AMTAtmospheric Measurement TechniquesAMTAtmos. Meas. Tech.1867-8548Copernicus PublicationsGöttingen, Germany10.5194/amt-10-315-2017Tropospheric temperature measurements with the pure rotational Raman lidar
technique using nonlinear calibration functionsZuevVladimir V.GerasimovVladislav V.gvvsnake@mail.ruPravdinVladimir L.PavlinskiyAleksei V.NakhtigalovaDaria P.Institute of Monitoring of Climatic and Ecological Systems SB RAS,
Tomsk, 634055, RussiaTomsk State University, Tomsk, 634050, RussiaTomsk Polytechnic University, Tomsk, 634050, RussiaVladislav V. Gerasimov (gvvsnake@mail.ru)27January20171013153321June201613June201629December20162January2017This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://amt.copernicus.org/articles/10/315/2017/amt-10-315-2017.htmlThe full text article is available as a PDF file from https://amt.copernicus.org/articles/10/315/2017/amt-10-315-2017.pdf
Among lidar techniques, the pure rotational Raman (PRR) technique is the
best suited for tropospheric and lower stratospheric temperature
measurements. Calibration functions are required for the PRR technique to
retrieve temperature profiles from lidar remote sensing data. Both
temperature retrieval accuracy and number of calibration coefficients depend
on the selected function. The commonly used calibration function (linear in
reciprocal temperature 1/T with two calibration coefficients) ignores all
types of broadening of individual PRR lines of atmospheric N2 and
O2 molecules. However, the collisional (pressure) broadening dominates
over other types of broadening of PRR lines in the troposphere and can
differently affect the accuracy of tropospheric temperature measurements
depending on the PRR lidar system. We recently derived the calibration
function in the general analytical form that takes into account the
collisional broadening of all N2 and O2 PRR lines (Gerasimov and
Zuev, 2016). This general calibration function represents an infinite series
and, therefore, cannot be directly used in the temperature retrieval
algorithm. For this reason, its four simplest special cases (calibration
functions nonlinear in 1/T with three calibration coefficients), two of
which have not been suggested before, were considered and analyzed. All the
special cases take the collisional PRR lines broadening into account in
varying degrees and the best function among them was determined via
simulation. In this paper, we use the special cases to retrieve tropospheric
temperature from real PRR lidar data. The calibration function best suited
for tropospheric temperature retrievals is determined from the comparative
analysis of temperature uncertainties yielded by using these functions. The
absolute and relative statistical uncertainties of temperature retrieval are
given in an analytical form assuming Poisson statistics of photon counting.
The vertical tropospheric temperature profiles, retrieved from nighttime
lidar measurements in Tomsk (56.48∘ N, 85.05∘ E; Western
Siberia, Russia) on 2 October 2014 and 1 April 2015, are presented as an
example of the calibration functions application. The measurements were
performed using a PRR lidar designed in the Institute of Monitoring of
Climatic and Ecological Systems of the Siberian Branch of the Russian Academy
of Sciences for tropospheric temperature measurements.
Introduction
The pure rotational Raman (PRR) technique is known to be the best suited for
lower atmosphere temperature measurements (Wulfmeyer et al., 2015). The
retrieval algorithm of vertical temperature profiles of the troposphere and
lower stratosphere from PRR lidar raw signals consists of four main steps:
PRR lidar raw data averaging to improve the signal-to-noise ratio and
decrease the statistical uncertainties;
lidar calibration, i.e.,
determination of the lidar calibration function coefficients by applying,
for example, the least square method to the reference radiosonde (or model) data and
previously averaged lidar data;
temperature profile retrieval by using the temperature retrieval function
derived from the selected calibration function;
estimation of the absolute and relative uncertainties of the temperature
retrieval and calculation of the difference between the reference
temperature (radiosonde, model) and temperature retrieved from lidar data.
The PRR lidar technique suggested by Cooney (1972) is based on the
temperature dependence of individual lines intensity of atmospheric N2
and O2 PRR spectra. The intensity I(T, λ) of a
single PRR line of the wavelength λ backscattered by excited N2
or O2 molecules can be expressed as (Penney et al., 1974)
I(λ,T)=PLβπ(λ,T),
where P is the incident laser-beam power, L is the length
of the scattering volume, and βπ (λ, T) is the
backscatter cross section (atmospheric backscatter coefficient). The
backscattered signals of the Stokes and/or anti-Stokes branches of the
spectra can be used for temperature determination. The intensities of
individual PRR lines, corresponding to low and high rotational quantum
numbers J of the initial states of the PRR transitions, are of
opposite temperature dependence (Behrendt, 2005). Namely, the intensity of
each N2 PRR line with Jlow≤8
(Jlow≤9 for O2 PRR lines) decreases with
increasing temperature and, conversely, the intensity of N2 PRR lines
with Jhigh≥9 (Jhigh≥11 for
O2 PRR lines) increases with increasing temperature in both branches of
the spectra (Fig. 1). Note that only odd lines beginning with odd J
exist in O2 PRR spectrum (Wandinger, 2005). A ratio of backscattered
signal intensities from two PRR-spectrum bands with opposite temperature
dependence is required for air temperature T determination. However,
the PRR lidar theory (Cooney, 1972) gives the exact temperature dependence
only for intensity ratios of two individual PRR lines corresponding to
certain Jlow and JhighQindiv.(T)=I(Jlow,T)I(Jhigh,T)=βπ(Jlow,T)βπ(Jhigh,T)=expα+βT,
where the constants α and β are completely
defined from the theory.
Equidistant PRR spectra of N2 and O2 linear
molecules, schematic drawing of the IMCES lidar monochromator transmission
functions (MTF), and envelopes of N2 PRR spectrum at different
temperatures. The red and blue envelopes correspond to the temperature of 280
and 220 K, respectively. The laser-beam wavelength is 354.67 nm. 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. All PRR line intensities are
normalized to the intensity of N2 PRR line with J= 6 of
the anti-Stokes branch at T= 220 K.
