Abstract

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

1. Introduction

One of the most effective lidar techniques for vertical temperature profiling in the troposphere and lower stratosphere is known [1] to be the pure rotational Raman (PRR) lidar technique originally proposed by Cooney in 1972 [2]. 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 [9], 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).

 figure: Fig. 1

Fig. 1 a) The equidistant PRR spectra of N2 and O2 linear molecules. The index over a spectral line denotes the rotational quantum number J of the initial state of the transition. The spectral line number and number J are the same for the Stokes branch. b) A schematic drawing of gratings transmission functions (GTF) and envelopes of N2 PRR spectrum at different temperatures. The gratings transmission bands (GTB) are under the corresponding GTFs.

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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 dependence

Qindiv.(T)=IASt(Jlow,T)IASt(Jhigh,T)=exp(α+βT),
where α and β are the constants completely defined from the theory [1–3, 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 [8], one should consider the following expression
QΣ(T)=IlowΣ(T)IhighΣ(T)=[JN2IN2(JN2,T)+JO2IO2(JO2,T)]low[JN2IN2(JN2,T)+JO2IO2(JO2,T)]high,
where IN2(JN2,T) and IO2(JO2,T) are the intensities of N2 and O2 individual PRR lines, respectively; IlowΣ(T) and IhighΣ(T) are the overall intensities of the PRR lines which enter the corresponding lidar PRR channels; indexes “low” and “high” show that summations in the numerator and denominator refer to the corresponding PRR-spectrum bands with Jlow and Jhigh. Note that DGs and IFs transmission functions (curves) are different for different PRR lines. The curves for IFs can be found, e.g., in [10–24].

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 fcΣ(T) (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 [1]. 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) [3] 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 form

QΣ(T)fcΣ(T)=exp(A+BT)orlnfcΣ(T)=A+BTy=A+Bx,
where A and B are the calibration constants determined by applying the least square method to lidar remote sensing (or simulation) and reference radiosonde (or model) data; x = 1/T is the reciprocal temperature; the symbol ⇔ denotes the equivalence of expressions. This linear in x calibration function is applied by the majority of lidar researchers and, at the same time, yields significant temperature errors ( ± 1 K) [1] at the standard deviation of 0.38 K [10].

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 [10]. 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) [12]

lnQΣlnfcΣ(T)=A+BT+CT2y=A+Bx+Cx2,
which is shown in [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 [15],
lnQΣlnfcΣ(T)=A+BT+CT2+DT3y=A+Bx+Cx2+Dx3
can yield the less temperature errors compared to 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 n
lnQΣlnfcΣ(T)=A+BT+CT2+DT3+ET4+y=A+Bx+Cx2+Dx3+Ex4+
Equation (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 [3]. 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) [18] 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) [11] and 3 K for Eq. (4) [14]. 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 [10], 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 [29]. We suggest considering such a broadening during tropospheric temperature measurements using PRR lidars, especially in the atmospheric boundary layer [30].

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 ν˜low and ν˜high 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 [28] (or Cabannes [31]) 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 [29]. 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 ratio

Qall(T)=Ilowall(T)Ihighall(T)=i=N2,O2Ji[Ii(Ji,T)Xlowi(ν˜lowν˜i,T)]i=N2,O2Ji[Ii(Ji,T)Xhighi(ν˜highν˜i,T)].
Here ν˜i=ν˜(Ji) is the wavenumber of i PRR line with Ji; Ilowall(T) and Ihighall(T) are the overall intensities detected in the corresponding PRR channels with allowance for collisional broadening of all PRR lines; the functions Xlowi and Xhighi describe the parts of i PRR line signal falling within the PRR channels bands ν˜low and ν˜high, respectively. These functions depend on the distance between the bands central lines ν˜low and ν˜high and i PRR line ν˜i. The function Xlowi can be written as follows
Xlowi(ν˜lowν˜i,T)=+F(ν˜low,ν˜)V(ν˜i,ν˜,T)dν˜,
where V(ν˜i,ν˜,T) is the Voigt profile of i broadened PRR line shape, and F(ν˜low,ν˜) is the filter or grating transmission function of the PRR channel ν˜low, as shown in Fig. 2(a). Similarly, for the function Xhighi we have
Xhighi(ν˜highν˜i,T)=+F(ν˜high,ν˜)V(ν˜i,ν˜,T)dν˜.
Comparing Eqs. (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).

 figure: Fig. 2

Fig. 2 a) A schematic drawing (not to scale) of the filter or grating transmission functions (dashed), PRR channels transmission functions approximated by unity (solid), and broadened PRR lines profiles (color). b) The transmission function approximated by a piecewise-constant (staircase) function.

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In order to obtain an analytical approximation (calibration) function fcall(T) 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 L(ν˜i,ν˜,T) for a PRR line shape description instead of V(ν˜i,ν˜,T), 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 [30]. 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 ν˜1 to ν˜2, i.e. F(ν˜low,ν˜)=F(ν˜1,ν˜2)=F. Moreover, for the same reason, we can put F = 1 without loss of generality. The interval ν˜1 to ν˜2 is responsible for transmission of the bulk of the backscattered signal intensity in PRR channel ν˜low. The same is valid for F(ν˜high,ν˜) with the transmission band ν˜3 to ν˜4, as shown in Fig. 2(a). Thus, instead of Eqs. (8) and (9) we can write

Xlowi(T)=Xlowi(ν˜1,ν˜2,T)=ν˜1ν˜2L(ν˜i,ν˜,T)dν˜,Xhighi(T)=Xhighi(ν˜3,ν˜4,T)=ν˜3ν˜4L(ν˜i,ν˜,T)dν˜.

Following Arshinov et al. [3], we consider the overall intensity of PRR lines entered the transmission band ν˜1 to ν˜2 as the intensity IlowΣ(T) of an averaged line with ν˜low, as shown in Fig. 2(a). In this case, the backscattered signal intensity transmitted by IF or DG of the PRR channel ν˜low can be expressed as follows

Ilowall(T)=IlowΣ(T)[1XlowDSL(T)]+k,Jk[Ik(Jk,T)Xlowk(T)],
where XlowDSL(T) describes the signal falling outside the band ν˜1 to ν˜2, i.e. the loss of desired signal due to the collisional broadening; the second term (the sum) in the right part of Eq. (11) is responsible for the parasitic signal of the broadened PRR lines lying outside the band. It is convenient to rewrite Eq. (11) as
Ilowall(T)=IlowΣ(T){1XlowDSL(T)+k,Jk[Ik(Jk,T)IlowΣ(T)Xlowk(T)]}=IlowΣ(T)[1XlowDSL(T)+kYlowk(T)]=IlowΣ(T)[1XlowDSL(T)+YlowPS(T)],
where Ylowk(T) and YlowPS(T) are defined by the given equations. Similarly, for the intensity Ihighall(T) we obtain
Ihighall(T)=IhighΣ(T){1XhighDSL(T)+k,Jk[Ik(Jk,T)IhighΣ(T)Xhighk(T)]}=IhighΣ(T)[1XhighDSL(T)+kYhighk(T)]=IhighΣ(T)[1XhighDSL(T)+YhighPS(T)].
Substituting Eqs. (12) and (13) into Eq. (7), we obtain for the intensity ratio
Qall(T)fcall(T)=Ilowall(T)Ihighall(T)=IlowΣ(T)IhighΣ(T)[1XlowDSL(T)+YlowPS(T)][1XhighDSL(T)+YhighPS(T)].
To further simplify Eq. (14), we can replace the ratio IlowΣ(T)/IhighΣ(T)=fcΣ(T) by its temperature dependence from Eq. (6). Moreover, as each of the functions XlowDSL(T), YlowPS(T), XhighDSL(T), and YhighPS(T) is much less than unity, we can use an approximation (1+z)11z with z2 << 1 [34] and also retain only the first-order terms, i.e.
Qall(T)fcall(T)=fcΣ(T)[1XlowDSL(T)+YlowPS(T)][1+XhighDSL(T)YhighPS(T)]=exp(A+BT+CT2+)[1+XhighDSL(T)XlowDSL(T)+YlowPS(T)YhighPS(T)].
Thus, in order to obtain the analytical calibration function one should determine the temperature dependence of XlowDSL(T), YlowPS(T), XhighDSL(T), and YhighPS(T) in 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 XlowDSL(T) and XhighDSL(T). The function XlowDSL(T) represents the probability that a backscattered signal wavenumber will be less than ν˜1 or more than ν˜2, i.e. outside the transmission band (ν˜1;ν˜2), at the temperature T. Taking into account Eq. (45) in Appendix A, we can write

XlowDSL(T)=1π+γL(xν˜low)2+γL2dx1πν˜1ν˜2γL(xν˜low)2+γL2dx=11π[arctan(ν˜2ν˜lowγL)arctan(ν˜1ν˜lowγL)].
Otherwise speaking, XlowDSL(T) is the difference between the probability that a backscattered signal wavenumber will take any value within the whole spectrum (;+), i.e. unity, and the probability of the wavenumber to be within the band (ν˜1;ν˜2), i.e. the difference of arctangents in Eq. (16). Consider the expansion of arctan(z) in a series [34]
arctan(z)=π21z+13z315z5+17z7withz>1.
Since z=(ν˜ν˜low)/γL1 (the band edges ν˜1 and ν˜2are sufficiently far from ν˜low), we get
arctan(ν˜ν˜lowγL)=π2γLν˜ν˜low+13(γLν˜ν˜low)313(γLν˜ν˜low)5+.
Subtracting term by term Eq. (18) at ν˜=ν˜1 from its own expression at ν˜=ν˜2, we can reduce Eq. (16) to
XlowDSL(T)=11π[(1Δν˜21Δν˜1)γL+13(1Δν˜231Δν˜13)γL3]=1(a1γL+a2γL3+a3γL5+a4γL7+),
where Δν˜1=ν˜1ν˜low, Δν˜2=ν˜2ν˜low, and ai are the constants defined by the given equation. Substituting Eq. (48) from Appendix A into Eq. (19) and combining similar terms, we obtain for the temperature dependence of XlowDSL(T)
XlowDSL(T)=1[a1(AT+BT)+a2(AT+BT)3+a3(AT+BT)5+]=+b2TT+b1T+1+b1T+b2TT+=1+n=1(bnTn1/2+bnTn1/2),
where bn and bn are the constants defined by the given equation. Applying the same procedure to XhighDSL(T) with the transmission band (ν˜3;ν˜4) instead of (ν˜1;ν˜2), one can obtain
XhighDSL(T)=1+n=1(cnTn1/2+cnTn1/2),
where cn and cn are the constants defined similarly to bn and bn in Eq. (20). Subtracting Eq. (20) from Eq. (21), we have for the temperature dependence of XhighDSL(T)XlowDSL(T) in Eq. (15)
XhighDSL(T)XlowDSL(T)=n=1(dnTn1/2+dnTn1/2),
where dn = cnbn and dn = cnbn.

