AMTAtmospheric Measurement TechniquesAMTAtmos. Meas. Tech.1867-8548Copernicus GmbHGöttingen, Germany10.5194/amt-8-4645-2015Evaluation of methods for gravity wave extraction from middle-atmospheric lidar temperature measurementsEhardB.benedikt.ehard@dlr.dehttps://orcid.org/0000-0003-0517-8134KaiflerB.https://orcid.org/0000-0002-5891-242XKaiflerN.https://orcid.org/0000-0002-3118-6480RappM.https://orcid.org/0000-0003-1508-5900Deutsches Zentrum für Luft- und Raumfahrt, Institut für Physik der Atmosphäre, Oberpfaffenhofen, GermanyB. Ehard (benedikt.ehard@dlr.de)5November20158114645465512August20152September201529October201530October2015This 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/8/4645/2015/amt-8-4645-2015.htmlThe full text article is available as a PDF file from https://amt.copernicus.org/articles/8/4645/2015/amt-8-4645-2015.pdf
This study evaluates commonly used methods of extracting
gravity-wave-induced temperature perturbations from lidar measurements. The
spectral response of these methods is characterized with the help of
a synthetic data set with known temperature perturbations added to
a realistic background temperature profile. The simulations are
carried out with the background temperature being either constant or
varying in time to evaluate the sensitivity to temperature
perturbations not caused by gravity waves. The different methods are
applied to lidar measurements over New Zealand, and the performance of
the algorithms is evaluated. We find that the Butterworth filter
performs best if gravity waves over a wide range of periods are to be
extracted from lidar temperature measurements. The running mean method
gives good results if only gravity waves with short periods are to be analyzed.
Introduction
Atmospheric gravity waves are well known to have a strong impact on
the middle-atmospheric circulation
e.g.,. By transporting energy and momentum
from the lower atmosphere into the middle atmosphere, they are
responsible for the formation of the cold polar summer mesopause
e.g.,. Although some processes related to gravity
waves are believed to be well understood, there are still open
questions. For example, to what extent gravity wave excitation,
propagation and forcing is affected by a changing climate remains an
open question cf..
Lidar technology has been used to study gravity waves in the
middle atmosphere for the last 3 decades
e.g.,. Hence,
lidar studies can potentially be used to infer long-term
trends in gravity wave activity. Furthermore, lidars have the
advantage of providing measurements throughout the entire
middle atmosphere with high temporal and vertical resolution
of typically 1 h and 1 km. However, lidars
generally provide one-dimensional profiles, and no information
on the horizontal structure and the intrinsic properties of
atmospheric waves can be retrieved. Exceptions are
measurements from airborne lidars and multi-beam lidars.
Gravity waves are usually determined from lidar measurements
by separating an estimated background temperature (density)
profile from the measured profiles in order to derive
temperature (density) perturbation profiles. Several methods
have been developed and used over the last decades. For
example , and
calculate a nightly mean profile and subtract it from the
(time-resolved) individual profiles.
remove a background profile determined by a temporal running
mean (in addition to vertical filtering). Perturbation
profiles obtained through a fit of polynomial functions to the
measured profiles are examined by, e.g.,
, and
. apply a variance method in
order to determine perturbation profiles, while
use spectral filtering.
All of these methods are most sensitive to different parts of
the gravity wave spectrum. Thus, results from different lidar
studies become hardly comparable because one cannot
distinguish between variations that are caused by a different
methodology and variations that are geophysically
induced. compared values of gravity wave
potential energy density (GWPED) from different studies to
their results. Due to potential methodological biases it
remained unclear whether the differences were in fact of
geophysical origin. Hence, they expressed the need for
a standardized method to extract gravity wave amplitudes from
lidar measurements.
To our knowledge, no literature is yet available which
characterizes and evaluates the most commonly used methods to
extract information on gravity waves from lidar
profiles. Thus, we will evaluate and compare four methods in
detail: subtraction of the nightly mean profile, subtraction
of temporal running mean profiles, the sliding polynomial fit
method proposed by and the application of
a Butterworth filter. While the first two methods rely on
filtering in time, the latter two methods apply a filter in
space to determine wave-induced temperature perturbations.
This paper is structured as follows: the four methods are
described in detail in Sect. . The performance
is studied in terms of their spectral response to synthetic
data in Sect. . The results are then applied to
measurement data in Sect. . Finally, the
characteristics of the four methods as well as their
suitability for extracting gravity-wave-induced temperature
perturbations are discussed in Sect. , and
conclusions are drawn in Sect. .
