SCATTERING AND ABSORPTION OF MICROWAVES IN ATMOSPHERIC FORMATIONS (RAIN, SNOW, CLOUD COVER, FOG) AND RADAR

D. M. Vysokovskii

INTRODUCTION

Some time ago it was established that radar stations operating in the centimeter range are capable of detecting reflected signals arising from the scattering of microwaves by atmospheric formations, especially by heavy rains and thunderstorm clouds[^1-5].

For a theoretical treatment of the processes of scattering and absorption of microwaves in atmospheric formations consisting of spherical water droplets, it proved possible to use the theory, developed in its time, of the scattering of electromagnetic waves by spherical particles[^6-7]. On the basis of this theory and elementary radar calculations, one can estimate the order of magnitude of the amplitudes of the reflected signals. The results of such calculations are in agreement with observational data from radar stations.

Nevertheless, observations of reflected signals do not provide a direct means of precisely determining the content of liquid droplets in the atmosphere, since the dependence of the amplitude of the reflected signal on the character of the distribution of droplet sizes may lead to the result that, for example, a rain with a smaller total amount of precipitation but consisting of large drops can produce a larger amplitude of the reflected signal than a rain with greater precipitation but consisting of smaller drops. Therefore, for the correct interpretation of radar data, it has proved exceptionally important, in connection with the experimental study of these phenomena, to take into account the statistical picture of atmospheric formations and precipitation with respect to the distribution of droplet sizes and their fall velocities, the presence of fluctuations in droplet concentration in time and space, etc.

With a combined theoretical and experimental investigation both of the atmospheric formations themselves and of the processes of absorption and scattering of microwaves in them, it is possible to establish na

...the existence of a correlation between the amplitude of the reflected signals and the content of liquid droplets in the atmosphere or the magnitude of rainfall precipitation, and to use it for meteorological purposes. The use of radar thus makes it possible to investigate a number of problems of meteorology which until now have been difficult to access by direct experiment and observation.

These include such questions as the dependence of the distribution of the concentration of droplets in atmospheric formations on altitude, the structure and development of rains, the passage of precipitation through the level of the zero isotherm, the formation and motion of fronts, etc.8, 9, 10, 11. All this opens up great prospects for the application of radar in meteorology12, 13.

In the present survey the main results are presented of theoretical and experimental work on the study of the scattering and absorption of microwaves by macroscopic particles of spherical form and on the application of radar to the study of atmospheric formations.

1. THEORY OF SCATTERING AND ABSORPTION OF MICROWAVES BY SPHERICAL PARTICLES

The principal results of the theory of scattering and absorption of electromagnetic waves by spherical particles make it possible to calculate the magnitudes of the effective cross sections of absorption, total scattering, and radar scattering, which are of great importance in the theory of radar of atmospheric formations.

In the theory of scattering*) one considers the scattering and absorption of a plane electromagnetic wave by a sphere of radius \(a\). The electric and magnetic fields of a plane wave incident along the \(z\)-axis and polarized along the \(x\)-axis have the form:

\[ \mathbf{E}^{i}=\mathbf{E}_{x}=\mathbf{a}_{x}E_{0}\cdot e^{-ik_{2}z+i\omega t}, \]

\[ \mathbf{H}^{i}=\mathbf{H}_{y}=\frac{1}{\eta_{2}}\cdot \mathbf{a}_{y}E_{0}\cdot e^{-ik_{2}z+i\omega t}, \tag{1} \]

where

\[ k_{2}=(\mu_{2}\varepsilon_{2}\omega^{2}-i\mu_{2}\sigma_{2}\omega)^{\frac{1}{2}} \tag{2} \]

is the complex wave number for the external medium with parameters \(\mu_{2}, \varepsilon_{2}, \sigma_{2}\),

\[ \eta_{2}=\sqrt{\frac{\mu_{2}}{\varepsilon_{2}}}=377\ \text{ohm} \tag{3} \]

is the impedance of the medium, and \(\mathbf{a}_{x}, \mathbf{a}_{y}, \mathbf{a}_{z}\) are unit vectors in the positive directions of the axes.

) The theory of scattering of electromagnetic waves by spherical particles is set forth in detail in the recently published book by K. S. Shifrin, Scattering in a Turbid Medium*14a.

The material of the sphere is characterized by the parameters \(\varepsilon_1,\ \mu_1,\ \sigma_1\). If the wave vectors of the plane wave \(a_x e^{-ik_2 z}\), \(a_y e^{-ik_2 z}\) are expanded in terms of the wave vectors \(\mathbf m\) and \(\mathbf n\) of a spherical wave corresponding to the point with spherical polar coordinates \(r,\ \vartheta,\ \varphi\) (see, for example, \({}^{14}\), p. 365), then the expressions for the fields of the incident wave will be (omitting the time factor \(e^{i\omega t}\)) of the form

\[ \left. \begin{aligned} \mathbf E^i &= E_0 \sum_{n=1}^{\infty} (-i)^n \frac{2n+1}{n(n+1)} \left(\mathbf m_{0n}^{(1)} + i\mathbf n_{0n}^{(1)}\right), \\[4pt] \mathbf H^i &= -\frac{E_0}{\eta_2}\sum_{n=1}^{\infty} (-i)^n \frac{2n+1}{n(n+1)} \left(\mathbf m_{en}^{(1)} - i\mathbf n_{0n}^{(1)}\right), \end{aligned} \right\} \tag{4} \]

where the functions \(\mathbf m\) and \(\mathbf n\) are expressed in the form

\[ \left. \begin{aligned} \mathbf m_{0n}^{(\alpha)} &= \frac{1}{\sin\vartheta}\, Z_n^{(\alpha)}(kr) P_n^1(\cos\vartheta)\cos\varphi\,\mathbf i_2 - Z_n^{(\alpha)}(kr)\frac{dP_n^1}{d\vartheta}\sin\varphi\,\mathbf i_3, \\[4pt] \mathbf m_{en}^{(\alpha)} &= -\frac{1}{\sin\vartheta}\, Z_n^{(\alpha)}(kr) P_n^1(\cos\vartheta)\sin\varphi\,\mathbf i_2 - Z_n^{(\alpha)}(kr)\frac{dP_n^1}{d\vartheta}\cos\varphi\,\mathbf i_3, \end{aligned} \right\} \tag{5} \]

\[ \left. \begin{aligned} \mathbf n_{0n}^{(\alpha)} &= n(n+1)\frac{Z_n^{(\alpha)}(kr)}{kr} P_n^1(\cos\vartheta)\sin\varphi\,\mathbf i_1 \\ &\quad + \frac{1}{kr}\frac{\partial}{\partial r}\!\left[rZ_n^{(\alpha)}(kr)\right] \frac{dP_n^1}{d\vartheta}\sin\varphi\,\mathbf i_2 \\ &\quad + \frac{1}{kr\sin\vartheta}\frac{\partial}{\partial r}\!\left[rZ_n^{(\alpha)}(kr)\right] P_n^1(\cos\vartheta)\cos\varphi\,\mathbf i_3, \\[6pt] \mathbf n_{en}^{(\alpha)} &= n(n+1)\frac{Z_n^{(\alpha)}(kr)}{kr} P_n^1(\cos\vartheta)\cos\varphi\,\mathbf i_1 \\ &\quad + \frac{1}{kr}\frac{\partial}{\partial r}\!\left[rZ_n^{(\alpha)}(kr)\right] \frac{dP_n^1}{d\vartheta}\cos\varphi\,\mathbf i_2 \\ &\quad - \frac{1}{kr\sin\vartheta}\frac{\partial}{\partial r}\!\left[rZ_n^{(\alpha)}(kr)\right] P_n^1(\cos\vartheta)\sin\vartheta\,\mathbf i_3. \end{aligned} \right\} \tag{6} \]

In these formulas \(P_n^1(x)\) is the first associated Legendre polynomial of the first kind, and \(\mathbf i_1,\ \mathbf i_2,\ \mathbf i_3\) are unit vectors directed in the direction of increasing \(r,\ \vartheta\), and \(\varphi\) at the point \((r,\vartheta,\varphi)\) of a sphere of radius \(r\), with \(\mathbf i_2\) and \(\mathbf i_3\) respectively tangent to the meridian and to the parallel of the sphere (Fig. 1). The index \(\alpha\) takes the value 1 for the incident and transmitted wave and 3 for the scattered wave. The functions \(Z_n^{(1)}(x)\) and \(Z_n^{(3)}(x)\) are defined by the relations:

\[ Z_n^{(1)}(x)=\left(\frac{\pi}{2x}\right)^{\frac12} J_{n+\frac12}(x); \qquad Z_n^{(3)}(x)=\left(\frac{\pi}{2x}\right)^{\frac12} H_{n+\frac12}(x). \tag{7} \]

where \(J_{n+\frac{1}{2}}(x)\) is the Bessel function of the first kind of order \(n+\frac{1}{2}\), and

\(H^{(2)}_{n+\frac{1}{2}}(x)\) is the Hankel function of the second kind of order \(n+\frac{1}{2}\). The field of the scattered wave \(\mathbf{E}^{s}, \mathbf{H}^{s}\) is sought (for \(r>a\), where \(a\) is the radius of the scattering sphere) in the form

\[ \left. \begin{aligned} \mathbf{E}^{s} &= E_{0}\sum_{n=1}^{\infty}(-i)^{n}\, \frac{2n+1}{n(n+1)} \left(a_{n}^{s}\mathbf{m}_{0n}^{(3)}+i b_{n}^{s}\mathbf{n}_{en}^{(3)}\right),\\[6pt] \mathbf{H}^{s} &= -\,\frac{E_{0}}{\eta_{2}}\sum_{n=1}^{\infty}(-i)^{n}\, \frac{2n+1}{n(n+1)} \left(b_{n}^{s}\mathbf{m}_{en}^{(3)}-i a_{n}^{s}\mathbf{n}_{0n}^{(3)}\right). \end{aligned} \right\} \tag{8} \]

and by analogous expressions for \(\mathbf{E}^{i}, \mathbf{H}^{i}\). Inside the sphere \((r<a,\ k=k_{1}\) and \(\eta=\eta_{1})\) the field of the transmitted wave \(\mathbf{E}^{t}, \mathbf{H}^{t}\) is sought in the form

\[ \left. \begin{aligned} \mathbf{E}^{t} &= E_{0}\sum_{n=1}^{\infty}(-i)^{n}\, \frac{2n+1}{n(n+1)} \left(a_{n}^{t}\mathbf{m}_{0n}^{(1)}+i b_{n}^{t}\mathbf{n}_{en}^{(1)}\right),\\[6pt] \mathbf{H}^{t} &= -\,\frac{E_{0}}{\eta_{1}}\sum_{n=1}^{\infty}(-i)^{n}\, \frac{2n+1}{n(n+1)} \left(b_{n}^{t}\mathbf{m}_{en}^{(1)}-i a_{n}^{t}\mathbf{n}_{0n}^{(1)}\right). \end{aligned} \right\} \tag{9} \]

Thus, the problem of finding the field of the scattered and transmitted waves reduces to determining the amplitude coefficients \(a_{n}^{s}, b_{n}^{s}\) and \(a_{n}^{t}, b_{n}^{t}\) by using the boundary conditions. From the expressions for the fields \(\mathbf{E}^{i}, \mathbf{H}^{i}\), \(\mathbf{E}^{s}, \mathbf{H}^{s}\), and \(\mathbf{E}^{t}, \mathbf{H}^{t}\), one can find the components of these fields along the spherical coordinate axes.

The equality on the surface of the sphere of the tangential components of the fields outside and inside the sphere is expressed in the form

\[ \begin{gathered} E_{\theta}^{i}+E_{\theta}^{s}=E_{\theta}^{t},\\ H_{\varphi}^{i}+H_{\varphi}^{s}=H_{\varphi}^{t} \quad \text{at } r=a. \end{gathered} \tag{10} \]

Fig. 1. Spherical coordinate system.

After substitution of the expressions for the field components, the following system of equations is obtained for determining

coefficients \(a_n^s,\ b_n^s\) and \(a_n^t,\ b_n^t\):

\[ a_n^t Z_n^{(1)}(N\rho)-a_n^s Z_n^{(3)}(\rho)=Z_n^{(1)}(\rho), \]

\[ \mu_2 a_n^t \frac{d}{d(N\rho)}\left[N\rho Z_n^{(3)}(N\rho)\right] -\mu_1 a_n^s \frac{d}{d\rho}\left[\rho Z_n^{(3)}(\rho)\right] =\mu_1 \frac{d}{d\rho}\left[\rho Z_n^{(1)}(\rho)\right], \tag{11} \]

and

\[ \mu_2 N b_n^t Z_n^{(1)}(N\rho)-\mu_1 b_n^s Z_n^{(3)}(\rho)=\mu_1 Z_n^{(1)}(\rho), \]

\[ b_n^t \frac{d}{d(N\rho)}\left[N\rho Z_n^{(1)}(N\rho)\right] -N b_n^s \frac{d}{d\rho}\left[\rho Z_n^{(3)}(\rho)\right] =N\frac{d}{d\rho}\left[\rho Z_n^{(1)}(\rho)\right], \tag{12} \]

where

\[ N=\frac{k_1}{k_2}; \qquad \rho=k_2a. \]

Eliminating \(a_n^t\) from equations (11) and \(b_n^t\) from equations (12) makes it possible to find expressions for \(a_n^s\) and \(b_n^s\):

\[ a_n^s =- \frac{ \mu_1 Z_n^{(1)}(N\rho)\left[\rho Z_n^{(1)}(\rho)\right]' -\mu_2 Z_n^{(1)}(\rho)\left[N\rho Z_n^{(1)}(N\rho)\right]' }{ \mu_1 Z_n^{(1)}(N\rho)\left[\rho Z_n^{(3)}(\rho)\right]' -\mu_2 Z_n^{(3)}(\rho)\left[N\rho Z_n^{(1)}(N\rho)\right]' } \]

and

\[ b_n^s =- \frac{ \mu_1 Z_n^{(1)}(\rho)\left[N\rho Z_n^{(1)}(N\rho)\right]' -\mu_2 N^2 Z_n^{(1)}(N\rho)\left[\rho Z_n^{(1)}(\rho)\right]' }{ \mu_1 Z_n^{(3)}(\rho)\left[N\rho Z_n^{(1)}(N\rho)\right]' -\mu_2 N^2 Z_n^{(1)}(N\rho)\left[\rho Z_n^{(3)}(\rho)\right]' }, \tag{13} \]

where the prime denotes differentiation with respect to the argument of the Bessel function enclosed in brackets. In exactly the same way, eliminating \(a_n^s\) and \(b_n^s\), one can obtain expressions for \(a_n^t\) and \(b_n^t\).

