Retrieval of temperature from a multiple channel pure rotational Raman backscatter lidar using an optimal estimation method

Mahagammulla Gamage, Shayamila; Sica, Robert J.; Martucci, Giovanni; Haefele, Alexander

doi:https://doi.org/10.5194/amt-12-5801-2019

Articles | Volume 12, issue 11

https://doi.org/10.5194/amt-12-5801-2019

Articles | Volume 12, issue 11

Research article

05 Nov 2019

Research article |

| 05 Nov 2019

Retrieval of temperature from a multiple channel pure rotational Raman backscatter lidar using an optimal estimation method

Shayamila Mahagammulla Gamage, Robert J. Sica, Giovanni Martucci, and Alexander Haefele

Abstract

We present a new method for retrieving temperature from pure rotational Raman (PRR) lidar measurements. Our optimal estimation method (OEM) used in this study uses the full physics of PRR scattering and does not require any assumption of the form for a calibration function nor does it require fitting of calibration factors over a large range of temperatures. The only calibration required is the estimation of the ratio of the lidar constants of the two PRR channels (coupling constant) that can be evaluated at a single or multiple height bins using a simple analytic expression. The uncertainty budget of our OEM retrieval includes both statistical and systematic uncertainties, including the uncertainty in the determination of the coupling constant on the temperature. We show that the error due to calibration can be reduced significantly using our method, in particular in the upper troposphere when calibration is only possible over a limited temperature range. Some other advantages of our OEM over the traditional Raman lidar temperature retrieval algorithm include not requiring correction or gluing to the raw lidar measurements, providing a cutoff height for the temperature retrievals that specifies the height to which the retrieved profile is independent of the a priori temperature profile, and the retrieval's vertical resolution as a function of height. The new method is tested on PRR temperature measurements from the MeteoSwiss RAman Lidar for Meteorological Observations system in clear and cloudy sky conditions, compared to temperature calculated using the traditional PRR calibration formulas, and validated with coincident radiosonde temperature measurements in clear and cloudy conditions during both daytime and nighttime.

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Dates

Received: 19 Mar 2019 – Discussion started: 13 May 2019 – Revised: 30 Sep 2019 – Accepted: 04 Oct 2019 – Published: 05 Nov 2019

1 Introduction

High time and space resolution measurements of atmospheric temperature are necessary to improve our understanding of many atmospheric processes, both dynamical and chemical. Radiosounding is the most widely used method for temperature profiling in the troposphere and lower stratosphere and has the advantage of operation in most weather conditions, but is typically limited to two flights per day only at selected sites worldwide. Pure rotational Raman (PRR) lidars have excellent vertical and temporal resolution and can be combined with vibrational Raman channels to determine relative humidity (Mattis et al., 2002; Wang et al., 2011; Behrendt et al., 2002; Reichardt et al., 2012). Lidar temperature measurements can be assimilated with atmospheric models to improve weather forecasts, as recently demonstrated by Adam et al. (2016).

The traditional Raman lidar temperature retrieval method, introduced by Cooney (1972), uses the ratio of two PRR signals from the Stokes branch which have been corrected for saturation, background, and other instrumental effects as required. The PRR spectrum contains two branches: Stokes and anti-Stokes. Both branches have approximately the same intensity and they are positioned symmetrically in wavelength on either side of the excitation line. The traditional Raman lidar temperature retrieval algorithm requires the assumption of an analytic form of a lidar calibration function whose coefficients are usually determined with external measurements, such as radiosondes (Behrendt, 2005). The calibration function is an approximation of the relationship of the signal ratio and temperature and depends on two or more coefficients. Calibration errors exceeding 0.5 K can arise if the calibration data do not cover a sufficient temperature range (Behrendt, 2005).

Of primary importance is the calibration of the lidar returns to allow for absolute temperature measurements. In the traditional method, the ratio of the corrected photocounts is fit to a set of corresponding temperature data points usually obtained from radiosondes. Behrendt (2005) gives a comprehensive overview of the traditional rotational Raman lidar temperature calculation method. Over the years the traditional temperature method has been improved by advancements in the instrumentation capabilities and improvements to the estimation and calibration techniques. Below some examples of innovations in this area since the (Behrendt, 2005) review.