Methods
Lidar systems used for studies of the middle atmosphere measure the
Rayleigh backscatter signal which is proportional to atmospheric
density after range correction. The temperature is commonly retrieved by
integration assuming hydrostatic equilibrium
. Recently proposed a temperature
retrieval using optimal estimation methods. The derived temperature profiles typically range
between 30 and 80–90 km altitude depending on
signal-to-noise ratio. At the upper boundary, the temperature
retrieval is commonly initialized with satellite data
e.g., or resonance lidar measurements e.g.,.
The combination with a resonance lidar system extends the altitude
range of temperature measurements up to ≈ 105 km. Temperatures below 30 km altitude can be
retrieved by using a stratospheric Raman channel
e.g.,. The large altitude range allows for
studies of gravity wave propagation from the troposphere to the
mesosphere. Hence, we discuss the extraction of gravity waves from
temperature data rather than atmospheric density, although most of the
results can be applied to density measurements as well. For different methods
of extracting gravity waves from density measurements see, e.g., ,
and .
Lidar studies usually determine wave-induced temperature perturbations T′(z, t)
(which are a function of altitude z and time t) from the measured temperature
profile T(z, t) by subtracting a background temperature profile T0(z, t):
T′(z,t)=T(z,t)-T0(z,t).T0(z, t) ideally contains all contribution from radiative and
chemical heating and other large-scale effects such as planetary
waves and tides. Hence, the temperature perturbations T′(z, t)
should be solely caused by gravity waves. Estimation of
T0(z, t) is challenging due to the specific shape of the
temperature profile with its changes in vertical temperature
gradient, e.g., at the stratopause or mesopause.
The frequency range of gravity waves which may be present in
T′(z, t) can be inferred from the gravity wave dispersion relation
which states that the relation
N>|ω^|>f
between the intrinsic frequency ω^, the
Brunt–Väisälä frequency N and the Coriolis parameter f
must be fulfilled at all times. Using a typical stratospheric
value of N= 0.02 s-1 and a Coriolis parameter for
midlatitudes of f= 10-4 s-1, the intrinsic period
τ^=2πω^ ranges between
5 min and 17 h. It is important to note that the
lidar only detects the observed period τ which can be Doppler-shifted
to larger or smaller values, depending on local wind
conditions. Typical vertical wavelengths of gravity waves measured
by ground-based instruments vary between 1 and 17 km
seetheir Table 2. The spatial scales combined
with the temporal scales define the spectral requirements on the
methods of extracting gravity-wave-induced temperature perturbations.
Time-averaged background profiles
A widely applied method is the use of the nightly mean temperature
profile as a background temperature profile
e.g.,. It is then assumed
that the timescales of phenomena other than gravity waves
affecting the temperature profile are considerably larger and the
timescales of gravity waves are smaller than the measurement
period, which is typically in the range of 3–12 h.
Another common method is to determine background temperature
profiles by means of a running mean over a time window which is
typically on the order of 3 h
e.g.,. Temperature variations with
timescales larger than the window width are attributed to the
background temperature profiles and are therefore not included in
the extracted gravity wave spectrum.
Sliding polynomial fit
proposed a method of extracting temperature
perturbations based on a sliding polynomial fit in the spatial
domain. The method is sensitive to small vertical scales and
ignores the temporal evolution of waves. The method is based on
the assumption that temperature variations with large vertical
scales can be attributed to the climatological thermal
structure of the atmosphere (i.e., the different vertical
temperature gradients in the troposphere, stratosphere and
mesosphere), to the advection of colder or warmer air masses, or to
tides and planetary waves. Only variations with a spatial scale
smaller than a certain threshold are identified as gravity waves.
The sliding polynomial fit method was designed to produce
a background temperature profile which contains all perturbations
with vertical scales larger than 15 km. For each measured
temperature profile applied a series of overlapping
cubic polynomial fits to each range gate. Each fit was applied to
an altitude window with a width of Lf= 25 km.
A weighted average was computed to reconstruct the
background temperature profile from the individual polynomial fits
using the weighting function
w(z)i=expz-zc,i-δγifz≤zc,i-δ1ifzc,i-δ<z<zc,i+δexp-z-zc,i+δγifz≥zc,i+δ.
