This paper outlines a method for obtaining the relation between the singly scattered signal and the Fourier spectrum of medium dielectric permittivity fluctuations, with regard for the fact that the scattering volume is determined by antenna patterns and is not small. On the basis of this equation we obtained the radar equation relating the scattered signal spectrum to the spatial spectrum of fluctuations. Also, a statistical radar equation is obtained that relates the mean statistical power of the scattered signal to the spectral density of the dielectric permittivity fluctuations without a classical approximation of the smallness of the irregularities' spatial correlation radius. The work deals with the bistatic sounding case, when the exact forward scattering and exact backward scattering are absent and sounding signal has sufficiently narrow spectral band for scattered volume to change slowly on ranges of Fresnel radius order. The statistical radar equations obtained differ from the classical ones in the presence of coherent structures with big correlation radii, and so the received signal spectrum can differ from the intrinsic spectrum of irregularities.
 The method of radio wave backscattering due to dielectric permittivity fluctuations of the medium provides the basis for a wide variety of techniques for probing the ionosphere (the radio wave incoherent scattering method [Kofman, 1997]; scattering from artificial irregularities [Belikovich et al., 1986]), the atmosphere (mesospheric-stratospheric-tropospheric sounding [Woodman, 1989]), and other media. Central to these techniques is the radar equation that relates the mean spectral power of the scattered signal to the statistical characteristic of the medium dielectric permittivity fluctuations, their spectral density [Tatarsky, 1967; Woodman, 1989; Ishimaru, 1981]. A standard method for constructing the statistical radar equation involves constructing the spectral power (or a received signal autocorrelation function) using two approximations. One of them, namely, the single-scattering approximation, is applicable when the dielectric permittivity fluctuations are weak and the scattered field is significantly weaker than the incident field. The other approximation is the approximation of the irregularities' spatial correlation radius smallness in comparison to the Fresnel radius.
 The single-scattering problem of the electromagnetic wave has been reasonably well studied in situations where the receiver and the transmitter are in the far-field region of the scatterer. In this case it is possible to obtain a simple linear relation between the scattered signal and the spatial Fourier spectrum of irregularities without recourse to a statistical averaging [Newton, 1969; Ishimaru, 1981]. However, in remote diagnostics of media, the scattering volume size is determined by the antenna pattern crossing. Hence it is impossible to use a classical approximation of the sounding volume smallness [Ishimaru, 1981] to obtain the radar equation relating the scattered signal to the Fourier spectrum of dielectric permittivity fluctuations. Therefore, when obtaining the statistical radar equation, one has to average the received signal power and to use the approximation of the spatial correlation radius smallness, which makes it possible to generalize results derived from solving a classical problem of wave scattering from a single small irregularity to the problem of scattering from a set of uncorrelated small irregularities [Tatarsky, 1967; Ishimaru, 1981].
 Here we have obtained the radar equation relating the scattered signal spectrum to the spatial spectrum of dielectric permittivity fluctuations for bistatic sounding. This expression was obtained without a traditional limitation on the smallness of the scattering volume size and describes essentially the scattering from an extended scatterer. An analysis of the expression revealed a selectivity of the scattering similar to the widely known Wolf-Bragg condition.
 We obtained the statistical radar equation for bistatic sounding, which relates the scattered signal mean power to the dielectric permittivity fluctuations' spectral density without a classical approximation of the spatial correlation radius smallness. This equation has a more extensive validity range when compared to the well-known equation [Tatarsky, 1967; Ishimaru, 1981], obtained by assuming the smallness of the irregularities' spatial correlation radii.
 It was pointed out earlier [Doviak and Zrnic', 1984] that there are difficulties in obtaining the equation for correlation radii or scattering volume bigger than Fresnel radius without exact antenna patterns, irregularities, and sounder signals. The suggested method does not contain those limitations.
2. Starting Equation
 Consider a bistatic sounding within the single-scattering approximation. We use, as the starting equation, the well-known expression for a complex envelope of the received signal [Tatarsky, 1967; Ishimaru, 1981] (accurate to terms unimportant for the subsequent discussion):
and Ri,s = r − ri,s is the position of the point under investigation relative to ri,s (the locations of the transmitter and the receiver); o(t) and a(t) are the time window of reception and the emitted signal complex envelope, respectively, with the window and the signal being narrowband and having the bands ΔΩo, ΔΩa ≪ ω0; ω0 is the carrier frequency, k0 = ω0/c; is a unit vector in a given direction; and is the beam factor, where fi() and fs() are the patterns of the transmit and receive antennas, li and ls are their polarization factors; and ϵ (t, r) represents dielectric permittivity fluctuations. Expression (1) is obtained on the assumption that the scattering volume is in the far-field range of the receive and transmit antennas R ≫ D2/λ0, where D is a typical antenna size and λ0 is the wavelength of the emitted signal.
