Analytical calibration functions for the pure rotational Raman lidar technique

Vladislav V. Gerasimov; Vladimir V. Zuev

doi:10.1364/OE.24.005136

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 J_low and J_high) are of opposite temperature dependence, as shown in Fig. 1(b). The intensity of lines with J_low decreases with increasing temperature and, conversely, the intensity of lines with J_high 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).

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 J_low and J_high, e.g., from the anti-Stokes branch of the PRR spectrum, gives the exact temperature dependence

Q^{indiv .} (T) = \frac{I^{ASt} (J_{low}, T)}{I^{ASt} (J_{high}, T)} = \exp (α + \frac{β}{T}),

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

Q^{Σ} (T) = \frac{I_{low}^{Σ} (T)}{I_{high}^{Σ} (T)} = \frac{{[\sum_{J_{N_{2}}} I_{N_{2}} (J_{N_{2}}, T) + \sum_{J_{O_{2}}} I_{O_{2}} (J_{O_{2}}, T)]}_{low}}{{[\sum_{J_{N_{2}}} I_{N_{2}} (J_{N_{2}}, T) + \sum_{J_{O_{2}}} I_{O_{2}} (J_{O_{2}}, T)]}_{high}},

where

I_{N_{2}} (J_{N_{2}}, T)

and

I_{O_{2}} (J_{O_{2}}, T)

are the intensities of N₂ and O₂ individual PRR lines, respectively;

I_{low}^{Σ} (T)

and

I_{high}^{Σ} (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 J_low and J_high. 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 $f_{c}^{Σ} (T)$ (also called calibration function) for the ratio Q^Σ(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

Q^{Σ} (T) \approx f_{c}^{Σ} (T) = \exp (A + \frac{B}{T}) or \ln f_{c}^{Σ} (T) = A + \frac{B}{T} \Leftrightarrow y = A + B x,

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]

\ln Q^{Σ} \approx \ln f_{c}^{Σ} (T) = A + \frac{B}{T} + \frac{C}{T^{2}} \Leftrightarrow y = A + B x + C x^{2},

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],

\ln Q^{Σ} \approx \ln f_{c}^{Σ} (T) = A + \frac{B}{T} + \frac{C}{T^{2}} + \frac{D}{T^{3}} \Leftrightarrow y = A + B x + C x^{2} + D x^{3}

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

\ln Q^{Σ} \approx \ln f_{c}^{Σ} (T) = A + \frac{B}{T} + \frac{C}{T^{2}} + \frac{D}{T^{3}} + \frac{E}{T^{4}} + \dots \Leftrightarrow y = A + B x + C x^{2} + D x^{3} + E x^{4} + \dots

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 J_low and J_high 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 J_low and J_high from the backscattered light spectrum. The wavenumbers ${\tilde{ν}}_{low}$ and ${\tilde{ν}}_{high}$ 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

Q^{all} (T) = \frac{I_{low}^{all} (T)}{I_{high}^{all} (T)} = \frac{\sum_{i = N_{2} {,O}_{2}} \sum_{J_{i}} [I_{i} (J_{i}, T) \cdot X_{low}^{i} ({\tilde{ν}}_{low} - {\tilde{ν}}_{i}, T)]}{\sum_{i = N_{2} {,O}_{2}} \sum_{J_{i}} [I_{i} (J_{i}, T) \cdot X_{high}^{i} ({\tilde{ν}}_{high} - {\tilde{ν}}_{i}, T)]} .

