INTRODUCTION. Spatial and time correlations of the jitter of optical images usually need to be considered when the images are transmitted through a turbulent ...
ISSN 10248560, Atmospheric and Oceanic Optics, 2014, Vol. 27, No. 1, pp. 88–99. © Pleiades Publishing, Ltd., 2014. Original Russian Text © V.V. Nosov, V.P. Lukin, 2014, published in Optica Atmosfery i Okeana.
OPTICAL INSTRUMENTATION
Measurement Technique of Turbulence Characteristics from Jitter of Astronomical Images Onboard an Aircraft: Part 2. Accounting for Photoreceiver Response Time V. V. Nosov and V. P. Lukin V.E. Zuev Institute of Atmospheric Optics, pl. Akademika Zueva 1, Tomsk, 634021 Russia Received March 13, 2013
Abstract—The main theoretical relations required by the technique for measuring turbulence characteristics from the jitter of astronomical images onboard a flying aircraft are derived. Monostatic and differential (bistatic) receivers are compared. The analysis does not repeat the analysis of signals and errors of the com mon differential method. The technique suggested supposes operation from a moving carrier. It is shown that the maximum deviation of the bistatic response function from the monostatic response function is observed when the velocity vectors of the carrier and the channel spacing are collinear. The effects of the turbulence outer scale, the vector carrier velocity, sampling frequencies, and other parameters of the instrument scheme are estimated. In particular, it is found that the typical correlation time in a bistatic differential receiver depends on the transportation time of turbulent inhomogeneities between two receiving channels until the distance between them is less than the turbulence outer scale. DOI: 10.1134/S1024856014010096
wave statement in a general case; an equation for it was derived in this work that is applicable for sources with arbitrary coherence and divergence. It can be used for any turbulence intensity and considers the effect of turbulence outer scale [5, 6]. The wellknown theoret ical results [1, 3] follow from [4] in limiting cases. Conclusions drawn based on this equation agree with the experiments for laser beams along surface horizon tal paths [7]. The results from [4] were generalized to the case of inhomogeneous atmospheric paths of arbi trary geometry in [8]. Equations for spatial and time correlation func tions of image jitter [9, 8] and frequency spectra [10, 8] were derived based on the approach developed in [4]. In [4], the structure function of wave phase needed to calculate image shifts was specified based on the ray representation [11, 12]. As shown in [4], ray represen tations for the phase structure function are equivalent to the lognormality of wave intensity fluctuations; the latter has been confirmed experimentally [1, 2]. Some deviations from the lognormality in the saturation region are described by the exponential probability law [13] and relate to the region of low intensities (deep decays), where a difference between these laws is insig nificant [2]. Theoretical representations for a beam that is a normal to the average phase front of the wave are confirmed in experiments with a laser beam in a turbulent atmosphere [11]. The calculation of jitter dispersion using another approach [14, 15] confirmed the conclusions from [4] and forecasted the effect of deviations of the image analysis plane from the focal plane in a receiver.
INTRODUCTION Spatial and time correlations of the jitter of optical images usually need to be considered when the images are transmitted through a turbulent atmosphere con currently through several channels. Specific aircraft measurements assume the use of CCD arrays as photorecording devices. The speed of these arrays is small at present. Even good modern arrays provide a frame rate of no higher than 300 Hz at a high spatial resolution; therefore, parameters of tur bulence change along an optical path significantly for one frame (photoreceiver response time) due to a high speed of an aircraft. The process of image jitter is strongly averaged and, hence, the technique for mea suring turbulence parameters from measurements onboard a flying aircraft should consider the effect of a finite (nonzero) photoreceiver response time. 1. THEORETICAL EQUATIONS FOR TIME CORRELATION OF THE POINTSOURCE IMAGE JITTER FUNCTION Image jitter can be qualitatively described by fluc tuations of the angles of arrival [1, 2] on a base equal to the diameter of a receiving telescope. In a stricter wave statement, the dispersion of jitter of an image of a plane wave was first theoretically considered by V.I. Tatarskii. The first results of considering the spa tial limitation of waves [3] related to the dispersion of angles of arrival and were valid for conditions under which a beam was not broadened. In [4], the disper sion of lasersource image jitter was considered in the 88
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The position of a source image in the focal plane of a receiving telescope is characterized by the coordi nates of the center of gravity ρt(yt, zt) of image intensity distribution. Using the approach developed in [4, 9], the vector of random image deviation from its undis
respect to intensity) of the receiving lens of the tele scope. If a singlemode laser beam sloped to the Ox coor dinate axis at the angle α' (α' Ⰶ 1), the radiation aper ture of which is centered in the radiation plane (x' = 0)
turbed position ρt' = ρ − ρt is expressed as
and is ρ0' (y0' , z 0' ) apart from the Ox axis, is used as a source, then, according to [9], P(x', x, ρ) can be rep resented as
x
dx'∫ d ρt(ρ) I (x, ρ) ∇ ε ( x', P(x', x, ρ)) ∫ F 2
ρ 1
ρt' = −
t 0
∫ d ρt(ρ) I (x, ρ) 2
2
P(x', x, ρ) = ρ0' (1 − x' x) + ρ1'(x' x) + (ρ − ρ1')l(x'), P(x, x, ρ) = ρ,
, (1)
where x is the optical path length, ε1(x', ρ) is the filed of medium permittivity fluctuations (〈 ε1〉 = 0), P(x', x, ρ) is the trajectory of the average diffraction beam, 〈I(x, ρ)〉 is the average intensity of the source, and Ft and t(ρ) are the focal length and transmittance (with
where ρ1' = ρ1'(x) = ρ0' + n' xα' is the radiusvector of the beam optical axis in the receiving plane (x' = x), n' is the unit vector of projection on the plane y0z of a nor mal to the phase front at the center of the radiating aperture, and l(x') is the normalized trajectory of the average diffraction beam:
⎧ 1 ⎡⎣Ω −2t 2 − (xt F )(1 − xt F ) + 12σ12 5Ω −1t 16 5 ⎤⎦ ⎫⎪ ⎪ l (x') = exp ⎨− dt ⎬. −2 2 2 12 5 − 1 16 5 t ⎡⎣Ω t + (1 − xt F ) + 8σ Ω t ⎤⎦ ⎪ ⎪⎩ x' x ⎭
∫
