Temporal stability of calibration functions in the traditional pure rotational Raman lidar technique

Vladislav V. Gerasimov; Vladislav V. Gerasimov

doi:10.1364/OSAC.404945

1. Introduction

Several new techniques and algorithms for retrieving atmospheric temperature profiles from signals of pure rotational Raman (PRR) lidars have been presented over the past few years [1–6]. The technique proposed by Cooney in 1972 [7] is now called “traditional” one [2]. In the traditional PRR lidar technique, atmospheric temperature T is determined from an intensity ratio Q(T) of backscattered lidar signals extracted from two PRR-spectrum bands of N₂ and O₂ molecules [8]. For this purpose, various spectral filters are used: interference filters [9–23], diffraction gratings [2–4,24–30], fiber Bragg gratings [31,32], and Fabry–Perot interferometers [1,33–37]. The ratio Q(T) obtained from lidar measurements needs to be calibrated. For this reason, lnQ(T) is approximated by a calibration function (CF) f_c(T) that is often chosen as a first- or second-order polynomial in reciprocal temperature 1/T [8–30,38,39]. The coefficients of CFs (or calibration coefficients) are calculated by applying the least square method to the measured data Q(T) and reference temperature profile (this is also called “PRR lidar calibration”). A temperature profile of an atmospheric model or a profile obtained using a radiosonde may be taken as the reference one. Radiosondes are normally launched at the lidar site simultaneously with lidar measurements. To obtain temperature profiles from lidar measurement data, a temperature retrieval function (TRF) T = T(lnQ) derived from the selected CF f_c(T) is required in the traditional retrieval algorithm. The coefficients of both TRF and its initial CF are the same, but they are first determined for the CF. Note that a TRF and its initial CF may not be mutually inverse, the equation lnQ(T) = f_c(T) may not be solved with respect to T or its solution is cumbersome. These circumstances impose certain constraints on the functions that can be used as calibration ones [40–42].

New techniques [1,2] do not require periodic calibration of PRR lidars and launches of radiosondes, and simplify the procedure for obtaining temperature profiles. However, they do not show advantages in the temperature retrieval accuracy (at least, in the troposphere). Therefore, the traditional technique is still valid and recent studies have shown the possibility of improving the accuracy with new TRFs [40–42].

During a measurement campaign conducted over several days [17,20,43], there may be a situation when reference data are absent on one of the measurement days (day X) due to a premature burst of a radiosonde balloon. As a consequence, the following question arises: how will the accuracy of temperature retrieval from the lidar data obtained on day X be affected by the use of calibration coefficients determined from lidar and reference data for another day Y of the measurement campaign? A similar situation can also occur when several lidar measurements are performed during one night. In this case, reference data from a radiosonde launched during the first or last lidar measurements or the data from both radiosondes are usually used [22,23,29].

In this paper, we consider 1-week and 3-day measurement periods, on each day of which reference temperature profiles are removed from consideration. The calibration coefficients of TRFs are calculated from the coefficients determined on the remaining measurement days. Five nonlinear TRFs [42,44] are used for tropospheric (3–9 km) temperature retrievals. The temporal stability of the TRFs is analyzed on the example of nighttime temperature profiles retrieved from PRR lidar measurements of 1, 6, 7, and 8 April 2015 [45]. The measurements were performed in Tomsk (56.48°N, 85.05°E, Western Siberia, Russia) using a PRR lidar designed in the Institute of Monitoring of Climatic and Ecological Systems (IMCES) [41]. The function that retrieves temperature profiles with the smallest average errors during the measurement campaign will be considered as the most stable TRF.

2. Temperature retrieval functions

Inelastic N₂ and O₂ PRR lines are broadened mainly by the Doppler effect and molecular collisions in the atmosphere. The molecular collisions make a significant contribution to PRR line broadening in the troposphere [46,47] when using both 355 and 532 nm laser wavelengths for PRR lidar measurements. The CF that takes into account the collisional broadening of all N₂ and O₂ PRR lines during lidar signal reception and processing was derived in the general analytical form in Ref. [40]:

(1)$$\ln Q(T) \approx {f_\textrm{c}}(T) = \sum\limits_{n ={-} \infty }^\infty {{\alpha _n}{T^{\frac{n}{2}}}}, $$

where α_n are the calibration coefficients. As seen from Eq. (1), the general CF represents a series and, hence, can hardly be used in the temperature retrieval algorithm. Owing to this, nine simple CFs representing special cases of Eq. (1) were considered and intercompared in Refs. [42,47]. All the functions take the collisional broadening of PRR lines into account in varying degrees. Simulation and practical studies showed that five out of these nine CFs provide more accurate tropospheric temperature retrievals from lidar signals compared to the others.