Here

{\tilde{ν}}_{i} = \tilde{ν} (J_{i})

is the wavenumber of i PRR line with J_i;

I_{low}^{all} (T)

and

I_{high}^{all} (T)

are the overall intensities detected in the corresponding PRR channels with allowance for collisional broadening of all PRR lines; the functions

X_{low}^{i}

and

X_{high}^{i}

describe the parts of i PRR line signal falling within the PRR channels bands

{\tilde{ν}}_{low}

and

{\tilde{ν}}_{high}

, respectively. These functions depend on the distance between the bands central lines

{\tilde{ν}}_{low}

and

{\tilde{ν}}_{high}

and i PRR line

{\tilde{ν}}_{i}

. The function

X_{low}^{i}

can be written as follows

X_{low}^{i} ({\tilde{ν}}_{low} - {\tilde{ν}}_{i}, T) = \int_{- \infty}^{+ \infty} F ({\tilde{ν}}_{low}, \tilde{ν}) V ({\tilde{ν}}_{i}, \tilde{ν}, T) d \tilde{ν},

where

V ({\tilde{ν}}_{i}, \tilde{ν}, T)

is the Voigt profile of i broadened PRR line shape, and

F ({\tilde{ν}}_{low}, \tilde{ν})

is the filter or grating transmission function of the PRR channel

{\tilde{ν}}_{low}

, as shown in Fig. 2(a). Similarly, for the function

X_{high}^{i}

we have

X_{high}^{i} ({\tilde{ν}}_{high} - {\tilde{ν}}_{i}, T) = \int_{- \infty}^{+ \infty} F ({\tilde{ν}}_{high}, \tilde{ν}) V ({\tilde{ν}}_{i}, \tilde{ν}, T) d \tilde{ν} .

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).

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 $f_{c}^{all} (T)$ for the ratio Q^all(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 $L ({\tilde{ν}}_{i}, \tilde{ν}, T)$ for a PRR line shape description instead of $V ({\tilde{ν}}_{i}, \tilde{ν}, T)$ , 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 ${\tilde{ν}}_{1}$ to ${\tilde{ν}}_{2}$ , i.e. $F ({\tilde{ν}}_{low}, \tilde{ν}) = F ({\tilde{ν}}_{1}, {\tilde{ν}}_{2}) = F$ . Moreover, for the same reason, we can put F = 1 without loss of generality. The interval ${\tilde{ν}}_{1}$ to ${\tilde{ν}}_{2}$ is responsible for transmission of the bulk of the backscattered signal intensity in PRR channel ${\tilde{ν}}_{low}$ . The same is valid for $F ({\tilde{ν}}_{high}, \tilde{ν})$ with the transmission band ${\tilde{ν}}_{3}$ to ${\tilde{ν}}_{4}$ , as shown in Fig. 2(a). Thus, instead of Eqs. (8) and (9) we can write

\begin{array}{l} X_{low}^{i} (T) = X_{low}^{i} ({\tilde{ν}}_{1}, {\tilde{ν}}_{2}, T) = \int_{{\tilde{ν}}_{1}}^{{\tilde{ν}}_{2}} L ({\tilde{ν}}_{i}, \tilde{ν}, T) d \tilde{ν}, \\ X_{high}^{i} (T) = X_{high}^{i} ({\tilde{ν}}_{3}, {\tilde{ν}}_{4}, T) = \int_{{\tilde{ν}}_{3}}^{{\tilde{ν}}_{4}} L ({\tilde{ν}}_{i}, \tilde{ν}, T) d \tilde{ν} . \end{array}

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

I_{low}^{all} (T) = I_{low}^{Σ} (T) \cdot [1 - X_{low}^{DSL} (T)] + \sum_{k, J_{k}} [I_{k} (J_{k}, T) \cdot X_{low}^{k} (T)],

where

X_{low}^{DSL} (T)

describes the signal falling outside the band

{\tilde{ν}}_{1}

to

{\tilde{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

\begin{array}{l} I_{low}^{all} (T) = I_{low}^{Σ} (T) {1 - X_{low}^{DSL} (T) + \sum_{k, J_{k}} [\frac{I_{k} (J_{k}, T)}{I_{low}^{Σ} (T)} \cdot X_{low}^{k} (T)]} \\ = I_{low}^{Σ} (T) \cdot [1 - X_{low}^{DSL} (T) + \sum_{k} Y_{low}^{k} (T)] = I_{low}^{Σ} (T) \cdot [1 - X_{low}^{DSL} (T) + Y_{low}^{PS} (T)], \end{array}