For large distances from the center of the sphere (\(r\gg a\), or \(k_2 r\gg k_2 a\)), expressions (13) can be simplified. The coefficients \(a_n\) and \(b_n\) are small for \(n>k_2 a\), and the summation over \(n\) may be restricted to terms with number \(n<k_2 a\). In the case of large distances, \(k_2 r>n\), and the magnitude of a term is affected not so much by its ordinal number \(n\) as by the argument \(k_2 r\) of the spherical Bessel function. Under these conditions one may use the asymptotic representation of these functions:

\[ \left. \begin{aligned} Z_n^{(1)}(kr)&\simeq \frac{1}{kr}\cos\left(kr-\frac{n+1}{2}\pi\right),\\ Z_n^{(3)}(kr)&\simeq \frac{1}{kr}e^{-i\left(kr-\frac{n+1}{2}\pi\right)}. \end{aligned} \right\} \tag{14} \]

It is evident from this that the radial components of the fields may practically be neglected, since they are proportional to \(1/r^2\), whereas the components in \(\vartheta\) and \(\varphi\) are proportional to \(1/r\). Consequently, at large distances the field vectors will be perpendicular to the line of propagation (the wave zone).

Thus, for \(r \gg a\)

\[ E_r^s=H_r^s=0 \tag{15} \]

and

\[ \left. \begin{aligned} E_\vartheta^s=\eta_2 H_\varphi^s &= -\left(\frac{i}{kr}\right) E_0 e^{-ikr} \sum_{n=1}^{\infty}(-1)^n \frac{2n+1}{n(n+1)} \left( a_n^s \frac{P_n^1}{\sin\vartheta} + b_n^s \frac{dP_n^1}{d\vartheta} \right)\cos\varphi, \\[6pt] E_\varphi^s=-\eta_2 H_\vartheta^s &= -\left(\frac{i}{kr}\right) E_0 e^{-ikr} \sum_{n=1}^{\infty}(-1)^n \frac{2n+1}{n(n+1)} \left( a_n^s \frac{dP_n^1}{\sin\vartheta} + b_n^s \frac{P_n^1}{\sin\vartheta} \right)\sin\varphi . \end{aligned} \right\} \tag{16} \]

The resultant field at some point outside the sphere is the superposition of the fields of the incident and scattered waves

\[ \mathbf E=\mathbf E^i+\mathbf E^s; \qquad \mathbf H=\mathbf H^i+\mathbf H^s . \tag{17} \]

The radial component of the complex Umov–Poynting vector of the resultant field in the wave zone will be:

\[ S_c=\frac{1}{2}\left(E_\vartheta H_\varphi^* - E_\varphi H_\vartheta^*\right), \tag{18} \]

where the asterisk \(*\) denotes complex conjugation.
Substitution of (17) gives:

\[ \begin{aligned} S_c &=\frac{1}{2}\left(E_\vartheta^i H_\varphi^{i*}-E_\varphi^i H_\vartheta^{i*}\right) +\frac{1}{2}\left(E_\vartheta^s H_\varphi^{s*}-E_\varphi^s H_\vartheta^{s*}\right) \\ &\quad +\frac{1}{2}\left(E_\vartheta^i H_\varphi^{s*}+E_\vartheta^s H_\varphi^{i*} -E_\varphi^i H_\vartheta^{s*}-E_\varphi^s H_\vartheta^{i*}\right). \end{aligned} \tag{19} \]

The expression enclosed in the first parentheses is the energy flux density of the incident wave, while the expression in the second parentheses is the energy flux density of the scattered wave. Thus, the total magnitude of the scattered energy will be:

\[ P_s=\frac{1}{2}\operatorname{Re} \int_{0}^{2\pi}\int_{0}^{\pi} \left(E_\vartheta^s H_\varphi^{s*}-E_\varphi^s H_\vartheta^{s*}\right) r^2\sin\vartheta\,d\vartheta\,d\varphi, \tag{20} \]

where the integral is taken over the surface of a sphere of large radius. Substituting the values of \(E\) and \(H\) from (16), we find:

\[ P_s=\frac{1}{2\eta_2}\operatorname{Re} \int_{0}^{2\pi}\int_{0}^{\pi} \left(|E_\vartheta^s|^2+|E_\varphi^s|^2\right) r^2\sin\vartheta\,d\vartheta\,d\varphi . \tag{21} \]

The magnitude of the energy absorption is determined by the flux of energy into the closed surface surrounding this sphere:

\[ P_{ab}=\operatorname{Re}\int_{0}^{2\pi}\int_{0}^{\pi}(-S_c)\,r^3\sin\vartheta\,d\vartheta\,d\varphi . \tag{22} \]

Since the energy flux carried through a closed surface by the incident wave is zero, it follows from considerations of energy balance that the integral over the closed surface of the group of terms in expression (19) enclosed in the third parentheses must represent the sum of the energy absorbed and scattered by the sphere, i.e.

\[ P_t=P_{ab}+P_s= \]

\[ =-\frac{1}{2}\operatorname{Re}\int_{0}^{2\pi}\int_{0}^{\pi} \left(E_\vartheta^i H_\varphi^{s*}+E_\vartheta^s H_\varphi^{i*} -E_\varphi^i H_\vartheta^{s*}-E_\varphi^s H_\vartheta^{i*}\right) r^3\sin\vartheta\,d\vartheta\,d\varphi . \tag{23} \]

Substituting the values of the field components into (21) and (23), and taking into account that integration with respect to \(\varphi\) reduces to multiplication by \(\pi\), while the integrals of products of the associated Legendre polynomials \(P_n^{(1)}(x)\) are nonzero only when they enter in the form

\[ \int_{0}^{\pi}\left[\left(\frac{P_n^1}{\sin\vartheta}\right)^2+ \left(\frac{dP_n^1}{d\vartheta}\right)^2\right]\sin\vartheta\,d\vartheta = \frac{2}{2n+1}\,[n(n+1)]^2, \]

we obtain

\[ P_s=\frac{\pi E_0^2}{k_2^2\eta_2} \sum_{n=1}^{\infty}(2n+1)\left(|a_n^s|^2+|b_n^s|^2\right), \tag{24} \]

\[ P_t=-\frac{\pi E_0^2}{k_2^2\eta_3}\operatorname{Re} \sum_{n=1}^{\infty}(2n+1)(a_n^s+b_n^s). \tag{25} \]

To determine that part of the scattered energy, calculated per unit solid angle \(\omega\), which propagates in the backward direction \((\vartheta=\pi)\) and represents the energy that can be used for radiolocation purposes, we take the derivative of \(P_s\) with respect to \(\omega\)

\[ \left(\frac{dP_s}{d\omega}\right)_{\vartheta=\pi}, \]

where

\[ d\omega=\sin\vartheta\,d\vartheta\,d\varphi . \]

According to (21) and (16) we have:

\[ \left(\frac{dP_s}{d\omega}\right)_{\theta=\pi} = \frac{E_0^2}{8 k_2^2 \eta}\, \operatorname{Re} \sum_{n=1}^{\infty}\sum_{m=1}^{\infty} (-1)^{n+m}(2n+1)(2m+1) \times \]
\[ \times (a_n^s+b_n^s)(a_m^{s*}-b_m^{s*}). \tag{26} \]

The formulas given for \(P_t\), \(P_s\), and \(\left(\dfrac{dP_s}{d\omega}\right)_{\theta=\pi}\) make it possible to find expressions for the effective cross sections of attenuation, total scattering, and back (radar) scattering for a single spherical particle (drop).

The effective attenuation cross section \(Q_t\) is defined as the ratio of the power \(P_t\), by which the power of the incident wave is diminished as a result of absorption and scattering by the spherical drop, to the power of the incident wave calculated per unit area, i.e., to the energy-flux density of the incident wave.

The energy-flux density of the incident wave is expressed through the complex Umov–Poynting vector in the form

\[ S_{c,z}=\frac{E_0^2}{2\eta_2}. \tag{27} \]

Therefore, substituting the value \(P_t\) from (25), we obtain:

\[ Q=\frac{P_t}{S_{c,z}} = -\frac{\lambda^2}{2\pi}\, \operatorname{Re} \sum_{n=1}^{\infty}(2n+1)(a_n^s+b_n^s), \tag{28} \]

where \(\lambda=\dfrac{2\pi}{k_2}\) is the wavelength in the medium surrounding the drop, and \(k_2^2=\dfrac{4\pi^2}{\lambda^2}\). Similarly, the effective scattering cross section of a single spherical particle is determined, with allowance for (24), as

\[ Q_s=\frac{P_s}{S_{c,z}} = \frac{\lambda^2}{2\pi} \sum_{n=1}^{\infty}(2n+1)\left(|a_n^s|^2+|b_n^s|^2\right). \tag{29} \]

The differential effective cross section for scattering in the backward direction (or the radar effective cross section) is determined analogously from (26):

\[ \left(\frac{dQ_s}{d\omega}\right)_{\theta=\pi} = \sigma(\pi) = \left(\frac{\lambda}{4\pi}\right)^2 \operatorname{Re} \sum_{n=1}^{\infty}\sum_{m=1}^{\infty} (-1)^{n+m}(2n+1)(2m+1) \times \]
\[ \times \left[a_n^{s*}a_m^s+b_n^s b_m^{s*}-2a_n^s b_m^{s*}\right]. \tag{30} \]

Formulas (28), (29), and (30) may be used to compute the attenuation and scattering by spherical particles (drops), taking into account the sizes of the particles in relation to the wavelength.

2. SCATTERING AMPLITUDES \(a_n^s\) AND \(b_n^s\)

Let us consider a possible interpretation of the results obtained. The scattered field \(E^s, H^s\) outside the sphere and the field inside the sphere \(E^t, H^t\) are obtained as the result of forced oscillations of the sphere under the influence of the field of the incident wave \(E^i, H^i\). The fields \(E^s, H^s\) and \(E^t, H^t\) may be regarded as a superposition of the fields of electric and magnetic multipoles of order \(2^n\) (\(n = 1\) corresponds to a dipole, \(n = 2\) to a quadrupole, etc.) induced in the spherical particle. In the steady state these multipoles oscillate with the frequency of the incident field. It can be shown that the amplitudes \(a_n\) are connected with the oscillations of magnetic multipoles, while \(b_n\) are connected with the oscillations of electric multipoles. When the frequency of the incident field approaches the characteristic frequency of the free oscillations of the electric and magnetic multipoles, resonance should occur. The characteristic frequency of the free oscillations is determined from the condition that the denominators in the expressions for \(a_n\) and \(b_n\) vanish (hereafter everywhere \(a_n\) and \(b_n\) mean \(a_n^s b_n^s\), since there is no need to compute \(a_n^t b_n^t\)). However, the characteristic frequencies of the free oscillations in the present case are complex, and the denominators of the expressions for \(a_n\) and \(b_n\), although they decrease, never become equal to zero; therefore difficulties connected with resonance do not arise.

The very form of formulas (13) already shows the complicated character of the expressions for the amplitudes \(a_n\) and \(b_n\). Exact calculation of these coefficients is made difficult by the absence of tables of Bessel and Hankel functions of complex arguments for those values which are needed in particular calculations.