 Equation (1) defines the relation between the received signal and medium fluctuations and is essentially (in the sounding problem) the radar equation for signals unlike the classical one for power characteristics. The kernel H and the beam factor g are determined by transmitting and receiving system parameters. Specifically, the kernel H determines the region of fluctuations ϵ over time and space which contributes to the scattered signal. The beam factor g determines the region of space which contributes to the scattered signal.
3. Relation of the Scattered Signal to the Spatial Fourier Spectrum of Irregularities
3.1. Derivation of Radar Equations
 This section discusses the method of obtaining the relation between the scattered signal spectrum and the spatial spectrum of the medium dielectric permittivity fluctuations. The problem of obtaining a relation for the case of small scatterers was considered by Newton ; in this paper we have obtained a relation without a limitation on the size of the object being probed. The main idea of this method is the transition from the problem of scattering on spatial irregularities to the scattering on separate spatial Fourier harmonics of these irregularities. Their respective contributions are calculated by the stationary phase method and are summarized.
 Let us analyze expression (1), without performing a standard [Tatarsky, 1967; Ishimaru, 1981] transition to quadratic (in field) characteristics. Within Born's approximation the main physical mechanism for signal shaping is the scattering on certain Fourier harmonics ϵ. It is therefore convenient first to highlight the relation between the scattered signal spectrum and spectral characteristics of the medium.
 By going to the spectral representations for u and ϵ in (1) we obtain the following expression for received signal spectrum, fully equivalent to the initial equation (1):
where the integral I is determined by
The integral in (4) is proportional to the amplitude of the signal scattered on a separate spatial harmonic and contains a rapidly oscillating function.
 The virtue of (3) is that one can interpret received signal without averaging, when we have the model of ϵ (ν, k), because usually we have only spectral form. The drawback of such representation (neglecting rapidly oscillating function under integral (4)) is that it is of selective Bragg scattering character, which is well known for the small scatterers (equation (4)). That is why we will transform this representation (equation (3)) to the form with this drawback removed.
 According to Newton  and Tatarsky , let us assume that the main mechanism is the Bragg scattering and that the largest contribution to the scattered signal is made by medium spatial harmonics, the wave number of which has the order of twice the incident wave number and so will be large. The distances from which the signal arrives range (determined by the crossing of the beam factor g and the weighting function H), are in the antennas' far-field range, and also will be large. Therefore the phase in (5), the product of the wave number by the distance, is a large parameter kr ≫ (D/λ0)2 ≫ 1, which makes it possible to evaluate this integral by the stationary phase method (SPM) [Fedoryuk, 1987], whose applicability conditions will be discussed below.
 The expression under the integral sign in (4) has a stationary point r0(k) which makes the main contribution to the integral. Its location is defined by the equation
which is a modified Wolf-Bragg condition for the scattering from nonstationary spatial arrays.
 As a first approximation, the integral in (4) is therefore equal to the contribution from the stationary point:
where ξ = ν /(ck0) is usually small because the received signal is a narrowband signal.
 The quantity V has the meaning of a “cophasal” region spatial volume that makes the main contribution to the amplitude of scattering on the spatial harmonic in (4). Thus the main contribution to the integral in (4) comes from cophasal regions of spatial harmonic arrays having a nearly ellipsoidal shape. The direction toward these regions is determined by a modified Wolf-Bragg condition (6), and their volume is defined by (8). It is evident that the linear dimension of the cophasal region has the Fresnel radius order RF = (λr)1/2.
 The scattered signal is the superposition of contributions from separate spatial harmonics. Therefore, by substituting (7) into (3), we obtain the radar equation relating the received signal spectrum to the medium irregularity spectrum:
If, instead of the coordinate system tied to wave vectors k, we use a system of spatial coordinates r0, this would amount to calculating the transition Jacobian, which turns out to be
Thus, in a spatial coordinate system, the radar equation (9) may be written as
 The two expressions for the received signal, (9) and (11), are different representations for the scattered signal written in terms of integrals over space and over the wave vector space, and they relate the received signal spectrum to spectral characteristics of irregularities.