Here, Ω = ka2/x is the Fresnel number of the source (k = 2π/λ is the wave number, a is the radius of radiat ing aperture), F is the initial curvature of the phase front of the beam, and σ2 = 0.77C ε2 k 7 6 x 11 6 (C ε2 if the structure characteristic of the random field ε1). Con sider the wellknown equation for the average intensity of this beam [16]
I (x, ρ) =
⎪⎧ [ρ − ρ1'(x)]2 ⎪⎫ I 0a 2 exp ⎨− ⎬, aef2 (x) aef2 (x) ⎭⎪ ⎩⎪
where aef(x) = a[(1 – x/F)2 + Ω–2 + 8σ12/5Ω–1]1/2 is the effective beam radius at a distance x and I0 is the inten sity at the center of the radiating aperture, and use the Gaussian approximation of the transmittance of the receiving lens
{
t(ρ) = t 0 exp −[ρ − ρ2' ]
2
}, t
2 at
0
= const,
x
F 2 2 2 ρt' = ρt'(ρ0', ρ1', ρ2' ) = − t 2 dx' d ρ exp(−ρ a0 ) 2πa0
∫ ∫ 0
× ∇ ρε1(x', R(x', ρ0', ρ1', ρ2' ) + ρ l(x'));
(4)
R(x', ρ0', ρ1', ρ2' ) = ρ0' (1 − x' x) + ρ1'(x' x) + (ρ2' − ρ1')l(x')(a02 at2 ). ATMOSPHERIC AND OCEANIC OPTICS
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The parameter a0 in Eq. (4) is defined by the relation ship a0−2 = a0−2(x) = at−2 + aef−2(x) and characterized by the minimum of at (receiver radius) and aef(x) (beam radius in the receiving plane). Below, we consider a point source (spherical wave). A source can be considered point if its lateral dimen sions are much smaller than the radius of the first Fresnel zone, ka2/x Ⰶ 1. In this case, the beam radius should equal zero, a → 0. Then, Ω → 0 and aef(x) → ∞, which corresponds to the equality a0 = at. The parameter l(x') in Eq. (2) is also simplified as follows: l(x') = x'/x. As a result, from Eq. (4), we derive x
ρt' = ρt'(ρ0', ρ2' ) = −
Ft dx' d 2ρ exp(−ρ2 at2 ) 2 2πat
∫ ∫ 0
× ∇ ρε1(x', R(x', ρ0', ρ2' ) + ρ x' x),
(3)
where at and ρ2' are the radius and radiusvector of the center (in the plane x' = x) of receiving lens of the tele scope. After integration in Eq. (1), we have
(2)
(5)
R(x', ρ0', ρ2' ) = ρ0' (1 − x' x) + ρ2' (x' x). As can be seen from Eq. (5), the vector of random image deviation ρt' is independent of ρ1' in the case of a point source. Let us choose the second point source. Its center is characterized by the radius vector ρ0'' in the radiation plane x' = 0. The radiation of this source is received by the second receiver centered at point ρ2'' (in the receiv ing plane x' = x). Radii and focal lengths of both receivers are considered equal. The equation for the vector of random deviation of an image of the second
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source ρt'(ρ0'', ρ2'') coincides with Eq. (5) if the vectors ρ0'' and ρ2'' are substituted for the vectors ρ0' and ρ2', respec tively.
Time correlation functions of pointsource image jitter (B for monostatic and B d for bistatic differential receivers) are defined as
B(ρ0', ρ2', t', t'') = ρt'(ρ0', ρ2', t')ρt'(ρ0', ρ2', t'') ,
We consider two types of optical receivers, i.e., monostatic and bistatic differential. A monostatic receiver is a telescope in the focus of which (at a pho torecording device) a source is imaged. When measur ing the jitter, an output signal of the monostatic receiver is proportional to the image shift vector in the telescope focal plane ρt'(ρ0', ρ2' ). A bistatic photoreceiver consists of two monostatic photoreceivers spaced in a plane normal to the optical path. In the general case, images of different sources are formed in each channel. The signal in each chan nel is also proportional to the vector of image shift in the focal plane: ρt'(ρ0', ρ2' ) in the first channel and ρt'(ρ0'', ρ2'') in the second channel. Finally, in a bistatic differential receiver, the signal is proportional to the difference in vectors of image shifts that occur in each channel as follows: ρt'(ρ0', ρ2' ) – ρt'(ρ0'', ρ2''). Let us now study the time correlation functions of the pointsource image jitter. For this, it is necessary to know the dependence of the image shift vector on time t. The time dependence of statistical parameters is usually derived using the Taylor hypothesis of frozen turbu lence, which is widely used in studies of optical wave propagation in a turbulent atmosphere. There are many confirmations of this hypothesis [1, 2, 11, 17] obtained from a comparison of conclusions on its basis with experimental data. According to the theory of frozen turbulence, the following equality is true for the spatiotemporal field of medium permittivity fluctuations ε1 (x', ρ, t) [1, 2]: ε1(x, ρ, t + t 0) = ε1(x', ρ − v ⊥t, t 0),
where v ⊥ = (v1, v 2) is the component of the vector of mean wind speed normal to the wave propagation direction and t0 is a certain fixed (initial) time point. This equality follows from the motion of the spatial pattern of the field ε1(x', ρ, t0) in a lateral plane. Let us note that the lateral wind speed in a general case is a function of the longitudinal coordinate along the path x', v⊥ = v⊥(x'). Substituting the last relationship in Eq. (5), we obtain x
ρt' = ρt'(ρ0', ρ2', t ) = −
Ft dx' d 2ρ exp(−ρ2 at2 ) 2 2πat
∫ ∫ 0
× ∇ ρε1(x', R(x', ρ0', ρ2' ) + ρ x' x − v ⊥t, t 0 ), R(x', ρ0', ρ2' ) = ρ0' (1 − x' x) + ρ2' (x' x).
(6)
(7a)
d
B (ρ0', ρ2', ρ0'', ρ2'', t', t'') = ⎡ρt'(ρ0', ρ2', t') − ρt'(ρ0'', ρ2'', t')⎤ ⎣⎢ ⎦⎥
(7b)
× ⎡ρt'(ρ0', ρ2', t'') − ρt'(ρ0'', ρ2'', t'')⎤ . ⎣⎢ ⎦⎥ Let us carry out the averaging in Eqs. (7a) and (7b). In the approximation of deltacorrelated permittivity fluctuations, the correlation function of field ε1(x', ρ, t0) can be represented as follows [18, 19]:
ε1(x', ρ1, t 0)ε1(x'', ρ2, t 0) = δ(x' − x'')A(x', ρ1 − ρ2 ),
(8)
∫
A(x', ρ) = 2π d 2κΦ ε(x', κ)exp(iκρ),
where Φε(x', κ) is the 3D spectrum of the field ε1. Using Eq. (6) in Eqs. (7a) and (7b) and considering Eq. (8), we obtain
πF B(τ) = t 2
2
x
∫ dx'(x' x) ∫ d κκ Φ (x', κ) 2
2
2
ε
0
{
(9a)
}
× exp −κ 2at2 (x' x)2 2 + i κ v ⊥τ , πF B (τ, ρ0, ρ2) = t 2 d
× exp
2
x
∫ dx'(x' x) ∫ d κκ Φ (x', κ) 2
0 2 2 −κ at (x'
{
2
2
ε
}
x)2 2 + i κv ⊥τ
(9b)
× [2 – exp {i κR(x', ρ0, ρ2)} − exp {−i κR(x', ρ0, ρ2)}] , where τ = t'' − t'; ρ0 = ρ0'' − ρ0'; ρ2 = ρ2'' − ρ2'. Equations (9a) and (9b) implies that the correla tion function B(τ) for a monostatic receiver is inde pendent of the positions of source and receiver lateral to the path (vectors ρ0' and ρ2' ), and the correlation func tion Bd(τ, ρ0, ρ2) for a bistatic differential receiver depends only on lateral distances between sources (vector ρ0 = ρ0'' − ρ0' ) and receivers (vector ρ2 = ρ2'' − ρ2' ). For a more comprehensive comparison of the types of receivers used, we give the calculation results simul taneously for monostatic and bistatic differential receivers at each step of the calculations. Taking into account the significant effect of the tur bulence outer scale L0 on image jitter [4, 9, 20, 21], we can define the 3D spectrum of field ε1 as follows [20]: −11 3
{
}
⎡1 − exp −κ 2 κ 0(x')2 ⎤ , ⎣ ⎦ A0 = 0.033, κ 0(x') = 2π L0(x').
