1. Introduction

Scintillation of optical and infrared signals can have a strong impact on the performance of many systems that employ links propagating through the atmosphere [1]. For example, as is well known in a free-space optical (FSO) communication system it is necessary to design the acquisition-tracking channel such that the received optical signal is above threshold for the tracking control loop. When designing a FSO communication system it is necessary, therefore, to understand the effects of turbulence and be able to accurately predict a suitable averaging time provided by the temporal integration process of the acquisition sensor in order to reduce the (raw) scintillation variance to some prescribed level. This leads to an improved fade margin, which can be allocated to the tracking channel link budget.

An important remote sensing application where temporal averaging can lead to improved system performance is differential absorption lidar (DIAL) or column integrated differential spectroscopy (DOAS). These have been standard methods for decades to measure concentrations of trace chemical species in the atmosphere [2–5]. In the case of column integrated DIAL, the method relies upon measuring the light intensity transmitted from an optical transmitter to a receiver through an open atmospheric path for at least two wavelengths, one resonant with the molecular species of interest and another off resonance. The difference in transmitted intensity at the two wavelengths provides information regarding the trace species concentration. One limitation on the precision of this type of measurement is fluctuations of the detected irradiance levels associated with atmospheric turbulence. This is particularly a problem if the two wavelengths cannot be transmitted and received simultaneously, which is very often the case. In order to improve the precision and accuracy of the measurements the received irradiance must be time averaged to allow an accurate mean value of transmitted intensity to be determined. It is of importance when designing a differential absorption measurement therefore to understand the effects of turbulence and be able to accurately predict an optimum averaging time to provide the required level of precision. Here we consider a novel application of differential absorption measurement for the purpose of measuring column-integrated concentrations of various so-called greenhouse gas (GHG) atmospheric components such as CO₂, CH₄, H₂O, and N₂O that has not been utilized previously. The concept involves forming a narrow differential absorption path through the entire atmosphere by transmitting a laser beam from a ground-based transmitter to a geosynchronous orbiting (GEO) space sensor [5]. The concept is to track a GEO optical sensor with a ground-based laser transmitter alternately transmitting laser light tuned on and off a molecular species resonance. A small portion of the transmit laser beam is split off the main beam before transmission to allow monitoring of the transmit laser power. The laser wavelength is alternately switched between a series of on and off resonance wavelengths, the time of switching is time-tagged to a synchronized time base between the sensor and the transmitter to allow for post data analysis. In order to reduce beam wander effects the laser transmitter is purposely diverged to at least 100 μrad_{full-angle. This provides the tracking source with margin in terms of pointing accuracy and stability and atmospherically induced beam wander. For many climate-modeling applications it is necessary to measure the GHG constituent concentrations to a precision of better than 1%. Therefore the GEO sensor must average the transmitted laser signals for some period of time to reduce the fluctuations to a prescribed value of the standard deviation to mean value ratio. The theoretical treatment presented below provides a sound basis for predicting the time needed to average the signals on the detector in order to allow the precision required for the measurement.}

For the purpose of illustration, we consider the measurement of methane in the atmosphere. The monitoring of atmospheric methane and its sources and sinks is of increasing importance with the realization that methane is a thirty times more potent GHG than is CO₂ and even modest leaks from gas production facilities can offset the advantage of burning methane rather than coal for electrical generation [6]. Additionally methane sequestered as chlathrates in the oceans and Polar Regions pose a potential positive feedback system of global warming as higher air and water temperatures release this trapped methane, in turn increasing the warming of the atmosphere. Methane can be conveniently measured by the ground-to-space differential absorption method at a wavelength of 3168 nm. This spectral region has very little absorption due to other gases such as water, while having a sharp, substantial resonance for methane. It is not so strong as to be optically thick from the ground to space.