Here δ= 0.5 Lf-Lw, Lw is
the width of the weighting window, zc,i the center altitude of
the individual fit and γ the e-folding width which defines
how fast the weighting function decreases. used
a weighting window length Lw=Lf3 and
γ= 3 km.
smoothed the resulting background temperature
profiles with a 1.5 km boxcar mean. These profiles were
then subtracted from the corresponding measured temperature
profiles according to Eq. (), yielding the
temperature perturbation profiles.
In this study the following set of parameters is used: a fit length
Lf= 20 km, a weighting window length
Lw= 3 km and an e-folding width
γ= 9 km. These parameters are chosen because they
yield the flattest spectral response for the altitude resolution
used in this study (see Sect. for further
details). The boxcar smoothing proved to have a negligible
effect. Hence, it is not applied in this study.
Spectral filter
Another method which can be applied to vertical profiles is
spectral filtering e.g.,. By applying a high-pass
filter to individual temperature profiles, temperature
perturbations can be retrieved. In order to yield perturbations
caused by gravity waves, a filtering function has to be chosen
which has an adequate spectral response.
In this study we use a 5th-order Butterworth high-pass filter with
a cutoff wavelength λc= 15 km and the transfer function
Hλz=1+λzλc2n-12,
where n is the order of the filter and λz is the
vertical wavelength. The Butterworth filter is chosen due to its
flat frequency response in the passband. The filter itself is
applied in Fourier space. As the Fourier transformation assumes
a cyclic data set, the upper and lower end of the measured
temperature profile are internally connected. This creates an
artificial discontinuity which introduces a broad range of
frequencies including frequency components that are in the passband
of the filter. These frequency components contribute to temperature
perturbations at the upper and lower end of the analyzed altitude
window and thus artificially enhance gravity wave signatures. In
order to mitigate this effect, the data set is mirrored at the
lowest altitude bin and attached to the original data set before the
filtering process. The data set can be cyclically extended
without discontinuities at the lower end, where temperature
perturbations are smallest and therefore artificial enhancements
produce the largest relative errors. After the filtering only the
original half of the resulting perturbation profile is retained.
Application to synthetic data
In order to characterize the different methods regarding their
ability to extract temperature perturbations from middle-atmospheric
temperature profiles, we apply them to a synthetic
data set with known temperature perturbations. These perturbations
are added to a fixed, realistic background temperature profile
T0(z). The latter is derived from the mean temperature profile
above Lauder, New Zealand, (45.0∘ S, 169.7∘ E)
measured with the Temperature Lidar for Middle Atmosphere research (TELMA)
from July until the end of September (black line in
Fig. a). The particular choice of the background
temperature profile does not affect the results as long as the
background temperature profile is realistic, is smooth and does not
contain contributions from gravity waves. For example, with
a climatological or a model temperature profile, similar results
can be derived.
Sinusoidal temperature perturbations with exponentially increasing
amplitude were added to the background temperature profile according to
Ts(z,t)=T0(z)+Ts′(z,t),withTs′(z,t)=Acos2πzλz+2πtτexpz-z02H,
with the amplitude A, the vertical wavelength λz, the
observed period τ, the scale height H and the lowest altitude
of the analyzed altitude range z0. An example of the perturbed
background profile Ts can be seen in Fig. a (red
line), and the corresponding temperature perturbations Ts′ in Fig. b.
(a) Background temperature profile T0 used for the simulations
(black) and perturbed temperature profile T (red). (b) The temperature
perturbations T′ added to T0. Temperature perturbations in both panels
were constructed using Eq. () with the following set of
parameters: t= 4 h, A= 1.2 K, λz= 6 km, τ= 1.9 h,
H= 12 km.
For each method, the spectral response Rm(z) was calculated from
the ratio between the time averaged absolute values of the
determined temperature perturbations |Tm′(z,t)|‾ and
the synthetic temperature perturbations |Ts′(z,t)|‾ as
Rm(z)=|Tm′(z,t)|‾|Ts′(z,t)|‾⋅100%.
A spectral response larger than 100 % indicates an
overestimation of gravity wave amplitude, while a value below
100 % indicates an underestimation of gravity wave amplitude.