Below we analyze only five TRFs required for our study without specifying their initial nonlinear CFs. Each TRF i and its calibration coefficients A_i, B_i, C_i, D_i are numbered for convenience (i = 3, 4, 7, 8, and 9) in full accordance with those of Refs. [42,47]. The first two functions (TRFs 3 and 4) each have three calibration coefficients (A_i, B_i, C_i):

(2)$$T = \frac{{{C_3}}}{{{{(\ln Q)}^2} + {B_3}\ln Q + {A_3}}}, $$

(3)$$T = \frac{{\ln Q}}{{{B_4}{{(\ln Q)}^2} + {A_4}\ln Q + {C_4}}}. $$

The next three TRFs 7, 8 and 9 are the functions with four calibration coefficients:

(4)$$T = \frac{{{D_7}}}{{{{(\ln Q)}^3} + {C_7}{{(\ln Q)}^2} + {B_7}\ln Q + {A_7}}}, $$

(5)$$T = \frac{{{{(\ln Q)}^2}}}{{{B_8}{{(\ln Q)}^3} + {A_8}{{(\ln Q)}^2} + {C_8}\ln Q + {D_8}}}, $$

(6)$$T = \frac{{\ln Q}}{{{C_9}{{(\ln Q)}^3} + {B_9}{{(\ln Q)}^2} + {A_9}\ln Q + {D_9}}}. $$

The initial CFs for specified TRFs 3, 4, 7–9 can be found in Ref. [42]. Note that the CFs linear [17,20,43] or quadratic [22,23,29] in 1/T (not shown here) are usually used in measurement campaigns being conducted over several days or throughout the night.

3. Temperature measurements with the IMCES PRR lidar

This section describes the main technical parameters of the IMCES PRR lidar and special features of the temperature retrieval algorithm from its signals.

3.1 IMCES PRR lidar

Raw lidar signals for studying the TRF temporal stability were obtained using the IMCES PRR lidar designed for tropospheric temperature measurements [41]. An unseeded frequency-tripled Nd:YAG laser Solar LS LQ529B operating at a wavelength of 354.67 nm with ∼1 cm⁻¹ spectral line width is used as the lidar transmitter. The laser pulse repetition rate, pulse energy, and pulse duration are 20 Hz, 105 mJ, and 13 ns, respectively. The backscattered signals (photons) are collected by a prime-focus receiving telescope with a mirror diameter of 0.5 m and a focal length of 1.5 m. The extraction of desired lines from N₂ and O₂ PRR spectra in lidar temperature channels is performed with a double-grating monochromator. The lidar optical layout and other technical parameters of its transmitting, receiving, and data acquisition systems are presented in detail in Ref. [41].

3.2 Reference temperature points for lidar calibration

Air temperature points obtained by a radiosonde being launched simultaneously with measurements at a lidar station are most often used as reference points for PRR lidar calibration [3–6,8–30]. However, not all lidar stations (which include the IMCES station) have the possibility to launch such radiosondes. In these instances, a PRR lidar can be calibrated using temperature points from radiosondes being launched at the meteorological stations nearest to the PRR lidar location. Meteorological stations normally launch radiosondes twice a day (at 00:00 and 12:00 UTC).This imposes restrictions on the possible time for lidar calibration and nighttime measurements, because the measurements should be performed close in time to the radiosonde launches. The local time (LT) in Tomsk, corresponding to 00:00 and 12:00 UTC, is 06:00 and 18:00, respectively. Thus, the period of nighttime temperature measurements at the IMCES station lasts from October to April.

