DESIGN CONCEPTS OF THE HOLOGRAPHIC SUBSURFACE RADAR

Radiophysics and Quantum Electronics, Vol. 43, No. 3, 2000 DESIGN CONCEPTS OF THE HOLOGRAPHIC V. V. K o p e i k i n a n d A . V . P o p o v SUBSU...
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Radiophysics and Quantum Electronics, Vol. 43, No. 3, 2000

DESIGN

CONCEPTS

OF THE HOLOGRAPHIC

V. V. K o p e i k i n a n d A . V . P o p o v

SUBSURFACE

RADAR

UDC 524.1:550.383

We discuss general concepts of design and signal-processing algorithms of the holographic subsurface radar. We consider a model example of the holographic image of a discrete point scatterer. An estimate of the scattered-signal level in the problem of subsurface radio sounding is given.

i.

INTRODUCTION

Most radars of subsurface sounding (georadars) use a short pulse having only a few oscillations without the carrier as a sounding signal. The frequency band of such a pulse can lie in the range from megahertz to tens of gigahertz. As a rule, georadars have weakly directional antennas and determine the position of a scattering object by the delay of a reflected signal, using various scanning schemes. According to the operating band, georadars can be low-frequency (1-50 MHz), medium-frequency (50-500 MHz), and high-frequency (greater than 500 MHz). Each of these radar classes has its own field of use [1]. The low-frequency radars have a large sounding depth (up to hundreds of meters) but a poor depth resolution (decimeters to meters). The main field of their ~lse is geology. The medium-frequencyradars allow for sounding at a depth of a few to tens of meters with a resolution from a few to tens of centimeters. These parameters enable one to use such radars in various fields such as engineering geology, archaeology, ecological research, urban economy, etc. [2, 3]. The high-frequency radars have a maximum resolution (centimeters to millimeters) but a small sounding depth, which, as a rule, does not exceed a few decimeters. These radars can be used for paving monitoring, detecting the steel in reinforced concrete walls, etc. All the above-mentioned types of georadars use antenna systems which are located in the near vicinity of the sounded surface. Most frequently, they are merely placed on the surface. Along with practical inconveniences (positioning difficulties or a possible danger of scanning), such a "short-sightedness" of a georadar leads to a number of fundamental difficulties: 1) The radar radiation pattern becomes uncontrollable and strongly depends on the parameters of the underlying surface. 2) Changes in the properties of a medium in the antenna near zone affect the antenna input characteristics, which makes it impossible to match electrical parameters of the output circuits of a receiver and a transmitter with those of the antenna. 3) The spatial resolution of the radar in the antenna near zone is determined by the antenna aperture but not by the wavelength. This gives rise to a contradiction if one tries to provide maximum resolution in the near and far zones. When deMing with the low-frequency and medi,,m-frequency radars, one has to reckon with such a situation. This is due to the fact that it is impossible to use antennas of other types whose clearance limits are such that they c~nnot be realized in practice. For the high-frequency radars, there exists the possibility of "detaching~' the antenna system from the sounded surface and of providing good angular resolution

Institute for Terrestrial Magnetism, Ionosphere, and Radio Wave Propagation, Russian Academy of Sciences, Troitsk, Moscow region, Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 43, No. 3, pp. 224~-233, March, 2000. Original article submitted November 17, 1998. 202