In practice, diffraction gratings (DGs) or interference filters (IFs)
extract several adjacent PRR lines in the lidar temperature channels from
backscattered light. IFs extract PRR lines from the anti-Stokes branches of
N2 and O2 PRR spectra (Behrendt and Reichardt, 2000; Behrendt et
al., 2002, 2015; Alpers et al., 2004; Di Girolamo et al., 2004; Radlach et al.,
2008; Achtert et al., 2013; Newsom et al., 2013; Li
et al., 2015). DGs extract PRR lines from both the Stokes and anti-Stokes
branches of the spectra (Ansmann et al., 1999; Kim et al., 2001; Chen et
al., 2011; Jia and Yi, 2014). Thus, one should consider the following
expression (Arshinov et al., 1983)
QΣ(T)=IlowΣ(T)IhighΣ(T)=∑JN2βπ(JN2,T)+∑JO2βπ(JO2,T)low∑JN2βπ(JN2,T)+∑JO2βπ(JO2,T)high,
where βπ(JN2,T) and βπ(JO2,T) are the backscatter coefficients corresponding to
N2 and O2 individual PRR lines, respectively;
IlowΣ(T) and IhighΣ(T) are the
overall intensities of the PRR lines which enter the corresponding lidar
temperature channels; indexes “low” and “high” show that summations in
the numerator and denominator refer to the corresponding PRR-spectrum bands
with Jlow and Jhigh. The ratio
QΣ (T) in Eq. (3) has a complicated temperature
dependence and cannot be expressed as a simple function of T. For
this reason, an approximation (calibration) function fcΣ(T) for the ratio QΣ (T) is required to retrieve
temperature profiles from lidar remote sensing data (Behrendt, 2005). The
temperature retrieval accuracy and number of calibration coefficients depend
on the selected calibration function.
Assuming that each PRR line profile represents the Dirac function, the
general calibration function can be written in a natural logarithm form as
follows (Gerasimov and Zuev, 2016):
lnQΣ(T)≈lnfcΣ(T)=A+BT+CT2+DT3+⋯⇔y=A+Bx+Cx2+Dx3+⋯,
where A, B, C, D, etc. are the
calibration (fit) coefficients determined by applying the least square method
to lidar remote sensing (or simulation) data and reference radiosonde (or
model) data; the symbol ⇔ denotes the equivalence of
expressions; x=1/T is the reciprocal temperature. The
n order in x polynomial is assumed to retrieve temperature
profiles with any desired accuracy depending on n (Di Girolamo et
al., 2004). The linear in x special case of Eq. (4) with two
calibration coefficients A and B (Arshinov et al., 1983)
and the second-order in x polynomial with three calibration
coefficients A, B, and C (Behrendt and Reichardt,
2000) are usually used by lidar researchers for temperature retrievals in the
troposphere and lower stratosphere. However, N2 and O2 PRR lines
are broadened by the Doppler and molecular collision effects. Hence, their
backscatter profiles are described by a Voigt function, which is a
convolution of certain Gaussian and Lorentzian functions (Nedeljkovic et al.,
1993). As the molecular collision effect dominates over the Doppler effect in
the troposphere (Ivanova et al., 1993), one can consider the Lorentzian
function for a PRR line shape description instead of the Voigt one (Ginzburg,
1972). Therefore, all collisionally broadened PRR lines contribute to the
signals detected in both lidar temperature channels due to the long
Lorentzian tails of the line profiles (Measures, 1984), and the general
calibration function takes on the form (Gerasimov and Zuev, 2016)
IMCES lidar optical layout (see also Table 1): PC & DAS indicates
personal computer and data acquisition system; PhC is the photon counter;
PMT1–PMT3 are photomultiplier tubes; F0–F3 are optical fibers; FB is the four fiber
bundle, connecting two monochromator blocks; DGM is the double-grating
monochromator; L1 and L2 are lenses; DG1 and DG2 are diffraction gratings; BE is the beam
expander with expansion factor of 10; M is the mirror; SM is the stepping motor.
lnQall(T)=⋯+A-2T+A-1T+A0+A1T+A2T+⋯=∑n=-∞∞AnTn2,
where An are the calibration coefficients and Eq. (4)
represents a special case of Eq. (5). All the calibration functions mentioned
above are valid only when the parasitic elastic signal backscattered by
atmospheric aerosols and molecules is sufficiently suppressed in the lidar
temperature channels. The state-of-the-art narrow-band IFs and DGs provide
the suppression of the parasitic signal intensity in the channels up to 8–10
orders of magnitude (Achtert et al., 2013; Hammann and Behrendt, 2015;
Hammann et al., 2015).
IMCES lidar data taken between 03:45 and 05:15 LT on 1
April 2015 (31 March, 21:45–23:15 UTC). (a) Raw photocounts
NL and NH detected in the lidar channels with
Jlow and Jhigh, respectively,
together with the single-averaged ones N‾L and
N‾H. (b) Raw photocounts ratio
Q=NL/NH, single-averaged
photocounts ratio Q=N‾L/N‾H,
and additionally averaged ratio QI=Q‾=N‾L/N‾H‾.
In order to take into account the atmospheric extinction of backscattered
signals and their losses in the lidar transmitting and receiving optics, one
should consider the lidar equation (Measures, 1984)
N(λ,z,T)=ηN0G(λ,z)cτ02ξ(λ)Az2βπ(λ,z,T)Θ2(λ,z),
where N (λ,z,T) is the number of
backscattered photons (photocounts) detected by a photomultiplier tube (PMT)
in a lidar temperature channel, N0 is the number of emitted
photons, η is the PMT quantum efficiency, G (λ,
z) is the laser-beam receiver-field-of-view overlap, τ0 is
the laser pulse duration, c is the speed of light, ξ(λ) is the transmittance of the lidar receiving optical
system,
A is the receiver telescope area, z is the scattering
region altitude, and Θ (λ, z) is the transmission
coefficient through the atmosphere between the scattering region and the
lidar. Taking Eqs. (5) and (6) into account, the ratio of the
background-subtracted photocounts NL and NH from
two spectrally close bands involving several N2 and O2 PRR lines
with Jlow and Jhigh becomes (Newsom
et al., 2012; Newsom et al., 2013)
Q(T,z)=NL(T,z)NH(T,z)=GL(z)GH(z)exp∑n=-∞∞BnTn2=O(z)exp∑n=-∞∞BnTn2,
where Bn are the calibration coefficients and
O(z) =GL(z) /GH(z)
is the laser-beam receiver-field-of-view overlap function. At the complete
overlap altitudes (usually above the atmospheric boundary layer), where
O(z) = 1, Eq. (7) goes over into the calibration
function like Eq. (5):
lnQ(T)=∑n=-∞∞BnTn2.