Surprisingly, given the large number of published technical papers related to spatial aperture averaging of scintillation (see Chap. 10 of [1] and references therein) there has been virtually nothing published related to the quantitative aspects of temporal averaging. Toyoshima and Araki [7] present a formal expression for a “time-averaging factor”, but only evaluate a numerical fit from empirical data obtained from the Engineering Test Satellite (ETS-VI) [8]. Recently, Shen, Yu, and Fan [9] purport to analyze “time-averaging effects on scintillation in the frequency domain”, but no quantitative results are presented. In contrast, here based on the Kolmogorov spectrum in the regime of weak scintillation conditions we present quantitative results for the temporal averaging scintillation reduction factor for plane, spherical waves and beam waves, and give illustrative examples for both horizontal and for a DIAL related ground-to-space propagation condition. Weak scintillation conditions covers a broad range of applications that includes, but not limited to, optical and infrared terrestrial propagation near the surface on relatively short paths and both up and down links through the atmosphere unless the propagation path is close to the horizon. To the best of our knowledge no quantitative “first principle” analysis of time-averaged scintillation is available in the literature, and the work presented here is an initial attempt to fill this gap.

Thus, this paper should be of interest to both the FSO laser communication and remote sensing communities and is organized as follows. Assuming weak scintillations conditions, we derive in Sec. 2 a general expression based on the Rytov approximation for the reduction of the variance of irradiance due to temporal averaging from which we obtain a temporal averaging factor. In Sec. 3, based on the Kolmogorov spectrum, we consider horizontal propagation and derive both exact and highly accurate elementary analytic approximations for the temporal averaging factor for plane, spherical, and beam wave propagation. In Sec. 4 we present corresponding results for ground-to-satellite propagation paths. As a novel ground-to-satellite differential absorption application we address the question of how much averaging time is required provide a required level of precision. In particular, we present results for the required averaging time necessary to reduce the standard deviation of irradiance to some given fraction of the mean irradiance. Finally, in Sec. 5 we present some concluding remarks.

2. Temporal averaging of irradiance

Consider an optical signal that has propagated through the atmosphere impinging on a collecting aperture. Here we assume that the collecting aperture is sufficiently small that negligible (spatial) aperture averaging occurs (e.g., ground-to-satellite propagation). Because the corresponding received signal power equals the product of the irradiance and the receiving aperture area, the results presented here in terms of signal irradiance are identical to that for signal power. The received irradiance averaged over a time T is defined by

Here we only consider weak scintillation conditions where the variance of the irradiance fluctuations are much less the square of the mean irradiance, and the fluctuations obey log-normal statistics. In this regime it is well known that ${\sigma }_I^2=\exp \left[4{\chi }^2\right]-1\cong 4{\chi }^2$, and the spatial covariance of irradiance is given by$C_1\left(\rho \right)=\exp \left[4B_{\chi }\left(\rho \right)\right]\cong 4B_{\chi }\left(\rho \right)$, where χ² and B_χ(ρ) are the log-amplitude variance and log-amplitude correlation function, respectively. Assuming Taylor’s “frozen in” hypothesis the corresponding temporal covariance for isotropic turbulence is obtained from the log-amplitude spatial covariance in the Rytov approximation as [10]

In the Appendix we show after substituting Eq. (5) into Eq. (3) and performing the integrations over t₁ and t₂ that the time averaged irradiance variance can be expressed as

 

Fig. 1 The finite averaging time spatial wave number weighting function F_T(K;s) defined by (2.8) as a function of its argument.

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Alternatively, an equivalent expression for ${\sigma }_I^2$ can be obtained by expressing $C_I(\tau )={\displaystyle \underset{-\infty }{\overset{\infty }{\int }}W(\omega )\exp (i\omega \tau )d\omega /2\pi },$where ω denotes angular frequency, and W(ω) is the irradiance power spectral density. Performing the elementary temporal integrals indicated in Eq. (3) and simplifying yields

We note that spatial aperture averaging effects of a circular collecting aperture can be included directly in the analysis by introducing an additional low-pass filter function in the integrand of Eq. (7) given by [2J₁(Ka)/ Ka]² where a is the aperture radius, and J₁(∙) is the Bessel function of the first kind of order unity [10]. In order to identify the irradiance temporal averaging factor we express Eq. (7) as

I(T) is expected to still obey log-normal statistics. This is because the distribution of the sum of log-normal variates has been shown to be very accurately represented by a log-normal distribution instead of a normal distribution that might be expected from the central-limit theorem [13].

