We present a calibration function in the general analytical form for the tropospheric temperature retrievals using pure rotational Raman (PRR) lidars. The function is derived within the framework of the semiclassical theory and takes into account the collisional broadening of all PRR lines. We analyze via simulation its four simplest nonlinear (three-coefficient) special cases to determine the function that yields the least error, and therefore, is the best-suited for the temperature retrievals. Two of them are proposed for the first time. The comparative analysis of temperature errors showed that all the special cases yield errors less than 0.1 K in modulus, and therefore, can be applied for the tropospheric temperature retrievals. The best function yields the maximum error less than 0.002 K in modulus and five times smaller compared to the commonly used nonlinear calibration function.
© 2016 Optical Society of America
One of the most effective lidar techniques for vertical temperature profiling in the troposphere and lower stratosphere is known  to be the pure rotational Raman (PRR) lidar technique originally proposed by Cooney in 1972 . The possibility of air temperature measuring using the PRR lidar technique follows from the temperature dependence of individual lines intensity of N2 and O2 molecules PRR spectra. The backscattered signals of the Stokes or anti-Stokes branches of the spectra, as shown in Fig. 1(a), are used for temperature determination. One can use the signals from both the branches [3–8]. The intensities of individual PRR lines corresponding to low and high rotational quantum numbers J (i.e. with Jlow and Jhigh) are of opposite temperature dependence, as shown in Fig. 1(b). The intensity of lines with Jlow decreases with increasing temperature and, conversely, the intensity of lines with Jhigh increases with increasing temperature. Diffraction gratings (DGs) [4–8], fiber Bragg gratings , and/or interference filters (IFs) [9–24] are applied to extract the PRR signals from backscattered light in the lidar temperature channels (called also PRR channels).
In order to determine the air temperature T, the ratio Q(T) of backscattered signal intensities from two PRR-spectrum bands with opposite temperature dependence is required. The intensity ratio of two individual PRR lines corresponding to certain Jlow and Jhigh, e.g., from the anti-Stokes branch of the PRR spectrum, gives the exact temperature dependence1–3, 25–28]. However, since IFs and DGs pass several adjacent PRR lines in both the PRR channels, as shown in Fig. 1(b) for DGs , one should consider the following expression10–24].
The ratio in Eq. (2) has a complicated temperature dependence, and hence, cannot be expressed as a simple function of T. Therefore, a simple approximation function (also called calibration function) for the ratio QΣ(T) is required to make it possible in practice to retrieve temperature profiles from lidar remote sensing or simulation data . Otherwise speaking, a calibration function represents a function approximation to the intensity ratio when each lidar temperature channel extracts several PRR lines. The calibration functions play the key role in the temperature retrieval algorithm. The temperature retrieval accuracy and the number of calibration coefficients depend on the selected calibration function.
Arshinov et al. (1983)  suggested considering the overall intensity of PRR lines extracted by any PRR channel as the intensity of an individual PRR line. The line corresponds to a transition from a level with energy equal to the value of energy averaged over the group of levels. Such an approach gives a calibration function in the form of Eq. (1) or in its natural logarithm form1] at the standard deviation of 0.38 K .
The difference between the ideal model and retrieved by Eq. (3) temperature profiles (i.e. the temperature error) behaves like a polynomial function of degree 2 . Therefore, in order to reduce the approximation errors, one can apply the second-order in x polynomial as a calibration function with three calibration constants (coefficients) 1] to yield the least temperature errors ( ± 0.03 K). When using Eq. (4) as a calibration function, the temperature errors behave like a polynomial function of degree 3 (see subsection 5.2). Hence, the use of the third-order in x polynomial, as it was proposed in ,Eq. (4). Thus, it is reasonable to assume that the n-order in x polynomial can retrieve temperature profiles with any desired accuracy depending on nEquation (6) and Eqs. (3)–(5) can be called as the general calibration function and its special cases, respectively. Other calibration functions in different forms are presented in [10, 11, 16].