0033-8443/00/4303-0202525.00

(~) 2000 Kluwer Academic/Plenum Publishers

since the required antenna dimensions are now realizable. As such antennas, multielement antenna arrays which allow one to register the amplitude-phase structure of a reflected signal are the most promising. This information makes it possible to reconstruct a volume distribution of scatterers using the well-known holographic concept [4]. As applied to radiolocation, the idea of holography has evolved into the concept of a synthetic-aperture radar (see, for e~r~trnple, [5]) that allows one to eliminate constraints on physically realizable dimensions of antenna arrays. Some examples of successful application of this approach to underground sounding can be found in the literature [6]. The holographic data-processing algorithm allows one to significantly increase the spatial resolution of a georadar and to synthesize three-dimensional images of underground objects. Along with the advantages of the synthetic-aperture method, in [6] Halman et.al, discuss some difficulties which decrease its efficiency in subsurface sounding problems: time-consuming data sampling from an area, antenna positioning errors, difficulties in accounting for the diffraction effects in a pulse sounding scheme, and the possible inhomogeneity of the medium in which a sounded object is located. These dii~iculties, which cannot be avoided for the medium-frequency radars, are overcome to a considerable degree if one uses higher frequencies in the monochromatic or multifrequency regimes. In this case, an increase in the sounding volume and wavelength allows one to realize a multielement antenna array required for the formation of a holographic image. In particular, to increase a rate of paving monitoring, linear antenna arrays are used [7]. The development of the volume holographic subsurface radar would produce invaluable advantages for civil engineering and archaeological investigations. Technical realization of the holographic radar requires solving a n,,mber of fundamental problems including the choice of the operating frequency and antenna array parameters, determination of the dynamic range and sensitivity of a receiver, etc. In this paper we only give general estimates of the diffraction effects and the scattered-signal level, which allows us to determine the spatial resolution and sensitivity of the subsurface radar. We discuss the algorithm of reconstruction of the scattered-field spatial structure by the parabolic equation method and the use of a multifrequency regime for estimating the parameters of the medium. A more detailed description of the computation procedure and results of numerical simulations will be presented in a separate paper (see also [8]). 2.

SINGLE-FREQUENCY

HOLOGRAPHIC

RADAR

The main purpose of the holographic radar is to give a visible spatial image of the object in the radiofrequency band on the basis of information on the amplitude and phase distributions of a monochromatic field over the antenna aperture. In this respect, the operational concept of this device is analogous to that of optical holography [4]. It is evident that, when using the simplest signal-processing algorithms, there should be some distortions of the true dimensions of the object parts immersed in transparent or semitransparent media, in the same way as in optical holography. We note that similar distortions are typical of visual perception in inhomogeneous media, which, however, does not hamper the visual detection and identification of objects. The structural diagram of a single-frequency holographic radar is shown in Fig. 1. The radar consists of a transmitter 1 integrated structurally with an antenna 2, which irradiates an object 3 in a certain angular interval. The signal scattered by the object is registered by an antenna array 4. Measurements of the quadrature components, i.e., the real and imaginary parts of the signal complex amplitude, are made by each array element, which may be the usual electric dipole in the simplest case. The receiving unit 5 sequentially samples the dipoles and records the amplitude and phase of the received signal. The signal-processing unit 6 uses this information for mathematical reconstruction of the spatial structure of the scattered field. The obtained radio image of the scattering object is shown on a display of the computer 7 with the possible use of stereoscopic-vision equipment. Let us consider the simplest algorithm of reconstruction of the object image. First, we make some assumptions. Let no information on the shape and material of the object be known. We assume that the object consists of a large number of elements commensurate with the wavelength and that each element 203

reradiates a spherical wave propagating in free space and carrying some fraction of t h e scattered signal to the plane of the antenna aperture. For simplicity, we do not consider characteristic features of the subsurface sounding problems and ignore the effects of refraction at the boundary of the sounded medium. We note that the condition of free-space propagation of a spherical wave is valid only for outer parts of the object and is not satisfied for its inner parts, which leads to distortions of their true dimensions. We also note that multiple reflections from an inhomogeneous object structure can cause an additional illumination of its parts, i.e., some distortion of their brightness, which is well known in optical holography and the usual photography. Let us introduce Cartesian coordinates x, y, and z and let the plane z = 0 coincide with the antenna-aperture surface, which is a rectangle specified by the inequalities - X < x ~ X and - Y ~ y _< Y. Let (xo, Yo, zo) be coordinates of a node of the spatial grid in the bulk of the object. It is evident that one can reconstruct the object parts which are commensurate with the wavelength A in the medium. Hence the grid spacing should be of order A. Also, we introduce another grid on the antenna aperture with nodes put at the positions of the receiving dipoles. /,'/ /./ /// ///// Let (~, ~, 0) be the coordinates of an element of the antenna array. Let us assume that each node of the spatial grid is Fig. 1. Structural diagram of the holographic an elementary source of a spherical wave with an amplitude radar. A which should be estimated. The spatial distribution of the scattering amplitude A(xo, Y0, zo) (in the grid nodes) is the desired radio image of the object. Let us define the distance R0 between the nodes of both grids, which is a function of five coordinates ~, ~/, xo, Y0, and z0, as Ro =

- x 0 ) 2 + (V - y 0 ) 2 + "~0" -~

(1)