Φ ε(x', κ) = A0Cε (x')κ 2
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As shown in [4, 16, 20, 21], the effect of the turbu lence inner scale can be neglected when the receiver radius is significantly larger than the inner scale. This condition is usually true. Substituting the equation for spectrum into Eqs. (9a) and (9b) and calculating the corresponding integrals, we find final representations for the correla tion functions:
91
2. ANALYSIS AND COMPARISON OF CORRELATION FUNCTIONS OF POINTSOURCE IMAGE JITTER FOR MONOSTATIC AND BISTATIC RECEIVERS Let us analyze and compare the derived correlation functions of pointsource image jitter for monostatic and bistatic receivers [Eqs. (10a) and (10b)].
x
B(τ) =
2 σt 0
First, as the simplest case, we consider horizontal optical paths along which characteristics of the atmo spheric turbulence are constant:
∫ dx'(x' x) C (x') 2
2 ε
0
2 2 ⎧ ⎛ v (x') τ ⎞ × ⎨(x' x)−1 31 F1 ⎜ 1 ,1, − ⊥2 ⎟ 2at (x' x)2 ⎠ ⎩ ⎝6 2 − ⎡⎣(x' x) + β(x')⎤⎦
Cε2(x') = Cε2 = const, L0(x') = L0 = const (β(x') = β = const ) , v ⊥(x') = v ⊥ = const.
(10a)
−1 6
2 2 ⎛ ⎞⎫⎪ v (x') τ ⎟⎬ , × 1 F1 ⎜ 1 ,1, − 2 ⊥ 2 ⎜6 ⎟ ⎡ ⎤ a x x x 2 ( ) ( ) ' + β ' t ⎝ ⎣ ⎦⎠⎭⎪
Then, we consider the correlation functions of jitter of astronomical images along inclined paths. Let us assume that both monostatic and bistatic receivers are located onboard a flying aircraft. Let us derive theoreti cal equations for the correlation functions of astronom ical image jitter on the basis of results from [27] with accounting for a finite time of photoreceiver (CCD array) response. Let us derive relationships between the dispersions of astronomical image jitter in receivers with a finite and zero response times (for both types of receiv ers considered).
x
∫
B d (τ, ρ0, ρ2 ) = σt20 dx'(x' x)2 Cε2(x'){(x' x)
−1 3
A1(x')
0
−1 6
A2 (x')} , − ⎡⎣(x' x)2 + β(x')⎤⎦ ⎛ v (x')2 τ2 ⎞ A1(x') = 21 F1 ⎜ 1 ,1, − ⊥2 ⎟ 2at (x' x)2 ⎠ ⎝6 2 ⎛ v (x')τ + R(x', ρ0, ρ2) ⎞ − 1F1 ⎜ 1 ,1, − ⊥ ⎟ 2at2(x' x)2 ⎝6 ⎠ 2 ⎛ v (x')τ − R(x', ρ0, ρ2) ⎞ − 1F1 ⎜ 1 ,1, − ⊥ ⎟, 2at2(x' x)2 ⎝6 ⎠
(10b)
⎛ ⎞ v (x')2 τ2 ⎟ A2(x') = 21 F1 ⎜ 1 ,1, − 2 ⊥ 2 ⎜6 ⎤⎟ 2at ⎣⎡(x' x) + β(x')⎦⎠ ⎝ 2 ⎛1 v ⊥(x')τ + R(x', ρ0, ρ2) ⎞ ⎟ − 1F1 ⎜ ,1, − 2 2 ⎜6 2at ⎣⎡(x' x) + β(x')⎦⎤ ⎠⎟ ⎝ 2 ⎛ v (x')τ − R(x', ρ0, ρ2) ⎞ ⎟. − 1F1 ⎜ 1 ,1, − ⊥ 2 ⎜6 2at ⎣⎡(x' x)2 + β(x')⎦⎤ ⎠⎟ ⎝ The following designations are accepted in Eqs. (10a) and (10b):
2.1. Horizontal Optical Paths Let C ε2(x') = Cε2, β(x') = β, and v⊥(x') = v⊥ in Eqs. (10a) and (10b). These parameters are constant along horizontal paths. As a result, after the variable change (x' = ξx) and introduction of dimensionless normalized parameters
l = 2 −1 2 v ⊥τ at , r0 = 2 −1 2 ρ0 at , r = 2 −1 2 ρ2 at , we find 1
B(l ) =
σt20Cε2 x
2 ⎧ ⎛ ⎞ d ξξ 2 ⎨ξ −1 31F1 ⎜ 1 ,1, − l 2 ⎟ ξ ⎠ ⎩ ⎝6 0
∫
⎛ ⎞⎫ − (ξ 2 + β)−1 61 F1 ⎜ 1 ,1, − 2l ⎟⎬ , 6 ξ + β ⎠⎭ ⎝
σt20 = A0π2 2 −5 6 Γ(1 6)Ft 2at−1 3 2 −5 6
⎡⎣ A0π 2
B d (l, r0, r)
Γ(1 6) = 1.017⎤⎦ ,
1
=
β(x') = 2 [ κ 0(x')at ] = [ L0(x') (πat )] . 2
2
σt20Cε2 x
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∫ dξξ {ξ 2
−1 3
2 ⎛ ⎞ A1(ξx) = 21 F1 ⎜ 1 ,1, − l 2 ⎟ ξ ⎠ ⎝6
2014
}
A1(ξx) − (ξ 2 + β)−1 6 A2(ξx) ,
0
Comparing Eq. (10a) for the time correlation func tion of image jitter with those given in [9, 22], it is easy to see that they coincide for a point sources. ATMOSPHERIC AND OCEANIC OPTICS
(11a)
2
92
NOSOV, LUKIN Bd(l, 0, r)/Bd(0, 0, r) 1.2 1.0 0.8 5, B(l)/B(0)
0.6 0.4 0.2
5
0 –0.2 1 –0.4
2
4 3
–2 0 2 4 6 8 10 12 14 16 18 20 22 24
l
Fig. 1. Time correlation coefficient in the bistatic differen tial receiver for different distances ρ2 between the receiving channels at ϕ = 0°, at = 5 cm, v⊥ = 2 m/s, and L0 = 0.8 m: ρ2 = 0.7–7 (1), 27 (2), 57 (3), 157 cm (4) (r = 0.1–1 (1), 3.82 (2), 8.06 (3), and 22.2 (4)); monostatic receiver (5).