All the calibration functions mentioned above are valid only when parasitic signals are sufficiently suppressed. During the last several decades the significant efforts of lidar researches were aimed to suppress the parasitic elastic signals backscattered by atmospheric aerosols and molecules. The signals can leak into PRR channels even if the atmosphere is cloud-free . The state-of-the-art narrow-band IFs and DGs provide the suppression of the parasitic signal intensity in the channels up to 7–8 orders of magnitude. Behrendt et al. (2002)  suggested a way to make a correction to the elastic backscattered signal leakage into the nearest (to the laser line) PRR channel in the presence of cirrus clouds. Nevertheless, for some PRR lidar systems the difference between radiosonde and retrieved by PRR lidar temperatures in the troposphere can reach more than 10 K using Eq. (3)  and 3 K for Eq. (4) . On the other hand, PRR lines are broadened by the Doppler and molecular collision effects, and hence, their backscatter profiles are described by a Voigt function , whereas Eqs. (3)–(6) were obtained assuming that the PRR lines profiles represent the Dirac functions. So, each broadened PRR line gives a contribution to both the lidar temperature channels with Jlow and Jhigh due to the long Lorentzian tails of the line profile . We suggest considering such a broadening during tropospheric temperature measurements using PRR lidars, especially in the atmospheric boundary layer .
In this paper, we theoretically derive the calibration function in the general analytical form that takes into account the collisional broadening of all PRR lines. We also analyze via simulation its simplest nonlinear (three-coefficient) special cases to determine the function that yields the least error, and therefore, is the best-suited for temperature retrievals.
2. Problem statement
Let us consider only the anti-Stokes branch of the PRR spectrum for definiteness. IFs or DGs of PRR channels extract two portions of N2 and O2 PRR lines with Jlow and Jhigh from the backscattered light spectrum. The wavenumbers and are shown in Fig. 2(a) to correspond to the central lines of IF or DG transmission bands. Inelastic PRR lines are broadened by both the Doppler and molecular collision effects in the troposphere, whereas the elastic Mie line and Rayleigh  (or Cabannes ) line are broadened only due to the Doppler effect. Therefore, the backscatter profiles of PRR lines are described by a Voigt function, while the backscatter profiles of Mie and Rayleigh lines are governed by a Gaussian (normal) distribution (see Appendix A for details) [10, 30, 32, 33]. The Voigt profile is known to be very close to the pure Gaussian profile shape near the center of a spectral line, and to the pure Lorentzian profile shape in its wings . Owing to the long Lorentzian tails, all the PRR lines contribute to the signals detected in both the PRR channels, and instead of Eq. (2) we can write for the intensity ratioFig. 2(a). Similarly, for the function we haveEqs. (2) and (7), one can see that Eq. (7) is more complicated than Eq. (2), and hence, cannot be expressed as a simple function of T as well as Eq. (2).
In order to obtain an analytical approximation (calibration) function for the ratio Qall(T) taking into account only the pure effect of collisional broadening of PRR lines, we have to make some simplifying assumptions. First, we assume that the Mie and Rayleigh lines intensities are sufficiently suppressed to be neglected. Second, we also assume that the temperature sensitivity of a PRR lidar is low and its ambient temperature does not change significantly, so that there are no spectral shifts in IF or DG transmission bands. Third, we use the Lorentzian function for a PRR line shape description instead of , because the molecular collisional effect dominates over the Doppler one in the troposphere (see section 5). Note that the Doppler broadening can be neglected below 4 km under the standard atmosphere conditions . Fourth, as shown in Fig. 2(b), an IF or DG transmission function (described usually by a Gaussian) can be approximated by a piecewise-constant (staircase) function with any desired accuracy. We can also further simplify the calculations by using a rectangle function instead of the staircase one. The last assumption is reasonable because there will not be any change in the temperature dependence of the integral in Eq. (8) in both cases. So, the transmission function can be approximated by a constant in the transmission band to , i.e. . Moreover, for the same reason, we can put F = 1 without loss of generality. The interval to is responsible for transmission of the bulk of the backscattered signal intensity in PRR channel . The same is valid for with the transmission band to , as shown in Fig. 2(a). Thus, instead of Eqs. (8) and (9) we can write
Following Arshinov et al. , we consider the overall intensity of PRR lines entered the transmission band to as the intensity of an averaged line with , as shown in Fig. 2(a). In this case, the backscattered signal intensity transmitted by IF or DG of the PRR channel can be expressed as followsEq. (11) is responsible for the parasitic signal of the broadened PRR lines lying outside the band. It is convenient to rewrite Eq. (11) asEqs. (12) and (13) into Eq. (7), we obtain for the intensity ratioEq. (14), we can replace the ratio by its temperature dependence from Eq. (6). Moreover, as each of the functions , , , and is much less than unity, we can use an approximation with z2 << 1  and also retain only the first-order terms, i.e.Eq. (15).