 In the radar equations obtained (equations (9) and (11)), unlike the initial equations (1) and (3), the Bragg character of scattering is emphasized (equation (6)); that is why they look more useful for analysis than the initial equations (1) and (3).
3.2. Validity Range of the Expressions Obtained
 The validity range of the resulting expressions for the scattered signal (9) and (11) is determined by the region where we can use the first approximation of the initial integral by expanding it in asymptotic series by the SPM technique and not taking into account next members of series. In accordance with SPM theory one can take into account next members, but the equations obtained are too difficult. That is why we use a pretty simple criterion [Fedoryuk, 1987] for estimation of the validity of first-order equations. According to this criterion, next members of series can be neglected if the following condition is satisfied:
Here l is the Cartesian coordinate xl number, and ΔRl2 is the square of linear size in the lth coordinate direction of the cophasal region (or effective scattering volume, as we call it (equation (8))), which makes the main contribution into the signal scattered from exact spatial harmonic of ϵ.
 Let us consider the limitations in the case of sounding with the infinite signal when H(ω − ν, r) do not depend on r. We will consider the Cartesian coordinate system linked with effective scattering volume, two basis orthes of which lie on the plane determined by the vectors Rs(r0)), Ri(r0), where one orth is parallel to the direction of the transmitter-receiver. In this coordinate system the vector ΔRl2 will have the following scales:
The second derivative in (12) determined by the derivatives of spatial factor g(r) and geometrical factor 1/[Rs(r)Ri(r)] has components of the following order:
where ΔΘ = λ/D is the antenna pattern angle width and φ = π − β, where is the scattering angle.
 Taking into account that for distances r from the center of the transmit and receive antenna system to the scattered volume that are small in comparison with a(the distance between the receiver and the transmitter a = |Rs − Ri|), sin φ ≈ r/a, cos (φ/2) ≈ r/(2a), and Ri, s ≈ a and that for large distances r, sin φ ≈ a/r, cos (φ/2) ≈ 1, and Ri, s ≈ r, from condition (12) we have the validity range for (9) and (11) of
The first condition is coincident with validity limitations of initial equations (1) and (3): The scattering volume must be in the far-field range of receive and transmit antennas, and that is why the first condition does not make any additional limitations in comparison with the initial equations (1) and (3). The second condition requires that both the forward scattering case (the condition is not satisfied for forward scattering when r → 0) and the backscattering case (for the monostatic experiment when a ≈ D, the condition a2/λ ≫ r is not satisfied, effective scattering volume (8) degenerates into infinity, and the formulas need appropriate modification [Berngardt and Potekhin, 2000]) be excluded from analysis. Usually, in the bistatic case, those conditions are satisfied.
 For the length of the sounding signal and the receiving window, which determine the form of the kernel H(ω, r), the requirement (12) implies that the signal and the receiving window change little within distances of the Fresnel radius order:
This corresponds to using narrowband signals and windows:
Thus the conditions (13) and (14) determine the validity range of (9) and (11).
3.3. Selective Properties of the Radar Equations Obtained
 Let us consider in greater detail the properties of the resulting radar equations (9) and (11). These equations establish a linear relation between the scattered signal spectrum and the spatial spectrum of dielectric density fluctuations. They clearly show selective properties of the scattering determining the region of Fourier harmonics, which make the main contribution to the scattering.
 Expression (7) is useful for determining the signal scattered on some spatial harmonic. In real situations, however, the medium involves different harmonics. Furthermore, as has been shown above in section 3.1, for each spatial harmonic the greatest contribution to the scattering will be made by the region of effective scattering (RES), whose location is determined by the Wolf-Bragg condition (6). Also, the contribution to the scattering from those spatial harmonics whose RESs lie outside the region of beam crossing is small. If the beams are considered to be cones (with angles ΔΘi and ΔΘs for the transmitter and the receiver, respectively), then it is possible to obtain an upper estimate (assuming that the sounding signal and the receiving window are long enough) of the selectivity using wave vectors from geometrical considerations.
 The spread of wave vectors involved in the scattering along directions in the plane passing through the receiver and the transmitter is
The spread of wave vectors in absolute value is determined by the expression
where kmid = 2k0 cos (β0/2) is the wave vector corresponding to the scattering from the weight volume center. The wave vector spread along directions in the plane normal to the receiver-transmitter axis is estimated by the formulas
where H is the height at which the scattering volume is located above the ground.
