AMTAtmospheric Measurement TechniquesAMTAtmos. Meas. Tech.1867-8548Copernicus PublicationsGöttingen, Germany10.5194/amt-10-3325-2017Optimal estimation of water vapour profiles using a combination of Raman lidar and microwave radiometerFothAndreasandreas.foth@uni-leipzig.dehttps://orcid.org/0000-0002-1164-3576PospichalBernhardhttps://orcid.org/0000-0001-9517-8300Leipzig Institute for Meteorology, University of Leipzig, Leipzig, GermanyInstitute for Geophysics and Meteorology, University of Cologne, Cologne, GermanyAndreas Foth (andreas.foth@uni-leipzig.de)12September20171093325334420March20178May201710July201724July2017This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/3.0/This article is available from https://amt.copernicus.org/articles/10/3325/2017/amt-10-3325-2017.htmlThe full text article is available as a PDF file from https://amt.copernicus.org/articles/10/3325/2017/amt-10-3325-2017.pdf
In this work, a two-step algorithm to obtain water vapour profiles from
a combination of Raman lidar and microwave radiometer is presented. Both
instruments were applied during an intensive 2-month measurement campaign
(HOPE) close to Jülich, western Germany, during spring 2013. To retrieve
reliable water vapour information from inside or above the cloud a two-step
algorithm is applied. The first step is a Kalman filter that extends the
profiles, truncated at cloud base, to the full height range (up to 10 km) by
combining previous information and current measurement. Then the complete
water vapour profile serves as input to the one-dimensional variational
(1D-VAR) method, also known as optimal estimation. A forward model simulates
the brightness temperatures which would be observed by the microwave
radiometer for the given atmospheric state. The profile is iteratively
modified according to its error bars until the modelled and the actually
measured brightness temperatures sufficiently agree. The functionality of the
retrieval is presented in detail by means of case studies under different
conditions. A statistical analysis shows that the availability of Raman lidar
data (night) improves the accuracy of the profiles even under cloudy
conditions. During the day, the absence of lidar data results in larger
differences in comparison to reference radiosondes. The data availability of
the full-height water vapour lidar profiles of 17 % during the 2-month
campaign is significantly enhanced to 60 % by applying the retrieval. The
bias with respect to radiosonde and the retrieved a posteriori uncertainty of
the retrieved profiles clearly show that the application of the Kalman filter
considerably improves the accuracy and quality of the retrieved mixing ratio
profiles.
Introduction
In accordance with the latest report of the Intergovernmental Panel on
Climate Change (IPCC), water vapour plays a key role in the description of
the thermodynamic state of the atmosphere and is
the most important greenhouse gas . Its amount in the
atmosphere is controlled mostly by the air temperature, rather than by
emissions . Therefore, tropospheric water vapour
is considered as a feedback agent more than a forcing to climate
change . The water vapour amount is highly variable in
space and time, since it can considerably increase due to evaporation or
decrease due to condensation and precipitation .
Furthermore, the latent heat strongly influences the energy cycle. The
typical residence time of water vapour in the atmosphere amounts to 10
days . Due to its spatio-temporal variability and
its involvement in many atmospheric processes (e.g. cloud formation) it is
difficult to properly implement water vapour in climate
models .
In the last decades, the resolution of atmospheric circulation models has
been improved, more atmospheric processes have been incorporated and the
parametrizations of physical processes have been improved
. In order to evaluate and improve model forecasts,
parametrization schemes and satellite retrievals, the observations need to be
enhanced. Uncertainties in both observations and modelling of water vapour
strongly affect the representation of clouds and precipitation in climate
models and predictions. For that reason the German research project High
Definition Clouds and Precipitation for advancing Climate Prediction
(HD(CP)2) was initiated aiming to improve cloud and precipitation
representation in models and to quantify the errors associated. One part
within the HD(CP)2 initiative was the intensive observation campaign
HD(CP)2 Observational Prototype Experiment (HOPE) in Jülich
. Data from this campaign will be used in this work
which presents a retrieval of water vapour profiles from ground-based remote
sensing. During HOPE, different remote sensing instruments to measure water
vapour, both active and passive, were deployed.
An active method is given by the Raman lidar technique . Water vapour mixing ratio has been
determined for several decades using this technique . With advancing technology Raman lidars enabled high
vertical resolution measurements of water vapour and extended their range to
the whole troposphere , during daytime or
automatically . However, water vapour Raman
lidars should be calibrated using simultaneous and collocated measurements
from for example a microwave radiometer (MWR) or radiosonde
(RS) . Until now, Raman lidars were
mostly used as research instruments that did not work unattended or
automatically on a routine basis. Another major drawback of Raman lidars is
that they do not provide any water vapour information from inside the cloud
or above due to the strong signal attenuation, especially in liquid clouds.
Hence, these measurements are limited from the surface to the cloud base.
Furthermore, daytime measurements are limited in height due to the presence
of scattered solar radiation .
Another approach is to use passive remote sensing to sound the thermodynamic
state of the atmosphere. Passive microwave radiometry can provide atmospheric
water vapour observations with high temporal resolution, but limited vertical
information . However, the integrated
water vapour (IWV) can be retrieved very accurately. Microwave radiometers
can be operated during all weather conditions except for precipitation
. As with many remote sensing techniques accurate
calibrations are crucial for obtaining precise
measurements .
By contrast to the already presented remote sensing observations, water vapour
profiles can be measured in situ using RS . Routine
RS launches are mostly performed by national weather services usually twice
a day at special locations. Therefore, both horizontal and temporal
resolution of routine measurements are rather low. However, these profiles
can serve as reference for remote sensing observations.
As described above, it is a challenge to provide continuous high-resolution
water vapour profiles with a single instrument. In recent years, the Leipzig
Aerosol and Cloud Remote Observations System (LACROS) ,
installed a combination of ground-based remote sensing systems. The synergy
of complementary information from both active and passive instruments can
provide a more comprehensive understanding of atmospheric
processes . From
a combination of radar reflectivities and liquid water path from MWR,
successfully derived liquid water content (LWC)
profiles. presented a method based on a Kalman
filter that incorporates current and past
measurements followed by a statistical inversion that combines the lidar with
the radiometric and climatological data. The Cloudnet project is comprised of
a number of algorithms for the continuous analysis of cloud properties by
means of remote sensing with lidar, MWR and cloud radar
. The instruments synergy allows for a continuous
evaluation of the representation of clouds in climate and weather forecast
models . Additionally, the data
set enables the development and validation of new cloud remote sensing
synergy algorithms.
developed the so-called integrated
profiling technique (IPT) that integrates a ground-based MWR, a cloud radar
and a priori information, e.g. from RS. This approach enables the derivation
of temperature, humidity and liquid water content profiles
and their associated error estimates. The IPT is based on a variational
scheme, also known as optimal estimation .
as well as used a similar approach
as but with background information from a short-range
numerical weather prediction model instead of RS climatology.
The synergy of Raman lidar and MWR is beneficial for continuously observing
the vertical water vapour distribution. When both Raman lidar and MWR are
measuring collocated and simultaneously, continuous water vapour profiles can
be obtained operationally . However,
the Raman lidar needs to be calibrated on a routine basis. A calibration
method that is based on the IWV from MWR is suited for this
issue . In previous approaches the total precipitable water
from MWR in combination with RS has been used to calibrate the water vapour
profiles . Calibration methods only based on RS
are often inappropriate for
continuous monitoring of the tropospheric water vapour with Raman lidar
because of their low temporal resolution and the requirement of regular RS
launches.