All simulations conducted for this study use the realistic set of
parameters A= 1.2 K, H= 12 km and z0= 25 km. A
height resolution of Δz= 0.1 km
was used, while the altitude interval ranged from 25 to
90 km. A time interval of 8 h, corresponding to the
length of an average nighttime measurement period, with a resolution
of Δt= 0.5 h was used. For each simulation either
λz or τ was kept constant, while the other was
varied. The vertical wavelength λz was varied from 0.6 to
20 km in steps of 0.2 km, while τ was varied from 0.15 to 14.95 h in steps of 0.1 h.
Constant background temperature
As a first step, simulations were carried out with a constant
background temperature profile T0(z). In order to reduce
aliasing effects caused by even multiples of the analyzed time
window (8 h), the period of simulated gravity waves was set
to τ= 1.9 h while the vertical wavelength λz
was varied. Figure depicts the spectral response of
the different methods as a function of vertical wavelength.
Spectral response of different methods of determining temperature
perturbations as a function of vertical wavelength λz: nightly
mean (a), 3 h running mean (b), sliding polynomial fit (c) and Butterworth
filter (d). (e) and (f) depict mean extracted temperature perturbations
in the ranges 30–40 km (e) and 50–60 km (f) as well as the simulated temperature
perturbations (blue line). The different methods are color-coded as follows:
nightly mean – green; 3 h running mean – orange; sliding polynomial fit –
red; Butterworth filter – black. Please note that the blue line in this case
lies exactly underneath the green line. All simulations were carried out with
τ= 1.9 h and a background temperature profile constant in time.
The nightly mean method (Fig. a) and the 3 h
running mean method (Fig. b) both exhibit an almost
uniform spectral response at all altitudes and
wavelengths. However, the running mean slightly overestimates the
extracted temperature perturbations, which is due to the choice of a
specific period of τ= 1.9 h (cf. Fig. e). The sliding polynomial fit
method (Fig. c) shows a reduced spectral response for
vertical wavelengths larger than ≈ 13 km. For
shorter vertical wavelengths the spectral response is close to
100 % at most altitudes. Vertical wavelengths of
≈ 9 km show a slight reduction in spectral response over the
entire altitude range. At the upper and lower 5 km of the
analyzed altitude window, vertical wavelengths larger than
5 km are strongly damped. The spectral response of the
Butterworth filter (Fig. d) is very similar to the
sliding polynomial fit. The main difference is that the Butterworth
filter exhibits no underestimation of temperature perturbations at
9 km vertical wavelength.
Same as Fig. but as a function of period τ. All
simulations were carried out with a fixed vertical wavelength of 6 km and a
background temperature profile constant in time. Note that the blue and black
lines in (e) and (f) are lying on top of each other.
Figure e and f show mean extracted temperature
perturbations. The blue line (here underneath the green line)
depicts the original temperature perturbations added to the
background temperature profile. As evident from
Fig. e, the sliding polynomial fit method
underestimates temperature perturbations at vertical wavelengths
around 9 km. In agreement with the filter design both
vertical filtering methods, the sliding polynomial fit and the
Butterworth filter, show a decrease in extracted temperature
perturbations for vertical wavelengths larger than
13 km. This decrease is almost linear with increasing
vertical wavelength. As a consequence, amplitudes are effectively
reduced by a factor of 3 at λz= 20 km.
In the first simulation setup the vertical wavelength λz
was varied, while the period τ was kept constant. We now
proceed by varying the period τ with a fixed λz= 6 km
(Fig. ). The spectral response of the
nightly mean method (Fig. a) is close to
100 % at all altitudes. Temperature perturbations with
periods larger than 10 h are damped, and periods around
6 h are slightly underestimated. For τ= 15 h
the reduction in amplitude is ≈ 20 % (green line in
Fig. e and f). Like the nightly mean method, the
3 h running mean (Fig. b) exhibits a uniform
spectral response at all altitudes. However, waves with periods
longer than 3.5 h are strongly damped. At a period of
6 h temperature perturbations are underestimated by
a factor of 2, and for τ= 2.5 h amplitudes are
overestimated by ≈ 20 % (orange line in
Fig. e and f). The spectral response of the filter
for waves with shorter periods oscillates between over- and
underestimation as τ approaches zero. In contrast, the sliding
polynomial fit method (Fig. c) and the Butterworth
filter (Fig. d) both exhibit an almost uniform
spectral response for most periods. Only for very long periods do the
spectral response oscillate between over- and underestimation with
increasing altitude, indicating a slight phase delay between
simulated and extracted temperature perturbations. This oscillation is not
seen in Fig. e and f due to the vertical averaging over 10 km.