2 ⎛1 l + r0(1 − ξ) + rξ ⎞ − 1F1 ⎜ ,1, − ⎟ ξ2 ⎝6 ⎠ 2 ⎛ l − r0(1 − ξ) − rξ ⎞ − 1F1 ⎜ 1 ,1, − ⎟, ξ2 ⎝6 ⎠
(11b)
2 ⎛ ⎞ A2(ξx) = 21 F1 ⎜ 1 ,1, − 2l ⎟ ξ + β⎠ ⎝6 2 ⎛ l + r0(1 − ξ) + rξ ⎞ − 1F1 ⎜ 1 ,1, − ⎟ 2 ξ +β ⎝6 ⎠ 2 ⎛ l − r0(1 − ξ) − rξ ⎞ − 1F1 ⎜ 1 ,1, − ⎟. 2 ξ +β ⎝6 ⎠ Let us now consider the case when both channels in a bistatic receiver are adjusted to one source. In this case, the distance ρ0 between two sources is zero (ρ0 = 0) and, hence, r0 = 0 in Eq. (11b). Designating the angle between the vectors l and r by ϕ (coinciding with the angle between vectors v⊥ and ρ2), the parameters |l + rξ|2 and |l – rξ|2in arguments of confluent hypergeometric functions can be written as
|l + rξ|2 = l 2 + r2 ξ2 + 2lrξcosϕ, |l – rξ|2 = l 2 + r2ξ2 – 2lrξcosϕ. Let us study the behavior of the correlation func tions B(l) and Bd(l, 0, r) for typical values of the parameters at, v⊥, and β. Let us assume that the receiver radius at = 5 cm and the modulus of wind speed lateral to the path v⊥ = 2 m/s. Let the horizontal optical path be at an altitude of 2 m from a plane
underlying surface, h0 = 2 m. Using a simple altitude model for the turbulence outer scale, L0 = 0.4h0 [1, 2, 17, 23–25], we find L0 = 0.8 m and β = 12.9. Figure 1 shows the plots of time correlation coeffi cients B d (l, 0, r)/B d (0, 0, r) of pointsource image jit ter in the bistatic differential receiver as functions of the time separation τ (depending on l). The data in Fig. 1 correspond to ϕ = 0°. The parameter r, which characterizes the distance between the receivers (channels), varies in the range 0.1–22.2. For at = 5 cm, the range of variations in the parameters l and r in Fig. 1 correspond to variations in the time separation τ from 0 to 0.91 s and distances between the receivers ρ2 from 0.7 cm to 1.6 m. The time correlation coefficient of pointsource image jitter in the monostatic receiver B(l)/B(0) is also shown in Fig. 1 for comparison (curve 5). As can be seen from Fig. 1, the time correlation coef ficient in the bistatic (differential) receiver can take both positive and noticeable negative values, in con trast to the monostatic receiver. If the characteristic correlation time τ0 is found from a drop of the correla tion coefficient to zero, then τ0v⊥ ≈ ρ2 follows from Fig. 1 at short distances between the channels when r ⱗ 1 (ρ2 ⱗ at). The correlation time τ0 increases with the distance between the channels ρ2 (approximately proportional to ρ2), saturating to τ0k, which corre sponds to the monostatic receiver. An equation for τ0k is easily derived from Eq. (11a) based on the results from [9]. Thus, τ0k = (L0/v⊥)π–1 Γ–3(5/6) if β Ⰷ 1, which is usually true in practice. The characteristic correlation time in the bistatic differential receiver is determined by transport of tur bulent inhomogeneities through the space between two receiving channels with the speed v⊥, until the dis tance is smaller than the turbulence outer scale L0 (ρ2 ⱗ L0, τ0 ∼ ρ2/v⊥). If the distance between the channels exceeds the outer scale (ρ2 > L0), then the correlation time is determined by transport of inhomogeneities through the turbulence outer scale (τ0 ∼ τ0k ∼ L0/v⊥). The above results are valid if the vector of average wind speed v⊥ and the vector ρ2, which connects the channel centers, are collinear (ϕ = 0°). If the vectors rotate relative to each other then the situation changes. Figure 2 shows the time correlation coefficient B d(l, 0, r)/B d(0, 0, r) of pointsource image jitter in the bistatic differential receiver as a function of the angle ϕ between the vectors v⊥ and ρ2. It can be seen that the deviation of the bistatic cor relation coefficient from the monostatic one is maxi mal when the vectors v⊥ and ρ2 are collinear (ϕ = 0°). As the angle ϕ increases, the bistatic coefficient approaches the monostatic one (curve 5). The bistatic coefficient almost coincides with the monostatic one, when the vectors v⊥ and ρ2 are perpendicular (ϕ = 90°). Hence, in this case, the effects caused by the
ATMOSPHERIC AND OCEANIC OPTICS
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MEASUREMENT TECHNIQUE OF TURBULENCE CHARACTERISTICS FROM JITTER Bd(l, 0, r)/Bd(0, 0, r)
Bd(l, 0, r)/B(l)
1.2
2 1
1.0 0
0.8 5, B(l)/B(0)
0.6
5
0.4
–2 4
2
–4
0.2 3
0
–6 2
–0.2
3 –8
1
–0.4 –1
93
0
1
2
3
4
5
6
7
8
9 l
–2
Fig. 2. Correlation coefficient in the bistatic differential receiver as a function of the angle ϕ between the vectors v⊥ and ρ2: at = 5 cm, ρ2 = 27 cm (r = 3.82), L0 = 0.8 m, and v⊥ = 2 m/c; ϕ = 0° (1), 30° (2), 60° (3), and 90° (4); mono static receiver (5).
It is clear that random coordinates of images become independent random variables at a sufficiently large distance between the channels. As is known, the variance of the difference (or sum) between two inde pendent random variables is equal to the sum of vari ances of each parameter. We have chosen parameters Vol. 27
2
4
6
8
10
12
14
r
Fig. 3. Ratio of correlation functions in bistatic and mono static receivers as a function of the distance ρ2 between the receiving channels at at = 5 cm, L0 = 0.8 m, and v⊥ = 2 m/s: l = 0 (τ = 0 s) (1), 2.0 (τ = 0.07 s) (2), and 2.8 (τ = 0.1 s) (3).
spacing of the channels in the bistatic differential receiver weaken significantly and can vanish. This result is clear if we consider that the same tur bulence is transported by wind sequentially through both receiving channels for collinear vectors v⊥ and ρ2, thus increasing the contribution into the correlation function. In the case of perpendicular vectors v⊥ and ρ2, the same inhomogeneity (especially if its size is less than the distance ρ2) is mainly transported through only one channel, and even does not appear in the field of view of the second channel. Figure 3 shows the ratio of the correlation function in the bistatic receiver B d(l, 0, r) to the correlation function in the monostatic receiver B(l). This ratio is considered to be a function of the distance ρ2 between the receiving channels (function of r) at certain values of the time separation τ (parameter l). Data from the Fig. 3 supplement the results in Fig. 1. As can be seen from Fig. 3, the correlation function in the bistatic receiver tends to the doubled correlation function in the monostatic receiver as the distance between the channels (parameter r) increases at any values of τ. Hence, the image jitter dispersion in the bistatic receiver (obtained from the correlation func tion at τ = 0, curve 1 in Fig. 3) tends to the doubled image jitter dispersion in the monostatic receiver with an increase in the distance ρ2.