3. Terrestrial links: horizontal propagation

Here we employ the Kolmogorov spectrum and thus neglect the effects of both the inner and outer scale of turbulence, where ${\Phi }_n\left(K;s\right)=0.033C_n^2\left(s\right)/K^{11/3}$, and $C_n^2\left(s\right)$ is the index structure constant along the propagation path. For illustrative examples (and analogous to the historical development of aperture averaging of scintillation in the literature [1]) we consider plane, spherical and beam wave horizontal propagation for both constant cross wind speed and turbulence conditions. We present below various exact analytic expressions containing multiple argument hypergeometric functions. These expressions are reproduced here primarily for documentation and reference purposes only and are not recommended to be used in practice. Rather, the primary intent and purpose of this paper is to provide the reader with the corresponding highly accurate engineering approximations for predicting the averaging times required to obtain an arbitray value of the standard deviation to the mean.

3.1. Plane waves

The scintillation induced on terrestrial links was first analyzed for plane wave propagation and constant turbulence conditions [11]. Here, because of mathematical considerations, analytic and numerical results are more readily obtained from (2.12) than from (2.13). Substituting the Kolmogorov spectrum into (2.11) for constant values of both $C_n^2$ and V yields ${\sigma }_I^2\left(0\right)=1.23k^{7/6}{C}_n^2{L}^{11/6}$ and

 

Fig. 2 The temporal averaging factor for plane, spherical and beam waves for constant turbulence conditions. Plane and spherical waves correspond to the limit $\text{}{F}_N\to \infty ,$ and 0, respectively, while the blue and red curves pertain to the exact and approximate results of Sec. 3.

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3.2. Spherical waves

Testing terrestrial scintillation predictions for plane waves required the use of large collimators, while by contrast simple measurements could be made with point-source transmitters [11–15]. Again, because of mathematical considerations analytic and numerical results are more readily obtained from Eq. (12) than from Eq. (13). Substituting the Kolmogorov spectrum into Eq. (10) for constant values of both $C_n^2$ and V yields ${\sigma }_l^2\left(0\right)=0.496k^{7/6}{C}_n^2{L}^{\frac{11}{6}}$ and

For large values of T it follows from the results presented above that for both plane and spherical waves the time averaged variance ${\sigma }_I^2(0)A(T)~{\sigma }_I^2(0)/T_N~C_n^2{L}^{7/3}/TV{\lambda }^{2/3}$. In contrast, for a circular receiving aperture of radius a that is large compared to the Fresnel length the corresponding aperture averaged variance is wavelength independent and varies as $C_n^2{L}^3/a^{7/3}$ (see [10], Chap. 3).

3.3. Beam waves

Laser development provided the practical means of producing coherent spatially confined optical beams, and we next consider the propagation of a TEM₀₀ Gaussian beam, which is the standard case used to describe beam waves in the literature (see [1], Chap. 4). Of particular interest is the Rytov irradiance variance for Gaussian-beam waves with an initial field distribution profile given by

For definiteness and illustrative purposes we only consider the irradiance variance on or near the center line of the beam. Based on the results presented in Sec. 3.2.5 of [10] the off-axis scintillation variance is within 9% of the on-axis scintillation variance for $\rho /W\left(L\right)\le 0.1$, where ρ is the radial distances from the centerline and W(L) is the 1 / e² beam radius at the receiver. It can be shown that by generalizing the results of Ishimaru [16] that the (raw) on the optic-axis beam-wave variance is given by

 

Fig. 3 The ratio of the Rytov beam wave irradiance variance to that of a spherical wave as a function of the transmitter Fresnel number.

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Evaluation of the irradiance covariance for beam waves represents a formidable challenge due to the mathematical complexity involved (see Ref. 10, Sec. 4.1.6). We can, however, readily obtain results in the frequency domain. That is we use, in the beam wave case, Eq. (13) to obtain results rather than Eq. (12).

For weak scintillation conditions the PSD for Gaussian-beam waves and slant range propagation can be expressed as [17]

From Eq. (13) the temporal averaging factor can be expressed as

The functions Q_1,2 can be expressed analytically in terms of hypergeometric functions given by

As expected physically and shown in Fig. 2, A_B(T) ~ 1 / T_{Nfor large TN and consequently we find that an accurate engineering approximation for the beam wave irradiance temporal averaging factor can be expressed as}

 

Fig. 4 A comparison between the numerical and analytic fit for the temporal scale T₀ as a function of Fresnel number.