In order to determine the temperature dependence of YlowPS(T) in Eq. (12), we can use the results obtained above. First, since we consider IlowΣ(T) as the intensity of a single line, each intensity ratio Ik(Jk,T)/IlowΣ(T) in Eq. (12) can be defined as well as the ratio in Eq. (1), i.e.

Ik(Jk,T)IlowΣ(T)=exp(Ck+EkT)=exp(Ck)exp(EkT)=Dkexp(EkT),
where Ck, Dk, and Ek are the constants corresponding to k intensity ratio. Second, the temperature dependence of each function Xlowk(T) is defined by Eq. (10), and hence, by the difference of arctangents in Eq. (16). Taking Eq. (20) into account, we get
Xlowk(T)=n=1(fnkTn1/2+fnkTn1/2),
where fnk and fnk are the constants defined similarly to bn and bn in Eq. (20). Expanding the exponential function in Eq. (23) in Taylor series, we obtain for YlowPS(T) in Eq. (12)
YlowPS(T)=kYlowk(T)=k,Jk[Ik(Jk,T)IlowΣ(T)Xlowk(T)]=k[Dk(1+EkT+Ek22T2+)n=1(fnkTn1/2+fnkTn1/2)]=n=1(gnTn1/2+gnTn1/2),
where gn and gn are defined by the given equation, and Xlowk(T) is defined by Eq. (24). The same is valid for the function YhighPS(T), i.e.
YhighPS(T)=n=1(hnTn1/2+hnTn1/2),
where hn and hn are the constants defined similarly to gn and gn in Eq. (25).

It is easy to see that all the functions XlowDSL(T), YlowPS(T), XhighDSL(T), and YhighPS(T) 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 [30]. Therefore, combining Eqs. (22), (25), and (26) yields the same dependence

Z(T)=XhighDSL(T)XlowDSL(T)+YlowPS(T)YhighPS(T)=n=1(lnTn1/2+lnTn1/2).
where ln and ln are the constants.

Taking the natural logarithm of both sides of Eq. (15), we obtain

lnQall(T)lnfcall(T)=A+BT+CT2+DT3++ln[1+Z(T)].
Since Z(T) << 1 in Eq. (27), the logarithm in Eq. (28) can be expanded in a series [34]
ln(1+z)=zz22+z33z44+with1<z1.
Substituting z = Z(T) from Eq. (27) into Eq. (29), we have
ln[1+Z(T)]=n=1(lnTn1/2+lnTn1/2)12[n=1(lnTn1/2+lnTn1/2)]2+=+m4T2+m3TT+m2T+m1T+m0+m1T+m2T+m3TT+m4T2+,
where the constants m–n and mn are responsible for both the loss of desired signal and parasitic signal leakage due to collisional broadening. Substituting Eq. (30) into Eq. (28) and combining similar terms, we obtain the PRR lidar calibration function in the general form
lnQall(T)+C+m4T2+m3TT+B+m2T+m1T+(A+m0)+m1T+m2T+=+A4T2+A3TT+A2T+A1T+A0+A1T+A2T+A3TT+=n=AnTn2,
where An are the lidar calibration coefficients. Setting Z(T) = 0 in Eq. (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.

lnQ(T)+B2T2+B1T+B0+B1T+B2T2+=n=BnTn,
where Bn are the calibration coefficients (here and elsewhere, we write Q for the intensity ratio). In this section we consider the linear and simplest nonlinear (three-coefficient) calibration functions and their corresponding temperature retrieval functions.

The most-used linear in x = 1/T calibration function expressed by Eq. (3) is a special case of Eq. (32). The temperature retrieval function can be simply derived from Eq. (3)

T=B0lnQA0,
where A0 and B0 are the calibration constants determined by the least square method.

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

T=2C1B1+B12+4C1(lnQA1),
where A1, B1, and C1 are the calibration constants.

The next three-coefficient special case of Eq. (32) we consider is the following nonlinear function with the hyperbolic in x = 1/T term

lnQ=A2+B2T+C2Ty=A2+B2x+C2x,
where A2, B2, and C2 are the calibration constants. In this case the temperature retrieval function derived from Eq. (35) is

T=2B2(lnQA2)+(lnQA2)24B2C2.

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 c

x=a+by+cy2.
The temperature profile can be simply retrieved via
1/T=a+blnQ+c(lnQ)2T=[c(lnQ)2+blnQ+a]1,
or
T=C3(lnQ)2+B3lnQ+A3,
where A3 = a/c, B3 = b/c, and C3 = 1/c. Note that Eq. (39) was applied in [11]. 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 gives

x=A4+B4y+C4/y1/T=A4+B4lnQ+C4/lnQ,
where A4, B4, and C4 are the calibration constants. In this case, inverting Eq. (40) yields

T=1A4+B4lnQ+(C4/lnQ)=lnQB4(lnQ)2+A4lnQ+C4.

5. Simulation

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 ν˜low and ν˜high 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 (ν˜1;ν˜2) = (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 (ν˜3;ν˜4) = (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.

 figure: Fig. 3

Fig. 3 a) The U.S. Standard tropospheric temperature profile. b) The FWHMs of the Gaussian, Lorentzian, and Voigt profiles of N2 broadened PRR line (with J = 6 of the anti-Stokes branch).

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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.

 figure: Fig. 4

Fig. 4 The calculated dependences of altitude (a) and temperature (b) on the intensity ratio Q.

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 figure: Fig. 5

Fig. 5 A comparative analysis of temperature errors yielded by using the different temperature retrieval functions.

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Tables Icon

Table 1. Calibration coefficients for temperature retrieval functions.

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.

6. Summary

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.

Appendices

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]

G(ν˜i,ν˜,T)=1γGi2πexp[(ν˜μi)22(γGi)2].
Here μi=ν˜i (cm–1) is the mathematical expectation (or the central wavenumber of i broadened line profile); γGi (cm–1) is the standard deviation defined by the equation
γGi=ν˜ikBTmairc2,
where c is the speed of light, mair is the average mass of air molecules, kB is the Boltzmann constant. For example, if an emitting laser wavelength λ0 = 354.67 nm, for N2 PRR line (J = 6) of the anti-Stokes branch we have: λJ = 6 = 354.12 nm and ν˜J=6 ≈28238.994 cm–1. The full width at half maximum (FWHM) of the Gaussian profile is defined as

Δν˜i,GFWHM=22ln2γGi.

The backscatter profile of a PRR line homogeneously broadened by only the collision effect is governed by a Lorentzian (Cauchy) distribution [10, 29]

L(ν˜i,ν˜,T)=1πγL(ν˜ν˜i)2+γL2,
where γL (cm–1) specifies the half-width at half-maximum (HWHM) of the Lorentzian profile. In other words, γL = 1/τ, where τ is the average lifetime of a molecular excited state under molecular collisions [29]. The FWHM of the Lorentzian profile is defined as Δν˜LFWHM=2γL. 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
γL=p12nairπd128kTπμ1c2+2p1p2nairπd1228kTπμ12c2+p22nairπd228kTπμ2c2,
where p1 = 0.7809 and p2 = 0.2095 are the probabilities to find N2 and O2 molecules in the homosphere, respectively; m1 and m2 are the N2 and O2 masses, respectively; nair is the air molecular number density; d1 and d2 are the effective optical collision diameters in N2–N2 and O2–O2 collisions, respectively; d12 = (d1 + d2)/2 is the effective optical collision diameter in N2–O2 collisions; μ1 = m1/2 and μ2 = m2/2 are the reduced masses of colliding molecules in N2–N2 and O2–O2 collisions, respectively; and μ12 = m1m2/(m1 + m2) is the reduced mass of colliding molecules in N2–O2 collisions.

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 [35]

di2(T)=di,2(1+φiRT),
where the constant di,∞ is the effective optical collision diameter at T → ∞; φi is the constant having the dimension of energy/mol; and R is the molar gas constant. By substituting Eq. (47) into Eq. (46) and then combining similar terms, we obtain for the temperature dependence of γL at given molecular number density nair
γL=AT+BT,
where A and B are the constants defined by Eqs. (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 [32], φ1/R = 105 K, φ2/R = 125 K, φ12/R = 115 K [35].

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 effects

V(ν˜i,ν˜,T)=G(T,ν˜i,ν˜*)L(T,ν˜,ν˜*)dν˜*.
By substituting Eqs. (42) and (45) into Eq. (49), the Voigt profile can generally be expressed in the following form
V(ν˜i,ν˜,T)=u0yπexp(t2)dt(xt)2+y2,
where
u0=2Δν˜i,GFWHMln2π,y=Δν˜LFWHMΔν˜i,GFWHMln2,x=ν˜ν˜iΔν˜i,GFWHM2ln2.
The integral in Eq. (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 [37]
Δν˜i,VFWHM0.5346Δν˜LFWHM+(Δν˜i,GFWHM)2+0.2166(Δν˜LFWHM)2.
This approximation has an error of about 0.000305% percent for the pure Lorentzian profile and is exactly correct for the pure Gaussian profile.

B. Equation (39) as a special case of the general calibration function

Here we show that the initial calibration function of Eq. (39) (i.e. the T-dependence of lnQ) represents a special case of Eq. (31) or (32). Solving quadratic Eq. (39) we get

lnQ=B32±(B32)2A3+C3Ty=B32±(B32)2A3+C3x.
In order to analyze Eq. (53) in two limiting cases, one can rewrite it as follows
y=B32±|C3|B324C3A3C3+x=α+βγ+x=α+β|γ|1+xγ.
If x >> γ, then yα+βx, and Eq. (53) can be expressed as a special case of Eq. (31)
lnQ=A0+A1T.
If x << γ, we can use the following expansion in a series in z = x/γ << 1 [34]
1+z=1+12z18z2+116z3withz21.
Substituting Eq. (56) into Eq. (54), for the calibration function we obtain
lnQ=α+β|γ|(1+x2γx28γ2+x316γ3)=B0+B1T+B2T2+,
i.e. a special case of Eq. (32).