Each element of the scattering object is a source of a spherical wave producing the field distribution F

Eo(~, 7) = ~ exp (ikRo) zto

(2)

in the antenna plane, where F(x0, Y0, z0) = A exp (i@) is the complex amplitude of scattering of this element and k = 2~r/A is the wave number. Hereafter we drop the factor e x p ( - i w t ) . In reality, determination of the unknown scattering amplitude A(xo, y0, z0) is reduced to the statistical estimation of the measured function E0(G r/) = E0(G r/, xo, I/0, z0) in the additive mixture of the signal and white noise: /~ = E0 + Enoise. Assuming that the number of elementary re-emitters is sufficiently large and taking the central limit theorem into account, we can consider the resulting field E as a two-dimensional, &correlated random function with the normal distribution [9]. In the case of an infinite antenna aperture, the problem of determination of the scattering amplitude is solved using the maximum-likelihood method for continuous functions, and the solution is expressed via the two-dimensional integral -boo -boo

(3) Here E ( ~ , ~ , x , y , z ) = e x p ( i k R ) / R is the field excited by an elementary point source with coordinates (x, y, z) on the receiving aperture and the asterisk stands for the complex conjugate. 204

The following sum can play the role of a discrete analog of Eq. (3):

A(x,y,z)=

N ~_,

~

E,n,mEn,m(X,y,z) Wn,m .

(4)

n=-N m=-M

Here n = ~/Ax is the node number of the antenna-aperture grid along the x axis, m = ~7/Ay is the number of the grid node along the y axis, Ax and Ay are spacings of the grid along t h e x and y axes, respectively, and Wn,m(~,rl) is a certain real two-dimensional weight function ( ' ~ i n d o w function") which smooths out the boundary effects related to finite dimensions of the antenna aperture. Equation (3) can also be applied to more complicated cases if one h a s additional information on the object. For example, let a concrete wall with a known dielectric permittivity ~ be located at a distance z = L. For an elementary point source located at a distance z > L, one should n o w calculate a more realistic field pattern in the antenna plane and use this pattern as a trial function E' instead of a simple spherical wave. In this case, it is possible to reconstruct the true geometric dimensions o f the exterior contour of an object immersed in the wall. The simplest, though not the most accurate, m e t h o d of taking a medium into account is to extend the spatial grid by a factor of v/~ along the z axis for all z > L. The algorithm for obtaining the image (4) of point scatterers can also b e modified to detect larger objects with known scattered fields, e.g., when searching for metallic or dielectric objects in soft against the impeding background of natural irregularities. Using the known scattering pattern as the function E', we obtain the maximum correlation at the point in which the sought-for o b j e c t is located. In this case, as for any optimum filtering, the spatial image of the object shape will be distorted. However, using the distribution of the scattering amplitude, we will be able to make an optimum decision on the presence and the spatial position of the object in the maximum-likelihood sense. 3.

MODEL EXAMPLE OF THE RECONSTRUCTION

OF A H O L O G R A P H I C

IMAGE

Let us demonstrate the implementation of the proposed algorithm, using t h e simplest model example of a single point scatterer in free space. Let (x0, Y0, z0) be coordinates of the scatterer irradiated by a plane w a v e e ikz. The scattered spherical wave

eik(zo+Ro) E0 (~, rl) =

R

'

(5)

where R0 = ~/(~ x0) 2 + (rl Y0)2 + z02 ~ zo + [(~ x0) 2 + (rl yo)2]/(2zo), is measured in the plane z = 0 on the receiving-antenna aperture specified by the inequalities - a < ~ < a and - b < rl < b. The holographic method is aimed at reconstructing the scattering-object image which, ideally, should be a focus with coordinates (x0, Y0, z0). The above-described algorithm is reduced to evaluation of the integral -

-

-

a

-

-

b

z(x y,z)= / f

E*(e,,)ded,,

(6)

--a -b

which is the convolution of the measured field (5) and the trial spherical-wave field with the origin at a certain point (x, y, z): eik(z+R)

R

(7)

Here R ---- %/(~ - x) 2 + (rl - y)2 + z z ~ z + [(~ - x) 2 + ( r / - y)2]/(2z). In what follows, the absolute value A(x, y, z) = II(x, y, z)l of the integral (6), which depends on the coordinates (x, y, z) of the trial source, will be called the diffraction image of the scattering object. For the model considered, this integral is evaluated 205

in explicit form:

e2ik(zo--z) a b

{

- xo) 2 + ( 7 - yo)

2z0

--a --b

-

+ (,7- y)2] } 2z

d7.