where

Y_{low}^{k} (T)

and

Y_{low}^{PS} (T)

are defined by the given equations. Similarly, for the intensity

I_{high}^{all} (T)

we obtain

\begin{array}{l} I_{high}^{all} (T) = I_{high}^{Σ} (T) {1 - X_{high}^{DSL} (T) + \sum_{k, J_{k}} [\frac{I_{k} (J_{k}, T)}{I_{high}^{Σ} (T)} \cdot X_{high}^{k} (T)]} \\ = I_{high}^{Σ} (T) \cdot [1 - X_{high}^{DSL} (T) + \sum_{k} Y_{high}^{k} (T)] = I_{high}^{Σ} (T) \cdot [1 - X_{high}^{DSL} (T) + Y_{high}^{PS} (T)] . \end{array}

Substituting Eqs. (12) and (13) into Eq. (7), we obtain for the intensity ratio

Q^{all} (T) \approx f_{c}^{all} (T) = \frac{I_{low}^{all} (T)}{I_{high}^{all} (T)} = \frac{I_{low}^{Σ} (T)}{I_{high}^{Σ} (T)} \cdot \frac{[1 - X_{low}^{DSL} (T) + Y_{low}^{PS} (T)]}{[1 - X_{high}^{DSL} (T) + Y_{high}^{PS} (T)]} .

To further simplify Eq. (14), we can replace the ratio

I_{low}^{Σ} (T) / I_{high}^{Σ} (T) = f_{c}^{Σ} (T)

by its temperature dependence from Eq. (6). Moreover, as each of the functions

X_{low}^{DSL} (T)

,

Y_{low}^{PS} (T)

,

X_{high}^{DSL} (T)

, and

Y_{high}^{PS} (T)

is much less than unity, we can use an approximation

{(1 + z)}^{- 1} \approx 1 - z

with z² << 1 [34] and also retain only the first-order terms, i.e.

\begin{array}{l} Q^{all} (T) \approx f_{c}^{all} (T) = f_{c}^{Σ} (T) \cdot [1 - X_{low}^{DSL} (T) + Y_{low}^{PS} (T)] \cdot [1 + X_{high}^{DSL} (T) - Y_{high}^{PS} (T)] \\ = \exp (A + \frac{B}{T} + \frac{C}{T^{2}} + \dots) \cdot [1 + X_{high}^{DSL} (T) - X_{low}^{DSL} (T) + Y_{low}^{PS} (T) - Y_{high}^{PS} (T)] . \end{array}

Thus, in order to obtain the analytical calibration function one should determine the temperature dependence of

X_{low}^{DSL} (T)

,

Y_{low}^{PS} (T)

,

X_{high}^{DSL} (T)

, and

Y_{high}^{PS} (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 $X_{low}^{DSL} (T)$ and $X_{high}^{DSL} (T)$ . The function $X_{low}^{DSL} (T)$ represents the probability that a backscattered signal wavenumber will be less than ${\tilde{ν}}_{1}$ or more than ${\tilde{ν}}_{2}$ , i.e. outside the transmission band $({\tilde{ν}}_{1}; {\tilde{ν}}_{2})$ , at the temperature T. Taking into account Eq. (45) in Appendix A, we can write

\begin{array}{l} X_{low}^{DSL} (T) = \frac{1}{π} \int_{- \infty}^{+ \infty} \frac{γ_{L}}{{(x - {\tilde{ν}}_{low})}^{2} + γ_{L}^{2}} d x - \frac{1}{π} \int_{{\tilde{ν}}_{1}}^{{\tilde{ν}}_{2}} \frac{γ_{L}}{{(x - {\tilde{ν}}_{low})}^{2} + γ_{L}^{2}} d x \\ = 1 - \frac{1}{π} [arc \tan (\frac{{\tilde{ν}}_{2} - {\tilde{ν}}_{low}}{γ_{L}}) - arc \tan (\frac{{\tilde{ν}}_{1} - {\tilde{ν}}_{low}}{γ_{L}})] . \end{array}