Radlach et al. (2008) and Weng et al. (2018) introduce changes to the their Raman lidar systems to improve the temperature measurements. Radlach et al. (2008) introduced a new high-resolution rotational Raman lidar system with a receiving system that uses multicavity interference filters in a sequential setup to improve the efficiency of the elastic and rotational Raman signal separation. Together with the filter adjustments they have made noontime temperature measurements with uncertainties less than 1 K up to 1 km for 1 min integration time. Weng et al. (2018) introduced a new PRR lidar system that effectively detects two isolated nitrogen molecule PRR line signals and elastic backscatter signals. With this new system, temperatures at any given time can be obtained without a calibration.

The accuracy of the traditional Raman temperature estimations is highly sensitive to the estimation of the calibration function and calibration coefficients. Zuev et al. (2017) has investigated the use of nonlinear calibration functions to improve the accuracy of the traditional Raman temperature estimation and relaxing the assumption of sampling two single PRR lines. The results showed the $\frac{1}{T}$ term expressed in a form of a quadratic function of log ratio of the PRR measurements as best for practical use. He et al. (2018) proposed a new calibration method for PRR lidar temperature profiling based on the different temperature sensitivities of Stokes and anti‐Stokes PRR lines. They reconstruct the expression of the differential backscatter cross section according to the temperature dependencies of each component and form both a temperature factor and a calibration factor in the intensity ratio. This new method reduces the temperature error by 50 % compared with commonly used calibration methods in conditions of low signal‐to‐noise ratio (SNR).

Yan et al. (2019) has proposed an iterative method for determining Raman temperature. Their method allows independent alternating solutions to the high- and low-quantum-number PRRs separately, where high-quantum-number PRR lidar returns are used to solve for the channel constant, while low-quantum-number PRR returns with higher SNR are used for retrieving temperature profiles in an iterative fashion. Their results showed that the effective temperature retrieval height for their system improved from 17 to 25 km.

The application of the optimal estimation method (OEM) for temperature retrievals using pure rotational Raman (PRR) lidar measurements has several advantages over the traditional method, including the ability to find temperature without assuming an analytic form of the temperature or count ratio relation. Our OEM retrieval does not use the ratio of the counts. Rather we use a forward model which includes complete physics to describe the raw count profiles. For calibration, the ratio of the lidar constants, here referred to as coupling constant, needs to be determined. The coupling constant can in principle be estimated at a single point, such as a nearby flux tower or surface measurement. Our OEM retrieval has other important benefits over the traditional method as it can directly retrieve ancillary parameters (in addition) to temperature, such as geometrical overlap, particle extinction, and the lidar constant. Our OEM is also capable of providing a full uncertainty budget, including both random and systematic uncertainties on a profile-by-profile basis, including the systematic uncertainty introduced in the retrieved temperature by the estimation of the coupling constant. The OEM is an inverse method and is a standard tool in the retrieval of geophysical parameters from passive atmospheric remote sensing instruments. Recent studies including Povey et al. (2014); Sica and Haefele (2015, 2016), and Farhani et al. (2019) have shown that OEM can be used to retrieve atmospheric aerosol, water vapour mixing ratio, middle and upper atmospheric temperature, and ozone using lidar measurements.

In Sect. 2 a brief description of the instrument and the measurements used in this study is presented. Section 3 presents the development of the PRR lidar equation.

The development of a forward model for application of the OEM to PRR temperature retrieval is given in Sect. 4. The OEM-retrieved temperature results from the PRR measurements for different atmospheric conditions are shown in Sect. 5. A discussion of these results is presented in Sect. 6, followed by conclusions.