Acknowledgments

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).

References and Links

1. A. Behrendt, “Temperature measurement with Lidar,” in Lidar Range-Resolved Optical Remote Sensing of the Atmosphere (C. Weitkamp, 2005), Ch. 10.

2. J. Cooney, “Measurement of atmospheric temperature profiles by Raman backscatter,” J. Appl. Meteorol. 11(1), 108–112 (1972). [CrossRef]  

3. Y. F. Arshinov, S. M. Bobrovnikov, V. E. Zuev, and V. M. Mitev, “Atmospheric temperature measurements using a pure rotational Raman lidar,” Appl. Opt. 22(19), 2984–2990 (1983). [CrossRef]   [PubMed]  

4. D. Kim, H. Cha, J. Lee, and S. Bobronikov, “Pure rotational Raman lidar for atmospheric temperature measurements,” J. Korean Phys. Soc. 39(5), 838–841 (2001).

5. S. Chen, Z. Qiu, Y. Zhang, H. Chen, and Y. Wang, “A pure rotational Raman lidar using double-grating monochromator for temperature profile detection,” J. Quant. Spectrosc. Rad. Transf. 112(2), 304–309 (2011). [CrossRef]  

6. T. Dinoev, V. Simeonov, Y. Arshinov, S. Bobrovnikov, P. Ristori, B. Calpini, M. Parlange, and H. van den Bergh, “Raman lidar for meteorological observations, RALMO – Part 1: Instrument description,” Atmos. Meas. Tech. 6(5), 1329–1346 (2013). [CrossRef]  

7. F. Liu and F. Yi, “Lidar-measured atmospheric N₂ vibrational-rotational Raman spectra and consequent temperature retrieval,” Opt. Express 22(23), 27833–27844 (2014). [CrossRef]   [PubMed]  

8. J. Jia and F. Yi, “Atmospheric temperature measurements at altitudes of 5-30 km with a double-grating-based pure rotational Raman lidar,” Appl. Opt. 53(24), 5330–5343 (2014). [CrossRef]   [PubMed]  

9. J. Mao, L. Hu, D. Hua, F. Gao, and M. Wu, “Pure rotational Raman lidar with fiber Bragg grating for temperature profiling of the atmospheric boundary layer,” Opt. Appl. 38(4), 715–726 (2008).

10. D. Nedeljkovic, A. Hauchecorne, and M. L. Chanin, “Rotational Raman lidar to measure the atmospheric temperature from the ground to 30 km,” IEEE Trans. Geosci. Rem. Sens. 31(1), 90–101 (1993). [CrossRef]  

11. R. B. Lee III, “Tropospheric temperature measurements using a rotational Raman lidar,” doctoral dissertation (Hampton University, 2013), http://gradworks.umi.com/35/92/3592881.html.

12. A. Behrendt and J. Reichardt, “Atmospheric temperature profiling in the presence of clouds with a pure rotational Raman lidar by use of an interference-filter-based polychromator,” Appl. Opt. 39(9), 1372–1378 (2000). [CrossRef]   [PubMed]  

13. P. Achtert, M. Khaplanov, F. Khosrawi, and J. Gumbel, “Pure rotational-Raman channels of the Esrange lidar for temperature and particle extinction measurements in the troposphere and lower stratosphere,” Atmos. Meas. Tech. 6(1), 91–98 (2013). [CrossRef]  

14. M. Radlach, “A scanning eye-safe rotational Raman lidar in the ultraviolet for measurements of tropospheric temperature fields,” doctoral dissertation (University of Hohenheim, 2009), https://opus.uni-hohenheim.de/volltexte/2009/345/pdf/Radlach_doctoral_thesis_engl_LINKS.pdf.

15. P. Di Girolamo, R. Marchese, D. N. Whiteman, and B. B. Demoz, “Rotational Raman Lidar measurements of atmospheric temperature in the UV,” Geophys. Res. Lett. 31(1), L01106 (2004). [CrossRef]  

16. R. K. Newsom, C. Sivaraman, and S. A. McFarlane, “Raman lidar profiles–temperature (RLPROFTEMP) value-added product,” https://www.arm.gov/publications/tech_reports/doe-sc-arm-tr-120.pdf.

17. R. K. Newsom, D. D. Turner, and J. E. M. Goldsmith, “Long-term evaluation of temperature profiles measured by an operational Raman lidar,” J. Atmos. Ocean. Technol. 30(8), 1616–1634 (2013). [CrossRef]  

18. A. Behrendt, T. Nakamura, M. Onishi, R. Baumgart, and T. Tsuda, “Combined Raman lidar for the measurement of atmospheric temperature, water vapor, particle extinction coefficient, and particle backscatter coefficient,” Appl. Opt. 41(36), 7657–7666 (2002). [CrossRef]   [PubMed]  

19. A. Behrendt, T. Nakamura, and T. Tsuda, “Combined temperature lidar for measurements in the troposphere, stratosphere, and mesosphere,” Appl. Opt. 43(14), 2930–2939 (2004). [CrossRef]   [PubMed]  

20. M. Alpers, R. Eixmann, C. Fricke-Begemann, M. Gerding, and J. Höffner, “Temperature lidar measurements from 1 to 105 km altitude using resonance, Rayleigh, and rotational Raman scattering,” Atmos. Chem. Phys. 4(3), 793–800 (2004). [CrossRef]  

21. M. Radlach, A. Behrendt, and V. Wulfmeyer, “Scanning rotational Raman lidar at 355 nm for the measurement of tropospheric temperature fields,” Atmos. Chem. Phys. 8(2), 159–169 (2008). [CrossRef]  

22. A. Behrendt, V. Wulfmeyer, E. Hammann, S. K. Muppa, and S. Pal, “Profiles of second- to fourth-order moments of turbulent temperature fluctuations in the convective boundary layer: first measurements with rotational Raman lidar,” Atmos. Chem. Phys. 15(10), 5485–5500 (2015). [CrossRef]  

23. E. Hammann, A. Behrendt, F. Le Mounier, and V. Wulfmeyer, “Temperature profiling of the atmospheric boundary layer with rotational Raman lidar during the HD(CP)2 observational prototype experiment,” Atmos. Chem. Phys. 15(5), 2867–2881 (2015). [CrossRef]  

24. E. Hammann and A. Behrendt, “Parametrization of optimum filter passbands for rotational Raman temperature measurements,” Opt. Express 23(24), 30767–30782 (2015). [CrossRef]   [PubMed]  

25. G. Placzek, Rayleigh-Streuung und Raman effekt,” in Handbuch der Radiologie, E. Marx, ed. (Akademischer Verlag, 1934).

26. W. R. Fenner, H. A. Hyatt, J. M. Kellam, and S. P. S. Porto, “Raman cross section of some simple gases,” J. Opt. Soc. Am. 63(1), 73–77 (1973). [CrossRef]  

27. M. Penney, R. L. St. Peters, and M. Lapp, “Absolute rotational Raman cross sections for N2, O2, and CO2,” J. Opt. Soc. Am. 64(5), 712–716 (1974). [CrossRef]  

28. U. Wandinger, “Raman lidar,” in Lidar Range-Resolved Optical Remote Sensing of the Atmosphere (C. Weitkamp, 2005), Ch. 9.

29. R. M. Measures, Laser Remote Sensing, Fundamentals and Applications (Wiley, 1984).

30. I. D. Ivanova, L. L. Gurdev, and V. M. Mitev, “Lidar technique for simultaneous temperature and pressure measurement based on rotation Raman scattering,” J. Mod. Opt. 40(3), 367–371 (1993). [CrossRef]  

31. C.-Y. She, “Spectral structure of laser light scattering revisited: bandwidths of nonresonant scattering lidars,” Appl. Opt. 40(27), 4875–4884 (2001). [CrossRef]   [PubMed]  

32. P. A. Bazhulin, “The study of rotation and vibration-rotation spectra of gases by the method of combination (Raman) scattering of light,” Sov. Phys. Usp. 5(4), 661–666 (1963). [CrossRef]  

33. V. L. Ginzburg, “Line width in the spectrum of scattered light,” Sov. Phys. Usp. 15(1), 114–120 (1972). [CrossRef]  

34. H. B. Dwight, Tables of Integrals and Other Mathematical Data, 4th ed. (The Macmillan Company, 1961).

35. Y. I. Gerasimov, Course in Physical Chemistry, Vol. 2 (Khimiya, 1973).

36. V. S. Matveev, “Approximate representations of absorption coefficient end equivalent widths of lines with Voigt profile,” J. Appl. Spectrosc. 16(2), 228–233 (1972).

37. J. J. Olivero and R. L. Longbothum, “Empirical fits to the Voigt line width: a brief review,” J. Quantum Spectrosc. Radiat. Transf. 17(2), 233–236 (1977). [CrossRef]  

38. S. R. Drayson, “Rapid computation of the Voigt profile,” J. Quantum Spectrosc. Radiat. Transf. 16(7), 611–614 (1976). [CrossRef]  

References

  • View by:

  1. A. Behrendt, “Temperature measurement with Lidar,” in Lidar Range-Resolved Optical Remote Sensing of the Atmosphere (C. Weitkamp, 2005), Ch. 10.
  2. J. Cooney, “Measurement of atmospheric temperature profiles by Raman backscatter,” J. Appl. Meteorol. 11(1), 108–112 (1972).
    [Crossref]
  3. Y. F. Arshinov, S. M. Bobrovnikov, V. E. Zuev, and V. M. Mitev, “Atmospheric temperature measurements using a pure rotational Raman lidar,” Appl. Opt. 22(19), 2984–2990 (1983).
    [Crossref] [PubMed]
  4. D. Kim, H. Cha, J. Lee, and S. Bobronikov, “Pure rotational Raman lidar for atmospheric temperature measurements,” J. Korean Phys. Soc. 39(5), 838–841 (2001).
  5. S. Chen, Z. Qiu, Y. Zhang, H. Chen, and Y. Wang, “A pure rotational Raman lidar using double-grating monochromator for temperature profile detection,” J. Quant. Spectrosc. Rad. Transf. 112(2), 304–309 (2011).
    [Crossref]
  6. T. Dinoev, V. Simeonov, Y. Arshinov, S. Bobrovnikov, P. Ristori, B. Calpini, M. Parlange, and H. van den Bergh, “Raman lidar for meteorological observations, RALMO – Part 1: Instrument description,” Atmos. Meas. Tech. 6(5), 1329–1346 (2013).
    [Crossref]
  7. F. Liu and F. Yi, “Lidar-measured atmospheric N₂ vibrational-rotational Raman spectra and consequent temperature retrieval,” Opt. Express 22(23), 27833–27844 (2014).
    [Crossref] [PubMed]
  8. J. Jia and F. Yi, “Atmospheric temperature measurements at altitudes of 5-30 km with a double-grating-based pure rotational Raman lidar,” Appl. Opt. 53(24), 5330–5343 (2014).
    [Crossref] [PubMed]
  9. J. Mao, L. Hu, D. Hua, F. Gao, and M. Wu, “Pure rotational Raman lidar with fiber Bragg grating for temperature profiling of the atmospheric boundary layer,” Opt. Appl. 38(4), 715–726 (2008).
  10. D. Nedeljkovic, A. Hauchecorne, and M. L. Chanin, “Rotational Raman lidar to measure the atmospheric temperature from the ground to 30 km,” IEEE Trans. Geosci. Rem. Sens. 31(1), 90–101 (1993).
    [Crossref]
  11. R. B. Lee III, “Tropospheric temperature measurements using a rotational Raman lidar,” doctoral dissertation (Hampton University, 2013), http://gradworks.umi.com/35/92/3592881.html .
  12. A. Behrendt and J. Reichardt, “Atmospheric temperature profiling in the presence of clouds with a pure rotational Raman lidar by use of an interference-filter-based polychromator,” Appl. Opt. 39(9), 1372–1378 (2000).
    [Crossref] [PubMed]
  13. P. Achtert, M. Khaplanov, F. Khosrawi, and J. Gumbel, “Pure rotational-Raman channels of the Esrange lidar for temperature and particle extinction measurements in the troposphere and lower stratosphere,” Atmos. Meas. Tech. 6(1), 91–98 (2013).
    [Crossref]
  14. M. Radlach, “A scanning eye-safe rotational Raman lidar in the ultraviolet for measurements of tropospheric temperature fields,” doctoral dissertation (University of Hohenheim, 2009), https://opus.uni-hohenheim.de/volltexte/2009/345/pdf/Radlach_doctoral_thesis_engl_LINKS.pdf .
  15. P. Di Girolamo, R. Marchese, D. N. Whiteman, and B. B. Demoz, “Rotational Raman Lidar measurements of atmospheric temperature in the UV,” Geophys. Res. Lett. 31(1), L01106 (2004).
    [Crossref]
  16. R. K. Newsom, C. Sivaraman, and S. A. McFarlane, “Raman lidar profiles–temperature (RLPROFTEMP) value-added product,” https://www.arm.gov/publications/tech_reports/doe-sc-arm-tr-120.pdf .
  17. R. K. Newsom, D. D. Turner, and J. E. M. Goldsmith, “Long-term evaluation of temperature profiles measured by an operational Raman lidar,” J. Atmos. Ocean. Technol. 30(8), 1616–1634 (2013).
    [Crossref]
  18. A. Behrendt, T. Nakamura, M. Onishi, R. Baumgart, and T. Tsuda, “Combined Raman lidar for the measurement of atmospheric temperature, water vapor, particle extinction coefficient, and particle backscatter coefficient,” Appl. Opt. 41(36), 7657–7666 (2002).
    [Crossref] [PubMed]
  19. A. Behrendt, T. Nakamura, and T. Tsuda, “Combined temperature lidar for measurements in the troposphere, stratosphere, and mesosphere,” Appl. Opt. 43(14), 2930–2939 (2004).
    [Crossref] [PubMed]
  20. M. Alpers, R. Eixmann, C. Fricke-Begemann, M. Gerding, and J. Höffner, “Temperature lidar measurements from 1 to 105 km altitude using resonance, Rayleigh, and rotational Raman scattering,” Atmos. Chem. Phys. 4(3), 793–800 (2004).
    [Crossref]
  21. M. Radlach, A. Behrendt, and V. Wulfmeyer, “Scanning rotational Raman lidar at 355 nm for the measurement of tropospheric temperature fields,” Atmos. Chem. Phys. 8(2), 159–169 (2008).
    [Crossref]
  22. A. Behrendt, V. Wulfmeyer, E. Hammann, S. K. Muppa, and S. Pal, “Profiles of second- to fourth-order moments of turbulent temperature fluctuations in the convective boundary layer: first measurements with rotational Raman lidar,” Atmos. Chem. Phys. 15(10), 5485–5500 (2015).
    [Crossref]
  23. E. Hammann, A. Behrendt, F. Le Mounier, and V. Wulfmeyer, “Temperature profiling of the atmospheric boundary layer with rotational Raman lidar during the HD(CP)2 observational prototype experiment,” Atmos. Chem. Phys. 15(5), 2867–2881 (2015).
    [Crossref]
  24. E. Hammann and A. Behrendt, “Parametrization of optimum filter passbands for rotational Raman temperature measurements,” Opt. Express 23(24), 30767–30782 (2015).
    [Crossref] [PubMed]
  25. G. Placzek, Rayleigh-Streuung und Raman effekt,” in Handbuch der Radiologie, E. Marx, ed. (Akademischer Verlag, 1934).
  26. W. R. Fenner, H. A. Hyatt, J. M. Kellam, and S. P. S. Porto, “Raman cross section of some simple gases,” J. Opt. Soc. Am. 63(1), 73–77 (1973).
    [Crossref]
  27. M. Penney, R. L. St. Peters, and M. Lapp, “Absolute rotational Raman cross sections for N2, O2, and CO2,” J. Opt. Soc. Am. 64(5), 712–716 (1974).
    [Crossref]
  28. U. Wandinger, “Raman lidar,” in Lidar Range-Resolved Optical Remote Sensing of the Atmosphere (C. Weitkamp, 2005), Ch. 9.
  29. R. M. Measures, Laser Remote Sensing, Fundamentals and Applications (Wiley, 1984).
  30. I. D. Ivanova, L. L. Gurdev, and V. M. Mitev, “Lidar technique for simultaneous temperature and pressure measurement based on rotation Raman scattering,” J. Mod. Opt. 40(3), 367–371 (1993).
    [Crossref]
  31. C.-Y. She, “Spectral structure of laser light scattering revisited: bandwidths of nonresonant scattering lidars,” Appl. Opt. 40(27), 4875–4884 (2001).
    [Crossref] [PubMed]
  32. P. A. Bazhulin, “The study of rotation and vibration-rotation spectra of gases by the method of combination (Raman) scattering of light,” Sov. Phys. Usp. 5(4), 661–666 (1963).
    [Crossref]
  33. V. L. Ginzburg, “Line width in the spectrum of scattered light,” Sov. Phys. Usp. 15(1), 114–120 (1972).
    [Crossref]
  34. H. B. Dwight, Tables of Integrals and Other Mathematical Data, 4th ed. (The Macmillan Company, 1961).
  35. Y. I. Gerasimov, Course in Physical Chemistry, Vol. 2 (Khimiya, 1973).
  36. V. S. Matveev, “Approximate representations of absorption coefficient end equivalent widths of lines with Voigt profile,” J. Appl. Spectrosc. 16(2), 228–233 (1972).
  37. J. J. Olivero and R. L. Longbothum, “Empirical fits to the Voigt line width: a brief review,” J. Quantum Spectrosc. Radiat. Transf. 17(2), 233–236 (1977).
    [Crossref]
  38. S. R. Drayson, “Rapid computation of the Voigt profile,” J. Quantum Spectrosc. Radiat. Transf. 16(7), 611–614 (1976).
    [Crossref]

2015 (3)

A. Behrendt, V. Wulfmeyer, E. Hammann, S. K. Muppa, and S. Pal, “Profiles of second- to fourth-order moments of turbulent temperature fluctuations in the convective boundary layer: first measurements with rotational Raman lidar,” Atmos. Chem. Phys. 15(10), 5485–5500 (2015).
[Crossref]

E. Hammann, A. Behrendt, F. Le Mounier, and V. Wulfmeyer, “Temperature profiling of the atmospheric boundary layer with rotational Raman lidar during the HD(CP)2 observational prototype experiment,” Atmos. Chem. Phys. 15(5), 2867–2881 (2015).
[Crossref]

E. Hammann and A. Behrendt, “Parametrization of optimum filter passbands for rotational Raman temperature measurements,” Opt. Express 23(24), 30767–30782 (2015).
[Crossref] [PubMed]

2014 (2)

2013 (3)

T. Dinoev, V. Simeonov, Y. Arshinov, S. Bobrovnikov, P. Ristori, B. Calpini, M. Parlange, and H. van den Bergh, “Raman lidar for meteorological observations, RALMO – Part 1: Instrument description,” Atmos. Meas. Tech. 6(5), 1329–1346 (2013).
[Crossref]

P. Achtert, M. Khaplanov, F. Khosrawi, and J. Gumbel, “Pure rotational-Raman channels of the Esrange lidar for temperature and particle extinction measurements in the troposphere and lower stratosphere,” Atmos. Meas. Tech. 6(1), 91–98 (2013).
[Crossref]

R. K. Newsom, D. D. Turner, and J. E. M. Goldsmith, “Long-term evaluation of temperature profiles measured by an operational Raman lidar,” J. Atmos. Ocean. Technol. 30(8), 1616–1634 (2013).
[Crossref]

2011 (1)

S. Chen, Z. Qiu, Y. Zhang, H. Chen, and Y. Wang, “A pure rotational Raman lidar using double-grating monochromator for temperature profile detection,” J. Quant. Spectrosc. Rad. Transf. 112(2), 304–309 (2011).
[Crossref]

2008 (2)

J. Mao, L. Hu, D. Hua, F. Gao, and M. Wu, “Pure rotational Raman lidar with fiber Bragg grating for temperature profiling of the atmospheric boundary layer,” Opt. Appl. 38(4), 715–726 (2008).