Otherwise speaking,

X_{low}^{DSL} (T)

is the difference between the probability that a backscattered signal wavenumber will take any value within the whole spectrum

(- \infty; + \infty)

, i.e. unity, and the probability of the wavenumber to be within the band

({\tilde{ν}}_{1}; {\tilde{ν}}_{2})

, i.e. the difference of arctangents in Eq. (16). Consider the expansion of arctan(z) in a series [34]

arc \tan (z) = \frac{π}{2} - \frac{1}{z} + \frac{1}{3 z^{3}} - \frac{1}{5 z^{5}} + \frac{1}{7 z^{7}} - \dots with z > 1 .

Since

z = (\tilde{ν} - {\tilde{ν}}_{low}) / γ_{L} ≫ 1

(the band edges

{\tilde{ν}}_{1}

and

{\tilde{ν}}_{2}

are sufficiently far from

{\tilde{ν}}_{low}

), we get

arc \tan (\frac{\tilde{ν} - {\tilde{ν}}_{low}}{γ_{L}}) = \frac{π}{2} - \frac{γ_{L}}{\tilde{ν} - {\tilde{ν}}_{low}} + \frac{1}{3} {(\frac{γ_{L}}{\tilde{ν} - {\tilde{ν}}_{low}})}^{3} - \frac{1}{3} {(\frac{γ_{L}}{\tilde{ν} - {\tilde{ν}}_{low}})}^{5} + \dots .

Subtracting term by term Eq. (18) at

\tilde{ν} = {\tilde{ν}}_{1}

from its own expression at

\tilde{ν} = {\tilde{ν}}_{2}

, we can reduce Eq. (16) to

\begin{array}{l} X_{low}^{DSL} (T) = 1 - \frac{1}{π} [- (\frac{1}{Δ {\tilde{ν}}_{2}} - \frac{1}{Δ {\tilde{ν}}_{1}}) γ_{L} + \frac{1}{3} (\frac{1}{Δ {\tilde{ν}}_{2}^{3}} - \frac{1}{Δ {\tilde{ν}}_{1}^{3}}) γ_{L}^{3} - \dots] \\ = 1 - (a_{1} γ_{L} + a_{2} γ_{L}^{3} + a_{3} γ_{L}^{5} + a_{4} γ_{L}^{7} + \dots), \end{array}

where

Δ {\tilde{ν}}_{1} = {\tilde{ν}}_{1} - {\tilde{ν}}_{low}

,

Δ {\tilde{ν}}_{2} = {\tilde{ν}}_{2} - {\tilde{ν}}_{low}

, and a_i 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

X_{low}^{DSL} (T)

\begin{array}{l} X_{low}^{DSL} (T) = 1 - [a_{1} (A \sqrt{T} + \frac{B}{\sqrt{T}}) + a_{2} {(A \sqrt{T} + \frac{B}{\sqrt{T}})}^{3} + a_{3} {(A \sqrt{T} + \frac{B}{\sqrt{T}})}^{5} + \dots] \\ = \dots + \frac{b_{- 2}}{T \sqrt{T}} + \frac{b_{- 1}}{\sqrt{T}} + 1 + b_{1} \sqrt{T} + b_{2} T \sqrt{T} + \dots = 1 + \sum_{n = 1}^{\infty} (\frac{b_{- n}}{T^{n - 1 / 2}} + b_{n} T^{n - 1 / 2}), \end{array}

where b_–_n and b_n are the constants defined by the given equation. Applying the same procedure to

X_{high}^{DSL} (T)

with the transmission band

({\tilde{ν}}_{3}; {\tilde{ν}}_{4})

instead of

({\tilde{ν}}_{1}; {\tilde{ν}}_{2})