2 The RAman Lidar for Meteorological Observations

PRR measurements from the RAman Lidar for Meteorological Observations (RALMO), located in Payerne, Switzerland (46^∘48^′ N, 6^∘56^′ E), are used for the OEM temperature profiling. RALMO is a fully automated lidar built at the École Polytechnique Fédérale de Lausanne and operated by MeteoSwiss (Dinoev et al., 2013). It is dedicated to operational meteorology, validating models and satellite measurements, and climate studies. RALMO has been operating nearly continuously since 2008, with an average data availability of 50 %. Data gaps are due to rain and low clouds (approximately 30 % of the time), maintenance (1–2 d per month), and other occasional technical problems. RALMO consists of a frequency-tripled Q-switched Nd:YAG laser of 354.7 nm producing up to 400 mJ of emission energy at a 30 Hz repetition rate. The pulse duration is 8 ns. The laser is operated at 300 mJ of energy per pulse to extend the lifetime of the flash lamps from 20 to approximately 60 million shots. The RALMO receiver uses four parabolic mirrors, each with 1 m focal length and 30 cm diameter, and it is fiber coupled to a two-stage grating polychromator. The data acquisition system consists of photomultipliers and analog or photon-counting transient recorders from Licel. The system obtains a measurement by adding together 1800 laser shots (every minute) at a vertical resolution of 3.75 m. For a detailed description of the lidar and a detailed validation of the temperature measurements, the reader is referred to Dinoev et al. (2013).

The returns of the Raman-shifted backscatter arising from rotational energy state transitions of nitrogen and oxygen molecules due to the excitation at the laser wavelength at 354.7 nm are detected in analog or photon-counting mode. The high-quantum-number channel (JH) of RALMO is assigned to the backscattered signals from the energy exchange that occurs in the high-quantum-number states for both the Stokes (355.77–356.37 nm) and anti-Stokes (353.07–353.67 nm) branches. The low-quantum-number channel (JL) is assigned to the signals from the energy exchange occurring in the low-quantum-number states in the Stokes (355.17–355.76 nm) and anti-Stokes (353.67–354.25 nm) branches.

3 The PRR lidar equation

The backscattered PRR signal is given by the Raman lidar equation:

\begin{matrix} (1) & \begin{aligned} N_{RR, t} (z) & = \frac{C_{RR}}{z^{2}} O (z) n (z) Γ_{atm}^{2} (z) \\ (\sum_{i = O_{2}, N_{2}} \sum_{J_{i}} η_{i} {(\frac{d σ}{d Ω})}_{π}^{i} (J_{i})) + B_{RR} (z), \end{aligned} \end{matrix}

where the true backscattered PRR signal, N_RR,t, is a function of height z, C_RR is the lidar constant, n(z) is the number density of the air molecules, O(z) is the geometrical overlap, η_i is the volume mixing ratio of nitrogen and oxygen, Γ_atm(z) is the atmospheric transmission, ${(\frac{d σ}{d Ω})}_{π}^{i} (J_{i})$ is the attenuated differential backscatter cross section for each RR (rotational Raman) line J_i, and B_RR(z) is the background of the measured signal. For different lidar systems the background can either be a constant or vary with height. Since air below 80 km is a constant mixture of oxygen and nitrogen, η_i is a constant. The lidar constant C_RR depends on the number of transmitted photons, detector efficiency, and the area of the telescope. The attenuated differential backscatter cross section for Stokes and anti-Stokes line pairs of equal quantum number of the PRR spectrum is expressed as (Penney et al., 1974):

\begin{matrix} (2) & \begin{aligned} {(\frac{d σ}{d Ω})}_{π}^{i} (J_{i}) = \frac{112}{15} . \frac{π^{4} g_{i} (J_{i}) h c B_{0, i} (v_{o} + Δ v_{i} (J_{i}))^{4} {ζ_{i}}^{2}}{(2 I_{i} + 1)^{2} k T} \\ \times (X^{+} (J_{i}) τ^{+} (J_{i}) + X^{-} (J_{i}) τ^{-} (J_{i})) \exp (\frac{- E_{rot, i} (J_{i})}{k T}), \end{aligned} \end{matrix}

where for the Stokes branch

\begin{matrix} (3) & X^{+} (J) = \frac{(J + 1) (J + 2)}{2 J + 3} for J = 0, 1, 2, \dots \end{matrix}

and for the anti-Stokes branch

\begin{matrix} (4) & \begin{aligned} X^{-} (J) = & \frac{J (J - 1)}{2 J - 1} for J = 2, 3, 4, \dots and X^{-} (J) = 0 \\ for J = 0, 1 . \end{aligned} \end{matrix}

τ⁺(J_i) and τ⁻(J_i) are the transmissions of the receiver for the Stokes and anti-Stokes lines J_i, respectively. g_i(J) is the statistical weighting factor, which depends on the nuclear spin I_i for each atmospheric constituent, h is Planck's constant, c is the velocity of light, k is Boltzmann's constant, B_0,i is the ground-state rotational constant, v₀ is the frequency of the incident light, and ζ_i is the anisotropy of the molecular polarizability. The rotational energy E_rot,i(J) for each Stokes and anti-Stokes branch is estimated based on the assumption of a homonuclear diatomic molecule in the quantum state J for nitrogen and oxygen molecules with no electronic momentum coupled to the scattering (Behrendt, 2005).

