), equal to the mean value of the quantity \(k_1\) over the interval \(T\),

\[ k_{1\mathrm{cp}}=\frac{1}{T}\int_{t_0}^{t_0+T} k_1(t)\,dt. \tag{9} \]

At the instant \(t\), the quantity \(k_1\) differs from \(k_{1\mathrm{cp}}\) by the amount of the increment \(\Delta k_1(t)\),

\[ k_1(t)=k_{1\mathrm{cp}}+\Delta k_1(t). \tag{10} \]

Substituting (10) into (6), we have

\[ N_{T\mathrm{cp}}= k_{1\mathrm{cp}}\int_{t_0}^{t_0+T}\varepsilon(t)^{-\frac{s-1}{2}}\,dt +\int_{t_0}^{t_0+T}\Delta k_1(t)\varepsilon(t)^{-\frac{s-1}{2}}\,dt. \tag{11} \]

The second term on the right-hand side of (11), which characterizes the mutual correlation of two independent random processes (one of which, \(\Delta k_1(t)\), has zero mean), tends to zero as \(T\to\infty\). For sufficiently large \(T\) this term may be neglected. Then we have:

\[ N_{T\mathrm{cp}}= k_{1\mathrm{cp}}\int_{t_0}^{t_0+T}\varepsilon(t)^{-\frac{s-1}{2}}\,dt. \tag{12} \]

Let us rewrite (1) in the form

\[ N(t)=k_2(a+h)^\alpha, \tag{13} \]

where

\[ a=\frac{T_a(t)}{T_0},\qquad h=F_c-1,\qquad \alpha=-\frac{s-1}{2},\qquad k_2=(kT_0\Delta f\rho)^{-\frac{s-1}{2}}. \]

In the wavelength range usually used for meteor radar, cosmic noise exceeds internal noise, \(T_a(t)>T_0\), \(a>h\). Under these conditions, the Taylor series for \(N(t)\) is

\[ N(t)=a^\alpha+h\alpha a^{\alpha-1} +\frac{1}{2}h^2\alpha(\alpha-1)a^{\alpha-2}+\ldots \tag{14} \]

converges rapidly. Since \(\alpha < 0\), with an error smaller in absolute value than \(\dfrac{h^2 \alpha(\alpha-1)}{a^{2-\alpha}}\), one may write

\[ (a+h)^\alpha = a^\alpha + \alpha h \frac{1}{a^{1-\alpha}} . \tag{15} \]

Substituting (15) into (12), we obtain

\[ N_{T_{\mathrm{av}}} = k_2 k_{1\mathrm{av}} \left( \frac{1}{T_o^\alpha} \int_{t_o}^{t_o+T} T_a^\alpha(t)\,dt + \frac{\alpha h}{T_o^{\alpha-1}} \int_{t_o}^{t_o+T} T_a^{\alpha-1}(t)\,dt \right). \tag{16} \]

Represent the function \(T_a(t)\) in the form

\[ T_a(t)=T_{a\mathrm{av}}+\Delta T_a(t), \tag{17} \]

where

\[ T_{a\mathrm{av}}=\frac{1}{T}\int_{t_o}^{t_o+T}T_a(t)\,dt . \tag{18} \]

Of course,

\[ \int_{t_o}^{t_o+T}\Delta T_a(t)\,dt=0 . \tag{19} \]

Taking (17) into account, we expand the functions \(T_a^\alpha(t)\) and \(T_a^{\alpha-1}(t)\) in a Taylor series and, assuming that \(\Delta T_a < T_{a\mathrm{av}}\), discard terms of order \(\dfrac{\Delta T_a^3}{T_{a\mathrm{av}}^{3-\alpha}}\) and higher. Then, taking (19) into account, after simple transformations we obtain:

\[ N_{T_{\mathrm{av}}} = k_2 k_{1\mathrm{av}} \left( \frac{T_{a\mathrm{av}}}{T_o} \right)^\alpha T \left[ 1+\frac{1}{2}\alpha(\alpha-1)\sigma^2+\frac{\alpha}{b} \right]. \tag{20} \]

Here

\[ \sigma^2=\frac{1}{T_{a\mathrm{av}}^2\cdot T}\int_{t_o}^{t_o+T}\Delta T_a^2(t)\,dt \]

is the normalized variance characterizing the degree of scatter of \(T_a(t)\) relative to the mean;

\[ b=\frac{T_{a\mathrm{av}}}{T_o}-\frac{1}{h} \]

characterizes the relative excess of cosmic noise over internal noise. The expression in square brackets in (20) gives the correction caused by the different influence of cosmic noise on the number of detected meteors as a function of the variance of the enveloping curve and the mean level of the noise power. The term \(\dfrac{\alpha}{b}\) takes into account, in particular, the change in the influence of noise (both external and internal) when the wavelength changes. When the wavelength is increased, \(\dfrac{\alpha}{b}\) decreases, which leads to an increase in the correction factor.