However, these expressions are simplified in the case when the parameter

\[ \rho = \frac{2\pi a}{\lambda} \ll 1. \]

In these cases one may apply the expansion of the functions \(Z_n^{(1)}(\rho)\) and \(Z_n^{(3)}\rho\) in series in increasing powers of \(\rho\) in the form

\[ Z_n^{(1)}(\rho) = 2^n \cdot \rho^n \sum_{m=0}^{\infty} \frac{(-1)^m \cdot (n+m)!}{m!(2n+2m+1)!}\rho^{2m}, \]

\[ Z_n^{(1)}(\rho) = 2^n \cdot \rho^n \sum_{m=0}^{\infty} \frac{(-1)^m \cdot (n+m)!}{m!(2n+2m+1)!}\rho^{2m} + \frac{i}{2^n \rho^{n+1}} \sum_{m=0}^{n} \frac{(2n-2m)!}{m!(n-m)!}\rho^{2m}. \tag{31} \]

Substituting these expressions into (13), putting \(\mu_1 = \mu_2\) and retaining

the first several terms, we obtain expressions for \(a_n\) and \(b_n\):

\[ a_n=-i2^{2n}\left(\frac{n!}{(2n+1)!}\right)^2\cdot\frac{N^2-1}{2n+3}\cdot \rho^{2n+3}\times \]

\[ \times\left[1+\rho^2\left(\frac{N^2-1}{2n+1}-\frac{N^2+1}{2(2n+5)}\right)+\ldots\right], \tag{32} \]

\[ b_n=-i2^{2n}\left(\frac{n!}{(2n+1)!}\right)^2\cdot \frac{(2n+1)(n+1)(N^2-1)}{nN^2+n+1}\rho^{2n+1}\times \]

\[ \times\left[1+\rho^2\frac{(2n+1)[(2n-1)N^2-n-1]}{(2n+3)(2n-1)(nN^2+n+1)}+\ldots\right]= \]

\[ =-i2^{2n}\left(\frac{n!}{(2n+1)!}\right)^2\cdot \frac{(2n+1)(n+1)(N^2-1)}{nN^2+n+1}\rho^{2n+1}+\ldots \tag{33} \]

Neglecting powers of \(\rho\) higher than the sixth, we obtain:

\[ a_1=-\frac{i}{45}(N^2-1)\rho^5; \]

\[ b_1=-\frac{2}{3}i\frac{N^2-1}{N^2+2}\rho^3 \left(1-\frac{3}{5}\frac{N^2-2}{N^2+2}\rho^2-\frac{2}{3}i\frac{N^2-1}{N^2+2}\rho^3\right); \tag{34} \]

\[ b_2=-\frac{i}{15}\frac{N^2-1}{2N^2+3}\rho^5. \]

The quantity \(N\) for the material of the sphere (drop) is related to the complex dielectric constant \(\varepsilon_c\) by the formula

\[ N^2=\varepsilon_c=\varepsilon_r-i\varepsilon_i . \tag{35} \]

Substituting (35) into (33), we obtain:

\[ a_1=\frac{1}{45}[-\varepsilon_i-i(\varepsilon_r-1)]\rho^5, \]

\[ \operatorname{Re} b_1= \frac{-2\varepsilon_i}{(\varepsilon_r+2)^2+\varepsilon_i^2}\rho^3 -\frac{2}{5}\varepsilon_i \frac{[(\varepsilon_r+2)(7\varepsilon_r-10)+7\varepsilon_i^2]} {[(\varepsilon_r+2)^2+\varepsilon_i^2]^2}\rho^5- \]

\[ -\frac{4}{9} \frac{(\varepsilon_r-1)^2(\varepsilon_r+2)^2+\varepsilon_i^2[2(\varepsilon_r-1)(\varepsilon_r+2)-9]+\varepsilon_i^4} {[(\varepsilon_r+2)^2+\varepsilon_i^2]^2}\rho^6, \]

\[ \operatorname{Im} b_1= -\frac{2}{3}\frac{(\varepsilon_r-1)(\varepsilon_r+2)+\varepsilon_i^2} {3(\varepsilon_r+2)^2+\varepsilon_i^2}\rho^3- \]

\[ -\frac{2}{5}\cdot \frac{(\varepsilon_r-1)(\varepsilon_r-2)(\varepsilon_r+2)^2+\varepsilon_i^2[2(\varepsilon_r+1)^2-(3\varepsilon_r+20)]+\varepsilon_i^4} {[(\varepsilon_r+2)^2+\varepsilon_i^2]^2}\rho^5+ \]

\[ +\frac{8}{3}\cdot \frac{\varepsilon_i[(\varepsilon_r-1)(\varepsilon_r+2)+\varepsilon_i^2]} {[(\varepsilon_r+2)^2+\varepsilon_i^2]^2}\rho^6, \]

\[ b_2=\frac{1}{2}\cdot \frac{-\varepsilon_i-\frac{i}{5}\left[(\varepsilon_r-1)(2\varepsilon_r+3)+2\varepsilon_i^2\right]} {(2\varepsilon_r+3)^2+4\varepsilon_i^2}. \tag{36} \]

Table I

\(\lambda\) (cm)	1	1.26	2.00	3.00	5.00	8.00	10.00	15.00
\(\varepsilon_r\)	24.2	32.5	50.6	63.6	73.8	78.00	79.00	81.00
\(\varepsilon_i = 60\sigma\lambda\)	35.6	38.6	38.5	32.7	22.7	15.1	12.3	8.4
\(\sigma\), mho/m	59.3	51.1	32.0	18.1	7.56	3.15	2.05	0.93

The quantities \(\varepsilon_r\) and \(\varepsilon_i\) entering into these formulas depend on the wavelength \(\lambda\), and also on the temperature, the latter dependence being still little studied.

Fig. 2. Dependence of the real \((\varepsilon_r)\) and imaginary \((\varepsilon_i)\) parts of the complex dielectric constant of water at \(18^\circ\text{C}\) on the wavelength \(\lambda\).

Experimental data on the quantities \(\varepsilon_r\) and \(\varepsilon_i\), as well as \(\sigma\), for water at \(t = 18^\circ\text{C}\), as a function of \(\lambda^{15}\), are given in Table I and in Fig. 2.

The temperature dependence of \(\varepsilon_r\) and \(\varepsilon_i\) for water and ice at \(\lambda = 1.25\) cm is given in Table II.

Table II

Temperature in degrees C	3	25	60	−15
\(\varepsilon_r\)	27	35	44	3.3
\(\varepsilon_i\)	27	23	14	0.011

3. ATTENUATION OF MICRORADIO WAVES BY SUSPENSIONS OF SPHERICAL WATER DROPLETS IN THE ATMOSPHERE

The intensity of the electric or magnetic field of a plane wave propagating along the \(z\)-axis in a medium with parameters \(\varepsilon, \mu, \tau\) is expressed by the formula (omitting the factor \(e^{-i\omega t}\))

\[ F = F_0 e^{-i\beta z-\alpha z}, \tag{37} \]

where the real numbers \(\beta = \dfrac{\omega}{c} n\) (the phase constant) and \(\alpha = \dfrac{\omega}{c} \chi\) (the attenuation constant) are related to the complex wave number by the relation

\[ ik = \alpha + i\beta , \tag{38} \]

(\(\chi\) is the absorption index of the medium, \(n\) is its refractive index). Through the parameters of the medium and the frequency \(\omega\) they are expressed as follows:

\[ \beta = \omega \left[\frac{\mu \varepsilon}{2} \left(\sqrt{1+\frac{\sigma^2}{\varepsilon^2 \omega^2}}+1\right)\right]^{\frac12}, \tag{39} \]

\[ \alpha = \omega \left[\frac{\mu \varepsilon}{2} \left(\sqrt{1+\frac{\sigma^2}{\varepsilon^2 \omega^2}}-1\right)\right]^{\frac12}. \tag{40} \]

When a wave propagates in such a medium over a path of length \(\dfrac{1}{\alpha}\) meters, the field intensity decreases by a factor of \(\dfrac{1}{e}=0.368\), while the power per unit surface (the Umov–Poynting vector) decreases by the same amount over a distance \(\dfrac{1}{2\alpha}\) meters.

In the practical system of units (MKS), the attenuation coefficient will be \(a\ \dfrac{\text{neper}}{\text{m}}\), and the power absorption is \(20a \lg e\, F = 8.686a\ \text{db}/\text{m}\).

In our case, the attenuation of microradio waves is studied in a medium that is not homogeneous and isotropic, since it is a suspension of water droplets in the atmosphere and therefore cannot be characterized by the parameters \(\varepsilon, \mu, \sigma\).

It can be shown, however, that in such a medium, under the condition of incoherence of scattering by individual droplets, the total magnitude of the attenuation is the sum of the attenuations by the individual droplets. We shall also denote the total attenuation by the droplets by \(\alpha\). Then it can be determined in the form (for droplets of the same size)

\[ \alpha = \frac{1}{2} N Q_t, \tag{41} \]

where \(N\) is the total number of droplets per unit volume; \(Q_t\) is the effective attenuation cross section of one droplet.

Using expression (28) for \(Q_t\) and measuring the attenuation in \(db/km\), we find:

\[ a=-0.4343\cdot 10^6\,\frac{N\lambda^2}{2\pi}\operatorname{Re}\sum_{n=1}^{\infty}(2n+1)(a_n+b_n)\ db/km. \tag{42} \]

Using approximation (36), one can obtain:

\[ a=0.4343\cdot 10^6\,\frac{3\pi NV}{\lambda}\left(c_1+c_2\rho^3+c_3\rho^3+\ldots\right)\ db/km, \tag{43} \]

where \(V\) is the volume of one droplet, \(D\) is its diameter, and \(\rho=\dfrac{\pi D}{\lambda}\). The coefficients \(c_1,\ c_2,\ c_3\) are determined from:

\[ c_1=\frac{6\varepsilon_i}{(\varepsilon_r+2)^2+\varepsilon_i^2};\quad c_2=\frac{\varepsilon_i}{15}+\frac{5}{3}\frac{\varepsilon_i}{(2\varepsilon_r+3)^2+4\varepsilon_i^2}+ \]

\[ +\frac{6}{5}\frac{\varepsilon_i\left[(\varepsilon_r+2)(7\varepsilon_r-10)+7\varepsilon_i^2\right]} {(\varepsilon_r+2)^2+\varepsilon_i^2}; \]

\[ c_3=\frac{4}{3}\cdot \frac{(\varepsilon_r-1)(\varepsilon_r+2)^2+\varepsilon_i\left[2(\varepsilon_r-1)(\varepsilon_r+2)-9\right]+\varepsilon_i^4} {\left[(\varepsilon_r+2)^2+\varepsilon_i^2\right]}. \tag{44} \]

Taking into account that \(NV\) is the volume of liquid contained in the form of droplets in \(1\ cm^3\), and denoting the mass of water contained in \(1\ m^3\) by \(m=10^6NV\), we find:

\[ a=4.092\cdot\frac{m}{\lambda}\left(c_1+c_2\rho^3+c_3\rho^3+\ldots\right)\ db/km. \tag{45} \]

For \(\rho\ll 1\), all terms in (45) are small in comparison with \(c_1\), and

\[ a_{\rho\ll 1}=\frac{4.092mc_1}{\lambda} =\frac{24.55}{\lambda}\cdot\frac{m\varepsilon_i}{(\varepsilon_r+2)^2+\varepsilon_i^2}\ db/km. \tag{46} \]

Consequently, when \(D\ll\lambda\) (for example, for fogs and ordinary cloudiness), the attenuation depends only on the total mass of liquid water contained in a unit volume of air, but does not depend on the droplet size. The attenuation per \(1\ g/m^3\) is \(0.28\ db/km\) for waves of \(1.25\ cm\), \(0.049\ db/km\) for waves of \(3.2\ cm\), and \(0.0045\ db/km\) for waves of \(10\ cm\).

Expression (46) is valid with an accuracy up to \(10\%\), if

\[ c_2\rho^3\leq \frac{c_1}{10}, \tag{47} \]

provided that the droplet diameter is less than

\[ D_c=\frac{\lambda}{10}\left(\frac{c_1}{c_2}\right)^{\frac12}. \tag{48} \]

The quantities \(c_1,\ c_2\), and \(D_c\) for different \(\lambda\) are given in Table III.

Table III

\(\lambda\) (cm)	1	1.26	2	3	5
\(c_1\)	0.109	0.0862	0.0543	0.0365	0.0217
\(c_2\)	2.53	2.69	2.64	2.23	1.54
\(D_c\) (cm)	0.0656	0.0713	0.0906	0.121	0.187
\(\lambda\) (cm)	8	10	15	25	50
\(c_1\)	0.0137	0.011	0.00724	0.00437	0.00219
\(c_2\)	1.01	0.835	0.570	0.342	0.171
\(D_c\) (cm)	0.293	0.363	0.534	0.892	0.78

The values of \(c_3\) change hardly at all (\(c_3 = 1.224\) for \(\lambda = 1\) cm and \(c_3 = 1.239\) for \(\lambda = 100\) cm).

For drops with diameter greater than \(D_c\), one may use equalities (43) or (45). For \(\rho \simeq 1\), however, these expressions do not ensure an accurate determination of \(Q_t\) and \(\alpha\). In Table IV the values of \(Q_t\) for \(\lambda = 1.25\) and 3 cm are calculated from the exact formulas (42), and for \(\lambda > 5\) cm from formula (43). This table gives the effective attenuation cross sections for values of \(\rho\) from 0.0016 to 1.4 for different \(\lambda\), as functions of \(D\).

To compute attenuation by various forms of atmospheric formations, it is necessary to know the size distribution of the drops and their concentration. The total attenuation is equal to the sum of the attenuations for groups of drops of different diameter

\[ \alpha_{\mathrm{tot}}=\sum_{k=0}^{s}\alpha_k =0.4343\cdot 10^6\sum_{k=0}^{s}N_k\cdot Q_{t,k}\quad \text{db/km}, \tag{49} \]

where \(N_k\) is the number of drops of diameter \(D_k\) cm in \(1\ \mathrm{cm}^3\), and \(Q_{t,k}\) is the effective attenuation cross section (expressed in \(\mathrm{cm}^2\)) of drops of diameter \(D_k\).