The nearest to Tomsk (56.48°N, 85.05°E) stations launching radiosondes twice a day are situated in Novosibirsk (55.02°N, 82.92°E) and Kolpashevo (58.32°N, 82.92°E) at distances of 210 and 242 km, respectively [48]. Therefore, the temperature data from these stations cannot be directly used for PRR lidar calibration in Tomsk. To calibrate the IMCES lidar, a method for obtaining several reference points using constant pressure altitude charts (CPACs) was proposed in Ref. [41]. The 700, 500, 400, and 300 hPa CPACs with isotherms plotted on them were used to retrieve four reference points in the 3–9 km altitude range. Six additional reference points for this study were obtained taking into account the stability of the troposphere between the atmospheric boundary layer (≥ 3 km) and tropopause (< 9–10 km) for middle latitudes. Such reference points do not allow retrieving temperature profiles with high accuracy. However, it is possible to make a comparative analysis of temperature measurement errors yielded by using different TRFs under the same initial conditions. The temperature profiles obtained in Novosibirsk and Kolpashevo using radiosondes launched at 06:00 LT (00:00 UTC) on 7 April 2015 along with reference points for the IMCES lidar calibration are shown in Fig. 1 as an example. The other reference data for the measurements of 1, 6, and 8 April 2015 are given in Supplement 1 (Figs. S1–S3).

Fig. 1. (a) Temperature profiles (3–9 km) from radiosondes launched in Novosibirsk and Kolpashevo at 06:00 LT (00:00 UTC) on 7 April 2015, temperature points over Tomsk retrieved using CPACs, and additional points obtained by linear interpolation. (b) Reference points for the IMCES lidar calibration.

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3.3 IMCES lidar measurement data

The meteorological conditions over Tomsk in April 2015 allowed us to perform nighttime measurements on 1, 6, 7, and 8 April. The lidar data of 1 April were taken from 03:45 to 05:15 LT, i.e., within 90 min integration time (108 000 laser shots). The lidar data of 6, 7, and 8 April were taken in the 03:20–05:20, 03:07–05:07, and 03:00–05:00 LT periods, respectively, with 120 min integration time (144 000 laser shots) each. Such a long signal integration time is a necessary measure in our case due to the low laser power and lack of thermal stabilization of the lidar receiving system. It was experimentally established that the 1–2 hour integration time provides the required number of photocounts to minimize both absolute and relative statistical uncertainties.

Raw and averaged signals obtained on 1, 6, 7, and 8 April 2015 are presented in Supplement 1 (Figs. S4–S7). The corresponding tropospheric temperature profiles (3–9 km) retrieved from the IMCES lidar data using TRFs 3, 4, 7–9 are provided in Figs. S8–S11. The calibration coefficients A₄, B₄, and C₄ of TRF 4, calculated by applying the least square method to the lidar data and reference temperature points of 1, 6, 7, and 8 April, are presented in Fig. 2. The coefficients A_i, B_i, C_i, D_i of TRFs 3, 7–9 are shown in Figs. S12–S15.

Fig. 2. Calibration coefficients A₄, B₄, and C₄ of TRF 4 calculated for 1, 6, 7, and 8 April.

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4. Intercomparison of TRFs

4.1 Two measurement situations

Let us consider the following two situations to analyze the temporal stability of TRFs. The first situation assumes that appropriate reference temperature points for lidar calibration are available for each measurement day (1, 6, 7, and 8 April). This makes it possible to assess to what extent the TRF coefficients determined on each day of the measurement campaign may be used to retrieve temperature on the remaining days. Figure 3(a) shows four temperature profiles obtained from the lidar measurement data of 1 April (Fig. S4). Profiles “1 Apr”, “6 Apr”, “7 Apr”, and “8 Apr” were retrieved by TRF 4 with calibration coefficients A₄, B₄, and C₄ (Fig. 2) determined for 1, 6, 7, and 8 April, respectively. As a consequence, profile “1 Apr” is closer to the reference points than the others. Similarly, each of Figs. 4(a), 5(a), and 6(a) shows four profiles retrieved by TRF 4 from the lidar data of 6, 7, and 8 April (Fig. S5–S7), respectively. It is notable that profile “1 Apr” is significantly spaced away from the group of the other profiles (“6 Apr”, “7 Apr”, and “8 Apr”) in all the four figures. This indicates that already a 5-day time interval between the determination and use of the calibration coefficients leads to large errors in temperature retrievals.