(8)

It is evident that integral (8) splits into two onecllmeusional integrals (I = IxI~) which can be reduced to Fresnel integrals:

, 4 ~ ( x ~ ~ Loo .._ / 0.75~ 0 _ ~ ~

t+ =7)

klz u- zol tf(9)

wheret•

(x~ zo :-x zz~ =]=a) ~ k l z2zzo -

. Thecorrespond-

ing expression for I~ is obtained from Eq. (9) if x and a are replaced by y and b, respectively.

40--~,

Figure 2 shows the results of computations of the function A~ = ]Ix(x,z)] for the model two-dimensional example: wavelength A = 3 cm, half-width of the aperFig. 2. Holographic image of a linear scatterer. ture a = 50 cm, and coordinates of the linear scatterer Wavelength ,~ -- 3 cm and coordinates of the x0 -15 cm and z0 = 100 cm. It is seen that the function scatterer x0 = 15 cm and zo--100 cm. The Ax has a m a x i m u m at the point (x0, z0) corresponding to receiving aperture is specified as - 5 0 cm< x < the position of the scattering object. Smearing of the max50 cm, and W(~) = {1 if I~l < a; 0 if I~l > a}. imum is determined by the diffraction effects and is mainly due to the finite dimensions of the receiving aperture. Note that the maximum becomes more sharp in a three-dlmensional problem due to multiplication of the functions Ax(x, z) and A~(x, z). The boundary effects result in the appearance of sharp crests joining the source and the aperture boundaries. The values of the crests and the sharpness of the maximum can be controlled by choosing a suitable weight function. Let us give two examples. Figure 3 corresponds to the weight function W(f) = cos (Trf/(2a)). One sees that the border crests disappear in this case, but some deterioration of the object location in depth occurs. Therefore, in some cases an opposite approach can be useful in which the weight function concentrated near the aperture boundaries is applied. As is seen from Fig. 4 plotted for the weight function W(~) = 1 - cos(Tr~/(2a)), the maximum of the function Ax(x, z) becomes sharpened in this case, which implies an increase in the spatial resolution of the device. A practical algorithm for reconstruction of the radio image can be developed using numerical methods for solving the wave equation. Of considerable universality and computational efficiency is the LeontovichFock parabolic equation [10, 11]. In the above example, we actually used this approximation. Indeed, one can easily see that the kernel of the integral tran.qformation (8) is the Green's function of the parabolic equation

2ik

02u

02u

(10)

which describes propagation of a rather narrow wave-packet E --- u(x, y, z) e ikz. Thus, formula (8) makes an approximate reconstruction (the so-called "migration") of the spatial structure of the scattered field from the measured field distribution E0(~, 77) on the antenna aperture. Numerical approaches to solving the parabolic equation are well developed (see, for example, [1215]). An important advantage of them is possible generalization to the case of an inhomogeneous background medium. To simulate propagation in unbounded space, one can use the boundary conditions of 206

O.O_

-.

0 02_'~ I , ~~ 0 ~ O

~"' '- .~

~

~

~., 0 r

4(' ~

- ' ~ - ~ i ~ m ~ z SO

X~ ~

J~ c m

Fig. 3. Holographic image of a linear scatterer for W(~) ----cos(Tr~/(2a)) under the same conditions as for Fig. 2.

Fig. 4. Holographic image of a linear scatterer for I~(~) = 1 -- cos(Tr~/(2a)) under the same conditions as for Fig. 2.

tran.qparency [16] around the edges of the aperture of the receiving antenna. If necessary, one can use a wide-angle modification of the parabolioequation [17], which eliminates the paraxial constraint on the wave packet. The parabolic equation is convenient when dealing with real data, and it allows one to readily take the inhomogeneity of a sounded medium into account. 4.