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of the receiving channels equal; therefore, the vari ances of each independent random variable are also equal. This implies the doubling of the bistatic corre lation function compared to the monostatic function under significant spacing of the receiving channels. 2.2. Inclined Optical Paths Let us now study the correlation functions of astro nomical image jitter along inclined paths. Let mono static and bistatic receivers be located onboard a high flying aircraft. The parameters of a turbulent atmosphere in Eqs. (10a) and (10b) are not constant along inclined paths. They are functions of the altitude h measured from the underlying surface: Cε2(h), L0(h), v⊥(h). In turn, the altitude h in the case of inclined paths is a function of coordinate x', which flows along the path from a source to a receiver, h = h(x'). If the curvature of the Earth’s surface is neglected, then, according to [2, 8, 11, 25], h(x') = h0 + (x – x')cosθ, where θ is the zenith angle of the source and h0 is the receiver altitude measured from the underlying surface (h0 is the altitude of a flying aircraft for a receiver mounted onboard the aircraft). As was ascertained in [25], the Earth’s curvature should be taken into account in calculations of astronomical image jitter only for paths close to flat, when θ > 89°. Therefore, the above equation for the current altitude h(x') is valid almost in the whole range of zenith angles 0° ≤ θ ≤ 89°, which is of interest.
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Let us change two integration variables in Eqs. (10a) and (10b): x' → x – x'', x'' → (t – h0)secθ. The integra tion interval (0, x) does not vary in the first change and becomes (h0, h0 + xcosθ) in the second change. Again, for h(x') and x'/x in Eqs. (10a) and (10b), we obtain
As a result, from Eqs. (10a) and (10b), we obtain ∞
2 2 ⎧ ⎛ v (t ) τ ⎞ 2 2 B(τ) = σ t 0 sec θ dtC ε (t ) ⎨ 1 F1 ⎜ 1 ,1, − ⊥ 2 ⎟ 2at ⎠ ⎩ ⎝6 h
∫
0
⎛ v (t )2 τ 2 ⎞⎫ − [1 + β(t )]−1 61 F1 ⎜ 1 ,1, − 2⊥ ⎟⎬ , 2at [1 + β(t )]⎠⎭ ⎝6
h(x') → h0 + x''cosθ → t, x'/x → 1 – x''/x → 1 – (t – h0)secθ/x. Hence, after these changes, the altitude profiles of atmo spheric parameters, written in Eqs. (10a) and (10b) as Cε2(x'), L0(x'), and v⊥(x'), become functions of the only variable t:
Cε2(x') = Cε2(h(x')) → Cε2(t), L0(x') = L0(h(x')) → L0(t ), v ⊥(x') = v ⊥(h(x')) →v ⊥(t). The parameter β(x') is a function of the profile of outer scale L0(x'); therefore, β(x') → β(t). The vector R(x', ρ0, ρ2) in arguments of hypergeo metric functions in Eqs. (10a) and (10b) is defined by Eq. (6). Changing the variables, we have
R(x', ρ0, ρ2 ) = ρ0(1 − x' x) + ρ2(x' x) → r0(x'' x) + r2(1 − x'' x) → ρ0(t − h0 )sec θ x + ρ2[1 − (t − h0 )sec θ x]. If each channel in the bistatic differential receiver is adjusted to its own astronomical source, then the lin ear distance between sources ρ0 can be expressed in terms of the angle α, at which both sources are seen from the land, and the astronomical path length x. Again, if the sources are arranged symmetrically about the longitudinal coordinate axis (Ox axis), then the vector ρ0 can be represented as ρ0 = 2nxtan(α/2), where n = ρ0/ρ0 is the unit vector along the vector ρ0 that connects the sources. After the change in integration variables, the inte grands in Eqs. (10a) and (10b) include the multiplier Cε2(t). As is known [26], this parameter characterizes the turbulence intensity and noticeably differs from zero only in the optically active air layer of several kilo meters in depth h*. Above the active layer, Cε2(t) rapidly decreases with an increase in the altitude (increase in t). Hence, the region significant for integration over t is mainly bounded by h , t ⱗ h . * * The length of an astronomical path x exceeds sig nificantly the depth of the active air layer; therefore, we can change the upper limit in the integrals in Eqs. (10a) and (10b) to an infinite one at x → ∞, and, according to the above said, set x'/x = 1. In this case, R(x', ρ0, ρ2) → R(t, α, ρ2) = 2 tan(α/2)(t – h0)sec θ n +ρ2.
B d (τ, α, ρ2 ) ∞
=
2 σt 0
∫
−1 6
sec θ dtCε (t){A1(t) − [1 + β(t )] 2
A2(t )},
h0
⎛ v (t)2 τ2 ⎞ A1(t) = 21 F1 ⎜ 1 ,1, − ⊥ 2 ⎟ 2at ⎠ ⎝6 2 ⎛1 v ⊥(t)τ + R(t, α, ρ2 ⎞ − 1F1 ⎜ ,1, − ⎟ 2 2at ⎝6 ⎠
(12)
2 ⎛1 v ⊥(t)τ − R(t, α, ρ2 ⎞ − 1F1 ⎜ ,1, − ⎟, 2 2at ⎝6 ⎠
2 2 ⎛ v (t) τ ⎞ A2(t) = 21 F1 ⎜ 1 ,1, − 2⊥ ⎟ 2at [1 + β(t)]⎠ ⎝6 2 ⎛ v (t)τ + R(t, α, ρ2 ⎞ − 1F1 ⎜ 1 ,1, − ⊥ 2 ⎟ 2at [1 + β(t)] ⎠ ⎝6 2 ⎛ v (t)τ − R(t, α, ρ2 ⎞ − 1F1 ⎜ 1 ,1, − ⊥ 2 ⎟. 2at [1 + β(t)] ⎠ ⎝6 Equations (12) can be simplified when taking into account the rapid decrease in the turbulence intensity Cε2(t) with an increase in the altitude t. In this case, the region significant for integration in Eq. (12) is located near the lower limit of integration t = h0. In this region, L0(t) ∼ L0(h0), β(t) ∼ β(h0), and v⊥(t) ∼ v⊥(h0). Further estimates can be made with accounting for specific values of atmospheric parameters at the altitude h0. For receivers mounted onboard a flying aircraft, the altitude h0 is usually several kilometers. For the esti mates, let us assume h0 to be within the range 4–7 km. At present, there are different models of altitude profiles of the turbulence outer scale (see, e.g., [5, 6, 23–25]). These models describe altitude variations in the outer scale versus the type of underlying surface. How ever, in most cases that can be implemented in the atmo sphere, the turbulence outer scale L0(h) growths quite rapidly with the altitude h above the land [23–25]. There fore, the simplest model L0(h) = κKh (κK = 0.4 is the Karman constant) [1, 17] can be used for the esti mates. In this case, for the typical receivers’ radius at = 5 cm and the aircraft altitude from the range 4 ≤ h0 ≤ 7 km, we have
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5.2 × 10+7 ≤ β(h0) ≤ 1.2 × 10+8,
the refractivity index In by a simple relationship (due to the wellknown equality Cε2 = 4C n2 ):
4.3 × 10–2 ≤ [1 + β(h0)]–1/6 ≤ 5.2 × 10–2.