The aim of this study is to present a two-step algorithm that combines
a Raman lidar and a MWR by using an optimal estimation approach. The
retrieval can be seen as an extension of the IPT by .
also generated a variational retrieval based on
these two instruments. At first glance, both approaches seem to be similar,
but they are fundamentally different with regard to the optimal estimation
method. used both (Raman lidar and MWR) as part
of the observation vector. Because the water vapour profiles from Raman lidar
are strongly disturbed by clouds, they are truncated at the cloud base. In
the present work, the truncated Raman lidar profiles are extended to the full
height range by using a Kalman filter in a first step. Then the
Kalman-filtered profiles serve as input to the optimal estimation. This
approach is based on studies of . Additionally,
the focus of the presented work is to develop a method that enables routine
retrieval of a continuous time series of water vapour profiles and their
error estimates during all non-precipitating conditions.
Instrumentation
In the framework of the HD(CP)2 initiative HOPE was conducted around
Jülich in western Germany during April and
May 2013 . The goal of HOPE was to probe the
atmosphere with a specific focus on boundary layer development and the
development of clouds and precipitation. Two observatories were set up in
addition to JOYCE . The LACROS site
was temporarily built up in Krauthausen,
which is about 4 km south of JOYCE. Both JOYCE and LACROS
observatories are equipped with a set of active and passive remote sensing
instruments such as lidars and MWRs which allow the application of the
proposed retrieval. Radiosondes were launched at the KIT
station in Hambach, which is about 4 km away from JOYCE and LACROS.
Furthermore, a 120 m tower provide surface meteorological data as
pressure, temperature and humidity.
Raman lidar PollyXT
At LACROS, the lidar measurements were conducted with the fully automatic
portable multiwavelength Raman and polarization lidar
PollyXT by the Leibniz Institute
for Tropospheric Research (TROPOS). PollyXT measures
backscattered light at wavelengths of 355, 532 and 1064 nm and Raman
scattered light at 387, 407 and 607 nm wavelengths. From that, water vapour
profiles can be determined . In
the lowermost heights the overlap of the laser beam with the receiver
field of view of the bistatic system is incomplete. However, the overlap of
both Raman channels is assumed to be identical and for that reason the
overlap effect should be negligible regarding water vapour measurements.
Nevertheless, there are some uncertainties in the lowermost 600 m. Therefore,
the signal ratio is set constant to account for the overlap problem.
Additionally, the mixing ratio error is artificially increased resulting in
less impact of erroneous profiles near the surface to enlarge the influence
of both Kalman filter and optimal estimation. During daytime, no water vapour
measurements can be performed due to the high daylight background and the
weak signal from Raman scattering. The PollyXT raw data (30 m
and 30 s) are processed and calibrated to mixing ratio profiles as explained
in . The vertical and temporal resolution of the calibrated
profiles amounts to 90 m and 5 min to decrease the measurement noise and to
retrieve water vapour from higher altitudes. The calibrated water vapour
profiles are then used for the proposed retrieval.
An overview of the area of operation and the automated measurement capabilities of Polly systems all
over the world is extensively introduced by .
Microwave radiometer HATPRO
The humidity and temperature profiler (HATPRO), built by Radiometer Physics
GmbH, Germany, is a passive instrument that measures atmospheric emission at
two frequency bands in the microwave spectrum. Seven channels are along the
22.235 GHz H2O absorption line. From these observations humidity
information can be retrieved. The seven channels of the other band from 51 to
58 GHz along the O2 absorption complex contain the vertical
temperature profile information. The fully automatic microwave radiometer
HATPRO makes it possible to derive temperature and humidity profiles as well as
integrated quantities such as integrated water vapour (IWV) and liquid water
path (LWP) with a high temporal resolution up to 1 s .
Their uncertainties are 0.5 kgm-2 for IWV
and 22 gm-2 for low LWP values and increase up to
45 gm-2 for LWP values higher than 500 gm-2,
respectively . Observations are possible during nearly all
weather conditions except precipitation.
Statistical algorithms were used to retrieve temperature profiles, IWV and
LWP from the measured brightness temperatures by means of a multi-linear
regression between modelled brightness temperatures and atmospheric profiles.
That algorithm is based on a long-term data set of De Bilt
radiosondes .
Weighting functions are well suited to describe the ability for humidity
profiling. Figure shows the weighting functions for the
seven HATPRO frequencies along the H2O absorption band. Generally,
the measured brightness temperatures do not originate from an isolated height
level. The weighting functions describe the contribution of a certain height
to the observed signal. Ideally, the weighting functions are peaked functions
and several frequencies contribute information from different height levels.
Three weighting functions (22.24, 23.04 and 23.84 GHz) differ considerably
from each other. The higher frequencies have a similar shape as the
atmosphere is optically thin at these frequencies. For that reason they add
only little information and the vertical distribution of humidity is limited.
The usage of the 31 GHz channel caused unrealistic results. The reason for
that behaviour was not identified but might be induced by the forward model
or a faulty calibration.
The MWR was also equipped with a standard meteorological weather station
measuring temperature, pressure and relative humidity. These values are only
used to calculate the pressure profile that is used in the forward model. The
surface values needed for the optimal estimation originate in the surface
tower measurement which is much more accurate. Arising pressure uncertainties
result in negligible deviation in the modelled brightness temperatures.
Absolute humidity weighting function for the HATPRO frequencies for
a cloud-free model atmosphere.
Radiosondes
During HOPE, radiosondes (RSs) were launched a minimum twice a day (11:00 and 23:00 UTC) and more
often during intensive observation periods (IOPs) at the KITCube site in Hambach. The RS (type Graw
DFM-09) measures temperature, humidity, pressure and wind
velocity . Due to the vicinity of the RS station to an
open-cast mining with a depth of nearly 400 m, horizontal inhomogeneities between the RS launch
station and LACROS are likely .
Retrieval methodology
The focus of this work is to retrieve a continuous time series of water
vapour profiles from a combination of ground-based remote sensing with Raman
lidar and MWR in a straightforward way to offer a broad application. Most of
this section has already been described and presented
in without explicit citation. The retrieval is
a two-step algorithm that combines the Raman lidar mixing ratio profile with
the MWR brightness temperatures. The Kalman filter (first step) eliminates
measurement disruptions (e.g. clouds) to provide a full-height mixing ratio
profile that serves as input to the one-dimensional variational assimilation
(optimal estimation method). The retrieval can be applied to raw data (photon
counts) using the calibration method based on or using
already calibrated profiles.
Figure gives a brief overview of the retrieval
framework. It starts with the latest analysed state x^k-1,
which is advanced to the estimated state xkE, with k
being the time index. This state is then combined with the current lidar
measurement yk to obtain the filtered state
xkF using the Kalman filter.
xkF is then used as the a priori input to the
one-dimensional variational assimilation. The a priori profile is modified
such that the modelled brightness temperature matches those measured with the
microwave radiometer (MWR) zk, resulting in the most probable
estimated state x^k, which is again projected in time in the
consecutive step. Inverse methods for atmospheric sounding are well described
in . For clarity the same notation is used.
Sketch of the retrieval scheme. Details are given in the text. This
figure is adapted from .
Definition of quantities
In this section the state vector and the two measurement vectors are
described. The first measurement vector contains the mixing ratio profile
from the lidar measurement. It is used in the first retrieval step (Kalman
filter). The second measurement vector consists of the brightness
temperatures from the MWR measurement and a surface mixing ratio from a
standard meteorological station. This vector is used in the optimal
estimation.