MULTIFI~QUENCY

HOLOGRA.PIIIC

RADAR

We note that the initial phases of the scattered wave and hence the values of the two-dimensional integral in Eq. (3) or the double sum in Eq. (4) are complex quantities determined by the following conditions of illumination: the distance from the source to the re-emitter, the refractive index of a m e d i u m , and the presence of multiple reflections. However, the phase of the illumination source (transmitting antenna) can be considered constant. This circumstance is determined by the desigm of a device in which the quadrature components of received signals are measured with respect to the signal of the transmitter carrier frequency. This allows one to measure the value of the scattered-signal phase, which contains additional information about the scatterer and the medium of propagation. It is evident that the convolution of the measured field /~(~, 77) and the elementary trial wave E(~, ~), which was discussed in Sect. 2, allows us to estimate not only the amplitude A but also the phase @ of the elementary scatterer (2). This, in turn, allows us to solve the inverse problem of radio sounding, i.e., the problem of reconstruction of both the true object geometry in transparent media and the refractive index of the sounded media, if one makes measurements at a number of close frequencies (at least two frequencies). The scheme of the multifrequency holographic radar is the same as above (see Fig. 1), but now we have the possibility of changing the frequency of the device. The algorithm of solving the inverse problem is based on the fact that the time r of signal propagation from the transmitter to the receiver can be determined via the derivative of the phase @ with respect to the frequency w: 0@ = Ow

(II)

On the other hand, the time of propagation through an inhomogeneous m e d i u m is determined by the following expression in the geometric-optical approximation:

v=-

1/ n(l)dl, c

(12) 207

where n is the refractive index of the medium, I is the distance along the ray path, and c is the velocity of light in free space [18]. These formulas relate the phase variations to the medium properties along the propagation path. For the case of a locally plane-layered medium, the inverse problem can be solved using elementary geometric-optical formulas. Let us consider the simplest example where the sounded medium have no frequency dispersion in the operating band. This condition ensures invariance of the object radio-image with respect to a small frequency shift. For each element of the image at each of the used frequencies, we determine the complex amplitude of the scattered signal as N

M

n=-N

m=-M

'~ =

n, E:,mWn,m-

(13)

The time delay of the signal for each element of the image is calculated via the phase difference A ~ for the frequency interval Aw as r = A r We now calculate the true positions of two elements of the image. One of the elements is located on the medium surface, while the radio image of the other is located inside the medium at an apparent distance A~. The fact that the wave from the first source propagates in free space is immediately evident from the measured time delay, which will be equal to r = l/c. Hence the position of the first source on the radio image is true. The true position of the second source with respect to the medium surface is found from Shell's law as A z = n A S . The wave travels this distance during the time A r = 2 A S n 2 / c , whence it follows t h a t the refractive index of the medium is n=

V 2A~"

For objects of more complex shape, more complicated calculations in the direct problem are evidently needed as compared with this simplest example. 5.

A N E S T I M A T E OF T H E S C A T T E R E D - S I G N A L LEVEL

We now give an energetic estimate of the level of a signal scattered by a subsurface dielectric object. Let the source of power P0 illnminate an area of radius B on the ground (see Fig. 5). The energy flux incident on the scattering object 2 of radius r, which is located at a small depth d in soil with 4 2a ), dielectric permittivity el and conductivity az, is equal to r 2

P1 = Po T2e-2~d B-"~ ,

tt

(15)

h where T1 = 2 cos 80/(cos 00 + ~/el - sin ~ 00) is the Fresnel tran.~mi.~gion coefficient and ~ -- 2 1 r a l / ( c x / ~ ) is the damping rate of radio waves in the soil. The power scattered by object 2 to the upper half-space can be estimated as

81) O"I

~2rq

Fig. 5. Diagram illustrating the formation of the scattered signal.

208

t>2 = P 1 1 ~ ~2e-2za,

(16)

where R2 is the reflection coefficient from the interface between media with electric parameters sl and e2 (e2 is the dielectric permittivity of the object) and T1 is the backward tran~mi~ion coefficient from soil to air.

The reflected-signal power, which is incident on an element dA of the antenna array, depends on the scattering pattern of the dielectric object 2. A qualitative estimate can be obtained for mlnlmunl object dimensions of the order of the wavelength when scattering may be considered isotropic in the upper half-space. T h e n the fraction of energy flux per element d A N 2A is dA/(2h) P3 = P2

I

sin8d8 ~ ~ 2

,

(17)

-dA/(2h) where h is the altitude of the array above the Earth's surface. The final formula is 1 ,r2 r~2~2o-4Zd ; r A ,~2 P0.