Same as Fig. but with a varying background
temperature (see Sect. 3.2 for details).
Varying background temperature
While in the previous section the simulated background temperature
was kept constant, we now examine the influence of a time-dependent
variation of the background temperature on the different
methods. Slow variations of the form
T0′(z,t)=αtsin2πz-z060kmexpz-z0H0
were added to Eq. (), where α= 0.5 K h-1 is the heating/cooling rate and
H0= 65 km is the scale height of the background
temperature variation. This results in a warming of the
stratosphere and a cooling of the mesosphere over time,
representing a very simplified effect of a propagating planetary
wave with a vertical wavelength of 60 km. All other
parameters are the same as before.
Filter characteristics are shown for a varying vertical wavelength
in Fig. . Compared to the steady background simulations
(e.g., Fig. ), the nightly mean method exhibits an
enhanced spectral response around 35 and 65 km altitude
(Fig. a). From Fig. e it can be determined
that the nightly mean method overestimates temperature
perturbations by roughly 25 % between 30 and 40 km
altitude. No change in spectral response is detected for the
3 h running mean method (Fig. b), the sliding
polynomial fit method (Fig. c) and the Butterworth
filter (Fig. d).
The filters exhibit similar characteristics if the gravity wave
period is varied instead of the vertical wavelength. The nightly
mean method (Fig. a) overestimates temperature
perturbations in the same altitude bands as shown for the
simulations with varying vertical wavelength
(cf. Fig. a). The filter characteristics of the
3 h running mean method (Fig. b), the sliding
polynomial fit method (Fig. c) and the Butterworth
filter (Fig. d) are not affected by the varying
background temperature.
Application to measurement data
Rayleigh lidar measurements at Lauder, New Zealand,
(45.0∘ S, 169.7∘ E) were obtained with the TELMA
instrument from mid-June to mid-November 2014
. We use temperature data with a temporal
resolution of 10 min and a vertical resolution of
100 m. The effective vertical resolution of the temperature
data is 900 m due to smoothing of the raw data before
processing. Measurement uncertainties are typically on the order of
2–3 K at 70 km altitude and generally lower than 1 K below 60 km altitude.
Case study: 23 July 2014
A detailed analysis with the four different methods of extracting
temperature perturbations is shown for the data set obtained on
23 July 2014 in Fig. . This case was chosen because
the gravity wave analysis depicts many previously noted
characteristics of the four methods.
The main features of the mean temperature profile
(Fig. b) are the stratopause between 45 and
55 km altitude with T≈ 245 K and the
temperature minimum of approximately 200 K at 73 km
altitude below a mesospheric inversion layer. The time evolution of
the temperature measurements (Fig. a) shows an
increase of the temperature at the stratopause and a jump in
stratopause height around 08:00 UTC. Afterwards, the
stratopause descends slowly. The structure of the mesospheric
inversion layer varies also over time, with the minimum temperature
below the inversion layer reaching ≈ 175 K around 14:00 UTC.
The temperature perturbations as determined by the nightly mean
method (Fig. c) exhibit a vertically broad maximum
descending from about 80 km altitude down to 50 km
altitude over the 12 h measurement period. Temperature
perturbations within this descending maximum reach values of up to
±20 K. Below 50 km altitude temperature
perturbations are generally on the order of ±5 K.
The 3 h running mean method on the other hand
(Fig. d) shows strongly tilted patterns. Below
50 km altitude the phase lines tend to be steeper than
above. The magnitude of the temperature perturbations generally
increases with altitude from approximately ±5 K below
60 km altitude to approximately ±15 K above 60 km altitude.
Same as Fig. but with a varying background
temperature (see Sect. 3.2 for details).
Temperature (a), mean temperature profile (b) and derived
temperature perturbations obtained by different methods (c–f) over Lauder,
New Zealand, (45.0∘ S, 169.7∘ E) on 23 July 2014. The following
methods were used for the different panels: nightly mean (c), 3 h running
mean (d), sliding polynomial fit (e), and Butterworth filter (f).
Time is given in UTC.
The sliding polynomial fit method (Fig. e) and the
Butterworth filter (Fig. f) extract almost identical
patterns of temperature perturbations, with the Butterworth filter
inferring slightly larger amplitudes. The phase lines in
Fig. e and f decrease more slowly in altitude
compared to the 3 h running mean method. Below
60 km altitude temperature perturbations are below
±10 K for both filters and increase to ±15 K
above 60 km altitude.