The response of the photomultiplier tubes operated in the photon-counting mode can become nonlinear at high count rates. In the case of RALMO, the true and observed counts are related by the equation for non-paralyzed systems:

\begin{matrix} (5) & N_{observed} = \frac{N_{true}}{1 + N_{true} γ}, \end{matrix}

where γ is the counting system dead time.

4 Application of the OEM for PRR temperature retrieval

4.1 Brief review of the optimal estimation method

The OEM is an inverse method that uses the measurements y to estimate the state (retrieval) variables x of a system via a forward model. The forward model F contains all the atmospheric and instrumental physics that describe the measurements. The forward model can include model parameters b, which are assumed and not retrieved, and their effect on the retrieved quantity uncertainties can be calculated.

The measurements are related to the forward model by

\begin{matrix} (6) & y = F (x, b) + ϵ, \end{matrix}

where ϵ represents measurement noise. Under the assumption that all parameters have Gaussian probability density functions Bayes’ theorem can be applied to determine the cost function,

\begin{matrix} (7) & \begin{aligned} cost & = [y - F (x, b)]^{T} S_{y}^{- 1} [y - F (x, b)] + [x - x_{a}]^{T} \\ S_{a}^{- 1} [x - x_{a}], \end{aligned} \end{matrix}

where x_a is the a priori of the retrieval parameters, S_y is the measurement covariance, which describes the random measurement uncertainty, and S_a is the a priori covariance. The cost function measures the goodness of fit for a solution, and for good models the cost is on the order of unity. For nonlinear forward models, the Marquardt–Levenberg method can be used iteratively to minimize the cost of the retrieval (see Sect. 5.7 in Rodgers, 2000, for details).

The uncertainty budget is determined from the measurement and model parameter covariance matrices (Rodgers, 2000). The total error covariance, S_total, is

\begin{matrix} (8) & S_{total} = S_{m} + S_{F}, \end{matrix}

where S_m is the retrieval covariance due to measurement noise and S_F is the retrieval covariance due to the forward model parameter uncertainty. The retrieval covariance due to measurement noise, S_m, is

\begin{matrix} (9) & S_{m} = {GS}_{y} G^{T}, \end{matrix}

where G is the gain matrix, which is the sensitivity of the retrieval to the measurements. The retrieval covariance due to the forward model parameters, S_F, is

\begin{matrix} (10) & S_{F} = {GK}_{b} S_{b} K_{b}^{T} G^{T}, \end{matrix}

where K_b and S_b are the forward model parameter Jacobian and covariance matrices, respectively. The model parameter Jacobians, K_b, can be estimated analytically or numerically for each model parameter. To construct S_b we require the uncertainties in the model parameters. We recommend Rodgers (2000) for more details of the OEM.

4.2 The forward model for a PRR lidar

The forward model describes the measurement as a function of both the state of the atmosphere and instrumental parameters. The core of our forward model is the lidar equation as presented in Sect. 3. It is called four times to generate the measurements corresponding to high and low quantum numbers, i.e. JH and JL, with photon-counting and analog detection. Analog detection is assumed to be linear.

The pressure, P(z), and temperature, T(z), can be taken from either a radiosonde measurement or an atmospheric model. The background noise, B_RR, is in general a function of height, z, but is constant with height for RALMO. Unlike all the other existing forward models for lidar except Povey et al. (2012) (which was designed specifically to determine overlap), we retrieve O(z) the geometrical overlap function in addition to temperature. The overlap functions of JH and JL channels were assumed to be the same (Dinoev et al., 2010).