M. Radlach, A. Behrendt, and V. Wulfmeyer, “Scanning rotational Raman lidar at 355 nm for the measurement of tropospheric temperature fields,” Atmos. Chem. Phys. 8(2), 159–169 (2008).
[Crossref]

2004 (3)

A. Behrendt, T. Nakamura, and T. Tsuda, “Combined temperature lidar for measurements in the troposphere, stratosphere, and mesosphere,” Appl. Opt. 43(14), 2930–2939 (2004).
[Crossref] [PubMed]

M. Alpers, R. Eixmann, C. Fricke-Begemann, M. Gerding, and J. Höffner, “Temperature lidar measurements from 1 to 105 km altitude using resonance, Rayleigh, and rotational Raman scattering,” Atmos. Chem. Phys. 4(3), 793–800 (2004).
[Crossref]

P. Di Girolamo, R. Marchese, D. N. Whiteman, and B. B. Demoz, “Rotational Raman Lidar measurements of atmospheric temperature in the UV,” Geophys. Res. Lett. 31(1), L01106 (2004).
[Crossref]

2002 (1)

2001 (2)

D. Kim, H. Cha, J. Lee, and S. Bobronikov, “Pure rotational Raman lidar for atmospheric temperature measurements,” J. Korean Phys. Soc. 39(5), 838–841 (2001).

C.-Y. She, “Spectral structure of laser light scattering revisited: bandwidths of nonresonant scattering lidars,” Appl. Opt. 40(27), 4875–4884 (2001).
[Crossref] [PubMed]

2000 (1)

1993 (2)

D. Nedeljkovic, A. Hauchecorne, and M. L. Chanin, “Rotational Raman lidar to measure the atmospheric temperature from the ground to 30 km,” IEEE Trans. Geosci. Rem. Sens. 31(1), 90–101 (1993).
[Crossref]

I. D. Ivanova, L. L. Gurdev, and V. M. Mitev, “Lidar technique for simultaneous temperature and pressure measurement based on rotation Raman scattering,” J. Mod. Opt. 40(3), 367–371 (1993).
[Crossref]

1983 (1)

1977 (1)

J. J. Olivero and R. L. Longbothum, “Empirical fits to the Voigt line width: a brief review,” J. Quantum Spectrosc. Radiat. Transf. 17(2), 233–236 (1977).
[Crossref]

1976 (1)

S. R. Drayson, “Rapid computation of the Voigt profile,” J. Quantum Spectrosc. Radiat. Transf. 16(7), 611–614 (1976).
[Crossref]

1974 (1)

1973 (1)

1972 (3)

V. L. Ginzburg, “Line width in the spectrum of scattered light,” Sov. Phys. Usp. 15(1), 114–120 (1972).
[Crossref]

V. S. Matveev, “Approximate representations of absorption coefficient end equivalent widths of lines with Voigt profile,” J. Appl. Spectrosc. 16(2), 228–233 (1972).

J. Cooney, “Measurement of atmospheric temperature profiles by Raman backscatter,” J. Appl. Meteorol. 11(1), 108–112 (1972).
[Crossref]

1963 (1)

P. A. Bazhulin, “The study of rotation and vibration-rotation spectra of gases by the method of combination (Raman) scattering of light,” Sov. Phys. Usp. 5(4), 661–666 (1963).
[Crossref]

Achtert, P.

P. Achtert, M. Khaplanov, F. Khosrawi, and J. Gumbel, “Pure rotational-Raman channels of the Esrange lidar for temperature and particle extinction measurements in the troposphere and lower stratosphere,” Atmos. Meas. Tech. 6(1), 91–98 (2013).
[Crossref]

Alpers, M.

M. Alpers, R. Eixmann, C. Fricke-Begemann, M. Gerding, and J. Höffner, “Temperature lidar measurements from 1 to 105 km altitude using resonance, Rayleigh, and rotational Raman scattering,” Atmos. Chem. Phys. 4(3), 793–800 (2004).
[Crossref]

Arshinov, Y.

T. Dinoev, V. Simeonov, Y. Arshinov, S. Bobrovnikov, P. Ristori, B. Calpini, M. Parlange, and H. van den Bergh, “Raman lidar for meteorological observations, RALMO – Part 1: Instrument description,” Atmos. Meas. Tech. 6(5), 1329–1346 (2013).
[Crossref]

Arshinov, Y. F.

Baumgart, R.

Bazhulin, P. A.

P. A. Bazhulin, “The study of rotation and vibration-rotation spectra of gases by the method of combination (Raman) scattering of light,” Sov. Phys. Usp. 5(4), 661–666 (1963).
[Crossref]

Behrendt, A.

E. Hammann, A. Behrendt, F. Le Mounier, and V. Wulfmeyer, “Temperature profiling of the atmospheric boundary layer with rotational Raman lidar during the HD(CP)2 observational prototype experiment,” Atmos. Chem. Phys. 15(5), 2867–2881 (2015).
[Crossref]

A. Behrendt, V. Wulfmeyer, E. Hammann, S. K. Muppa, and S. Pal, “Profiles of second- to fourth-order moments of turbulent temperature fluctuations in the convective boundary layer: first measurements with rotational Raman lidar,” Atmos. Chem. Phys. 15(10), 5485–5500 (2015).
[Crossref]

E. Hammann and A. Behrendt, “Parametrization of optimum filter passbands for rotational Raman temperature measurements,” Opt. Express 23(24), 30767–30782 (2015).
[Crossref] [PubMed]

M. Radlach, A. Behrendt, and V. Wulfmeyer, “Scanning rotational Raman lidar at 355 nm for the measurement of tropospheric temperature fields,” Atmos. Chem. Phys. 8(2), 159–169 (2008).
[Crossref]

A. Behrendt, T. Nakamura, and T. Tsuda, “Combined temperature lidar for measurements in the troposphere, stratosphere, and mesosphere,” Appl. Opt. 43(14), 2930–2939 (2004).
[Crossref] [PubMed]

A. Behrendt, T. Nakamura, M. Onishi, R. Baumgart, and T. Tsuda, “Combined Raman lidar for the measurement of atmospheric temperature, water vapor, particle extinction coefficient, and particle backscatter coefficient,” Appl. Opt. 41(36), 7657–7666 (2002).
[Crossref] [PubMed]

A. Behrendt and J. Reichardt, “Atmospheric temperature profiling in the presence of clouds with a pure rotational Raman lidar by use of an interference-filter-based polychromator,” Appl. Opt. 39(9), 1372–1378 (2000).
[Crossref] [PubMed]

Bobronikov, S.

D. Kim, H. Cha, J. Lee, and S. Bobronikov, “Pure rotational Raman lidar for atmospheric temperature measurements,” J. Korean Phys. Soc. 39(5), 838–841 (2001).

Bobrovnikov, S.

T. Dinoev, V. Simeonov, Y. Arshinov, S. Bobrovnikov, P. Ristori, B. Calpini, M. Parlange, and H. van den Bergh, “Raman lidar for meteorological observations, RALMO – Part 1: Instrument description,” Atmos. Meas. Tech. 6(5), 1329–1346 (2013).
[Crossref]

Bobrovnikov, S. M.

Calpini, B.

T. Dinoev, V. Simeonov, Y. Arshinov, S. Bobrovnikov, P. Ristori, B. Calpini, M. Parlange, and H. van den Bergh, “Raman lidar for meteorological observations, RALMO – Part 1: Instrument description,” Atmos. Meas. Tech. 6(5), 1329–1346 (2013).
[Crossref]

Cha, H.

D. Kim, H. Cha, J. Lee, and S. Bobronikov, “Pure rotational Raman lidar for atmospheric temperature measurements,” J. Korean Phys. Soc. 39(5), 838–841 (2001).

Chanin, M. L.

D. Nedeljkovic, A. Hauchecorne, and M. L. Chanin, “Rotational Raman lidar to measure the atmospheric temperature from the ground to 30 km,” IEEE Trans. Geosci. Rem. Sens. 31(1), 90–101 (1993).
[Crossref]

Chen, H.

S. Chen, Z. Qiu, Y. Zhang, H. Chen, and Y. Wang, “A pure rotational Raman lidar using double-grating monochromator for temperature profile detection,” J. Quant. Spectrosc. Rad. Transf. 112(2), 304–309 (2011).
[Crossref]

Chen, S.

S. Chen, Z. Qiu, Y. Zhang, H. Chen, and Y. Wang, “A pure rotational Raman lidar using double-grating monochromator for temperature profile detection,” J. Quant. Spectrosc. Rad. Transf. 112(2), 304–309 (2011).
[Crossref]

Cooney, J.

J. Cooney, “Measurement of atmospheric temperature profiles by Raman backscatter,” J. Appl. Meteorol. 11(1), 108–112 (1972).
[Crossref]

Demoz, B. B.

P. Di Girolamo, R. Marchese, D. N. Whiteman, and B. B. Demoz, “Rotational Raman Lidar measurements of atmospheric temperature in the UV,” Geophys. Res. Lett. 31(1), L01106 (2004).
[Crossref]

Di Girolamo, P.

P. Di Girolamo, R. Marchese, D. N. Whiteman, and B. B. Demoz, “Rotational Raman Lidar measurements of atmospheric temperature in the UV,” Geophys. Res. Lett. 31(1), L01106 (2004).
[Crossref]

Dinoev, T.

T. Dinoev, V. Simeonov, Y. Arshinov, S. Bobrovnikov, P. Ristori, B. Calpini, M. Parlange, and H. van den Bergh, “Raman lidar for meteorological observations, RALMO – Part 1: Instrument description,” Atmos. Meas. Tech. 6(5), 1329–1346 (2013).
[Crossref]

Drayson, S. R.

S. R. Drayson, “Rapid computation of the Voigt profile,” J. Quantum Spectrosc. Radiat. Transf. 16(7), 611–614 (1976).
[Crossref]

Eixmann, R.

M. Alpers, R. Eixmann, C. Fricke-Begemann, M. Gerding, and J. Höffner, “Temperature lidar measurements from 1 to 105 km altitude using resonance, Rayleigh, and rotational Raman scattering,” Atmos. Chem. Phys. 4(3), 793–800 (2004).
[Crossref]

Fenner, W. R.

Fricke-Begemann, C.

M. Alpers, R. Eixmann, C. Fricke-Begemann, M. Gerding, and J. Höffner, “Temperature lidar measurements from 1 to 105 km altitude using resonance, Rayleigh, and rotational Raman scattering,” Atmos. Chem. Phys. 4(3), 793–800 (2004).
[Crossref]

Gao, F.

J. Mao, L. Hu, D. Hua, F. Gao, and M. Wu, “Pure rotational Raman lidar with fiber Bragg grating for temperature profiling of the atmospheric boundary layer,” Opt. Appl. 38(4), 715–726 (2008).

Gerding, M.

M. Alpers, R. Eixmann, C. Fricke-Begemann, M. Gerding, and J. Höffner, “Temperature lidar measurements from 1 to 105 km altitude using resonance, Rayleigh, and rotational Raman scattering,” Atmos. Chem. Phys. 4(3), 793–800 (2004).
[Crossref]

Ginzburg, V. L.