The atmospheric state is described by the state vector
x=[q1,…,qn]T,
which contains the humidity variable q at different height levels from 0 to height n
(e.g. 10 km). The vertical resolution originates from the lidar measurements and is equal to 90 m.
The humidity variable q is given as the natural logarithm of water vapour mixing ratio. The
benefit of using the logarithm is the limited range of variation and the prevention of negative
unphysical values resulting in a lower amount of unrealistic states .
The lidar measurement vector of length myy=[q1,…,qmy]T
contains the water vapour mixing ratio at each height level from ground up to
a possible cloud base. The lidar profiles y and the associated errors
ϵy are usually given in mixing ratio. For the
reasons mentioned above, both have to be transformed into q values. The
transformed errors define the diagonal elements of the lidar measurement
covariance matrix Sy. The off-diagonal elements are assumed to
be zero which means that no correlation exists between the errors at
different height levels.
The second measurement vector, called from now on observation vector, is given as
z=TB,1,…,TB,mν,qsT
with the dimension mz. It contains the brightness temperature
TB at a certain frequency ν and the surface mixing
ratio qs from a standard meteorological station. In this study
only zenith observations and frequencies along the water vapour absorption
band are chosen. The combined measurement and forward model covariance matrix
Sz contains the errors from the MWR observation, from the
surface mixing ratio measurement and from the forward model. The errors from
the MWR observation are the radiometric noise. Its variance is set to
0.25 K2 at each frequency. The off-diagonal elements are set to
0.01 K2, meaning small covariances between the
frequencies . The determination of the forward
model error is described in Sect. . Forward model uncertainties that
occur due to assumptions in the LWC profiles are illustrated in
Sect. . The measurement uncertainty of the surface
mixing ratio is roughly assumed to be 0.1 gkg-1. However, the
uncertainty is increased due to the distance between the measurement site and
the surface humidity sensor (see Sect. ) and is assumed
to be 0.3 gkg-1.
First-guess profiles and errors are created for the HOPE campaign. Usually
they are formed by a certain number of RS. Therefore the covariance matrix is
sometimes called RS climatology. For the HOPE campaign 211 RS that were
launched during April and May 2013, were used to calculate a mean profile
that serves as a first-guess profile and is used after a long measurement
disruption. Additionally, the correlation and covariance matrices are
determined (Fig. ). Here, the humidity variable
is interpolated to the state grid space (lidar height grid) and is
transformed to the natural logarithm before calculating the matrices. Both
clearly illustrate the correlations between water vapour at different heights
in the atmosphere. Naturally, the correlation is close to 1 near the main
diagonal and is smaller for off-diagonal terms. Due to well-mixed conditions
the correlation in the lowest 1.5 km is higher. These matrices are similar
to those from previous studies .
Correlation (a) and covariance matrix (b) derived
from 211 radiosondes for HOPE. Both are shown for the natural logarithm of
the mixing ratio (ln(MR)) as a function of height with a resolution of 90 m.
Kalman filter
In the presence of clouds, the lidar profile is truncated at the cloud base due to the strong
attenuation within the cloud. We use the Kalman filter to expand the truncated lidar profile to the
full height range using previous information. The Kalman filter is based on the following two
equations:
yk=Hkxk+ϵy,k,xk+1=Mkxk+ϵt,k.
The evolution operator (e.g. forward model) Hk projects the
state into measurement space (Eq. ). Since xk
and yk use the same humidity variable, the forward model matrix
Hk equals the unity matrix with dimension my×n.
Equation () describes the transition of the state vector
at time step k to time step k+1. The transition matrix Mk
is assumed to be the unity matrix due to the lack of an atmospheric model.
The square of the transition error ϵt,k forms the
diagonal elements of the covariance matrix St,k. For the
calculation of St,k the Schneebeli method can be applied
. Schneebeli generated a time series of synthetic
profiles from a combination of consecutive radiosondes and ground values.
St,k is finally calculated from an ensemble of these
consecutive profiles. A similar approach is described by .
After a large number of time steps, it might happen that the correlations
between layers get lost which can result in unrealistic profiles.
Additionally, the retrieval tends to be unstable with either unphysical
solutions or even be non-convergent when using the transition error. Another
possibility is to start with the RS climatology covariance
(Sclim) as previous covariance matrix
(S^k-1) at every consecutive time step. Using this
approach the addition of the transition covariance matrix
(St,k) can be skipped. In this application the latter approach
is used which is much more stable.
Using Eq. () and the assumptions explained above, the last
analysed state x^k-1 and its covariance matrix
S^k-1 are propagated as follows:
xkE=Mkx^k-1=x^k-1,SkE=MkS^k-1MkT+St,k=Sclim,
where xkE and SkE are the estimated state and its
covariance matrix, respectively. These are then combined with the lidar measurement at time step k
to obtain the filtered state:
xkF=xkE+GkKyk-HkxkE,
with GkK being the Kalman gain matrix:
GkK=SkEHkTHkSkEHkT+Sy,k-1.
The covariance matrix of the filtered state is determined by
SkF=SkE-GkKHkSkE.
Finally, xkF and SkF serve as input to the optimal
estimation.
The application of this technique for linear filtering and prediction
problems was first described by and .
Forward model
In the optimal estimation framework microwave brightness
temperatures (TB) at given
frequencies (ν) are modelled from the a priori atmospheric
profiles and are compared to those that are measured. However, in this work
only zenith observations are used. Based on ,
F(x) models the non-scattering microwave radiative transfer
using gas absorption by Rosenkranz and liquid water absorption by Liebe
for each height level of the retrieval
grid (90 m). The Rosenkranz gas absorption model is corrected for the water
vapour continuum absorption according to . The humidity
information (q) of the a priori profile originates from the
Kalman-filtered state, whereas the temperature profiles (T) are
provided by statistical retrievals from MWR observations
(Sect. ). The pressure profiles (p) are calculated by
surface pressure observations from MWR and the barometric formula. Because
the retrieval grid is limited to 10 km, the thermodynamic state between 10
and 30 km is taken from a RS climatology above Essen, which is in the
vicinity of the HOPE area. The restriction to the troposphere up to 10 km
would lead to errors of around 1 K in the calculation of the brightness
temperatures. Assumptions about the liquid water content (LWC) and
its determination are described in Sect. . The forward
modelling of the surface mixing ratio is trivial. It is a one-to-one
translation to the lowest level of the state vector x. In conclusion,
F(x) is of the following form:
F(x)=RTO(T,q,p,LWC,ν1)⋮RTO(T,q,p,LWC,νmν)q1,
with RTO being the radiative transfer operator.
The forward model error is calculated as covariance of the difference between
brightness temperatures modelled by two different absorption codes of
Rosenkranz and Liebe applied to a long-term
data set of radiosondes from Lindenberg, Germany. The diagonal elements of
its covariance matrix are shown in Table . One has to consider
that there are significant off-diagonal terms. This error is part of the
combined observation and forward model covariance Sz. The
uncertainties of the gas absorption models cause biased mixing ratio profiles
(see Sect. ).
Forward model error for each frequency due to different absorption
codes. Uncertainties are given as square root of the diagonal elements of the
covariance matrix.