Statistical performance
A quantity often used as a proxy for gravity wave activity is the
GWPED per mass:
Ep=12g2N2T′T02‾,withN2=gT0‾dT0‾dz+gcp,
where g denotes the acceleration due to gravity and cp the heat
capacity of dry air under constant pressure, in addition to the
previously defined variables. The mean GWPED is determined as the
average over one measurement period – typically 5–12 h in
our case – which is denoted by the overline in
Eq. (). Due to the decrease in density with altitude,
GWPED per mass increases exponentially with altitude in the case of
conservative wave propagation. For a more detailed description and
physical interpretation of the GWPED see, e.g., and .
From TELMA observations above New Zealand over the period 1 July
to 30 September 2014 we determined the mean GWPED per mass
using the four methods of gravity wave extraction discussed in this
study (Fig. ). Relative uncertainties of the GWPED for all
methods are on the order of 0.5 % in the stratosphere and increase to
approximately 5 % at 80 km altitude, which is considerably smaller
than the variations of the GWPED due to the geophysical variability.
The absolute value of the GWPED varies by
as much as 1 order of magnitude depending on which method is
used. The largest relative deviations appear in the lower
stratosphere between the 3 h running mean method and the
Butterworth filter. Above 65 km altitude all methods produce
similar results. A distinct feature of Fig. is the
larger growth of GWPED with altitude if the running mean method is
used instead of the vertical filtering methods. Additionally, the
3 h running mean method yields the lowest GWPED values. If
a 4 h running mean is used instead, the GWPED profile is
shifted towards slightly larger values. Below 45 km altitude
the nightly mean method produces values comparable to the sliding
polynomial fit and the Butterworth filter. Above 45 km
altitude the nightly mean method shows the largest values of all
methods. The sliding polynomial fit and the Butterworth filter
produce generally similar results, with the Butterworth filter
yielding a slightly larger GWPED. Another striking feature in
Fig. is the enhanced GWPED below 35 km
altitude which is detected by both vertical filtering methods. This
enhancement is not detected by the running mean method.
Mean gravity wave potential energy density (GWPED) per mass over
Lauder, New Zealand, (45.0∘ S, 169.7∘ E) between 1 July and
30 September 2014. The methods used to determine the GWPED are color-coded. The
profiles were smoothed by a vertical running mean with a window width of 3 km.
DiscussionTemporal filters
The nightly mean method has been applied in many studies
e.g.,. The major
disadvantage is that a varying length of measurement periods
results in a variation of the sensitivity to different
timescales. This effect is clearly demonstrated in
Fig. e, showing that gravity waves with periods larger
than 10 h are significantly underestimated if an
8 h long time series is used. If the time series is
shortened, the cutoff period is smaller as well (not shown) and the
spectral response for long-period waves is reduced even
further. Strictly speaking, this implies that gravity wave analyses
of time series of different length cannot be compared.
In practice measurement periods vary typically in length between
a few hours up to a whole night as weather conditions can change
rapidly during an observational period. Moreover, there is
a seasonal dependency because most middle-atmospheric lidars are
capable of measuring in darkness only. This results in shorter
measurement periods in summer and longer measurement periods in
winter. Hence, the nightly mean method is sensitive to different
parts of the gravity wave spectrum depending on weather conditions
as well as season. For example, compared winter
and summer measurements of gravity wave activity determined by the
nightly mean method. They resolved gravity waves with periods of 1.5–12 h
during winter and 1.5–3.5 h during summer. Hence,
limited their analysis to 3–5 h long
measurement periods in order to reduce the variation of the spectral response.
The use of the nightly mean method in gravity wave analysis is
further complicated by the fact that there are processes besides
gravity waves which occur on similar timescales. For example tides
with periods of 8, 12 and 24 h are within the sensitivity
range of this method. In the analysis of radar data, the removal of
tidal signals is a standard procedure
e.g.,. With lidar data, however, this is
problematic due to generally shorter and often intermitted
measurement periods. Figure c shows an example of
a tidal signal extracted with the nightly mean method. The broad
descending maximum in temperature perturbations is caused by the
semidiurnal tide, which was confirmed by a composite analysis over several
days (not shown). Note that the nightly mean method is not a suitable
method for tidal analysis. Tidal signals are generally
extracted from lidar measurements by means of the previously mentioned
composite analysis e.g.,.