The formula for attenuation (49) may be transformed. If \(p_k\) is the partial amount of precipitation from drops of diameter \(D_k\) cm

Table IV

Effective attenuation cross section \(Q_i\) \((\mathrm{cm}^2)\)

\(D\) (cm) / \(\lambda\) (cm)	1.25	3	5	8	10	15
0.05	\(6.19\cdot10^{-5}\)	\(9.19\cdot10^{-6}\)	\(2.84\cdot10^{-6}\)	\(1.09\cdot10^{-6}\)	\(6.9\cdot10^{-7}\)	\(2.98\cdot10^{-7}\)
0.10	\(9.6\cdot10^{-4}\)	\(1.52\cdot10^{-4}\)	\(2.75\cdot10^{-5}\)	\(9.49\cdot10^{-6}\)	\(5.84\cdot10^{-6}\)	\(2.45\cdot10^{-6}\)
0.15	\(5.66\cdot10^{-3}\)	\(1.3\cdot10^{-3}\)	\(1.2\cdot10^{-4}\)	\(3.65\cdot10^{-5}\)	\(2.16\cdot10^{-5}\)	\(8.66\cdot10^{-6}\)
0.20	\(1.89\cdot10^{-2}\)	\(5.53\cdot10^{-3}\)	\(3.79\cdot10^{-4}\)	\(1.02\cdot10^{-4}\)	\(5.76\cdot10^{-5}\)	\(2.18\cdot10^{-5}\)
0.25	\(5.04\cdot10^{-2}\)	\(1.63\cdot10^{-2}\)	\(9.85\cdot10^{-4}\)	\(2.4\cdot10^{-4}\)	\(1.46\cdot10^{-4}\)	\(4.59\cdot10^{-5}\)
0.30	\(1.13\cdot10^{-1}\)	\(3.73\cdot10^{-2}\)	\(2.24\cdot10^{-3}\)	\(4.98\cdot10^{-4}\)	\(2.59\cdot10^{-4}\)	\(8.65\cdot10^{-5}\)
0.35	\(2.15\cdot10^{-1}\)	\(6.65\cdot10^{-2}\)	\(4.59\cdot10^{-3}\)	\(9.63\cdot10^{-4}\)	\(4.81\cdot10^{-4}\)	\(1.51\cdot10^{-4}\)
0.40	\(3.66\cdot10^{-1}\)	\(1.08\cdot10^{-1}\)	\(8.68\cdot10^{-3}\)	\(1.74\cdot10^{-3}\)	\(8.44\cdot10^{-4}\)	\(2.51\cdot10^{-4}\)
0.45	\(5.66\cdot10^{-1}\)	\(0.152\)	\(1.54\cdot10^{-2}\)	\(2.97\cdot10^{-3}\)	\(1.4\cdot10^{-3}\)	\(3.98\cdot10^{-4}\)
0.50	\(7.62\cdot10^{-1}\)	\(0.215\)	\(2.59\cdot10^{-2}\)	\(4.85\cdot10^{-3}\)	\(2.25\cdot10^{-3}\)	\(6.1\cdot10^{-4}\)
0.55	\(1.01\)	\(0.272\)	\(4.18\cdot10^{-2}\)	\(7.63\cdot10^{-3}\)	\(3.47\cdot10^{-3}\)	\(9.06\cdot10^{-4}\)

in rain with total precipitation \(p\), then

\[ p=\sum_{k=0}^{s} p_k, \tag{50} \]

where \(D_s\) is the diameter of the largest drops.

Since

\[ P_k=3.6\cdot 10^6 \cdot V_k \cdot v_k N_k \ \text{mm/hour}, \tag{51} \]

where \(V_k\) is the volume of a drop of diameter \(D_k\), and \(v_k\) is the terminal velocity of the drop in \(\text{m/sec}\). Substituting \(N_k\) from (51) into (49), we find the attenuation in rain with precipitation \(P\):

\[ \alpha_{\lambda,p}=\sum_{k=0}^{s}\alpha_{\lambda}(p_k)=\frac{0.4343}{3.6}\sum_{k=0}^{s}\frac{P_k Q_{t,k}}{V_k v_k}. \tag{52} \]

For a given wavelength \(\lambda\),

\[ \frac{Q_{t,k}}{V_k v_k}=\mathrm{const}=q_k \]

and

\[ \alpha_p=0.126\sum_{k=0}^{s}p_k q_k. \tag{53} \]

This expression shows that the attenuation in rain with precipitation \(p\) mm/hour depends linearly on the partial precipitation values \(p_k\) for the various groups of drop sizes composing the given rain, but there is no direct dependence of the attenuation on the total precipitation. Points plotted from observations on the \((\alpha, p)\) plane cover a definite region of this plane, and a curve drawn through these points is meaningful only insofar as it permits one to predict the probable value of the attenuation for rain with a given precipitation value, since such a curve does not represent any exact physical dependence between \(\alpha\) and \(p\).

Experimental measurements of attenuation for \(\lambda=3.2\) cm in rain\({}^{16}\) showed a scatter of attenuation values for rains with the same precipitation, which confirms the above statements.

The average attenuation value for 3.2-cm waves proved to be 0.031 db/km per 1 mm/hour of rainfall. Theoretical values\({}^{17}\), calculated with allowance for the distribution of drop sizes, proved to be substantially smaller. Thus, for example, the attenuation in light rain (1.25 mm/hour) was 0.012 db/km, in moderate rain (5.00 mm/hour) 0.074 db/km, and in heavy rain (12.5 mm/hour) 0.24 db/km.

4. SCATTERING OF MICRORADIO WAVES BY SPHERICAL WATER DROPLETS

The effective scattering cross section by spherical particles is determined by equation (29).

Using the approximate expressions for the amplitudes (34) and (36), and applying the notation \(\alpha_1^{(5)}, \bar{\alpha}_1^{(5)}, \beta_1^{(3)}, \bar{\beta}_1^{(3)}\), etc., representing the real and imaginary parts of the coefficients at \(\rho^5\) in \(a_1\) and \(\rho^3\) in \(b\), we obtain the following expression for the effective scattering cross section:

\[ Q_s=\frac{\lambda^2}{2\pi}\rho^6\left\{ 3|\beta_1^{(3)}|^2 +6\left[\beta_1^{(3)}\beta_1^{(5)}+\bar{\beta}_1^{(3)}\bar{\beta}_1^{(5)}\right]\rho^2 +\right. \]

\[ \left. +6\left[\beta_1^{(3)}\beta_1^{(6)}+\bar{\beta}_1^{(3)}\bar{\beta}_1^{(6)}\right]\rho^3 +\left[3\left(|\alpha_1^{(5)}|^2+|\beta_1^{(5)}|^2\right)+5|\beta_2^{(5)}|^2\right]\rho^4 +\right. \]

\[ \left. +6\left[\beta_1^{(5)}\beta_1^{(6)}+\bar{\beta}_1^{(5)}\bar{\beta}_1^{(6)}\right]\rho^5 +3|\beta_1^{(6)}|^2\rho^6+\ldots \right\}\ \text{cm}^2, \tag{54} \]

where \(|\beta_1^{(3)}|=(\beta_1^{(3)})^2+(\bar{\beta}_1^{(3)})^2\), etc.

For values \(\rho \ll 1\), one may neglect \(\rho^2\) and higher powers in the brackets and, applying the expanded expressions for \(\beta_1^{(3)}\) and \(\bar{\beta}_1^{(3)}\), obtain:

\[ Q_{s,\rho\ll 1}= \frac{128\pi^5a^6}{3\lambda^4}\times \]

\[ \times \frac{(\varepsilon_r-1)^2(\varepsilon_r+2)^2+\varepsilon_i[2(\varepsilon_r-1)(\varepsilon_r+2)+9]+\varepsilon_i^4} {\left[(\varepsilon_r+2)^2+\varepsilon_i^2\right]^2} \ \text{cm}^2. \tag{55} \]

For \(\varepsilon_i \to 0\),

\[ Q_{s,\rho\ll 1\,\varepsilon_i\to 0} = \frac{128\pi^5a^6}{3\lambda^4} \left(\frac{n^2-1}{n^2+2}\right)^2, \tag{56} \]

where \(n^2=\varepsilon_r\).

This is the well-known expression for the effective scattering cross section derived by Rayleigh\({}^{18}\).

Table V gives the values of \(Q_s\) for \(\rho\) from 0.00157 to 0.576, i.e., for droplets of diameter \(0.05\)—\(0.50\ \text{cm}\) for different \(\lambda\). At large \(\rho\), the actual cross section, according to (56), is always greater than the Rayleigh one. For \(\rho<0.10\), the effective cross section, according to (55), is given with an accuracy of up to several percent.

Calculation by the exact formula (54) is difficult, since the coefficients at \(\rho\) themselves depend on \(D\) and \(\lambda\).

Knowledge of the effective scattering cross section \(Q_s\) and the effective attenuation cross section \(Q_t\) makes it possible to calculate the relative

Table V

Effective scattering cross sections \(Q_s\) \((\text{cm}^2)\)

\(D\) (cm) \(\backslash\) \(\lambda\) (cm)	3	5	8	10	15	30	50
0.05	\(3.62\cdot10^{-8}\)	\(4.7\cdot10^{-9}\)	\(7.23\cdot10^{-10}\)	\(2.93\cdot10^{-10}\)	\(5.8\cdot10^{-11}\)	\(3.62\cdot10^{-12}\)	\(4.65\cdot10^{-13}\)
0.10	\(2.35\cdot10^{-6}\)	\(3.04\cdot10^{-7}\)	\(4.64\cdot10^{-8}\)	\(1.88\cdot10^{-8}\)	\(3.75\cdot10^{-9}\)	\(2.32\cdot10^{-10}\)	\(2.99\cdot10^{-11}\)
0.15	\(2.74\cdot10^{-5}\)	\(3.51\cdot10^{-6}\)	\(5.35\cdot10^{-7}\)	\(2.15\cdot10^{-7}\)	\(4.33\cdot10^{-8}\)	\(2.66\cdot10^{-9}\)	\(3.44\cdot10^{-10}\)
0.20	\(1.58\cdot10^{-4}\)	\(1.97\cdot10^{-5}\)	\(2.98\cdot10^{-6}\)	\(1.21\cdot10^{-6}\)	\(2.43\cdot10^{-7}\)	\(1.49\cdot10^{-8}\)	\(1.91\cdot10^{-9}\)
0.25	\(6.06\cdot10^{-4}\)	\(7.56\cdot10^{-5}\)	\(1.14\cdot10^{-5}\)	\(4.62\cdot10^{-6}\)	\(9.03\cdot10^{-7}\)	\(5.66\cdot10^{-8}\)	\(7.29\cdot10^{-9}\)
0.30	\(1.98\cdot10^{-3}\)	\(2.32\cdot10^{-4}\)	\(3.44\cdot10^{-5}\)	\(1.38\cdot10^{-5}\)	\(2.7\cdot10^{-6}\)	\(1.69\cdot10^{-7}\)	\(2.18\cdot10^{-8}\)
0.35	\(5.36\cdot10^{-3}\)	\(5.97\cdot10^{-4}\)	\(8.72\cdot10^{-5}\)	\(3.50\cdot10^{-5}\)	\(6.8\cdot10^{-6}\)	\(4.11\cdot10^{-7}\)	\(5.51\cdot10^{-8}\)
0.40	\(1.31\cdot10^{-2}\)	\(1.36\cdot10^{-3}\)	\(1.96\cdot10^{-4}\)	\(7.85\cdot10^{-5}\)	\(1.52\cdot10^{-5}\)	\(9.51\cdot10^{-7}\)	\(1.22\cdot10^{-7}\)
0.45	\(2.96\cdot10^{-2}\)	\(2.86\cdot10^{-3}\)	\(4.01\cdot10^{-4}\)	\(1.59\cdot10^{-4}\)	\(3.1\cdot10^{-5}\)	\(1.92\cdot10^{-6}\)	\(2.48\cdot10^{-7}\)
0.50	\(6.36\cdot10^{-2}\)	\(5.61\cdot10^{-3}\)	\(7.65\cdot10^{-4}\)	\(3.01\cdot10^{-4}\)	\(5.87\cdot10^{-5}\)	\(3.62\cdot10^{-6}\)	\(4.65\cdot10^{-7}\)

the probability of scattering \(\bar{\omega}_s\) and absorption \(\bar{\omega}_{\text{abs}}\) of a wave by a drop:

\[ \bar{\omega}_s=\frac{Q_s}{Q_t}, \tag{57} \]

\[ \bar{\omega}_{\text{abs}}=1-\bar{\omega}_s. \tag{58} \]

Calculations show that, with the exception of the very shortest waves and large drops, the probability of absorption is much greater than the probability of scattering. The smaller the diameter of the drops, the greater the probability of absorption, since for small drops \(Q_s \sim \dfrac{D^6}{\lambda^4}\), whereas \(Q_t \sim Q_{\text{abs}} \sim \dfrac{D^3}{\lambda}\). In our case \(D\) is always smaller than \(\lambda\).

5. BACK (RADAR) SCATTERING

The effect of attenuation of microwaves in the atmosphere and in atmospheric formations is important both for communications and for radar. The phenomenon of back scattering is important only for radar, since it is precisely this phenomenon that makes it possible to detect and study certain atmospheric formations by radar methods.