V. L. Ginzburg, “Line width in the spectrum of scattered light,” Sov. Phys. Usp. 15(1), 114–120 (1972).
[Crossref]

Goldsmith, J. E. M.

R. K. Newsom, D. D. Turner, and J. E. M. Goldsmith, “Long-term evaluation of temperature profiles measured by an operational Raman lidar,” J. Atmos. Ocean. Technol. 30(8), 1616–1634 (2013).
[Crossref]

Gumbel, J.

P. Achtert, M. Khaplanov, F. Khosrawi, and J. Gumbel, “Pure rotational-Raman channels of the Esrange lidar for temperature and particle extinction measurements in the troposphere and lower stratosphere,” Atmos. Meas. Tech. 6(1), 91–98 (2013).
[Crossref]

Gurdev, L. L.

I. D. Ivanova, L. L. Gurdev, and V. M. Mitev, “Lidar technique for simultaneous temperature and pressure measurement based on rotation Raman scattering,” J. Mod. Opt. 40(3), 367–371 (1993).
[Crossref]

Hammann, E.

E. Hammann and A. Behrendt, “Parametrization of optimum filter passbands for rotational Raman temperature measurements,” Opt. Express 23(24), 30767–30782 (2015).
[Crossref] [PubMed]

E. Hammann, A. Behrendt, F. Le Mounier, and V. Wulfmeyer, “Temperature profiling of the atmospheric boundary layer with rotational Raman lidar during the HD(CP)2 observational prototype experiment,” Atmos. Chem. Phys. 15(5), 2867–2881 (2015).
[Crossref]

A. Behrendt, V. Wulfmeyer, E. Hammann, S. K. Muppa, and S. Pal, “Profiles of second- to fourth-order moments of turbulent temperature fluctuations in the convective boundary layer: first measurements with rotational Raman lidar,” Atmos. Chem. Phys. 15(10), 5485–5500 (2015).
[Crossref]

Hauchecorne, A.

D. Nedeljkovic, A. Hauchecorne, and M. L. Chanin, “Rotational Raman lidar to measure the atmospheric temperature from the ground to 30 km,” IEEE Trans. Geosci. Rem. Sens. 31(1), 90–101 (1993).
[Crossref]

Höffner, J.

M. Alpers, R. Eixmann, C. Fricke-Begemann, M. Gerding, and J. Höffner, “Temperature lidar measurements from 1 to 105 km altitude using resonance, Rayleigh, and rotational Raman scattering,” Atmos. Chem. Phys. 4(3), 793–800 (2004).
[Crossref]

Hu, L.

J. Mao, L. Hu, D. Hua, F. Gao, and M. Wu, “Pure rotational Raman lidar with fiber Bragg grating for temperature profiling of the atmospheric boundary layer,” Opt. Appl. 38(4), 715–726 (2008).

Hua, D.

J. Mao, L. Hu, D. Hua, F. Gao, and M. Wu, “Pure rotational Raman lidar with fiber Bragg grating for temperature profiling of the atmospheric boundary layer,” Opt. Appl. 38(4), 715–726 (2008).

Hyatt, H. A.

Ivanova, I. D.

I. D. Ivanova, L. L. Gurdev, and V. M. Mitev, “Lidar technique for simultaneous temperature and pressure measurement based on rotation Raman scattering,” J. Mod. Opt. 40(3), 367–371 (1993).
[Crossref]

Jia, J.

Kellam, J. M.

Khaplanov, M.

P. Achtert, M. Khaplanov, F. Khosrawi, and J. Gumbel, “Pure rotational-Raman channels of the Esrange lidar for temperature and particle extinction measurements in the troposphere and lower stratosphere,” Atmos. Meas. Tech. 6(1), 91–98 (2013).
[Crossref]

Khosrawi, F.

P. Achtert, M. Khaplanov, F. Khosrawi, and J. Gumbel, “Pure rotational-Raman channels of the Esrange lidar for temperature and particle extinction measurements in the troposphere and lower stratosphere,” Atmos. Meas. Tech. 6(1), 91–98 (2013).
[Crossref]

Kim, D.

D. Kim, H. Cha, J. Lee, and S. Bobronikov, “Pure rotational Raman lidar for atmospheric temperature measurements,” J. Korean Phys. Soc. 39(5), 838–841 (2001).

Lapp, M.

Le Mounier, F.

E. Hammann, A. Behrendt, F. Le Mounier, and V. Wulfmeyer, “Temperature profiling of the atmospheric boundary layer with rotational Raman lidar during the HD(CP)2 observational prototype experiment,” Atmos. Chem. Phys. 15(5), 2867–2881 (2015).
[Crossref]

Lee, J.

D. Kim, H. Cha, J. Lee, and S. Bobronikov, “Pure rotational Raman lidar for atmospheric temperature measurements,” J. Korean Phys. Soc. 39(5), 838–841 (2001).

Liu, F.

Longbothum, R. L.

J. J. Olivero and R. L. Longbothum, “Empirical fits to the Voigt line width: a brief review,” J. Quantum Spectrosc. Radiat. Transf. 17(2), 233–236 (1977).
[Crossref]

Mao, J.

J. Mao, L. Hu, D. Hua, F. Gao, and M. Wu, “Pure rotational Raman lidar with fiber Bragg grating for temperature profiling of the atmospheric boundary layer,” Opt. Appl. 38(4), 715–726 (2008).

Marchese, R.

P. Di Girolamo, R. Marchese, D. N. Whiteman, and B. B. Demoz, “Rotational Raman Lidar measurements of atmospheric temperature in the UV,” Geophys. Res. Lett. 31(1), L01106 (2004).
[Crossref]

Matveev, V. S.

V. S. Matveev, “Approximate representations of absorption coefficient end equivalent widths of lines with Voigt profile,” J. Appl. Spectrosc. 16(2), 228–233 (1972).

McFarlane, S. A.

R. K. Newsom, C. Sivaraman, and S. A. McFarlane, “Raman lidar profiles–temperature (RLPROFTEMP) value-added product,” https://www.arm.gov/publications/tech_reports/doe-sc-arm-tr-120.pdf .

Mitev, V. M.

I. D. Ivanova, L. L. Gurdev, and V. M. Mitev, “Lidar technique for simultaneous temperature and pressure measurement based on rotation Raman scattering,” J. Mod. Opt. 40(3), 367–371 (1993).
[Crossref]

Y. F. Arshinov, S. M. Bobrovnikov, V. E. Zuev, and V. M. Mitev, “Atmospheric temperature measurements using a pure rotational Raman lidar,” Appl. Opt. 22(19), 2984–2990 (1983).
[Crossref] [PubMed]

Muppa, S. K.

A. Behrendt, V. Wulfmeyer, E. Hammann, S. K. Muppa, and S. Pal, “Profiles of second- to fourth-order moments of turbulent temperature fluctuations in the convective boundary layer: first measurements with rotational Raman lidar,” Atmos. Chem. Phys. 15(10), 5485–5500 (2015).
[Crossref]

Nakamura, T.

Nedeljkovic, D.

D. Nedeljkovic, A. Hauchecorne, and M. L. Chanin, “Rotational Raman lidar to measure the atmospheric temperature from the ground to 30 km,” IEEE Trans. Geosci. Rem. Sens. 31(1), 90–101 (1993).
[Crossref]

Newsom, R. K.

R. K. Newsom, D. D. Turner, and J. E. M. Goldsmith, “Long-term evaluation of temperature profiles measured by an operational Raman lidar,” J. Atmos. Ocean. Technol. 30(8), 1616–1634 (2013).
[Crossref]

R. K. Newsom, C. Sivaraman, and S. A. McFarlane, “Raman lidar profiles–temperature (RLPROFTEMP) value-added product,” https://www.arm.gov/publications/tech_reports/doe-sc-arm-tr-120.pdf .

Olivero, J. J.

J. J. Olivero and R. L. Longbothum, “Empirical fits to the Voigt line width: a brief review,” J. Quantum Spectrosc. Radiat. Transf. 17(2), 233–236 (1977).
[Crossref]

Onishi, M.

Pal, S.

A. Behrendt, V. Wulfmeyer, E. Hammann, S. K. Muppa, and S. Pal, “Profiles of second- to fourth-order moments of turbulent temperature fluctuations in the convective boundary layer: first measurements with rotational Raman lidar,” Atmos. Chem. Phys. 15(10), 5485–5500 (2015).
[Crossref]

Parlange, M.

T. Dinoev, V. Simeonov, Y. Arshinov, S. Bobrovnikov, P. Ristori, B. Calpini, M. Parlange, and H. van den Bergh, “Raman lidar for meteorological observations, RALMO – Part 1: Instrument description,” Atmos. Meas. Tech. 6(5), 1329–1346 (2013).
[Crossref]

Penney, M.

Porto, S. P. S.

Qiu, Z.

S. Chen, Z. Qiu, Y. Zhang, H. Chen, and Y. Wang, “A pure rotational Raman lidar using double-grating monochromator for temperature profile detection,” J. Quant. Spectrosc. Rad. Transf. 112(2), 304–309 (2011).
[Crossref]

Radlach, M.

M. Radlach, A. Behrendt, and V. Wulfmeyer, “Scanning rotational Raman lidar at 355 nm for the measurement of tropospheric temperature fields,” Atmos. Chem. Phys. 8(2), 159–169 (2008).
[Crossref]

Reichardt, J.

Ristori, P.

T. Dinoev, V. Simeonov, Y. Arshinov, S. Bobrovnikov, P. Ristori, B. Calpini, M. Parlange, and H. van den Bergh, “Raman lidar for meteorological observations, RALMO – Part 1: Instrument description,” Atmos. Meas. Tech. 6(5), 1329–1346 (2013).
[Crossref]

She, C.-Y.

Simeonov, V.

T. Dinoev, V. Simeonov, Y. Arshinov, S. Bobrovnikov, P. Ristori, B. Calpini, M. Parlange, and H. van den Bergh, “Raman lidar for meteorological observations, RALMO – Part 1: Instrument description,” Atmos. Meas. Tech. 6(5), 1329–1346 (2013).
[Crossref]

Sivaraman, C.

R. K. Newsom, C. Sivaraman, and S. A. McFarlane, “Raman lidar profiles–temperature (RLPROFTEMP) value-added product,” https://www.arm.gov/publications/tech_reports/doe-sc-arm-tr-120.pdf .

St. Peters, R. L.