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4. Ground-to-satellite propagation

As alluded to in the introduction we next consider novel application of differential absorption along a propagation path from ground-to- GEO, where the beam divergence of the transmitted beam is sufficiently large such that it can be assumed to emanate from a point source. As an illustrative example we employ both the often-used Hufnagel-Valley 5/7-turbulence model [18] and the Bufton wind speed height profile [19]

is the height above ground in kilometers. If the wind blows from an azimuth angle θ₀ at speed V₀, and the line of sight is at elevation angle E and azimuth angle θ, the component of the wind normal to the line of sight is given by

(35)$$V=V_0\sin E{\left[1+{\cot }^{2}E{\sin }^2\left(\theta -{\theta }_0\right)\right]}^{1/2}.$$

When θ = θ₀, this reduces to V = V₀ sin E (i.e., when looking directly into or away from the wind); when θ = θ₀ ± 90°, V = V₀. For definiteness we consider an uplink directed toward the equator and because the prevailing wind is predominately in the west-east direction over CONUS V = V₀.

For slant range propagation through the atmosphere the irradiance variance in the absence of time averaging is given by

(36)$${\sigma }_I^2\left(0\right)=2.25k^{7/6}{\displaystyle \underset{0}{\overset{L}{\int }}ds{C}_n^2\left(h\left[s\right]\right)s^{5/6}{\left(1-s/L\right)}^{5/6}}$$

where for a flat earth h = s sin E. Integrating over both spatial wave number in the numerator and denominator and over propagation distance in the denominator of Eq. (12), and simplifying yields

(37)$$A\left(T\right)=\frac{{\displaystyle \underset{0}{\overset{L}{\int }}ds{C}_n^2\left[h\left(s\right)\right]}{s}^{5/6}{\left(1-s/L\right)}^{5/6}R\left(T;s\right)}{{\displaystyle \underset{0}{\overset{L}{\int }}ds{C}_n^2\left[h\left(s\right)\right]}{s}^{5/6}{\left(1-s/L\right)}^{5/6}},$$

where the path weighting function R(T;s)

(38)$$\begin{array}{l}R\left(T;s\right)=\frac{9\left(1+\sqrt{3}\right)r^{5/3}\Gamma \left(-\frac{5}{6}\right)}{352\pi }+{}_{3}F{}_4\left(-\frac{5}{12},\frac{1}{12},\frac{1}{4};\frac{1}{2},1,\frac{5}{4},\frac{3}{2};-\frac{r^4}{256}\right)\\ +\frac{5}{288}\left(2+\sqrt{3}\right)r^2{}_{3}F{}_4\left(\frac{1}{12},\frac{7}{12},\frac{3}{4};\frac{3}{2},1,\frac{7}{4},2;-\frac{r^4}{256}\right)\end{array}$$

where $r=TV\left(s\right)\sqrt{2k/s\left(1-s/L\right)}$_. As illustrated in Fig. 5, an accurate elementary approximation is given by

 

Fig. 5 A comparison of the exact and approximate path weighting function given by Eq. (38) and Eq. (39), respectively.

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(39)$$R\left(T;s\right)=\frac{1}{\sqrt{1+0.154r^2}}$$

An elementary engineering expression can be obtained for the temporal averaging factor by noting that for sufficiently large T, $R\left(T;s\right)=\frac{1.80}{TV\left(s\right)\sqrt{2k/s\left(1-s/L\right)}}$. Substituting this value in Eq. (38) and integrating yields $A\left(T\right)\to \frac{8.66\sqrt{\csc E/k}}{T}=\frac{3.45\sqrt{\lambda \csc E}}{T}$, where the units of λ and T are in m and s, respectively. Based on this asymptotic result an elementary engineering approximation for all values of T for the HV-5/7 profile and the Bufton wind model can be expressed as

(40)$$A\left(T\right)\cong \frac{1}{\sqrt{1+{\left(\frac{T}{3.45\sqrt{\lambda \csc E}}\right)}^2}}$$