The amplitude of the reflected signal can be calculated from formula (30) for the differential effective cross section of a drop \(\sigma(\pi)\), corresponding to the case of back scattering (scattering angle \(\pi\)).

According to equation (21), the energy scattered by a spherical particle per unit solid angle \(\omega\) in the direction \((\vartheta,\varphi)\) is

\[ \left(\frac{dP_s}{d\omega}\right)_{\vartheta,\varphi} = \frac{1}{2\eta_2} \left[\,|E_\vartheta^s|^2+|E_\varphi^s|^2\,\right]r^2. \tag{59} \]

Using equations (15) and (16) and knowing that the energy incident on unit area is equal to \(\dfrac{E_0^2}{2\eta_2}\), we obtain the following expressions for the differential effective cross section:

\[ \left(\frac{dQ_s}{d\omega}\right)_{\vartheta,\varphi} = \sigma(\vartheta,\varphi) = \frac{\lambda}{2\pi}\operatorname{Re} \sum_{n=1}^{\infty}\sum_{m=1}^{\infty} \frac{(2n+1)(2m+1)}{n(n+1)\,m(m+1)} \times \]

\[ \times \left[ a_n a_m^{*} \left( \frac{P_n^1 P_m^1}{\sin^2\vartheta}\cos^2\varphi + \frac{dP_n^1}{d\vartheta}\cdot\frac{dP_m^1}{d\vartheta}\sin^2\vartheta \right) + \right. \]

\[ \left. + b_n b_m^{*} \left( \frac{P_n^1 P_m^1}{\sin^2\vartheta}\sin^2\varphi + \frac{dP_n^1}{d\vartheta}\cdot\frac{dP_m^1}{d\vartheta}\cos^2\varphi \right) + \right. \]

\[ \left. + 2a_n b_m^{*} \left( \frac{P_n^1}{\sin\vartheta}\cdot\frac{dP_m^1}{d\vartheta}\cos^3\varphi + \frac{P_m^1}{\sin\vartheta}\cdot\frac{dP_n^1}{d\vartheta}\sin^3\varphi \right) \right] \ \text{cm}^2. \tag{60} \]

(For \(\vartheta=\pi\) this equation becomes equation (26) for the radar effective cross section.)

Restricting ourselves to the approximation in which only the influence of the electric dipole and quadrupole (\(b_1\) and \(b_2\)) and the magnetic dipole (\(a_1\)) is taken into account, and using the expression for the associated Legendre polynomials, we obtain:

\[ \left(\frac{dQ_s}{d\omega}\right)_{\vartheta,\varphi} =\sigma(\vartheta,\varphi)= \left(\frac{\lambda}{4\pi}\right)^2 \operatorname{Re}\left[ 9|b_1|^2(\sin^2\varphi+\cos^2\vartheta\cos^2\varphi)+ \right. \]
\[ \left. +9|a_1|^2(\cos^2\varphi+\cos^2\vartheta\sin^2\varphi) +25|b_2|^2(\cos^2\vartheta\sin^2\varphi+ \right. \]
\[ \left. +\cos^2(2\vartheta)\cos^2\varphi) +18a_1 b_1^{*}\cos\vartheta +30b_1 b_2^{*}(\cos\vartheta\sin^2\varphi+ \right. \]
\[ \left. +\cos(2\vartheta)\cos^2\varphi) +30a_1 b_2^{*}[\cos^2\vartheta\sin\varphi+\cos(2\vartheta)\cos^2\varphi] \right]\ \text{cm}^2 . \tag{61} \]

In this expression the first term in brackets gives the action of the electric dipole, the second—the magnetic dipole, and the third—the electric quadrupole, while the last three terms correspond to the interaction between them.

The differential effective cross section for backscattering is obtained from (61) at \(\vartheta=\pi\). Using the expanded expressions (36) for the amplitudes \(a_1\), \(b_1\), and \(b_2\), we obtain the radar effective cross section

\[ \sigma(\pi)= \left(\frac{\lambda}{4\pi}\right)^2 \rho^6\left(A_0+A_2\rho^2+A_3\rho^3+A_4\rho^4+A_5\rho^5+A_6\rho^6+\ldots\right)\ \text{cm}^2, \tag{62} \]

where the coefficients \(A_n\) are defined as

\[ A_0=9|\beta_1^{(3)}|^2;\quad A_2=18[\beta_1^{(3)}\beta_1^{(5)}+\bar{\beta}_1^{(3)}\bar{\beta}_1^{(5)} -\alpha_1^{(5)}\beta_1^{(3)}-\bar{\alpha}_1^{(5)}\bar{\beta}_1^{(3)}]- \]

\[ -30\ \beta_1^{(3)}\beta_2^{(5)}+\bar{\beta}_1^{(3)}\bar{\beta}_2^{(5)}]; \quad A_3=18[\beta_1^{(3)}\beta_1^{(6)}-\bar{\beta}_1^{(3)}\bar{\beta}_1^{(6)}]; \]

\[ A_4=9[|\alpha_1^{(5)}|^2+|\beta_1^{(5)}|^2] -18[\alpha_1^{(5)}\beta_1^{(5)}+\bar{\alpha}_1^{(5)}\bar{\beta}_1^{(5)}]- \]

\[ -30[\beta_1^{(5)}\beta_2^{(5)}+\bar{\beta}_1^{(5)}\bar{\beta}_2^{(5)} -\alpha_1^{(5)}\beta_2^{(5)}-\bar{\alpha}_1^{(5)}\bar{\beta}_2^{(5)}] +25|\beta_2^{(5)}|^2; \]

\[ A_5=18[\beta_1^{(5)}\beta_1^{(6)}+\bar{\beta}_1^{(5)}\bar{\beta}_1^{(6)} -\alpha_1^{(5)}\beta_1^{(6)}-\bar{\alpha}_1^{(5)}\bar{\beta}_1^{(6)}]- \]

\[ -30[\beta_1^{(6)}\beta_2^{(5)}+\bar{\beta}_1^{(6)}\bar{\beta}_2^{(5)}]; \]

\[ A_6=9|\beta_1^{(6)}|^2 . \tag{63} \]

In formulas (63) the quantities \(\alpha\) and \(\beta\) are taken from (54). Thus, \(\sigma(\pi)\) is not a function only of \(\rho\), since the coefficients at \(\rho\) also depend on \(\lambda\). The calculated values of \(\sigma(\pi)\) for drops with diameter \(0.05\)—\(0.55\ \text{cm}\) are given in Table VI.

The values of the effective cross sections for small drops at any \(\lambda\) and for any drops at \(\lambda>15\ \text{cm}\) are practically equal to the Rayleigh effective cross section, since in this case only the first term remains in (62).

The character of the variation of \(\sigma(\pi)\) as a function of \(D\) and \(\lambda\) is approximately the same as the character of the variation of the total effective cross section

Table VI

Radar scattering cross section (cm²)

\(D\) (cm) / \(\lambda\) (cm)	3	5	8	10	15	20	50
0.05	\(4.25\cdot10^{-9}\)	\(5.55\cdot10^{-10}\)	\(8.63\cdot10^{-11}\)	\(3.5\cdot10^{-11}\)	\(6.96\cdot10^{-12}\)	\(2.18\cdot10^{-12}\)	\(5.6\cdot10^{-14}\)
0.10	\(2.64\cdot10^{-7}\)	\(3.52\cdot10^{-8}\)	\(5.47\cdot10^{-9}\)	\(2.24\cdot10^{-9}\)	\(4.44\cdot10^{-10}\)	\(1.4\cdot10^{-10}\)	\(3.59\cdot10^{-12}\)
0.15	\(2.88\cdot10^{-6}\)	\(3.97\cdot10^{-7}\)	\(6.28\cdot10^{-8}\)	\(2.54\cdot10^{-8}\)	\(5.1\cdot10^{-9}\)	\(1.6\cdot10^{-9}\)	\(4.12\cdot10^{-11}\)
0.20	\(1.48\cdot10^{-5}\)	\(2.15\cdot10^{-6}\)	\(3.45\cdot10^{-7}\)	\(1.42\cdot10^{-7}\)	\(2.84\cdot10^{-8}\)	\(8.94\cdot10^{-9}\)	\(2.29\cdot10^{-10}\)
0.25	\(5.02\cdot10^{-5}\)	\(7.42\cdot10^{-6}\)	\(1.30\cdot10^{-6}\)	\(5.34\cdot10^{-7}\)	\(1.07\cdot10^{-7}\)	\(3.42\cdot10^{-8}\)	\(8.72\cdot10^{-10}\)
0.30	\(1.34\cdot10^{-4}\)	\(2.25\cdot10^{-5}\)	\(3.8\cdot10^{-6}\)	\(1.57\cdot10^{-6}\)	\(3.19\cdot10^{-7}\)	\(1.2\cdot10^{-7}\)	\(2.62\cdot10^{-9}\)
0.35	\(2.48\cdot10^{-4}\)	\(5.4\cdot10^{-5}\)	\(9.37\cdot10^{-6}\)	\(3.91\cdot10^{-6}\)	\(8.01\cdot10^{-7}\)	\(2.58\cdot10^{-7}\)	\(6.53\cdot10^{-9}\)
0.40	\(5.04\cdot10^{-4}\)	\(1.12\cdot10^{-4}\)	\(2.03\cdot10^{-5}\)	\(8.55\cdot10^{-6}\)	\(1.77\cdot10^{-6}\)	\(5.75\cdot10^{-7}\)	\(1.46\cdot10^{-8}\)
0.45	\(7.76\cdot10^{-4}\)	\(2.12\cdot10^{-4}\)	\(3.99\cdot10^{-5}\)	\(1.7\cdot10^{-5}\)	\(3.55\cdot10^{-6}\)	\(1.16\cdot10^{-6}\)	\(3.00\cdot10^{-8}\)
0.50	\(9.91\cdot10^{-4}\)	\(3.65\cdot10^{-4}\)	\(7.3\cdot10^{-5}\)	\(3.14\cdot10^{-5}\)	\(6.63\cdot10^{-6}\)	\(2.18\cdot10^{-6}\)	\(5.6\cdot10^{-8}\)
0.55	—	\(5.82\cdot10^{-4}\)	\(1.24\cdot10^{-4}\)	\(5.44\cdot10^{-5}\)	\(1.16\cdot10^{-5}\)	\(3.87\cdot10^{-6}\)	\(9.98\cdot10^{-8}\)

scattering \(Q_s\) by (54), except for large drops and small \(\lambda\). For these cases expression (62) gives only the order of magnitude, since series (62) converges too slowly (more slowly than for \(Q_s\)). To calculate the radar attenuation \(a\), i.e., the ratio of the field strength of the reflected signal to the field strength of the incident wave, one may use equation (41), and for \(N_k\) drops of diameter \(D_k\) cm we obtain:

\[ 2_{\pi,k} = \frac{1}{2} N_k \cdot \sigma_k(\pi)\ \frac{\text{neper}}{\text{cm}}, \tag{64} \]

and, for a given distribution of drop sizes,

\[ \sigma_{\pi}=\sum_{k=0}^{s} a_{\pi,k} = \frac{1}{2}\sum_{k=0}^{s} N_k \cdot \sigma_k(\pi)\ \frac{\text{neper}}{\text{cm}} . \tag{65} \]

Since the coefficient of radar attenuation of power \(2a_{\pi}\) in backscattering also represents that fraction of the incident energy which is scattered backward by a unit thickness of the scattering medium, the data from Table VI may be used to determine the magnitude of the reflected signal at the radar under given conditions.

The magnitude of the reflected energy is determined for a layer of thickness \(\Delta x\) from the equation

\[ \Delta P_{\pi} = -2a_{\pi}\cdot P_i\cdot \Delta x, \tag{66} \]

where \(P_i\) is the incident energy.

The fraction of energy reflected by a layer of thickness \(\Delta x=1\) km will be:

\[ \Delta P_{\pi}=2a_{\pi}\cdot P_i \tag{67} \]

or

\[ 10\lg \frac{\Delta P_{\pi}}{P_i} = -10\lg(2a_{\pi})\ \text{db}. \tag{68} \]

6. SOME PHYSICAL DATA ON ATMOSPHERIC FORMATIONS

As already indicated, to calculate attenuation and scattering by various forms of atmospheric formations it is necessary to know the distribution of drops by size and their concentration. For drops of very small size \((D \ll \lambda)\), the attenuation values, according to (46), do not depend on the drop sizes, but only on the total mass of liquid water per unit volume of the atmosphere.

Experimental data show that the water content in clouds of the ordinary type (except rain clouds) is \(0.15—0.50\ \text{g}/\text{m}^3\), and in fog \(0.006—0.01\ \text{g}/\text{m}^3\), with an average drop size from 6 to 15 microns\(^{19}\).

Formulas (62) for the radar effective cross section show the presence of a strong dependence of \(\sigma(\pi)\) on drop size.

In the theory of radar detection of atmospheric formations it will be shown that the magnitude of the reflected signal is approximately proportional to the quantity \(N\overline{a^6}\), where \(\overline{a^6}\) is the mean value of the sixth power of the radius and \(N\) is the number of drops per unit volume. For cloudiness of the usual type the quantity \(N\overline{a^6}\) is of the order of \(10^{-17}\)—\(10^{-16}\), and for rains, \(10^{-11}\)—\(10^{-9}\). In practice, clouds of the usual type (with the exception of rain clouds) are not detected by modern radar stations even directly above the station. Therefore the question of the distribution of drops by size is of no significance for the radar detection of such atmospheric formations as clouds, since radar detection of atmospheric formations is possible only when the sizes of the drops exceed \(0.5\) mm.