Channel numberFrequencyHATPRO uncertainty(GHz)(K)122.240.07223.040.2323.840.42425.440.56526.240.55627.840.53731.400.51Liquid water assumption
Since liquid water strongly affects the absorption in the microwave spectrum,
its amount and height have to be known. However, from MWR only the integral
value can be derived, and not its vertical distribution. In order to
determine LWC profiles, the cloud boundaries have to be determined. The cloud
base of a liquid water cloud is identified by the gradient method based on
the 1064 nm channel from lidar which has been shown to
be a more robust method for the automatic detection of the cloud base than
the wavelet covariance transform . However,
a threshold value has to be chosen carefully to distinguish between thin
liquid water clouds and optically thick aerosol layers below liquid water
clouds. Additionally, liquid water clouds are only detected if the LWP is
larger than a narrow threshold of 5 gm-2.
The LWC is calculated from the modified adiabatic assumption
:
LWC=LWCad1.239-0.145ln(h),
where h indicates the height above cloud base in m and h within the range
of 1–5140 m. The adiabatic LWCad is calculated using the
temperature and pressure profiles and is corrected for effects of dry air
entrainment, freezing drops or precipitation. The LWC is integrated over all
layers until the calculated LWP equals the LWP measured with MWR. This height
is finally defined as cloud top. However, any profile is treated as single-layer cloud with this method.
(a) Brightness temperature difference as a function of LWP
(dots) using two different LWC assumptions. The colours indicate the
according frequencies (top right). The mean and the standard deviation per
bin size are indicated by coloured lines and error bars, respectively. The
bin size amounts to 0.05 kgm-2. The number of occurrences is
given in grey bars at the top. (b) Exemplary covariance matrix for
an LWP between 0.45 and 0.5 kgm-2. The channel numbers
correspond with the HATPRO frequencies given in (a), which means, for example, that 1
refers to 22.24 GHz.
Usual approaches to diagnose LWC profiles from radiosonde are based on
a threshold method . Cloud bases or tops are identified when
the relative humidity exceeds or falls below 95 %, respectively. Within the
cloud the LWC is calculated using the modified adiabatic assumption
. The uncertainty that results in the assumption of
single-layer clouds is estimated by comparing both mentioned methods. This is
done for a long-term data set of radiosondes from Lindenberg, Germany. For
these radiosonde profiles, brightness temperatures are modelled at the HATPRO
frequencies using both LWC profile assumptions. The brightness temperature
difference as a function of LWP is illustrated in Fig. a. As
can be seen, the means and standard deviations (coloured lines and error
bars) increases with increasing LWP. In addition, the difference increases
from the 22.24 to 31.4 GHz. Naturally, there is no difference for single-layer clouds indicated by the dots at 0 K. The number of occurrences
decreases with increasing LWP (grey bars on the top). However, only clouds
with an LWP larger than 0.02 kgm-2 are considered.
Figure b shows an exemplary covariance matrix for an LWP
between 0.45 and 0.5 kgm-2. These uncertainties contain
significant off-diagonal terms and are larger for the channels that are more
sensitive to liquid water (31.4 GHz). According to the observed LWP the
corresponding covariance is added to the combined observation and forward
model covariance matrix Sz to account for the assumption of
single-layer liquid water clouds.
Optimal estimation method (OEM)
A schematic overview over the optimal estimation is given in Fig. . In basic terms,
the forward model simulates what the MWR would observe given an arbitrary state. The problem is that
several different states may produce the same measurement. This is a so-called ill-posed problem. To
constrain the state space a priori information as lidar profiles are needed. In the proposed
retrieval the lidar profiles are Kalman filtered as mentioned above. Finally, the optimal estimation
finds the most probable solution (mixing ratio profile) from a class of solutions. The theory of
inverse modelling based on optimal estimation methods is briefly introduced in this section and
described in more detail in .
The optimal estimation of an atmospheric state by a given observation vector
z and an a priori state xa=xF can
be found by minimizing the cost J(x^) function of the form
J(x^)=Ja(x^)+Jz(x^)+Jsup(x^).
Here Ja(x^) indicates the a priori costs,
Jz(x^) the observation costs and
Jsup(x^) is a penalty term to avoid supersaturation.
Since both liquid and ice phase can occur in clouds at temperatures between
-38 and -5 ∘C
, the
saturation mixing ratio is defined as follows:
qsat=qliqsat:-5∘C<ϑqlinsat:-38∘C<ϑ<-5∘Cqicesat:ϑ<-38∘C,
where qliqsat and qicesat
are the saturation mixing ratios above liquid water and ice, respectively. The
qlinsat denotes a linear function that describes
the transition from qliqsat to
qicesat. The related uncertainty is defined as
the difference between qliqsat and
qlinsat and between
qlinsat and qicesat,
respectively. It amounts to a maximum of 0.23 gkg-1 at
-8 ∘C and decreases with decreasing temperature which usually means
increasing height.
Jsup(x^) adds a penalty if the retrieval produces
supersaturation all over the profile
. This function is defined by
Jsup(x^)=∑jnJsup(xj),Jsup(xj)=0:qj⩽qjsatζqj-qjsat3:qj>qjsat.
The constant ζ=106 drives the strictness of the constraint. The larger ζ, the more
strict is the constraint. Here, a large value is set to avoid supersaturation all over the profile.
However supersaturation is not completely avoided due to the uncertainties in the temperature
profiles from the MWR that are the basis of the saturation mixing ratio qsat.
Retrieved mixing ratio profiles both with (red) and without (green)
supersaturation constraint on 23 April 2013, 01:02 UTC. Cloud base is
indicated by the dashed line. The saturation mixing ratio is illustrated by
the dotted grey line. The a priori profile (blue) for both scenarios is the
same.
Figure illustrates the benefit of the
supersaturation constraint on 23 April 2013, 01:02 UTC. The disregard of the
constraint results in too large mixing ratio values in altitudes above 5 km.
This overestimation corresponds to a supersaturation of 200 up to 300 %
relative humidity. The application of the constraint prevents the
overestimation of humidity. The resulting values are in good agreement with
the saturation mixing ratio with relative humidity values not exceeding
115 %, which is more realistic.
The implementation of a constraint that prohibits subsaturation within clouds is not beneficial in
this application. The assumption of single-layer liquid water clouds and the uncertainties in the
temperature profile would result in uncertain saturation mixing ratio profiles and finally lead to
wrong retrievals.
With each term written out Eq. () becomes
J(x^)=x^-xaTSa-1x^-xa+z-F(x^)TSz-1z-F(x^)+Jsup(x^).
For clarity the time index is omitted here. x^ is the optimal
estimate of the atmospheric state. Sa and Sz
denote the covariance matrices of the a priori state and the observation,
respectively. The optimum solution can be found iteratively using the
Levenberg–Marquardt method:
xi+1=xi+1+γSa-1+KiTSz-1Ki+J¨sup-1×KiTSz-1z-Fxi+Sa-1xi-xa+J˙sup,
with i being the iteration index. The dots above J indicate the
first and the second derivative, respectively. The Levenberg–Marquardt
parameter γ is increased by a factor of 10 if
J(x^i+1)⩾J(x^i) and reduced by a
factor of 2 if J(x^i+1)<J(x^i). In this work
the initial value of γ=2. It was found that the Levenberg–Marquardt
method does not reach convergence faster but more reliably than the
Gauss–Newton approach (γ=0)
. If γ→∞,
the step tends towards the steepest descent of the cost function, allowing
for leaving a local minimum towards a global minimum .
Ki denotes the weighting function matrix, also known as
Jacobian or kernel (hence K), but from now on Jacobian. It is
defined as
Illustration of the optimal estimation method. Details are given in
the text.