Tsuda, T.

Turner, D. D.

R. K. Newsom, D. D. Turner, and J. E. M. Goldsmith, “Long-term evaluation of temperature profiles measured by an operational Raman lidar,” J. Atmos. Ocean. Technol. 30(8), 1616–1634 (2013).
[Crossref]

van den Bergh, H.

T. Dinoev, V. Simeonov, Y. Arshinov, S. Bobrovnikov, P. Ristori, B. Calpini, M. Parlange, and H. van den Bergh, “Raman lidar for meteorological observations, RALMO – Part 1: Instrument description,” Atmos. Meas. Tech. 6(5), 1329–1346 (2013).
[Crossref]

Wang, Y.

S. Chen, Z. Qiu, Y. Zhang, H. Chen, and Y. Wang, “A pure rotational Raman lidar using double-grating monochromator for temperature profile detection,” J. Quant. Spectrosc. Rad. Transf. 112(2), 304–309 (2011).
[Crossref]

Whiteman, D. N.

P. Di Girolamo, R. Marchese, D. N. Whiteman, and B. B. Demoz, “Rotational Raman Lidar measurements of atmospheric temperature in the UV,” Geophys. Res. Lett. 31(1), L01106 (2004).
[Crossref]

Wu, M.

J. Mao, L. Hu, D. Hua, F. Gao, and M. Wu, “Pure rotational Raman lidar with fiber Bragg grating for temperature profiling of the atmospheric boundary layer,” Opt. Appl. 38(4), 715–726 (2008).

Wulfmeyer, V.

A. Behrendt, V. Wulfmeyer, E. Hammann, S. K. Muppa, and S. Pal, “Profiles of second- to fourth-order moments of turbulent temperature fluctuations in the convective boundary layer: first measurements with rotational Raman lidar,” Atmos. Chem. Phys. 15(10), 5485–5500 (2015).
[Crossref]

E. Hammann, A. Behrendt, F. Le Mounier, and V. Wulfmeyer, “Temperature profiling of the atmospheric boundary layer with rotational Raman lidar during the HD(CP)2 observational prototype experiment,” Atmos. Chem. Phys. 15(5), 2867–2881 (2015).
[Crossref]

M. Radlach, A. Behrendt, and V. Wulfmeyer, “Scanning rotational Raman lidar at 355 nm for the measurement of tropospheric temperature fields,” Atmos. Chem. Phys. 8(2), 159–169 (2008).
[Crossref]

Yi, F.

Zhang, Y.

S. Chen, Z. Qiu, Y. Zhang, H. Chen, and Y. Wang, “A pure rotational Raman lidar using double-grating monochromator for temperature profile detection,” J. Quant. Spectrosc. Rad. Transf. 112(2), 304–309 (2011).
[Crossref]

Zuev, V. E.

Appl. Opt. (6)

Atmos. Chem. Phys. (4)

M. Alpers, R. Eixmann, C. Fricke-Begemann, M. Gerding, and J. Höffner, “Temperature lidar measurements from 1 to 105 km altitude using resonance, Rayleigh, and rotational Raman scattering,” Atmos. Chem. Phys. 4(3), 793–800 (2004).
[Crossref]

M. Radlach, A. Behrendt, and V. Wulfmeyer, “Scanning rotational Raman lidar at 355 nm for the measurement of tropospheric temperature fields,” Atmos. Chem. Phys. 8(2), 159–169 (2008).
[Crossref]

A. Behrendt, V. Wulfmeyer, E. Hammann, S. K. Muppa, and S. Pal, “Profiles of second- to fourth-order moments of turbulent temperature fluctuations in the convective boundary layer: first measurements with rotational Raman lidar,” Atmos. Chem. Phys. 15(10), 5485–5500 (2015).
[Crossref]

E. Hammann, A. Behrendt, F. Le Mounier, and V. Wulfmeyer, “Temperature profiling of the atmospheric boundary layer with rotational Raman lidar during the HD(CP)2 observational prototype experiment,” Atmos. Chem. Phys. 15(5), 2867–2881 (2015).
[Crossref]

Atmos. Meas. Tech. (2)

T. Dinoev, V. Simeonov, Y. Arshinov, S. Bobrovnikov, P. Ristori, B. Calpini, M. Parlange, and H. van den Bergh, “Raman lidar for meteorological observations, RALMO – Part 1: Instrument description,” Atmos. Meas. Tech. 6(5), 1329–1346 (2013).
[Crossref]

P. Achtert, M. Khaplanov, F. Khosrawi, and J. Gumbel, “Pure rotational-Raman channels of the Esrange lidar for temperature and particle extinction measurements in the troposphere and lower stratosphere,” Atmos. Meas. Tech. 6(1), 91–98 (2013).
[Crossref]

Geophys. Res. Lett. (1)

P. Di Girolamo, R. Marchese, D. N. Whiteman, and B. B. Demoz, “Rotational Raman Lidar measurements of atmospheric temperature in the UV,” Geophys. Res. Lett. 31(1), L01106 (2004).
[Crossref]

IEEE Trans. Geosci. Rem. Sens. (1)

D. Nedeljkovic, A. Hauchecorne, and M. L. Chanin, “Rotational Raman lidar to measure the atmospheric temperature from the ground to 30 km,” IEEE Trans. Geosci. Rem. Sens. 31(1), 90–101 (1993).
[Crossref]

J. Appl. Meteorol. (1)

J. Cooney, “Measurement of atmospheric temperature profiles by Raman backscatter,” J. Appl. Meteorol. 11(1), 108–112 (1972).
[Crossref]

J. Appl. Spectrosc. (1)

V. S. Matveev, “Approximate representations of absorption coefficient end equivalent widths of lines with Voigt profile,” J. Appl. Spectrosc. 16(2), 228–233 (1972).

J. Atmos. Ocean. Technol. (1)

R. K. Newsom, D. D. Turner, and J. E. M. Goldsmith, “Long-term evaluation of temperature profiles measured by an operational Raman lidar,” J. Atmos. Ocean. Technol. 30(8), 1616–1634 (2013).
[Crossref]

J. Korean Phys. Soc. (1)

D. Kim, H. Cha, J. Lee, and S. Bobronikov, “Pure rotational Raman lidar for atmospheric temperature measurements,” J. Korean Phys. Soc. 39(5), 838–841 (2001).

J. Mod. Opt. (1)

I. D. Ivanova, L. L. Gurdev, and V. M. Mitev, “Lidar technique for simultaneous temperature and pressure measurement based on rotation Raman scattering,” J. Mod. Opt. 40(3), 367–371 (1993).
[Crossref]

J. Opt. Soc. Am. (2)

J. Quant. Spectrosc. Rad. Transf. (1)

S. Chen, Z. Qiu, Y. Zhang, H. Chen, and Y. Wang, “A pure rotational Raman lidar using double-grating monochromator for temperature profile detection,” J. Quant. Spectrosc. Rad. Transf. 112(2), 304–309 (2011).
[Crossref]

J. Quantum Spectrosc. Radiat. Transf. (2)

J. J. Olivero and R. L. Longbothum, “Empirical fits to the Voigt line width: a brief review,” J. Quantum Spectrosc. Radiat. Transf. 17(2), 233–236 (1977).
[Crossref]

S. R. Drayson, “Rapid computation of the Voigt profile,” J. Quantum Spectrosc. Radiat. Transf. 16(7), 611–614 (1976).
[Crossref]

Opt. Appl. (1)

J. Mao, L. Hu, D. Hua, F. Gao, and M. Wu, “Pure rotational Raman lidar with fiber Bragg grating for temperature profiling of the atmospheric boundary layer,” Opt. Appl. 38(4), 715–726 (2008).

Opt. Express (2)

Sov. Phys. Usp. (2)

P. A. Bazhulin, “The study of rotation and vibration-rotation spectra of gases by the method of combination (Raman) scattering of light,” Sov. Phys. Usp. 5(4), 661–666 (1963).
[Crossref]

V. L. Ginzburg, “Line width in the spectrum of scattered light,” Sov. Phys. Usp. 15(1), 114–120 (1972).
[Crossref]

Other (9)

H. B. Dwight, Tables of Integrals and Other Mathematical Data, 4th ed. (The Macmillan Company, 1961).

Y. I. Gerasimov, Course in Physical Chemistry, Vol. 2 (Khimiya, 1973).

R. K. Newsom, C. Sivaraman, and S. A. McFarlane, “Raman lidar profiles–temperature (RLPROFTEMP) value-added product,” https://www.arm.gov/publications/tech_reports/doe-sc-arm-tr-120.pdf .

U. Wandinger, “Raman lidar,” in Lidar Range-Resolved Optical Remote Sensing of the Atmosphere (C. Weitkamp, 2005), Ch. 9.

R. M. Measures, Laser Remote Sensing, Fundamentals and Applications (Wiley, 1984).

R. B. Lee III, “Tropospheric temperature measurements using a rotational Raman lidar,” doctoral dissertation (Hampton University, 2013), http://gradworks.umi.com/35/92/3592881.html .

M. Radlach, “A scanning eye-safe rotational Raman lidar in the ultraviolet for measurements of tropospheric temperature fields,” doctoral dissertation (University of Hohenheim, 2009), https://opus.uni-hohenheim.de/volltexte/2009/345/pdf/Radlach_doctoral_thesis_engl_LINKS.pdf .

G. Placzek, Rayleigh-Streuung und Raman effekt,” in Handbuch der Radiologie, E. Marx, ed. (Akademischer Verlag, 1934).

A. Behrendt, “Temperature measurement with Lidar,” in Lidar Range-Resolved Optical Remote Sensing of the Atmosphere (C. Weitkamp, 2005), Ch. 10.

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Figures (5)

Fig. 1
Fig. 1 a) The equidistant PRR spectra of N2 and O2 linear molecules. The index over a spectral line denotes the rotational quantum number J of the initial state of the transition. The spectral line number and number J are the same for the Stokes branch. b) A schematic drawing of gratings transmission functions (GTF) and envelopes of N2 PRR spectrum at different temperatures. The gratings transmission bands (GTB) are under the corresponding GTFs.
Fig. 2
Fig. 2 a) A schematic drawing (not to scale) of the filter or grating transmission functions (dashed), PRR channels transmission functions approximated by unity (solid), and broadened PRR lines profiles (color). b) The transmission function approximated by a piecewise-constant (staircase) function.
Fig. 3
Fig. 3 a) The U.S. Standard tropospheric temperature profile. b) The FWHMs of the Gaussian, Lorentzian, and Voigt profiles of N2 broadened PRR line (with J = 6 of the anti-Stokes branch).
Fig. 4
Fig. 4 The calculated dependences of altitude (a) and temperature (b) on the intensity ratio Q.
Fig. 5
Fig. 5 A comparative analysis of temperature errors yielded by using the different temperature retrieval functions.