Fig. 3. (a) Temperature profiles retrieved by TRF 4 from the 1 April lidar data. Reference points (RP) are given for 1 April. The absolute differences: (b) |T_ref – T_i|; (c) |T₁ – T_j| (see description in Section 4.2).

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Fig. 4. (a) Temperature profiles retrieved by TRF 4 from the 6 April lidar data. Reference points (RP) are given for 6 April. The absolute differences: (b) |T_ref – T_i|; (c) |T₆ – T_j|.

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Fig. 5. (a) Temperature profiles retrieved by TRF 4 from the 7 April lidar data. Reference points (RP) are given for 7 April. The absolute differences: (b) |T_ref – T_i|; (c) |T₇ – T_j|.

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The second situation assumes that reference points are absent on one of the measurement days. Therefore, the TRF coefficients on such a day can be calculated using the coefficients over the remaining days of the measurement campaign. Both arithmetic and weighted average values of the coefficients determined in these remaining days were used for this purpose (Table S2).

Figure 7(a) shows three profiles obtained from the lidar measurement data of 1 April. The first profile “1 Apr” is the same that in Fig. 3(a), i.e., it was retrieved by TRF 4 with A₄, B₄, and C₄ calculated using the 1 April reference data. The second profile designated in Fig. 7(a) as “ar.av(6,7,8)” was retrieved using TRF 4 coefficients arithmetically averaged over their values for 6, 7, and 8 April. The third profile “w.av(6,7,8)” was identically retrieved using the weighted average values of the coefficients for 6–8 April. Each of Figs. 8(a), 9(a), and 10(a) shows three profiles similarly obtained from the lidar data of 6, 7, and 8 April, respectively.

Fig. 6. (a) Temperature profiles retrieved by TRF 4 from the 8 April lidar data. Reference points (RP) are given for 8 April. The absolute differences: (b) |T_ref – T_i|; (c) |T₈ – T_j|.

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Fig. 7. (a) Temperature profiles retrieved by TRF 4 from the 1 April lidar data. Arithmetic and weighted average values of TRF 4 coefficients were used to obtain profiles “ar.av(6,7,8)” and “w.av(6,7,8)”, respectively. The absolute differences: (b) |T_ref – T_i| and (c) |T₁ – T_j| (see description in Section 4.2).

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Fig. 8. (a) Temperature profiles retrieved by TRF 4 from the 6 April lidar data. Arithmetic and weighted average values of TRF 4 coefficients were used to obtain profiles “ar.av(1,7,8)” and “w.av(1,7,8)”, respectively. The absolute differences: (b) |T_ref – T_i| and (c) |T₆ – T_j|.

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Fig. 9. (a) Temperature profiles retrieved by TRF 4 from the 7 April lidar data. Arithmetic and weighted average values of TRF 4 coefficients were used to obtain profiles “ar.av(1,6,8)” and “w.av(1,6,8)”, respectively. The absolute differences: (b) |T_ref – T_i| and (c) |T₇ – T_j|.

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4.2 Numerical characteristics of TRF stability

In order to assess the temporal stability of TRFs, we consider two simple numerical characteristics. The first characteristic shown in Fig. 3(b) is the absolute differences |T_ref – T_i| between the 1 April reference points T_ref and corresponding points (in altitude) of temperature profiles T_i retrieved using TRF 4. The index i takes the values 1, 6, 7, and 8, i.e., T₁, T₆, T₇, and T₈ represent profiles “1 Apr”, “6 Apr”, “7 Apr”, and “8 Apr” shown in Fig. 3(a). Similarly, the index i takes the values 1, “ar.av”, and “w.av” in Fig. 7(b), which means that T₁, T_ar.av, and T_w.av correspond to profiles “1 Apr”, “ar.av(6,7,8)”, and “w.av(6,7,8)” presented in Fig. 7(a). For example, the largest differences both in Figs. 3(b) and 7(b) occur at an altitude of 8.4 km and take the following values: |T_ref – T₈| = 6.0 K and |T_ref – T_ar.av| = 4.5 K. When using the reference points T_ref of 6, 7, and 8 April, the corresponding differences |T_ref – T_i| presented in Figs. 4(b), 5(b), 6(b), and 8(b), 9(b), 10(b) are calculated in the same way as those given in Figs. 3(b) and 7(b), respectively.