K=∂F(x^)∂x^
and calculated by perturbing the state vector at each height level by
ln(0.1 gkg-1). Equation () is iterated
until the following criterion is fulfilled:
F(xi+1)-F(xi)Sδz-1F(xi+1)-F(xi)≪m,
with Sδz being the covariance matrix between the
measurement and F(x^):
Sδz=SzKSaKT+Sz-1Sz.
Finally, the covariance matrix of the resulting analysed state vector (a posteriori) is
calculated as
S^=KTSz-1K+Sa-1-1.
Since the retrieval might converge to a false minimum it is necessary to check the retrieval for
correct convergence. Therefore, the χ2 test for consistency of the optimal retrieval
(xop) with the observation (zobs) is introduced:
χ2=F(xop)-zobsTSδz-1F(xop)-zobs.
Here, the forward modelled state F(xop) and
the observation vector zobs are compared with the error
covariance matrix Sδz. The test is usually used to look
for outliers, i.e. cases where the χ2 value is larger than
a threshold value (χthr). χthr is
calculated for a probability of 5 % that χ2 is greater than the
threshold for a theoretical χ2 distribution with mz degrees of
freedom. All retrieved profiles with a χ2 value that exceeds the
threshold are marked as untrustworthy. The χ2 values of all retrieved
profiles are analysed and discussed in Sect. .
The averaging kernel matrix A gives the sensitivity of the retrieval to the true state:
A=∂x^∂x=KTSz-1K+Sa-1-1KTSz-1K.
The rows aiT of A are the averaging kernels. In an
ideal inverse method, A would be a unity matrix. Generally the
averaging kernels are peaked functions which indicate the smearing of
information across multiple levels. In this work, the averaging kernels are
not peaked functions, because the MWR observation does not provide enough
vertical information. This issue is covered in detail in
Sect. . The averaging kernel has an area
aarea, which is a measure of fraction that comes from the
observation, rather than the a priori. The area of ai is the sum
of its elements and can be calculated as Au where u
is a vector with unit elements.
The information content of a measurement can be expressed by the degree of freedom
(d), which is the trace of A. d is a measure of how many
independent quantities are measured. One has to consider that the larger the a priori uncertainty,
the larger d and the larger the retrieved a posteriori uncertainty .
In summary, the retrieval is strongly driven by the a priori uncertainty which constrains the
subspace in which the retrieval must lie. The larger the off-diagonal elements of this covariance,
that means the higher the correlations and the smaller is the subspace. For that reason the a priori
covariance has to be estimated very carefully. In the proposed retrieval the a priori covariance is
strongly decreased by the application of the Kalman filter that reduces the subspace of possible
solutions.
Retrieval applicationCloud-free conditions
Overview of a mostly cloud-free case on 5 May 2013.
(a) liquid water path (LWP). (b) Height–time display of the
mixing ratio measured by the Raman lidar. (c) Height–time display of
the retrieved optimal estimated mixing ratio. The solid line indicates the
height where the Raman lidar profiles are truncated. The dotted line defines
the cloud base height determined by the lidar.
In this section the general functionality of the retrieval of water vapour
profiles and basic parameters such as averaging kernels and degree of freedom
are introduced using a straightforward cloud-free case.
Figure gives an overview of a mostly cloud-free
day (5 May 2013). It shows the LWP, the height–time display of the mixing
ratio measured by the Raman lidar PollyXT and the height–time
display of the retrieved profiles after applying the two-step algorithm. The
vertical and temporal resolution of the Raman lidar mixing ratio profiles is
90 m and 5 min, respectively. In the early morning up to 03:00 UTC the
mixing ratio could be measured very well by the lidar
(Fig. b). With the rising
sun the profiles are more and more
noisy such that even the lowermost values are disturbed. For that reason the
lidar profiles can be no longer used; they serve as an input to the OEM only
if they are available. At 05:00 UTC the water vapour channel is
automatically switched off and usually switched on again at 18:00 UTC. The
noise decreases after sunset allowing an undisturbed water vapour observation
from 20:00 UTC on. An automated depolarization calibration produces a gap
around 22:00 UTC. The cloud base height indicates the development of
boundary layer clouds which can also be seen in the LWP values during daytime
(Fig. a). Although there are no lidar profiles
during the day, a complete time series of mixing ratio profiles can be
retrieved (Fig. c). In the following, the retrieval
application of two different conditions, with full height and without mixing
ratio profiles from lidar, are distinguished.
Overview of cloud-free scene on 5 May 2013, 23:02 UTC. Mixing ratio
(MR) profiles from the Raman lidar and the estimated (a), the Kalman-filtered (b) and the optimally estimated state (c).
Additionally, the mixing ratio of the radiosonde (RS) is shown (c).
Error bars are added to the profiles at the different states of the
processing. (d) Averaging kernel for a subset of 10 levels
indicated by the coloured numbers. (e) Accumulated degree of freedom
dacc (solid) and the area of the averaging kernel
Aarea (dotted).
Figure illustrates the algorithm processing in the
presence of full-height calibrated Raman lidar profiles on 5 May 2013,
23:02 UTC. The last analysed state (from 5 min ago) is propagated in time
to the estimated state (Fig. a). The propagation is a
1 : 1 translation. Its uncertainty is small because it originates in the
last analysed state that was also driven by a lidar profile. The plotted
uncertainties are the square roots of the diagonal elements of the
corresponding covariance matrix. The Kalman filter combines the current lidar
measurement and the estimated state to the filtered state that is more driven
by the estimated state than by the lidar measurement
(Fig. b). The filtered profile serves as input
(a priori) to the optimal estimation (Fig. c). The small
uncertainties of the a priori forces the retrieval to resemble the filtered
state with similar uncertainties. The ability of the lidar to perform precise
water vapour measurements results in small differences to the reference RS.
The comparison to RS is discussed in detail in the next paragraph.
Figure d shows the averaging kernels for a subset of 10
levels. They demonstrate how the information in one retrieved bin is derived
from an average of those around it. Ideally the averaging kernels are peaked
functions. However, the vertical humidity information at the HATPRO
frequencies is limited, which results in smooth functions that are similar to
each other. The area of the averaging kernels aarea
describes the sensitivity to a unit perturbation. It gives an indication of
where the MWR observation is sensitive to the true state and where the final
information originates. aarea values around unity or
differing from unity indicate that the information originates in the
observation (z) or in the a priori, respectively. In
Fig. e, aarea is close to zero up to
6 km and increases to values around 1.8 for higher altitudes. This means that
the MWR observation is not sensitive to the true state, caused by small
a priori (Kalman filtered) uncertainty. In this case the retrieved profile is
driven by the accurate a priori state that originates in the lidar
measurement. The information content that comes from the observation is given
by the degree of freedom d. Figure e represents the
accumulated degree of freedom dacc which maximally amounts to
∼0.4. That means that 0.4 independent pieces of information are added
by the observation (MWR and surface value).
(a) Comparison of mixing ratio profiles on 5 May 2013
around 23:00 UTC: retrieved profile (OEM, red), retrieved profile with RS
climatology as a priori (OEMMWR, blue), profile from the MWR
statistical retrieval (green), the Raman lidar measurement (grey) and RS
(black) as reference. Error bars are added to the optimally estimated
profiles (red, blue, grey). Absolute (b) and relative (c)
difference from the reference RS.