In the radar detection of rains and rain clouds, the question of the distribution of drop sizes is of great importance for determining the functional dependence between the magnitude of the reflected signal and the total amount of rainfall. Determination of this dependence is connected with determining the terminal velocity of fall of drops in the atmosphere and their instantaneous concentration.

The terminal velocity of fall of a drop \(v\) is determined from the condition of equilibrium between the force of gravity and the resistance of the air \(^{20}\):

\[ \frac{1}{6}\pi D^3 \rho_w g = \frac{1}{4} C_d \cdot \pi D^2 \cdot \rho v^2, \tag{69} \]

where \(\rho_w\) is the density of water, \(g\) is the acceleration due to gravity, \(C_d\) is the drag coefficient of a spherical drop, and \(\rho\) is the density of air.

From (69) one can obtain the dependence of the terminal velocity on the diameter of the drop:

\[ v^3 = k^3 D, \tag{70} \]

where

\[ k^3=\frac{2}{3}\frac{\rho_w g}{\rho C_d}. \]

According to experimental data, equation (70) is valid up to values \(D = 4\) mm. At larger drop sizes, air resistance causes their deformation and an increase in resistance, as a result of which the velocity for large drops already changes almost not at all with the growth of their sizes \(^{22}\) (Fig. 3). The fraction \(R_D\), contributed by drops of each size to the total precipitation, is the volume of water contained in all drops of this size that fall through a horizontal cross-section per unit time, expressed as the depth of a layer of water over the area of the cross-section. It is equal to the volume of water corresponding to all drops of diameter \(D\) in a unit volume, multiplied by the velocity of fall of these drops:

\[ R_D=\frac{1}{6}\pi N D^3 v = k_1 N D^{\frac{7}{2}}. \tag{71} \]

where

\[ k_1=\frac{1}{6}\pi k=0.745\cdot 10^3\ \mathrm{cm}^{1/2}\,\mathrm{sec}^{-1}. \]

The distribution of drops by size in rains with different precipitation intensity \(p\) mm/hour was studied experimentally, and the results are given in Table VII as percentages of the total number of drops\(^{23}\).

Fig. 3. Dependence of the established velocity of raindrops on their diameter.

Table VII

\(D\) (cm)	\(p\) (mm/hour) 0.25	1.25	2.5	12.5	25	50	100	150
0.05	28.00	10.9	7.3	2.6	1.7	1.2	1.00	1.00
0.10	50.1	37.1	27.8	11.5	7.6	5.4	4.6	4.1
0.15	18.2	31.3	32.8	24.5	18.5	12.5	8.8	7.6
0.20	3.00	13.5	19.00	25.4	23.9	19.9	13.9	11.7
0.25	0.7	4.9	7.9	17.3	19.9	20.9	17.1	13.9
0.30		1.5	3.3	10.1	12.8	15.6	18.4	17.7
0.35		1.6	1.1	4.3	8.2	10.9	15.00	16.1
0.40		0.2	0.6	2.3	3.5	6.7	9.00	11.9
0.45			0.2	1.2	2.1	3.3	5.8	7.7
0.50				0.6	1.1	1.8	3.00	3.6
0.55				0.2	0.5	1.1	1.7	2.2
0.60					0.3	0.5	1.0	1.2
0.65						0.2	0.7	1.00
0.70								0.3

On the basis of these data an experimental curve was constructed for the percentage distribution of the volume of rain falling on a horizontal surface over intervals \(\delta D\) of the diameters of the drops forming this rain. From this curve one can derive the general law of distribution of drops by size in the form of the function

\[ \ln \frac{\bar D R_D}{R}=k_0^2 u^3, \tag{72} \]

where \(\bar D\) is the mean value of the drop diameter, more precisely that value of it such that one half of the precipitation is formed by drops of smaller and one half by drops of larger diameter, \(k_0\) is a constant, the same for all rains, and

\[ u=\left(\frac{D}{\bar D}\right)^{\frac12}-1. \]

Fig. 4. Distribution of drops in rain by size. The points are experimental data; the solid line corresponds to the formula

\[ \ln \frac{\bar D R_D}{R}=k_0^2 u^3 \quad \text{for} \quad k_0^2=11.5. \]

Equation (72) gives the distribution of drops by size, since \(R_D\) can be expressed in terms of \(N\) and \(D\) from (71). The dependence \(\ln \frac{\bar D R_D}{R}\) on \(u^3\), calculated from the data of Table VII, is given in Fig. 4. From

it is seen from the figure that the linear dependence (72) is obeyed fairly well for the value \(k_0^2=11.5\). This result indicates that rain can be described with sufficient completeness by the mean drop diameter and the magnitude of the total precipitation.

Let us now try to obtain the relation between the quantity \(Z=\Sigma N D^6\cdot \delta D\), which determines the magnitude of the reflected signal, and the magnitude of the rainfall \(R\).

From (71) and (72) one can obtain

\[ ND^{\frac{7}{2}}=k_1^{-1}R\overline{D}^{-1}e^{-k_0^2u^2}, \tag{73} \]

\(Z\) can be expressed in the form of the integral

\[ Z=\int_0^\infty ND^6\cdot \delta D, \tag{74} \]

where the limits of integration 0 and \(\infty\) have been chosen for convenience of calculation, since this gives a very small error, because drops whose diameters are less than \(0.5\) mm have almost no effect on the magnitude of the reflected signal, while drops with diameters exceeding \(7\) mm are very few in number. According to (73), (74) can be rewritten in the form

\[ Z=k_1^{-1}R\overline{D}^{-1}\int_0^\infty D^{\frac{5}{2}}e^{-k_0^2u^2}\cdot \delta D, \]

or, since \(D=\overline{D}(u+1)^2\),

\[ Z=2k_1^{-1}R\overline{D}^{-\frac{5}{2}}I_6, \tag{75} \]

where the integral

\[ I_n=\int_{-1}^{\infty}(u+1)^n e^{-k_0^2u^2}\,du \]

can be expressed through

\[ I_0=\int_{-1}^{\infty} e^{-k_0^2u^2}\,du \]

by integration by parts.

For the rainfall intensity \(R=\sum_D R_D\), from (71) we obtain:

\[ R=k_1\sum ND^{\frac{7}{2}}\cdot \delta D=k_1\overline{N}\overline{D}^{\frac{7}{2}}, \tag{76} \]

where \(\overline{N}\) is the number of drops with mean diameter \(\overline{D}\), giving the same total precipitation as the actual rain.

Eliminating \(\overline{D}\) from (75) and (76), we find:

\[ Z=\left(2k_1^{-\frac{12}{7}}\cdot I_6\cdot \overline{N}^{-\frac{5}{7}}\right)\cdot R^{\frac{12}{7}}. \tag{77} \]

According to experimental data \(^{34}\),

\[ Z=190R^{1.72}, \tag{78} \]

where \(Z\) is expressed in \(\text{mm}^6/\text{m}^3\) and \(R\) in \(\text{mm}/\text{hour}\).

If (78) is converted to CGS units and the exponent \(12/7\) is used instead of 1.72, then

\[ Z=1.24\cdot 10^{-2}\cdot R^{\frac{12}{7}} . \tag{79} \]

Calculation of \(I_0\) for \(k_0^2=11.5\) and comparison of (77) with the empirical expression (78) make it possible to determine the value of the mean number of drops in \(1\ \mathrm{cm}^3\): \(\overline{N}=1.7\cdot 10^{-4}\), which is constant for any rains. To verify the correctness of the value \(\overline{N}\), one may use the independently found empirical relation\({}^{24}\) connecting the precipitation intensity \(R\) with the mass of water \(M\) contained in a unit volume of the atmosphere:

\[ M=\frac{1}{6}\pi\rho_w \sum ND^3\cdot \delta D=80 R^{0.83}, \tag{80} \]

where \(M\) is expressed in \(\mathrm{mg}/\mathrm{m}^3\) and \(R\) in \(\mathrm{mm}/\mathrm{hour}\).

If this expression is converted to CGS units, then the quantity \(V=\sum ND^3\cdot \delta D\) will be equal to

\[ V=1.24\cdot 10^{-3}\cdot R^{0.83}. \tag{81} \]

If the summation is replaced by an integral, then, according to (73),

\[ V=k_1^{-1}R\overline{D}^{-1}\int_{0}^{\infty} D^{-\frac{1}{2}} e^{-k_0^2u^2}\,\delta D, \]

or, using the notation from (75),

\[ V=2k_1^{-1}R\cdot \overline{D}^{-\frac{1}{2}}\cdot I_0 . \tag{82} \]

By (76),

\[ \overline{D}^{-\frac{1}{2}}=k_1^{\frac{1}{7}}\overline{N}^{\frac{1}{7}}R^{-\frac{1}{7}}, \]

whence

\[ V=\left(2k_1^{-\frac{6}{7}}N^{-\frac{2}{7}}I_0\right)R^{\frac{6}{7}} . \tag{83} \]

Applying the previously obtained value \(\overline{N}=1.7\cdot 10^{-4}\), we finally obtain:

\[ V=\sum ND^3\cdot \delta D=1.00\cdot 10^{-3}\cdot R^{\frac{6}{7}} . \tag{84} \]

This expression is to be compared with (81).

If equation (81) had been derived for \(R^{6/7}\) instead of \(R^{0.83}\), then the coefficient of the equation would have had to be reduced by 10%. Thus, (84) and (87) give sufficiently good agreement.

In order to describe rain completely, it is necessary, in addition to the distribution function (72), to know \(\overline{N}\) as a function of \(\overline{D}\). Concerning this dependence there are contradictory data. Experimental data\({}^{23}\) gave a dependence between

with the mean diameter of the drops and the rainfall rate in the form

\[ \overline{D}=2.23\cdot R^{0.182}, \tag{85} \]

where \(\overline{D}\) is expressed in millimeters and \(R\) in inches/hour. If (85) is expressed in CGS units and the exponent \(0.182\) is replaced by \(2/11\), then

\[ R=2.7\,\overline{D}^{\frac{11}{2}}. \tag{86} \]

Substituting into (76), we obtain:

\[ \overline{N}=2.7\,k^{-1}\overline{D}^{3}=3.63\cdot 10^{3}\cdot \overline{D}^{3}. \tag{87} \]

On the other hand, experimental data, dating back to 1904\({}^{25}\), show the constancy of the number of drops for sufficiently long rains, independently of the mean diameter of the drops. These data are given in Table VIII.

Table VIII

Rain characteristic	Number of drops in \(1\,m^{3}\)	Rainfall rate in mm/hour
Ordinary rain	428	5.4
Ordinary rain	605	3.6
Rain with interruptions (with sun)	109	6.6
Beginning of a short rain	119	3.0
Sudden heavy rain from a small cloud	81	19.2
Heavy rain (downpour)	472	43.00
Heavy prolonged rain	475	34.00
Less intense prolonged rain	508	20.4
End of a prolonged heavy rain	97	15.6

Thus, for sufficiently long, established rains, the number of drops contained in \(1\,m^{3}\) is approximately 500, independently of the rainfall rate.

The empirical relations derived on the basis of observations between the quantity \(Z=\sum ND^{6}D\) (in \(\text{mm}^{6}/\text{m}^{3}\)), which determines the amplitude of the reflected signal, and the rainfall rate \(R\) (mm/hour) make it possible to make an assumption about the proportionality of \(\overline{N}\) and \(\overline{D}\).

In Table IX the principal empirical dependences of \(Z\) on \(R\) are given.

Table IX

Author	Year	Place of observation
Laws and Parsons	1943	USA	\(320\,R^{1.44}\)
Marshall and Palmer	1948	Canada	\(220\,R^{1.60}\)
Anderson	1947	Hawaiian Islands	\(208\,R^{1.53}\)
Best	1947	England	\(224\,R^{1.54}\)

If we assume the validity of the relation \(\overline N=c\overline D\), then, eliminating \(\overline N\) from (76) and (77), we obtain:

\[ Z=(2k_1^{-1.55}c^{-0.55}I_6)R^{1.55}. \tag{88} \]

Thus, the assumption of proportionality of \(\overline N\) and \(\overline D\) apparently best agrees with the various experimental relations. The quantity \(Na^6\) can in each case be calculated from the law of distribution of drop sizes. If the terminal velocity of the \(i\)-th group of drops is \(v_i\), then the number of drops per unit volume of air will be \(\dfrac{n_i}{v_i}\), where \(n_i\) is the number of drops of the \(i\)-th group falling on a unit area of the ground per unit time. Then the function \(Na^6\) is determined by summing the products \(\dfrac{n_i a_i^6}{v_i}\) over all size groups.

7. THEORY OF RADAR DETECTION OF ATMOSPHERIC FORMATIONS \({}^{30}\)

In radar theory, the radar equation is derived, expressing the dependence of the power of the received reflected signal on the distance to the reflector, as well as on the parameters of the radar, the target, and the medium.