The running mean method e.g., tries to
compensate for some of the shortcomings of the nightly mean
method. The spectral response is limited to timescales on the order
of the window width of the running mean – which is typically
3 h – resulting in the suppression of tides and planetary
waves. However, due to this limitation, only a very small part of
the gravity wave spectrum is retained in the analysis
(e.g., Fig. e). As stated previously, gravity wave
periods can range from about 5 min to 17 h. Thus
the limitation to short timescales excludes a major part of the
gravity wave spectrum. Figure shows that, as the length
of the running mean window increases, the GWPED increases as
well. Still, gravity waves with long periods are
suppressed. Additionally, the running mean method overestimates periods
slightly shorter than the chosen window width (Fig. e). The
strongly oscillating spectral response of the running mean method for short
periods (Fig. e) arises due to the coarse temporal resolution
of 0.5 h used in the simulations, which is a typical temporal resolution of
lidar measurements. If the temporal resolution of the simulations is
increased, these sharp peaks for periods shorter than 1 h vanish (not shown).
The beginning and the end of the measurement period pose an
additional problem for the application of the running mean
method. At the beginning of the measurement period, a centered
running mean of 3 h lacks the first 1.5 h of
observations necessary for determining the background
temperature. Thus, if in the beginning of the measurement only
1.5 h of data are available for averaging, the spectral
response differs at the beginning of the measurement period
compared to later times when 3 h of measurements are
available. The same is true at the end of the measurement period as
well as in the presence of measurement gaps. Thus, when requiring
the same spectral response at all times, the “spin-up” time of the
running mean method would have to be discarded. However, this would
result in a significantly reduced data set because one window width
of data would have to be discarded from each measurement period, in
addition to another window width for each measurement gap.
Note that the resolved high-frequency range of the gravity wave
spectrum is limited by the sampling frequency of the lidar system
which ranges typically between 10 min and 1 h,
depending on lidar performance. This is a fundamental limitation to
the extractable part of the gravity wave spectrum which affects all
methods of extracting gravity-wave-induced temperature
perturbations in the same way. The same holds true for the
effective vertical resolution of the temperature profiles.
Spatial filters
Filtering in the spatial domain, by using either the sliding
polynomial fit or the Butterworth filter, has the advantage that
the spectral response in the time domain is independent of the
length of the measurement period and the presence of measurement
gaps. This makes it possible to derive temperature perturbations
associated with gravity waves from observational periods which are
too short to yield meaningful results if temporal filtering methods
are applied. In addition, both spatial filtering methods are
capable of detecting waves with periods larger than 12 h
(Fig. c and d). One disadvantage of both spatial filtering
methods is the dampening of vertical wavelengths larger than 5 km
at the upper and lower edge of the analyzed altitude window due to edge effects.
The sliding polynomial fit has been applied in several studies
e.g.,. Different authors use
temperature data with different altitude resolutions and slightly
different parameter setups for Lf, Lw and
γ. The fit length Lf determines the cutoff
wavelength of the spectral response. The weighting window length
Lw and the e-folding width γ must be adapted to
the altitude resolution of the data used. For example, the
parameter setup γ= 3 km and Lw=Lf/3
used by results in a flat spectral
response for their altitude resolution of Δz= 2 km
and fit length Lf= 25 km. If a different
altitude resolution is chosen, a different set of parameters is
needed in order to achieve a flat spectral response in the
passband. For the altitude resolution of Δz= 0.1 km used in this study, a flat spectral response was
found for γ= 9 km and Lw= 3 km. However, vertical wavelengths of
≈ 9 km are still slightly underestimated with this parameter
set. The fit length of Lf= 20 km was
chosen following . Additional high-pass filtering,
as applied by or , was found
to be unnecessary because the long vertical wavelengths are already
strongly suppressed by the sliding polynomial fit itself.
The sliding polynomial fit method is sensitive to large changes of
the temperature gradient and may falsely overestimate temperature
perturbations for example in the presence of mesospheric inversion
layers (not shown). The Butterworth filter tends to overestimate
sudden changes in the temperature gradient of the measured
temperature profile as well. However, the magnitude of the
overestimation is generally lower than for the sliding polynomial
fit method. Furthermore, the Butterworth filter has the advantage
that it can be easily adjusted if a different cutoff wavelength is desired.