As mentioned above, the retrieved optimal profile (OEM) fits well with the RS
profile. A more intense comparison is illustrated in
Fig. a. Instead of feeding the retrieval
with lidar data, one can only use the MWR data as well. In this way, the improvement of applying Kalman-filtered lidar
profiles as a priori is emphasized. In such cases (OEMMWR) the
Kalman filter is completely skipped. The profile corresponding to d=2 is
added to Fig. a. The uncertainties are
larger over the whole profile in comparison to the OEM. Both the
OEMMWR and the MWR profiles from the statistical retrieval
(MWRstat) are unable to distinguish vertical structures as
indicated by the OEM and RS. For that reason, their absolute differences to
the RS are larger than those from the
OEM (Fig. b). Furthermore, in this
application the OEMMWR clearly overestimates the humidity below
1 km. The OEM profile fits best and the zero line (no difference) is within
the error bars over nearly the whole profile. The OEM is slightly more
accurate especially near the surface and with smaller uncertainties over the
whole profile. The relative differences (to RS) are smaller below 4 km and
large for altitudes where the mixing ratio from RS is
small (Fig. c). In summary, the OEM
profile fits best with small uncertainties and differences referred to the
RS. However, in cases with full-height lidar profiles the optimal estimation
is not necessary, because the Raman lidar profiles already contain
nearly all information. But full-height lidar profiles are only available
18 % of the time during HOPE and by applying the OEM the data set is
extended to 60 % coverage (see Sect. ).
In contrast to 23:02 UTC there is no mixing ratio profile from lidar
available at 07:02 UTC (Fig. a). Due to the missing lidar
profiles the estimated and the filtered profiles as well as their
uncertainties are the same (Fig. b). The difference between
the filtered and the optimal estimated profile is very small since the
atmospheric changes within a 5 min step are quite small. However, the
uncertainty decreases near the ground. This is not only caused by the MWR but
by the surface measurement which is also part the observation vector
(z). The optimally estimated profile is very smooth, since the HATPRO
frequencies do not provide enough information to distinguish fine vertical
structures. This can be seen in the difference between the optimal estimated
profile and the RS profile which is used as reference. The corresponding
averaging kernels (Fig. d) are smooth functions that are
similar to each other, because the vertical humidity information at the
HATPRO frequencies is limited. The area of the averaging kernels
aarea is around unity (Fig. e). This means
that the MWR observation is sensitive to the true state and most information
(nearly all) originates in the observation (z). The accumulated
degree of freedom dacc maximally amounts to ∼ 1.9, meaning
that 1.9 independent pieces of information can be retrieved.
used RS climatology as a priori for different locations
and found d values around 2 for humidity profiling with HATPRO. In
contrast, one has to consider that here the observation vector is
supplemented by the surface humidity which also adds information. The
difference might be explained by different a priori covariance
matrices Sa.
In summary, the presence of a lidar measurement results in more accurate
retrievals, whereas retrievals without water vapour profiles from lidar are
mainly driven by the MWR observation for example during daytime. However, the
two-step algorithm makes it possible to retain structures from high vertically resolved
lidar data to use for periods without lidar data.
As Fig. but on 5 May 2013 07:02 UTC.
Cloudy conditions
As introduced in Sect. , liquid water strongly affects
the absorption in the microwave region. Therefore, the operation of the
retrieval in the presence of clouds containing liquid water has to be treated
separately. Figure shows an overview of a
cloudy day, 21 April 2013. In the course of the day the LWP increases to
a maximum of 600 gm-2 (Fig. a).
Between 00:00 and 03:30 UTC the measured lidar profiles reach from ground up
to the cloud base between 2.5 and 3.5 km. Referring to the rather low LWP
the cloud seems to be an ice cloud. During the day, the mixing ratio is
determined on the basis of the MWR observation only disturbed by five short
interruptions that are caused by missing cloud base detection by lidar. From
19:30 UTC on the lidar profiles are truncated at the cloud base at around
1.5 km. The LWP shows that these clouds contain liquid water. The possible
content of ice water is not relevant for the radiative transfer in the
considered spectrum. However, ice clouds as well as all other clouds disturb
the precise determination of water vapour with Raman lidar. For that reason
the profile is only considered up to cloud base. The problem of truncated
profiles is solved by the application of the Kalman
filter (Sect. ). It enhances the profiles up to 10 km by the
combination of previous information and the respective truncated lidar profile
such that a full-height profile can serve as input to the optimal estimation.
As Fig. but on 21 April 2013.
A comparison between the retrieved profiles (OEM), the retrieved profiles
based on climatology (OEMMWR), the MWR profiles from the
statistical retrieval (MWRstat) and the RS is shown in
Fig. a. There is a cloud with
LWP= 242 gm-2 between 1.3 and 2.4 km. Both
OEMMWR and MWRstat are unable to distinguish the
vertical structure inside the cloud given by the RS. Furthermore, they show
large differences to the RS profile below and slightly above the cloud
(Fig. b). The OEM profile shows a good
agreement with the RS profile below the cloud based on available lidar data.
The associated uncertainties are small. Within the cloud the uncertainty
increases. The profile approximates to the RS. Above the cloud, the OEM
uncertainties are in the same range than the OEMMWR profile,
whereas the difference to the RS profile is smaller. Over nearly the whole
range the RS profile is within the uncertainty range of the OEM profile.
However, for the most part, the RS profile is also within the
OEMMWR uncertainty. The corresponding relative differences with
the RS profile are plotted in Fig. c. Up to
4 km the relative difference of the OEM profile is less than 25 %. Above
this height the relative difference increases. The OEMMWR and
MWRstat have larger relative differences to the RS. In summary,
the OEM fits best the RS with lowest differences in and above the cloud.
As Fig. but on 21 April 2013
around 23:00 UTC. The grey area indicates the cloud with an LWP of
242 gm-2.
Statistical analysis
In the previous section (Sect. ) the functionality of the
retrieval is introduced based on clear-sky and cloudy cases during HOPE.
A statistical analysis of the retrieved water vapour profiles during the
whole HOPE campaign is presented in the following section. Here, also
profiles from RS and the OEMMWR (without lidar) are used as
reference.
First, an overview over the calibrated water vapour profiles observed by
PollyXT during HOPE is given in Fig. a.
The grey area indicates regions without lidar data (up to 6 km) due to cloud
attenuation (17 %) and during the day (65 %). The well-resolved vertical
profiles enable the determination of distinct water vapour structures or
inversions that can be seen, for example, at around 1 km during the night between 26 and
27 May 2013.
As introduced in the previous sections, one can use the covariance of the RS
climatology as uncertainty from the previous state, instead of lidar data.
However, the cloud base height determined by the lidar is necessary. This
approach (OEMMWR) is only based on the observation with MWR and
surface humidity and is similar to that proposed by . The
corresponding height–time display is illustrated in
Fig. b. The gaps (40 %) are caused by rain, MWR
malfunctions, flagged MWR data, the absence of cloud base height from lidar
or that no solution was found by the retrieval. Nevertheless, the profile
availability is 60 %. Although the data availability for OEMMWR
is larger than for the Raman lidar (Fig. a), the
vertical resolution is coarser. This can be seen clearly by comparing to the
lidar profiles (Fig. a) of the night between 26 and
27 May 2013.
Figure c shows the retrieved mixing ratio profiles
(OEM) based on the method that was described in the previous sections. The
data coverage is nearly the same as for OEMMWR. However, the OEM
is able to retrieve fine water vapour structures by means of the lidar
profiles. The OEM enables not only the distinction between dry (e.g.
beginning of April) and more humid (e.g. middle of April) periods but also
the vertical distribution of water vapour especially from within and above
a cloud.