 The wave vectors that make the main contribution to the scattering lie within a region near the middle wave vector kmid, corresponding to the Bragg scattering from the center of the volume covered by the beam (6). A selectivity of the scattering process for the scattering by a small scatterer was shown by Newton ; in a statistical setting of the problem it was estimated by Tatarsky  and Ishimaru , but in a linear setting of the scattering by an arbitrary extended scatterer, this scattering process selectivity has not yet been established to date. Thus the resulting radar equations (9) and (11) make it possible to establish selective properties of the scattering and to determine the region of wave vectors involved in the scattering, both in direction and in absolute value. They can be used in the analysis of scattered signals, rather than their statistical characteristics only.
4. Statistical Radar Equations
 In sections 4.1 and 4.2 the radar equation is obtained for arbitrary spatial correlation radii of irregularities, and limitations on its applicability and its limiting cases are considered.
4.1. Obtaining the Radar Equations
 Using the starting equation (1), let us develop the expression for the scattered signal mean statistical spectral power (up to constant factors and with the transition to a spectral representation from difference arguments):
W(τ, S, ΔS) = ∫ o(t)a(t − S/c)o*(t − τ)a*[t − τ − (S − ΔS)/c]dt is a weighting function dependent on the signal waveform and on the receiving window only, and is a stationary correlation function of dielectric permittivity fluctuations. Its arguments are the mean statistical distance r, the correlation radius ρ, and the correlation time τ [Rytov et al., 1978]. It is apparent that the integrals appearing in (18) and (19) are analogous to the integral in (4) considered above (in section 3.1) and contain a rapidly oscillating function under the integral. We now apply the procedure described in section 3 to expressions (18) and (19). It is seen that the integral over R (equation (19)) can be evaluated by a three-dimensional SPM, as done in section 3. The integral over r can be evaluated using this method on the assumption that the dielectric permittivity fluctuations' spectral density Φ (ν, r, k) (the spatiotemporal spectrum of their correlation function) changes slowly with r. Criteria for weak variability will be presented below. Thus, by integrating (18) and (19) over R and r, respectively, by the stationary phase method, we get
 Furthermore, the stationary points r0 and R0 depend on the wave vector k, the wave number K, and the frequency ν and are defined by equations similar to the modified Wolf-Bragg condition (6):
whence it follows that the stationary points in (21) and (22) are coincident:
In view of (21), (22), (6), and (24) we obtain a radar equation for root-mean-square quantities in the form
where r0 is defined by (23).
 The radar equation obtained here relates the spectral power of the scattered signal to the fluctuations' spectral density in the form of an integral over the space of wave vectors. In a manner like obtaining (11) from (9), one can obtain the radar equation relating these two functions in terms of an integral over space, for which purpose it suffices merely to take into account the transition Jacobian from the coordinate system k to the coordinate system r0(10):
 Here k(r0) is determined by the modified Wolf-Bragg condition (23). The validity range of the resulting expressions (25) and (26) is constrained, in addition to the limitations pointed out in section 3, by media, for which the fluctuations' spectral density changes little with a change of the direction in the parameter r by the angle (λ/r)1/2. This implies that mean statistical properties of the medium change little in r within distances of the Fresnel radius order.
 Selective properties of the scattering are also pronounced in the resulting radar equation. The beam factor g(r) determines selection both from wave vectors, which make the main contribution to the scattering, and from spatial regions, with the wave vector and the spatial location of the diagnosed region being related by condition (23). This corresponds to the local fulfillment of the Wolf-Bragg conditions at each point of the diagnosed medium. The effective weight volume W(ω, S, k), which is determined solely by the sounding signal and receiving window forms, determines also the selective properties of the scattering for wave numbers, distances, and frequencies. Its width in the spatial variable S determines the region of transmitter-irregularity-receiver optical paths for irregularities contributing to the scattering, while the width in the wave variable extends further the region of fluctuations' spectral density wave numbers participating in the scattering. The width of the wave number region in absolute values and distances was considered earlier in section 3.3, having the order of (15)–(17), and their values are concentrated near the wave vector corresponding to the fulfillment of the Wolf-Bragg condition for the center of the diagnosed volume. In the frequency variable the effective volume is convoluted with the spectral density. For that reason, the scattered signal spectrum is mostly broader than the frequency spectrum of the spectral density of fluctuations, and this broadening is determined by the properties of the effective weight volume.