Note that the same result can be obtained on the assumption that the
collisionally broadened elastic backscattered signal leaks into the nearest
(to the laser line) lidar temperature channel (Gerasimov et al., 2015).
In our recent Optic Express paper, we considered the physics of our approach,
derived mathematically the general calibration function that takes into
account the collisional broadening of all N2 and O2 PRR lines,
analyzed four nonlinear three-coefficient special cases of Eq. (8) via
simulation to be used in the temperature retrieval algorithm, and determined
the best function among them. In this paper, we apply these calibration
functions to real lidar remote sensing data. The calibration function
best suited for tropospheric temperature retrievals (for our PRR lidar
system) is determined from the comparative analysis of temperature
uncertainties yielded by using these functions.
Temperature profiles from radiosondes launched on 1 April 2015 at
06:00 LT (00:00 UTC) in Novosibirsk (station 29634) and Kolpashevo
(station 29231) as well as temperature points over Tomsk retrieved from
constant pressure altitude charts (CPACs).
(1 April 2015) Temperature profile retrieved using the
temperature retrieval function (Eq. 11) derived from the standard linear
calibration function (Eq. 10, Arshinov et al., 1983). The absolute and
relative uncertainties ΔT=ΔT‾‾ and
(ΔT/T)=(ΔT‾‾/T) are calculated by
Eqs. (A21) and (A22), respectively. The values TCPAC over Tomsk
are retrieved from the 700, 500, 400, 300, and 200 hPa constant pressure
altitude charts (CPACs). The radiosonde data from the nearest station in
Novosibirsk and Kolpashevo are given for comparison.
(1 April 2015) Temperature profile retrieved using the
temperature retrieval function (Eq. 13) derived from the standard calibration
function suggested by Behrendt and Reichardt (2000). The uncertainties
ΔT and (ΔT/T) are calculated by
Eqs. (A27) and (A28), respectively.
(1 April 2015) Temperature profile retrieved using the
temperature retrieval function (Eq. 15) derived from the calibration function
suggested by Gerasimov and Zuev (2016). The uncertainties ΔT
and (ΔT/T) are calculated by Eqs. (A33) and (A34),
respectively.
(1 April 2015) Temperature profile retrieved using the
temperature retrieval function (Eq. 18) derived from the calibration function
suggested by Lee III (2013). The uncertainties ΔT and
(ΔT/T) are calculated by Eqs. (A40) and (A41),
respectively.
Special cases of the general calibration function
The general calibration function expressed by Eq. (8) represents an infinite
series and, hence, the temperature retrieval function
T=T(Q) cannot be obtained in an analytical
form from this series. Therefore, one can use, for example, some special cases of
the integer power approximation of Eq. (8), i.e.,
lnQ(T)≈⋯+C-2T2+C-1T+C0+C1T+C2T2+⋯=∑n=-∞∞CnTn,
where Cn are the calibration coefficients which can differ from
Bn in Eq. (8). Here we consider the linear and four simplest
nonlinear (in reciprocal temperature 1/T) calibration functions and their
corresponding temperature retrieval functions. Since Eq. (9) is a special
case of Eq. (8), any special case of Eq. (9) automatically represents a
special case of Eq. (8). The absolute and relative uncertainties of indirect
temperature measurements are obtained in an analytical form in Appendices A,
A1–A5.
The frequently used calibration function linear in x=1/T
(Arshinov et al., 1983) is a special case of Eq. (9):
lnQ=A0+B0T⇔y=A0+B0x.
Its corresponding temperature retrieval function is
T=B0lnQ-A0,
where A0 and B0 are the commonly designated
calibration constants.
The most used nonlinear calibration function (Behrendt and Reichardt, 2000),
containing the term quadratic in x=1/T, also represents a
special case of Eq. (9), i.e.,
lnQ=A1+B1T+C1T2⇔y=A1+B1x+C1x2,
where A1, B1, and C1 are the
calibration constants. The corresponding temperature retrieval function is
simply derived from Eq. (12)
T=2C1-B1+B12+4C1lnQ-A1.
Another three-coefficient special case of Eq. (9) can be written as follows
(Gerasimov and Zuev, 2016):
lnQ=A2+B2T+C2T⇔y=A2+B2x+C2x,
where A2, B2, and C2 are the
calibration constants. Solving Eq. (14), we have for the temperature
retrieval function
T=2B2(lnQ-A2)+(lnQ-A2)2-4B2C2.
As it follows from the PRR lidar theory (Cooney, 1972),
y= lnQ is a linear function of reciprocal temperature
x=1/T (Arshinov et al., 1983). Conversely, the reciprocal
temperature represents a linear function of lnQ, i.e.,
x=a+by. In order to take nonlinear
effects into account, we consider the function
x=a+by+cy2⇔1T=a+blnQ+c(lnQ)2,
where a, b, and c are some constants. Thus, a
temperature profile can simply be retrieved via
T=c(lnQ)2+blnQ+a-1
or
T=C3(lnQ)2+B3lnQ+A3,
where A3=a/c,
B3=b/c, and
C3= 1 /c. Equation (18) was first applied to real
lidar data by Lee III (2013). Note that Eq. (16) represents a special case of
Eq. (8), as we showed in our 2016 paper.
(1 April 2015) Temperature profile retrieved using the
temperature retrieval function (Eq. 20) derived from the calibration function
suggested by Gerasimov and Zuev (2016). The uncertainties ΔT
and (ΔT/T) are calculated by Eqs. (A47) and (A48),
respectively.