I ε = 4I n,
The absolute value of the function A2(t) in Eq. (12) does not exceed a constant of about unity; therefore, the second term in the integral for the function B d(τ, α, ρ2) in Eq. (12), which includes the multiplier [1 + β(t)]–1/6, turns out to be significantly less than the first one in the region significant for integration. Hence, it can be neglected. Additional numerical analysis shows that this neglect is legal, even in the regions of variations in external parameters in the function B d(τ, α, ρ2) where the arguments in hypergeometric functions in A1(t) become large, while these arguments in A2(t) are still small (due to a high value of [1 + β(h0)]). Again, due to aircraft motion, the altitude profile of the velocity of transverse wind v⊥(h) is composed by the aircraft velocity v⊥C and the transverse wind speed v⊥A(h), caused by the presence of natural air mass transport in the atmosphere, v⊥(h) = v⊥C + v⊥A(h). In the region significant for the integration in Eq. (12), v⊥A(t) ∼ v⊥A(h0). As follows from [26], the absolute val ues of each component of the vector v⊥A(h0) are usually not higher than several meters per second in the alti tude range 4 ≤ h0 ≤ 7 km. The typical speed of an air craft is higher than 500 km/h, which is about 140 m/s. Therefore, the atmospheric component of the vector of transverse speed can be neglected during integration in Eq. (12) compared to the aircraft speed, i.e., we can take v⊥(t) = v⊥C. To further simplify the results, let us assume that each channel in a bistatic differential receiver is adjusted to the same astronomical source. This corre sponds to zero angle α in Eq. (12). Again, accounting for the simplifications made, all multipliers in the integrands in Eq. (12), except for Cε2(t), turn out to be independent of the integration variable. Finally, from Eq. (12), we obtain
⎛ v 2 τ2 ⎞ 2 B(τ) = σto sec θI ε1F1 ⎜ 1 ,1, − ⊥C 2 ⎟ , 2at ⎠ ⎝6 ⎡ ⎛ v 2 τ2 ⎞ B d (τ,0, ρ2 ) = σto2 sec θI ε ⎢21 F1 ⎜ 1 ,1, − ⊥C 2 ⎟ 2at ⎠ ⎣ ⎝6
(13)
2 2 ⎛ ⎛ v τ+ρ ⎞ v τ − ρ ⎞⎤ − 1F1 ⎜ 1 ,1, − ⊥C 2 2 ⎟ −1 F1 ⎜ 1 ,1, − ⊥C 2 2 ⎟⎥ . 2at 2at ⎝6 ⎠ ⎝6 ⎠⎦
Here, Iε designates the integral value of the structure characteristic of atmospheric air permittivity fluctua tions (along the whole optical path); it is connected with the integral value of structure characteristics of ATMOSPHERIC AND OCEANIC OPTICS
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∞
∫
I ε = dhCε2(h), h0
∞
∫
I n = dhC n2(h). h0
Comparing Eqs. (13), which correspond to inclined paths, with Eqs. (11a) and (11b) derived for horizontal paths, we can conclude that main regulari ties of time correlation functions of image jitter revealed along horizontal paths are also true for inclined paths, when receivers are mounted onboard a highflying aircraft. Thus, for example, Eqs. (13) imply the following representations for the variances of image shifts in monostatic and bistatic receivers derived from the cor relation functions at τ = 0: B(0) = σ t20 sec θI ε,
(
)
(14) B d (0,0, ρ2 ) = 2σ t20 sec θI ε ⎡1 −1 F1 1 ,1, −r 2 ⎤ , ⎣⎢ ⎦⎥ 6 r = 2 –1 2 ρ 2 at . Therefore, the ratio of these variances is small at small distances between the receiving channels in the bistatic receiver (ρ2 Ⰶ at) and tends to 2 at large dis tances (ρ2 Ⰷ at):
B d (0,0, ρ2) B(0) = r 2 3, r Ⰶ 1;
(15) B d (0,0, ρ2) B(0) → 2, r Ⰷ 1. These relationships are fulfilled in the case of inclined paths. A similar result takes place for horizontal paths at quite high values of the outer scale. Let us now derive theoretical equations for the cor relation functions of astronomical image jitter along inclined paths with accounting for a finite response time of a photodetector (CCD array). For this, let us use the results from [27]. It is shown here that the pro cess of averaging over the ensemble of realizations can be changed to time averaging when constructing sta tistical characteristics of random functions. The esti mates of the rate of convergence of the variance of deviation of the timeaverages from the ensemble average provides for a required convergence in proba bility. If a measuring device used has nonzero response time, then the time averaging is discretecontinuous with the time of partial averaging Δt and the length of discretization interval ΔT, Δt ≤ ΔT. The process of dis cretecontinuous averaging can be reduced to finding the arithmetic mean of a sufficiently long sequence of partially averaged (with the continuous partial averag
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ing time Δt) empirical values of a random function separated by ΔT. The length of discretization interval ΔT is a time lag between starting points of the process of continuous partial time averaging. If the length of discretization interval is equal to the time of partial averaging Δt = ΔT then each new value in the above discrete sequence appears just after termination of the partial averaging process. As shown in [27, Eq. (36)], the equation that con nects the correlation functions of nonaveraged (Bu, zero response time of device, Δt = 0) and partially aver aged (Bf, nonzero response time, Δt ≠ 0) random pro cesses is 1
∫
B f (τ) = dt(1 − t ) [ Bu (t Δt + τ) + Bu (t Δt − τ)] ,
(16)
0
Δt ≤ ΔT , where ΔT is the length of discretization interval and Δt is the partial averaging time (response time of record ing device). According to Eq. (16), accounting for a finite response time of a photodetector can be reduced to calculating the correlation function Bf (τ) from the known function Bu(τ). Time correlation functions (13) should be used in Eq. (16) as the nonaveraged (zero response time) correlation function Bu(τ) for a ran dom process of astronomical image jitter along inclined paths. Let us consider only the variance of a partially aver aged random process, i.e., the correlation function at zero Bf (0). Assuming τ = 0 in Eq. (16), we obtain 1
∫
B f (0) = 2 dt(1 − t )Bu (t Δ t ).
(17)
0
Let us substitute correlation functions (13) for Bu(tΔt) in Eq. (17). Let Bf(0, L) designate the variances of the partially averaged random process of astronom ical image jitter along inclined paths in the monostatic receiver, and B df (0, L, r) designate the variances in the bistatic differential receiver as follows: 1
∫
B f (0, L) = 2 dt(1 − t )B(t Δt ), 0
(18)
1 d B f (0, L, r)
∫
= 2 dt(1 − t )B (t Δt,0, ρ2 ), d
0
2–1/2v⊥CΔt/at
and r = 2–1/2ρ2/at are normal where L = ized dimensionless parameters similar to the normal ized parameters in case of horizontal paths (see Eqs. (11a) and (11b)). The parameter L is propor tional to the response time of a recording device Δt; hence, it specifies the degree of partial averaging of the
random jitter process; L = 0 corresponds to the absence of averaging. Again, from Eq. (18), we have B f (0,0) = B(0), B df (0,0, r) = B d (0,0, ρ2).