Application to measurement data
All the previously discussed characteristics influence the gravity
wave spectrum which is extracted from lidar temperature
measurements. This becomes visible if the mean GWPED of a set of
measurements is computed using different methods as shown in
Fig. . The running mean method extracts only a small
part of the gravity wave spectrum and thus shows the lowest GWPED
values. The GWPED increases if the window width of the running mean
is increased. The nightly mean method yields the largest GWPED
values at higher altitudes. This can be attributed to the
insufficient suppression of tides and other processes unrelated to
gravity waves which happen on longer timescales. In the lower
stratosphere the sliding polynomial fit method and the Butterworth
filter yield the largest GWPED values. This is most likely caused
by the inclusion of long-period waves such as quasi-stationary
mountain waves. These waves have the largest impact on GWPED in the
lower stratosphere above Lauder during winter
. Above 30 km altitude GWPED values are
reduced. A possible mechanism is that mountain waves with very
large amplitudes become unstable at these altitudes and break. This
has for example been observed by , who detected
a self-induced critical layer around 30 km altitude caused
by a strong mountain wave event above northern Scandinavia.
The fact that the Butterworth filter exhibits a lower growth rate
of GWPED compared to the running mean method (Fig. ) may
be evidence that short-period gravity waves can propagate more
easily to higher altitudes than gravity waves with long
periods. This complicates the comparison and interpretation of
GWPED growth rates (generally expressed in terms of scale heights)
of different studies. For example deduced a GWPED
scale height of 9–11 km with the nightly mean method for
a midlatitude site. On the other hand, reported
a GWPED scale height of approximately 7 km determined with
the sliding polynomial fit method for measurements conducted at
Antarctica. A large part of the difference in retrieved scale
height can be attributed to different wave propagation conditions
at the two sites. However, it remains an open question to what extent
the results are affected by the use of different methods to extract gravity waves.
Conclusions
We evaluated four commonly used methods of extracting
gravity-wave-induced temperature perturbations from lidar
measurements. A widely used method – the nightly mean method – relies
on filtering in time by subtraction of the nightly mean
temperature. Thus, it is sensitive to all temperature changes
occurring on the timescale of the measurement period, including
temperature changes induced by planetary waves and tides. Because
measurement periods can vary substantially in length and the
spectral response of the nightly mean method depends on the length
of the measurement period, the extracted gravity wave spectrum can
vary from observation to observation. This makes the nightly mean
method an improper choice for compiling gravity wave statistics if a
data set with a varying length of observational periods is analyzed.
The second method which relies on filtering in time, the running
mean method, provides a more stable spectral response with regard
to a varying length of the measurement period. However, it
extracts only a small fraction of the gravity wave spectrum, with
long-period waves being strongly suppressed. Moreover, the running
mean method exhibits a variation in the spectral response at the
beginning and end of a measurement period as well as in the
presence of measurement gaps.
The sliding polynomial fit method is not only capable of
extracting waves over a broad range of temporal scales but also
suppresses tides and planetary waves due to their large vertical
wavelengths. In addition, it is unaffected by measurement
gaps. However, the parameters used for the sliding polynomial fit
need to be adjusted to the altitude resolution of the measured
temperature profiles in order to provide a flat spectral response
in the passband.
The Butterworth filter provides an alternative to the sliding
polynomial fit method which is not only easy to implement but also
easily adjustable to a desired cutoff wavelength. Also, the filter
is largely independent of the altitude resolution while providing
all the advantages of the sliding polynomial fit
method. Furthermore, sudden changes in the background temperature
gradient affect the Butterworth filter less than the sliding
polynomial fit method.
Based on the results presented here, two methods are recommended
for gravity wave extraction from lidar temperature measurements covering a large altitude range:
the running mean method is the most suitable method if the
analysis is focused on short-period gravity waves with large
vertical wavelengths. On the other hand, if a broad passband is
desired which covers a large part of the gravity wave spectrum,
the Butterworth filter is the method of choice. Additional
advantages are the insensitivity to measurement gaps, a varying
length of observational periods and the altitude resolution of the
measured temperature profile.
Acknowledgements
We thank the NIWA personnel at Lauder station for their support
during the DEEPWAVE measurement campaign. This work was supported by
the project “Processes and Climatology of Gravity Waves” (PACOG)
in the framework of the research unit “Mesoscale Dynamics of
Gravity Waves” (MS-GWaves) funded by the German Research Foundation.
The article processing charges for this open-access publication
were covered by a Research Centre of the Helmholtz Association.
Edited by: L. Hoffmann
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