, one can obtain

X_{high}^{DSL} (T) = 1 + \sum_{n = 1}^{\infty} (\frac{c_{- n}}{T^{n - 1 / 2}} + c_{n} T^{n - 1 / 2}),

where c_–_n and c_n are the constants defined similarly to b_–_n and b_n in Eq. (20). Subtracting Eq. (20) from Eq. (21), we have for the temperature dependence of

X_{high}^{DSL} (T) - X_{low}^{DSL} (T)

in Eq. (15)

X_{high}^{DSL} (T) - X_{low}^{DSL} (T) = \sum_{n = 1}^{\infty} (\frac{d_{- n}}{T^{n - 1 / 2}} + d_{n} T^{n - 1 / 2}),

where d_–_n = c_–_n – b_–_n and d_n = c_n – b_n.

In order to determine the temperature dependence of $Y_{low}^{PS} (T)$ in Eq. (12), we can use the results obtained above. First, since we consider $I_{low}^{Σ} (T)$ as the intensity of a single line, each intensity ratio $I_{k} (J_{k}, T) / I_{low}^{Σ} (T)$ in Eq. (12) can be defined as well as the ratio in Eq. (1), i.e.

\frac{I_{k} (J_{k}, T)}{I_{low}^{Σ} (T)} = \exp (C_{k} + \frac{E_{k}}{T}) = \exp (C_{k}) \cdot \exp (\frac{E_{k}}{T}) = D_{k} \exp (\frac{E_{k}}{T}),

where C_k, D_k, and E_k are the constants corresponding to k intensity ratio. Second, the temperature dependence of each function

X_{low}^{k} (T)

is defined by Eq. (10), and hence, by the difference of arctangents in Eq. (16). Taking Eq. (20) into account, we get

X_{low}^{k} (T) = \sum_{n = 1}^{\infty} (\frac{f_{- n}^{k}}{T^{n - 1 / 2}} + f_{n}^{k} T^{n - 1 / 2}),

where

f_{- n}^{k}

and

f_{n}^{k}

are the constants defined similarly to b_–_n and b_n in Eq. (20). Expanding the exponential function in Eq. (23) in Taylor series, we obtain for

Y_{low}^{PS} (T)

in Eq. (12)

\begin{array}{l} Y_{low}^{PS} (T) = \sum_{k} Y_{low}^{k} (T) = \sum_{k, J_{k}} [\frac{I_{k} (J_{k}, T)}{I_{low}^{Σ} (T)} \cdot X_{low}^{k} (T)] \\ = \sum_{k} [D_{k} (1 + \frac{E_{k}}{T} + \frac{E_{k}^{2}}{2 T^{2}} + \dots) \cdot \sum_{n = 1}^{\infty} (\frac{f_{- n}^{k}}{T^{n - 1 / 2}} + f_{n}^{k} T^{n - 1 / 2})] = \sum_{n = 1}^{\infty} (\frac{g_{- n}}{T^{n - 1 / 2}} + g_{n} T^{n - 1 / 2}), \end{array}

where g_–_n and g_n are defined by the given equation, and

X_{low}^{k} (T)

is defined by Eq. (24). The same is valid for the function

Y_{high}^{PS} (T)

, i.e.

Y_{high}^{PS} (T) = \sum_{n = 1}^{\infty} (\frac{h_{- n}}{T^{n - 1 / 2}} + h_{n} T^{n - 1 / 2}),

where h_–_n and h_n are the constants defined similarly to g_–_n and g_n in Eq. (25).

It is easy to see that all the functions $X_{low}^{DSL} (T)$ , $Y_{low}^{PS} (T)$ , $X_{high}^{DSL} (T)$ , and $Y_{high}^{PS} (T)$ 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) = X_{high}^{DSL} (T) - X_{low}^{DSL} (T) + Y_{low}^{PS} (T) - Y_{high}^{PS} (T) = \sum_{n = 1}^{\infty} (\frac{l_{- n}}{T^{n - 1 / 2}} + l_{n} T^{n - 1 / 2}) .

where l_–_n and l_n are the constants.