(18)

For numerical estimates, we consider the normal-incidence case in which ]Tl:F1 ] "~ 1 and the coefficient ]R2] ( v / ~ - vre2)/(vrg~+ v~2). For typical values of electric parameters (el = 9, ~2 = 4, and al - 10 -2 S/m (108 CGS units)) and geometric dimensions (b = 100 cm, d = 10 cm, r = 3 cm, A = 3 cm, and h = 200 cm), we have R2 = 0.2 and ~3 = 7" 10 -4, whence finally P3 = 10-SP0. Clearly, the possible irrhomogeneity of the subsurface medium and the roughness of the interface need be taken into account in practical applications. In particular, these problems were considered in [19], where it was shown that, under typical conditions, the power of a signal scattered by the air-ground interface is approximately 20 dB smaller than the radar response of the sounded object. The presented estimate of the level of the scattered signal shows t h a t the power of the sounding transmitter may not exceed a few watts if the receiver sensitivity is of the order o f 10-s W. These parameters axe technically realizable when using standard hardware components. The problem of protection of the receiver from direct transmitter radiation can be solved by using a narrow-beam transmitting antenna and absorbing screens and by providing a large d y n a m i c range of the input circuits of both the receiver and analog-to-digital converter. 6.

CONCLUSIONS

The existing high-frequency pulse radars are rather inconvenient for m a n y applications. The measurement results yielded by detailed interface scanning are represented in the form of a two- or three-dimensional radar pattern whose deciphering is a complicated and ambiguous problem. Moreover, such a scanning procedure is not always possible in practice and is even dangerous in some cases. A high-frequency holographic radar allows us to make the object whoUy visible from a distance of a few of meters, and its stereoscopic image, which is the most habitual form of representing information on the surrounding medium for a human, can strongly facilitate a practical solution of the inverse problem. This work was supported by the Russian Foundation for Basic Research (project No. 98-05-64626). REFERENCES

1. L. Peters, M. Poirier, and M. Barnes, in: 4th Int. Con/. on Ground Penetrating Radar, Geological Survey of Finland (1992), p. 7. 2. V . V . Kopeikin et al., in: 11th Int. Microwave Con/., Vol. 2, Warsaw (1996), p. 509. 3.

V . V . Kopeikin et al., in: 6th Int. Con/. on Ground Penetrating Radars, Sendai (1996), p. 515.

4.

M. Born and E. Wolf, Principles of Optics, Pergamon Press, Oxford (1970).

5. V . A . Zverev, Physical Principles o/Image Formation by Wave Fields [in Russian], Inst. Appl. Phys., Russian Acad. Sci. Press, Nizhny Novgorod (1998). 209

6. J. I. Halman, K. A. Shubert, and G. T. Ruck, IEEE Trans. Antennas Propagat., AP-46, No. 7, 1023 (1998). 7. D.J. Daniels, Surface-Penetrating Radar, IEE, London (1996), p. 300. 8. V.A. Vinogradov and A. V. Popov, in: 19th All-Russian Conf. "Radio Wave Propagation" [in Russian], Kazan' (1999). 9. S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarsldi, Introduction to Statistical Radiophysics, Pt. 2. Random Waves [in Russian], Nauka, Moscow (1978). 10. M.A. Leontovich, Selected Papers. Theoretical Physics [in Russian], Nauka, Moscow (1985). 11. V. A. Fock, Problems of Diffraction and Electromagnetic Wave Propagation [in Russian], Sovetskoe Radio, Moscow (1970). 12. G.D. Malyuzhinets, Usp. Fiz. Nauk, 69, No. 2, 321 (1959). 13. A.V. Popov, Zh. Vychisl. Mat. Mat. Fiz., 8, No. 5, 1140 (1968). 14. F. Tappert, Wave Propagation and Underwater Acoustics [Russian translation], Mir, Moscow (1980). 15. V. Yu. Zavadsky, Modeling of Wave Processes [in l~ussian], Nauka, Moscow (1991). 16. A.V. Popov, Radio Sci., 31, No. 6, 1781 (1996). 17. J.F. Clairbout, Theoretical Fundamentals of Geophysical Information Processing [Russian tran.~lation], Nedra, Moscow (1981). 18. Yu. A. Kravtsov and Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media, Springer-Verlag, Berlin-Heidelberg (1990). 19. T. Dogaru and L. Carin, IEEE Trans. Antennas Propagat., 46, No. 3, 360 (1998).

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