Three different height–time displays of mixing ratio profiles during
HOPE: (a) calibrated Raman lidar profiles, (b) optimal
estimated profiles based only on MWR (and surface humidity) without any Raman
lidar mixing ratio profile (OEMMWR) and (c) optimal
estimated profiles based on Kalman-filtered Raman lidar mixing ratio a priori
profiles (OEM).
Overview of the different situations depending on Raman lidar
mixing ratio (RL MR) profile availability and truncation height
(htr) where the RL MR profile is truncated (due to clouds). The
three columns on the right indicate the sample size used for the comparison
with radiosonde (RS), to validate the retrieved profiles, and all cases.
Furthermore, the profiles that are used for the comparison with RS are
separated between those passing and failing the χ2 test based on
a threshold χthr2. The temporal resolution of the
retrieved profiles amounts to 5 min.
For a more comprehensive investigation of the quality of the profiles
a differentiation between three situations based on certain initial
conditions is helpful. These situations are in accordance with the case
studies presented in the previous section (Sect. ). The first
situation includes cases where a full-height lidar profile is available
(minimum up to 8 km). Such a case is presented in Sect.
especially in Fig. . Referring to the statistical
analysis these profiles are marked in blue unless stated otherwise. The
second group includes cases with lidar profiles which are truncated between
0 and 8 km mostly due to clouds. Such cases were introduced in
Sect. in Fig. and are
marked in green from now on. The last group contains all cases without lidar
profiles as introduced in Fig. shown in red. An overview is
given in Table . The table also lists the sample size for
all profiles and those that are used for comparisons with RS. These are also
distinguished between profiles passing and failing the χ2 test that
is discussed later in this section. Additionally, the OEMMWR is
used as reference and is marked in grey.
To assess the accuracy of a water vapour profile, reference profiles from
RS and OEMMWR profiles are used. In this work the bias and the root
mean square error (RMSE) between the retrieved profiles and those from RS are
applied to evaluate the quality of the retrieved profiles. For this
comparison retrieved profiles that are between RS launch time and 1 h
after launch time are used. This results in a maximum of 12 profiles for one
sounding. Only cases which pass the χ2 test are considered for the
comparison. Figure a shows the bias for the
specified situations and for the OEMMWR. The blue line illustrates
the retrieved profiles that are based on lidar profiles in minimum up to
8 km (clear sky). It has a maximum value of 0.5 gkg-1 near the
surface and it decreases close to zero above 1.5 km. However, the bias is
positive, which means that the retrieved profiles have larger values than the
RS profiles. Above 6 km the retrieved profiles show higher values than the
RS. This bias needs to be investigated in further studies and is beyond the
scope of this study.
Statistical analysis of the synergy improvement: mean difference
(bias) between the retrieval and the RS (a), root mean square error
(RMSE) to RS (b) and a posteriori uncertainty (c). It
distinguishes four situations according to Table . The
sample size is given by the numbers in the middle panel. Only profiles
between RS launch time and 1 h after are considered.
The bias of the situations where the lidar profiles are truncated below 8 km
is shown in green (Fig. a). The values are in
maximum around 0.6 gkg-1 and are largest in the planetary boundary layer. Above
2.5 km the bias is around zero. The bias of the situations where no lidar
profiles are available and of the OEMMWR show a similar behaviour
to each other. Both curves show an overestimation of the retrieved mixing
ratio within the boundary layer up to 2 km. Between 2 and 5 km the
retrieval underestimates the mixing ratio by around
-0.4 gkg-1. Additionally, the small amount of vertical
information that comes from the MWR observation might not be able to
compensate this misbehaviour and to resemble the profile given by the
reference. This effect can also be seen in the presented clear-sky case study
in Fig. . Nevertheless, situations where no lidar profiles
are available show a bias closer to zero than the OEMMWR. These
cases benefit from the night cases whose vertical structure is propagated
into the day cases. The positive biases of all four curves seem to have
a systematic difference that might be explained by some sources of
uncertainty in the RS profiles. The different locations of the platform in
Krauthausen and the RS launch station and drifts of the balloon might result
in the observation of different air masses . Naturally, the
forward model itself is a source of uncertainty. The modelled brightness
temperatures strongly depend on the assumed absorption line
shapes . Figure illustrates a comparison
of forward models using two different gas absorption models:
(1998, R98) and (1993,
L93). The differences are the line shape parameters of the 22.235 GHz water
vapour line, as well as the water vapour continuum absorption. Both models
are corrected for water vapour continuum absorption according to
. All other parameters, e.g. cloud absorption, are the
same. Both forward models were performed under two different a priori states,
both without lidar. The first uses the a priori profile and the a priori
covariance from RS climatology. It simulates the theoretical uncertainty
(theor.) only induced by the different absorption models. In the other case
the a priori profile is propagated (prop.) from the previous state as used in
the original retrieval. Here, the a priori uncertainty is also taken from
the RS climatology. The bias to RS in the second case is larger because the
theoretical uncertainty is propagated from each previous state resulting in
an increase in uncertainty (Fig. a). It can be seen that the
L93 model has a smaller bias below 1 km. Above 2.5 km the R98 model
simulations better fit the RS with a bias around -0.3 gkg-1
and a bias close to 0 gkg-1 above 5 km. The retrieved
uncertainty, the so-called a posteriori uncertainty, of the R98 simulations
are smaller than those from the L93. The uncertainty of the L93 runs is also
largest in heights above 3 km. Finally, the R98 gas absorption model seems
to be more suitable for the presented retrieval. Nevertheless, the forward
model is a major source of uncertainty.
Mean difference (bias) between the retrieval and the RS (a)
and a posteriori uncertainty (b) for two different absorption codes:
Rosenkranz, (R98, grey) and Liebe (L93, orange). The retrievals shown are
based only on MWR but with different a priori states. On the one hand, both
a priori profile and a priori uncertainty are taken from the RS climatology
(theor.) and on the other hand the a priori profile is propagated (prop.)
from the previous step while the uncertainty is taken from the RS climatology
(red cases in the figures above). The sample size is given by the numbers.
Only profiles between RS launch time and 1 h after launch time are
considered.
The RMSE between OEM and RS is illustrated in
Fig. b. It gives an indication of the
statistic error. The RMSE of all four curves decreases with height. In
addition, the RMSE is smaller for cases with lidar profiles as a priori and
larger for those without. The RMSE of the HOPE RS profiles is larger than any
RMSE of the retrieved profiles, which is basically the variance of mixing
ratio in the whole period.
Figure c illustrates the a posteriori
uncertainty of the mixing ratio profiles (see Eq. ). The
black line indicates the uncertainty of the RS climatology which is the square
root of the diagonal elements of its covariance matrix. It can clearly be
seen that the retrieved a posteriori uncertainty is smaller for all
situations. The curves of the cases without lidar profiles and the
OEMMWR are nearly in agreement. In both cases the Kalman filter is
skipped due to the absence of lidar profiles. Therefore, both use the same
a priori uncertainty and their retrievals are solely driven by the MWR and
surface humidity observation. The presence of lidar data (full height or
truncated) results in much lower uncertainties. Their small a posteriori
uncertainties underline the synergy improvement.
In summary, Fig. clearly shows that the application of Kalman-filtered lidar profiles enormously improves the accuracy and quality of the retrieved mixing ratio
profiles.
Comparison of optimal estimated (OEM) and radiosonde (RS) mixing
ratio profiles for the four situations (panels a–d) given in Table .
The black solid line indicates the regression line.