There exists another way to represent collisional PRR lines broadening (and,
therefore, nonlinear effects). Adding a term hyperbolic in
y= lnQ to the linear calibration function of the form
x=a+by gives
x=A4+B4y+C4y⇔1T=A4+B4lnQ+C4lnQ,
where A4, B4, and C4 are the
calibration constants. Solving Eq. (19) yields
T=1A4+B4lnQ+(C4/lnQ)=lnQB4(lnQ)2+A4lnQ+C4.
All the nonlinear calibration (or temperature retrieval) functions
considered here take into account in varying degrees the collisional PRR
lines broadening.
The IMCES lidar setup
The IMCES PRR lidar was developed in the Institute of Monitoring of Climatic
and Ecological Systems of the Siberian Branch of the Russian Academy of
Sciences (IMCES SB RAS) for nighttime tropospheric temperature measurements.
A frequency-tripled Nd:YAG laser operating at a wavelength of 354.67 nm with
105mJ pulse energy at a pulse repetition rate of 20 Hz is used as the lidar
transmitter. The backscattered signals (photons) are collected by a
prime-focus receiving telescope with a mirror diameter of 0.5 m. The IMCES
lidar optical layout is shown in Fig. 2. The selection of spectrum bands
containing PRR lines with Jlow and
Jhigh from both the Stokes and anti-Stokes branches of
N2 and O2 PRR spectra (Fig. 1) is performed via a double-grating
monochromator (DGM). The DGM design and arrangement of optical fibers
connecting both DGM blocks are the same as suggested by Ansmann et
al. (1999). The main technical parameters of the IMCES lidar transmitting,
receiving, and data acquisition systems are summarized in Table 1. The
spectral selection parameters of the DGM channels are listed in Table 2.
Main technical parameters of the IMCES lidar transmitting,
receiving, and data acquisition systems.
Transmitting system LaserTypeUnseeded frequency-tripled Nd:YAGModelSolar LS LQ529BWavelength354.67 nmSpectral line width∼ 1 cm-1Pulse repetition rate20 HzPulse energy105 mJPulse duration13 nsBeam divergence0.3 mradExpansion factor10Receiving system TelescopeTypePrime-focusReceiving mirror diameter0.5 mFocal length1.5 mField of view0.4 mradOptical fibersF0 input fiber diameter0.55 mm (FG 550 UER)F1 output fiber diameter0.6 mm (FT 600 UMT)FB intermediate fibers diameter0.6 mm (FT 600 UMT)F2 and F3 output fibers diameter1.5 mm (FT 1.5 UMT)Double-grating monochromator Lens L1, L2Diameter130 mmFocal length300 mmDiffraction gratings DG1, DG2Grooves mm-12100Diffraction order2Diffraction angle48.151∘Data acquisition system Photomultiplier tubes PMT1–PMT3Hamamatsu R7207-01PMTs quantum efficiency25 %Photon counterPHCOUNT_4 (IMCES SB RAS)Number of channels4 (3 in use)Counting rateUp to 200 counts s-1Initial vertical resolution24 m
Spectral selection parameters of the DGM channels (central
wavelength (CWL) and full width at half maximum (FWHM)).
(1 April 2015) Comparative analysis of the absolute
temperature uncertainties yielded by using Eqs. (A27), (A33), (A40), and
(A47) and of the difference in modulus between temperature values retrieved
from the CPACs and IMCES lidar data.
IMCES lidar data taken between 20:21 and 21:21 LT on 2 October 2014
(13:21–14:21 UTC). (a) Raw photocounts
NL and NH detected in the lidar channels with
Jlow and Jhigh, respectively,
together with the single-averaged ones N‾L and
N‾H. (b) Raw photocounts ratio
Q=NL/NH, single-averaged
photocounts ratio Q=N‾L/N‾H,
and additionally averaged ratio QI=Q‾=N‾L/N‾H‾.
Temperature measurement example (1 April 2015)
In this section we consider an example of nighttime tropospheric temperature
measurements performed with the IMCES lidar on 1 April 2015 in Tomsk
(56.48∘ N, 85.05∘ E; Western Siberia, Russia). The lidar
data were taken from 03:45 to 05:15 LT (or 31 March, 21:45–23:15 UTC),
i.e., within 90 min integration time (108 000 laser shots). In order to
determine the best calibration function that yields the minimum temperature
retrieval uncertainties, we compare and analyze five vertical tropospheric
temperature profiles retrieved from the lidar data using Eqs. (11), (13),
(15), (18), and (20).
(2 October 2014) Temperature profile retrieved using
Eq. (11). The absolute and relative uncertainties ΔT=ΔT‾‾ and (ΔT/T)=(ΔT‾‾/T)
are calculated by Eqs. (A21) and (A22), respectively.
(2 October 2014) Temperature profiles retrieved using
Eqs. (13) and (18).
Raw lidar data averaging
In order to improve the signal-to-noise ratio, raw lidar data
(background-subtracted photocounts NL and NH
detected by PMTs in the DGM channels) should be averaged. We tested more than
dozens of different data-averaging methods including the equal-sized and
variable sliding-window averaging ones presented in various papers (Behrendt
and Reichardt, 2000; Behrendt et al., 2002; Alpers et al., 2004; Di Girolamo
et al., 2004; Radlach et al., 2008; Radlach, 2009; Jia and Yi, 2014). The
optimal data-averaging method for our lidar system is the following. The
IMCES lidar raw data with vertical resolution of
Δz= 24 m are averaged with a variable sliding average
window (Appendix A). Having an initial size of
n= 2k+ 1 = 3 (k= 1), the
sliding window is increased by one point on either side of the central point
for every 10 data points. Otherwise, starting with an initial length of
Δz‾=nΔz=72 m in the lidar to 240 m altitude
range, the sliding window is increased above and below by 24 m for every
240 m increase in altitude (see Fig. 3a). For example, the sliding window
size and length (or averaged data resolution) are of n= 27
(k= 13) and Δz‾=648 m at an altitude of
3 km and n= 85 (k= 42) and Δz‾=2040 m at an altitude of 10 km, respectively. Note that similar
lidar-data-averaging procedure was used, e.g., in Lee III (2013). Due to low
power of the IMCES lidar laser, the ratio of single-averaged signals (i.e.,
Q=N‾L/N‾H) was additionally
slightly averaged with a small equal-sized sliding window
(l= 5, and m= 11 in Eq. A7) to reduce signal
statistical fluctuations (Fig. 3b; see also the Supplement). For example, the
double-averaged data resolution becomes Δz‾‾=[2(k+l)+1]Δz=2280 m (k= 42, l= 5) at an
altitude of 10 km, but both absolute and relative statistical uncertainties
additionally decrease by m=11 times (Appendix A). For any
other lidar system, the optimal data-averaging method can differ from the
method we used.