If we denote by \(P_t\) the power of the pulse emitted by the radar station, then at a distance \(R\) (at the target) the energy density for isotropic radiation would be

\[ \frac{P_t}{4\pi R^2}. \]

Taking into account the directional action of the radar antenna, expressed by the power gain \(G_t\), the energy density at the target will be:

\[ \frac{P_tG_t}{4\pi R^2}. \]

If the effective cross section of the target is denoted by \(T\), then the power radiated by the target in the backward direction toward the radar is given by the expression

\[ \frac{T P_t G_t}{4\pi R^2}. \]

The density of the energy reflected by the target at the receiving antenna of the radar is equal to

\[ \frac{T P_t G_t}{(4\pi R^2)^2}. \]

The total amount of energy received at the output of the receiving antenna with effective area \(A\) is then determined by the quantity

\[ P_r = A \frac{T P_t G_t}{(4\pi R^2)^2}. \tag{89} \]

If one takes into account that in radar antennas, used simultaneously both as transmitting and as receiving antennas, the quantities \(G_t\) and \(A\) are related by

\[ G_t = \frac{4\pi A}{\lambda^2}, \tag{90} \]

where \(\lambda\) is the wavelength, and the value of the effective area of the receiving antenna \(A\) is related to its actual area (aperture) \(A_p\) by

\[ A = \frac{2}{3} A_p, \]

then

\[ P_r = \frac{P_t A_p^2}{9\pi R^4 \lambda^2}\, T. \tag{91} \]

Formula (91) is the general radar equation for the case of a single target and a nonabsorbing medium.

In the radar equation for atmospheric formations it is necessary to take into account the multiplicity of targets and the propagation of radio waves in an absorbing medium.

To account for the multiplicity of targets, it is necessary to find an expression for the total effective cross section of radar scattering of all targets (drops) located in the antenna irradiation zone during the pulse duration.

If \(V\) is the volume of space from which, at the given time, scattered radiation from one pulse can arrive at the radar, and \(\eta\) is the total effective cross section of radar scattering of all drops in a unit volume of the atmosphere, then the val—

quantity \(T\) in (91) will be equal to

\[ T = V \cdot \eta . \tag{92} \]

Denoting by \(h\) the length of the wave train of the pulse and by \(A\) the width of the antenna radiation pattern in degrees, we obtain\({}^{10}\), for the volume of a spherical layer of thickness \(h/2\) from which scattered radiation will reach the radar during the duration of one pulse, the value

\[ V=\pi\left[\left(\frac{\pi}{180}\right)\frac{AR}{2}\right]^2 \frac{h}{2}. \tag{93} \]

The quantity \(\eta\) is determined from

\[ \eta=\sum_D N(D)\cdot \sigma(D,\lambda), \tag{94} \]

where \(\sigma(D,\lambda)\) is the effective radar cross section of a droplet of diameter \(D\) for wavelength \(\lambda\), and \(N(D)\) is the number of droplets of diameter \(D\) per unit volume.

To take into account attenuation during propagation in an absorbing medium, one may introduce a coefficient \(x\), related to the attenuation coefficient \(a(R,\lambda)\) by the equation

\[ x=10^{-0.2\int_0^R a(R,\lambda)\,dR}; \tag{95} \]

where \(a(R,\lambda)\) is the attenuation (in decibels) per unit path length at a distance \(R\) from the transmitter in the direction of the beam.

Taking (92), (93), (94), and (95) into account, the radar equation for atmospheric formations will be written in the form

\[ P_r=\frac{P_t\cdot A_p^2}{9\pi\lambda^2 R^4}\cdot \pi\left[\left(\frac{\pi}{180}\right)\cdot\frac{AR}{2}\right]^2 \frac{h}{2}\sum_D N(D)\cdot\sigma(D,\lambda)\cdot x. \tag{96} \]

This formula is suitable for all cases of the application of radar to the study of atmospheric formations.

One can obtain a simplified expression for approximate calculations (neglecting attenuation) if one uses Rayleigh’s formula (56) for the effective scattering cross section and takes the mean value of the sixth power of the radii of the droplets, \(a^6\). Then

\[ \eta=N\cdot\frac{128}{3}\cdot\frac{\pi^5 a^6}{\lambda^4} \left(\frac{n^2-1}{n^2+2}\right)^2; \tag{97} \]

Substituting (97) into (91), we have:

\[ P_t=\frac{128}{3}\pi^4 P_t A_p^2 \left(\frac{n^2-1}{n^2+2}\right)^2 \frac{VNa^6}{9R^4\lambda^6}. \tag{98} \]

If the width of the directional pattern \(\theta\) is expressed in radians, then

\[ V=\frac{\pi R^{2}\theta^{2}\cdot h}{8}. \tag{99} \]

and

\[ P_r=\frac{16}{3}\pi S_t\cdot A_p^{2} \left(\frac{n^{2}-1}{n^{2}+2}\right)^{2} \left(\frac{\upsilon^{3}hN\alpha^{6}}{9R^{2}\lambda^{6}}\right). \tag{100} \]

For an antenna with a paraboloid reflector,

\[ \theta \cong 0.85\,\frac{\lambda}{d}, \]

where \(d\) is the diameter of the paraboloid,

\[ A_p=\frac{\pi d^{2}}{4}; \qquad \upsilon^{3}=\left(\frac{0.85\lambda}{p}\right)^{2} =0.85^{3}\left(\frac{\pi\lambda^{2}}{4A_p}\right). \tag{101} \]

Substituting \(\upsilon^{3}\) into (100), we obtain:

\[ \frac{P_r}{P_t} =0.11\pi^{6} \left(\frac{n^{2}-1}{n^{2}+2}\right)^{2} \left(\frac{A_p hN\alpha^{6}}{R^{2}\lambda^{4}}\right). \tag{102} \]

For wavelengths from 3 to 12 cm,

\[ \left(\frac{n^{2}-1}{n^{2}+2}\right)^{2}\cong 0.9 \]

and

\[ \frac{P_r}{P_t}\cong 0.1\pi^{6}A_p\cdot h \left(\frac{N\alpha^{6}}{R^{2}\lambda^{4}}\right). \tag{103} \]

It is evident from this formula that the reflected signal received from an atmospheric formation will have a power inversely proportional to the square of the distance and to the fourth power of the wavelength.

If not the entire directional pattern is intercepted by the rain (at a large distance or with a small rain area), then the volume \(V\) will be only a part of the maximum volume according to (99) and (98); in this case the received power will be inversely proportional to the fourth power of the distance and to the sixth power of the wavelength.

Neglecting the attenuation \(\chi\) in (96), it can be seen that, in order to obtain long ranges and detect weak atmospheric formations (rain), a radar station must have a short wavelength \(\lambda\), high power, a sufficiently long pulse without reducing the peak power, and an antenna with high gain.

When choosing the band of a radar station intended for the study of atmospheric formations, it is important to estimate the effect of attenuation on the detection range, since in some cases the longer waves of the 10-cm band may provide a greater detection range than, for example, waves of the 3-cm band, despite the inverse proportionality of the received-signal power to the fourth power of the wavelength.

For the 10-cm range one may almost always, except for the heaviest showers, neglect attenuation. For waves in the 3-cm range, attenuation can be neglected only for light rains. For still shorter waves, 1.25 cm, attenuation cannot be neglected at all.

In calculations of detection range and in attenuation measurements it is necessary to take into account the presence of molecular absorption of microwaves in oxygen and water vapor[^27],[^28],[^29]. The total amount of molecular absorption for a humidity of \(7.5\ \mathrm{g/m^3}\) is shown in Fig. 5 by curve 1. Absorption bands stand out at \(\lambda = 0.25\ \mathrm{cm}\), \(0.5\ \mathrm{cm}\) (oxygen), and \(1.25\ \mathrm{cm}\) (water vapor). The same figure gives graphs of absorption in rains of various types. It is evident from the figure that, for waves in the range \(1\text{–}5\ \mathrm{cm}\), attenuation in rain is of greater significance than attenuation in atmospheric gases. Attenuation in atmospheric gases predominates for waves whose lengths are less than \(1\ \mathrm{cm}\) and greater than \(5\ \mathrm{cm}\). For still longer waves \((\lambda > 10\ \mathrm{cm})\) attenuation no longer has practical significance because of its small absolute magnitude. In designing special apparatus for studying atmospheric formations and meteorological phenomena, it is necessary to avoid the use of ranges falling in the region of selective absorption by oxygen and water vapor.

Fig. 5. Attenuation of microwaves in the atmosphere.
1 — oxygen and water vapor at a humidity of \(7.5\ \mathrm{g/m^3}\) and a temperature of \(20^\circ\mathrm{C}\).
2 — moderate rain (\(6\ \mathrm{mm/hour}\)).
3 — heavy rain (\(22\ \mathrm{mm/hour}\)).
4 — shower (\(44\ \mathrm{mm/hour}\)).

Calculations were carried out of the magnitude \(\dfrac{P_r}{P_t}\) as a function of distance, taking into account atmospheric attenuation, for three types of idealized rain: light (precipitation \(1.25\ \mathrm{mm/hour}\), mean drop diameter \(1.3\ \mathrm{mm}\)), moderate (precipitation \(5\ \mathrm{mm/hour}\), mean drop diameter \(1.65\ \mathrm{mm}\)), and heavy (precipitation \(12.5\ \mathrm{mm/hour}\), mean ...

diameter of the drops \(1.95\ \text{mm}\) for a radar operating in the range \(\lambda = 3.2\ \text{cm}\).

The results of these calculations are given in Fig. 6. From this figure it is clear that at certain distances heavy rain will be detected, whereas light and moderate rain will not. With the ratio

\[ \frac{P_r}{P_t}=10^{-15}, \]

light rain will be detected at a distance of about \(65\ \text{km}\), moderate rain at a distance of \(155\ \text{km}\), and heavy rain up to \(240\ \text{km}\).

Fig. 6. Detection of rain at \(\lambda = 3.2\ \text{cm}\) (without taking attenuation in rain into account).

Fig. 7. Detection of rain through continuous rain at \(\lambda = 3.2\ \text{cm}\) (taking attenuation in rain into account).

Stronger rains, producing precipitation exceeding \(12.5\ \text{mm/hour}\), will also be detected at greater distances, provided only that they are at a sufficiently great height within direct visibility from the station.

For modern radar stations, whose values of \(P_r\) and \(P_t\) are of the order of \(10^{-13}\ \text{W}\) and \(100\ \text{kW}\), respectively, such rain-detection ranges are quite realistic.

A calculation was also made of \(\frac{P_r}{P_t}\) as a function of distance for the case of detecting rain by a radar operating in the range \(\lambda = 3.2\ \text{cm}\) during propagation of microwaves through the same continuous rain, taking into account theoretical values of attenuation (Fig. 7). From this figure it is evident that at distances up to \(75\ \text{km}\) moderate rain gives a stronger reflection than light rain. At distances exceeding \(75\ \text{km}\) (because of the more significant magnitude of attenuation in moderate rain), the stronger ...

reflection is produced by light rain. In exactly the same way, heavy rains produce stronger reflected signals at the radar than moderate rains at distances up to 10 km, and weaker ones at greater distances.

The large attenuation in heavy rain gives a sharp decrease in the amplitude of the received signals with increasing distance. For

\[ \frac{P_r}{P_t}=10^{-15} \]

light rain will be detected through light rain at distances up to 50 km, moderate rain through moderate rain up to 65 km, and heavy rain through heavy rain only up to 25 km.

Attenuation in rain for waves of the 10-cm band is considerably less than for waves of the 3-cm band; therefore, despite the law of inverse proportionality of the power of the reflected signal to the fourth power of the wavelength, in some cases it may prove advantageous to use, for detecting rains, a station operating at wavelength \(\lambda = 10\) cm. Calculations show that, with identical parameters of the radar stations, the power of the signals reflected from heavy rain through the same rain will be greater for \(\lambda = 3.2\) cm up to 41 km. At distances exceeding 41 km, because of the greater attenuation in propagation of waves of the 3.2-cm band, the power of the reflected signals will be greater for \(\lambda = 10\) cm. Already for moderate rain this will hold only at distances greater than 124 km.

Despite the indicated advantages of stations of the 10-cm band when operating at large distances through moderate and heavy rains, for general purposes of rain detection, when the station itself is located outside the rain zone, it is apparently better to use stations of the 3-cm band.

8. METHODOLOGY AND RESULTS OF INVESTIGATING ATMOSPHERIC FORMATIONS BY MEANS OF RADAR

Apparatus used\(^{2,12,13}\)

Three types of radar stations convenient for use in meteorology may be indicated:

1. Powerful stations of the 10-cm band (with pulse power on the order of 1000 kW and more) are most effective for detecting atmospheric formations at large distances, of the order of 150–300 km.

2. For investigating the vertical structure of atmospheric formations at medium distances up to 40–50 km, stations of the 3-cm band of medium or high power are more suitable. Stations of the 10-cm band may also be used for these purposes, but with less success.

1. For a detailed study of the fine structure of atmospheric formations at distances of several kilometers, chiefly for scientific purposes, apparatus in the 1-cm range has a number of advantages.

It is desirable that the radar apparatus be specially adapted for meteorological purposes. To ensure a sufficient range, the station must have high power and good directivity. The choice of pulse length depends on the specific conditions. The longer the pulse, the greater the amplitude of the signal reflected from the atmospheric formation. It is therefore advantageous to use the longest pulses permissible under the conditions of normal pulsed operation of the generator tubes. On the other hand, to obtain high resolving power the shortest possible pulse is necessary. It is therefore desirable to have an adjustable pulse length, if this is possible under the other conditions. The pulse repetition frequency is determined by the required range of the station.