3. PRR lidar general calibration function
In this section we derive the PRR lidar calibration function in the general form. First of all, let us determine the temperature dependence of and . The function represents the probability that a backscattered signal wavenumber will be less than or more than , i.e. outside the transmission band , at the temperature T. Taking into account Eq. (45) in Appendix A, we can writeEq. (16). Consider the expansion of arctan(z) in a series Eq. (18) at from its own expression at , we can reduce Eq. (16) toEq. (48) from Appendix A into Eq. (19) and combining similar terms, we obtain for the temperature dependence of Eq. (20). Subtracting Eq. (20) from Eq. (21), we have for the temperature dependence of in Eq. (15)
In order to determine the temperature dependence of in Eq. (12), we can use the results obtained above. First, since we consider as the intensity of a single line, each intensity ratio in Eq. (12) can be defined as well as the ratio in Eq. (1), i.e.Eq. (10), and hence, by the difference of arctangents in Eq. (16). Taking Eq. (20) into account, we getEq. (20). Expanding the exponential function in Eq. (23) in Taylor series, we obtain for in Eq. (12)Eq. (24). The same is valid for the function , i.e.Eq. (25).
It is easy to see that all the functions , , , and have the same temperature dependence. This is not surprising, because this is a direct consequence of collisional broadening of PRR lines adequately described by the Lorentzian profile in the troposphere . Therefore, combining Eqs. (22), (25), and (26) yields the same dependence
Taking the natural logarithm of both sides of Eq. (15), we obtainEq. (27), the logarithm in Eq. (28) can be expanded in a series Eq. (27) into Eq. (29), we haveEq. (30) into Eq. (28) and combining similar terms, we obtain the PRR lidar calibration function in the general formEq. (28) or m–n = mn = 0 in Eq. (31), we get the general calibration function expressed by Eq. (6).
4. Special cases of the general calibration function
The problem of obtaining the temperature retrieval function T(Qall) from the general calibration function expressed by Eq. (31) is insoluble. Therefore, it is reasonable to use special cases of the integer power approximation of Eq. (31), i.e.
The best known nonlinear calibration function is the function containing the parabolic in x = 1/T term and defined by Eq. (4). The corresponding temperature retrieval function is
The next three-coefficient special case of Eq. (32) we consider is the following nonlinear function with the hyperbolic in x = 1/T termEq. (35) is
Since y = lnQ expressed by Eq. (3) is a linear function of x = 1/T, the reciprocal temperature x is also a linear function of lnQ, i.e. x = a + by. In order to take nonlinear effects into account, one can introduce a parabolic in y term with some constant cEq. (39) was applied in . Equation (39) represents a special case of Eq. (31) or (32) (see Appendix B for details).
There is another way to take nonlinear effects into account. Adding a hyperbolic in y term with some constant C4 to the linear calibration function givesEq. (40) yields
In order to determine the best three-coefficient calibration function that yields the least error in temperature retrievals, we performed a simulation making the simplifying assumptions described in section 2.
5.1 Initial conditions for the simulation
In our simulation we used the U.S. Standard Atmosphere (1976) data. The U.S. Standard tropospheric temperature profile presented in Fig. 3(a) was used as a reference one. We considered a narrow-linewidth (∼0.003–0.01 cm–1) laser with the wavelength of λ0 = 354.67 nm. Such a linewidth can be ignored compared to the widths of PRR lines broadened by the Doppler effect (∼0.06 cm–1). The FWHMs of the Gaussian, Lorentzian, and Voigt profiles of N2 broadened PRR line (with J = 6) are presented in Fig. 3(b), as an example. We took into account the contribution to both PRR channels and from the first strongest 28 lines of the anti-Stokes branch of N2 and O2 PRR spectra. These PRR lines include 17 N2 lines with J = 2, 3, 4, …, 18, and 11 O2 lines with J = 3, 5, 7, …, 23. Note that only odd lines beginning with odd J exist in O2 molecule PRR spectrum. Five PRR lines (3 N2 lines with J = 5, 6, and 7; and 2 O2 lines with J = 7 and 9) directly fall inside the transmission band = (30; 55) cm–1. Ten PRR lines (6 N2 lines with J = 12, 13, …, 17; and 4 O2 lines with J = 17, 19, 21, and 23) directly fall inside the band = (85; 135) cm–1. To calculate the contribution of all mentioned PRR lines more precisely under tropospheric conditions, we used in Eqs. (10)–(16) the Voigt profile HWHM (given by Eq. (52) in Appendix A) instead of HWHM of the Lorentzian profile. The characteristics (atom masses, collision diameters, etc.) required for the calculation of the Gaussian, Lorentzian, and Voigt FWHMs (and HWHMs) are given in Appendix A.