The second characteristic of the TRF temporal stability shown in Fig. 3(c) represents the absolute differences |T₁ – T_j| between temperature profile T₁ and profiles T₆, T₇, and T₈ presented in Fig. 3(a) (index j = 6, 7, and 8). Similarly, the index j takes the values “ar.av” and “w.av” in Fig. 7(c). For example, |T₁ – T_ar.av| < 3.5 K and |T₁ – T_w.av| < 2.9 K in the 3–9 km altitude range. Profile T₁ is considered as the reference one in the differences |T₁ – T_j| in Figs. 3(c) and 7(c), because it was retrieved using the reference points obtained simultaneously with the 1 April lidar measurements.

The analogous differences presented in Figs. 4(c), 5(c), 6(c), and 8(c), 9(c), and 10(c) are calculated in the same way as those given in Figs. 3(c) and 7(c), respectively. Note that profiles T₆, T₇, and T₈ are the reference ones in the corresponding differences |T₆ – T_j|, |T₇ – T_j|, and |T₈ – T_j|.

The characteristics considered in Figs. 3–10 do not allow us to clearly assess the degree to which the temperature profiles differ from each other. For example, when intercomparing the profiles in Fig. 8, the difference |T₆ – T_ar.av| shows smaller values compared to |T₆ – T_w.av| in the altitude ranges of 3.0–4.0 and 7.2–9.0 km, while |T₆ – T_w.av| < |T₆ – T_ar.av| at altitudes between 4.0 and 7.2 km. The same is valid for the differences presented in the other figures. For this reason, we propose to use the averaged values of the characteristics to be able to distinguish between the profiles, intercompare TRFs, and determine the most stable function among them. Figure 11 shows <|T_ref – T_i|>, i.e., the |T_ref – T_i| values averaged over the number of reference points. Figure 12 shows the average values <|T₁ – T_j|>, <|T₆ – T_j|>, <|T₇ – T_j|>, and <|T₈ – T_j|> of the corresponding differences. This averaging is performed over the 3–9 km altitude range. Both characteristics of the TRF temporal stability and their averaged values (shown in Figs. 3–12 for TRF 4) are presented in Figs. S16–S55 for the other TRFs 3, 7–9.

Fig. 10. (a) Temperature profiles retrieved by TRF 4 from the 8 April lidar data. Arithmetic and weighted average values of TRF 4 coefficients were used to obtain profiles “ar.av(1,6,7)” and “w.av(1,6,7)”, respectively. The absolute differences: (b) |T_ref – T_i| and (c) |T₈ – T_j|.

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Fig. 11. Average differences <|T_ref – T_i|> (i = 1, 6, 7, 8, “ar.av”, and “w.av”) obtained using the lidar measurement data of: (a) 1 April, (b) 6 April, (c) 7 April, and (d) 8 April. The solid and dotted horizontal lines correspond to i = “ar.av” and i = “w.av”, respectively.

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Fig. 12. Average differences: (a) <|T₁ – T_j|> (j = 6, 7, 8, “ar.av”, and “w.av”), (b) <|T₆ – T_j|> (j = 1, 7, 8, “ar.av”, and “w.av”), (c) <|T₇ – T_j|> (j = 1, 6, 8, “ar.av”, and “w.av”), and (d) <|T₈ – T_j|> (j = 1, 6, 7, “ar.av”, and “w.av”). The solid and dotted horizontal lines correspond to j = “ar.av” and j = “w.av”, respectively.

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5 One-week stability of TRFs

A temperature profile retrieved from lidar data and reference points for the same date should normally be more accurate (closer to the reference) than a profile retrieved using reference points for any other date. This is primarily due to instability in the optical and electronic systems of PRR lidars, because of which regular recalibration of the lidars is required [8]. As a consequence, the average values <|T_ref – T₁|> [Fig. 11(a)], <|T_ref – T₆|> [Fig. 11(b)], <|T_ref – T₇|> [Fig. 11(c)], and <|T_ref – T₈|> [Fig. 11(d)] are unsurprisingly the smallest on the corresponding measurement dates.