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

\ln Q^{all} (T) \approx \ln f_{c}^{all} (T) = A + \frac{B}{T} + \frac{C}{T^{2}} + \frac{D}{T^{3}} + \dots + \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) = z - \frac{z^{2}}{2} + \frac{z^{3}}{3} - \frac{z^{4}}{4} + \dots with - 1 < z \leq 1 .

Substituting z = Z(T) from Eq. (27) into Eq. (29), we have

\begin{array}{l} \ln [1 + Z (T)] = \sum_{n = 1}^{\infty} (\frac{l_{- n}}{T^{n - 1 / 2}} + l_{n} T^{n - 1 / 2}) - \frac{1}{2} {[\sum_{n = 1}^{\infty} (\frac{l_{- n}}{T^{n - 1 / 2}} + l_{n} T^{n - 1 / 2})]}^{2} + \dots \\ = \dots + \frac{m_{- 4}}{T^{2}} + \frac{m_{- 3}}{T \sqrt{T}} + \frac{m_{- 2}}{T} + \frac{m_{- 1}}{\sqrt{T}} + m_{0} + m_{1} \sqrt{T} + m_{2} T + m_{3} T \sqrt{T} + m_{4} T^{2} + \dots, \end{array}

where the constants m_–n and m_n 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

\begin{array}{l} \ln Q^{all} (T) \approx \dots + \frac{C + m_{- 4}}{T^{2}} + \frac{m_{- 3}}{T \sqrt{T}} + \frac{B + m_{- 2}}{T} + \frac{m_{- 1}}{\sqrt{T}} + (A + m_{0}) + m_{1} \sqrt{T} + m_{2} T + \dots \\ = \dots + \frac{A_{- 4}}{T^{2}} + \frac{A_{- 3}}{T \sqrt{T}} + \frac{A_{- 2}}{T} + \frac{A_{- 1}}{\sqrt{T}} + A_{0} + A_{1} \sqrt{T} + A_{2} T + A_{3} T \sqrt{T} + \dots = \sum_{n = - \infty}^{\infty} A_{n} T^{\frac{n}{2}}, \end{array}

where A_n are the lidar calibration coefficients. Setting Z(T) = 0 in Eq. (28) or m_–n = m_n = 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(Q^all) 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.

\ln Q (T) \approx \dots + \frac{B_{- 2}}{T^{2}} + \frac{B_{- 1}}{T} + B_{0} + B_{1} T + B_{2} T^{2} + \dots = \sum_{n = - \infty}^{\infty} B_{n} T^{n},

where B_n 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 = \frac{B_{0}}{\ln Q - A_{0}},

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 = \frac{2 C_{1}}{- B_{1} + \sqrt{B_{1}^{2} + 4 C_{1} (\ln Q - A_{1})}},

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

\ln Q = A_{2} + \frac{B_{2}}{T} + C_{2} T \Leftrightarrow y = A_{2} + B_{2} x + \frac{C_{2}}{x},

where A₂, B₂, and C₂ are the calibration constants. In this case the temperature retrieval function derived from Eq. (35) is

T = \frac{2 B_{2}}{(\ln Q - A_{2}) + \sqrt{{(\ln Q - A_{2})}^{2} - 4 B_{2} C_{2}}} .

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 + b y + c y^{2} .

The temperature profile can be simply retrieved via

1 / T = a + b \ln Q + c {(\ln Q)}^{2} \Leftrightarrow T = {[c {(\ln Q)}^{2} + b \ln Q + a]}^{- 1},

or

T = \frac{C_{3}}{{(\ln Q)}^{2} + B_{3} \ln Q + A_{3}},

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 = A_{4} + B_{4} y + C_{4} / y \Leftrightarrow 1 / T = A_{4} + B_{4} \ln Q + C_{4} / \ln Q,

where A₄, B₄, and C₄ are the calibration constants. In this case, inverting Eq. (40) yields

T = \frac{1}{A_{4} + B_{4} \ln Q + (C_{4} / \ln Q)} = \frac{\ln Q}{B_{4} {(\ln Q)}^{2} + A_{4} \ln Q + C_{4}} .