4.2. Limiting Cases of the Radar Equations Obtained
 Let us illustrate the implications of the resulting radar equation for two limiting cases of scattering media. In the case of a time-independent isotropic medium with a small spatial correlation radius, the irregularities' spatial spectrum is sufficiently broad in all directions k, is uniform in r, and has the form of a δ function in frequency ν. For such a model of the medium an integration can be performed in (26) over these parameters to give a standard radar equation for the power of the scattered signal for small spatial correlation radii [Tatarsky, 1967; Ishimaru, 1981]:
where W1(S) = ∫ W(ν, S, K) dK(dν/2π).
 As a further example, we consider another limiting case, which corresponds to the case of large spatial correlation radii. Let the sounding be performed with an infinitely long impulse with reception by an infinitely long window, which corresponds to W(ν, S, K) = δ (ν)δ (K). Let there exist dielectric permittivity irregularities in the medium which have the form of a nonstationary spatial statistically isotropic harmonic array with broad frequency spectrum. Then Φ(ω, r, k) = Φ0δ (k − k1), and the radar equation (25) becomes
It is evident that the spectral power of the scattered signal at each frequency will be determined by a small region whose location is determined by the fulfillment of an analogue for the Wolf-Bragg condition
and by dimensions of Fresnel radius (the size of the cophasal region V(r)) order. Thus each spectral component of the received signal will arrive from its own point of space and with the amplitude determined by the other terms involved in (25). Thus, by virtue of the radar equation properties, the spectrum of the received signal will differ from the spectral density frequency spectrum and will be determined solely by the beam. This distortion of the received signal spectrum when compared with the medium frequency spectrum in the case of scattering from a separate harmonic array is not described by a standard radar equation [Tatarsky, 1967; Ishimaru, 1981], and one has to consider consequent models for exact experiment conditions for explanation of the phenomena observed experimentally (for example, for radioacoustic sounding this leads to the significant difference between expected Doppler shift and observed frequency shift of received signal [Kon and Tatarsky, 1980]).
 In this paper we have outlined the method for obtaining the relation between scattered signal and spectral characteristics of the medium, which implies essentially the transition to a consideration of the scattering on medium spatial harmonics and to the stationary phase method implementation for calculating the contributions from separate harmonics.
 By using the proposed method we have obtained two equivalent radar equations relating the Fourier spectrum of the scattered signal to the spatial Fourier spectrum of dielectric permittivity fluctuations (9) and (11) for the case where the receiver and the transmitter are not in the far-field range of the scattering volume. These equations show explicitly the selective character of the scattering process: The main contribution to the scattering is made by spatial harmonics of the medium, for which conditions similar to the Wolf-Bragg conditions are satisfied (equation (6)). The validity range of the expression obtained in this study is virtually coincident with that of the starting expression in a spatiotemporal representation (equation (1)), provided that sufficiently narrowband sounding signals are used (equation (14)) and the cases of exactly backward (considered by Berngardt and Potekhin ) and exactly forward soundings are excluded (equation (13)). Thus the resulting expressions (9) and (11) are an analogue for (1) in a spectral region in most bistatic experiments on remote probing of media. The radar equation obtained in this study is a generalization of formulas for small scatterers [Newton, 1969] to scatterers of an arbitrary size. In problems of media diagnostics, (9) and (11) can be used to analyze scattering signals as such, rather than their statistical characteristics alone.
 The proposed method has been used to obtain two equivalent, statistical radar equations for arbitrary radii of spatial correlation (equations (25) and (26)) which hold true for media whose mean statistical parameters change little within distances of the Fresnel radius order. This limitation is usually true for measurements of real media, because mean statistical parameters of the medium (scattering cross section, drift velocity, etc.) usually vary smoothly throughout the diagnosed volume. It has been shown that these radar equations (25) and (26) can be useful for obtaining both a standard radar equation for small spatial correlation radii of irregularities (equation (27)) and radar equations for other models of scattering media.
 One should use the statistical radar equations (25) and (26) instead of the standard equation in the cases where spatial correlation radii are compared to or greater than Fresnel radius in some direction. Situations can arise, for example, in case of scattering from ionospheric irregularities elongated with Earth's magnetic field and in case of scattering from anisotropic air turbulence and do arise in case of atmosphere radioacoustic sounding. The nonstatistical radar equations obtained (equations (9) and (11)) may be useful for analysis of scattered signal without averaging.