The atmospheric transmission, Γ_atm(z) in Eq. (1), includes both molecular and particle scattering.

\begin{matrix} (11) & Γ_{atm} (z) = \exp [- \int_{0}^{z} (α_{mol} + α_{par}) d z], \end{matrix}

where α_mol is the molecular extinction coefficient and α_par is the particle extinction coefficient. The molecular extinction can be expressed using the Rayleigh cross section σ_Ray and air density n_air as

\begin{matrix} (12) & α_{mol} = σ_{Ray} . n_{air}, \end{matrix}

where σ_Ray is calculated using the expressions given by Nicolet (1984).

For each channel the subscript RR is replaced by JL and JH, the high- and low-quantum-number PRR channels. Then C_RR, B_RR, and J_i have different values.

RALMO detects multiple Stokes and anti-Stokes lines from both nitrogen and oxygen in a PRR spectrum. Therefore, to determine the attenuated backscatter cross-section in the forward model, we require knowledge of the exact number of quantum states detected by each RALMO PRR channel. From the JH and JL channel characteristics we can calculate the range of frequency shifts for each channel relative to the elastic wavelength 354.7 nm. Then using the equations given by Herzberg (2013) we can determine the quantum numbers J_i for both nitrogen and oxygen molecules. This calculation process is repeated to determine the J_i numbers for the JL channel of the RALMO. A summary of the findings is given in Table 1.

Table 1The respective quantum lines from both nitrogen and oxygen in a PRR spectrum detected by the RALMO temperature polychromator. Note the given quantum lines are valid for both the Stokes and anti-Stokes branches.

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In order to establish absolute calibration, we define the coupling constant R as the ratio of the lidar constants C_JL and C_JH,

\begin{matrix} (13) & R = \frac{C_{JH}}{C_{JL}}, \end{matrix}

and use the substitution C_JH=RC_JL. The coupling constant is height independent and can be determined with no assumptions at, if desired, a single altitude using the following equation derived from Eq. (1).

\begin{matrix} (14) & R = (\frac{N_{t, JH} - B_{JH}}{N_{t, JL} - B_{JL}}) / (\frac{σ_{JH} (T, z)}{σ_{JL} (T, z)}) . \end{matrix}

The differential cross section terms are determined by applying temperature from a coincident reference temperature, typically from a radiosonde. For a well designed lidar system the coupling constant should be stable over weeks. Unlike the fitting of an analytic calibration function to the data as in the traditional method, R can be estimated at a specific height or range of heights, which can be over a narrow range of temperature without introducing extrapolation errors. We extensively tested this assertion using both synthetic and real measurements. The results show that the estimation of R is indeed height independent. The value of R is only affected by the measurement noise. Hence, we recommend using a range of heights or a specific height where the photocounts have a high signal-to-noise ratio.

Using R in the forward model allows us to retrieve only one lidar constant, while constraining the two channels to vary so as to satisfy Eq. (13). We will see in the next section that any variations or uncertainty in the determination of R introduces an uncertainty on the order of 0.2 K to the retrieved temperature profile.

The retrieval parameters (Table 2) in our OEM algorithm are temperature, background signals (including photomultiplier shot noise, sky background, and offset for analog channels), the lidar constants, dead times of the photon-counting systems, geometrical overlap, and particle extinction as a function of height. In OEM we can retrieve parameters on a height grid where the resolution can be the same or different than the vertical resolution of the height grid that the measurements obtained. If the retrieval grid is coarser than the measurement grid, we use linear interpolation on retrieved quantities when they are required in the forward model.

Table 2Values and associated uncertainties for the OEM retrieval and forward model parameters.

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4.3 Implementation of the RR temperature retrieval

The OEM solver in the Qpack software package is used for our retrieval (Eriksson et al., 2005). The solver requires the following as inputs: the measurements from each lidar channel and their covariances, a priori values for the retrieval parameters and their covariances, model (b) parameters and their covariances, and the Jacobians of the forward model.

The lidar measurements from the photon-counting channels follow Poisson counting statistics in the range where the counts are linear. Thus, the measurement variance S_y is equal to the number of photons in each height bin, assuming no correlation between the height bins (that is, the off-diagonal terms in the S_y matrix are zero). The autocorrelation function method of Lenschow et al. (2000) is used to estimate the measurement covariance of the analog and photon-counting measurements in the nonlinear region. For both analog and photon-counting channels, the a priori backgrounds and their variances are taken as the mean and the variance of the measurements above 50 km height.