(2 October 2014) Temperature profiles retrieved using
Eqs. (15) and (20).
(2 October 2014) Comparative analysis of the absolute
temperature uncertainties yielded by using Eqs. (A27), (A33), (A40), and
(A47) and of the difference in modulus between temperature values retrieved
from the CPACs and IMCES lidar data.
Reference temperature points for the lidar calibration
One of the problems we face during temperature measurements is as follows.
Unfortunately, we do not have our own radiosondes and, therefore, we have no
possibility to launch a radiosonde simultaneously with lidar remote sensing
at the lidar site. The two nearest to Tomsk meteorological stations launching
radiosondes twice a day are situated in Novosibirsk (55.02∘ N,
82.92∘ E) and Kolpashevo (58.32∘ N, 82.92∘ E).
Both towns are at a distance of more than 250 km from Tomsk. Hence, we
cannot directly use vertical temperature profiles from these radiosondes as
reference data points, which are known to be required for PRR lidars
calibration. However, we solved this problem by using temperature
and altitude data from the 925, 850, 700, 500, 400, 300, 200, and 100 hPa
constant pressure altitude charts (CPACs) as “reference” data to obtain
several points over Tomsk for the IMCES lidar calibration. Six CPACs are
presented in the Supplement as an example. Original CPACs can be found at
http://gpu.math.tsu.ru/maps/. Assuming the uncertainty to be half of
the least significant digit, the required points were determined by linear
interpolation (Saucier, 2003) with the temperature accuracy of 0.5 K and the
vertical accuracy of 5 m. It is clear that the CPAC points are not quite
suitable for use as the reference points to calibrate lidars and
retrieve temperature profiles with high accuracy (for this purpose the local
radiosonde data are required). Nevertheless, the accuracy of these points
(0.5 K, 5 m) is sufficient to make the comparative analysis of temperature
uncertainties, yielded by using different calibration functions, and
determine the best-suited function (among them) for our lidar system. Two
temperature profiles from radiosondes, launched on 1 April 2015 at 06:00 LT
(00:00 UTC) in Novosibirsk and Kolpashevo, together with temperature points
over Tomsk retrieved from the CPACs are shown in Figs. 4–9. The radiosonde
data are presented only for comparison and can be found on the web page
http://weather.uwyo.edu/upperair/sounding.html?region=np of the
University of Wyoming (Novosibirsk and Kolpashevo station numbers are 29634
and 29231, respectively).
Temperature profiles retrieved with different calibration
functions
Here we compare nighttime temperature profiles retrieved using five
calibration functions considered in Sect. 2 from the altitude where the
laser-beam receiver-field-of-view overlap is complete (∼ 3 km) to
13 km (i.e., slightly above the local tropopause). Figure 5 presents a
tropospheric temperature profile retrieved using the temperature retrieval
function (Eq. 11) derived from the standard linear calibration function
(Eq. 10). The absolute statistical uncertainty ΔT‾‾ of temperature retrieval is calculated by Eq. (A21),
whereas the relative uncertainty (ΔT‾‾/T) is
calculated by Eq. (A22). The difference in modulus TCPAC-T between temperature values retrieved from the CPACs and IMCES
lidar data is also presented in Fig. 5. The nearest radiosonde data are
given for comparison. Figures 6–9 show temperature profiles retrieved using
the temperature retrieval functions expressed by Eqs. (13), (15), (18), and
(20), respectively. These functions are derived from the corresponding
nonlinear calibration functions, i.e., Eqs. (12), (14), (16), and (19).
Comparing all five profiles among themselves, one can see that, despite the
lowest values of both the statistical uncertainties in the 3–12 km altitude
region (ΔT‾‾ < 0.7 K,
(ΔT‾‾/T) < 0.004) yielded by using
Eq. (11), the difference TCPAC-T can reach
∼ 5.5 K (Fig. 5). For the nonlinear functions in the same altitude
region, the maximum difference TCPAC-T is less
than 2.2 and ∼ 0.9 K when using Eq. (13) and Eq. (20), respectively,
as seen in Figs. 6, 9, and 10 (see also the Supplement). Similarly, for both
the uncertainties we have ΔT‾‾ < 2.3
K, (ΔT‾‾/T) < 0.011 when applying
Eq. (13), and ΔT‾‾ < 1 K,
(ΔT‾‾/T) < 0.005 for Eq. (20). Note
that the peaks of curves ΔT‾‾ and
(ΔT‾‾/T) near 11 km altitude in Figs. 6 and 7
are caused by the problem with square roots in Eqs. (13) and (15) described
in Appendices A2 and A3. There is no such problem in the case of Eqs. (18) and
(20) without square roots. The tropopause is also located near 11 km
altitude. Taking into account all three parameters ΔT‾‾, (ΔT‾‾/T), and TCPAC-T, we can conclude that Eqs. (13), (15), (18), and
(20) retrieve the tropospheric temperature much better compared to Eq. (11).
Moreover, the functions expressed by Eqs. (18) and (20) yield the smallest
uncertainties and TCPAC-T values among
considered nonlinear functions and, therefore, they are the best suited for
tropospheric temperature retrievals with the IMCES PRR lidar.
Temperature measurement example (2 October 2014)
Let us consider another example of nighttime tropospheric temperature
measurements performed with the IMCES PRR lidar on 2 October 2014 in Tomsk.