In the absence of reference points on the first or last day of a measurement campaign, the use of reference points for the nearest neighboring day leads to the best result. Indeed, the values <|T_ref – T₆|> in Fig. 11(a) and <|T₁ – T₆|> in Fig. 12(a) are smaller than the other values <|T_ref – T_i|> and <|T₁ – T_j|> for i and j = 7, 8, “ar.av”, and “w.av”. This means that the profile retrieved from the lidar data of 1 April using the calibration coefficients (A₄, B₄, and C₄) of 6 April is more accurate compared to the profiles retrieved using averaged coefficients or coefficients determined on the other measurement days. Similarly, the profile retrieved from the lidar data of 8 April using A₄, B₄, and C₄ determined from the reference data of 7 April has the smallest values <|T_ref – T₇|> (without considering <|T_ref – T₈|>) in Fig. 11(d) and <|T₈ – T₇|> in Fig. 12(d).

In order to determine the most stable function among TRFs 3, 4, 7–9, we performed a comparative analysis of the smallest average values of two characteristics considered above (Table 1). As seen in Table 1, TRFs 3, 4, and 7 are more stable compared to TRFs 8 and 9. The most stable function in time (for a 1-week measurement period) is TRF 4 with the four smallest average values out of eight, while TRFs 3 and 7 each have only two such values. In support of this conclusion, we also analyzed the total deviations of the retrieved profiles both from reference points and reference profiles. Table 2 lists for each TRF the sum ∑₁ of the values <|T_ref – T₆|> (1 April), <|T_ref – T_w.av|> (6 and 7 April), and <|T_ref – T₇|> (8 April) presented in Table 1; sum ∑₂ of the values <|T₁ – T₆|>, <|T₆ – T_w.av|>, <|T₇ – T_w.av|>, and <|T₈ – T₇|>; total sum ∑ = ∑₁ + ∑₂; and averaged value ∑/4. It is easy to see that all three sums ∑₁, ∑₂, and ∑ are smaller for TRFs 3 and 4 than for TRFs 7–9, and TRF 4 is the best (most stable) function among considered TRFs.

Table 1. The smallest average values of the 1-week (1, 6–8 April) stability characteristics for TRFs 3, 4, 7–9. The smallest value in each column is in bold.

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Table 2. Total deviations from the reference data (for 1, 6–8 April). The smallest value in each column is in bold.

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6. Three-day stability of TRFs

A time interval of five days or longer between the determination and use of calibration coefficients results in large errors in temperature retrievals (Figs. 3–12). The profiles, retrieved from the 1 April lidar data using the calibration coefficients determined from the reference data for 6, 7, or 8 April, have much larger values of <|T_ref – T_i|> (i = 6, 7, and 8) compared to <|T_ref – T₁|> [Fig. 11(a)]. Conversely, the profiles retrieved from the lidar data of 6, 7, or 8 April using the coefficients determined from the 1 April reference data, have larger values of <|T_ref – T₁|> than <|T_ref – T_i|> in each case [Figs. 11(b), 11(c), and 11(d)]. Therefore, it makes sense to consider the short-term (3-day) stability of TRFs too.

The temperature profiles retrieved using TRFs 3, 4, 7–9 separately for the 3-day measurement period (6–8 April) along with both characteristics and their averaged values are given in Supplement 1 (Figs. S56–S78). A comparative analysis of the profiles, in general, leads us to similar conclusions as in the case of the 1-week period. In the absence of reference points on the first (6 April) and last (8 April) days of the 3-day campaign, the use of reference points for the intermediate day (7 April) leads to the best result (Figs. 13 and 14). By contrast with the results for the 1-week period, the use of the weighted average values of TRF 4 coefficients [Figs. 13(a) and 13(c)] provides almost the same accuracy of temperature retrieval. The difference between <|T_ref – T₇|> and <|T_ref – T_w.av|> is about 0.04 K in Fig. 13(a) and less than 0.02 K in Fig. 13(c). Similarly to the results for the 1-week period, if reference points are absent on the intermediate day (7 April), the use of the weighted average values of A₄, B₄, and C₄ determined from the reference data for 6 and 8 April leads to the smallest temperature errors [Figs. 13(b) and 14(b)]. Note that the weighted average values are equal to the arithmetic ones (1/2) in this case (Table S3). Thus, when retrieving temperature by TRF 4, we can recommend the use of weighted average values of calibration coefficients in the absence of reference data on any day of a 3-day measurement campaign.