Fig. 8. Movement of a thunderstorm across the screen of a circular-scan tube.

To ensure sufficiently detailed observations and measurements, it is desirable that the station be equipped with indicators of several types. An indicator of type A, with linear sweep and amplitude marking, together with a calibrated signal generator, makes it possible to measure directly the amplitudes of signals reflected from atmospheric formations. In the radar observation of atmospheric formations on a type-A indicator, reflec-

...as lines of noise, differing from ordinary noise in that individual noise peaks have a longer duration and there is often a considerable distance between them.

The plan-position indicator (PPI) with brightness marking makes it possible to form an idea of the location (azimuth and range) of atmospheric formations within the radius of action of the station, and also gives an idea of the size, shape, and motion of these formations (Fig. 8). Reflections from atmospheric formations on indicators with brightness modulation usually occupy a larger area than do reflections from such objects as aircraft, aerostats, etc.

The range–height indicator (elevation angle) (HPI), when the antenna is rocked in the vertical plane, gives on the screen of the tube a distorted vertical section of the irradiated space with a horizontal coordinate corresponding to slant range, and a vertical coordinate corresponding to height (or elevation angle). Images of reflected signals on this indicator with brightness marking are also obtained in the form of illuminated areas with a shape depending on the type of atmospheric formations. The study of images on the HPI indicator makes it possible to investigate the vertical structure of atmospheric formations.

Experimental work in the application of radar to the study of atmospheric formations was carried out along the following principal lines:

1. Establishing the presence of a correlation between the amount of rainfall \(R\), the power of the reflected signal \(P_r\), and the quantity
   \[ Z=\frac{\Sigma D^6}{V}, \]
   which determines the scattering of energy in a unit volume of air containing droplets.

2. Studying the vertical structure of atmospheric formations.

3. Investigating fluctuations of signals reflected from atmospheric formations.

Correlation between the amount of rainfall \(R\), the amplitude of the reflected signal \(P_r\), and the quantity
\[ Z=\frac{\Sigma D^6}{V}, \]
which determines the scattering of energy in a unit volume of air containing droplets\(^{24,\ 20,\ 21}\).

The quantity \(Z\) was determined experimentally by measuring the diameters of the traces left by raindrops on filter paper treated with a special dye\(^{24}\). The diameters of these traces are an experimentally determined function of the diameters of the drops. At the same time the amount of rainfall \(R\) was determined with a rain gauge. The quantities \(Z\) and \(R\) were measured at the earth’s surface under the same rain cloud that was being observed by radar, taking into account the time of fall of the drops from the place

passage of the station beam to the earth’s surface. The quantity \(P_r\) was measured on an indicator of type \(A\) by maintaining the amplitude of the reflected signal at a constant level by means of gain control, previously calibrated with a signal generator. For the measurements, such a meteorological situation was chosen in which, over a period of less than one hour, the rain continuously changed the amount of precipitation \(R\) from 1 to 38 mm/hour (\(Z\) from 350 to 64,000 \(\mathrm{mm}^6/\mathrm{m}^3\)).

Fig. 9. Correlation of the quantities \(P_r\), \(R\), and \(Z\). Dots correspond to measurements of \(R\); crosses, to measurements of \(Z\).

The measurement data are shown in Fig. 9. From the graph it is seen that \(P_r\) is proportional to the first power of \(Z\) (or, more precisely, \(Z^{1.1}\)) and to the square of the rainfall amount \(R^2\).

If one plots \(\lg Z\) as a function of \(\lg R\) for all cases of rain, then the resulting experimental dependence (Fig. 10) is well expressed by the formula

\[ Z = 190 \cdot R^{1.72} \quad \text{or} \quad R = 0.048 Z^{0.58}, \tag{104} \]

where \(Z\) is expressed in \(\mathrm{mm}^6/\mathrm{m}^3\) and \(R\) in mm/hour.

Thus, it has been experimentally established that there is a distinct correlation between the quantity determining the amplitude of the signal reflected from an atmospheric formation—rain—and the amount of rainfall, which makes it possible to measure rainfall by means of radar. In the experimental determination of the quantity \(Z\), an empirical dependence was also found between

by the rainfall amount \(R\) and the mass of raindrops per unit volume of air \(M\) in the form

\[ M = R^{0.88}, \tag{105} \]

where \(M\) is expressed in \(\mathrm{mg}/\mathrm{m}^3\) and \(R\) in \(\mathrm{mm}/\mathrm{hour}\).

Taking (104) into account, the dependence between \(Z\) and \(M\) is obtained:

\[ Z = 0.020\, M^{2.08}. \tag{106} \]

Thus, the power of the reflected signal is approximately

Fig. 10. Correlation of the quantities \(Z\) and \(R\).

proportional to \(M^2\) (moreover, \(M\) does not take into account small drops that do not participate in forming the reflected signal).

This relation makes it possible to estimate approximately the content of droplet liquid in rain clouds from which rain is not yet falling, but from which reflected signals are obtained.

Study of the Vertical Structure of Atmospheric Formations31,32

For studying the vertical structure of atmospheric formations, it is most convenient, as has been said, to use stations with a height–slant-range indicator (HRI). Observations with such an indicator have shown the presence of two types of reflections. One of them

(fig. 11) is associated with prolonged steady rain, the other (fig. 12) with showers. In prolonged rain the amplitude of the reflected signal (the brightness of the image) gradually decreases with increasing distance. Changes in brightness with height are insignificant up to the zero isotherm. Reflections from a band 200–400 m wide, located near the zero isotherm, are obtained as considerably stronger than from the regions lying

Fig. 11. Steady rain on the screen of an HPI tube.

below. Above this bright band the reflections become very weak and indistinct. The appearance of a bright band in the region of the zero isotherm is associated with the melting of snow or ice crystals. Observations from aircraft have shown that above the level of the zero isotherm there is snow, which, owing to the small value of the dielectric coefficient of ice, gives an insignificant reflection. On entering the band of the zero isotherm, the flakes or crystals stick together into lumps of considerable size, which simultaneously begin to melt and acquire the reflecting ability of water, while retaining for a certain distance the low fall velocity of snow; this causes a large concentration of particles in this region and

leads to a very high reflectivity of the layer of the zero isotherm.

In showers, the magnitude of the reflection depends strongly on distance, with good vertical homogeneity. The image on the indicator screen consists of separate vertical bands, the inclination of which is explained by the presence of wind. The upper part of the bands is very varied in height, but for showers it is usually located above 3000 m.

Fig. 12. Shower on the screen of the HPI tube.

Since in showers, as a rule, there are strong vertical air currents, accumulations of large quantities of liquid droplets may form in the upper regions. However, sufficiently long observations (about 1 hour) show that such accumulations are short-lived.

For a detailed study of the dependence of the concentration of liquid droplets on height, the method of calibrated regulation of the receiver gain was applied.^31

If \(n\) is the reduction of gain in decibels, then the intensity of the signal after amplification will be the same as if a signal \(10^{0.1 n}\) times smaller were subjected to normal amplification. Denot...

the power of such an equivalent signal \(P_m\), we have:

\[ P_m = 10^{-0.1n}\cdot P_r, \tag{107} \]

where \(P_r\) is the power of the received signal. Since

\[ P_r \approx \frac{M^2}{r^2} \tag{108} \]

(where \(r\) is the distance to the atmospheric formation and \(M\) is the mass of water contained in a unit volume of the atmosphere in the form of droplets whose diameter exceeds \(0.5\ \mathrm{mm}\)), we have:

\[ P_m \approx \frac{10^{-0.1n}\cdot M^2}{r^2}. \tag{109} \]

Let \(M_{\min}\) be the minimum amount of water (concentration) which, at a distance \(r\), gives the lowest detectable signal power \(P_{m(\min)}\); then

\[ P_{m(\min)}=\mathrm{const}\cdot 10^{-0.1n}\frac{(M_{\min})^2}{r^2}. \tag{110} \]

By means of comparative radar and meteorological measurements for a radar operating at full power \((n=0)\), it was established that a concentration of \(40\ \mathrm{mg}/\mathrm{m}^3\) could still be detected at a distance of approximately \(8\ \mathrm{km}\), whence

\[ 10^{-0.1n}\frac{(M_{\min})^2}{r^2}=\left(\frac{40}{8}\right)^2, \]

\[ M_{\min}=5\cdot 10^{0.05n}\cdot r, \tag{111} \]

where \(M_{\min}\) is expressed in \(\mathrm{mg}/\mathrm{m}^3\) and \(r\) in \(\mathrm{km}\).

The gain of the radar receiver could be automatically reduced by changing the bias on the grids of the second and third tubes of the intermediate-frequency amplifier (by switching in calibrated potentiometers). This reduction of receiver gain was carried out in steps of \(10\ \mathrm{db}\) (starting from full gain) at the end of each up-and-down antenna sweep and, consequently, of each sweep cycle on the HPI and PPI tubes, and was continued successively until the reflected signals on the tube finally disappeared.

The images on the HPI and PPI indicators were photographed with motion-picture cameras in such a way that each frame of the film corresponded to the picture on the indicator during the time of one sweep (one antenna oscillation). On each frame the time, date, and series number were recorded, and on photographs from the HPI indicator also the azimuth and the magnitude of the receiver gain. Observations were made by successive photographing of the screen as the magnitude changed—

of the receiver gain in steps of 10 db (one frame per gain step), from full gain until the image disappears on the screen (Fig. 13). From the individual photographs one can obtain data for studying the structure of cloudiness. For the reflection to disappear under successive reduction of the gain, three or four gain-reduction steps are usually required, giving the same

Fig. 13. Change in the pattern observed on the HPI tube with successive reduction of the receiver gain.

number of photographs (frames). Each such photograph gave the contour of a region corresponding to the value of the minimum signal still detected on the indicator at the given receiver gain. A group of photographs yielded a series of several contours. These contours, obtained in height—slant-range coordinates, were then transformed into height—horizontal-range coordinates. Such a system of contours in natural coordinates gave an undistorted section of atmospheric formations in a vertical plane at a specified azimuth, with contours of equal values of received power.

To convert the magnitude of the received power into the concentration of drops per unit volume \(M\) \((\mathrm{mg}/\mathrm{m}^3)\), equation (111) was used. The diagrams obtained, with lines of equal concentration (Fig. 14), are very convenient for studying the vertical structure

Fig. 14. Diagrams of the dependence of the concentration of droplet liquid on height and distance.

atmospheric formations. A detailed study of such diagrams makes it possible to draw the following conclusions:

a) Within the region of space occupied by the rain, the values of \(M\) may change by a factor of 100 over a distance of the order of \(1.5\)—\(2\) km.

b) At certain moments in time the values of \(M\) may be greater aloft than near the earth’s surface.

c) Areas with maximum \(M\) tend to be arranged vertically.

d) Where an area with a large \(M\) is moving, the center of gravity of the area always moves in the direction toward the earth.

e) The contours corresponding to the minimum values \(M=100\) \((\text{mg}/\text{m}^3)\), during the time covered by one series of photographs, remain at approximately one and the same height, whereas the contours corresponding to large \(M\) may descend considerably during this time.

f) The detailed vertical diagram of rain changes rapidly with time and may change completely, beyond recognition, in 5—6 minutes.

Investigation of fluctuations of signals reflected from atmospheric formations*²

Signals reflected from atmospheric formations are characterized by considerable fluctuations of various origins. Fluctuations arise mainly because of interference resulting from the relative motions of a large number of drops in the illumination zone of the radar.

The study of the frequency spectrum of fluctuations of the amplitudes of signals lying in the audio range can provide information for the study of atmospheric turbulence in the region under investigation.

The theory of fluctuations of reflections from extended targets containing a large number of independent reflectors with random phases is based on consideration of the resultant signal. If the phases of the components vary slightly, for example because of the relative motion of the reflectors, then the resultant may also change.

In the theory of fluctuations a formula is derived for determining the probability \(P(I)\) that a given reflected signal will have amplitude \(I\) to within \(dI\), i.e., the fraction of reflected signals having amplitude \(I\) to within \(dI\):

\[ P(I)\,dI=e^{-\frac{I}{I_0}}\cdot \frac{dI}{I_0}, \tag{112} \]

where \(I_0\) is the mean amplitude of the reflected signal. The rapidity of fluctuations is determined mainly by changes in the reflections and the relative velocity of the reflectors. The fluctuation frequency increases linearly with increasing radio frequency. For experimental investi-

... fluctuations, it is necessary to measure the amplitudes of individual reflected pulses, and not the average amplitude over a considerable interval of time, as is obtained on the radar screen. This is achieved by photographing each individual sweep trace on an indicator of type \(A\) with a tube of blue fluorescence.

For photographing, a 16-mm cine camera driven by a motor with a large number of revolutions was used. At the same receiver gain, a pulse from a calibrator with a known amplitude is also photographed, for the comparative measurement of the amplitudes of the reflected pulses obtained on the film.

From these experimental data one can calculate the probability-distribution curve.

Fig. 15. Probability distribution \(P(I)\) that a signal reflected from rain has amplitude \(I\).

In Fig. 15, such a distribution, obtained from the measurement of 1000 reflected pulses from atmospheric formations (rain) during operation with a radar of the 10-cm band, is given in the form of a step curve. In the same figure, the distribution according to the theoretical formula (112) is given in the form of a smooth curve.

By carrying out a harmonic analysis of the experimental curves obtained, one can also determine the frequency spectrum of these fluctuations.

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