In the particular case of a constant noise level \(T_a(t)=T_{a\mathrm{av}}\), the correction factor is

\[ p_n=1-\frac{s-1}{2}\,\frac{T_o(F_c-1)}{T_a}. \]

When comparing observational results under different conditions, we have

\[ \frac{k_{1\mathrm{av}1}}{k_{1\mathrm{av}2}} = \frac{N_{T\mathrm{av}1}}{N_{T\mathrm{av}2}} \cdot \frac{T_{a\mathrm{av}2}}{T_{a\mathrm{av}1}} \cdot \frac{p_2}{p_1}, \tag{21} \]

where

\[ \alpha_i=-\frac{s_i-1}{2},\qquad p_i=1+\frac{1}{2}\alpha_i(\alpha_i-1)\sigma_i^2+ \frac{\alpha_i T_o h_i}{T_{a\mathrm{av}i}}. \]

Thus, it is necessary to know such characteristics of cosmic noise as the mean power level over the interval \(T\) and the variance of the envelope relative to the mean level. In general, for this purpose it is sufficient that the noise power be specified as a random function of time. In practice, for different antenna directivities and pointing directions, this is hardly possible.\(^{1)}\) Therefore it is necessary to record the noise power level during the observations and then determine the required quantities \(\sigma^2\) and \(T_{a\mathrm{av}}\). It is also possible to perform direct averaging of two quantities proportional, respectively, to the noise power and to the square of the noise power. In the latter case the variance is calculated as

\[ \sigma^2=\frac{(T_a^2)_{\mathrm{av}}}{T_{a\mathrm{av}}^2}-1, \tag{22} \]

where

\[ (T_a^2)_{\mathrm{av}}=\frac{1}{T}\int_{t_o}^{t_o+T} T_a^2(t)\,dt \]

is the mean square of the antenna temperature.

It should be noted that direct averaging of the quantity

\[ T_a^{-\frac{s-1}{2}} \]

can give significant errors, because the value of \(s\), even under conditions of an experiment coincident in time, and still more so in experiments at different times, may be different.

From (7) and (20) we obtain:

\[ \sigma(N_T)\sim T_{a\mathrm{av}}^{\frac{\alpha}{2}} \sqrt{ 1+\frac{1}{2}\alpha(\alpha-1)\sigma^2+\frac{\alpha}{b} }. \tag{23} \]

Thus, the possible deviation of the number of detected meteors from \(N_{T\mathrm{av}}\) increases with increasing \(T_{a\mathrm{av}}\) and \(\sigma^2\).

In addition to determining the mean value \(k_{1\mathrm{av}}\), it is of interest to find \(k_1(t)\) (the most probable realization, according to the observational results). This can be done by determining the correlation between \(N(t)\) and \(T_a(t)\). Consideration of this question lies beyond the scope of the present work.

\(^{1)}\) In particular, at meter wavelengths, for one and the same antenna the radiation pattern changes as a function of elevation angle due to the influence of the Earth.

Conclusions

1. The process of recording meteors by radar may be characterized as a special kind of nonstationary “Poisson-like” process.

2. The mean number of registrations of meteor radio echoes over a given time interval depends, to a first approximation, on the mean (over the given interval) level of noise power and on the variance of the noise envelope.

3. The variance of the number of registrations over a given time interval (under given conditions) also depends on the characteristics of the noise. This makes it possible to estimate the degree of scatter of the total number of registrations relative to the mean due to purely random causes.

4. To introduce a correction for the influence of noise and to take account of the different action of noise when comparing results obtained under different conditions (or at different times), it is necessary to know the mean noise-power level over the observing interval and the variance of the noise envelope. To obtain these characteristics, it is necessary to record the noise level during the observation or directly average the power and the square of the noise power over time.

References

1. E. I. Fialko and L. P. Serafinovich. On the influence of the intensity of cosmic interference on the hourly number of meteors detected by radio means, Izvestiya TsPI, vol. 105, 1960.

2. T. R. Kaiser. Radio-echo studies of meteor ionization, J. Adv. Phys., 1953, V. 2, No. 8, 495.

3. V. I. Siforov. Radio Receiving Devices, Oborongiz, 1954.

4. B. R. Levin. The Theory of Random Processes and Its Application in Radio Engineering, [[unclear: publication details not visible]].

5. V. S. Pugachev. The Theory of Random Functions and Its Application to Problems of Automatic Control. Fizmatgiz, 1960.

6. E. I. Fialko. Some statistical characteristics of meteor radio echoes, Bulletin of the Commission on Comets and Meteors of the Astronomical Council of the Academy of Sciences of the USSR, 1959, No. 4, p. 9.

7. E. I. Fialko. On the distribution of intervals between meteor radio echoes (in press).