Figure 6 is a plot of the ratio of standard deviation of irradiance to the mean at 3168 nm for GEO observation for a line of sight at θ = θ₀ ± 90°, as a function of averaging time for various values of elevation angle, where the 1/T^1/2 scaling dependence is clearly shown by the red curve. Examination of this figure yields that a 1% value of this quantity is obtained for averaging times of about 2.2, 3.7, and 8.3 s for elevation angles of 60, 45 and 30 degrees, respectively. Indeed, one can obtain an order of magnitude estimate of the normalized standard deviation of irradiance for averaging times large compared to the scintillation correlation time from the following physical argument. The correlation time $t_c\sim \sqrt{2h_0\csc E/k}/V\left(h_0\right)$, where h₀ is the effective scale height that produces scintillation [20], and V(h₀) is the normal component of the wind speed to the line of sight at altitude h₀. Hence, for large T the number of uncorrelated irradiance samples N ~ t_c. For the HV-5/7 turbulence and the Bufton wind speed profile h₀ ~ 8 km, and, from which it follows, for example at 3168 nm and an elevation angle of 45°, that N ~468T_. From Eq. (36) the corresponding (raw) irradiance standard deviation is 0.22, and an estimate of the time averaged standard deviation ~0.22/N^1/2, from which it follows that the averaging time required to obtain a value of 1% is about 1.1 s, as compared to the corresponding rigorous value of 3.7 s indicated in Fig. 6. These considerations show that averaging time required to obtain a given level is proportional to both the square root of cosecant of the elevation angle, and the wavelength, and inversely proportional to the wind speed at high altitudes.

 

Fig. 6 The standard deviation of irradiance normalized to the mean as a function of averaging time. The horizontal black line is at the 1% level, and the red curve indicates the 1 / T^1/2 scaling dependence.

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5. Conclusion

The analysis above presents, for what we believe is the first time, a theoretical basis within the frame work of the Rytov theory of smooth perturbations for the often-observed temporal averaging behavior of the irradiance fluctuations of an optical beam due to atmospheric turbulence. It provides a simple analytical result for predicting the averaging time required to obtain an arbitrary value of the ratio of the standard deviation to the mean irradiance of an optical beam transmitted through an arbitrary path in the atmosphere. The application of interest to us is the possibility of performing an accurate column integrated differential absorption measurement of GHG concentrations in the atmosphere by utilizing a ground-based laser transmitter and a GEO-based optical sensor. The results of the analysis above indicate that a relatively short averaging time, on the order of a few seconds, is required to reduce the irradiance fluctuations to a value precise enough for GHG measurements of value to climate related studies.

Appendix: evaluation of temporal integrals

After substituting (2.5) into (2.3) and changing time variables fromm t₁ and t₂ to μ₁ = t₁ / T_{and μ2 = t2 / T, respectively we obtain that the temporal integrals in (2.3) is expressed as}

(41)$$\frac{1}{T^2}{\displaystyle \underset{0}{\overset{T}{\int }}d{t}_1}{\displaystyle \underset{0}{\overset{T}{\int }}d{t}_2{J}_0\left(KV\left(s\right)\left(t_1-t_2\right)\right)}={\displaystyle \underset{0}{\overset{1}{\int }}d{\mu }_1}{\displaystyle \underset{0}{\overset{1}{\int }}d{\mu }_2{J}_0\left(KTV\left(s\right)\left({\mu }_1-{\mu }_2\right)\right)}$$

To facilitate this integration we note that the zero order Bessel function in Eq. (41) can be expressed as [12]

(42)$$J_0\left(KTV\left(s\right)\left({\mu }_1-{\mu }_2\right)\right)=\frac{1}{2\pi }{\displaystyle \underset{0}{\overset{2\pi }{\int }}d\varphi \cos \left[KTV\left(s\right)\left({\mu }_1-{\mu }_2\right)\cos \varphi \right]}$$

Substituting Eq. (42) into Eq. (41), performing the elementary integrations over μ₁ and μ₂, and simplifying yields

(43)$$\frac{1}{T^2}{\displaystyle {\int }_0^{T}d{t}_1}{\displaystyle {\int }_0^{T}d{t}_2{J}_0\left(\gamma \left(t_1-t_2\right)\right)=}\frac{1}{2\pi }{\displaystyle {\int }_0^{2\pi }d\varphi 4{\sec }^2\varphi {\sin }^2\left(\frac{\gamma }{2}\cos \varphi \right)/{\gamma }^2}={}_{2}F{}_1\left(\frac{1}{2},\frac{3}{2};2;-\frac{{\gamma }^2}{4}\right),$$

where the integral over ϕ is obtained from [12] and γ = KTV(s).

Acknowledgments

This work was supported by The Aerospace Corporation under its Sustained Experimentation and Research Applications program.

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