The lidar data were taken from 20:21 to 21:21 LT (13:21–14:21 UTC), i.e.,
within 60 min integration time (72 000 laser shots). The raw and averaged
IMCES lidar signals together with raw and averaged signal ratios are
presented in Fig. 11. Here also we compare five temperature profiles
retrieved using Eqs. (11), (13), (15), (18), and (20). The temperature
retrieval algorithm is the same as was applied to the IMCES lidar data dated
1 April 2015. For the lidar calibration, we retrieved temperature points over
Tomsk using the corresponding CPACs. Two temperature profiles from
radiosondes, launched on 2 October 2014 at 19:00 LT (12:00 UTC) in
Novosibirsk and Kolpashevo, are also given for comparison.
Figure 12 shows a temperature profile retrieved using Eq. (11). For this
profile in the 3–12 km altitude region we have ΔT‾‾ < 1 K, (ΔT‾‾/T) < 0.005, and TCPAC-T < 6.5 K. Figure 13 shows temperature profiles retrieved
using Eqs. (13) and (18). The temperature profiles retrieved using Eqs. (15)
and (20) are presented in Fig. 14. As seen, e.g., in Fig. 14, ΔT‾‾ < 1.8 K, (ΔT‾‾/T) < 0.009, and TCPAC-T < 2.9 K when applying Eq. (15); and ΔT‾‾ < 1.3 K, (ΔT‾‾/T) < 0.007, and TCPAC-T < 1.8 K for Eq. (20) in the 3–12 km altitude region. The
comparative analysis of the parameters is presented in Fig. 15. The
tropopause is located near 12.3 km altitude. Comparing pairwise all the
retrieved profiles for both measurement examples, one can see that ΔT‾‾, (ΔT‾‾/T), and TCPAC-T values in case of the second example
(2 October 2014) are higher than that for the first one (1 April 2015,
Sect. 4.3). This is due to the smaller number of laser shots (and,
therefore, photocounts detected in both DGM channels) leading to the higher
absolute and relative statistical uncertainties, as seen from Eqs. (A9) and
(A10) in Appendix A. The two best-suited functions for temperature retrievals
are seen in Figs. 13 and 14 to be the same as in the previous example
(1 April 2015). The large difference between the CPAC and lidar temperature
values in 2 to 3 km altitude region (Figs. 5 and 12; see also Lee III, 2013)
is, perhaps, due to the incomplete laser-beam receiver-field-of-view overlap
in the region. We also cannot exclude that any of the nonlinear calibration
functions are able to somehow correct for this incomplete overlap in the
atmospheric boundary layer.
The calibration coefficients of all the calibration functions used in both
the temperature measurement examples can be found in the Supplement.
Summary and outlook
We have considered and used the linear and four nonlinear (three-coefficient)
in x=1/T calibration functions in the tropospheric
temperature retrieval algorithm. The corresponding temperature retrieval
functions were applied to the nighttime temperature measurement data obtained
with the IMCES PRR lidar on 2 October 2014 and 1 April 2015. We have also
derived and used the absolute and relative statistical uncertainties of
indirect temperature measurements in an analytical form (Appendices A,
A1–A5).
For the case of the IMCES PRR lidar system, the comparative analysis of three
parameters ΔT‾‾, (ΔT‾‾/T), and TCPAC-T showed the following:
the nonlinear functions expressed by Eqs. (13), (15), (18), and (20)
retrieve the tropospheric temperature much better compared to the linear
function (Eq. 11);
Eqs. (18) and (20) give the almost equally best-suited functions for the
tropospheric temperature retrievals (although Eq. (20) is slightly better
than Eq. 18);
the function given by Eq. (18) is the best from both practical (real lidar
data) and theoretical (simulation) points of view (Gerasimov and Zuev, 2016).
As it was mentioned previously (Sect. 4.2), the CPAC points can hardly be used as
the reference data to reliably calibrate PRR lidars and retrieve accurate
temperature profiles. Nevertheless, the results suggest that the best-suited
calibration function for temperature retrievals can depend on the lidar
system (e.g., based on DGs or IFs for PRR lines extracting), which can take
into account the collisional broadening of PRR lines in varying degrees.
Indeed, the calibration errors depend on the spectral characteristics of the
lidar receiver such as the central wavelength, shape and width of the
transmission functions, and whether just the anti-Stokes (IFs) or both
branches of the PRR spectrum (DGs) are used to extract the PRR signals from
backscattered light. Therefore, it is reasonable to check all the mentioned
nonlinear functions against lidar data obtained with different lidar systems
to determine the best function in each specific case. Furthermore, the
stability of the calibration functions coefficients during long-time lidar
measurements is one of the crucial aspects in determination of the best
function. Hence, it would be a good thing to study the coefficients stability
during a night (Jia and Yi, 2014; Li et al., 2015), week, month, etc. as it
was done in Lee III (2013) for the linear calibration function coefficients.
Data availability
The radiosonde data are available on the web page
http://weather.uwyo.edu/upperair/sounding.html?region=np of the
University of Wyoming. Original CPACs can be found at
http://gpu.math.tsu.ru/maps/. The IMCES PRR lidar raw and averaged
signals (signal ratios) together with the processed CPACs are presented in
the Supplement.
Absolute and relative uncertainties of temperature retrieval
Each value T of a temperature profile retrieved from raw lidar data
is known to be within the confidence interval [T-ΔT; T+ΔT], where ΔT > 0. Assuming Poisson statistics of photon
counting, the 1–σ absolute statistical uncertainty ΔT
of indirect temperature measurements is defined in the general form as
follows (Behrendt, 2005; Radlach, 2009):
ΔT=dTdQΔQ2=dTdQQ1NH+1NL,
where the temperature retrieval function
T=T(Q) is derived from any required
calibration function (see Sect. 2);
Q=NL/NH is the ratio of the
background-subtracted photocounts NL and NH
registered in the lidar temperature channels with Jlow
and Jhigh, respectively. Consequently, the relative
statistical uncertainty (ΔT/T) of indirect
temperature measurements is simply derived from Eq. (A1):
ΔTT=dTdQQT1NH+1NL.