The U.S. Standard Atmosphere (NASA, 1976) model temperature profile is normalized to the surface temperature from the coincident sonde temperature and then used as the a priori temperature profile in our retrievals. Due to the high variability in the temperature, a standard deviation of 35 K for all heights is used to specify the covariance matrix for a priori values. Other choices of a priori temperature profile are possible but as an operational fully automated lidar system RALMO retrievals must be performed automatically every 30 min, so the choice of the U.S. Standard Atmosphere with this covariance simplifies this procedure. As discussed in Eriksson et al. (2005), the elements of the retrieval and model parameters are often correlated, and some of the covariance matrices should have off-diagonal elements. Off-diagonal elements are parameterized with the correlation length and an appropriate analytical function describing the decay of the correlation. For this study, we used a tent function with a 1 km correlation length for temperature retrievals (Eriksson et al., 2005).

Molecular and particle extinction terms occur in the atmospheric transmission term of Eq. (11). An a priori particle extinction profile is estimated based on the following expression:

\begin{matrix} (15) & α_{par} = LR \cdot β_{par} = LR \cdot β_{mol} \cdot (ℜ_{β} - 1), \end{matrix}

where LR is the lidar ratio and β_par is the particle backscatter coefficient. β_par is related to the backscatter ratio ℜ_β as (Whiteman, 2003):

\begin{matrix} (16) & ℜ_{β} = \frac{(β_{mol} + β_{par})}{β_{mol}} . \end{matrix}

The backscatter ratio ℜ_β is estimated using the RALMO PRR and elastic measurements. Bucholtz (1995) gives a method for calculating β_mol using pressure, temperature, and Rayleigh cross sections. The Rayleigh extinction cross sections required for β_mol estimation are computed using the formula of Nicolet (1984). Calculated Rayleigh extinction cross sections are also used to estimate the air density profile used as a b parameter in the forward model, assuming an uncertainty of 1 % for the standard deviation.

The lidar ratio LR is chosen based on the ℜ_β values for the given measurements. Typically ℜ_β values inside clouds are greater than 2. Thus, for this study the height at which ℜ_β is first equal to 2 is considered the height of the cloud base or the height of an aerosol layer (cloud or aerosol layer base height). The cloud or aerosol layer base height is later used to determine the transition height that constrains the range of the geometrical overlap and the particle extinction retrievals. In cloud-free sky conditions that we will refer to as clear sky conditions from here on (that is if ℜ_β does not exceed 2), LR is assumed to be 80 sr inside the boundary layer and 50 sr elsewhere. In cloudy conditions, LR is assumed to be 20 sr within the clouds present below 6 km. If the cloud is above 6 km, LR is assumed to be 15 sr within the cloud. These choices for lidar ratios are taken from Ansmann et al. (1992 a) and Pappalardo et al. (2004). Accurate LR is not crucial, as it is used to estimate an a priori particle extinction profile. However, we can calculate a LR profile using the OEM-retrieved α_par and compare it with the initially chosen LR values to evaluate how good a choice of the initial value is.

The effect of geometrical overlap and particle extinction on the signals is coupled and hence retrieving both parameters simultaneously is not possible unless at least one of the effects is highly constrained. The particle extinction is indefinitely measured using the backscatter ratio outside clouds, and overlap is well known above the height of full overlap, i.e. above 6 km (Dinoev et al., 2010). We use this knowledge to define a transition height, in clear skies 6 km or at the cloud base height, whatever is lower. Below this height overlap is retrieved and above this height particle extinction is retrieved. The a priori overlap function is estimated from measurements in clear-sky conditions. A 50 % standard deviation is used for geometrical overlap below the transition height and a constant standard deviation of 10⁻³ is used above this height, constraining the geometrical overlap to the a priori values above the transition height. For particle extinction, a standard deviation of $10^{- 6} {km}^{- 1}$ is used below the transition height to constrain the retrieval, then a 50 % standard deviation is used above this height, allowing the OEM to retrieve exclusively the particle extinction. The a priori covariance matrices for both particle extinction and geometrical overlap are determined using a tent function with a 100 m correlation length.