Another possibility to evaluate the accuracy of the retrieved profiles is to
analyse the bias as a function of the mixing
ratio (Fig. ). The slope of the regression line is
smaller than the one-to-one line. This means that larger differences occur
for larger mixing ratios. Figure also indicates
the correlation between retrieved and RS mixing ratios. The squared
coefficient of correlation R2 is largest for those situations with
full-height lidar profiles and amounts to 0.97 (Fig. a).
The R2 of the OEM based on truncated lidar profiles (panel b) is
slightly smaller (0.96). In situations without lidar data and the
OEMMWR have still smaller values of 0.92 and 0.91, respectively.
Nevertheless, all cases show a better agreement with RS than the
OEMMWR. This illustration also demonstrates the synergy improvement
by implementing the lidar data with a Kalman filter before applying the OEM.
To assess the quality of retrieved profiles a statistical test for correct
convergence of the solution is applied. The modelled state
F(xop) and the observation vector
zobs are compared with the error covariance matrix
Sδz (see Eq. ) to check if the
retrieval is consistent with the observation. Figure shows
the χ2 test statistics for all mentioned situations. The χ2
test was introduced in Sect. . It can be seen that 29 profiles
are rejected in the situations with full-height lidar profiles because their
χ2 value exceeds the 5 % threshold value of 14
(Fig. a). The amount of untrustworthy profiles is similar to
the situations with truncated lidar profiles. In both cases the smaller
a priori uncertainty prevents an adjustment of the modelled brightness
temperatures to those measured by MWR. For that reason, their difference is
larger resulting in a larger χ2 value. The χ2 test rejects
a smaller relative amount of profiles for the daytime cases (panel c) and at
the OEMMWR (panel d). Their larger a priori uncertainty enables
a better match between the modelled and the measured brightness temperatures.
However, all situations show a peak at small values that originates in a very
good agreement between the forward modelled optimal state and the observation
vector. Admittedly, the test is very strict and rejects all failing profiles
although they might be realistic atmospheric states. Nevertheless, it
enhances the confidence of the retrieved profiles.
Histograms of the χ2 test for the four situations given in
Table . The dotted lines indicate the theoretical
χ2 distribution with my degree of freedom. Dashed lines
indicate the 5 % threshold value of 14. The absolute number of cases below
and above the threshold value is given to the left and to the right of
the dashed line, respectively.
(a) Degree of freedom as a function of truncation height for
different situations introduced in Table .
(b) Frequency distribution of the degree of freedom. The symbols and
error bars correspond to the related mean and standard deviation,
respectively. The numbers indicate the sample size of the considered
profiles; full height (blue), truncated (green), no lidar (red) and
OEMMWR (grey).
A good measure for the proportion of information that comes from the
observation is given by the degree of freedom. It describes the number of
independent pieces of information that is added by the retrieval and has
already been introduced in Sects. and .
Figure a illustrates the degree of freedom as a function
of truncation height. It clearly demonstrates that the lower the truncation
height the higher the degree of freedom. This is caused by the larger
a priori uncertainty in cases with truncated or without lidar mixing ratio
profiles. The sample size is much higher than in the comparisons above
because here all profiles can be used and not only those around the RS launch
time. Most of the grey crosses are not visible because they are covered by
the red diamonds. The related frequency distributions are shown in
Fig. b. Both the OEMMWR and the daytime cases
are very similar to each other. Even their mean values and standard
deviations are nearly identical, with values of 1.9±0.22. These values
are in good agreement with those found by for a similar
approach. The situations with the truncated lidar profiles show a wide range
of values from 0.3 to 2.1. The green distribution also has the largest
standard deviation, which amounts to 0.34. The situations with full-height
lidar profiles have the smallest mean and standard deviation with values of
0.45±0.17. These cases are mostly driven by the a priori information
and not by the observation. The variation within each situation is caused by
different atmospheric conditions. Figure illustrates the
degree of freedom as a function of IWV. It shows an increase in d with
increasing IWV caused by a stronger emission of water vapour. For higher IWV,
the MWR is able to add more information to the retrieval. Finally, the
behaviour of the degree of freedom and especially its dependence on
truncation height and hence a priori uncertainty agrees well with similar
studies .
Degree of freedom as a function of IWV for the situations introduced
in Table . The lines indicate the according regression
lines.
Conclusions
A good knowledge of the water vapour distribution is essential for the
description of the thermodynamic state of the troposphere. Since the
continuous observation of water vapour profiles with a single instrument is
challenging, the synergy of complementary information from active and passive
remote sensing has become more important in recent years.
In this study we present a two-step retrieval combining the Raman lidar water
vapour profiles with the MWR brightness temperatures. The Kalman-filtered
water vapour profile serve as input (a priori) to the one-dimensional
variational approach, also known as optimal estimation. In addition to the
water vapour profile, its uncertainty is retrieved.
The retrieval enables the observation of a continuous time series of water
vapour profiles with known uncertainties. During HOPE, the availability of
full-height water vapour profiles from lidar amounts to 17 % excluding
all cloudy and daytime cases. By applying the retrieval, the availability of
water vapour profiles can be enlarged to 60 %. The bias with respect to
RS and the retrieved a posteriori uncertainty of the retrieved profiles
clearly show that the application of the Kalman filter considerably improves
the accuracy and quality of the retrieved mixing ratio profiles. In the
presence of full-height Raman lidar profiles, the MWR does not add much
information to the retrieved profiles. However, cases without Raman lidar
profiles are dominated by the MWR information with a larger degree of
freedom. The lower the truncation height of the lidar profiles, the higher the
importance of the MWR.
Furthermore, the retrieval can be applied to raw data (photon counts) using
the calibration method based on or using already calibrated
profiles.
In future steps, the precipitation evaporation can be assessed by means of
observed or retrieved temperature and humidity profiles. This information can
be used to improve model parametrization of physical processes with water
vapour participation and finally to improve weather and climate predictions.
The retrieval will be implemented into the Cloudnet processing. A better
knowledge of the water vapour distribution and the collocated and
simultaneous monitoring of cloud microphysics within Cloudnet might improve
the understanding of cloud formation, precipitation, evaporation and
entrainment rates. The application of this algorithm might help to decrease
uncertainties in the area of cloud and precipitation formation as well as
cloud dissipation, as mentioned in the latest IPCC
report .
The quality-controlled MWR, RS and tower data used in this
work are archived in a common format to the HD(CP)2 data archive centre for
Standardized Atmospheric Measurement Data (SAMD). All the data are publicly
available at https://icdc.cen.uni-hamburg.de/index.php?id=samd.
For more information on data availability and on data policy of the PollyXT
raw measurement data please contact the Polly team via the website
(http://polly.tropos.de).
The authors declare that they have no conflict of
interest.
This article is part of the special issue “HD(CP)2
Observational Prototype Experiment (ACP/AMT inter-journal SI)”. It is not
associated with a conference.
Acknowledgements
The present study was conducted within the research programme “High
Definition Clouds and Precipitation for Climate Prediction – HD(CP)2”. This
project is funded by the German Federal Ministry of Education and Research
within the framework programme “Research for Sustainable Development
(FONA)”, www.fona.de, under the numbers 01LK1209D and 01LK1504C.
The authors also acknowledge the LACROS team from the Leibniz Institute for
Tropospheric Research (TROPOS) for the Raman lidar and microwave radiometer
data, the Karlsruhe Institute for Technology (KIT) for the radiosonde
launches, and the Research Centre Jülich for the in situ
observations on the 120 m tower. Edited by:
Brian Kahn Reviewed by: two anonymous referees
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