Fig. 13. Average differences <|T_ref – T_i|> (i = 6, 7, 8, and “w.av”) obtained using the lidar data of: (a) 6 April, (b) 7 April, and (c) 8 April. The dotted horizontal line corresponds to i = “w.av”.

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Fig. 14. Average differences: (a) <|T₆ – T_j|> (j = 7, 8, and “w.av”), (b) <|T₇ – T_j|> (j = 6, 8, and “w.av”), and (c) <|T₈ – T_j|> (j = 6, 7, and “w.av”). The dotted horizontal line corresponds to j = “w.av”. Figure 14.

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The analysis also revealed that the profiles retrieved using TRFs 7–9 (Figs. S64–S66, S69–S71, and S74–S76) are strongly distorted at the boundaries of the 3–9 km altitude range. This indicates the weak temporal stability of these functions already for the 3-day measurement period. Table 3 shows the averaged values of both TRF stability characteristics. One can see that TRFs 3 and 4 are more stable compared to TRFs 7–9 and TRF 4 is the most stable function among considered TRFs. These conclusions are also confirmed by the analysis of total deviations of the retrieved profiles from reference data. Table 4 lists the sum ∑₁ of <|T_ref – T_w.av|> for 6–8 April presented in Table 3; sum ∑₂ of <|T₆ – T_w.av|>, <|T₇ – T_w.av|>, and <|T₈ – T_w.av|>; total sum ∑ = ∑₁ + ∑₂; and averaged value ∑/3. Thus, the conclusions drawn for the 3-day period are in full accordance with those drawn for the 1-week period. Comparing the average values ∑/4 in Table 2 with the values ∑/3 in Table 4, we can conclude that the accuracy of temperature retrieval for the 3-day period is higher when excluding data for 1 April.

Table 3. Averaged values of the 3-day TRF stability characteristics (6–8 April). The smallest value in each column is in bold.

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Table 4. Total deviations from the reference data (for 6–8 April). The smallest value in each column is in bold.

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7. Conclusion

Based on the example of nighttime temperature measurements performed with the IMCES PRR lidar on 1, 6, 7, and 8 April 2015, we have studied the 1-week and 3-day temporal stabilities of TRFs 3, 4, 7–9 for the first time. To assess the stability of these functions, a simple way has been proposed. The way consists in a comparative analysis of two numerical characteristics and their averaged values revealing the degree of difference between the retrieved profiles and corresponding reference points. The way can be used to intercompare different CFs (TRFs) and determine the most stable one of them for any PRR lidar systems and short-term measurement campaigns.

The assessment of the temporal stability of TRFs 3, 4, 7–9 allows us to draw the following conclusions:

(1) Three-coefficient TRFs 3 and 4 are more stable than four-coefficient TRFs 7–9 both for 1-week (Table 2) and 3-day (Table 4) measurement periods.
(2) TRF 4 is the most stable function in time for both considered periods.
(3) TRFs 8 and 9 are the least stable and, therefore, not recommended for use during measurement campaigns, on one or several days of which reference data are not available.
(4) If reference points are absent on the first or last day of a 1-week measurement period (or a period lasting more than 3 days), we recommend the use of calibration coefficients determined from lidar and reference data for the nearest neighboring day for accurate temperature retrieval with TRFs 3 and 4 [Figs. 11(a), 11(d), 12(a), and 12(d)]. In the absence of reference points on an intermediate day, we recommend the use of weighted average values of calibration coefficients determined in the remaining days of a measurement campaign [Figs. 11(b), 11(c), 12(b), and 12(c)].
(5) In the absence of reference points on any day of a 3-day measurement campaign, the use of weighted average values of calibration coefficients leads to the smallest temperature errors (Figs. 13 and 14).

Due to the variety of designs and technical features of different PRR lidar systems, the most stable TRF may differ for each specific lidar system.

Acknowledgments

The author thanks V.V. Zuev, V.L. Pravdin, A.V. Pavlinskiy, and D.P. Mordus (Nakhtigalova) for help in obtaining the IMCES PRR lidar data and reference CPAC points for the lidar calibration.