Substituting functions (13) into Eq. (18), we find
(
1
)
B f (0, L) = 2σt20 sec θI ε dt(1 − t )1 F1 1 ,1, −L2t 2 , 6
∫ 0 1
(
)
B df (0, L, r) = 2σt20 sec θI ε dt(1 − t ) ⎡21 F1 1 ,1, −L2t 2 (19) ⎢⎣ 6
∫
(
0
) (
)
2 2 − 1F1 1 ,1, − Lt + r −1 F1 1 ,1, − Lt − r ⎤ . ⎥⎦ 6 6 As for horizontal paths, here, we can introduce the angle ϕ between the vectors L and r, which coincides with the angle between the vectors v⊥C and ρ2. Again, we can write |Lt + r|2 and |Lt – r|2 in terms of the argu ments of degenerate hypergeometric functions as
|Lt + r| 2 = L2 t 2 + r 2 + 2rLt cos ϕ, |Lt – r| 2 = L2 t 2 + r 2 – 2 rLt cos ϕ. 2.3. Correlation between Dispersions of Astronomical Image Jitter in Receivers with Finite and Zero Response Times Let us now find relationships between the disper sions of astronomical image jitter in monostatic and bistatic differential receivers with finite and zero response times. For this, we consider the response function of inertial recording equipment F, which characterizes the degree of partial averaging of a ran dom jitter process. This function can be represented for monostatic and differential receivers as
F (L) = B f (0, L) B f (0,0), F d (L, r) = B df (0, L, r) B df (0,0, r),
(20)
respectively. The response function shows how the dispersion of image jitter decreases when using inertial recording equipment (Δt ≠ 0, L ≠ 0) compared with the jitter dis persion recorded by memoryless equipment (Δt = 0, L = 0). Let the response functions F(L) and F d(L, r) be known, and the variances of partially averaged random process of astronomical jitter Bf (0, L) and B df (0, L, r) be measured from a flying aircraft. Then, considering Bf (0, 0) = B(0) and B df (0,0, r) = B d (0,0, ρ2), from Eqs. (19), (20), and (14), we derive final relationships of the measurement technique of integral turbulence param eters from aircraft observations of astronomical image jitter:
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Iε =
B f (0, L) F (L)σt20 sec θ
Fd(L, r) 1.0
,
d (21) 2B f (0, L, r) . d 2 2 F (L, r) ⎡1 − 1 F1 1 ,1, −r ⎤ σt 0 sec θ ⎢⎣ ⎥ ⎦ 6 To define the response function, it is necessary to know the time of partial averaging (response time) of measuring equipment. If a CCD array is used as a pho todetector, the output of which is a time sequence of frames, then the time of partial averaging Δt (response time of the CCD array) is the length of a frame. In this case, the frame length Δt is equal to the length of dis cretization interval ΔT. For a CCD array with the frame rate fM, the response time Δt = 1/fM; for exam ple, Δt = 0.0033 s for fM = 300 Hz. Figure 4 shows the response function of a bistatic differential receiver versus the partial averaging time Δt, which is equivalent to the dependence on the parameter L = 2–1/2v⊥CΔt/at. The response function has been calculated by Eqs. (19) and (20) for different distances between the receiving channels and corre sponds to a bistatic receiver mounted onboard a high flying aircraft. For the data in Fig. 4, the vectors v⊥C and ρ2 are col linear, i.e., the angle between them ϕ = 0°. The parameter r, which characterizes the distance between the channels, varies in the range 0.5–1704. For at = 5 cm, the variation ranges of the parameters L and r in Fig. 4 correspond to variations in the partial aver aging time Δt from 0 to 0.05 s and in the distance between the channels ρ2 from 3.5 cm to 120.5 m. The partial averaging time Δt and the distance between the channels ρ2 decrease with the radius of receiving channel at proportionally at the same values of the parameters L, r, and v⊥C. The response function of a monostatic receiver is shown in Fig. 4 for comparison (curve 9). As can be seen from Fig. 4, at small values of the parameter r (r ≤ 2.4, curves 1–3), the response function of the bistatic differential receiver decreases signifi cantly (by e times) at L ≈ 5. This means that this decrease occurs at partial averaging times Δt ≥ 0.00254 s for at = 5 cm, v⊥C = 500 km/h, and 3.5 ≤ ρ2 ≤ 17 cm. The response time Δt = 0.00254 s corresponds to the frame rate fM ≈ 393 Hz in the CCD array. Thus, for accepted values of the parameters, the dispersion of image jitter in the bistatic differential receiver weakens significantly (by e times) at a frame rate of lower than 393 Hz. However, the standard devi ations decreases only by e1/2 ≈ 1.65 times, which is about 60% of the value corresponding to zero response time. The dispersion (response time) weakens by an order of magnitude (by 10 times) already at L ≈ 9–11. This corresponds to the partial averaging time Δt ≥
Iε =
(
)
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0.8 9, F(L) 0.6
9
0.4 6 0.2
3
8
5
2
7
4
1
0 0
20
40
60
80
L
Fig. 4. Response function of a bistatic differential receiver mounted onboard a highflying aircraft for different dis tances ρ2 between the receiving channels at ϕ = 0°, at = 5 cm, and v⊥C = 500 km/h: ρ2 = 3.5 (r = 0.495) (1), 7 (0.99) (2), 17 (2.4) (3), 47 cm (6.65) (4); ρ2 = 1.47 (r = 20.79) (5), 2.47 (34.93) (6), 12.47 (176.3) (7), and 120.47 m (1703.7) (8); response function of an aircraft monostatic receiver (9).
(0.0046–0.0056) s or a frame rate of 218–178 Hz in the CCD array. The standard deviation of the jitter is about 32% of the value corresponding to zero response time in this case, which is insignificant in the measure ments. As follows from data of Fig. 4, to decrease the effect of partial averaging (to increase the response func tion), we should increase the distance between the receiving channels in the bistatic differential receiver. Thus, e.g., at ρ2 = 47 cm, the response function decreases by an order of magnitude at L ≈ 19. This cor responds to the partial averaging time Δt ≈ 0.0097 s or the frame rate fM ≈ 103 Hz. Thus, an allowable increase in the distance between the receiving chan nels enables one to get rid of excessively large partial averaging (a large decrease in the response function) in the bistatic differential receiver. A comparison of the response function of the bistatic receiver F d(L, r) with the response function of the monostatic receiver F(L), shown in Fig. 4, shows that F(L) is a limit of F d(L, r) under an infinite increase in r: F d(L, r) → F(L) at r → ∞. It is clear that this behavior of the function F d(L, r) corresponds to the independence of random processes of image jitter in both receiving channels of the bistatic receiver at a fairly long distance between the channels (r → ∞). The results of Fig. 4 relates to the case of collinearity (ϕ = 0°) of the vector of aircraft velocity v⊥C and the vector ρ2 connecting the channel centers. The response function changes when the vectors rotate rel ative to each other. The dependence of the response
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averaging (a large decrease in the response function) in a bistatic differential receiver even with the use of slow CCD arrays.