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 ${\tilde{ν}}_{low}$ and ${\tilde{ν}}_{high}$ 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 $({\tilde{ν}}_{1}; {\tilde{ν}}_{2})$ = (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 $({\tilde{ν}}_{3}; {\tilde{ν}}_{4})$ = (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.

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 A_i, B_i, and C_i, determined by applying the least square method to the simulation and reference U.S. Standard tropospheric temperature data, are collected in Table 1.

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

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

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Table 1. Calibration coefficients for temperature retrieval functions.

View Table

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 ({\tilde{ν}}_{i}, \tilde{ν}, T) = \frac{1}{γ_{G}^{i} \sqrt{2 π}} \exp [- \frac{{(\tilde{ν} - μ_{i})}^{2}}{2 {(γ_{G}^{i})}^{2}}] .

Here

μ_{i} = {\tilde{ν}}_{i}

(cm^–1) is the mathematical expectation (or the central wavenumber of i broadened line profile);

γ_{G}^{i}

(cm^–1) is the standard deviation defined by the equation

γ_{G}^{i} = {\tilde{ν}}_{i} \sqrt{\frac{k_{B} T}{m_{air} c^{2}}},

where c is the speed of light, m_air is the average mass of air molecules, k_B 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: λ_J _{= 6} = 354.12 nm and

{\tilde{ν}}_{J = 6}

≈28238.994 cm^–1. The full width at half maximum (FWHM) of the Gaussian profile is defined as

Δ {\tilde{ν}}_{i, G}^{FWHM} = 2 \sqrt{2 \ln 2} \cdot γ_{G}^{i} .

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

L ({\tilde{ν}}_{i}, \tilde{ν}, T) = \frac{1}{π} \cdot \frac{γ_{L}}{{(\tilde{ν} - {\tilde{ν}}_{i})}^{2} + γ_{L}^{2}},

where γ_L (cm^–1) specifies the half-width at half-maximum (HWHM) of the Lorentzian profile. In other words, γ_L = 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

Δ {\tilde{ν}}_{L}^{FWHM} = 2 γ_{L} .

In a case of two-component gases (e.g., air consisting of > 99% of N₂ and O₂ molecules), the HWHM γ_L at each altitude z is defined by

γ_{L} = p_{1}^{2} n_{air} π d_{1}^{2} \sqrt{\frac{8 k T}{π μ_{1} c^{2}}} + 2 p_{1} p_{2} n_{air} π d_{12}^{2} \sqrt{\frac{8 k T}{π μ_{12} c^{2}}} + p_{2}^{2} n_{air} π d_{2}^{2} \sqrt{\frac{8 k T}{π μ_{2} c^{2}}},

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; n_air 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]

d_{i}^{2} (T) = d_{i, \infty}^{2} (1 + \frac{φ_{i}}{R T}),

where the constant d_i_,∞ is the effective optical collision diameter at T → ∞; φ_i 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 γ_L at given molecular number density n_air

γ_{L} = A \sqrt{T} + \frac{B}{\sqrt{T}},

where A and B are the constants defined by Eqs. (46) and (47). The characteristics required for our simulation are the following: m_air = 4.81 × 10⁻²⁶ kg, m₁ = 4.65 × 10⁻²⁶ kg, m₂ = 5.31 × 10⁻²⁶ kg, d_1,∞ = 3.51 × 10⁻¹⁰ m, d_2,∞ = 3.52 × 10⁻¹⁰ m, d_12,∞ ≈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 ({\tilde{ν}}_{i}, \tilde{ν}, T) = \int_{- \infty}^{\infty} G (T, {\tilde{ν}}_{i}, \tilde{ν} *) L (T, \tilde{ν}, \tilde{ν} *) d \tilde{ν} * .