Disclosures

The author declares no conflicts of interest.

Supplemental document

See Supplement 1 for supporting content.

References

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	1 April	6 April	7 April	8 April
	<\|T_ref – T₆\|> (K)	<\|T_ref – T_w.av\|> (K)	<\|T_ref – T_w.av\|> (K)	<\|T_ref – T₇\|> (K)
TRF 3	2.09	0.84	0.96	0.92
TRF 4	2.08	0.82	0.76	0.96
TRF 7	2.09	0.62	1.10	1.09
TRF 8	2.09	0.73	1.60	1.07
TRF 9	2.09	0.74	1.58	1.08
	<\|T₁ – T₆\|> (K)	<\|T₆ – T_w.av\|> (K)	<\|T₇ – T_w.av\|> (K)	<\|T₈ – T₇\|> (K)
TRF 3	2.20	0.63	0.58	0.77
TRF 4	2.21	0.63	0.47	0.72
TRF 7	2.24	0.52	0.74	1.10
TRF 8	2.24	0.56	0.89	1.07
TRF 9	2.24	0.55	0.90	1.09

	∑₁ (K)	∑₂ (K)	∑ = ∑₁ + ∑₂ (K)	∑/4 (K)
TRF 3	4.81	4.18	8.99	2.25
TRF 4	4.62	4.03	8.65	2.16
TRF 7	4.90	4.60	9.50	2.38
TRF 8	5.49	4.76	10.25	2.56
TRF 9	5.49	4.78	10.27	2.57

	6 April	7 April	8 April
<\|T_ref – T_w.av\|> (K)
TRF 3	0.96	0.95	0.89
TRF 4	0.91	0.75	0.94
TRF 7	0.73	1.09	1.05
TRF 8	0.83	1.61	1.05
TRF 9	0.84	1.59	1.05
	<\|T₆ – T_w.av\|> (K)	<\|T₇ – T_w.av\|> (K)	<\|T₈ – T_w.av\|> (K)
TRF 3	0.72	0.54	0.82
TRF 4	0.70	0.44	0.77
TRF 7	0.71	0.68	1.12
TRF 8	0.61	0.88	1.10
TRF 9	0.60	0.89	1.11

	∑₁ (K)	∑₂ (K)	∑ = ∑₁ + ∑₂ (K)	∑/3 (K)
TRF 3	2.80	2.08	4.88	1.63
TRF 4	2.60	1.91	4.51	1.50
TRF 7	2.87	2.51	5.38	1.79
TRF 8	3.49	2.59	6.08	2.03
TRF 9	3.48	2.60	6.08	2.03

	1 April	6 April	7 April	8 April
	<\|T_ref – T₆\|> (K)	<\|T_ref – T_w.av\|> (K)	<\|T_ref – T_w.av\|> (K)	<\|T_ref – T₇\|> (K)
TRF 3	2.09	0.84	0.96	0.92
TRF 4	2.08	0.82	0.76	0.96
TRF 7	2.09	0.62	1.10	1.09
TRF 8	2.09	0.73	1.60	1.07
TRF 9	2.09	0.74	1.58	1.08
	<\|T₁ – T₆\|> (K)	<\|T₆ – T_w.av\|> (K)	<\|T₇ – T_w.av\|> (K)	<\|T₈ – T₇\|> (K)
TRF 3	2.20	0.63	0.58	0.77
TRF 4	2.21	0.63	0.47	0.72
TRF 7	2.24	0.52	0.74	1.10
TRF 8	2.24	0.56	0.89	1.07
TRF 9	2.24	0.55	0.90	1.09

Temporal stability of calibration functions in the traditional pure rotational Raman lidar technique

Abstract

1. Introduction

2. Temperature retrieval functions

3. Temperature measurements with the IMCES PRR lidar

3.1 IMCES PRR lidar

3.2 Reference temperature points for lidar calibration

3.3 IMCES lidar measurement data

4. Intercomparison of TRFs

4.1 Two measurement situations

4.2 Numerical characteristics of TRF stability

5 One-week stability of TRFs

6. Three-day stability of TRFs

7. Conclusion

Acknowledgments

Disclosures

Supplemental document

References

Supplementary Material (1)

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Figures (14)

Tables (4)

Equations (6)

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