However, Eqs. (A1) and (A2) are valid only for unaveraged (raw) lidar data
NL and NH. In practice, raw data are previously
averaged to improve the signal-to-noise ratio. One of the most simple and
used data-averaging methods is the equal-sized (or variable) sliding-window
averaging (Behrendt and Reichardt, 2000; Behrendt et al., 2002; Alpers et
al., 2004; Di Girolamo et al., 2004; Radlach et al., 2008; Radlach, 2009; Lee
III, 2013). The averaged data N‾(z) and their variance
Var‾(z) are related to the corresponding unaveraged data
N(z) and variance Var(z) as follows (El'nikov et al.,
2000):
N‾j(z)=12k+1∑i=-kkNj+i=12k+1∑i=-kkN(z+iΔz)=1nN(z-kΔz)+⋯+N(z)+⋯+N(z+kΔz),Var‾(z)=Var(z)/n,
where Δz is the vertical resolution of raw lidar data
(initial vertical resolution); k is the number of data points on
either side of the central point Nj; and
n= 2k+ 1 is the sliding average window size, i.e.,
the number of raw lidar data points determining the sliding average window
length or data resolution after averaging (Otnes and Enochson, 1978). The
weighting coefficients of the raw data points in Eq. (A3) are the same and
equal to 1/(2k+1). The vertical resolution of the averaged data
series {Nj‾} is Δz‾=nΔz=(2k+1)Δz. As the variance decreases by n times, the
absolute uncertainty ΔN‾(z) of averaged data decreases by
n times. Therefore, for the absolute uncertainty of temperature
retrieval from the averaged lidar data (photocounts)
N‾H and N‾L we have
ΔT‾=ΔTn=dTdQQn1N‾H+1N‾L,
where
Q=N‾L/N‾H. Hence, the
confidence interval of the retrieved temperature profile is [T-ΔT‾;T+ΔT‾], and the relative uncertainty is given
by
ΔTT‾=dTdQQTn1N‾H+1N‾L.
In some cases, the second-order averaging of raw data (or/and their ratio) is
required and more preferable than the first-order one (see, e.g., El'nikov et
al., 2000). In such cases, the double-averaged data N‾‾(z) and their variance Var‾‾(z) are related to
the corresponding single-averaged data N‾(z), and variances
Var‾(z) and Var(z) as follows:
N‾‾j(z)=12l+1∑i=-llN‾j+i=1m∑i=-llN‾(z+iΔz),Var‾‾(z)=Var‾(z)/m=Var(z)/(nm),
where l is the number of the single-averaged data points on either
side of the central point N‾j and m= 2l+ 1 is the sliding average window size. The confidence interval of a
retrieved temperature profile is [T-ΔT‾‾;T+ΔT‾‾], where ΔT‾‾=ΔT/nm.
There are two ways to average previously averaged PRR lidar data. The first
way is to average the ratio Q=N‾L/N‾H of the single-averaged data
N‾H and N‾L. In this case, the
absolute and relative uncertainties of temperature retrieval from the
averaged ratio QI=Q‾=N‾L/N‾H‾ are
given by
ΔT‾‾=dTdQIQInm1N‾H+1N‾L,ΔTT‾‾=dTdQIQITnm1N‾H+1N‾L.
The second way is to average the single-averaged data
N‾H and N‾L. The absolute and
relative uncertainties of temperature retrieval from the double-averaged
lidar data N‾‾H and N‾‾L (and for QII=N‾‾L/N‾‾H) are determined by
ΔT‾‾=dTdQIIQIInm1N‾‾H+1N‾‾L,ΔTT‾‾=dTdQIIQIITnm1N‾‾H+1N‾‾L.
The vertical resolution of the double-averaged data series
{N‾‾j} and for both ways of the second-order
averaging becomes
Δz‾‾=pΔz=[2(k+l)+1]Δz.
If the window size n (and/or m) varies with altitude
z, both the uncertainties should be estimated separately for each
altitude interval where n= const (and/or
m= const). To determine the weighting coefficients of the raw
data points in Eq. (A7), it is necessary to consider three possible simple
cases of the second-order averaging.
Let k >l
(n > m);
i.e., the sliding average window size for the first-order
averaging is larger than that for the second-order one. ThenN‾‾j=1(2l+1)(2k+1)(2l+1)Nj+∑i=1k-lNj-i+Nj+i+∑i=k-l+1k+l(k+l+1-i)Nj-i+Nj+i.
The weighting coefficients can be determined from Eq. (A14) of the following
form:N‾‾j=12k+1Nj+∑i=1k-lNj-i+Nj+i+∑i=k-l+1k+lk+l+1-i(2l+1)(2k+1)Nj-i+Nj+i.
Let l > k
(m > n);
i.e., the window size for the second-order averaging is larger
than that for the first-order one. ThenN‾‾j=1(2k+1)(2l+1)(2k+1)Nj+∑i=1l-kNj-i+Nj+i+∑i=l-k+1l+k(l+k+1-i)Nj-i+Nj+i.
The corresponding weighting coefficients are determined similar to case (1).
Let l=k (m=n); i.e., the
window size for the second-order averaging is equal to that for the
first-order one. ThenN‾‾j=1(2k+1)2(2k+1)Nj+∑i=12k(2k+1-i)Nj-i+Nj+i.
The weighting coefficients are determined similar to cases (1) and (2). The
vertical resolution of the double-averaged data series
{N‾‾j} in case (3) is Δz‾‾=pΔz=(4k+1)Δz (El'nikov et al., 2000).
Linear calibration function
As we applied the first way of the second-order averaging of the IMCES lidar
raw data (see Appendix A and Sect. 4.1), we use Eqs. (A9) and (A10) to derive
the absolute and relative uncertainties in an analytical form. In case of the
first-order averaging of lidar raw data, one can use Eqs. (A5) and (A6),
respectively.
In order to obtain both the uncertainties for the linear calibration
function, let us differentiate the temperature retrieval function derived
from Eq. (10), i.e. (see Sect. 2)
T=B0lnQ-A0.
The first-order derivative of the function is
dTdQ=-B0Q(lnQ-A0)2.
Substituting Eq. (A19) into Eq. (A9), for the absolute uncertainty we get
ΔT‾‾=B0(lnQI-A0)2nm1N‾H+1N‾L.