Tables (1)

Tables Icon

Table 1 Calibration coefficients for temperature retrieval functions.

Equations (57)

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Q indiv. (T)= I ASt ( J low ,T) I ASt ( J high ,T) =exp( α+ β T ),
Q Σ (T)= I low Σ (T) I high Σ (T) = [ J N 2 I N 2 ( J N 2 ,T) + J O 2 I O 2 ( J O 2 ,T) ] low [ J N 2 I N 2 ( J N 2 ,T) + J O 2 I O 2 ( J O 2 ,T) ] high ,
Q Σ (T) f c Σ (T)=exp( A+ B T )orln f c Σ (T)=A+ B T y=A+Bx,
ln Q Σ ln f c Σ (T)=A+ B T + C T 2 y=A+Bx+C x 2 ,
ln Q Σ ln f c Σ (T)=A+ B T + C T 2 + D T 3 y=A+Bx+C x 2 +D x 3
ln Q Σ ln f c Σ (T)=A+ B T + C T 2 + D T 3 + E T 4 +y=A+Bx+C x 2 +D x 3 +E x 4 +
Q all (T)= I low all (T) I high all (T) = i= N 2 ,O 2 J i [ I i ( J i ,T) X low i ( ν ˜ low ν ˜ i ,T) ] i= N 2 ,O 2 J i [ I i ( J i ,T) X high i ( ν ˜ high ν ˜ i ,T) ] .
X low i ( ν ˜ low ν ˜ i ,T)= + F( ν ˜ low , ν ˜ )V( ν ˜ i , ν ˜ ,T)d ν ˜ ,
X high i ( ν ˜ high ν ˜ i ,T)= + F( ν ˜ high , ν ˜ )V( ν ˜ i , ν ˜ ,T)d ν ˜ .
X low i (T)= X low i ( ν ˜ 1 , ν ˜ 2 ,T)= ν ˜ 1 ν ˜ 2 L( ν ˜ i , ν ˜ ,T)d ν ˜ , X high i (T)= X high i ( ν ˜ 3 , ν ˜ 4 ,T)= ν ˜ 3 ν ˜ 4 L( ν ˜ i , ν ˜ ,T)d ν ˜ .
I low all (T)= I low Σ (T)[ 1 X low DSL (T) ]+ k, J k [ I k ( J k ,T) X low k (T) ] ,
I low all (T)= I low Σ (T){ 1 X low DSL (T)+ k, J k [ I k ( J k ,T) I low Σ (T) X low k (T) ] } = I low Σ (T)[ 1 X low DSL (T)+ k Y low k (T) ]= I low Σ (T)[ 1 X low DSL (T)+ Y low PS (T) ],
I high all (T)= I high Σ (T){ 1 X high DSL (T)+ k, J k [ I k ( J k ,T) I high Σ (T) X high k (T) ] } = I high Σ (T)[ 1 X high DSL (T)+ k Y high k (T) ]= I high Σ (T)[ 1 X high DSL (T)+ Y high PS (T) ].
Q all (T) f c all (T)= I low all (T) I high all (T) = I low Σ (T) I high Σ (T) [ 1 X low DSL (T)+ Y low PS (T) ] [ 1 X high DSL (T)+ Y high PS (T) ] .
Q all (T) f c all (T)= f c Σ (T)[ 1 X low DSL (T)+ Y low PS (T) ][ 1+ X high DSL (T) Y high PS (T) ] =exp( A+ B T + C T 2 + )[ 1+ X high DSL (T) X low DSL (T)+ Y low PS (T) Y high PS (T) ].
X low DSL (T)= 1 π + γ L (x ν ˜ low ) 2 + γ L 2 dx 1 π ν ˜ 1 ν ˜ 2 γ L (x ν ˜ low ) 2 + γ L 2 dx =1 1 π [ arctan( ν ˜ 2 ν ˜ low γ L )arctan( ν ˜ 1 ν ˜ low γ L ) ].
arctan(z)= π 2 1 z + 1 3 z 3 1 5 z 5 + 1 7 z 7 withz>1.
arctan( ν ˜ ν ˜ low γ L )= π 2 γ L ν ˜ ν ˜ low + 1 3 ( γ L ν ˜ ν ˜ low ) 3 1 3 ( γ L ν ˜ ν ˜ low ) 5 +.
X low DSL (T)=1 1 π [ ( 1 Δ ν ˜ 2 1 Δ ν ˜ 1 ) γ L + 1 3 ( 1 Δ ν ˜ 2 3 1 Δ ν ˜ 1 3 ) γ L 3 ] =1( a 1 γ L + a 2 γ L 3 + a 3 γ L 5 + a 4 γ L 7 +),
X low DSL (T)=1[ a 1 ( A T + B T )+ a 2 ( A T + B T ) 3 + a 3 ( A T + B T ) 5 + ] =+ b 2 T T + b 1 T +1+ b 1 T + b 2 T T +=1+ n=1 ( b n T n1/2 + b n T n1/2 ) ,
X high DSL (T)=1+ n=1 ( c n T n1/2 + c n T n1/2 ) ,
X high DSL (T) X low DSL (T)= n=1 ( d n T n1/2 + d n T n1/2 ) ,
I k ( J k ,T) I low Σ (T) =exp( C k + E k T )=exp( C k )exp( E k T )= D k exp( E k T ),
X low k (T)= n=1 ( f n k T n1/2 + f n k T n1/2 ) ,
Y low PS (T)= k Y low k (T) = k, J k [ I k ( J k ,T) I low Σ (T) X low k (T) ] = k [ D k ( 1+ E k T + E k 2 2 T 2 + ) n=1 ( f n k T n1/2 + f n k T n1/2 ) ] = n=1 ( g n T n1/2 + g n T n1/2 ) ,
Y high PS (T)= n=1 ( h n T n1/2 + h n T n1/2 ) ,
Z(T)= X high DSL (T) X low DSL (T)+ Y low PS (T) Y high PS (T)= n=1 ( l n T n1/2 + l n T n1/2 ) .
ln Q all (T)ln f c all (T)=A+ B T + C T 2 + D T 3 ++ln[ 1+Z(T) ].
ln(1+z)=z z 2 2 + z 3 3 z 4 4 +with1<z1.
ln[ 1+Z(T) ]= n=1 ( l n T n1/2 + l n T n1/2 ) 1 2 [ n=1 ( l n T n1/2 + l n T n1/2 ) ] 2 + =+ m 4 T 2 + m 3 T T + m 2 T + m 1 T + m 0 + m 1 T + m 2 T+ m 3 T T + m 4 T 2 +,
ln Q all (T)+ C+ m 4 T 2 + m 3 T T + B+ m 2 T + m 1 T +(A+ m 0 )+ m 1 T + m 2 T+ =+ A 4 T 2 + A 3 T T + A 2 T + A 1 T + A 0 + A 1 T + A 2 T+ A 3 T T += n= A n T n 2 ,
lnQ(T)+ B 2 T 2 + B 1 T + B 0 + B 1 T+ B 2 T 2 += n= B n T n ,
T= B 0 lnQ A 0 ,
T= 2 C 1 B 1 + B 1 2 +4 C 1 ( lnQ A 1 ) ,
lnQ= A 2 + B 2 T + C 2 Ty= A 2 + B 2 x+ C 2 x ,
T= 2 B 2 (lnQ A 2 )+ (lnQ A 2 ) 2 4 B 2 C 2 .
x=a+by+c y 2 .
1/T =a+blnQ+c (lnQ) 2 T= [ c (lnQ) 2 +blnQ+a ] 1 ,
T= C 3 (lnQ) 2 + B 3 lnQ+ A 3 ,
x= A 4 + B 4 y+ C 4 /y 1/T = A 4 + B 4 lnQ+ C 4 / lnQ ,
T= 1 A 4 + B 4 lnQ+( C 4 / lnQ ) = lnQ B 4 (lnQ) 2 + A 4 lnQ+ C 4 .
G( ν ˜ i , ν ˜ ,T)= 1 γ G i 2π exp[ ( ν ˜ μ i ) 2 2 ( γ G i ) 2 ].
γ G i = ν ˜ i k B T m air c 2 ,
Δ ν ˜ i,G FWHM =2 2ln2 γ G i .
L( ν ˜ i , ν ˜ ,T)= 1 π γ L ( ν ˜ ν ˜ i ) 2 + γ L 2 ,
γ L = p 1 2 n air π d 1 2 8kT π μ 1 c 2 +2 p 1 p 2 n air π d 12 2 8kT π μ 12 c 2 + p 2 2 n air π d 2 2 8kT π μ 2 c 2 ,
d i 2 (T)= d i, 2 ( 1+ φ i RT ),
γ L =A T + B T ,
V( ν ˜ i , ν ˜ ,T)= G(T, ν ˜ i , ν ˜ *)L(T, ν ˜ , ν ˜ *)d ν ˜ * .
V( ν ˜ i , ν ˜ ,T)= u 0 y π exp( t 2 )dt (xt) 2 + y 2 ,
u 0 = 2 Δ ν ˜ i,G FWHM ln2 π ,y= Δ ν ˜ L FWHM Δ ν ˜ i,G FWHM ln2 ,x= ν ˜ ν ˜ i Δ ν ˜ i,G FWHM 2 ln2 .
Δ ν ˜ i,V FWHM 0.5346Δ ν ˜ L FWHM + (Δ ν ˜ i,G FWHM ) 2 +0.2166 (Δ ν ˜ L FWHM ) 2 .
lnQ= B 3 2 ± ( B 3 2 ) 2 A 3 + C 3 T y= B 3 2 ± ( B 3 2 ) 2 A 3 + C 3 x .
y= B 3 2 ± | C 3 | B 3 2 4 C 3 A 3 C 3 +x =α+β γ+x =α+β | γ | 1+ x γ .
lnQ= A 0 + A 1 T .
1+z =1+ 1 2 z 1 8 z 2 + 1 16 z 3 with z 2 1.
lnQ=α+β | γ | ( 1+ x 2γ x 2 8 γ 2 + x 3 16 γ 3 )= B 0 + B 1 T + B 2 T 2 +,

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