Fd(L, r) 1.0 0.8 0.6
5, F(L)
5
0.4
4
0.2
3
2 1
0 0
10
20
30
40
50
60 L
Fig. 5. Response function of a bistatic differential receiver mounted onboard a highflying aircraft versus the angle ϕ between the vectors v⊥C and ρ2: ρ2 = 47 cm, at = 5 cm (r = 6.65), and v⊥C = 500 km/h; ϕ = 0° (1), 30° (2), 60° (3), and 90° (4); response function of an air craft monostatic receiver (5).
function of the bistatic differential receiver on the angle ϕ between the vectors v⊥C and ρ2 is shown in Fig. 5. It can be seen that the deviation of bistatic response function from monostatic is maximal when the vectors v⊥C and ρ2 are collinear (ϕ = 0°). The bistatic response function approaches the monostatic one with an increase in ϕ (curve 5). A similar situation is observed in Fig. 2 for the time correlation coefficient along hor izontal paths, when the bistatic coefficient almost coincides with monostatic at ϕ = 90°. However, in the case of inclined paths, though the bistatic response function approaches the monostatic one with an increase in ϕ, but does not coincide with it when the vectors v⊥C and ρ2 are perpendicular (ϕ = 90°). The bistatic response function at ϕ = 90° decreases more rapid than the monostatic response function with an increase in the partial averaging time (increase in the parameter L). As follows from Fig. 5, to decrease the effect of par tial averaging (to increase the response function), it is necessary to increase the angle between the vectors v⊥C and ρ2. The best situation is observed when v⊥C and ρ2 are perpendicular (ϕ = 90°). If the distance between the receiving channels is 47 cm (ρ2 = 47 cm), then, as can be seen from Fig. 5, the response function decreases by an order of magni tude at L ≈ 19 for the collinear vectors (ϕ = 0°) and at L ≈ 60 at perpendicular vectors (ϕ = 90°). This corre sponds to the partial averaging times Δt ≈ 0.0097 s and 0.031 s and limiting frame rates fM ≈ 103 Hz and 33 Hz. Thus, the choice of perpendicular vectors v⊥C and ρ2 allows for the elimination of excessively large partial
CONCLUSIONS In this work, we have consequently compared monostatic and differential (bistatic) receivers. The analysis performed does not repeat the analysis of sig nals and errors for a common differential technique [28]. The main feature of the technique suggested consists in measurements from a movable carrier. We have shown that the maximal deviation of the bistatic response function from the monostatic function is observed in the case of collinear vectors of carrier velocity and channel spacing. The characteristic cor relation time in a bistatic differential receiver is deter mined by the time of turbulent inhomogeneity trans portation through the space between two receiving channels, until the distance between the channels is shorter than the turbulence outer scale L0. If this distance exceeds the outer scale, then the correlation time is determined by the time of inhomogeneity transportation through the turbulence outer scale (∼L0 /v⊥). We will use these results to design a differential detector based on a twochannel bistatic detector. The equations derived connect the measurement data on the mutual correla tion function of lightsource image jitter with the inte gral level of turbulence along a propagation path. The effect of the turbulence outer scale, the carrier velocity vector, the sampling rate, and other parameters of the detector circuit have been estimated. To construct a detector prototype, the path turbulencelevel meter (TLM) designed earlier at the Institute of Atmo spheric Optics, Siberian Branch, Russian Academy of Sciences can be used [29, 30]. In addition, when analyzing the effect of aircraft vibrations on the measurement errors in a bistatic (dif ferential) receiver, a difference in spatial frequencies of vibrations in longitudinal and lateral directions should be taken into account. This can pose some restrictions on the choice of the angle between the vectors of car rier velocity and receivers spacing, as well as on the distance between the receiving channels. REFERENCES 1. V. I. Tatarskii, Wave Propagation in a Turbulent Atmo sphere (Nauka, Moscow, 1967) [in Russian]. 2. A. S. Gurvich, A. I. Kon, V. L. Mironov, and S. S. Khmelevtsov, Laser Radiation in a Turbulent Atmosphere (Nauka, Moscow, 1976) [in Russian]. 3. A. I. Kon and V. I. Tatarskii, “Parameter fluctuations of a spacelimited light beam in a turbulent atmosphere,” Radiophys. Quant. Electron. 8 (5), 617–620 (1965). 4. V. L. Mironov, V. V. Nosov, and B. N. Chen, “Quivering of optical images of laser sources in a turbulent atmo sphere,” Radiophys. Quant. Electron. 23 (4), 319–325 (1980).
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17. A. S. Monin and A. M. Yaglom, Statistical Hydrome chanics (Nauka, Moscow, 1965), Vol. 2 [in Russian]. 18. V. I. Tatarskii, Light propagation in a medium with ran dom refractive index inhomogeneities in the Markov random process approximation,” JETP 29 (6), 1133– 1138 (1969). 19. V. I. Klyatskin, Stochastic Equations and Waves in Ran domly Nonuniform Media (Nauka, Moscow, 1980) [in Russian]. 20. V. L. Mironov and V. V. Nosov, “Concerning the effect of the external scale of atmospheric turbulence on the space correlation of random displacements of light beams,” Radiophys. Quant. Electron. 17 (2), 187–190 (1974). 21. V. L. Mironov and V. V. Nosov, “On the theory of spa tially limited light beam displacements in a randomly in homogeneous medium,” J. Opt. Soc. Amer. 67 (8), 1073–1080 (1977). 22. E. I. Gel’fer, “Correlation of displacement of point source images,” Radiophys. Quant. Electron. 17 (8), 905–908 (1974). 23. V. P. Lukin, Adaptive Atmospheric Optics (Nauka, Novosibirsk, 1986) [in Russian]. 24. V. P. Lukin, E. V. Nosov, and B. V. Fortes, “The efficient outer scale of atmospheric turbulence,” Atmos. Ocean. Opt. 10 (2), 100–106 (1997). 25. V. V. Nosov, V. P. Lukin, and E. V. Nosov, “Influence of the underlying terrain on the jitter of astronomic images,” Atmos. Ocean. Opt. 17 (4), 321–328 (2004). 26. M. S. Belen’kii, G. O. Zadde, B. C. Komarov, G. M. Krekov, V. V. Nosov, A. A. Pershin, V. I. Khama rin, and V. G. Tsverava, An optical Model of the Atmo sphere, Ed. by V.E. Zuev and V.V. Nosova (Publishing House of Siberian Branch of the Academy of Sciences of USSR, Tomsk, 1987) [in Russian]. 27. V. V. Nosov and V. P. Lukin, “Method of measurements of turbulence characteristics from jitter of astronomical images from onboard an aircraft. Part 1. Main ergodic theorems,” Atmos. Ocean. Opt. 27 (1), 75–87 (2014). 28. A. Tokovinin, “From differential image motion to see ing,” Publications of the Astronomical Society of the Pacific 114, 1156–1166 (2002). 29. L. V. Antoshkin, N. N. Botygina, O. N. Emaleev, L. N. Lavrinova, and V. P. Lukin, “Differential optical meter of the parameters of atmospheric turbulence,” Atmos. Ocean. Opt. 11 (11), 1046–1050 (1998). 30. V. P. Lukin, “Differential turbulence meter,” Fotonika, No. 5, 52–59 (2010). Translated by O. Ponomareva
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