One can rewrite Eq. (A20) in more simple form by substituting the expression
lnQ – A0=B0/T derived
from Eq. (A18)
ΔT‾‾=T2B0nm1N‾H+1N‾L.
Consequently, substituting Eq. (A19) into Eq. (A10), for the relative
uncertainty we have
ΔTT‾‾=1lnQI-A0nm1N‾H+1N‾L=TB0nm1N‾H+1N‾L.
Calibration function quadratic in x=1/T
The temperature retrieval function derived from Eq. (12) is written as (see
Sect. 2)
T=2C1-B1±B12+4C1lnQ-A1.
The sign “+” instead of “±” should be chosen in the denominator
of Eq. (A23), if Q=NL/NH. When
applying Eq. (A23) for temperature retrievals, one should take into account
the constraint coming from the square root. Namely, the expression under the
square root should be nonnegative, i.e., B12+4C1lnQ(z)-A1≥0 or lnQ(z)≤(B12/4C1)-A1.
Hence, Eq. (A23) can retrieve the temperature profile T only at
altitudes z where this condition holds.
The first-order derivative of the function is
dTdQ=-4C12-B1+B12+4C1lnQ-A1-2QB12+4C1lnQ-A1.
It is clear that the expressions for both absolute and relative uncertainties
will be cumbersome and poorly adapted for use after substitution of this
derivative in Eqs. (A9) and (A10). However, Eq. (A24) can be put in a more
convenient form by substituting the expressions which follow from Eq. (A23):
-B1+B12+4C1lnQ-A1=2C1T,B12+4C1lnQ-A1=2C1T+B1.
After substitution of Eqs. (A25) into Eq. (A24), we can write instead of
Eq. (A24)
dTdQ=-T3Q(2C1+B1T).
Substituting Eq. (A26) into Eqs. (A9) and (A10), we obtain correspondingly
for the absolute and relative uncertainties
ΔT‾‾=T32C1+B1Tnm1N‾H+1N‾L,ΔTT‾‾=T22C1+B1Tnm1N‾H+1N‾L.
Calibration function hyperbolic in x=1/T
The temperature retrieval function in the general form derived from Eq. (14)
represents (see Sect. 2)
T=2B2(lnQ-A2)±(lnQ-A2)2-4B2C2.
For the case of
Q=NL/NH, the sign “+” instead
of “± ” should also be chosen in the denominator of Eq. (A29). Note
that Eq. (A29) can retrieve the temperature T only at altitudes
z where the following condition holds: [lnQ(z)-A2] 2-4B2C2≥0 or lnQ(z)≥A2+2B2C2
(with B2C2≥ 0).
The derivative of the temperature retrieval function is
dTdQ=2B2Q(lnQ-A2)+(lnQ-A2)2-4B2C22×1+lnQ-A2(lnQ-A2)2-4B2C2.
Equation (A30) can be put in a more convenient form by substituting the
expressions which follow from Eqs. (A29) and (14), respectively
(lnQ-A2)+(lnQ-A2)2-4B2C2=2B2/T,lnQ-A2=B2/T+C2T.
After substitution of Eqs. (A31) into Eq. (A30), we get for the derivative
dTdQ=T2QB2-C2T2.
Then substituting Eq. (A32) into Eqs. (A9) and (A10), we obtain for both the
uncertainties
ΔT‾‾=T2B2-C2T2nm1N‾H+1N‾L,ΔTT‾‾=TB2-C2T2nm1N‾H+1N‾L.
Calibration function quadratic in y= lnQ
The first-order derivative of the temperature retrieval function, obtained
from Eq. (16) (see Sect. 2)
T=C3(lnQ)2+B3lnQ+A3,
is simply expressed as
dTdQ=-C3(2lnQ+B3)Q(lnQ)2+B3lnQ+A32.
Substituting Eq. (A36) into Eq. (A9), for the absolute uncertainty we get
ΔT‾‾=C3(2lnQI+B3)(lnQI)2+B3lnQI+A3 2nm1N‾H+1N‾L.
Using the expression derived from Eq. (A35), i.e.,
(lnQ)2+B3lnQ+A3=C3/T,
for the relative uncertainty we obtain
ΔTT‾‾=2lnQI+B3(lnQI)2+B3lnQI+A31nm1N‾H+1N‾L.
In order to estimate both the uncertainties, one can also use Eqs. (A37) and
(A39) in a more simple form. Substituting Eq. (A38) in Eqs. (A37) and (A39),
we obtain the following equations containing both lnQI and
retrieved temperature T:
ΔT‾‾=2lnQI+B3C3T2nm1N‾H+1N‾L,ΔTT‾‾=2lnQI+B3C3Tnm1N‾H+1N‾L.
Calibration function hyperbolic in y= lnQ
Tropospheric temperature profiles are mentioned in Sect. 2 can also be
retrieved via the function
T=lnQB4(lnQ)2+A4lnQ+C4,
the first-order derivative of which is defined as
dTdQ=C4-B4(lnQ)2QB4(lnQ)2+A4lnQ+C4 2.
Substituting Eq. (A43) in Eq. (A9), we obtain the absolute uncertainty
containing only lnQ:
ΔT‾‾=C4-B4(lnQI)2B4(lnQI)2+A4lnQI+C4 2nm1N‾H+1N‾L.
Using the expression derived from Eq. (A42), i.e.,
B4(lnQ)2+A4lnQ+C4=(lnQ)/T,
for the relative uncertainty we get
ΔTT‾‾=C4-B4(lnQI)2B4(lnQI)3+A4(lnQI)2+C4lnQI1nm1N‾H+1N‾L.
Similarly, using Eq. (A45), one can rewrite Eqs. (A44) and (A46) in a
practically useful form:
ΔT‾‾=C4(lnQI)2-B4T2nm1N‾H+1N‾L,ΔTT‾‾=C4(lnQI)2-B4Tnm1N‾H+1N‾L.
The Supplement related to this article is available online at doi:10.5194/amt-10-315-2017-supplement.
Acknowledgements
We thank S. M. Bobrovnikov for helpful 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).Edited by: R. Sica
Reviewed by: A. Hauchecorne and one anonymous referee
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