By substituting Eqs. (42) and (45) into Eq. (49), the Voigt profile can generally be expressed in the following form

V ({\tilde{ν}}_{i}, \tilde{ν}, T) = \frac{u_{0} y}{π} \int_{- \infty}^{\infty} \frac{\exp (- t^{2}) d t}{{(x - t)}^{2} + y^{2}},

where

u_{0} = \frac{2}{Δ {\tilde{ν}}_{i, G}^{FWHM}} \sqrt{\frac{\ln 2}{π}}, y = \frac{Δ {\tilde{ν}}_{L}^{FWHM}}{Δ {\tilde{ν}}_{i, G}^{FWHM}} \sqrt{\ln 2}, x = \frac{\tilde{ν} - {\tilde{ν}}_{i}}{Δ {\tilde{ν}}_{i, G}^{FWHM}} 2 \sqrt{\ln 2} .

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]

Δ {\tilde{ν}}_{i, V}^{FWHM} \approx 0.5346 Δ {\tilde{ν}}_{L}^{FWHM} + \sqrt{{(Δ {\tilde{ν}}_{i, G}^{FWHM})}^{2} + 0.2166 {(Δ {\tilde{ν}}_{L}^{FWHM})}^{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 $\ln Q$ ) represents a special case of Eq. (31) or (32). Solving quadratic Eq. (39) we get

\ln Q = - \frac{B_{3}}{2} \pm \sqrt{{(\frac{B_{3}}{2})}^{2} - A_{3} + \frac{C_{3}}{T}} \Leftrightarrow y = - \frac{B_{3}}{2} \pm \sqrt{{(\frac{B_{3}}{2})}^{2} - A_{3} + C_{3} x} .

In order to analyze Eq. (53) in two limiting cases, one can rewrite it as follows

y = - \frac{B_{3}}{2} \pm \sqrt{| C_{3} |} \sqrt{\frac{B_{3}^{2}}{4 C_{3}} - \frac{A_{3}}{C_{3}} + x} = α + β \sqrt{γ + x} = α + β \sqrt{| γ |} \sqrt{1 + \frac{x}{γ}} .

If x >> γ, then

y \approx α + β \sqrt{x}

, and Eq. (53) can be expressed as a special case of Eq. (31)

\ln Q = A_{0} + \frac{A_{- 1}}{\sqrt{T}} .

If x << γ, we can use the following expansion in a series in z = x/γ << 1 [34]

\sqrt{1 + z} = 1 + \frac{1}{2} z - \frac{1}{8} z^{2} + \frac{1}{16} z^{3} - \dots with z^{2} \leq 1 .

Substituting Eq. (56) into Eq. (54), for the calibration function we obtain

\ln Q = α + β \sqrt{| γ |} (1 + \frac{x}{2 γ} - \frac{x^{2}}{8 γ^{2}} + \frac{x^{3}}{16 γ^{3}} - \dots) = B_{0} + \frac{B_{- 1}}{T} + \frac{B_{- 2}}{T^{2}} + \dots,

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).

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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]

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	Equation (33)	Equation (34)	Equation (36)	Equation (39)	Equation (41)
A_i	−1.4611165	−1.6852011	−1.0127842	26.0591977	0.00289499
B_i	495.6844325	607.7464374	439.9063493	16.5887743	0.00206899
C_i		−13898.2629	−0.0008937	8754.000685	0.00001183

Analytical calibration functions for the pure rotational Raman lidar technique

Abstract

1. Introduction

2. Problem statement

3. PRR lidar general calibration function

4. Special cases of the general calibration function

5. Simulation

5.1 Initial conditions for the simulation

5.2 Results of the simulation

6. Summary

Appendices

A. Gaussian, Lorentzian, and Voigt profiles

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

Acknowledgments

References and Links

Cited By

Figures (5)

Tables (1)

Equations (57)

Optics Express