5.2 Results of the simulation
The dependences of altitude and temperature on the intensity ratio Q, calculated under the conditions described above, are presented in Figs. 4(a) and 4(b), respectively. A comparative analysis of temperature errors yielded by using Eqs. (33), (34), (36), (39), and (41) in temperature retrievals is presented in Fig. 5. The corresponding calibration coefficients Ai, Bi, and Ci, determined by applying the least square method to the simulation and reference U.S. Standard tropospheric temperature data, are collected in Table 1.
One can see from Fig. 5 that all the nonlinear three-coefficient calibration functions (temperature retrieval functions) yield errors less than 0.1 K in modulus. Hence, all these functions can be successfully applied in the temperature retrieval algorithm.
The temperature retrieval function given by Eq. (39) yields the least temperature errors, and therefore, is the best-suited for the tropospheric temperature retrievals. As also seen in Fig. 5(d), Eq. (39) yields the maximum error five times smaller compared to the commonly used Eq. (34). The temperature instability in a PRR lidar optical system can lead to a spectral shift of the IF or DG transmission bands. Such a shift will not change the temperature dependence of calibration functions, but can change the calibration coefficients. This problem is solved via periodic lidar recalibration.
We have derived the PRR lidar calibration function in the general analytical form that takes into account the collisional broadening of all PRR lines, especially in the atmospheric boundary layer (0 to 1.5–2 km). We have also analyzed via simulation its four simplest nonlinear (three-coefficient) special cases to be used in the temperature retrieval algorithm. Two calibration functions given by Eqs. (35) and (40) are proposed for the first time. The comparative analysis of temperature errors showed that all the temperature retrieval functions expressed by Eqs. (34), (36), (39), and (41) yield errors less than 0.1 K in modulus, and therefore, can be applied for the tropospheric temperature retrievals. The best-suited function for temperature retrievals (Eq. (39)) yields the maximum error less than 0.002 K in modulus. However, to determine the best function, which can also depend on a selected lidar system, all the mentioned temperature retrieval functions should be applied to real lidar remote sensing data.
A. Gaussian, Lorentzian, and Voigt profiles
The backscatter profiles of PRR, Mie, and Rayleigh lines inhomogeneously broadened by only the Doppler effect are governed by a Gaussian (normal) distribution (assuming the Maxwellian distribution of molecular velocities) [10, 29]
The backscatter profile of a PRR line homogeneously broadened by only the collision effect is governed by a Lorentzian (Cauchy) distribution [10, 29]29]. The FWHM of the Lorentzian profile is defined as In a case of two-component gases (e.g., air consisting of > 99% of N2 and O2 molecules), the HWHM γL at each altitude z is defined by
The temperature dependence of the effective optical collision diameters can be described by Sutherland's semi-empirical formula with a reasonable degree of accuracy. Taking into account binary collisions of molecules, we can write for any i atmospheric gas Eq. (47) into Eq. (46) and then combining similar terms, we obtain for the temperature dependence of γL at given molecular number density nairEqs. (46) and (47). The characteristics required for our simulation are the following: mair = 4.81 × 10−26 kg, m1 = 4.65 × 10−26 kg, m2 = 5.31 × 10−26 kg, d1,∞ = 3.51 × 10−10 m, d2,∞ = 3.52 × 10−10 m, d12,∞ ≈3.515 × 10−10 m , φ1/R = 105 K, φ2/R = 125 K, φ12/R = 115 K .
The real PRR line backscatter profile is described by a Voigt profile [10, 29]. The Voigt profile is defined as the convolution of the Gaussian and Lorentzian profiles and takes into account the broadening due to both the Doppler and molecular collision effectsEqs. (42) and (45) into Eq. (49), the Voigt profile can generally be expressed in the following formEq. (50) can be calculated numerically [36–38]. One of the approximations for FWHM of the Voigt profile (with the accuracy of 0.02%) was proposed in 
B. Equation (39) as a special case of the general calibration functionEq. (53) in two limiting cases, one can rewrite it as followsEq. (53) can be expressed as a special case of Eq. (31)34]Eq. (56) into Eq. (54), for the calibration function we obtainEq. (32).
We thank V. L. Pravdin, Dr. S. M. Bobrovnikov, Dr. S. L. Bondarenko, and Dr. A. P. Shelekhov for fruitful discussions. This study was conducted in the framework of the Federal Targeted Programme «R&D in Priority Fields of S&T Complex of Russia for 2014–2020» in the Priority Field “Rational use of natural resources” (contract No. 14.607.21.0030, unique identifier ASR RFMEFI60714X0030).
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