The a priori lidar constants for the JL analog and JL photon-counting channels are estimated by fitting the measurements generated using the sonde temperature and pressure used in the forward model to the PRR measurements. For analog measurements, the fitting range is between 1.5 to 2 km height. For photon-counting measurements with clear conditions or cloud/aerosol presence above 8 km, 6 to 8 km is used as the fitting range to ensure the photocounts are linear. If the photon-counting measurements contain cloud or aerosols in the geometrical overlap region, a fitting range below this is used, typically 3.5–4 km height. The fitting uncertainty for each analog and photon-counting lidar constants is used as the variance of the a priori lidar constant.

The a priori dead times for the two photon-counting systems are considered to be 3.8 ns, consistent with estimations from previous studies for RALMO and with values specified by the manufacturer (Sica and Haefele, 2015, 2016; Dinoev et al., 2010). The uncertainty in the dead time is taken as 10 %. Coincident radiosonde pressure profiles are used assuming a 10 % standard deviation. The coupling constants for analog (R_a) and photon-counting (R) channels are estimated by fitting the ratio of PRR measurements with the ratio of the differential cross section (Eq. 14). The coupling constants are estimated using the same fitting range as the lidar constants. Table 2 gives a summary of the parameters and associated uncertainties in the retrieval and model parameters used in the forward model.

5 Results from the temperature retrieval

We present four different measurement situations which demonstrate the robust nature of our OEM temperature retrieval. Details of each case study are given in Table 3. The RALMO measurements used in the retrievals are added in time over 30 min and to 15 m in height. Analog measurements are used from the surface to 6 km height, while photon-counting measurements are used from 4 to 28 km. The retrieval grid has a vertical resolution of 60 m at all heights. For all the cases given in Table 3 we used radiosonde measurements that coincide within 1 h of the lidar measurements for comparison purposes to estimate the required a priori information and to determine the forward model b parameters (Table 2).

Table 3Details of the four cases in clear and cloudy sky conditions we present to demonstrate the flexibility of our OEM temperature retrieval.

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The traditional temperature profiles that will be shown later in this section are calculated using count profiles consisting of glued analog and photon-counting measurements, which are corrected for nonlinearity and background before processing. The vertical resolution of the traditional temperature profiles is 30 m at the lowest heights, increasing to 400 m by the upper heights to decrease the magnitude of the statistical uncertainty. A calibration function linear in 1∕T is used and the two coefficients have been determined with radiosonde data using the altitude range from the surface to 10 km. Hence, for this comparison the (traditional method) temperature profile is smoother than the OEM-retrieved temperature profile. The change in vertical resolution is based on the random uncertainty in the temperature profile at each height. The vertical resolution is decreased until the temperature uncertainty becomes less than a threshold value, set here as 1 K.

5.1 Case 1: nighttime with clear conditions

Figure 1 shows the RALMO 30 min coadded count measurements in the four PRR channels for case 1 given in Table 3. The analog signal is linear over its entire range. The photon-counting measurements are affected by nonlinearity below about 9 km for the JL channel (6 km for JH) by about 0.5 % (count rates of about 1 MHz) and as much as 10 % for the JL photon counts around 4 km altitude (Eq. 5).

Figure 2 shows the residuals, which are defined as the difference between the forward model and the measurements. For a good retrieval with cost on the order of unity, the residuals (blue curve) should be on the order of the standard deviation of the measurement noise (red curve), and indeed this is the case, hence the forward model is not over-fitting the measurements (e.g. cost much less than unity).

https://www.atmos-meas-tech.net/12/5801/2019/amt-12-5801-2019-f01

Figure 1Count rate for 30 min of coadded RALMO measurements from 23:00 UT on 9 September 2011, a clear night; (a) Photon-counting channels (blue curve, JL; red curve, JH); (b) Analog channels.

Retrieval of temperature from a multiple channel pure rotational Raman backscatter lidar using an optimal estimation method

4.1 Brief review of the optimal estimation method

4.2 The forward model for a PRR lidar

4.3 Implementation of the RR temperature retrieval

5.1 Case 1: nighttime with clear conditions

5.2 Case 2: daytime with clear conditions

5.3 Case 3: nighttime with cirrus cloud

5.4 Case 4: nighttime with lower level cloud